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[
[
"FluTO: Graded Multiscale Fluid Topology Optimization using Neural\n Networks"
],
[
"Abstract Fluid-flow devices with low dissipation, but high contact area, are of importance in many applications.",
"A well-known strategy to design such devices is multi-scale topology optimization (MTO), where optimal microstructures are designed within each cell of a discretized domain.",
"Unfortunately, MTO is computationally very expensive since one must perform homogenization of the evolving microstructures, during each step of the homogenization process.",
"As an alternate, we propose here a graded multiscale topology optimization (GMTO) for designing fluid-flow devices.",
"In the proposed method, several pre-selected but size-parameterized and orientable microstructures are used to fill the domain optimally.",
"GMTO significantly reduces the computation while retaining many of the benefits of MTO.",
"In particular, GMTO is implemented here using a neural-network (NN) since: (1) homogenization can be performed off-line, and used by the NN during optimization, (2) it enables continuous switching between microstructures during optimization, (3) the number of design variables and computational effort is independent of number of microstructure used, and, (4) it supports automatic differentiation, thereby eliminating manual sensitivity analysis.",
"Several numerical results are presented to illustrate the proposed framework."
],
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"0pt",
"8pt plus 4pt minus 2pt0pt plus 2pt minus 2pt nosep -10pt Fluid-flow devices with low dissipation, but high contact area, are of importance in many applications.",
"A well-known strategy to design such devices is multi-scale topology optimization (MTO), where optimal microstructures are designed within each cell of a discretized domain.",
"Unfortunately, MTO is computationally very expensive since one must perform homogenization of the evolving microstructures, during each step of the homogenization process.",
"As an alternate, we propose here a graded multiscale topology optimization (GMTO) for designing fluid-flow devices.",
"In the proposed method, several pre-selected but size-parameterized and orientable microstructures are used to fill the domain optimally.",
"GMTO significantly reduces the computation while retaining many of the benefits of MTO.",
"In particular, GMTO is implemented here using a neural-network (NN) since: (1) homogenization can be performed off-line, and used by the NN during optimization, (2) it enables continuous switching between microstructures during optimization, (3) the number of design variables and computational effort is independent of number of microstructure used, and, (4) it supports automatic differentiation, thereby eliminating manual sensitivity analysis.",
"Several numerical results are presented to illustrate the proposed framework.",
"Figure: Graphical abstract: Given a set of candidate microstructures and a fluid topology optimization problem, a neural network (NN) selects appropriate microstructures, optimizes their size and orientation to produce a graded multiscale design.Topology Optimization Stokes Flow Multiscale Neural Networks Fluid Microstructure"
],
[
"Introduction",
"Topology optimization is used extensively today in various disciplines [42].",
"When applied to fluid-flow problems, it addresses the following question [1]: \"Where should fluid flow within a design domain?\"",
"As an example, consider the simple diffuser problem posed in fig:singlescale(a) where fluid enters on the left and exits on the right, as illustrated.",
"The objective is to design the fluid flow path such that the dissipated power is minimized.",
"In addition, a volume constraint is often imposed on the surrounding solid material (or equivalently, on the fluid space).",
"Using topology optimization techniques described, for example in [9], one can solve this problem to arrive at the topology, i.e., fluid flow path, illustrated in fig:singlescale(b).",
"For such simple problems, one can argue that topology optimization is not needed since the solution is often a single flow path [1].",
"Figure: Classic fluid-flow topology optimization to minimize dissipated power.Topology optimization becomes more important when additional criteria come into play.",
"For example, maximizing the fluid-solid contact area is critical in many applications including bio-sensors for detecting tumour cells [37], microfluidic devices for cell sorting [18], [25], [28], [34], [15], micro-channel heat sinks [57], [24], [36], and other microfluidic devices involving heat transfer and mass transportation/mixing mechanisms [6], [7].",
"In such applications, a heuristic strategy to increase the contact surface is to use micro-pillar arrays [26], [32] as illustrated in fig:pillars.",
"While uniform cylindrical micro-pillars can increase the contact surface, they can also significantly increase the dissipated power loss [31], [8].",
"To balance the two, one can vary the the cylinder radii or, better still, use non-cylindrical micro-pillars, henceforth referred to as microstructures, with proper orientation for minimizing power loss and maximizing fluid-solid contact interface [53].",
"Optimizing the cross-section, size and orientation of the microstructures leads to a multi-scale fluid-flow topology optimization problem [53], the main focus of this paper.",
"Figure: Uniform cylindrical micro-pillars .In multi-scale topology optimization (MTO), optimal microstructures are designed in each cell of a discretized domain.",
"Unfortunately, this can be very expensive [52] since one must carry out homogenization of the evolving microstructures [56] during each step of the global optimization process.",
"Specifically, if $N_e$ is the number of elements (typically in the order of 1000s), $N$ is the number of global optimization steps (typically in the order of 100s), and $c$ is the cost of homogenization, then the MTO cost is at least $N * N_e * c$ .",
"In this paper, we propose a graded MTO (GMTO) strategy that retains much of the advantages of MTO, but exhibits a significantly lower computational cost.",
"The remainder of this paper is organized as follows.",
"In sec:RelWork we review the literature on topology optimization for fluid flow problems, and summarize the main contributions of this paper.",
"In sec:tecBack, we briefly review relevant technical background, followed by the proposed method in sec:method.",
"Numerical examples are illustrated in sec:experiments, and conclusions are provided in sec:conclusion."
],
[
"Related Work",
"In this section, we briefly review prior work on fluid-flow topology optimization methods.",
"In particular, while many different strategies have been proposed [1], [10], we focus here on density-based methods."
],
[
"Single-scale TO",
"Topology optimization for fluid-flow problems first appeared in 2003 in the seminal work of Borrvall and Petersson [9].",
"They presented the optimal layout of channel flows for minimal drag (or pressure drop), for Stokes equation with Brinkman–Darcy law equations under low Reynolds number (laminar flow conditions).",
"Gersborg-Hansen et al.",
"[19] continued this study and presented applications with low Reynolds numbers for microfluidic problems and micro-electro-mechanical devices.",
"Guest and Prévost [22] solved the formulation of the Stokes–Darcy problem numerically using stabilized finite element methods.",
"Wiker et al.",
"[51] used the viscosity as a dependent parameter and presented examples of channels in a tree-shaped structure for a pure Darcy problem and mixed Stokes–Darcy flow.",
"Pereira et al.",
"[41] reproduced the classical examples presented by [9] for optimal channels designs considering Stokes–Darcy flow using polygonal meshes and provided an educational software written in MATLAB.",
"Suárez et al.",
"[44] applied topology optimization to non-Newtonian flows in arbitrary domains, using a virtual element method."
],
[
"Multi-scale TO",
"Multiscale topology optimization (MTO) for fluid flow, as mentioned earlier, involves generating appropriate microstructures, i.e., generalization of micro-pillars, typically, in each finite element cell, for maximizing fluid permeability.",
"Guest and Prévost [23] maximised the permeability of porous microstructures using a Darcy–Stokes interpolation [22] subject to isotropic symmetry constraints.",
"This was extended in [21] to optimize microstructures for maximal stiffness and permeability.",
"Bio-mimicking techniques have been demonstrated for achieving this goal [27] but are not proven to be optimal [53].",
"MTO that minimizes energy loss and offers high contact area was demonstrated in [53].",
"However, the volume fraction of the microstructures was pre-determined, and the surface contact area was not explicitly controlled.",
"To address the high computational cost of MTO, graded MTO (GMTO) techniques have been proposed, in structural mechanics [12], [38], [54], [55], [46], [47], [49], [50], [48], but have not been extended to fluid flow problems."
],
[
"Paper Contributions",
"In this work we extend a particular GMTO framework proposed in [12] for structural problems, to fluid flow problems.",
"In structural GMTO, a total volume constraint is imposed, whereas here, a total contact area constraint is more critical, changing the type of microstructures one must choose.",
"Further, microstructure orientation is not typically considered in structural problems (due to potential loss in connectivity), but it plays an important role in fluid problems.",
"The proposed GMTO strategy is based on a neural-network based optimization framework for the the following reasons: (1) it implicitly guarantees the partition of unity, i.e., ensures that the net volume fraction of microstructures in each cell is unity, as described later on, (2) it supports automatic differentiation, and (3) the number of design variables is only weakly dependent on the number of pre-selected microstructures."
],
[
"Fluid Flow Governing Equations",
"We assume here a low-Reynolds in-compressible Stokes flow, i.e., $-2\\nabla .",
"[\\mu \\mathbf {\\epsilon (u)}]+\\mathbf {C^{-1}}\\mathbf {u}+\\nabla p= 0 \\text{ in $\\Omega $} \\\\\\nabla .\\mathbf {u}=0 \\text{ in $\\Omega $}\\\\\\mathbf {u}=\\mathbf {g} \\text{ over } \\partial \\Omega $ where $\\mathbf {u}$ and p are the velocity and pressure of the fluid, $\\mu $ is its viscosity, $\\mathbf {\\epsilon (u)} = (\\mathbf {\\nabla } \\mathbf {u} + \\mathbf {\\nabla ^T}\\mathbf {u})/2$ denotes the rate-of-strain tensor, $\\mathbf {C^{-1}}$ is the inverse permeability tensor; the fluid is assumed to be of unit density."
],
[
"Problem Formulation",
"We also assume that a set of microstuctures have been pre-selected.",
"Specifically, we consider the microstructures illustrated in fig:microstructures (selected from various sources [23], [22], [3], [35], [53], [4]) that exhibit a wide range of permeability and contact area.",
"Each of these microstructures can be scaled and oriented for optimal flow.",
"For future reference, they are named as follows: 1: Squircle, 2: Fish-body-1, 3: Fish-body-2, 4: Square, 5: Circle, 6: Ellipse, 7: Mucosa-10, and 8: Mucosa-20.",
"Such microstructures are often seen in nature; for example, mucosa-like structures are known to be present in the human intestine [4].",
"Figure: Pre-selected microstructures.A typical design domain with prescribed flow boundary conditions is illustrated in fig:fluidGMTO.",
"We seek to compute an optimal multiscale design where we determine, in each finite element cell, the appropriate microstructure, its size (gradation) and orientation.",
"The objective is to minimize the dissipated power subject to a total contact area (i.e., perimeter in 2D) constraint.",
"The strategy we adopt is to pre-compute the permeability and contact area of each of the microstructures as a function of its size and orientation, and exploit these for a global multi-scale fluid-flow optimization.",
"Figure: Graded multiscale TO.Towards this end, we introduce the following design variables.",
"The presence or absence of a microstructure at any element $e$ will be captured by the set of density variables $\\mathbf {\\rho }_e = \\lbrace \\rho _{e,1}, \\rho _{e,2}, ..., \\rho _{e,M} \\rbrace $ ; see fig:fluidGMTO.",
"Ideally $ \\rho _{e,m}$ should take a binary value $0/1$ subject to partition of unity constraint $\\sum \\rho _{e,m} = 1, \\forall e$ .",
"However, for gradient-based optimization, we will let $0 \\le \\rho _{e,m} \\le 1$ and drive it towards $0/1$ through penalization (once again subject to the partition of unity constraint).",
"We control the size of all the microstructures at any element $e$ by the scalar design variable $0 \\le s_e \\le 1$ (see fig:fluidGMTO), where 0 denotes completely filled with fluid, and 1 denotes maximum size of the microstructure(s).",
"Note that: (1) a single variable $s_e$ controls the size of all microstructures at that point, (2) $s_e=1$ means that the microstructures at that element are at their maximum size (that can fit into that cell); however, it does not imply that the cell is fully solid, and (3) as we drive $\\rho _{m,e}$ towards $0/1$ only one microstructure of that size will prevail at that element.",
"Finally, the orientation of all the microstructures at $e$ , with respect to the $x$ axis, will be denoted by $0\\le \\theta _e\\le 2\\pi $ (see fig:fluidGMTO).",
"Thus the design variables associated with each element will be denoted by $\\mathbf {\\zeta }_e = \\lbrace \\rho _{e,1}, \\rho _{e,2}, \\ldots , \\rho _{e,M}, s_e, \\theta _e \\rbrace $ ."
],
[
" Effective Permeability through Numerical Homogenization",
"Prior to carrying out global optimization, we pre-compute the $2 \\times 2 $ permeability tensor $\\mathbf {C}_m$ of each microstructure at discrete sizes.",
"The components of $\\mathbf {C}_m$ can be computed by posing two low-Reynolds in-compressible Stokes flow problems over a unit cell, with unit body forces $f_x = 1$ and $f_y = 1$ , respectively, as illustrated in fig:homogenization.",
"The boundary conditions for both problems are as follows: (a) boundaries 1 and 3 are coupled through periodic boundary conditions for velocity and pressure, and (b) boundaries 2 and 4 are similarly coupled.",
"The velocities obtained by solving the problem in fig:homogenizationa are denoted by $u_0(x,y)$ and $v_0(x,y)$ , while those obtained from fig:homogenizationb are denoted by $u_1(x,y)$ and $v_1(x,y)$ .",
"The components of the permeability tensor $\\mathbf {C}_m$ are then defined as [2], [30], [45]: $\\mathbf {C}_m =\\begin{bmatrix} C_m^{00} & C_m^{01} \\\\C_m^{10} & C_m^{11}\\end{bmatrix} =\\frac{1}{|V|} \\begin{bmatrix} \\int \\limits _{V}u_{0}dV & \\int \\limits _{V}v_{0}dV \\\\\\int \\limits _{V}u_{1}dV & \\int \\limits _{V}v_{1}dV \\end{bmatrix}$ where $V$ is the volume of the unit cell.",
"Observe that, at a default orientation, all the microstructures in fig:microstructures are symmetric about the two axis.",
"Therefore, the off-diagonals are zero, i.e., $C_m^{01} = C_m^{10} = 0$ .",
"Figure: Fluid flow problems to determine permeability tensor (a) a body force f x =1f_{x} = 1 to determine velocities {u 0 (x,y),v 0 (x,y)}\\lbrace u_{0}(x,y), v_{0}(x,y)\\rbrace (b) a body force f y =1f_{y} = 1 to determine velocities {u 1 (x,y),v 1 (x,y)}\\lbrace u_{1}(x,y), v_{1}(x,y)\\rbrace .We compute the two remaining components for each microstructure at a finite number of sizes, at default orientation, using the implementation provided in [2].",
"Then, two interpolating and positive polynomials [29] $C_m^{00}(s)$ and $C_m^{11}(s)$ are constructed from these samples.",
"As an example, the polynomials for a fish-body-type microstructure are illustrated in fig:permVsSizeHalfFishscale.",
"In this paper, we use a 5th degree polynomial.",
"The orientation is then accounted for via the following tensor operation [30]: $\\mathbf {C}_m(s,\\theta ) = \\begin{bmatrix} \\cos (\\theta ) & -\\sin (\\theta ) \\\\ \\sin (\\theta ) & \\cos (\\theta ) \\end{bmatrix}\\begin{bmatrix} C_m^{00}(s) & 0 \\\\0 & C_m^{11}(s) \\end{bmatrix}\\begin{bmatrix} \\cos (\\theta ) & -\\sin (\\theta ) \\\\ \\sin (\\theta ) & \\cos (\\theta ) \\end{bmatrix}^T$ Finally, since multiple microstructure can co-exist at any element, the following penalization scheme is proposed to ensure that we drive $\\rho _m$ towards $0/1$ (subscript $e$ has been suppressed for simplicity): $\\mathbf {C}(\\mathbf {\\rho }, s, \\theta ) = \\sum \\limits _{m=1}^M \\rho _m^p \\mathbf {C}_m(s,\\theta )$ where the penalization $p > 1$ discourages microstructure mixing.",
"Figure: Permeability components versus size of the fish-body-2 microstructure.Observe that the pre-computation cost is approximately $ S * M *c$ , where $S$ is the number of size samples (6 in this paper) , $M$ is the number of microstructures (8 in this paper) and $c$ is the cost of homogenization.",
"This is significantly lower than the cost for MTO."
],
[
"Contact area ",
"Observe that the contact area $\\Gamma _m$ of a microstructure is proportional to its size $s$ , and independent of orientation $\\theta $ .",
"Thus, it is sufficient to compute the maximum contact area $\\Gamma _m^{max}$ at $s =1$ for each microstructure.",
"The contact areas are then combined as follows (subscript $e$ has been suppressed for simplicity): $\\Gamma (\\mathbf {\\rho }, s) = \\sum \\limits _{m=1}^M \\rho _m s \\Gamma _m^{max}$"
],
[
"Volume Constraint",
"One can also impose a constraint on the total fluid volume allowed; this constraint is mainly used in validation experiments (please see sec:doublepipevalidation).",
"Similar to the contact area computation, the total volume occupied by the fluid is given by: $V(\\mathbf {\\rho }, s) = V_e(1 - \\sum \\limits _{m=1}^M \\rho _m s^{2} v_m^{max})$ where $v_m^{max}$ is the fraction of the volume occupied by a microstructure $m$ at maximum size."
],
[
"Fluid Flow Finite Element Analysis",
"For the global fluid flow analysis, we use quadrilateral Q2-Q1 (Taylor-Hood) elements.",
"The elemental stiffness matrix $\\mathbf {K_e}$ and degrees of freedom vector $\\mathbf {S_e}$ for the governing equation (see sec:Fluideqs) are given by (see [41] for details): $\\mathbf {K_e} =\\begin{bmatrix}\\mathbf {A_e} && \\mathbf {B_e} && \\mathbf {0} \\\\\\mathbf {B_e^T} && \\mathbf {0} && \\mathbf {a_e}\\\\\\mathbf {0} && \\mathbf {a_e^T} && \\mathbf {0}\\end{bmatrix} \\;, \\; \\mathbf {S_e}=\\begin{bmatrix}\\mathbf {U_e} \\\\\\mathbf {P_e} \\\\\\mathbf {\\lambda }\\end{bmatrix}$ where $& \\mathbf {A_e} = \\mathbf {A_e^\\mu }+\\mathbf {C_e}^{-1}\\mathbf {A_e^\\alpha }\\\\&\\mathbf {[A_e^\\mu ]}_{ij} = \\int _{\\Omega _e}2\\mu \\mathbf {\\epsilon (N_i)}:\\mathbf {\\epsilon (N_j)}d\\Omega \\\\&\\mathbf {[A_e^\\alpha ]}_{ij} = \\int _{\\Omega _e}\\mathbf {N_i}\\mathbf {N_j}d\\Omega \\\\&\\mathbf {[B_e]}_{ij} = \\int _{\\Omega _e}\\mathbf {L_j}\\nabla .\\mathbf { N_i}d\\Omega \\\\&\\mathbf {[a_e]}_{i} = \\int _{\\Omega _e}\\mathbf {L_i}d\\Omega $ where $\\mathbf {N_i}$ and $\\mathbf {L_i}$ are the velocity and pressure basis functions, $U_e$ and $P_e$ represent elemental velocity and pressure degrees of freedom respectively and $\\mathbf {C_e} $ is the element permeability matrix (described previously).",
"In order to uniquely define the pressure field, a zero mean condition is enforced."
],
[
"Optimization Problem",
"Consequently, one can pose the GMTO problem in a finite-element setting as: $& \\underset{\\overline{\\mathbf {\\zeta }} = \\lbrace \\mathbf {\\zeta }_1, \\mathbf {\\zeta }_2, \\ldots \\mathbf {\\zeta }_{N_e} \\rbrace }{\\text{minimize}}& J(\\overline{\\mathbf {\\zeta }}) &= \\sum \\limits _{e=1}^{N_e}\\frac{1}{2}\\mathbf {U_e}^T\\mathbf {[A_e^\\mu +A_e^\\alpha C_e^{-1}]U_e} \\\\& \\text{subject to}& \\mathbf {K}(\\overline{\\mathbf {\\zeta }})\\mathbf {S} & = \\mathbf {f}\\\\& & g_{\\Gamma } (\\overline{\\mathbf {\\zeta }}) & \\equiv 1 - \\frac{\\sum \\limits _{e=1}^{N_e} \\sum \\limits _{m=1}^M \\rho _{e,m} s \\Gamma _m^{max}}{\\Gamma ^*} \\le 0 \\\\& & \\text{(or)} \\quad g_{V} (\\overline{\\mathbf {\\zeta }}) & \\equiv \\frac{\\sum \\limits _{e=1}^{N_e}V_e( 1 -\\sum \\limits _{m=1}^M \\rho _{e,m} v_m^{max}s^{2})}{(\\sum \\limits _{e=1}^{N_e} V_e) v^*}-1 \\le 0 \\\\& & \\sum \\limits _{m=1}^M \\rho _{e,m} &= 1 \\; , \\; \\forall e \\\\& & 0 \\le \\rho _{e,m} &\\le 1 \\; , \\; \\forall e \\; , \\; \\forall m \\\\& & 0 \\le s_e &\\le 1 \\; , \\; \\forall e \\\\& & 0 \\le \\theta _e & \\le 2 \\pi \\; , \\; \\forall e $ where $J$ is the dissipated power, ${\\Gamma ^*}$ is the lower bound on the total contact area, $v^{*}$ is the upper bound on the total volume fraction, $V_e$ is the total unit cell volume, and $\\mathbf {K}(\\mathbf {\\zeta })$ , $\\mathbf {S}$ and $\\mathbf {f}$ are the global stiffness, degrees of freedom and the boundary conditions respectively.",
"In the optimization problem, either we impose volume constraint, eq:optimizationbasevolCons, for validation (please see sec:doublepipevalidation) or contact area constraint, eq:optimizationbaseperimCons, for other numerical experiments.",
"In the above direct formulation, the number of design variables is proportional to the mesh size.",
"Further, the partition of unity, and bound constraints must be strictly enforced over each element.",
"In this paper, we avoid these issues by indirectly controlling the design variables via a coordinate-based neural-network [14], [13] described next."
],
[
"Neural Network",
"The proposed neural-network (NN) architecture is illustrated in fig:neuralNetwork, and it consists of the following entities: Input Layer: The input to the NN are points $\\mathbf {x} \\in \\mathbf {R}^2$ within the domain.",
"Although these points can be arbitrary, they correspond here to the center of the elements.",
"Hidden Layers: The hidden layers consist of a series of dense fully connected LeakyReLU activated neurons.",
"Output Layer: The output layer consists of $M + 2$ neurons correspond to the design variables for each element $\\zeta = \\lbrace \\rho _1(\\mathbf {x}), \\rho _2(\\mathbf {x}), \\ldots , \\rho _M(\\mathbf {x}), s(\\mathbf {x}), \\theta (\\mathbf {x})\\rbrace $ .",
"Further, the neurons associated with the density variables are controlled by a softmax function such that the partition of unity $(\\sum \\rho _m = 1)$ and physical validity $0 \\le \\rho _m \\le 1$ are automatically satisfied.",
"The output neuron associated with the size parameter is controlled via a Sigmoid function $\\sigma (\\cdot )$ , ensuring that $0 \\le s \\le 1$ .",
"Finally, the output neuron associated with the orientation parameter is also controlled via a Sigmoid function and scaled as $\\theta \\leftarrow 2\\pi \\sigma (\\theta )$ .",
"Thus, no additional box constraints are needed.",
"NN Design Variables: The weights and bias associated with the NN, denoted by the $\\mathbf {w}$ , now become the primary design variables, i.e., we have $\\mathbf {\\rho }(\\mathbf {x}; \\mathbf {w}),$ $s(\\mathbf {x}; \\mathbf {w})$ and $\\theta (\\mathbf {x}; \\mathbf {w})$ .",
"Thus the strategy is to perform GMTO via the NN weights $\\mathbf {w}$ , i.e., eq:optimizationbaseEqn reduces to: $& \\underset{\\mathbf {w}}{\\text{minimize}}& &J(\\mathbf {w}) \\\\& \\text{subject to}& & \\mathbf {K}(\\mathbf {w})\\mathbf {S} = \\mathbf {f}\\\\& & & g_{\\Gamma } (\\mathbf {w}) \\equiv 1 - \\frac{\\sum \\limits _{e=1}^{N_e} \\sum \\limits _{m=1}^M \\rho _{e,m} (\\mathbf {w}) s (\\mathbf {w}) \\Gamma _m^{max}}{\\Gamma ^*} \\le 0 \\\\& & \\text{(or)} \\quad g_{V} (\\mathbf {w}) & \\equiv \\frac{\\sum \\limits _{e=1}^{N_e} V_e(1 -\\sum \\limits _{m=1}^M \\rho _{e,m} (\\mathbf {w}) v_m^{max} s^{2} (\\mathbf {w}))}{(\\sum \\limits _{e=1}^{N_e} V_e) v^*}-1 \\le 0 $ Observe that: (1) no additional constraint is needed since they are automatically satisfied, and (2) the number of design variables ($\\mathbf {w}$ ) is independent of the mesh size and the number of microstructures.",
"Figure: Neural network architecture."
],
[
"Loss Function",
"We now consider solving the NN-based optimization problem in eq:optimizationnnEqn.",
"Since neural networks are designed to minimize an unconstrained loss function, we convert the constrained minimization problem into a loss function minimization by employing the augmented Lagrangian scheme [5].",
"Specifically, the loss function is defined as $L(\\mathbf {w}) = \\frac{J(\\mathbf {w})}{J^0} + \\alpha g(\\mathbf {w})^2 + \\lambda g(\\mathbf {w})$ where the parameters $\\alpha $ and $\\lambda $ are updated during each iteration, making the enforcement of the constraint stricter as the optimization progresses (see discussion below).",
"Thus the overall framework is illustrated in fig:algoFlowChart.",
"Figure: Optimization loop of the proposed framework."
],
[
"Sensitivity Analysis",
"For minimizing the loss function in eq:lossFunction we use L-BFGS [33], a well-known optimization technique.",
"Since L-BFGS is a gradient-based optimizer, it requires sensitivities, i.e., derivative of the loss function in eq:lossFunction with respect to the design variables $\\mathbf {w}$ .",
"Fortunately, one can exploit modern automatic differentiation frameworks [11] to avoid manual sensitivity calculations.",
"In particular, we use PyTorch [40] as an implementation environment, resulting in an end-to-end differentiable framework."
],
[
"Algorithm",
"The algorithm for the proposed framework is summarized below.",
"First, we generate the dataset $\\mathbf {D_m}$ consisting of permeability matrices (at various sizes) and contact area values for the pre-selected microstructures (line 2).",
"Polynomials are constructed to fit the data (line 3).",
"Next the domain is discretized for finite element analysis, and the stiffness matrix components are computed (lines 4-5).",
"The mesh is sampled at the center of each element (line 6); these serve as inputs to the NN.",
"The augmented Lagrangian parameters $\\alpha $ , $\\lambda $ , the penalty parameter $p$ and NN weights $\\mathbf {w}$ are initialized (line 7).",
"In the main iteration, the design variables $\\overline{\\mathbf {\\zeta }}$ are computed using the NN (line 9), followed by the computation of the permeability matrices for each element (line 10).",
"These are then used to construct the stiffness matrix and to solve for the velocity and pressure (line 11 - line 12).",
"Then the objective and contact area constraint are computed (lines 13 - 14), leading to the loss function (line 15).",
"The sensitivities are computed in an automated fashion (line 16).",
"The weights $\\mathbf {w}$ are then updated using L-BFGS optimization scheme (line 17).",
"Finally the augmented Lagrangian multipliers and penalty parameters are updated (line 18 - 20).",
"The process is repeated until termination, i.e., until the relative change in loss is below a certain threshold or the iterations exceed a maximum value.",
"[] GM-Flow [1] GMFlow$\\Omega ^0$ , BC, $\\Gamma ^*$ $s \\rightarrow \\mathbf {D_m} \\quad s \\in [0,1]$ calculate permeability and contact area at size instances $\\mathbf {D_m} \\rightarrow \\mathbf {C}_m(s), {\\Gamma }_m(s)$ generate polynomial functions from dataset fig:permVsSizeHalfFishscale $\\Omega ^0 \\rightarrow \\Omega ^0_h$ discretize domain for FE sec:FluidFEA $\\Omega ^0_h \\rightarrow \\mathbf {A^\\mu },\\mathbf {A^\\alpha },\\mathbf {B}, \\mathbf {a}$ compute stiffness matrices eq:stiffnesstermdefn $\\mathbf {x} = \\lbrace x_e,y_e\\rbrace _{e \\in \\Omega ^0_h} \\quad \\mathbf {x} \\in \\mathbb {R}^{n_e \\times 2}$ elem centers; NN input epoch = 0; $\\alpha = \\alpha _0$ ; $p = p_0$ ; $\\mathbf {w} = \\mathbf {w}_0$ initialization optimization (Training) $NN(\\mathbf {x} ; \\mathbf {w}) \\rightarrow \\overline{\\mathbf {\\zeta }}(\\mathbf {x})$ fwd prop through NN $\\overline{\\mathbf {\\zeta }}(\\mathbf {x}) \\rightarrow \\mathbf {C}_m(\\mathbf {x}) $ Permeability tensor eq:effectiveCMatrix $ \\mathbf {C}_m(\\mathbf {x}) \\rightarrow \\mathbf {K}, \\mathbf {f} $ Stiffness matrix eq:StiffnessMatrix $\\mathbf {K}, \\mathbf {f} \\rightarrow \\mathbf {S}$ solve eq:optimizationnngovnEq $\\mathbf {K}, \\mathbf {S} \\rightarrow J$ Objective, eq:optimizationnnobjective $ \\overline{\\mathbf {\\zeta }}, \\Gamma ^* \\rightarrow g_{\\Gamma }$ Contact area constraint eq:effectivecontact area $J, g_{\\Gamma } \\rightarrow L$ loss from eq:lossFunction $AD(L, \\mathbf {w}) \\rightarrow \\nabla L $ sensitivity analysis via Auto.",
"Diff $\\mathbf {w} , \\nabla L \\rightarrow \\mathbf {w} $ BFGS optimizer step $ \\alpha + \\Delta \\alpha \\rightarrow \\alpha $ increment penalty $ \\lambda + 2 \\alpha g_\\Gamma \\rightarrow \\lambda $ increment Lagrange multiplier eq:lossFunction $ p + \\Delta p \\rightarrow p$ continuation $\\text{epoch}++$ $|| \\Delta L || < \\Delta L_c^*$ or epoch < max_epoch check for convergence"
],
[
"Numerical Experiments",
"In this section, we conduct several experiments to illustrate the method and algorithm.",
"All experiments are conducted on a MacBook Air M2, using the PyTorch [40] environment.",
"The default settings are as follows: Neural Network: The NN comprises of 2 LeakyReLU-activated hidden layers with 20 neurons in each layer.",
"This corresponds approximately to 4730 design variables.",
"The initial values for $\\mathbf {w}$ are determined via Xavier weight initialization [20], with a seed value of 77.",
"Candidate microstructures: A set of 8 predefined microstructures (fig:microstructures) is used in the experiments.",
"Numerical homogenization is performed at six uniformly spaced sample points.",
"A quintic polynomial is then used to interpolate the components of the sampled permeabiliity matrix.",
"Material Penalization: The penalization $p$ is incremented every iteration by 0.02 using the continuation scheme [43], starting from a value of 1, with a maximum value of 8.",
"Loss Function: The initial constraint penalty is $\\alpha _0$ = 0.05, and is increased by $\\Delta \\alpha = 0.15$ per epoch.",
"Optimizer: L-BFGS optimizer with a strong Wolfe line search function is used [39].",
"The maximum number of iterations (epochs) is set to 25.",
"For convergence of optimization, we set change in loss $\\Delta L_c^{*} = 10^{-5}$ .",
"For reference, the offline numerical homogenization of the 8 microstructures, at 6 different sizes, was performed in 16 seconds."
],
[
"Validation",
"In the first experiment, we validate the proposed method using the double-pipe problem considered in [9], and illustrated in fig:doublepipeval(a).",
"The objective is to find the optimal topology of $33\\%$ fluid volume fraction that minimizes the dissipated power; contact area constraint is not imposed.",
"The domain is discretized into 15x15 elements.",
"The authors of [9] report the topology illustrated in fig:doublepipeval(b), with an objective value of $J=25.7$ .",
"In the proposed method, we use a single square microstructure (see fig:microstructures), and arrive at the topology illustrated in fig:doublepipeval(c), with an objective of $J=27.4$ .",
"We observe partial infills in some of the cells since the size is not penalized towards $0,1$ , i.e., we allow for intermediate sizes.",
"Figure: Validation using the double pipe problem ."
],
[
"Impact of microstructures",
"Next we replace the square microstructure with three other microstructures, namely the circle, fish-body-2, and Mucosa-10 (see fig:microstructures), one at a time, and study the impact of the microstructure on the dissipated power; the desired contact area is kept constant at $\\Gamma ^* = 60$ .",
"However, no constraint is imposed on the volume fraction.",
"The resulting topologies are illustrated in fig:doublepipesinglemstr.",
"Note that for the same contact area, the permeability of the fish-body-2 is higher than that of circle.",
"This results in the fish-body-2 having a smaller dissipated power compared to the circle.",
"On the other hand, despite a lower-permeability, Mucosa-10 performs better than the fish-body-2 since its contact area is significantly higher.",
"Figure: Impact of microstructure on dissipated power with Γ * =60\\Gamma ^* = 60.We then reduced the surface constraint to $\\Gamma ^* = 30$ ; the results are illustrated in fig:doublepipesinglemstr30.",
"Now, fish-body-2 and circle perform better than Mucosa-10.",
"This experiment illustrates that the choice of a microstructure critically depends on the constraints imposed.",
"Figure: Impact of microstructure on dissipated power with Γ * =30\\Gamma ^* = 30."
],
[
"Validation",
"For the second set of experiments, we consider the diffuser problem discussed in [9], and illustrated in fig:0-1val(a).",
"In the first experiment, the objective is to find the optimal topology of $50\\%$ fluid volume fraction that minimizes the dissipated power; contact area constraint is not imposed.",
"The domain is discretized into 15x15 elements.",
"The authors of [9] report the topology illustrated in fig:0-1val(b), with an objective value of $J=30.6$ .",
"Next, in the proposed method, the square microstructure is once again used for optimization.",
"The final topology is illustrated in fig:0-1val(c), consistent with fig:0-1val(b), with an objective of $33.4$ [9].",
"The single scale optimization was performed in five seconds.",
"Figure: (a) Diffuser problem.",
"(b) Topology reported in , (c) Topology generated via proposed method."
],
[
"Multiple microstructures",
"Two central hypotheses of the current work is that one can achieve better designs with larger number of candidate microstructures, and that the framework is computationally insensitive to the number of candidates.",
"To validate, we consider again the problem in fig:0-1val(a), but instead of the volume constraint, we impose a contact area constraint of $\\Gamma ^* = 70$ .",
"We consider the first $m$ microstructures, where $m = 1, 3, 5, 8$ , and study the impact on the dissipated power and computational time.",
"The resulting topologies are illustrated in fig:diffusermstrsvar, as expected, the objective improves as we allow for larger number of microstructures.",
"The computational time was approximately 62 seconds, independent of the number of microstructures.",
"Figure: Designs with 1, 3, 5 and 8 microstructures respectively."
],
[
"Validation",
"Next, we consider the bent-pipe problem proposed in [53], and illustrated in fig:fixsizeval(a).",
"The domain is discretized into 20x60 elements.",
"In [53], a two-scale topology optimization was carried out to minimize the dissipated power, with a constraint that the optimal microstructure must occupy exactly 25 percent of each unit cell.",
"The reported topology is illustrated in fig:fixsizeval(b); the final dissipated power was not reported.",
"However, as noted in [53], the computed microstructures resemble the fish-body.",
"In the proposed method, the fish-body-2 microstructure was chosen a priori, and its size was fixed to occupy 25 percent of each unit cell, as in [53].",
"The orientation of each microstructure was optimized, resulting in the design illustrated in fig:fixsizeval(c) with the final dissipated power of 16.6.",
"Not surprisingly, the final topology is similar to fig:fixsizeval(b).",
"To improve on this design, we removed the constraint of the 25 percent unit-cell volume occupation, and instead imposed a total (global) volume constraint of 25 percent, i.e., we allowed the size of each microstructure to vary as well.",
"The resulting topology is illustrated in fig:fixsizeval(d).",
"As one can observe, the dissipated power further reduces to 13.8.",
"Figure: Size fixed geometry validation (a) Domain and velocity boundary conditions (b) Solution reported in (c) Topology generated via proposed method (d) solution via proposed method without rotation of microstructures."
],
[
"Fluid flow validation",
"Next, for the above problem, we compute the pressure predicted using the GMTO framework and compare it against full-scale fluid flow simulation using Ansys.",
"Due to challenges in importing the geometry into ANSYS, the domain was discretized using a coarser mesh of 8x24 elements.",
"The pressure prediction using the homogenization-based GMTO framework is illustrated in fig:3dprint(a); the total pressure drop is approximately 49.0 Pascals.",
"We then exported the optimized topology as an \".stl\" file and imported it into ANSYS [16] for full-scale fluid-flow simulation.",
"The pressure drop predicted using ANSYS is illustrated in fig:3dprint(b); the total pressure drop is approximately 51.17 Pascals.",
"The 3d printed part is illustrated in fig:3dprint(c).",
"Figure: Bent pipe pressure drop using: (a) Proposed homogenization approach (b) Ansys (c) 3D printed design with microstructures."
],
[
"Pareto Designs",
"Exploring the Pareto-front is critical in making design choices and understanding the trade-off between the objective (dissipated power) and constraint (contact area).",
"To illustrate this, we computed the optimal topologies for various values of contact areas for the bent-pipe problem, by considering all 8 microstructures; the results are illustrated in fig:ObjVsperim.",
"Observe that as expected, the dissipated power increases with the contact area.",
"Further, one can observe that regions with high fluid flow (see inset) are dominated by fish-body-2 (that exhibits high permeability), whereas regions with low fluid flow are dominated by Mucosa-20 (that exhibits high contact area).",
"Figure: Dissipated power versus contact area."
],
[
"Resampling ",
"A subtle but important aspect of the proposed framework is that since the design fields (density, orientation, and size) are represented globally using the neural-network, one can obtain high-resolution topologies with no additional cost.",
"To illustrate, suppose we have computed the optimal weights $\\mathbf {w^*}$ and optimal topology for a given mesh discretization.",
"We can then sample the domain at a higher resolution using the optimized weights $\\mathbf {w^*}$ , resulting in a more-detailed topology (Note that this is not a simple linear interpolation.)",
"This is illustrated in fig:high res bent where we optimize using a 8 × 24 mesh, and then re-sample using a 16 × 48 mesh.",
"Figure: Extraction of high resolution design through resampling."
],
[
"Conclusion",
"A graded multi-scale fluid flow topology optimization framework was proposed where homogenization is performed off-line, followed by global optimization.",
"Two-scale designs with high contact area and low dissipated power were generated through the framework.",
"Furthermore, the computational cost was found to be independent of the number of pre-selected microstructures.",
"The neural-network configuration ensured that the partition of unity constraint was automatically satisfied.",
"Finally, the PyTorch environment allowed for automated sensitivity computation.",
"Currently, the framework is limited to microstructures with a single-size parameter.",
"A contact area constraint, whose value was arbitrarily chosen, was imposed in the current work.",
"We plan to explore extensions to multi-physics problems such as convection-driven heat transfer problems [17] where the contact area constraint is determined through the underlying physics.",
"Furthermore, it will be interesting to combine the proposed framework with data-driven methods [47]."
],
[
"Compliance with ethical standards",
"The authors declare that they have no conflict of interest.",
"The Python code pertinent to this paper is available at github.com/UW-ERSL/FluTO.",
"The authors would like to thank the support of National Science Foundation through grant CMMI 1561899.",
"The authors acknowledge Subodh Subedi for helping with the 3D printing."
]
] | 2209.08168 |
[
[
"Furthur developements regarding Euler equation xyz(x+y+z)=a"
],
[
"Abstract In this paper, we derived the parametric solution of Euler and Elkies, xyz(x+y+z) = a, in an elementary manner.",
"In addition we proved there are infinitely many parametric solutions of Euler's and Elkies's family of solutions."
],
[
"In this paper, we derived the parametric solution of Euler and Elkies in an elementary manner.",
"In addition we proved there are infinitely many parametric solutions of Euler's and Elkies's family of solutions.",
"1.",
"Introduction According to Elkies[1], in 1749 Euler takes a look at $xyz(x + y + z) = a$ and says that he has found, with quite some effort.",
"His parametric solution is as follows.",
"$x &= \\frac{6ast^3(at^4-2s^4)^2}{(4at^4+s^4)(2a^2t^8+10as^4t^4-s^8)}\\\\[5pt]y &= \\frac{3}{2}\\frac{s^5(4at^4+s^4)^2}{t(at^4-2s^4)(2a^2t^8+10as^4t^4-s^8)}\\\\[5pt]z &= \\frac{2}{3}\\frac{(2a^2t^8+10as^4t^4-s^8)}{s^3t(4at^4+s^4)}$ We don't know how he got his solution.",
"In 2014, Elkies looked for a solution, and he got some solutions using algebraic geometry.",
"Simpler solution is as follows.",
"$x &= \\frac{(s^4-4a)^2}{2s^3(s^4+12a)}\\\\[5pt]y &= \\frac{2a(3s^4+4a)^2}{s^3(s4-4a)(s^4+12a)}\\\\[5pt]z &= \\frac{s(s^4+12a)}{2(3s^4+4a)})$ In 2022,inspired by the problem posted on MathStackExchange, it is related to our problem, so we decided to work on elementary derivation of Euler's solution.",
"We proved that there are infinitely many parametric solutions of Euler's family of solutions.",
"In addition we derived Elkies's solution in an elementary manner and proved that there are infinitely many parametric solutions of Elkies's family of solutions.",
"Moreover we showed the small positive solutions table for $a <100$ .",
"Furthermore, parametric solution of $wxyz(w+x+y+z)\\ =\\ a$ was shown in Appendix.",
"An infinitely many solutions are generated from infinite order point of elliptic curve using group law.",
"In this context, we call generated solutions are family of solutions of infinite order point.",
"2.",
"Derive Euler's solution According to Elkies, Euler found a parametric solution of $xyz(x+y+z) = a$ below.",
"$x &= \\frac{6ast^3(at^4-2s^4)^2}{(4at^4+s^4)(2a^2t^8+10as^4t^4-s^8)}\\\\[5pt]y &= \\frac{3}{2}\\frac{s^5(4at^4+s^4)^2}{t(at^4-2s^4)(2a^2t^8+10as^4t^4-s^8)}\\\\[5pt]z &= \\frac{2}{3}\\frac{(2a^2t^8+10as^4t^4-s^8)}{s^3t(4at^4+s^4)}$ We prove that Euler's family of solutions has an infinitely many parametric solutions where $a$ is arbitrary.",
"Proof.",
"$xyz(x+y+z)\\ =\\ a$ We define Euler's solution form as follows.",
"$x &= \\frac{c_1as^s_1t^t_1A^2}{BC}\\\\[5pt]y &= \\frac{c_2s^s_2t^t_2B^2}{AC}\\\\[5pt]z &= \\frac{c_3s^s_3t^t_3C}{B}$ Taking $(s1,s2,s3)=(1, 5, -3),\\ (t1,t2,t3)=(3, -1, -1),\\ (c1,c2,c3)=(6, \\frac{3}{2}, \\frac{2}{3})$ , then $x\\ &=\\ \\frac{6Ast^3A^2}{BC}\\\\y\\ &=\\ \\frac{3}{2}\\frac{s^5B^2}{tAC } \\\\z\\ &=\\ \\frac{2}{3}\\frac{C}{s^3tB}$ Hence $xyz(x+y+z)\\ =\\ \\frac{a(36As^4t^4A^3+9s^8B^3+4C^2A)}{C^2B}$ RHS of above equation must be a,then $36As^4t^4A^3+9s^8B^3+4C^2A=C^2B\\nonumber $ Since $C$ must be rational number then discriminant must be square number.",
"Let $U=\\frac{A}{B}$ we get quartic curve $V^2=-16at^4U^4+4at^4U^3-4s^4U+s^4$ Quartic is birationally equivalent to an elliptic curve.",
"$E: Y^2-4s^2YX+8s^2at^4Y = X^3-4s^4X^2+64at^4s^4X-256at^4s^8$ E has a point $P(X,Y)=(4s^4,\\ 16s^6-8s^2at^4)$ .",
"Hence we can obtain a point $2Q(U) = \\cfrac{at^4-2s^4}{4at^4+s^4}$ from $2P(X,Y)$ using group law.",
"Then we obtain $(A,B,C)=(at^4-2s^4, 4at^4+s^4, -s^8+10as^4t^4+2a^2t^8)$ Finally, we obtain an Euler's solution.",
"$x &= \\frac{6ast^3(at^4-2s^4)^2}{(4at^4+s^4)(2a^2t^8+10as^4t^4-s^8)}\\\\[5pt]y &= \\frac{3}{2}\\frac{s^5(4at^4+s^4)^2}{t(at^4-2s^4)(2a^2t^8+10as^4t^4-s^8)}\\\\[5pt]z &= \\frac{2}{3}\\frac{(2a^2t^8+10as^4t^4-s^8)}{s^3t(4at^4+s^4)}$ Similarly, we can obtain other new solutions using group law with $P(X,Y)=(4s^4, 16s^6-8s^2at^4)$ .",
"Hence Euler's family of solutions has an infinitely many parametric solutions.",
"For instance, we can obtain a new solution using $3P(X,Y)$ as follows.",
"$x &= \\frac{18as^5t^3(-2s^4+at^4)^2(-s^8+10at^4s^4+2a^2t^8)^2(a^2t^8-5s^8+14at^4s^4)}{(at^4+s^4)(a^3t^{12}+3s^4a^2t^8+111at^4s^8+s^{12})(a^6t^{24}+6s^4a^5t^{20}-255s^8a^4t^{16}-790a^3t^{12}s^{12}-2253a^2t^8s^{16}-264s^{20}at^4+s^{24})}\\\\[5pt]y &= \\frac{1}{6}\\frac{-(at^4+s^4)^2(a^3t^{12}+3s^4a^2t^8+111at^4s^8+s^{12})^2(a^2t^8-5s^8+14at^4s^4)}{s^3t(-2s^4+at^4)(-s^8+10at^4s^4+2a^2t^8)(a^6t^{24}+6s^4a^5t^{20}-255s^8a^4t^{16}-790a^3t^{12}s^{12}-2253a^2t^8s^{16}-264s^{20}at^4+s^{24})}\\\\[5pt]z &= \\frac{2s(a^6t^{24}+6s^4a^5t^{20}-255s^8a^4t^{16}-790a^3t^{12}s^{12}-2253a^2t^8s^{16}-264s^{20}at^4+s^{24})}{(a^2t^8-5s^8+14at^4s^4)t(at^4+s^4)(a^3t^{12}+3s^4a^2t^8+111at^4s^8+s^{12})}\\\\$ The proof is completed.",
"3.",
"Derive Elkies's solution According to Elkies, Elkies found a parametric solution of $xyz(x+y+z) = a$ below.",
"$x &= \\frac{1}{2}\\frac{(s^4-4a)^2}{s^3(s^4+12a)}\\\\[5pt]y &= \\frac{2a(3s^4+4a)^2}{s^3(s^4-4a)(s^4+12a)}\\\\[5pt]z &= \\frac{1}{2}\\frac{s(s^4+12a)}{3s^4+4a}$ We show Elkies's family of solutions has an infinitely many parametric solutions where $a$ is arbitrary.",
"Proof.",
"$xyz(x+y+z)\\ =\\ a$ Taking $x\\ &=\\frac{1}{2}\\frac{A^2}{s^3C}\\\\[5pt]y\\ &=\\frac{2aB^2}{s^3AC}\\\\[5pt]z\\ &=\\frac{1}{2}\\frac{sC}{B}$ Hence $xyz(x+y+z)\\ =\\ \\frac{1}{4}\\frac{a(A^3B+4aB^3+s^4C^2A)}{s^8C^2}$ RHS of above equation must be $a$ ,then $A^3B+4aB^3+s^4C^2A-4s^8C^2=0\\qquad \\mathrm {(1)}$ Since $C$ must be rational number then discriminant must be square number.",
"$v^2 = -4(A-4s^4)s^4B(A^3+4aB^2)$ Let $B = -A+4s^4$ then $V^2 &= A^3+4aB^2 \\nonumber \\\\&= A^3+4aA^2-32s^4aA+64s^8a$ In order to find a parametrization for $A, V$ , substitute $(A,V)=(s^4+p, s^6+qs^4+rs^2)$ to (2).",
"We obtain $(p,q,r)=(-4a, 0, 12a)$ then $(A,B,C)=(s^4-4a, 3s^4+4a, s^4+12a)$ .",
"Finally, we obtain Elkies's solution.",
"$x &= \\frac{1}{2}\\frac{(s^4-4a)^2}{s^3(s^4+12a)}\\\\[5pt]y &= \\frac{2a(3s^4+4a)^2}{s^3(s^4-4a)(s^4+12a)}\\\\[5pt]z &= \\frac{1}{2}\\frac{s(s^4+12a)}{3s^4+4a}$ From (2), we define elliptic curve E $E: V^2 = A^3+4aA^2-32s^4aA+64as^8$ Since E has a point $P(A,V)=(s^4-4a, s^6+12as^2)$ , we can obtain a point $2P(A) = \\cfrac{1}{4}\\cfrac{s^{16}-464s^{12}a+1632s^8a^2+768a^3s^4+256a^4}{s^4(s^4+12a)^2}$ .",
"According to Nagell-Lutz theorem, the point $P(A,V)$ is not a point of finite order.",
"Hence Elkies's family of solutions has an infinitely many parametric solutions.",
"We can obtain new solutions using group law with $P(A,V)=(s^4-4a, s^6+12as^2)$ .",
"For instance, we can obtain a new solution using $2P(A,V)$ as follows.",
"We obtain $A &= \\frac{1}{4}\\frac{(-4a+s^4)(s^{12}-460s^8a-208a^2s^4-64a^3)}{s^4(s^4+12a)^2}\\\\[5pt]B &= \\frac{1}{4}\\frac{(-4a+5s^4)(4a+3s^4)(s^8+56s^4a+16a^2)}{s^4(s^4+12a)^2}\\\\[5pt]C &= \\frac{-1}{8}\\frac{(-16a^2-32s^4a+s^8)(s^{16}+1136s^{12}a-928s^8a^2+1792a^3s^4+256a^4)}{s^8(s^4+12a)^3}$ $x &= \\frac{-1}{4}\\frac{(-4a+s^4)^2(s^{12}-460s^8a-208a^2s^4-64a^3)^2}{s^3(s^4+12a)(-16a^2-32s^4a+s^8)(s^{16}+1136s^{12}a-928s^8a^2+1792a^3s^4+256a^4)}\\\\[5pt]y &= \\frac{-4(s^4+12a)s(-4a+5s^4)^2(4a+3s^4)^2(s^8+56s^4a+16a^2)^2a}{(s^{16}+1136s^{12}a-928s^8a^2+1792a^3s^4+256a^4)(-16a^2-32s^4a+s^8)(-4a+s^4)(s^{12}-460s^8a-208a^2s^4-64a^3)}\\\\[5pt]z &= \\frac{-1}{4}\\frac{(-16a^2-32s^4a+s^8)(s^{16}+1136s^{12}a-928s^8a^2+1792a^3s^4+256a^4)}{s^3(-4a+5s^4)(4a+3s^4)(s^8+56s^4a+16a^2)(s^4+12a)}$ The proof is completed.",
"4.",
"Solution table Small positive solutions by brute force search with $a<100$ .",
"[c]rrrr Solutions of $xyz(x+y+z) = a$ a x y z 1 3/2 4/3 1/6 2 5/2 5/6 4/15 3 1 1 1 4 7/2 36/35 7/30 5 4 1/2 1/2 6 2 3/2 1/2 7 10/3 21/20 5/12 8 2 1 1 9 2 2 1/2 10 5/2 4/3 2/3 11 2 11/6 2/3 12 5/3 27/20 5/4 13 3/2 3/2 4/3 14 2 4/3 7/6 15 3 1 1 16 3 8/3 1/3 17 5/2 10/7 34/35 18 4 3/2 1/2 19 3 3 1/3 20 2 2 1 21 5/2 5/2 3/5 22 9/2 25/6 2/15 23 5 9/10 23/30 24 4 1 1 25 4 9/4 5/12 26 4 2 1/2 27 9/4 25/12 16/15 28 3 7/3 2/3 29 4 29/20 4/5 30 4 15/4 1/4 31 8/3 31/12 3/4 32 5 5/3 8/15 33 2 2 3/2 34 7/2 7/3 9/14 35 3 3/2 4/3 36 3 2 1 37 20 37/15 1/30 38 6 12/7 19/42 39 4 3/2 1 40 3 3 2/3 41 3 25/12 16/15 42 15/4 7/5 5/4 43 9/2 2 2/3 44 7 7/4 11/28 45 4 3 1/2 46 8 25/12 4/15 47 25/6 32/15 3/4 48 2 2 2 49 6 3/2 2/3 50 5 5/2 1/2 51 4 17/12 4/3 52 6 26/15 3/5 53 7/2 7/2 4/7 54 5 8/5 9/10 55 5 11/3 1/3 56 4 2 1 57 16/5 25/8 4/5 58 20/3 6/5 5/6 59 5 49/30 20/21 60 8 3/2 1/2 61 32/3 9/2 1/12 62 9/4 25/12 31/15 63 3 3 1 64 9 1 2/3 65 3 13/6 3/2 66 5/2 5/2 8/5 67 15/2 32/15 5/12 68 4 4 1/2 69 13/4 13/4 23/26 70 6 35/6 1/6 71 10 49/20 8/35 72 16 25/8 3/40 73 6 6 1/6 74 15/2 10/3 4/15 75 4 5/2 1 76 7/2 18/7 7/6 77 5 16/5 11/20 78 4 13/4 3/4 79 8 25/12 9/20 80 5 2 1 81 9/2 4 1/2 82 10/3 5/2 41/30 83 5 5/2 4/5 84 3 2 2 85 9 16/9 17/36 86 3 8/3 3/2 87 9 25/3 1/15 88 3 3 4/3 89 27/2 2/3 2/3 90 4 2 3/2 91 4 9/4 4/3 92 5 23/5 2/5 93 11/2 11/2 3/11 94 9/2 2 4/3 95 5 4 1/2 96 4 3 1 97 10/3 49/20 45/28 98 7/2 7/2 1 99 5/2 5/2 11/5 5.",
"Final remarks We used the Euler's solution form, combination of $(A,B,C)$ to derive Euler's parametric solution.",
"So, we think there must be parametric solutions other than Euler's form.",
"The same could be said of Elkies case in a similar way.",
"Furthermore, it might be interesting to construct the parametric solutions using $(A,B,C,D)$ .",
"Appendix We consider the extension of $xyz(x+y+z) = a$ and show $wxyz(w+x+y+z) = a$ has infinitely many parametric solutions where $a$ is arbitrary.",
"The problem $wxyz(w+x+y+z) = 1$ was posted on mathoverflow.net.",
"$\\textbf { An equation wxyz(w+x+y+z) = a}$ has an infinitely many parametric solutions where $a$ is arbitrary.",
"Proof.",
"$wxyz(w+x+y+z)=a$ $w &= \\frac{ac_1t^{t_1}}{ABC}\\\\x &= \\frac{c_2Bt^{t_2}}{A}\\\\y &= \\frac{c_3At^{t_3}}{C}\\\\z &= \\frac{c_4Ct^{t_4}}{B}$ Hence $wxyz(w+x+y+z) = \\frac{c_1t^{t_1}c_2t^{t_2}c_3t^{t_3}c_4t^{t_4}(ac_1t^{t_1}+c_2B^2t^{t_2}C+c_3A^2t^{t_3}B+c_4C^2t^{t_4}A)}{A^2B^2C^2}$ RHS of above equation must be a,then $&(c_1c_2c_3c_4^2t^{t_1+t_2+t_3+2t_4}A-A^2B^2)C^2\\\\&+c_1c_2^2c_3c_4t^{t_4+t_3+t_1+2t_2}B^2C\\\\&+ac_1^2c_2c_3c_4t^{t_4+t_3+t_2+2t_1}+c_1c_2c_3^2c_4t^{t_1+t_2+t_4+2t_3}A^2B=0$ Since $C$ must be rational number then discriminant must be square number.",
"$v^2 &= 4c_1c_2c_3^2c_4t^{t_4+t_2+t_1+2t_3}B^3A^4\\\\&-4c_1^2c_2^2c_3^3c_4^3Bt^{3t_4+3t_3+2t_2+2t_1}A^3\\\\&+a4c_1^2c_2c_3c_4t^{t_4+t_3+t_2+2t_1}B^2A^2\\\\&-a4c_1^3c_2^2c_3^2c_4^3t^{3t_4+2t_3+2t_2+3t_1}A\\\\&+c_1^2c_2^4c_3^2c_4^2B^4t^{2t_4+2t_3+4t_2+2t_1}$ Obviously, quartic is birationally equivalent to an elliptic curve.",
"For instance, take $(c_1,c_2,c_3)=(1,1,1),(t_1,t_2,t_3)=(1,1,1),B=1$ , then $v^2 = 4t^5A^4-4t^{10}A^3+4t^5aA^2-4t^{10}aA+t^{10}$ Quartic is transformed to an elliptic curve below.",
"$E: Y^2-4t^5aYX-8t^{15}Y = X^3+(4t^5a-4t^{10}a^2)X^2-16t^{15}X-64t^{20}a+64t^{25}a^2$ $E$ has a point $P(X,Y)=(-4t^5a+4t^{10}a^2,\\ -16t^{10}a^2+16t^{15}a^3+8t^{15})$ .",
"We obtain $2P(X)= \\frac{4((-a^2-2a^5+a^8)t^{20}+(-2a^4-a-4a^7)t^{15}+(1+6a^6+6a^3)t^{10}+(-2a^2-4a^5)t^5+a^4)}{((2a^3+1)t^5-2a^2)^2}$ According to Nagell-Lutz theorem, the point $P(X,Y)$ is not a point of finite order, hence we can obtain infinitely many parametric solutions.",
"Thus we can obtain a quartic point $2Q(A) = \\frac{-t^5(-2a^2+2t^5a^3+t^5)}{t^{10}a^4-2t^5a^3-t^5+a^2}$ Then we obtain $(A,C)$ $A &= \\frac{-t^5(-2a^2+2t^5a^3+t^5)}{t^{10}a^4-2t^5a^3-t^5+a^2}\\\\[5pt]C &= \\frac{t^{10}+a-2t^5a^2+t^{10}a^3}{t^{10}a^2-1}$ Finally we obtain $(w,x,y,z)$ $w &= \\frac{-a(t^{10}a^4-2t^5a^3-t^5+a^2)(t^5a+1)(t^5a-1)}{t^4(-2a^2+2t^5a^3+t^5)(t^{10}+a-2t^5a^2+t^{10}a^3)}\\\\[5pt]x &= \\frac{-(t^{10}a^4-2t^5a^3-t^5+a^2)}{t^4(-2a^2+2t^5a^3+t^5)}\\\\[5pt]y &= \\frac{-t^6(-2a^2+2t^5a^3+t^5)(t^5a+1)(t^5a-1)}{(t^{10}a^4-2t^5a^3-t^5+a^2)(t^{10}+a-2t^5a^2+t^{10}a^3)}\\\\[5pt]z &= \\frac{(t^{10}+a-2t^5a^2+t^{10}a^3)t}{(t^5a+1)(t^5a-1)}$ In this way we can obtain infinitely many parametric solutions.",
"The proof is completed.",
"MathStackExchange, https://math.stackexchange.com/questions/4496520/solving-the-diophantine-system-pqr-a4-pqr-b4 Mathoverflow.net, https://mathoverflow.net/questions/428053/equation-wxyzwxyz-1-in-mathbbq-4"
]
] | 2209.08157 |
[
[
"PlaneSLAM: Plane-based LiDAR SLAM for Motion Planning in Structured 3D\n Environments"
],
[
"Abstract LiDAR sensors are a powerful tool for robot simultaneous localization and mapping (SLAM) in unknown environments, but the raw point clouds they produce are dense, computationally expensive to store, and unsuited for direct use by downstream autonomy tasks, such as motion planning.",
"For integration with motion planning, it is desirable for SLAM pipelines to generate lightweight geometric map representations.",
"Such representations are also particularly well-suited for man-made environments, which can often be viewed as a so-called \"Manhattan world\" built on a Cartesian grid.",
"In this work we present a 3D LiDAR SLAM algorithm for Manhattan world environments which extracts planar features from point clouds to achieve lightweight, real-time localization and mapping.",
"Our approach generates plane-based maps which occupy significantly less memory than their point cloud equivalents, and are suited towards fast collision checking for motion planning.",
"By leveraging the Manhattan world assumption, we target extraction of orthogonal planes to generate maps which are more structured and organized than those of existing plane-based LiDAR SLAM approaches.",
"We demonstrate our approach in the high-fidelity AirSim simulator and in real-world experiments with a ground rover equipped with a Velodyne LiDAR.",
"For both cases, we are able to generate high quality maps and trajectory estimates at a rate matching the sensor rate of 10 Hz."
],
[
"Introduction",
"Simultaneous localization and mapping (SLAM) is a core functionality necessary for autonomous robots operating in unknown environments.",
"For robot navigation and obstacle avoidance, maps generated by SLAM may be used by downstream motion planning algorithms to plan collision-free paths through the environment, bridging the perception and planning layers of the autonomy stack.",
"Successful integration requires maps to be generated in real-time from sensor data, occupy minimal memory, and support fast and precise collision-checking.",
"LiDAR (Light Detection and Ranging) has emerged as a powerful sensor for spatial mapping and SLAM due to its ability to provide highly accurate depth information under a wide range of lighting and weather conditions.",
"However, operating directly on LiDAR point clouds can quickly become computationally expensive as the number of points increases, and point cloud maps created by LiDAR SLAM rapidly grow in memory as the robot explores more area.",
"One way to address this challenge is to leverage structure, and represent points which lie on the same plane with a single object, which saves memory and compute for later operations.",
"This strategy is particularly effective for man-made urban and indoor environments, which consist primarily of mutually orthogonal planes.",
"These environments can be modeled as Manhattan worlds [1] and are conducive to lightweight geometric map representations.",
"Maintaining such minimal memory maps is especially important for embedded or resource-constrained robotic applications and for multi-robot settings, in which maps must be shared between robots over limited communication bandwidth.",
"Furthermore, converting point clouds to planar representations is beneficial for operations such as collision checking, which are essential to motion planning.",
"Planes allow for continuous representation of boundaries in space, which can be described with concise mathematical expressions, whereas point clouds may visually form a surface, but mathematically are still only a discrete collection of points which, by themselves, do not define a boundary.",
"Traditionally, mapping and planning algorithms are designed separately, where maps produced by SLAM may not be compatible with motion planners, and planners may assume a map representation which is difficult to construct from real sensor data.",
"SLAM map representations which enable rapid collision checking allow for direct integration with downstream motion planning for autonomous mobile robots, and design of a fluid, real-time autonomy stack.",
"In this work, we present an approach for LiDAR-based SLAM in Manhattan worlds that extracts planes from point clouds for registration and map creation.",
"Contrary to most prior works in SLAM, our approach is designed with the task of motion planning in mind, as our geometric plane-based map representation enables fast and precise collision checking.",
"Furthermore, by registering sets of planes instead of dense point clouds, we speed up registration while maintaining accuracy, enabling faster odometry and loop closure.",
"We leverage the Manhattan world assumption to extract orthogonal planes, and by merging these planes sequentially, we are able to create lightweight maps based on geometric primitives which capture the overarching 3D structure of the environment while occupying significantly less storage than their point cloud counterparts (an example such map is shown in Fig.",
"REF ).",
"Finally, we integrate our plane-based frontend with a pose graph backend for loop closure and global consistency, and demonstrate real-time localization and mapping, as well as rapid motion planning within our map with an RRT example."
],
[
"Related Work",
"LiDAR SLAM has been well studied, and established methods such as LOAM [2] and its variations (LeGO-LOAM [3], F-LOAM [4]) rely on point clouds as a natural choice for registration and map representation.",
"These methods achieve accurate real-time localization and mapping, but the maps they produce are still point clouds, which grow rapidly in memory and are unsuited for direct use by motion planners.",
"Although point cloud maps are able to capture high levels of detail, for predominantly planar environments such as Manhattan worlds, they encode highly redundant information when compared to a direct plane representation.",
"Extracting planes from point clouds is not a new idea [5], [6], [7].",
"One general strategy is to search for best-fit planes to the point cloud, which can be accomplished via RANSAC [5], [8] or the Hough transform [6], [9].",
"Another strategy involves segmenting the point cloud into clusters of points which belong to the same plane, then extracting planes for each of these clusters.",
"Point cloud segmentation is commonly done using region growing [10], in which clusters of points are grown based on local surface normal similarity.",
"These region growing methods rely on computing surface normals of local neighborhoods via principal component analysis (PCA).",
"An alternative method for normal estimation is meshing the point cloud and computing normals for each triangle or polygon in the mesh.",
"Polylidar3D [7] introduced an algorithm for fast non-convex polygon extraction from depth sensor data which relies on triangulation and region growing.",
"One can also use meshes to construct lightweight scene representations, as in [11].",
"Prior works have also explored plane-based registration and SLAM.",
"Pathak et al.",
"[12] demonstrate plane-based registration and SLAM in real-world scenarios, generating maps that are orders of magnitude smaller than their point cloud equivalents.",
"However, their approach lacks solutions for merging planes together and is not suitable for real-time operation, as their core registration takes around 10 seconds.",
"More recently, Grant et al.",
"[13] introduced Velodyne SLAM using point and plane features, extracting planes when possible to lower computation, while leveraging raw points when necessary to fully constrain the registration problem.",
"Their approach generates maps consisting of polygonal planar patches and is shown to work well on a variety of indoor and outdoor environments, outperforming both Generalized ICP and LOAM, while running at rates between 2 and 10 Hz (depending on the number of points vs. planes used).",
"Favre et al.",
"[14] also present an approach for plane-based point cloud registration which is shown to outperform traditional point-based methods while running at 1-3 Hz on various datasets, but does not consider the problem of SLAM and map generation.",
"Additionally, a variety of approaches for generating plane-based maps from dense depth information exist [8], [15], [16].",
"However, prior works in plane-based SLAM only present their maps for visualization, and do not consider the problem of motion planning within their plane maps.",
"In contrast, our work highlights the advantage of plane maps for fast and efficient motion planning.",
"Additionally, through our plane extraction and merging approach, we are able to produce more structured and organized maps for Manhattan world environments."
],
[
"Problem Statement and Preliminaries",
"Given a robot equipped with a LiDAR sensor traveling through an unknown environment, our task is to accurately estimate the robot's trajectory while simultaneously constructing a sparse geometric map of the environment.",
"At frame $k$ our method takes as input an unorganized 3D point cloud $P_k = \\lbrace (x_1,y_1,z_1),\\dots ,(x_N,y_N,z_N)\\rbrace $ .",
"We represent the trajectory as a sequence of poses $(R_k,\\mathbf {t}_k)$ , where $R_k\\in \\mathbb {R}^{3\\times 3}$ is a rotation matrix describing orientation and $\\mathbf {t}_k\\in \\mathbb {R}^3$ is a translation vector describing position.",
"For our approach we represent the map as a collection of rectangular bounded planes (which we henceforth refer to simply as planes).",
"We parameterize a plane $\\mathcal {P}$ with center point $\\mathbf {c}$ and a pair of orthogonal vectors $\\mathcal {P}_x, \\mathcal {P}_y \\in \\mathbb {R}^3$ which span its area (see Fig.",
"REF ).",
"We define a plane's basis as $B = [\\mathbf {b}_x, \\mathbf {b}_y, \\mathbf {b}_z]$ , where $\\mathbf {b}_x = \\mathcal {P}_x/||\\mathcal {P}_x||$ , $\\mathbf {b}_y = \\mathcal {P}_y/||\\mathcal {P}_y||$ , and $\\mathbf {b}_z = \\mathbf {b}_x \\times \\mathbf {b}_y$ .",
"Note that $\\mathbf {b}_z$ is also the plane's normal vector $\\mathbf {n}$ .",
"This basis can be used to map the plane to 2D, as shown in Fig.",
"REF .",
"The 2D plane representation is essential to our plane extraction process, and for operations used later in plane merging and collision checking."
],
[
"Approach",
"As with most SLAM pipelines, our approach consists of a frontend which performs feature extraction and registration, and a backend which maintains and optimizes a pose graph and handles map generation.",
"Our approach overview is shown in Fig.",
"REF .",
"We now describe the components of our approach in detail."
],
[
"Plane Extraction",
"When we receive a new point cloud from our LiDAR sensor (Fig.",
"REF ), the first step is to extract planar features from it.",
"Our extraction approach uses region growing for point cloud clustering, with meshing for surface normal estimation and graph structure.",
"In order to mesh the point cloud, we first reparameterize each point in spherical coordinates relative to the scan-centered coordinate frame: $\\theta _i = \\operatorname{arctan2}(y_i,x_i), \\ \\phi _i = \\operatorname{arctan2}(z_i, \\sqrt{x_i^2 + y_i^2}).$ We then compute the Delaunay triangulation [17] of the $(\\theta _i, \\phi _i)$ points.",
"The result of the triangulation is a set of simplices $(i,j,k)$ , where $i,j,k \\in \\lbrace 1,\\dots ,N\\rbrace $ are indices of points which form a triangle in the triangulation.",
"Lastly, we prune the mesh to remove triangles with side lengths which exceed a certain threshold and smooth the mesh with a Laplacian filter [18] to reduce noise.",
"Fig.",
"REF shows an example of point cloud meshing."
],
[
"Clustering",
"Next, we cluster the triangles in the mesh according to their surface normal and locality.",
"We do this through breadth-first graph search over the mesh triangles.",
"First, we compute a surface normal vector for each triangle in the mesh by taking the cross product of two of its edge vectors $\\mathbf {n}= \\mathbf {e}_1 \\times \\mathbf {e}_2$ .",
"We then randomly select a root triangle to begin the cluster and initialize a search queue $S$ with its neighbors.",
"At each iteration, a new triangle is removed from $S$ , and its normal vector is compared to the root triangle's normal by taking their dot product.",
"If the normals are close enough (i.e., their dot product exceeds a threshold), the triangle is added to the current cluster, and its neighbors are added to $S$ .",
"The cluster is grown until $S$ becomes empty, indicating there are no more neighboring triangles with normals similar to the root's normal.",
"This process is repeated with a new unclustered root triangle until all triangles in the mesh have been clustered.",
"The mesh clustering process is summarized in Algorithm REF .",
"Mesh $M$ , normals $\\mathbf {n}$ Clusters $C$ poppop appendappend removeremove updateupdate lenlen $Q$ = {set of all Mesh indices} $C$ = [] $Q$ not empty root = $Q$ .",
"cluster = [root] $S$ = {neighbors of root also in $Q$ } $S$ not empty $i$ = $S$ .",
"$\\mathbf {n}[i]^\\mathsf {T}\\mathbf {n}[\\mathrm {\\textnormal {root}}] > thresh$ cluster.i $Q$ .i $S$ .neighbors of $i$ also in $Q$ $C$ .cluster Mesh clustering Once we have a clustering of the mesh triangles, we can use it to extract a clustering over the points themselves.",
"Fig.",
"REF shows an example of clustered points.",
"Figure: Plane extraction process.",
"First the point cloud () is triangulated to form a mesh ().",
"Then graph search is used to cluster the mesh triangles and corresponding points ().",
"Finally, planes () are extracted for each cluster of points."
],
[
"Extraction",
"Given clusters of points $\\lbrace P^{(1)},\\dots ,P^{(k)}\\rbrace $ with associated average normals $\\lbrace \\mathbf {n}^{(1)},\\dots ,\\mathbf {n}^{(k)}\\rbrace $ (computed as the normalized average of the normal vectors of all the mesh triangles in the cluster), we now aim to extract a plane for each cluster.",
"First we determine a basis $B = [\\mathbf {b}_x, \\mathbf {b}_y, \\mathbf {b}_z]$ for the ground plane.",
"Then, leveraging the Manhattan world assumption, all planes are extracted such that their normals aligns with either $\\mathbf {b}_x, \\mathbf {b}_y$ or $\\mathbf {b}_z$ , and the resulting set of extracted planes are mutually orthogonal.",
"To find $B$ , we first identify the ground plane as the largest cluster with normal closest to the $z$ direction (i.e., $(0,0,1)$ ), and set its normal to $\\mathbf {b}_z$ .",
"Next, the largest cluster with normal approximately orthogonal to the ground plane is found, and $\\mathbf {b}_x$ is chosen as the orthogonal projection of its normal onto the ground plane.",
"Finally $\\mathbf {b}_y = \\mathbf {b}_z \\times \\mathbf {b}_x$ .",
"Once we have the extraction basis, the plane extraction procedure for each cluster $P^{(i)}$ (with average normal $\\mathbf {n}^{(i)}$ ) is as follows.",
"First we project $P^{(i)}$ onto $B$ to obtain $P^{(i)}_B =~B^{-1}P^{(i)}$ .",
"Next we find the basis vector closest to $\\mathbf {n}^{(i)}$ and extract a 2D bounding box over the points in the remaining dimensions (e.g.",
"if $\\mathbf {n}^{(i)}$ is closest to $\\mathbf {b}_x$ , then the bounding box is extracted over $y$ and $z$ ).",
"Once the 2D bounding box over $P^{(i)}_B$ is found, the center point and the vectors which span the bounding box are projected back to the standard basis to obtain the parameters $\\mathbf {c}, \\mathcal {P}_x$ and $\\mathcal {P}_y$ for the final extracted plane.",
"The output of the extraction is a plane set $\\mathcal {P}^k =~\\lbrace \\mathcal {P}^k_1,\\dots ,\\mathcal {P}^k_n\\rbrace $ (example shown in Fig.",
"REF ).",
"For plane $\\mathcal {P}^k_i$ , $\\mathbf {n}^k_i$ is its normal vector, $\\mathbf {c}^k_i$ is its center point, and $d^k_i$ is its distance to origin (computed as $d^k_i = \\mathbf {n}^k_i{^\\mathsf {T}} \\mathbf {c}^k_i$ )."
],
[
"Plane Registration",
"Next, in order to perform odometry and loop closure, we must be able to perform registration between two plane sets.",
"Given a source plane set $\\mathcal {P}^s$ and target plane set $\\mathcal {P}^t$ with known correspondences denoted by the common index $i$ , we formulate the registration problem as $ \\begin{split} \\operatornamewithlimits{min.",
"}_{R, \\mathbf {t}} \\quad & \\sum _{i=1}^n ||R\\mathbf {n}^s_i - \\mathbf {n}^t_i||_2^2 + ||(R\\mathbf {n}^s_i)^\\mathsf {T}\\mathbf {t}+ d^s_i - d^t_i||_2^2 \\\\\\mathrm {\\textnormal {s.t.}}",
"\\quad & R\\in \\mathbf {SO}(3), \\ \\mathbf {t}\\in \\mathbb {R}^3\\end{split} $"
],
[
"Correspondences",
"In order to determine correspondences, we compute a similarity cost metric for each pair of source and target planes based on their normal vectors and center-to-center distance $c(\\mathcal {P}^s_i,\\mathcal {P}^t_j) = \\alpha ||\\mathbf {n}^s_i - \\mathbf {n}^t_j|| + \\beta ||\\mathbf {c}^s_i - \\mathbf {c}^t_j||$ where $\\alpha $ and $\\beta $ are user-specified scaling parameters.",
"We then form a cost matrix from the $c(\\mathcal {P}^s_i,\\mathcal {P}^t_j)$ , and select the minimum value from each row (i.e., lowest cost assignment for each source plane to a target plane).",
"Additionally, if different source planes are assigned to the same target plane, we choose only the lowest cost assignment.",
"In this way, each plane in both the source and target is uniquely corresponded, which we observed to reduce false correspondences."
],
[
"Registration",
"As shown in [12], [8], the rotation and translation components of (REF ) can be decoupled.",
"For rotation estimation, we define the rotation residual $r_i(R) = R\\mathbf {n}^s_i - \\mathbf {n}^t_i$ and use the Gauss-Newton method to solve $ \\begin{split}\\operatornamewithlimits{min.",
"}_{R} \\quad & \\sum _i^n ||r_i(R)||_2^2 \\\\\\mathrm {\\textnormal {s.t.}}",
"\\quad & R\\in \\mathbf {SO}(3).\\end{split} $ Since $R \\in \\mathbf {SO}(3)$ , we use the Lie algebra parameterization [19] to map the problem to an unconstrained optimization over $\\mathbb {R}^3$ .",
"The Gauss-Newton update proceeds as $r &= \\mathbf {r}(R^{(i)}) \\\\J &= \\mathbf {J}(R^{(i)}) \\\\d\\omega &= -\\mu (J^T J + \\lambda I)^{-1} J^T r \\\\R^{(i+1)} &= \\exp (d\\omega ) R^{(i)}$ where $\\mu $ and $\\lambda $ are the update step and regularization parameters respectively, $\\exp $ is the $\\mathbf {SO}(3)$ exponential map, $\\mathbf {r}$ is the stacked rotation residual, and $\\mathbf {J}$ is the Jacobian of $\\mathbf {r}$ $\\mathbf {r}(R) = \\begin{bmatrix} r_1(R) \\\\ \\vdots \\\\ r_n(R) \\end{bmatrix}, \\ \\mathbf {J}(R) = \\begin{bmatrix} -(R \\mathbf {n}^s_1)^\\wedge \\\\ \\vdots \\\\ -(R \\mathbf {n}^s_n)^\\wedge \\end{bmatrix}$ where $^\\wedge $ denotes the skew symmetric operator.",
"The derivation of $\\mathbf {J}$ can be achieved using chain rule with Equation 10.8 from [19].",
"After we obtain a rotation estimate $\\hat{R}$ from Gauss-Newton, we set $R = \\hat{R}$ and solve for $\\mathbf {t}$ $ \\begin{split}\\operatornamewithlimits{min.",
"}_{\\mathbf {t}} \\quad & \\sum _i^n ||(\\hat{R}\\mathbf {n}^s_i)^\\mathsf {T}\\mathbf {t}+ d^s_i - d^t_i||_2^2 \\\\\\mathrm {\\textnormal {s.t.}}",
"\\quad & \\mathbf {t}\\in \\mathbb {R}^3\\end{split} $ We can formulate this as a least squares problem with $A = \\begin{bmatrix} \\hat{R}\\mathbf {n}^s_1 \\\\ \\vdots \\\\ \\hat{R}\\mathbf {n}^s_n \\end{bmatrix}, \\ b = \\begin{bmatrix} d^t_1 - d^s_1 \\\\ \\vdots \\\\ d^t_n - d^s_n \\end{bmatrix}.$ The least squares solution is thus given by $\\hat{\\mathbf {t}} = (A^\\mathsf {T}A)^{-1} A^\\mathsf {T}b$ .",
"Note that the least squares problem is rank deficient unless there are at least three correspondences whose normals span $\\mathbb {R}^3$ .",
"Our current implementation assumes this condition is always satisfied, although this issue has been addressed in priors works [12]."
],
[
"Outlier Correspondence Removal",
"As the computed correspondences are not always perfect, there is a need to make our registration robust to faulty correspondences.",
"We employ an outlier removal method inspired by residual-based fault detection and exclusion [20], in which first the magnitude of the translation residual is computed to determine if a fault is present.",
"If so, then the correspondence associated with the largest component of the residual is iteratively removed until the optimization yields a solution under the desired threshold.",
"We employ the standard pose-graph formulation presented in [21], in which nodes of the graph represent the pose of the robot at different time frames, and edges between nodes represent measurements.",
"The pose graph is initialized with some initial pose $(R_0, \\mathbf {t}_0)$ .",
"At frame $k$ , a new LiDAR scan is taken and plane set $\\mathcal {P}^k$ is extracted, which is registered with $\\mathcal {P}^{k-1}$ to compute relative transformation $(R_{k-1}^k, \\mathbf {t}_{k-1}^k)$ .",
"This transformation is used to compute the new absolute pose estimate for current frame $k$ : $ \\begin{split}R_k &= R_{k-1}^k R_{k-1} \\\\\\mathbf {t}_k &= \\mathbf {t}_{k-1} + R_{k-1} \\mathbf {t}_{k-1}^k\\end{split} $ which is then used to initialize the new node $k$ , and $(R_{k-1}^k, \\mathbf {t}_{k-1}^k)$ is also used to create an odometry edge between node $k-1$ and $k$ .",
"After node $k$ is added to the graph, we search nearby nodes for loop closure.",
"If the distance between nonconsecutive nodes $k$ and $j$ is below our search radius, their plane sets are registered to compute the relative transformation for a loop closure edge.",
"Since in loop closure situations, scans can have large relative transformations, the absolute poses of the nodes are used to first transform $\\mathcal {P}^k$ and $\\mathcal {P}^j$ to the world frame to obtain correspondences, before the registration is run on the original planes in the local frame.",
"After a loop closure edge is added, the pose graph is optimized as in [21]."
],
[
"Plane Set Merging",
"Our map is formed through a recursive merging process, in which the first plane set is initialized as the map, and successive plane sets are merged in.",
"Once a plane set has been registered and its absolute pose estimated, it is transformed and merged with the current version of the map.",
"The merging process takes as input two plane sets $\\mathcal {P}^s$ and $\\mathcal {P}^t$ , and iterates through the planes in $\\mathcal {P}^s$ .",
"For each plane $\\mathcal {P}^s_i \\in \\mathcal {P}^s$ , we then iterate through each plane $\\mathcal {P}^t_j \\in \\mathcal {P}^t$ and check three conditions: The two planes are approximately coplanar.",
"The plane-to-plane distance is below a threshold.",
"The 2D projections of $\\mathcal {P}^s_i$ and $\\mathcal {P}^t_j$ onto $\\mathcal {P}^s_i$ 's basis overlap.",
"If these conditions are satisfied, then $\\mathcal {P}^t_j$ is added to $\\mathcal {P}^s_i$ 's correspondence set.",
"After we have iterated through all planes in $\\mathcal {P}^t$ , the corresponding planes are projected onto $\\mathcal {P}^s_i$ 's basis, and the 2D bounding box over $\\mathcal {P}^s_i$ and all its corresponding planes is found.",
"This box is then projected back to the standard basis to obtain the new merged plane.",
"In this way, all corresponding planes in $\\mathcal {P}^s$ and $\\mathcal {P}^t$ are merged to form a new overall merged plane set.",
"Following merging, we also clean up the map by removing planes with small area and fusing the edges of planes near each other.",
"In the event of loop closure and pose graph optimization, the map must be regenerated by transforming all plane sets $\\mathcal {P}^0,\\dots ,\\mathcal {P}^k$ based on the newly re-optimized trajectory and re-merging them all together."
],
[
"Collision Checking",
"We briefly describe a fast collision check procedure for a plane and line segment.",
"Although we focus on this specific example, which is used to generate an RRT later in our results, the plane representation also supports efficient collision checking with other types of trajectory representations and parameterizations.",
"Given a line segment $\\ell $ represented by two points ${\\mathbf {a}}, {\\mathbf {b}} \\in ~\\mathbb {R}^3$ and plane $\\mathcal {P}$ with normal $\\mathbf {n}$ and center $\\mathbf {c}$ , we wish to determine if $\\ell $ intersects with $\\mathcal {P}$ .",
"First we project both $\\ell $ and $\\mathcal {P}$ to $\\mathcal {P}$ 's basis.",
"Since the remainder of the intersection check process deals only with projected versions of $\\ell $ and $\\mathcal {P}$ , we will hereby refer to the projected versions as $\\ell $ and $\\mathcal {P}$ .",
"Next we compute the intersection of the infinite line formed by extending $\\ell $ with the infinite plane that $\\mathcal {P}$ lies in.",
"The infinite plane can be represented as $\\lbrace {\\mathbf {x}}\\ |\\ \\mathbf {n}^\\mathsf {T}{\\mathbf {x}} = \\mathbf {c}\\rbrace $ , and the infinite line can be represented as $\\lbrace c {\\mathbf {v}} + {\\mathbf {a}}\\ |\\ c\\in \\mathbb {R}\\rbrace $ where ${\\mathbf {v}} = {\\mathbf {b}} - {\\mathbf {a}}$ .",
"Thus we can find the intersection by solving for $c\\in \\mathbb {R}$ such that $\\mathbf {n}^\\mathsf {T}(c{\\mathbf {v}} + {\\mathbf {a}}) = \\mathbf {c}$ .",
"Furthermore, since we have projected to $\\mathcal {P}$ 's basis, we know that the projected plane lies in the $xy$ plane, and $\\mathbf {n}= (0,0,1)$ .",
"Thus we have $c v_3 + a_3 = {\\mathbf {c}}_3$ so $c = ({\\mathbf {c}}_3 - a_3) / v_3$ .",
"If $v_3 = 0$ then $\\ell $ is parallel to the $xy$ plane and there is no intersection.",
"Otherwise, we then proceed to check if the intersection point $c{\\mathbf {v}} + {\\mathbf {a}}$ lies within the line segment and plane boundaries.",
"To check the line segment constraint, we simply check if $0 \\le c \\le 1$ .",
"To check the plane constraint, we check if the 2D box defined by $xy$ dimensions of $p$ contains $c{\\mathbf {v}} + {\\mathbf {a}}$ .",
"If both of these conditions are satisfied, then $\\ell $ and $\\mathcal {P}$ intersect, otherwise, they do not intersect.",
"We evaluate our algorithm both in simulation and on hardware.",
"For simulation, we use AirSim [22] to simulate a drone flying through an environment and collecting LiDAR measurements.",
"For our hardware tests, we use a ground rover equipped with a Velodyne Puck LITE LiDAR shown in Fig.",
"REF .",
"Our method is implemented in python with documented code available onlinehttps://github.com/Stanford-NavLab/planeslam.",
"The python-graphslam library [23] is used for pose graph creation and optimization."
],
[
"AirSim Experiments",
"We use the “Blocks\" environment, shown in Fig.",
"REF , which consists of several rectangular blocks placed in an open space.",
"The AirSim LiDAR is configured with 16 channels, 60$$ vertical field of view (FOV), 360$$ horizontal FOV, 10 rotations per second, and 100,000 points per second, and the experiments are run on a 6-core, 3.6 GHz desktop with 32 GB RAM.",
"We show results for a trajectory in which the drone traverses the central area of the environment while LiDAR point clouds are collected at 10 Hz.",
"Fig.",
"REF shows the final map generated by our approach along with the ground-truth and estimated trajectories.",
"For trajectory of total length 603.6, the RMSE translational error is 1.3 with standard deviation (std.)",
"0.6, and the average rotational error (computed as the angle between the estimated and ground-truth rotational frames) is 0.30 degrees with std.",
"0.14 degrees.",
"Table REF shows a breakdown of the average runtime of each module of our method in milliseconds.",
"Note that loop closure has a high standard deviation as it is not run every iteration.",
"Our overall algorithm has an average runtime of 88.8 with standard deviation 24.6, which is comparable to LeGO-LOAM and F-LOAM, and is fast enough to be run at the LiDAR rate of 10 Hz.",
"Table: Average runtimes per iteration for each module of our method.",
"Our method can be run in real-time at 10 Hz and is comparable in speed to state-of-the-art approaches.We also compare the size of our plane map in Fig.",
"REF with the size of an equivalent point cloud map.",
"The plane map occupies 41.9 kB of memory, whereas a point cloud map generated from superimposing every 10th point cloud together occupies 34.5 MB of memory.",
"Thus we see that by using a plane map representation, we are able to reduce memory usage by approximately three orders of magnitude.",
"In order to demonstrate the suitability of our plane map for motion planning, we implement a naive straight-line RRT [24] using the collision check process described in Section REF .",
"An RRT with 1000 nodes is shown in Fig.",
"REF , and takes only 0.5 to generate.",
"Figure: RRT generated in 0.5, demonstrating the speed of collision checking within our plane-based map."
],
[
"Rover Experiments",
"For our hardware tests, we use a ground rover equipped with a Velodyne Puck LITE LiDAR, which has 16 channels, 30$$ vertical FOV, 360$$ horizontal FOV, and generates 300,000 points per second.",
"We manually drive the rover through an environment made up of cardboard boxes, shown in Fig.",
"REF , and ground-truth pose information is logged using motion capture.",
"The estimated map and trajectory from our algorithm are shown in Fig.",
"REF .",
"The RMSE translational error for the trajectory is 0.063 (std.",
"0.039), and the average rotational error is 0.71 degrees (std.",
"0.40 degrees).",
"All computation is done onboard the rover on an Intel NUC 7 mini PC (2-core, 3.5 GHz, 16 GB RAM), with total average runtime of 124 (std.",
"37).",
"Although the Velodyne LiDAR produces scans at 10 Hz, we run our algorithm at 5 Hz (every other scan) to ensure sufficient compute time available.",
"Figure: We validate our method with a ground rover equipped with Velodyne LiDAR in a Manhattan world environment.",
"Our method is able to estimate the rover's trajectory and construct a plane-based map of the environment in real-time.We observe that for real-world noisy LiDAR data, the frontend is susceptible to error, with some planes being slightly over-approximated.",
"Additionally, errors in the trajectory and map can be contributed to the rotation of the LiDAR during the rover's motion, which is not accounted for.",
"Nevertheless, our method is still able to reconstruct the overall structure of the cardboard box environment."
],
[
"Conclusion",
"In this paper, we present plane-based LiDAR SLAM for real-time lightweight map generation and localization.",
"Traditionally, SLAM algorithms are not designed with the requirements of downstream motion planning algorithms in mind, and motion planning algorithms are often designed assuming a map of the environment exists in a convenient form.",
"Our work takes a step towards bridging this gap, proposing a SLAM pipeline that can be tightly integrated with motion planning through its plane-based map representation.",
"Additionally, our map requires much less memory to store than a point cloud equivalent, enabling applications involving extensive mapping on resource-constrained systems, or multi-robot applications in which robots must communicate map information between each other over limited bandwidth."
]
] | 2209.08248 |
[
[
"Comparative density functional theory study for predicting oxygen\n reduction activity of single-atom catalyst"
],
[
"Abstract It has been well established that nitrogen coordinated transition metal, TM-N$_{4}$-C (TM$=$Fe and Co) moieties, are responsible for the higher catalytic activity for the electrochemical oxygen reduction reaction.",
"However, the results obtained using density functional theory calculations vary from one to another, which can lead to controversy.",
"Herein, we assess the accuracy of the theoretical approach using different class of exchange-correlation functionals, i.e., Perdew-Burke-Ernzerhof (PBE) and revised PBE (RPBE), those with the Grimme's semiempirical dispersion correction (PBE+D3 and RPBE+D3), and the Bayesian error estimate functional with the nonlocal correlation (BEEF-vdW) on the reaction energies of oxygen reduction reaction on TM-N$_{4}$ moieties in graphene and those with OH-termination.",
"We found that the predicted overpotentials using RPBE+D3 are comparable and consistent with those using BEEF-vdW.",
"Our finding indicates that a proper choice of the exchange-correlation functional is crucial to a precise description of the catalytic activity of this system."
],
[
"Introduction",
"Single-atom catalysts are an important type of the catalysis that have gain considerable attention for the electro-reduction of $\\mathrm {O_{2}}$ due to their potential to reduce the cost of electrochemical energy conversion devices such as fuel cells and metal-air batteries [1], [2], [3], [4].",
"Among explored catalyst formulations, non-precious metal, such as Fe and Co, embedded in N-doped graphene (Fe-N-C and Co-N-C) is the most representative of single-atom catalysts that have been shown to have remarkable catalytic activity and selectivity against oxygen reduction reaction (ORR) [5], [6], [7], [8], [9], [10], [11].",
"Recent experimental results report that the highest ORR catalytic activity of the Fe-N$_{}$ -C and Co-N$_{}$ -C catalysts measured from the half-wave potentials ($E_{\\mathrm {1/2}}$ ) is $\\sim $ 0.83 V and $\\sim $ 0.75 V, respectively [12], [13].",
"This class of materials exhibits unique structures and properties that can potentially match and compete with the performance of platinum.",
"The mechanism of ORR on these catalysts has been extensively investigated: Among different N/C environments surrounding the metal atom, a four-fold coordinated metal atom with pyridinic nitrogen atoms in graphene (such Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C moieties) is proposed to be the active site, due to their optimal binding strength with the chemical species involved in the ORR process [14], [15].",
"The reaction energies and the intermediate reactions on the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C sites have been extensively studied using periodic density functional theory (DFT) calculations based on the computational hydrogen electrode (CHE) model [16], [17].",
"The overpotential for ORR ($\\eta _{\\mathrm {ORR}}$ ) estimated from theoretical limiting-potential in the CHE model is one of the most useful metric for screening the catalysts for the ORR [18], [19].",
"However, despite of the high ORR catalytic activities of the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C catalysts observed in the experiments, the $\\eta _{\\mathrm {ORR}}$ predicted through the CHE model are scattered.",
"Kattel et al.",
"[17] and Li et al.",
"[20] estimated the $\\eta _{\\mathrm {ORR}}$ values of 0.91 V and 0.23 V for Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C, respectively, using the Perdew-Burke-Ernzherof (PBE) generalized gradient approximation (GGA) functional.",
"Sun et al.",
"estimated the $\\eta _{\\mathrm {ORR}}$ values of 0.79 V and 0.29 V for Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C catalysts, respectively, using PBE with the Grimme's semiempirical dispersion correction (DFT-D2)[21].",
"Wang et al.",
"included the solvation effect through continuum solvation model with PBE and DFT-D, and obtained the $\\eta _{\\mathrm {ORR}}$ values of the 0.72 V and 0.83 V for Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C, respectively [22].",
"In addition to the solvation effect, they also included the effect of OH-termination as suggested by Wang et al.",
"[23], resulting in lower $\\eta _{\\mathrm {ORR}}$ 's of 0.59 V and 0.74 V for Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C, respectively.",
"However, a previous paper by the same authors reported $\\eta _{\\mathrm {ORR}}$ 's of 0.47 V and 0.75 V for the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C through implicit solvation model with PBE, respectively [23].",
"Apparently, there is a discrepancy among the DFT studies besides the inclusion of dispersion correction as well as the solvation effect.",
"Theoretical studies based on the DFT method have been conducted using various approaches and have proven useful to elucidate the mechanisms of the reactions, especially in heterogeneous catalysis systems.",
"However, the accuracy and thus the success of DFT-based studies depend on the choice of exchange-correlation (XC) functional [24].",
"However, GGA functionals are prone to self-interaction errors for non-metallic system and calculation of accurate binding energies of the adsorbates on transition metal surfaces are a long-lasting challenge for the DFT XC functional [25], [26], [27], [28], [29].",
"Improper choice of the XC functional can result in inaccurate binding energies and thus, the incorrect reaction mechanism.",
"Moreover, it is known that the van der Waals (vdW) or dispersion interactions, which can be important in the catalytic reactions, cannot be captured properly by GGAs [30].",
"In this work, we performed systematic calculations of the reaction intermediate for the ORR on the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C systems and calculated the overpotential for the ORR by using different XC functionals, and investigate how the choice of the XC functional impacts the predicted ORR activities of these systems regardless of the solvation effect.",
"All the calculations were performed using the projector augmented wave (PAW)[31] method as implemented in the Quantum-ESPRESSO code [32].",
"The PAW potentials were adopted from the PSLIBRARY [33] version 1.0.0.",
"Wave functions and augmentation charge density were expanded in terms of a plane-wave basis set with the kinetic energy cutoffs of 80 and 800 Ry, respectively.",
"The Marzari-Vanderbilt cold smearing [34] with a smearing width of 0.02 Ry was used to treat the Fermi level.",
"In this work, we compare the performance of different XC functionals within GGA, specifically PBE [35] and revised PBE (RPBE) [36] functionals.",
"Grimme's semiempirical dispersion correction (DFT-D3) [37] is also include for both functionals namely, PBE+D3 and RPBE+D3 to further investigate the importance of the dispersion force in the systems considered in this work.",
"Finally, we include the Bayesian error estimations functional with nonlocal correlation (BEEF-vdW) [38] to further assess the accuracy of the GGA functionals.",
"The BEEF-vdW is designed specifically to address vdW forces reasonably well while maintaining an accurate description of chemisorption energies of molecule on surface.",
"Moreover, the BEEF-vdW itself has an error estimate built in, to not only yield an accurate energy difference in condensed matter studies, but also estimate the errors on computed quantities.",
"Therefore, BEEF-vdW serves as a benchmark and a baseline for comparison in this study.",
"To employ the error estimation capabilities in the BEEF-vdW, a probability distribution for model parameters for the exchange and correlation is randomly sampled through an ensemble of density functionals generated around the BEEF-vdW [38], [39].",
"In this work, we generated an ensemble of 2000 functionals for each adsorption system, isolated surface and adsorbate considered, to obtain a distribution of the adsorption energies ($E_{\\mathrm {ads}}$ ).",
"Then, the standard deviation of the ensemble for $E_{\\mathrm {ads}}$ around the BEEF-vdW value was used to estimate the error for each adsorption system.",
"In addition to $E_{\\mathrm {ads}}$ , the standard deviation was also used to estimate the uncertainty in the limiting potential and overpotential.",
"We note that we did not scale the error in such a way that it reproduces that for the benchmark adsorption energies, unlike Ref. [39].",
"Thus, our estimated error for both $E_{\\mathrm {ads}}$ and limiting potential/overpotential can be overestimated.",
"Throughout this study, we used PAW potentials generated using the PBE functional.",
"Detailed results of the optimized lattice constant and the adsorption energies of the ORR intermediates obtained using RPBE and RPBE+D3 functionals with PAW potentials generated using the RPBE functionals are included in Supplemental Materials [40] for further reference.",
"We note that it was suggested [24] that hybrid functions such as PBE0 [41], [42] and HSE06 [43] give more accurate results for ORR on nitrogen-doped graphene than GGA or SCAN meta-GGA[44], as compared with those obtained using the highly accurate coupled cluster theory.",
"Liu et al.",
"[45] used various XC functionals including HSE06 and SCAN and reported that the absolute value of the magnetic moment, which is suggested to be a good descriptor for ORR on Fe-N-C catalysts, varies depending on the functional, but the trend is the same (the authors do not report the reaction free energies with different functionals).",
"In this study, we limit ourselves to the GGA level of theory, in order to systematically and comprehensively study the roles of the XC functional and dispersion correction on not only the energetics, but also the vibrational contributions to the reaction free energies.",
"Table: Optimized lattice constant for pristine graphene obtained using each functionals compared to experimental value .",
"For RPBE and RPBE+D3 functionals, the optimized lattice constant are obtained using the PBE PAW potential.We considered Fe-N$_4$ and Co-N$_4$ moieties embedded in graphene.",
"The lattice constant for graphene was optimized in the primitive hexagonal unit cell using a 16$\\times $ 16 $\\mathbf {k}$ -point grid for each functional.",
"The DFT-optimized lattice constants are given in the Table REF .",
"For the calculation of the ORR intermediates, the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C models was constructed using a (5$\\times $ 5) surface unit cell of graphene, in which we removed two carbon atoms to anchor a single Fe/Co atom and replaced four-coordinated carbon atoms surrounding it with nitrogen atom [Fig.",
"REF (a)].",
"To eliminate the spurious electrostatic interaction with the periodic images, the effective screening medium method [47], [48] was employed, along with the vacuum spacing of 20 Å.",
"The geometries of all the reaction intermediate ($\\mathrm {^{\\ast }O_{2}}$ , $\\mathrm {^{\\ast }OOH}$ , $\\mathrm {^{\\ast }O}$ , and $\\mathrm {^{\\ast }OH}$ ) on both Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C moeties were fully optimized until the residual forces on the constituent atoms became smaller than $10^{-4}$ Ry/Bohr ($2.57 \\times 10^{-3}$ eV/Å).",
"A 4$\\times $ 4 $\\mathbf {k}$ -point grid was found to give reasonably accurate results (see Supplemental Materials [40]) and 8$\\times $ 8 $\\mathbf {k}$ -point grid was used for the electronic structure analyses.",
"The adsorption energy of the intermediate is defined by $E_{\\mathrm {ads}} = E_{\\mathrm {S/A}} - E_{\\mathrm {S}} - E_{\\mathrm {A}},$ where $E_{\\mathrm {S/A}}$ , $E_{\\mathrm {S}}$ , and $E_{\\mathrm {A}}$ , are the total energies of combined system, isolated surface structure, and isolated adsorbate, respectively.",
"All energies above were obtained under the same parameter settings."
],
[
"Free energy calculation",
"The change in adsorption energy with an applied electrode potential was calculated based on the CHE model proposed by Nørskov and co-workers [18].",
"In this study, we focus on the associative mechanism of the ORR process as it has been reported that it is energetically more favorable and that the activation barrier for dissociative O$_2$ adsorption is considerably high (1.16 eV)[23] in the cases of the Fe-N$_4$ -C and Co-N$_4$ -C active sites.",
"The elementary steps for ORR along the four-electron associative mechanisms can be written as follows: ${^{\\ast }} + \\mathrm {O_{2}} + \\mathrm {H}^{+} + e^{-} \\rightarrow {^{\\ast }}\\mathrm {OOH}\\\\$ ${^{\\ast }}{\\mathrm {OOH}} + \\mathrm {H}^{+} + e^{-} \\rightarrow {^{\\ast }{\\mathrm {O}}} + \\mathrm {H_{2}{O}} \\\\$ ${^{\\ast }}{\\mathrm {O}} + \\mathrm {H}^{+} + e^{-} \\rightarrow {^{\\ast }{\\mathrm {OH}}} \\\\$ ${^{\\ast }}{\\mathrm {OH}} + \\mathrm {H}^{+} + e^{-} \\rightarrow {^{\\ast }} + \\mathrm {H_{2}{O}} \\\\$ where $^{\\ast }$ denotes the adsorption site.",
"The DFT reaction energies of the $^{\\ast }\\mathrm {OOH}$ , $^{\\ast }\\mathrm {O}$ and $^{\\ast }\\mathrm {OH}$ are then defined relative to $\\mathrm {H_{2}{O}}$ (liquid phase) and $\\mathrm {H_{2}}$ (gas phase), to avoid the calculation of the $\\mathrm {O_{2}}$ molecule or the radical of $\\mathrm {OOH}$ and $\\mathrm {OH}$ , which is notoriously difficult to describe within the semilocal approximation to DFT [18] as: ${\\Delta {E}_{\\mathrm {OOH}}} = {E_{\\ast {\\mathrm {OOH}}}} - {E_{\\ast }} - ({2E_{\\mathrm {H_{2}O(l)}}} - {\\frac{3}{2}}E_{\\mathrm {H_{2}(g)}})$ ${\\Delta {E}_{\\mathrm {O}}} = {E_{\\ast {\\mathrm {O}}}} - {E_{\\ast }} - ({E_{\\mathrm {H_{2}O(l)}}} - {E_{\\mathrm {H_{2}(g)}}})\\\\$ ${\\Delta {E}_{\\mathrm {OH}}} = {E_{\\ast {\\mathrm {OH}}}} - {E_{\\ast }} - ({E_{\\mathrm {H_{2}O(l)}}} - {\\frac{1}{2}}E_{\\mathrm {H_{2}(g)}})\\\\$ where $E_{\\ast }$ is the total energy of the surface without adsorbates, $E_{\\ast {\\mathrm {OOH}}}$ , $E_{\\ast {\\mathrm {O}}}$ and $E_{\\ast {\\mathrm {OH}}}$ are total energies of the surface with adsorbates ($^{*}\\mathrm {OOH}$ , $^{*}\\mathrm {O}$ and $^{*}\\mathrm {OH}$ , respectively) bound to the surface, $E_{\\mathrm {H_{2}O(g)}}$ and $E_{\\mathrm {H_{2}(g)}}$ are the total energies of $\\mathrm {H_{2}O}$ and $\\mathrm {H_{2}}$ molecules in the gas phase, respectively.",
"The energy of H$_2$ O in the liquid phase ($E_{\\mathrm {H2O(l)}}$ ) is calculated at 0.035 bar (i.e.",
"$E_{\\mathrm {H_2O(l)}}$ = $E_{\\mathrm {H_2O(g)}}$ + $k_{\\mathrm {B}}T\\ln 0.035$ ), which correspond to equilibrium pressure of $\\mathrm {H_{2}O}$ at 298.15 K, and therefore this state corresponds to that of liquid water [49].",
"The DFT formation energies for the ORR intermediates are converted to the adsorption free energy by including corrections for the change in zero-point energy $\\Delta \\mathrm {ZPE}$ and entropy $T \\Delta S$ at 298.15 K estimated using finite displacement method to calculate the vibrational frequencies of the adsorbates through the use of the atomic simulation environment (ASE) [50] and $\\Delta G_{\\mathrm {pH}}$ are the pH values (pH was defined as 0 for acidic medium) as $\\Delta {G} = \\Delta E_{\\mathrm {DFT}} + \\Delta \\mathrm {ZPE} - T\\Delta {S} + \\Delta G_{\\mathrm {pH}}.$ This allows us to define the adsorption free energies of $^{*}\\mathrm {OOH}$ , $^{*}\\mathrm {O}$ and $^{*}\\mathrm {OH}$ (i.e.",
"along associative mechanism) relative to $\\mathrm {H_{2}O}$ and $\\mathrm {H_{2}}$ ${^{\\ast }} + {\\mathrm {H_{2}O}} {\\rightarrow } {^{*}{\\mathrm {O}}} + {\\mathrm {H_{2}}}, & &{\\Delta {G}(^{*}{\\mathrm {O})}}$ ${^{\\ast }} + {\\mathrm {H_{2}O}} {\\rightarrow } {^{*}{\\mathrm {OH}}} + {\\frac{1}{2}\\mathrm {H_{2}}}, & &{\\Delta {G}(^{*}{\\mathrm {OH})}}$ ${^{\\ast }} + 2{\\mathrm {H_{2}O}} {\\rightarrow } {^{\\ast }{\\mathrm {OOH}}} + {\\frac{3}{2}\\mathrm {H_{2}}}, & &{\\Delta {G}(^{\\ast }{\\mathrm {OOH})}}$ In the case of the adsorption free energy of O$_{2}$ ($^{\\ast }$ + $\\mathrm {O_{2}}$ $\\rightarrow $ $^{\\ast }{\\mathrm {O_{2}}}$ ), the adsorption energy is defined relative to an $\\mathrm {O}_2$ molecule in the gas phase ($\\mathrm {O_{2}}$ (g)).",
"By setting the reversible hydrogen electrode (RHE) as the reference electrode, the effect of the applied potential ($U$ $\\mathrm {V_{RHE}}$ ) on the Gibbs free energy can be approximated by $\\Delta G_{U} = \\Delta G - neU,$ where $n$ is the number of electrons transferred in each consecutive step and $e$ is the elementary charge of an electron.",
"The Gibbs free energy of the four elementary reactions steps in the associative mechanism (at $U$ = 0 $\\mathrm {V_{RHE}}$ ) can be written as $\\Delta {G_{1}} &= \\Delta {G(^{\\ast }{\\mathrm {OOH}})} - \\Delta {G_{\\mathrm {ORR}}}$ $\\Delta {G_{2}} &= \\Delta {G(^{\\ast }{\\mathrm {O}})} - \\Delta {G(^{\\ast }{\\mathrm {OOH}})}$ $\\Delta {G_{3}} &= \\Delta {G(^{\\ast }{\\mathrm {OH}})} - \\Delta {G(^{\\ast }{\\mathrm {O}})}$ $\\Delta {G_{4}} &= - \\Delta {G(^{\\ast }{\\mathrm {OH}})}$ where $\\Delta {G_{\\mathrm {ORR}}}$ is the overall formation free energy value of ORR through the relation (2$\\mathrm {H_{2}}$ + $\\mathrm {O_{2}}$ $\\rightarrow $ 2$\\mathrm {H_{2}{O}}$ ), defined experimentally ($\\Delta {G_{\\mathrm {ORR}}}$ = $-$ 4.92 eV) at 298.15 $\\mathrm {K}$ with pressure of $\\mathrm {O_{2}}$ and $\\mathrm {H_{2}}$ of 1 bar [51].",
"With this value, the thermodynamic equilibrium potential will be 1.23 $\\mathrm {V_{RHE}}$ .",
"Based on the calculations described above, the free energy diagram along the reaction path considered can be constructed.",
"The limiting-potential (${U_{\\mathrm {L}}}$ ) is estimated from Eq.",
"(REF ) by determining the minimum free energy change ($\\Delta {G_{\\mathrm {min}}}$ ) among all the electrochemical steps.",
"$U_{\\mathrm {L}} = \\mathrm {min}{\\lbrace {\\Delta {G_{1}},{\\Delta {G_{2}}},{\\Delta {G_{3}},{\\Delta {G_{4}}\\rbrace }/{e}}}}$ In the CHE model, the ${U_{\\mathrm {L}}}$ is also known as potential at which all the reaction steps are exothermic and by this definition, it can be used to compare with the experimental half-wave potential ($\\mathrm {E_{\\frac{1}{2}}}$ ) value [52].",
"Accordingly, the overpotential ($\\eta _{ORR}$ ) can be further derived from the (${U_{\\mathrm {L}}}$ ) as Eq.",
"(REF ), in which a lower overpotential implies a better ORR activity.",
"$\\eta _{\\mathrm {ORR}} = 1.23 - U{\\mathrm {_L}}$ Figure: (a) The structure of a clean TM-N 4 _4-C active site (TM is Fe or Co) Adsorption structures of (b) * O 2 \\mathrm {^{\\ast }O_{2}}, (c) * OOH \\mathrm {^{\\ast }OOH}, (d) * O\\mathrm {^{\\ast }O} and (e) * OH \\mathrm {^{\\ast }OH} on TM-N 4 _\\mathrm {4}.",
"The gold, brown, red, silver and white atoms represent transition metal (Fe or Co), C, O, N and H, respectively."
],
[
"Adsorption energies for the ORR intermediates",
"We first optimized the ORR intermediates ($\\mathrm {^{\\ast }O_{2}}$ , $\\mathrm {^{\\ast }OOH}$ , $\\mathrm {^{\\ast }O}$ , and $\\mathrm {^{\\ast }OH}$ ) on Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C active sites, and calculated their adsorption energies with PBE, PBE+D3, RPBE, RPBE+D3 and BEEF-vdW functionals (Fig.",
"REF ).",
"The calculated energies are summarizes in Table REF .",
"Figure: Calculated adsorption energies for the ORR intermediates on (a) Fe-N 4 \\mathrm {N_{4}}-C and (b) Co-N 4 \\mathrm {N_{4}}-C with difference functional considered.Table: Adsorption energies for the ORR adsorbates on the Fe-N 4 \\mathrm {N_{4}}-C and Co-N 4 \\mathrm {N_{4}}-C active sites with different functionals considered.",
"The unit of energy is eV.The present PBE results on both active sites are in a good agreement with the previous works by Kattel et al.",
"[17], [16].",
"In general, the adsorption energies obtained using PBE are larger in magnitude, and by the inclusion of dispersion correction, PBE+D3 gives adsorption energies larger in magnitude than those obtained using other functionals, and the energy differences are more significant in the case of Fe-$\\mathrm {N_{4}}$ -C active site.",
"In contrast, RPBE predicts more repulsive interactions and resulted in more positive interaction energies on both active sites.",
"However, RPBE+D3 improves the adsorption energies, which are larger in magnitude than those by RPBE, and the values are in good agreement with those obtained by using BEEF-vdW.",
"The results indicate that the contribution of dispersion interaction is significant within these two systems, and the inclusion of the dispersion correction in RPBE+D3 leads to the adsorption energies of all the ORR intermediates with similar accuracy to the BEEF-vdW.",
"We note that the estimated error in the BEEF-vdW adsorption energies is relatively large (0.2 - 0.3 eV for Fe-$\\mathrm {N_{4}}$ -C and 0.1 - 0.2 eV for Co-$\\mathrm {N_{4}}$ -C), but agrees well with the previously reported one for N-doped graphene [24].",
"We also note that PBE tends to overestimate the adsorption energy of chemisorption species, partly because the exchange in the PBE functional tends to lead too attractive interaction in molecular systems [36].",
"In addition, PBE can show spurious attractive interaction for weakly interacting systems from the exchange only.",
"The detail discussion can be found in Ref.",
"abidin2022interaction,hamada2010interaction,hamada2012adsorption."
],
[
"Reaction mechanism of the ORR",
"We next evaluated the catalytic activities of the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C for the ORR with different functionals.",
"The free energy diagrams for ORR along the four-electron associative mechanism on the Fe-$\\mathrm {N_{4}}$ -C and the Co-$\\mathrm {N_{4}}$ -C catalysts are calculated and displayed in Fig.",
"REF .",
"The calculated adsorption free energies and details theoretical limiting potentials and overpotentials on the Fe-$\\mathrm {N_{4}}$ -C and the Co-$\\mathrm {N_{4}}$ -C catalysts are summarized in Tables REF and REF , respectively.",
"Figure: Calculated free energy diagrams of ORR along the associative mechanism on (a) Fe-N 4 \\mathrm {N_{4}}-C and (b) Co-N 4 \\mathrm {N_{4}}-C catalysts at (UU = 0 V RHE _{\\mathrm {RHE}}).Table: Calculated adsorption free energies for * OOH \\mathrm {{^\\ast }OOH}, * O\\mathrm {{^\\ast }O} and * OH \\mathrm {{^\\ast }OH} on Fe-N 4 \\mathrm {N_{4}}-C and Co-N 4 \\mathrm {N_{4}}-C active sites with different functionals considered.",
"The unit of energy is eV.Table: Calculated theoretical limiting potential (U L {U_{\\mathrm {L}}}) and overpotential (η ORR \\eta _{\\mathrm {ORR}}) for the Fe-N 4 \\mathrm {N_{4}}-C and Co-N 4 \\mathrm {N_{4}}-C catalysts.The potential determining steps (PDS) for the Fe-N 4 \\mathrm {N_{4}}-C are H 2 _2O formation with PBE, PBE+D3, and OH formation with RPBE, RPBE+D3 and BEEF-vdW.",
"For the Co-N 4 \\mathrm {N_{4}}-C, all functionals predict the OOH formation as PDS.The unit of potential is volt.Figure: The partial density of state for Fe 3dd, Co 3dd and O 2pp of Fe-N 4 \\mathrm {N_{4}}-C and Co-N 4 \\mathrm {N_{4}}-C active sites (a), (b) without and (c), (d) with * O 2 \\mathrm {^{\\ast }O_{2}} adsorbate.",
"The origin of the energy is set to the Fermi level (E F E_{\\mathrm {F}}).",
"The positive densities of states are for the spin up channel, while negative, the spin down one.Our calculated ORR free energy diagrams for the associative mechanism (Fig.",
"REF ) show that RPBE and RPBE+D3 functionals predict a similar potential determining step (PDS) with BEEF-vdW, which is the O protonation to form OH, while both PBE and PBE+D3 predict the formation of H$_2$ O (last reaction steps) on the Fe-$\\mathrm {N_{4}}$ -C. On the other hand, all the functionals predict the OOH formation (initial reaction steps) as the PDS on the Co-$\\mathrm {N_{4}}$ -C catalyst.",
"The difference in the estimated PDSs in the Fe-$\\mathrm {N_{4}}$ -C catalyst originates from the significant difference in the ORR intermediate adsorption energies predicted by the PBE and PBE+D3 functionals.",
"The ${U_{\\mathrm {L}}}$ values for Fe-$\\mathrm {N_{4}}$ -C predicted by PBE+D3 and RPBE are in good agreement with that by BEEF-vdW with a difference of $\\sim $ 0.02 V, while PBE and RPBE+D3 is slightly overestimate it by $\\sim $ 0.08 V. It should be stressed that all the functionals predicts ${U_{\\mathrm {L}}}$ 's for Fe-$\\mathrm {N_{4}}$ -C, which seemingly agree well with that from BEEF-vdW.",
"However, PBE and PBE+D3 values are obtained for a different PDS, i.e., different mechanisms.",
"On the other hand, in the case of Co-$\\mathrm {N_{4}}$ -C, only RPBE+D3 predicts ${U_{\\mathrm {L}}}$ , which is comparable to that from BEEF-vdW, while PBE, PBE+D3, and RPBE underestimate it by $\\sim $ 0.25, $\\sim $ 0.10, and $\\sim $ 0.26 V, respectively.",
"We also estimated the $\\eta _{\\mathrm {ORR}}$ by using Eq.",
"REF (Table REF ).",
"The error bars for BEEF-vdW indicate a relatively large uncertainty (0.4 V) for the $\\eta _{\\mathrm {ORR}}$ on both active sites.",
"Nevertheless, only RPBE+D3 consistently estimated a similar PDS and comparable $\\eta _{\\mathrm {ORR}}$ to BEEF-vdW.",
"The inconsistent results on both catalysts obtained using PBE and PBE+D3 may be due to the too attractive exchange in PBE, as discussed above.",
"With respect to the $\\eta _{\\mathrm {ORR}}$ values (Table REF ), PBE+D3, RPBE+D3, and BEEF-vdW predict that the Co-$\\mathrm {N_{4}}$ -C catalyst is more reactive (lower $\\eta _{\\mathrm {ORR}}$ value), while PBE and RPBE estimate the catalytic performance of the Fe-$\\mathrm {N_{4}}$ -C and the Co-$\\mathrm {N_{4}}$ -C are comparable.",
"It is noted that our results in Table REF are limited to the vacuum condition.",
"It is also noted that we estimated the adsorption free energies using RPBE+D3 with the on-site Coulomb interaction ($U$ ) and report in Supplemental Materials [40], which are consistent with those reported in Ref. [23].",
"However, calculated adsorption energies and potential determining steps can depend on the choice of the $U$ value, which calls for further investigation."
],
[
"Electronic structures",
"To gain insight into their catalytic activities, we calculated the densities of states for clean and $\\mathrm {{^\\ast }O_{2}}$ adsorbed Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C sites using RPBE+D3 functional.",
"We found that a distinct spin polarization at the Fe-$\\mathrm {N_{4}}$ -C site compared to the Co-$\\mathrm {N_{4}}$ -C sites, which can be also confirmed by a larger local magnetic moment (1.61 $\\mu _{\\mathrm {B}}$ and 0.68 $\\mu _{\\mathrm {B}}$ for the former and the latter, respectively).",
"These results further explain stronger binding strength for all ORR intermediates that we obtained for the Fe-$\\mathrm {N_{4}}$ -C as compared to Co-$\\mathrm {N_{4}}$ -C site (Table REF ).",
"Besides, the spin down channels of $3d_{z^2}$ and $3d_{zy}$ in the Fe-$\\mathrm {N_{4}}$ -C are unoccupied and located above the Fermi level, while the Co-$\\mathrm {N_{4}}$ -C has the half-filled $3d_{z^2}$ spin down channel cross the Fermi level.",
"This indicates that the Co-N$_{4}$ -C active site has a half-metallic nature, and has less opportunities to form a bond with O.",
"Therefore, the Fe-$\\mathrm {N_{4}}$ -C site can form a stronger Fe$-$ O $\\sigma $ bond compare to Co$-$ O [55], [56], [57].",
"This behavior can be related to the Sabatier principle in which the strong bonding between active site and reactant effectively inhibits further reaction and results in decreasing catalytic activity [58], [59].",
"As expected, the moderate spin polarization and half-metallic properties shown by the Co-$\\mathrm {N_{4}}$ -C site ensures the moderate binding strength of the main ORR intermediates, and is responsible for the higher ORR catalytic activity [21]."
],
[
"Effect of OH termination of the active Fe/Co site",
"In a recent theoretical work by Wang et al.",
"[23], it was proposed that the intrinsic intermediate $\\mathrm {{^\\ast }OH}$ is co-adsorbed on the single-metal-atom active site, and is the origin of the improved activity of the ORR through the associative mechanism on the TM-$\\mathrm {N_{4}}$ -C catalyst.",
"The presence of the OH species was also confirmed by the experiment [60].",
"Following the aforementioned studies, we further investigate the effect of OH-termination on the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C active sites without the solvent effect (Fig.",
"REF ).",
"Here we employed RPBE+D3 functional, which predicts consistent results with BEEF-vdW and compares with the results obtained using PBE+D3.",
"We also include PBE in this comparison, because it was used in Ref. wang2019self.",
"The calculated energies are summarizes in Table REF Figure: The adsorption structures of (a) * O 2 \\mathrm {^{\\ast }O_{2}}, (b) * OOH \\mathrm {^{\\ast }OOH}, (c) * O\\mathrm {^{\\ast }O} and (d) * OH \\mathrm {^{\\ast }OH} on OH-terminated TM(OH)-N 4 _\\mathrm {4} active site (TM is Fe or Co).",
"The gold, brown, red, silver and white atoms represent transition metal (Fe or Co), C, O, N and H, respectively.Table: Calculated adsorption free energies for * OOH \\mathrm {{^\\ast }OOH}, * O\\mathrm {{^\\ast }O} and * OH \\mathrm {{^\\ast }OH} on Fe-N 4 \\mathrm {N_{4}}-C and Co-N 4 \\mathrm {N_{4}}-C active sites with and without OH termination of the Fe/Co sites.",
"The PBE, PBE+D3, and RPBE+D3 functionals was used and the unit of energy is eV.Figure: Free energy diagrams of ORR along associative pathway on the Fe-N 4 \\mathrm {N_{4}}-C and Co-N 4 \\mathrm {N_{4}}-C active sites with and without OH-termination obtained using (a) PBE, (b) PBE+D3, and (c) RPBE+D3 functionals.Here, with the OH-termination of both Fe-N$_4$ -C and Co-N$_4$ -C active sites, PBE+ D3 and RPBE+D3 predict the formation of $\\mathrm {H_{2}O}$ as the PDS, while PBE, the protonation of $\\mathrm {O_{2}}$ to form $\\mathrm {OOH}$ on the Fe-N$_4$ -C. In the case of Co-N$_4$ -C, all the functionals predicts the $\\mathrm {O_{2}}$ protonation to form $\\mathrm {OOH}$ as the PDS.",
"We further estimated the $\\eta _{\\mathrm {ORR}}$ for the detailed comparison (Table REF ).",
"The $\\eta _{\\mathrm {ORR}}$ values for Fe-N$_4$ -C predicted by PBE, PBE+D3 and RPBE+D3 show a noticeable improvement of the ORR catalytic activities (lowering of the overpotential) after the OH co-adsorbs on the Fe site by $\\sim $ 0.14, $\\sim $ 0.25, and $\\sim $ 0.17 V, respectively.",
"On the other hand, on the Co site, no significant change can be found by using each functionals.",
"The slight improvement of the ORR activity in the Fe-N$_4$ -C is mainly due to the formation of the $\\sigma $ -bonding between $2p_{z}$ and $3d_{z^2}$ orbitals upon termination of the $\\mathrm {{^\\ast }OH}$ on the Fe site (see Supplemental Materials [40]).",
"Accordingly, the proportion of $\\sigma $ -bonding state on the Fe(OH)-$\\mathrm {N_{4}}$ -C site decreased, and the interaction between the ORR intermediates is dominated by the $\\pi $ -bonding states (i.e., the filling degree of $3d_{zx}$ and $3d_{zy}$ ).",
"Therefore, the adsorption of ORR intermediates on the Fe(OH)-$\\mathrm {N_{4}}$ -C becomes weaker (see Supplemental Materials [40]), and the ORR performance is improved.",
"The same behavior is observed for the Co-$\\mathrm {N_{4}}$ -C with the OH-termination.",
"However, since all the orbitals of $3d_{z^2}$ , $3d_{zx}$ , and $3d_{zy}$ are almost filled after the $\\mathrm {{^\\ast }OH}$ co-adsorption at the Co site, and also due to the half-metallic nature of the Co-$\\mathrm {N_{4}}$ -C, no significant change is observed in the adsorption strength of the ORR intermediates as well as the ORR catalytic activity.",
"Table: Calculated limiting potential (U L {U_{\\mathrm {L}}}) and overpotential (η ORR \\eta _{\\mathrm {ORR}}) for the Fe-N 4 \\mathrm {N_{4}}-C and Co-N 4 \\mathrm {N_{4}}-C with and without OH termination of the Fe/Co active sites.",
"The PBE, PBE+D3, and RPBE+D3 functionals was used and the unit of potential is volt.The potential determining steps (PDS) for the Fe-N 4 \\mathrm {N_{4}}-C without OH termination are H 2 _2O formation with PBE, PBE+D3, and OH formation with RPBE+D3, while with OH termination, the PDS are OOH formation with PBE, and H 2 _2O formation with PBE+D3, RPBE+D3, respectively.",
"For the Co-N 4 \\mathrm {N_{4}}-C, all functionals predict the OOH formation as PDS, without and with OH termination.In this work, we have used five different GGA functionals (PBE, PBE+D3, RPBE, RPBE+D3, and BEEF-vdW) to assess the consistency and accuracy of predicted ORR activity of the Fe-$\\mathrm {N_{4}}$ -C and Co-$\\mathrm {N_{4}}$ -C catalysts.",
"It is found that the contribution of dispersion interaction is significant and that RPBE+D3 predicts binding energies, limiting potentials, and the potential determining steps with similar accuracy to BEEF-vdW on both Fe-N$_4$ -C and Co-N$_4$ -C active sites.",
"In addition to the pristine Fe-N$_4$ -C and Co-N$_4$ -C catalysts, we investigate the OH-terminated ones and found that there is a slight lowering of the predicted $\\eta _{\\mathrm {ORR}}$ value for the case of Fe-$\\mathrm {N_{4}}$ -C, while no significant change in Co-$\\mathrm {N_{4}}$ -C. Our results suggest that further investigation is required to clarify the role of the OH termination, which is supposed to improve the catalytic activity of Co-N$_4$ -C active site.",
"Other functionalization groups, different N and C coordinations/configurations, and defects in the vicinity of the active site should also be considered for comprehensive understanding of the single-atom catalysts.",
"We anticipate that our theoretical assessment will be useful in selecting the appropriate XC functional for studying catalytic systems and will serve as a basis for more realistic simulations including the solvent and potential by using the state-of-the-art hybrid DFT and implicit solvation theory [61], [62] and for multiscale simulations based on the microkinetic analysis [63], [23], [64] toward more accurate description of the catalytic activity of the single-atom catalysts.",
"This work was partly supported by Grant in Aid for Scientific Research on Innovative Areas \"Hydrogenomics\" (Grant No.",
"JP18H05519) and “Program for Promoting Researches on the Supercomputer Fugaku” (Fugaku battery & Fuel Cell Project) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan (MEXT).",
"A.F.Z.A.",
"acknowledges the financial support from MEXT.",
"A part of calculations was performed using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo, and of the Cybermedia Center, Osaka University."
]
] | 2209.08226 |
[
[
"Uncertainty Quantification of Collaborative Detection for Self-Driving"
],
[
"Abstract Sharing information between connected and autonomous vehicles (CAVs) fundamentally improves the performance of collaborative object detection for self-driving.",
"However, CAVs still have uncertainties on object detection due to practical challenges, which will affect the later modules in self-driving such as planning and control.",
"Hence, uncertainty quantification is crucial for safety-critical systems such as CAVs.",
"Our work is the first to estimate the uncertainty of collaborative object detection.",
"We propose a novel uncertainty quantification method, called Double-M Quantification, which tailors a moving block bootstrap (MBB) algorithm with direct modeling of the multivariant Gaussian distribution of each corner of the bounding box.",
"Our method captures both the epistemic uncertainty and aleatoric uncertainty with one inference pass based on the offline Double-M training process.",
"And it can be used with different collaborative object detectors.",
"Through experiments on the comprehensive collaborative perception dataset, we show that our Double-M method achieves more than 4X improvement on uncertainty score and more than 3% accuracy improvement, compared with the state-of-the-art uncertainty quantification methods.",
"Our code is public on https://coperception.github.io/double-m-quantification."
],
[
"introduction",
"Multi-agent collaborative object detection has been proposed to leverage the viewpoints of other agents to improve the detection accuracy compared with the individual viewpoint [1].",
"Recent research has shown the effectiveness of early, late, and intermediate fusion of collaborative detection, which respectively transmits raw data, output bounding boxes, and intermediate features [2], [3], [4], and the improved collaborative object detection results will benefit the self-driving decisions of connected and autonomous vehicles (CAVs) [5].",
"However, CAVs may still have uncertainties on object detection due to out-of-distribution objects, sensor measurement noise, or poor weather [6], [7], [8].",
"Even a slightly false detection can lead the autonomous vehicle's driving policy to a completely different action [9], [10].",
"For example, miss-detected paint on the road surface can confuse the lane-following policy and cause potential accidents [11].",
"Therefore, it is crucial to quantify the uncertainty of object detection for safety-critical systems such as CAVs.",
"Various uncertainty quantification methods have been proposed for object detection [8], [12], [13], [14].",
"Uncertainty resources can be decomposed into aleatoric and epistemic uncertainty [15], [16].",
"Direct modeling methods [17], [18], [19] focus on the aleatoric uncertainty (or data uncertainty) which represents inherent measurement noises from the sensor.",
"Monte-Carlo dropout [20], [21] and deep ensemble [22] methods focus on the epistemic uncertainty (or model uncertainty) which reflects the degree of uncertainty that a model describes an observed dataset with its parameters.",
"However, none of the above methods have investigated the uncertainty quantification of collaborative object detection.",
"Figure: Left figure shows detection results of intermediate collaboration in bird's eye view (BEV), and right figure zooms on a specific part to show the robust range of two detections.",
"Red boxes are predictions, and green boxes are ground truth.",
"The orange ellipse denotes the covariance of each corner.",
"The shadow convex hull shows the uncertainty set of the detected object.",
"The shadow convex hull covers the green bounding box in most cases, which helps the later modules in self-driving tasks, such as trajectory prediction with uncertainty propogation and robust planning and control , .",
"With our Double-M Quantification method, detected objects with low accuracy tend to have large uncertainties.In this paper, we propose a novel uncertainty quantification method for collaborative object detection, called Double-M Quantification (direct-Modeling Moving-block bootstrap Quantification), that only needs one inference pass to capture both the epistemic and aleatoric uncertainties.",
"The constructed uncertainty set for each detected object by our method helps the later modules in self-driving tasks, such as trajectory prediction with uncertainty propogation [23] and robust planning and control [24], [25].",
"From Fig.",
"REF , we can see that with our uncertainty quantification method, detected objects with low accuracy tend to have large uncertainties, and the constructed uncertainty set covers the ground-truth bounding box in most cases.",
"Compared with the state-of-the-arts [18], [19], our Double-M Quantification method achieves up to 4$\\times $ improvement on uncertainty score and up to 3.13% accuracy improvement on the comprehensive collaborative perception dataset, V2X-SIM [1].",
"The main contributions of this work are as follows: To the best of our knowledge, our proposed Double-M Quantification is the first attempt to estimate the uncertainty of collaborative object detection.",
"Our method tailors a moving block bootstrap algorithm to estimate both the epistemic and aleatoric uncertainties in one inference pass.",
"We design a novel representation format of the bounding box uncertainty in the direct modeling component to estimate the aleatoric uncertainty.",
"We consider each corner of the bounding box as one independent multivariant Gaussian distribution and the covariance matrix for each corner is estimated from one output header, while the existing literature mainly assumes a univariant Gaussian distribution for each dimension of each corner or a high dimensional Gaussian distribution for all corners.",
"We validate the advantages of the proposed methodology based on V2X-SIM [1] and show that our Double-M Quantification method reduces the uncertainty and improves the accuracy.",
"The results also validate that sharing intermediate feature information between CAVs is beneficial for the system in both improving accuracy and reducing uncertainty.",
"Collaborative object detection, in which multiple agents collectively perceive a scene via communication, is able to address several dilemmas in individual object detection [1].",
"Compared to individual detection, multi-agent collaboration introduces more viewpoints to solve the long-range data sparsity and severe occlusions.",
"The pioneer collaborative detectors employ early collaboration which shares raw data [3] or late collaboration which shares output bounding boxes [4].",
"To further improve the performance-bandwidth trade-off, recent research proposes intermediate collaboration which shares intermediate feature representations from a neural network.",
"Various intermediate collaboration strategies have been developed such as neural message passing [26], knowledge distillation [2], and attention [27].",
"However, existing works only focus on improving the performance of collaborative detection, no existing work investigates uncertainty quantification of collaborative object detection."
],
[
"Uncertainty Quantification on Object Detection",
"Different types of uncertainty quantification methods for object detection have been proposed.",
"For epistemic uncertainty, the Monte-Carlo dropout method utilizes the dropout-based neural network training to perform approximated inference in Bayesian neural networks [21].",
"The deep ensembles method estimates probability distribution by an ensemble of networks with the same architecture and different parameters [22], [28], [29].",
"Both methods require multiple runs of inference, which makes them infeasible for real-time critical tasks with high computational costs such as collaborative objection detection.",
"Moreover, they do not consider time series properties in the dataset, which is one important characteristic of the autonomous driving dataset.",
"In contrast, our uncertainty quantification method overcomes these problems by tailoring a moving block bootstrap [30] (MBB, an effective algorithm for time series analysis) algorithm and quantifies the uncertainty of collaborative object detection in one inference pass.",
"The direct modeling (DM) method is designed to estimate aleatoric uncertainty.",
"The main steps of DM are [8]: a) select one object detector; b) set a certainty probability distribution on outputs of the detector and design the corresponding loss function; c) add extra regression layers to predict the covariance; d) train the modified detector.",
"The work [18] proposes the DM method for image object detection, which assumes that the distribution of each bounding box variable is a single-variate Gaussian distribution and introduces one additional layer to estimate the variance of the bounding box.",
"[19] proposes the DM method with a high dimensional multivariate Gaussian distribution.",
"DM methods for point cloud object detection [17], [31] have been proposed.",
"Methods with both DM and MC dropout to estimate aleatoric and epistemic uncertainties in object detection have also been investigated [16], [32], [33].",
"All the above works only focus on individual object detection.",
"How to quantify the aleatoric and epistemic uncertainties in collaborative object detection remains challenging.",
"In our work, we tailor an MBB-based algorithm process to estimate both aleatoric and epistemic uncertainties of collaborative object detection, with an independent multivariate Gaussian distribution assumption for each corner of the bounding box to represent the uncertainty."
],
[
"Uncertainty Quantification Approach",
"In this section, we first define the problem of uncertainty quantification for collaborative object detection.",
"Then we describe the overview structure of our novel Double-M Quantification (direct-Modeling Moving-block bootstrap Quantification) method as in Fig.",
"REF , followed by the detailed algorithm process.",
"Finally, we define our loss function of the neural network model.",
"One major novelty is the first to tailor a moving block bootstrapping [30] (MBB) algorithm to address the uncertainty quantification challenge of collaborative object detection, and estimate the epistemic and aleatoric uncertainty with one inference pass in addition to the offline training process.",
"The algorithm does not rely on a specific neural network model or structure and can be used with different collaborative object detectors such as DiscoNet [1].",
"The corresponding loss function considers both the prediction accuracy and covariance as metrics.",
"Figure: Overview of our Double-M Quantification method on collaborative object detection.",
"(a) Early collaboration shares raw point cloud with other agents, and (b) intermediate collaboration shares intermediate feature representations with other agents.",
"(c) Double-M Quantification method estimates the multi-variate Gaussian distribution of each corner.Our Double-M Quantification method can be used on different collaborative object detection.",
"During the training stage, Double-M Quantification tailors a moving block bootstrapping algorithm to get the final model parameter, Σ a \\Sigma _{a} as the average aleatoric uncertainty of the validation dataset and Σ e \\Sigma _{e} as the covariance of all residual vectors for epistemic uncertainty.",
"During the inference stage, combine Σ a \\Sigma _a, Σ e \\Sigma _e and the predicted covariance matrix Σ ^\\hat{\\Sigma } from the object detector to compute the covariance matrix Σ ¯=Σ e +(1 2Σ a +1 2Σ ^)\\bar{\\Sigma }=\\Sigma _{e} + (\\frac{1}{2}\\Sigma _{a} + \\frac{1}{2}\\hat{\\Sigma }) of the distribution."
],
[
"Problem Description",
"We use $X$ to represent the point cloud data.",
"For the deterministic collaborative object detection, the agent $\\alpha \\in \\lbrace 1,\\ldots ,N \\rbrace $ creates a local binary bev map $\\mathcal {M} \\in \\lbrace 0, 1\\rbrace ^{W_m \\times L_m \\times H_m}$ from $X$ , where $N$ is the number of agents, and $W_m$ , $L_m$ , and $H_m$ indicate the width, length, and height of the local bev map respectively [34].",
"As shown in Fig.",
"REF (a), we consider homogeneous collaborative object detection, in which multiple agents share the same detector $f_\\theta $ which consists of an encoder denoted by $\\mathcal {E}$ , and an aggregator as well as a decoder that is collectively represented by $\\mathcal {D}$ , where $\\theta $ denotes the parameters of $f$ .",
"$\\mathcal {E}$ compresses the bev map into an intermediate feature map $\\mathbf {F}_\\alpha \\in \\mathbb {R}^{\\frac{W_m}{K_m} \\times \\frac{L_m}{K_m} \\times F_m} = \\mathcal {E}(\\mathcal {M})$ , where $K_m$ is the spatial downsampling scale, and $F_m$ is the feature dimension.",
"Afterwards, $\\mathbf {F}_\\alpha $ is sent to other agents, and the agent $\\alpha $ will use $\\mathcal {D}$ to generate a set of bounding boxes $\\hat{Y} = \\mathcal {D}(\\mathbf {F}_\\alpha , \\lbrace \\mathbf {F}_\\beta \\rbrace _{\\beta \\ne \\alpha })$ .",
"During training, $\\mathcal {E}$ and $\\mathcal {D}$ are jointly learned by minimizing the detection loss $\\mathcal {L}_{det}(Y, \\hat{Y})$ , where $Y$ denotes the set of ground-truth boxes.",
"$\\mathcal {L}_{det}$ includes a classification loss and a regression loss.",
"Note that we also consider early collaboration which shares raw point cloud as shown in Fig.",
"REF (b), we omit its notations for simplicity because the only thing that sets it apart from the intermediate collaboration is the timing of sharing information.",
"In each point cloud data $X$ , there are $J$ objects.",
"For each object $j \\in \\lbrace 1, \\ldots , J \\rbrace $ , we propose to predict $I$ corners of the bounding box.",
"Each corner $i \\in \\lbrace 1, \\ldots , I\\rbrace $ is represented by a $D$ -dimensional vector in the bev map.",
"The set of ground-truth bounding boxes $Y$ is represented as $ Y = \\lbrace c_j, \\lbrace y_{ij}\\rbrace _{i=1}^I \\rbrace _{j=1}^J$ where $c$ is the classification label and $y_{ij} \\in \\mathbb {R}^D, \\forall (i,j)$ .",
"The set of predicted bounding box $\\hat{Y}$ is represented as $\\hat{Y} = \\lbrace \\hat{p}_j, \\lbrace \\hat{y}_{ij}, \\hat{\\Sigma }_{ij}\\rbrace _{i=1}^I \\rbrace _{j=1}^J $ , where $\\hat{p}$ is the predicted classification probability and $\\hat{y}_{ij} \\in \\mathbb {R}^D$ is the predicted coordinate of corner $i$ of object $j$ , $\\hat{\\Sigma }_{ij}$ is a $D \\times D$ covariance matrix.",
"Finally, we model each corner as an independent multivariate Gaussian distribution with mean $\\hat{y}_{ij}$ and covariance matrix $\\bar{\\Sigma }_{ij}$ , where $\\bar{\\Sigma }_{ij}$ combines both epistemic and aleatoric uncertainties."
],
[
"Solution Overview",
"We design a novel uncertainty quantification method called mdm to estimate epistemic and aleatoric uncertainties by tailoring an MBB algorithm with the DM method.",
"The overview of mdm on collaborative object detection is shown in Fig.",
"REF .",
"During the training stage, we train the object detector on resampled moving blocks.",
"After $N$ bootstraps, we get the object detector $f_{\\hat{\\theta }}$ where $\\hat{\\theta }$ is the final model parameter, compute $\\Sigma _a$ as the average aleatoric uncertainty of the validation dataset, and compute $\\Sigma _e$ as the covariance of all residual vectors for epistemic uncertainty.",
"During the inference stage, with input point cloud $X$ , we combine $\\Sigma _a$ , $\\Sigma _e$ and the predicted covariance matrix $\\hat{\\Sigma }_{ij}$ from $f_{\\hat{\\theta }}(X)$ to compute the covariance matrix $\\bar{\\Sigma }_{ij}=\\Sigma _{e} + (\\frac{1}{2}\\Sigma _{a} + \\frac{1}{2}\\hat{\\Sigma }_{ij})$ of the multivariate Gaussian distribution.",
"In the following subsection REF , we propose our novel design, the main contribution of this work, the Double-M Quantification algorithm.",
"Then, in subsection REF , we introduce the loss function for our method."
],
[
"Double-M Quantification",
"Monte-Carlo dropout [21] and deep ensembles [22] have been proposed to estimate the epistemic uncertainty.",
"However, none of them consider the time-series features in the dataset while the temporal features are important for CAVs.",
"We design a novel uncertainty quantification method called mdm to estimate epistemic and aleatoric uncertainties when considering the temporal features in the dataset.",
"In particular, our design tailors a moving block bootstrapping [30] process on time-series data that captures the autocorrelation within the data by sampling data from the constructed data blocks in the training process.",
"Double-M Quantification - Training The training dataset $\\mathbf {D}_{K} = \\lbrace (X_{k}, Y_{k}) \\rbrace _{i=k}^{K}$ with $K$ frames, the validation dataset $\\mathbf {V}_{K^{\\prime }} = \\lbrace (X_{k}, Y_{k}) \\rbrace _{k=1}^{K^{\\prime }}$ with $K^{\\prime }$ frames, the block length $l$ , number of bootstraps $N$ , the loss function $\\mathcal {L}_{det}$ in Eq (REF ), the collaborative object detector $f_\\theta $ $\\hat{\\theta }$ , $\\Sigma _{e}$ , $\\Sigma _{a}$ $\\theta _0 \\leftarrow $ argmin $\\mathcal {L}_{det}(\\mathbf {D}_{K}, \\theta )$ Construct the overlapping moving-block collection $\\mathbf {B} = \\lbrace \\mathcal {B}_b\\rbrace _{b=1}^{K-l+1}$ from the time-series training dataset $\\mathbf {D}_K$ with each block $\\mathcal {B}_b = \\lbrace (X_{b}, Y_{b}), ..., (X_{b + l - 1}, Y_{b+ l - 1}) \\rbrace , b = 1, 2, ..., K - l + 1$ .",
"$n$ from 1 to $N$ Sample $M = \\lfloor K/l \\rfloor $ blocks with replacement from $\\mathbf {B}$ and get a sampled dataset $\\mathbf {S}_n = \\lbrace \\mathcal {B}_{b_m}\\rbrace _{m=1}^{M}$ where $b_m$ are iid uniform random variables on $\\lbrace 1, 2, ..., K-l+1\\rbrace $ .",
"Update $\\theta _n \\leftarrow $ argmin $\\mathcal {L}_{det}(\\mathbf {S}_n, \\theta _{n-1})$ $f_{\\theta _n}(\\mathbf {V}_{K^{\\prime }}) = \\lbrace \\lbrace \\hat{p}_{jk}, \\lbrace \\hat{y}_{ijk}, \\hat{\\Sigma }_{ijk}\\rbrace _{i=1}^I \\rbrace _{j=1}^J\\rbrace _{k=1}^{K^{\\prime }}$ Compute the residual vector $e_{ijk} = y_{ijk} - \\hat{y}_{ijk}$ $\\forall i\\in [1,I], j\\in [1,J], k\\in [1,K^{\\prime }]$ $\\hat{\\theta } \\leftarrow \\theta _{N}$ Compute $\\Sigma _{a}$ as the average aleatoric uncertainty of the validation dataset Compute $\\Sigma _{e}$ as the covariance of all residual vectors for epistemic uncertainty We show the training stage of our proposed Double-M Quantification method in Algorithm REF .",
"We first initialize $\\theta $ , the parameters of a collaborative object detector and pretrain the model using the training data set.",
"Then we construct the constant-length time-series block set $\\mathbf {B}$ from the time-series training dataset $\\mathbf {D}_K$ which contains $K$ frames.",
"Notice that the block set $\\mathbf {B}$ still keeps the temporal characteristic (see Line REF ) by maintaining the order of frames within the same block.",
"Then in every training iteration, we retrain the model using a sampled dataset that contains $M$ blocks sampled with replacement and uniform random probability from the block set $\\mathbf {B}$ (see Lines REF –REF ).",
"The sampled dataset still contains about $K$ frames since $M=\\lfloor K/l \\rfloor $ .",
"The floor function $\\lfloor \\cdot \\rfloor $ is to make sure $M$ is an integer.",
"At the final step in each training iteration $n$ , we test the retained model $f_{\\theta _n}$ on the validation dataset $\\mathbf {V}_{K^{\\prime }}$ (see Line REF ) and compute the residual vector as the difference between the ground-truth vector $y_{ijk}$ and the predicted mean vector $\\hat{y}_{ijk}$ , $\\forall i\\in [1,I], j\\in [1,J], k\\in [1,K^{\\prime }]$ (see Line REF ).",
"After $N$ iterations, we get the final model parameters $\\theta _N$ as $\\hat{\\theta }$ so that our mdm method could predict the covariance by model $f_{\\hat{\\theta }}$ .",
"Other than the final trained model, we estimate both the aleatoric and epistemic uncertainties by using the residuals and predicted covariance matrics of the validation dataset.",
"We first estimate the aleatoric uncertainty by computing $\\Sigma _{a}$ , the mean of all predicted covariance matrices.",
"To estimate the epistemic uncertainty, we then compute the covariance matrix of all residual vectors, which is denoted by $\\Sigma _{e}$ .",
"On the one hand, our Double-M Quantification method provides the bagging aleatoric uncertainty estimates through the aggregation over multiple models from the $N$ iterations on the validation dataset.",
"On the other hand, it approximates the distribution of errors from the residuals so that we can quantify the epistemic uncertainty.",
"Double-M Quantification - Inference $\\Sigma _{e}$ , $\\Sigma _{a}$ , input point cloud $X$ , the trained collaborative object detector $f_{\\hat{\\theta }}$ $\\bar{Y}$ $f_{\\hat{\\theta }}(X) = \\hat{Y} = \\lbrace \\lbrace \\hat{p}_{j}, \\lbrace \\hat{y}_{ij}, \\hat{\\Sigma }_{ij}\\rbrace _{i=1}^I \\rbrace _{j=1}^J\\rbrace $ j from 1 to J i from 1 to I $\\bar{\\Sigma }_{ij} = \\Sigma _{e} + (\\frac{1}{2}\\Sigma _{a} + \\frac{1}{2}\\hat{\\Sigma }_{ij})$ $\\bar{Y} = \\lbrace \\lbrace \\hat{p}_{j}, \\lbrace \\hat{y}_{ij}, \\bar{\\Sigma }_{ij}\\rbrace _{i=1}^I \\rbrace _{j=1}^J\\rbrace $ The Inference stage of our Double-M Quantification method is shown in Algorithm REF .",
"For the $i$ th corner of the $j$ th bounding box, we use $\\hat{y}_{ij}$ as the mean of the multivariate Gaussian distribution.",
"we estimate the covariance matrix $\\bar{\\Sigma }_{ij}$ by using (i) the predicted covariance matrix $\\hat{\\Sigma }_{ij}$ from the extra regression header and (ii) the estimated aleatoric and epistemic uncertainties $\\Sigma _{e}$ , $\\Sigma _{a}$ obtained from the training stage.",
"$\\bar{\\Sigma }_{ij}$ is calculated as the following (see Lines REF –REF ): $\\begin{aligned}\\bar{\\Sigma }_{ij} = \\Sigma _{e} + (\\frac{1}{2}\\Sigma _{a} + \\frac{1}{2}\\hat{\\Sigma }_{ij}).\\end{aligned}$"
],
[
"Loss Function",
"In our mdm method, we define the regression loss function of the object detector as the KL-Divergence between the predicted distribution and the ground-truth distribution.",
"Here we assume all corners are independent, and the distribution of each corner is a multivariate Gaussian distribution: $\\nonumber P_{\\theta }(\\bar{y}_i|\\hat{y}_i,\\hat{\\Sigma }_i) = \\frac{1}{\\sqrt{2\\pi |\\hat{\\Sigma }_i|}} \\exp {\\left(-\\frac{(\\bar{y}_i - \\hat{y}_i)^T \\hat{\\Sigma }_i^{-1} (\\bar{y}_i - \\hat{y}_i)}{2} \\right)},$ where $\\bar{y}_i$ is one possible vector for the $i$ -th corner, $\\hat{\\Sigma }_i$ is a symmetric positive definite $D\\times D$ covariance matrix predicted for the $i$ -th corner.",
"In the implementation, we utilize the Cholesky decomposition [35] to calculate a symmetric positive definite covariance matrix.",
"We compare this distribution with other distributions in Section and demonstrate our selected distribution is the best.",
"We assume the distribution of each corner for the ground-truth bounding box as a Dirac delta function [18]: $ P_G(\\bar{y}_i|y_i) = \\delta (\\bar{y}_i - y_i).$ Then, we define the regression loss function for the $i$ -th corner as the Kullback–Leibler (KL) divergence between $P_{\\theta }(\\bar{y}_i|\\hat{y}_i,\\hat{\\Sigma }_i)$ and $P_G(\\bar{y}_i|y_i)$ [36]: $\\begin{aligned}\\mathcal {L}^i_{KL}(y_i, \\hat{y}_i, \\hat{\\Sigma }_i) = & \\frac{1}{2} (y_i - \\hat{y}_i)^T \\hat{\\Sigma }_i^{-1} (y_i - \\hat{y}_i) + \\\\& \\frac{1}{2} \\log |\\hat{\\Sigma }_i| + \\frac{\\log (2\\pi )}{2} - H(P_G(\\bar{y}_i)),\\end{aligned} $ where $H(P_G(\\bar{y}_i))$ is the entropy of $P_{G}(\\bar{y}_i)$ .",
"The last two terms $\\frac{\\log (2\\pi )}{2}$ and $H(P_G(\\bar{y}_i))$ could be ignored in the loss function, for they are independent of the model parameters $\\theta $ .",
"The first term encourages increasing the covariance of the Gaussian distribution as the predicted mean vector diverges from the ground-truth vector.",
"The second regularization term penalizes high covariance.",
"We add one extra regression header to predict all covariance matrix $\\hat{\\Sigma }_i$ ($i \\in 1,...,I$ ) with a similar structure as the regression header for $\\hat{y}_i$ , based on the origin collaborative object detector.",
"With a given training dataset, the collaborative object detector $f_{\\theta }$ is trained to predict $\\hat{Y}$ , the classification probability and the regression distribution of all $J$ objects.",
"The classification loss is $\\mathcal {L}_{cls}(c, \\hat{p})$ .",
"The loss function of the object detector is: $\\mathcal {L}_{det} = \\sum _{j=1}^J(\\mathcal {L}_{cls}(c_{j}, \\hat{p}_{j}) + \\sum _{i=1}^I \\mathcal {L}^i_{KL}(y_{ij}, \\hat{y}_{ij}, \\hat{\\Sigma }_{ij})).$ To demonstrate our uncertainty quantification method for collaborative detection, we evaluated it on the V2X-Sim dataset [1] which contains 80 scenes for training, 10 scenes for validation, and 10 scenes for testing.",
"It is generated by CARLA simulation [37].",
"Each scene contains a 20-second traffic flow at a certain intersection with a 5Hz record frequency, which means each scene contains 100 time-series frames.",
"In each scene, 2-5 vehicles are selected as the connected vehicles and 3D point clouds are collected from LiDAR sensors on them.",
"For object detection, we consider Lower-bound, DiscoNet, and Upper-bound for benchmark as follows: lb [1]: The individual object detector without collaboration which only uses the point cloud data from one individual LiDAR.",
"dn [2]: The intermediate collaborative object detector which utilizes a directed graph with matrix-valued edge weight to adaptively aggregate features of different agents by repressing noisy spatial regions while enhancing informative regions.",
"It has shown a good performance-bandwidth trade-off by sharing a compact and context-aware scene representation.",
"ub [1]: The early collaborative object detector uses raw point cloud data from all connected vehicles, as shown in Fig.",
"REF (b).",
"It usually has excellent performance with lossless information, yet consumes high communication bandwidth.",
"lb is the traditional individual object detector.",
"dn and ub are collaborative object detectors.",
"The basic implementation of all three benchmarks is from the public code of [1].",
"It uses FaFNet [38], a classic anchor-based model containing a convolutional encoder, a convolutional decoder, and two output headers for classification and regression, as the backbone of all detectors.",
"We compare three uncertainty quantification methods, which are dm, mbb, and Double-M Quantification on accuracy and uncertainty.",
"Compared with mdm, the mbb method does not have the predicted covariance matrix during training and inference.",
"For dm and mdm, we add one output header for the covariance matrix and the regression loss is the kl loss in Eq (REF ).",
"For mbb, the regression loss is the smooth $L_1$ loss.",
"For all, the classification loss is the focal cross-entropy loss [39].",
"Table: Detection performance of different UQ methods on LB, DN, and UB.",
"Our mdm method improves up to 3.13% average precision."
],
[
"Accuracy Evaluation",
"We use ap at iou thresholds of 0.5 and 0.7 as the accuracy measurement, which is widely used in object detection [8].",
"Table REF shows the ap results of different uncertainty quantification (UQ) methods on the test dataset, where “None\" means the object detector is a deterministic one without any uncertainty quantification method.",
"Our mdm achieves better accuracy than others, especially for collaborative object detection.",
"Compared with deterministic detectors without the UQ method, it increases up to 3.13% ap, which means our proposed mdm method improves the accuracy of collaborative object detection.",
"Meanwhile, DiscoNet and Upper-bound achieve much higher accuracy than Lower-bound for sharing information between CAVs.",
"Table: NLL comparison of different UQ methods on LB, DN, and UB.",
"Our mdm method achieves up to 4×\\times improvement on NLL."
],
[
"Uncertainty Evaluation",
"We use nll at iou thresholds of 0.5 and 0.7 as the uncertainty measurement [8], [13].",
"nll is a widely used uncertainty score to measure the quality of predicted probability distribution on a test dataset [17], [18], [22], [23].",
"For the test dataset with $K$ frames, it is computed as: $ NLL = - \\frac{1}{I \\times J \\times K} \\sum _{k=1}^{K}\\sum _{j=1}^{J}\\sum _{i=1}^{I}\\log P(y_{ijk}|\\hat{y}_{ijk}, \\bar{\\Sigma }_{ijk}),$ where $\\hat{y}_{ijk}$ $\\bar{\\Sigma }_{ijk}$ are the mean and covariance of the multivariate Gaussian distribution.",
"For mbb, $\\bar{\\Sigma }_{ijk}$ is $\\Sigma _{e}$ .",
"For dm, it is the predicted covariance $\\hat{\\Sigma }_{ijk}$ from the additional regression header.",
"For mdm, it is computed with Eq (REF ).",
"Table REF shows the nll results of different uncertainty quantification methods on the test dataset.",
"From Table REF , we can see our proposed mdm always achieves much smaller nll than other uncertainty quantification methods, with up to 4$\\times $ improvement, which means our mdm method performs best on uncertainty quantification.",
"Fig.",
"REF shows the qualitative results of our mdm method on different scenes.",
"For different types of object detectors, DiscoNet always achieves the smallest nll.",
"From Table REF and Fig.",
"REF , we can see sharing intermediate feature information between CAVs could improve the uncertainty of object detection."
],
[
"Ablation Study on Uncertainty Distribution",
"We compare different probability distributions, which is the key step of the dm method and the mdm method, on accuracy.",
"In particular, we consider the following probability Gaussian distribution: Independent Multivariate Gaussian (IMG): Our uncertainty representation of independent Gaussian distribution.",
"All corners are independent, and the distribution of each corner is a multivariate Gaussian distribution.",
"Independent Single-variate Gaussian (ISG) [18]: Single-variant Gaussian distribution.",
"All corners are independent, and all dimensions of one corner are also independent.",
"We use a single-variate Gaussian distribution for each dimension of each corner.",
"Dependent Multivariate Gaussian (DMG) [19]: High dimensional Gaussian distribution.",
"All corners are dependent, and the distribution of all corners is a multivariate Gaussian distribution.",
"Table REF shows the ap results of dm method and mdm method for the Upper-bound detector, under different probability distributions.",
"From the table, we can see IMG achieves the best accuracy with up to 4.24% improvement, which means our uncertainty representation of independent Gaussian distribution for each corner outperforms single-variate Gaussian distribution and high dimensional Gaussian distribution formats.",
"The reason is that our design considers high dependence of all dimensions in one corner and low dependence of all corners.",
"Table: The accuracy comparison of Upper-bound under different probability distributions.",
"Our IMG distribution improves up to 4.24% average precision."
],
[
"Conclusion",
"This work proposes the first attempt to estimate the uncertainty of collaborative object detection.",
"We propose one novel uncertainty quantification method, called Double-M Quantification, to predict both the epistemic and aleatoric uncertainty with one inference pass.",
"The key novelties are the tailored moving block bootstrap training process, and the loss function design that estimates one independent multivariant Gaussian distribution for each corner of the bounding box.",
"We validate our uncertainty quantification method on different collaborative object detectors.",
"Experiments demonstrate that our method achieves better uncertainty estimation and accuracy.",
"In the future, we will apply our method to more collaborative perception datasets, and enhance the performance of trajectory prediction with uncertainty quantification."
]
] | 2209.08162 |
[
[
"CLAIRE -- Parallelized Diffeomorphic Image Registration for Large-Scale\n Biomedical Imaging Applications"
],
[
"Abstract We study the performance of CLAIRE -- a diffeomorphic multi-node, multi-GPU image-registration algorithm, and software -- in large-scale biomedical imaging applications with billions of voxels.",
"At such resolutions, most existing software packages for diffeomorphic image registration are prohibitively expensive.",
"As a result, practitioners first significantly downsample the original images and then register them using existing tools.",
"Our main contribution is an extensive analysis of the impact of downsampling on registration performance.",
"We study this impact by comparing full-resolution registrations obtained with CLAIRE to lower-resolution registrations for synthetic and real-world imaging datasets.",
"Our results suggest that registration at full resolution can yield a superior registration quality -- but not always.",
"For example, downsampling a synthetic image from $1024^3$ to $256^3$ decreases the Dice coefficient from 92% to 79%.",
"However, the differences are less pronounced for noisy or low-contrast high-resolution images.",
"CLAIRE allows us not only to register images of clinically relevant size in a few seconds but also to register images at unprecedented resolution in a reasonable time.",
"The highest resolution considered is CLARITY images of size $2816\\times3016\\times1162$.",
"To the best of our knowledge, this is the first study on image registration quality at such resolutions."
],
[
"Introduction",
"3D diffeomorphic image registration (also known as “image alignment” or “matching”) is a critical task in biomedical image analysis [42], [87].",
"For example, it enables the study of morphological changes associated with the progression of neurodegenerative diseases over time or in imaging studies of patient populations.",
"The process of image registration involves finding a spatial transformation which maps corresponding points in an image to those in another [42].",
"In mathematical notation, we are given two images $m_0(x)$ (the template/moving image) and $m_1(x)$ (the reference/fixed image; here $x\\in \\Omega \\subset \\mathbb {R}^3)$ and we seek a spatial transformation $y : \\mathbb {R}^{3} \\rightarrow \\mathbb {R}^{3}$ , such that the deformed template image $m_0(y(x))$ is similar to the reference image $m_1(x)$ for all $x$ (see Figure REF for an illustration) [72], [73].",
"Image registration methods can be categorized based on the parameterization for $y$ [72].",
"We seek a diffeomorphic map $y$ , i.e., $y$ is a differentiable bijection and has a differentiable inverse.",
"Methods that parameterize $y$ in terms of a smooth, time-varying velocity field $v : \\mathbb {R}^3 \\times [0,1] \\rightarrow \\mathbb {R}^3$ belong to a class of methods referred to as large-deformation diffeomorphic metric mapping (LDDMM) [11], [90], [99].",
"In this study, we consider a related class of methods that use stationary velocity fields $v : \\mathbb {R}^3 \\rightarrow \\mathbb {R}^3$ .",
"This diffeomorphic registration problem is expensive to solve because the problem is infinite-dimensional, and upon discretization results in a nonlinear system with millions of unknowns—even for stationary velocity fields.",
"For example, solving the registration problem for two images of resolution $256^3$ (a typical size for clinical scans) requires solving for approximately 50 million unknowns in space (three vector components per image grid point).",
"Furthermore, image registration is a highly nonlinear, ill-posed inverse problem [29], resulting in ill-conditioned inversion operators.",
"Consequently, running registration on multi-core high-end CPUs can take several minutes.",
"There exist various algorithms and software packages for fast registration of images at standard clinical resolution (e.g., $256^3$ ) [94], [93], [8], [98], [53], [38], [102], [37].",
"This includes CLAIRE, which can execute image registration in parallel on multi-node multi-core CPUs and GPUs [67], [35], [69], [16], [15], [14].",
"We note that there is little work on scalable image registration.",
"One application that requires this scalability is the registration of CLARITY images [22], [50], [55], [56], [89], [96] with a resolution in the order of $20\\text{\\,K}\\times 20\\text{\\,K}\\times 1\\text{\\,K}$ .",
"This corresponds to a problem with approximately 1.2 trillion unknowns.",
"In [14], we extended CLAIRE to support GPU-accelerated scalable image registration, which can process high resolution images using multiple GPUs.",
"We demonstrated the scalability of our solver using synthetic images with a resolution up to $2048^3$ and CLARITY mouse brain images of size $768\\times 768\\times 1024$ .",
"In this work, we scale registration to an even higher resolution, e.g., CLARITY images of size $2816\\times 3016\\times 1162$ .",
"This corresponds to an increase of 16$\\times $ in problem size.",
"In our previous work [63], [64], [67], [65], [69], [68], [36], [70], we have extensively studied the algorithmic side of image registration within the framework of CLAIRE.",
"In this paper, we pay closer attention to the quality of the registration results.",
"We study the effect of different input parameters, including the quality and resolution of the input images, on the accuracy of the registration.",
"Figure: Illustration of the image registration problem.",
"Panel (A): 3D rendering of an exemplary set of input images.",
"Panel (B): Image registration is the task of computing spatial correspondences between two images of the same object.",
"These correspondences, denoted as yy, are indicated by the red arrows for example points within a single axial slice of the template and reference image data shown in panel (A).",
"Before we execute CLAIRE, we compensate for the global mismatch between the considered images by performing an affine registration.",
"In panel (C), we show an axial slice of the volume shown in panel (A) after an affine pre-registration step has been carried out; we execute CLAIRE on these images.",
"Panel (D): CLAIRE outputs a diffeomorphic deformation map yy that matches each point in the template image m 0 m_0 to its corresponding point in the reference image m 1 m_1.",
"We show a typical deformation map yy in the leftmost image and the corresponding determinant of the deformation gradient (encodes volume change) in the second and third image from the left (axial and coronal slice).",
"In CLAIRE, we invert for a stationary velocity field vv that parameterizes yy (second and third figure from the right; color denotes orientation).",
"The last figure in panel (D) shows the point-wise residual after applying CLAIRE."
],
[
"Contributions",
"We build up on our prior work on scalable deformable image registration [63], [64], [65], [69], [67], [68], [70], [16], [14], [15] using CLAIRE and analyze the effect of image resolution on image registration accuracy.",
"The present work analyzes image registration performance.",
"We do not propose any major improvements in our methodology, with the exception of additional advice for hyperparameter tuning.",
"We outline our past contributions on formulations, algorithms, and their parallel implementation below.",
"Our major contributions of the present work are: We evaluate CLAIRE on high resolution synthetic and real imaging datasets.",
"We demonstrate that image registration when performed at native high resolution results in higher accuracy (measured in terms of the Dice coefficient of the labeled structured in the images).",
"We conduct experiments to show that downsampling the images and then registering them result in loss of registration accuracy.",
"We design scalable image registration experiments to explore the effect of solver parameters—the number of time steps $n_t$ in the semi-Lagrangian scheme, and regularization parameters $\\beta _v$ and $\\beta _w$ —on the registration performance.",
"We present an extension of the regularization parameter continuation scheme first presented in [63] by searching for $\\beta _w$ in addition to $\\beta _v$ , thereby removing the need for selecting an additional resolution-dependent solver parameter.",
"We study the performance of our scalable registration solver CLAIRE for applications in high resolution mouse and human neuroimage registration.",
"We perform image registration for two pairs of CLARITY mouse brain images at a resolution of $2816\\times 3016\\times 1162$ voxels.",
"To the best of our knowledge, images of this scale have not been registered before at full resolution in under 30 min."
],
[
"Related Work",
"The current work builds upon the open source framework CLAIRE [63], [64], [67], [65], [69], [68], [16], [14], [15], [36], [66].",
"Our formulation for diffeomorphic image registration has been described in [63], [64].",
"Our Newton–Krylov solver was originally developed in [63].",
"We proposed efficient numerical implementations for evaluating forward and adjoint operators in [65], [70], [67].",
"We designed various methods for preconditioning in [63], [65], [14], [69].",
"The computational kernels of the parallel CPU implementation of our solver were introduced in [36], [67], [69].",
"More recently, we ported CLAIRE to GPU architectures [16], [14].",
"In summary, our work paved the way towards real-time applications of diffeomorphic image registration and its deployment to high-resolution medical imaging application.",
"To the best of our knowledge, this is the only existing software for diffeomorphic image registration with these capabilities.",
"We have integrated our framework with biophysical modeling in [36], [68], [62], [80], [81].",
"None of these works explores registration performance in large-scale biomedical imaging applications.",
"Literature surveys of image registration and associated algorithmic developments can be found in [72], [87].",
"A recent overview of existing LDDMM methods can be found in [69].",
"Related LDDMM software packages include Demons [94], ANTs [7], [5], [6], DARTEL [4], deformetrica [12], [13], [30], [25], FLASH [101], LDDMM [11], [18], ARDENT [75], ITKNDReg [46], and PyCA [79].",
"Surveys of GPU-accelerated image registration solvers can be found in [32], [84], [27]; particular examples for various formulations are [17], [13], [23], [26], [28], [38], [37], [40], [39], [49], [52], [71], [86], [82], [83], [91], [92].",
"Multi-GPU LDDMM implementations for atlas construction are described in [40], [39], [91], [92].",
"Their setup is embarrassingly parallel in the sense that they solve many small registration problems independently on single GPUs.",
"In [40], [39], [49], the computation bottlenecks are the repeated solution of a Helmholtz-type PDE and trilinear scattered data interpolation to compute and apply the deformation map.",
"They use hardware acceleration for the trilinear interpolation kernel with 3D texture volume support.",
"The runtime for a single dataset of size $160\\times 192\\times 160$ is 20 on an NVIDIA Quadro FX 5600.",
"CLAIRE uses a multi-node multi-GPU framework with high computational throughput for single (large-scale) registration problems [14] which is no longer an embarrassingly parallel problem.",
"CLAIRE uses the Message Passing Interface (MPI) to parallelize the implementation.",
"None of the GPU-accelerated LDDMM methods mentioned above, except for CLAIRE [63], [64], [67], [65], [69], [68], [16], [14], [15], [36], [66], use second-order numerical optimization.",
"Many of the available methods solve the registration problem by reducing the number of unknowns either through a coarse parameterization or by using a coarse grid and use simplified algorithms.",
"These crude approximations and simplifications can result in inferior registration quality [69], [16].",
"The work in [55] focuses on annotating CLARITY brain images by registering them to the Allen Institute's Mouse Reference Atlas (ARA).",
"They use a “masked” LDDMM approach.",
"They also consider the registration of CLARITY-to-CLARITY brain images and compare different mismatch terms for the registrations.",
"However, they downscale the images to a lower resolution for conducting all experiments.",
"In [56], mutual information is used for the registration of CLARITY to the ARA dataset but at an approximately one hundred times downsampled resolution (at an original in-plane isotropic resolution 0.58 $\\mu m$ ).",
"The authors in [74] analyze registration performance on high-resolution mouse brain images of size $2560\\times 2160\\times 633$ were obtained using the CUBIC protocol [88].",
"They report results using different software packages including ANTs and Elastix.",
"They did not observe a relationship between registration accuracy at different resolutions.",
"For their high-resolution runs using ANTs, they report a wall clock time of over 200 hrs on a single compute node (2.66GHz 64bit Intel Xeon processor with 256GB RAM) while the same run with elastix [51] took approximately 30hrs.",
"The authors in [76] register high-resolution images of mouse brains to the ARA dataset [54].",
"They perform nonlinear registration using ANTs at coarse resolution ($10\\mu m$ for the ARA) and apply the deformation at high-resolution.",
"In the current work, we do not downsample high-resolution images but register them at the original resolution.",
"We can register CLARITY images of resolution $2816\\times 3016\\times 1162$ in less than 30min using 256 GPUs.",
"In addition to that, we study the effect of resolution on the registration quality."
],
[
"Outline",
"We summarize the overall formulation in §REF and the algorithms in §REF for completeness.",
"We note, that all of the material presented in §REF and §REF has been discussed in detail in [14], [69].",
"In §, we present our kernels and parallel algorithms and discuss key solver parameters.",
"We also introduce a new scheme to automatically identify adequate parameters of our solver for unseen data.",
"This scheme extends on our prior work in [69].",
"We conclude with the main scalability experiments in §, and present conclusions in §."
],
[
"Limitations",
"CLAIRE currently only supports mono-modal similarity measures, which limits our study to registrations for images acquired with the same imaging modality.",
"Moreover, CLAIRE only supports periodic boundary conditions, i.e., we require that the image data be embedded in a larger background domain.",
"In most medical imaging applications, the images are embedded in a zero background and, therefore, naturally periodic.",
"If the images are not periodic, they can be zero-padded and mollified.",
"CLAIRE uses stationary velocities, which improves computational efficiency, but it suboptimal from a theoretical point of view.",
"In [63], we found no qualitative differences in registration mismatch when registering two images using stationary velocities.",
"This observation is in line with the work of other groups using stationary velocity fields [43], [60].",
"Regarding computational performance, one issue is the memory requirement of our method.",
"We have optimized memory allocation for the core components of CLAIRE.",
"Additional optimizations by reusing and sharing memory across external libraries to further reduce the memory load remain subject to future work."
],
[
"Methods",
"Before discussing our enhancements in §, we shortly introduce the underlying mathematical formulation of the image registration problem utilized in CLAIRE as well as the discretization and the numerical algorithms.",
"The following exposition is included for completeness, only, and is based on material described in our prior work on efficient algorithms for diffeomorphic image registration [63], [64], [67], [65], [70], [69], [16], [14], [36].",
"Consequently, we keep this section brief.",
"Table: Notation and main symbols."
],
[
"Formulation",
"We summarize our notation in Table REF .",
"CLAIRE uses an optimal control formulation.",
"We parameterize the deformation map $y(x)$ through a smooth, stationary velocity field $v(x)$ .",
"The optimization problem is: Given two images $m_0(x)$ (template image; image to be deformed) and $m_1(x)$ (reference image), we seek a stationary velocity field $v(x)$ by solving $\\operatornamewithlimits{minimize}_{v, m}\\;\\;\\frac{1}{2}\\!\\int _{\\Omega }\\!",
"(m(x,1) - m_1(x))^2\\!\\mathrm {d}x+\\frac{\\beta _v}{2} \\operatorname{reg}_v(v) + \\frac{\\beta _w}{2} \\operatorname{reg}_w(w)$ subject to $\\partial _t m(x,t) + v(x) \\cdot \\nabla m(x,t)&= 0 &&\\text{in}\\;\\Omega \\times (0,1],\\\\m(x,t)&= m_0(x) &&\\text{in}\\;\\Omega \\times \\lbrace 0\\rbrace ,\\\\\\nabla \\cdot v&= w &&\\text{in}\\;\\Omega ,$ on a rectangular domain $\\Omega \\subset \\mathbb {R}^3$ with periodic boundary conditions on $\\partial \\Omega $ .",
"The first term in (REF ) is a squared $L^2$ image similarity metric, which measures the distance between the deformed template image $m(x,t=1)$ and the reference image $m_1(x)$ .",
"The objective functional (REF ) additionally consists of two regularization models that act on the controls $v$ and $w$ with regularization parameters $\\beta _v>0$ and $\\beta _w>0$ , respectively.",
"The regularization operators are introduced to prescribe sufficient regularity requirements on $v$ and its divergence $\\nabla \\cdot v$ .",
"Smoothness of the velocity guarantees that the computed map is diffeomorphic [11], [90], [99].",
"We refer to [64] for details about our regularization scheme.",
"The default configuration of CLAIRE is an $H^1$ -Sobolev-seminorm for $v$ and $H^1$ -Sobolev-norm for $w$ [64], [69].",
"The transport equation () represents the geometrical deformation of $m_0(x)$ by advecting the intensities forward in time.",
"To solve (REF ), we apply the method of Lagrange multipliers to obtain the Lagrangian functional $\\begin{aligned}\\mathcal {L}(\\phi ) \\mathrel {\\mathop :}=&\\frac{1}{2}\\int _{\\Omega }\\!",
"(m(x,1) - m_1(x))^2\\mathrm {d}x+\\frac{\\beta _v}{2} \\operatorname{reg}_v(v) + \\frac{\\beta _w}{2} \\operatorname{reg}_w(\\nabla \\cdot v)\\\\&+ \\int _0^1\\!\\!\\int _\\Omega \\lambda (x,t) (\\partial _t m + v \\cdot \\nabla m )\\mathrm {d}xṭ\\\\& + \\int _\\Omega \\lambda (x,0) (m(x,0) - m_0(x)) \\mathrm {d}x+ \\int _\\Omega p(x)(\\nabla \\cdot v - w) \\mathrm {d}x\\end{aligned} $ with state, adjoint, and control variables $(m,\\lambda ,p,v,w) \\mathrel {\\mathop :}=\\phi $ , respectively."
],
[
"Optimality Conditions & Reduced Space Approach",
"To derive the first order optimality conditions, we take the variations of $\\mathcal {L}$ with respect to the state variable $m$ , the adjoint variables $\\lambda $ and $p$ , and the control variable $v$ .",
"This results in a set of coupled, hyperbolic-elliptic PDEs in 4D (space-time).",
"CLAIRE uses a reduced-space approach, in which one iterates only on the reduced-space of $v$ .",
"We require $g(v^\\star ) = 0$ for an admissible solution $v^\\star $ , where $g(v) \\mathrel {\\mathop :}=\\beta _v \\mathcal {A}v(x) + \\mathcal {K} \\int _0^1 \\!\\!\\!\\lambda (x,t)\\nabla m(x,t)ṭ$ is the so-called reduced gradient.",
"The operator $\\mathcal {A}$ corresponds to the first variation of the regularization model for $v$ (i.e., $\\operatorname{reg}_v$ in (REF )) and the operator $\\mathcal {K}$ projects $v$ onto the space of near-incompressible velocity fields (see [64] for details).",
"To evaluate (REF ), we first solve the forward problem () and then the adjoint problem given by $-\\partial _t \\lambda (x,t) - \\nabla \\cdot \\lambda (x,t)v(x) = 0 \\quad \\text{in } \\Omega \\times [0,1)$ with final condition $\\lambda (x,t) = m_1(x) - m(x,t)$ in $\\Omega \\times \\lbrace 1\\rbrace $ and periodic boundary conditions on $\\partial \\Omega $ .",
"We discretize the forward and adjoint PDEs in the space-time interval $\\Omega \\times [0,1]$ , $\\Omega \\mathrel {\\mathop :}=[0,2\\pi )^3\\subset \\mathbb {R}^3$ , with periodic boundary conditions on $\\partial \\Omega $ , on a regular grid with $N = N_1N_2N_3$ grid points $x_{ijk}\\in \\mathbb {R}^3$ in space and $n_t+1$ grid points in time.",
"We use a semi-Lagrangian time-stepping method to solve the transport equations that materialize in the optimality system [65], [67].",
"Key computational subcomponents of this scheme are $2^{\\mathsf {nd}}$ -order Runge–Kutta time integrators and spatial interpolation kernels [65], [67], [14], [69].",
"To solve the transport equation (REF ) and to evaluate the reduced-gradient $ g $ (REF ), we need to apply gradient and divergence operators.",
"We use an $8^{\\mathsf {th}}$ finite difference (FD) scheme for these first-order differential operators [16], [14].",
"The reduced gradient (REF ) also involves the vector-Laplacian $\\mathcal {A}$ and the Leray-like operator $\\mathcal {K}$ (see [64]).",
"In spectral methods, inversion and application of higher-order differential operators come at the cost of two FFTs and one Hadamard product in Fourier space.",
"CLAIRE uses a Gauss–Newton–Krylov method globalized with an Armijo line search to solve the non-linear problem $g(v) = 0$ [63], [69].",
"The iterative scheme is given by $v_{k+1} = v_k + \\alpha _k \\tilde{v}_k,\\quad H\\tilde{v}_k = -g_k,\\quad k = 0,1,2,\\ldots ,$ where $H\\in \\mathbb {R}^{3N,3N}$ is the discretized reduced-space Hessian operator, $\\tilde{v}_k\\in \\mathbb {R}^{3N}$ the search direction, $g_k\\in \\mathbb {R}^{3N}$ a discrete version of the gradient in (REF ), $\\alpha _k>0$ a line search parameter, and $k\\in \\mathbb {N}$ the Gauss–Newton iteration index.",
"We have to solve the linear system in (REF ) at each Gauss–Newton step.",
"We do not form or assemble $H$ ; we use a matrix-free preconditioned conjugate gradient (PCG) method to solve $H\\tilde{v}_k = -g_k$ for $\\tilde{v}_k$ .",
"This only requires an expression for applying $H$ to a vector that we term Hessian matvec.",
"In continuous form, the Gauss–Newton approximation of this matvec is given by $\\mathcal {H}\\tilde{v} =\\beta _v\\mathcal {A}\\tilde{v}(x)+ \\mathcal {K} \\int _0^1\\tilde{\\lambda }(x,t)\\nabla m(x,t) ṭ.$ Similarly to the evaluation of the reduced gradient in (REF ), the application of the Hessian to a vector in (REF ) requires the solution of two PDEs to find the space-time field $\\tilde{v}$ (see [63], [64], [69] for details).",
"Consequently, solving the linear system with $H$ in (REF ) is the most expensive part of CLAIRE.",
"Preconditioning of the reduced-space Hessian system can be used to alleviate these computational costs.",
"In [14] we have introduced a zero velocity approximation for $H$ as a preconditioner.",
"This preconditioner can be applied at full resolution and through a two-level coarse grid approximation (see [14] for details).",
"The latter variant represents the default considered in the present work."
],
[
"Computational Kernels and Parallel Algorithms",
"At each Gauss–Newton step, we have to solve the forward and the adjoint equations for the reduced gradient and the Hessian matvecs.",
"The main computational cost in CLAIRE constitute FFTs for (inverse) differential operators, scattered data interpolation (IP) for the semi-Lagrangian solver, and FD for computing first order derivatives (see [65], [67], [16], [14] for a detailed description of these computational components).",
"The distributed memory CPU implementation of CLAIRE uses AccFFT [34], [33] for spectral operations [67], [36].",
"In the single GPU setup, we use the highly optimized 3D FFT operations provided by NVIDIA's cuFFT library.",
"In the multi-node multi-GPU setup, we use a 2D slab decomposition to leverage 2D cuFFT functions.",
"We decompose the spatial domain in $x_1$ direction, which is the outer-most dimension, and the spectral domain in the $x_2$ direction.",
"Let $p$ be the number of MPI tasks.",
"Then, each MPI task gets $(N_1/p) \\times N_2 \\times N_3$ grid points, where $N_1, N_2, N_3$ are the image dimensions.",
"We have discussed the implementation details and shown scalability of the FFT kernel in [14].",
"The parallel implementation of our IP kernel on CPUs was introduced in [67] and improved in [36].",
"In [16], we explored linear, cubic Lagrange, and cubic B-spline interpolation schemes for the interpolation kernel on a single GPU setup.",
"In [14], we ported these kernels to the multi-node multi-GPU setup and made several optimizations.",
"In the present study, we use linear interpolation to evaluate the image intensities at the off-grid points (also called characteristic points) in our semi-Lagrangian scheme.",
"Depending on the image data layout and the velocity field, the IP kernel requires scattered peer-to-peer communication of off-grid points between the owner and the worker processors.",
"The CPU version of CLAIRE uses FFTs for spatial derivatives [69], [67], [36].",
"In [16], we introduced the $8^{\\mathsf {th}}$ order FD kernel to evaluate first order derivatives, i.e., spatial gradients and divergence operators on a single GPU.",
"In [14], we ported the FD kernel to the multi-GPU setup.",
"We use the FD kernel for computing first order derivatives throughout the registrations performed in this paper.",
"CLAIRE uses CUDA-aware MPI in the multi-node multi-GPU setup, thereby avoiding unnecessary CPU-GPU communication and automatically utilizing the high-speed on-node NVLink interconnect bus between GPUs if it is available."
],
[
"Compute Hardware and Libraries",
"All runs reported in this study were executed on TACC's Longhorn system in single precision.",
"Longhorn hosts 96 NVIDIA Tesla V100 nodes.",
"Each node is equipped with four GPUs and 16 GB GPU RAM each (i.e., 64 GB per node) and two IBM Power 9 processors with 20 cores (40 cores per node) at 2.3 GHz with 256 GB memory.",
"Our implementation uses PETSc [9], [10] for linear algebra, the PETSc TAO package for nonlinear optimization, CUDA [77], thrust [44], cuFFT for FFTs [78], niftilib [31] for serial I/O for small images and PnetCDF [59] for parallel I/O for large scale images, IBM Spectrum MPI [2], and the IBM XL compiler [45]."
],
[
"Code availability",
"CLAIRE [66], [15] is available publicly for download on github at https://github.com/andreasmang/claire under the GNU General Public License v3.0."
],
[
"Key Solver Parameters",
"Here, we summarize the key parameters of CLAIRE and discuss their effect on the solver and previous strategies to choose suitable values.",
"In §REF , we present our algorithm to choose these parameters in a combined continuation approach.",
"$\\beta _v$ — regularization parameter for the velocity field $v$ .",
"Large values for $\\beta _v$ result in very smooth velocities and, thus, maps that are typically associated with a large final image mismatch.",
"Smaller values of $\\beta _v$ allow complex deformations but lead to a solution that might be close to being non-diffeomorphic due to discretization issues.",
"From a user application point of view, we are interested in computing velocity fields, for which the Jacobian determinant, i.e., the determinant of the deformation gradient $F \\nabla y $ , is strictly positive for every image voxel.",
"This guarantees a locally diffeomorphic transformation (subject to numerical accuracy).",
"In [63], [41], we determined the regularization parameter $\\beta _v$ based on a binary search algorithm.",
"The search is constrained by the bounds on $J = \\det F$ .",
"That is, we choose $\\beta _v$ such that $J$ is bounded from below by $J_{\\text{min}}$ and bounded from above by $1/J_{\\text{min}}$ , where $J_{\\text{min}}\\in (0,1)$ is a user-defined parameter.",
"The binary search is expensive because we solve the inverse problem repeatedly: For each trial $\\beta _v$ , we iterate until the convergence criteria for the Gauss–Newton–Krylov solver is met then use the previous velocity field as an initial guess for the next trial $\\beta _v$ .",
"$\\beta _w$ — regularization parameter for the divergence of the velocity field $w=\\nabla \\cdot v$ .",
"The choice of $\\beta _w$ , along with $\\beta _v$ , is equally critical.",
"Small values can result in extreme values of $J$ and make the deformations locally non-diffeomorphic.",
"As discussed above, in our previous work [63], we do parameter continuation in $\\beta _v$ and keep $\\beta _w$ fixed.",
"This is sub-optimal for two reasons: (i) Both $\\beta _v$ and $\\beta _w$ depend on the resolution, so keeping $\\beta _w$ fixed for all resolutions can result in deformations with undesirable properties, and (ii) doing continuation in $\\beta _v$ alone does not ensure we get close enough to the set Jacobian bounds.",
"Therefore, adding continuation in $\\beta _w$ , which also affects the Jacobian, is necessary.",
"$J_{\\text{min}}$ — lower bound for the determinant $J$ of the deformation gradient.",
"The choice of this parameter is typically driven by dataset requirements, i.e., one has to decide how much volume change is acceptable.",
"CLAIRE uses a default value of 0.25 [69].",
"Tighter bound on the Jacobian, i.e., $J_{\\text{min}}$ close to unity, will result in large $\\beta _v$ and $\\beta _w$ values leading to simple deformations and sub-par registration quality.",
"Relaxing the Jacobian bound in combination with our continuation schemes for $\\beta _v$ and $\\beta _w$ can result in very small regularization parameters and extremely complex deformations.",
"$n_t$ — number of time steps in the semi-Lagrangian scheme.",
"The semi-Lagrangian scheme is unconditionally stable and outperforms RK2 time integration schemes in terms of runtime for a given accuracy tolerance [65].",
"The choice of $n_t$ is based on the adjoint error, which is the error measured after solving () forward and then backward in time.",
"In [65], we conducted detailed experiments for 2D image registration and found, that even for problems of clinical resolution $n_x=256^2$ , $n_t=3$ (CFL=10) did not cause issues in solver convergence.",
"Increasing $n_t$ beyond a certain value will introduce additional discretization errors from the interpolation scheme.",
"Resolution of $v$.",
"We use the same spatial discretization for $v$ as given for the input images.",
"There exist image registration algorithms that approximate the registration deformation in a low-dimensional bandlimited space without sacrificing accuracy, resulting in dramatic savings in computational cost [102].",
"We have not explored this within the framework of CLAIRE.",
"Note that [102] uses higher order regularization operators, which leads to smoother velocities compared to the ones CLAIRE produces, therefore enabling a representation on a coarser mesh.",
"Moreover, CLAIRE uses a stationary velocity field, i.e., $v$ is constant in time.",
"In our previous work [63], we have demonstrated that stationary and time-varying velocity fields yield similar registration accuracy for registration between two real medical images of different subjects.",
"More precisely, we did not observe any practically significant quantitative differences in registration accuracy for a varying number of coefficient fields in the case of time-varying velocity fields.",
"Using a stationary velocity field is significantly cheaper and has a smaller memory overhead from a computational cost perspective."
],
[
"Parameter Identification",
"Our algorithm to choose solver parameters proceeds as follows.",
"Resolution-dependent choice of the interpolation order and $n_t$ .",
"As our GPU implementation is only available in single precision (unlike the CPU implementation [69], which is available both in single and double precision), we use cubic interpolation (B-splines/Lagrange polynomials) with $n_t=4$ ($n_t=8$ for linear interpolation) for resolutions up to $n_x=(256,256,256)$ .",
"For higher resolutions, we use linear interpolation to save computational cost and increase $n_t$ proportionately to $n_x$ to keep the CFL number fixed.",
"Parameter search scheme for $\\beta _v$ and $\\beta _w$ .",
"We perform a two-stage search scheme: (i) In the first part of the parameter search, we fix $\\beta _w = \\beta _{w,\\text{max}}$ ($\\beta _{w,\\text{init}} = 1\\text{e-}05$ ) and search for $\\beta _v$ .",
"The registration problem is first solved for a large value of $\\beta _v = \\beta _{v,\\text{max}}$ so that we under-fit the data.",
"In our experiments, we set $\\beta _{v,\\text{max}}=1$ .",
"Subsequently, $\\beta _v$ is reduced by one order of magnitude in every continuation step and the registration problem is solved again with the new $\\beta _v$ .",
"We repeat the reduction of $\\beta _v$ until we breach the Jacobian bounds $[J_{\\text{min}}, 1/J_{\\text{min}}]$ .",
"When this happens, we do a binary search for $\\beta _v$ between the last two values and terminate the binary search when the relative change in $\\beta _v$ is less than 10% of the previous valid $\\beta _v$ .",
"In addition, we put a lower bound $\\beta _{v,\\text{min}}=1\\text{e-}05$ on $\\beta _v$ .",
"This lower bound is set purely to minimize computational cost.",
"We denote the final value of $\\beta _v$ as $\\beta _v^\\star $ .",
"(ii) In the second part of the search, we do a simple reduction search for $\\beta _w$ by fixing $\\beta _v=\\beta _v^\\star $ .",
"Starting with a given value $\\beta _{w,\\text{max}}$ , we reduce $\\beta _w$ by one order of magnitude and repeat solving the registration problem with $\\beta _v^\\star $ and the respective value for $\\beta _w$ until we reach $J_{\\text{min}}$ .",
"We put a lower bound $\\beta _{w,\\text{min}}=1\\text{e-}07$ on $\\beta _w$ in order to minimize computational cost.",
"We take the last valid value of $\\beta _w$ , for which the Jacobian determinant was within bounds and denote it as $\\beta _w^\\star $ .",
"We fixed the value of $\\beta _{w,\\text{max}}=1\\text{e-}05$ for all experiments and resolutions.",
"We determined this value empirically by running image registration on a couple of image pairs at resolution $640\\times 880\\times 880$ and $160\\times 220\\times 220$ (see §REF for the images) for different values of $\\beta _{w,\\text{max}}$ .",
"We report these runs in Table REF (see Appendix ).",
"We evaluate the parameter search scheme for real world brain images and report the performance in §REF .",
"Furthermore, we use it as the default parameter search scheme for all the experiments presented in this paper.",
"Parameter continuation scheme for $\\beta _v$ and $\\beta _w$ .",
"If we want to use target $\\beta _v^\\star $ and $\\beta _w^\\star $ values for a new registration problem, we can perform a parameter continuation which is exactly like the parameter search except that we neither perform the binary search for $\\beta _v$ nor check for the bounds on $J$ .",
"In the first stage of the continuation, we solve the registration problem for successively smaller values of $\\beta _v$ starting from $\\beta _v = 1$ and reducing it by one order of magnitude until we reach $\\beta _v=1\\text{e}k $ where $k = \\lceil \\log _{10}(\\beta ^\\star _v)\\rceil $ .",
"Then we do an additional registration solve at $\\beta _v =\\beta _v^\\star $ .",
"We fix $\\beta _w=\\beta _{w,\\text{max}}$ in the first stage.",
"In the second stage, we fix $\\beta _v = \\beta _v^\\star $ and reduce $\\beta _w$ from $\\beta _{w,\\text{max}}$ to $\\beta _w^\\star $ in steps of one order of magnitude.",
"Whereas the expensive parameter search allows us to identify an optimal set of regularization parameters for unseen data, we use the parameter continuation scheme to speed up convergence.",
"The combination of both is particularly efficient, for example in cohort studies, where we identify optimal regularization parameters for one image pair in the cohort and use the obtained parameters for all the other images."
],
[
"Materials",
"We use publicly available image datasets for carrying out the image registration experiments in this paper (see §).",
"We summarize these datasets in Table REF .",
"We discuss these datasets in detail."
],
[
"MUSE",
"This dataset consists of five real brain $T_1$ -weighted MRIs of different individuals.",
"These images were segmented into 149 functional brain regions (see Figure REF ) in a semi-automated manner including manual corrections by expert radiologists [3].",
"These images are part of a bigger set of template images that were used for the development of the MUSE [24] segmentation algorithm.",
"The original image size is $ 256\\times 256\\times 256$ at a spatial resolution of 1 mm.",
"This dataset is available for download through the neuromorphometrics website [3].",
"[20] is a standardized repository for assessing registration accuracy that contains 16 $T_1$ -weighted MR neuroimaging datasets (na01–na16) of different individuals at an isotropic resolution of 1 mm.",
"The original image size is $256\\times 300\\times 256$ voxels.",
"We resample these images to an isotropic image size of $ 256\\times 256\\times 256 $ .",
"We use the images na01-na10 for our experiments.",
"This dataset is available for download through the GitHub link https://github.com/andreasmang/nirep.",
"We create four sets of synthetic template and reference images to assess image registration accuracy as a function of resolution.",
"We create a set of synthetic reference images $m_1$ by solving () using a given synthetic template image $m_0$ and a synthetic velocity field $v$ .",
"To construct the template image $m_0$ , we use a linear combination of high-frequency spherical harmonics.",
"To be precise, we define the template image $m_0(x)$ as $m_0(x) = \\sum _{i=1}^{10} g_i(x)\\quad \\text{with}\\qquad g_i(x) ={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } \\Vert x - \\hat{x}_i\\Vert _2 \\le | Y_l^m( \\theta + \\hat{\\theta }_i, \\phi + \\hat{\\phi }_i)|, \\\\0, & \\text{otherwise},\\end{array}\\right.",
"}$ and image coordinates $x (x,y,z) \\in (-\\pi ,\\pi ]^3$ .",
"In (REF ), $Y_l^m$ represents spherical harmonics of the form $Y_l^m(\\theta , \\phi ) = \\sqrt{\\frac{2l+1}{4\\pi } \\frac{(l-m)!}{(l+m)!}}",
"e^{im\\theta }P_l^m (\\cos (\\phi ))$ with parameters $m$ , $l$ , angular directions $\\theta \\in [0,\\pi ]$ and $\\phi \\in [0,2\\pi ]$ , and associated Legendre functions $P^m_l$ .",
"We choose $m=6$ , $l=8$ for our setup.",
"$\\hat{\\theta }_i$ and $\\hat{\\phi }_i$ are random perturbations in integer multiples of $\\pi /2$ and $\\hat{x}_i \\in [-0.4\\pi , 0.4\\pi ]^3$ is a random offset from the origin.",
"The reference image $m_1(x)$ is generated by solving () with initial condition $m_0(x)$ and velocity field $v(x) \\mathrel {\\mathop :}=(v_x(x), v_y(x), v_z(x))$ , $x = (x,y,z)$ , defined as $v_x = \\sum _{k=1}^{K} \\frac{1}{k^{0.5}}\\cos (ky)\\cos (kx), \\quad v_y = \\sum _{k=1}^{K} \\frac{1}{k^{0.5}}\\sin (kz)\\sin (ky), \\quad v_z = \\sum _{k=1}^{K} \\frac{1}{k^{0.5}}\\cos (kx)\\cos (kz),$ where $K = \\lbrace 4,8,12,16\\rbrace $ .",
"We set the template and the reference base image size to $n_x = n = (1024,1024,1024)$ .",
"It is important to note that $m_0$ and $m_1$ possess only the discrete intensities $i \\in \\lbrace 1,2,\\ldots ,10\\rbrace $ .",
"This allow us to naturally define ten labels $l_0^i$ and $ l_1^i $ corresponding to $m_0$ and $m_1$ , respectively, for all image voxels with intensity $i$ for each $i \\in \\lbrace 1,2,\\ldots ,10\\rbrace $ .",
"We show a 2D slice of the template $m_0$ and reference $m_1$ images for the case $K=4$ in Figure REF .",
"The scripts for generating the template image $m_0$ and the synthetic velocity field $v$ can be found at https://github.com/naveenaero/scala-claire.",
"The reference image $m_1$ can be generated using CLAIRE [66], [15].",
"[61] is an in-vivo 250 $\\mu $ m human brain MRI image which consists of a $T_1$ -weighted anatomical data acquired at an isotropic spatial resolution of 250 $\\mu m$ .",
"The original image size is $640\\times 880\\times 880$ voxels.",
"This image can be downloaded from [1].",
"We skull strip the dataset by downsampling it to $128\\times 128\\times 128$ using linear interpolation and then manually create the brain mask in ITK-SNAP [100].",
"We upsample this brain mask back to the original resolution and then apply it to the original image.",
"We use the tool fast [103] from the FSL toolkit [97], [85], [47] to segment the $T_1$ -weighted MRI into gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF) to be able to evaluate the registration performance using Dice score (see (REF )) between the image labels before and after registration.",
"We use the dataset from [21], [95], [55], [57] which consists of 12 mouse brains images acquired using CLARITY-Optimized Light-sheet Microscopy (COLM).",
"This dataset is available for download from [19].",
"These images have low contrast and are noisy.",
"The in-plane resolution is $0.585\\mu \\text{m} \\times 0.585\\mu \\text{m}$ and the cross-plane resolution is 5 to 8 $\\mu $ m. The images are stored at eight different resolution levels with level zero being the full resolution and level seven being the lowest resolution.",
"We use the images at resolution levels three and six in our experiments.",
"These levels correspond to an in-plane resolution of $4.68 \\mu \\text{m} \\times 4.68\\mu \\text{m}$ and $37.44\\mu \\text{m}\\times 37.44\\mu \\text{m}$ , respectively, which translates to images of size $n=(2816,3016)$ and $n/8=(328,412)$ voxels.",
"The cross-plane resolution is constant at all levels and corresponds to 1162 voxels.",
"We select Control182, Fear197 and Cocaine178 as the test images in our experiments.",
"Table: We list the image datasets we use in our scalable registration experiments (see §).",
"All the datasets are accessible publicly and further discussed in §.",
"We list the dataset name tag (which we use to refer to them throughout the rest of the paper), the imaging modality, the number of images, the spatial resolution and the image resolution in voxels.",
"For dataset with an isotropic spatial resolution, we only provide a single value.",
"For datasets with anisotropic spatial resolution, we list the resolution in all three dimensions.",
"For the SYN dataset, spatial resolution does not carry a physical meaning, so we only list the image resolution."
],
[
"Results and Discussion",
"We test the image registration on real-world (see §REF and §REF ) and synthetic registration problems (see §REF ).",
"The measures to analyze the registration performance are summarized in §REF .",
"We evaluate the parameter search scheme (see §REF ) on a set of real brain images and present the results in §REF .",
"Furthermore, we explore the following questions in the context of scalable image registration: Question Q1: Do we need large scale high resolution image registration?",
"Does the registration quality degrade when the registration is performed at a downsampled resolution when compared to performing registration at the original high resolution?",
"Question Q2: How does registration perform and scale for real, noisy and high resolution medical images of human and mouse brains?"
],
[
"Measures of Performance",
"In our experiments, we evaluate both runtime performance (in terms of solver wall clock time) and the registration quality in terms of accuracy.",
"For the latter, we use the following metrics:"
],
[
"Dice Score Coefficient $D$ .",
"Let $l_0$ and $l_1$ be the binary label maps associated with the images $m_0$ and $m_1$ , respectively.",
"Then the Dice score $D$ between the two label maps is given by $D(l_0, l_1) = \\frac{2 |l_0 \\cap l_1|}{|l_0| + |l_1|},$ where $|\\cdot |$ denotes the cardinality of a set, and $\\cap $ denotes the intersection of the two sets, respectively.",
"We define $D(l_0,l_1)$ to be the Dice score pre-registration and $D(l(t=1),l_1)$ post-registration, where $l(t=1)$ is the label map that corresponds to the deformed template image $m(t=1)$ .",
"Furthermore, for a set of discrete labels $l^i$ , $i=\\lbrace 1,2,\\ldots ,M\\rbrace $ , where $i$ corresponds to the label index, we define the volume fraction $\\alpha ^i = \\frac{|l^i|}{\\sum _{i=1}^{M} |l^i|}.$ Using this definition, we compute the following statistics for the Dice coefficient: The Dice coefficient average $D_a$ given by $D_a = \\frac{1}{M}\\sum _{i=1}^{M} D(l_0^i, l_1^i) ,$ the volume weighted average of the Dice coefficient given by $D_{vw} = \\frac{1}{\\sum _{i=1}^{M}|l_1^i|} \\sum _{i=1}^{M} |l_1^i| D(l_0^i, l_1^i),$ and the inverse of the volume weighted average Dice coefficient given by $D_{ivw} = \\frac{1}{\\sum _{i=1}^{M} 1/|l_1^i|} \\sum _{i=1}^{M} \\frac{D(l_0^i, l_1^i)}{|l_1^i|}.$ Note that $D_{vw}$ gives more weight to labels with higher volume fractions while $D_{ivw}$ gives more weight to labels with smaller volume fractions.",
"This metric corresponds to the ratio of the image mismatch before and after the registration.",
"It is given by $r = \\frac{||m(t=1) - m_1||^2_2}{||m_0 - m_1||^2_2}.$ For each image registration, we also report the regularization parameters and the obtained minimum and maximum values of the determinant of the deformation gradient $J \\mathrel {\\mathop :}=\\det \\mathbf {F}$ , i.e., the determinant of the Jacobian of the deformation map.",
"We visually support this quantitative analysis with snapshots of the registration results.",
"The registration accuracy can be visually judged from the residual image, which corresponds to the absolute value of the pointwise difference between $m(t=1)$ and $m_1$ .",
"The regularity of the deformations can be assessed from the pointwise maps of the determinant of the deformation gradient."
],
[
"Experiment 1: Evaluation of the Parameter Search Scheme",
"We evaluate the parameter search scheme on a set of real brain images and compare the registration performance with a state-of-the-art SyN deformable registration tool in the ANTs toolkit."
],
[
"Dataset",
"We use the MUSE dataset (see §) for this experiment.",
"After registration of the original $T_1$ -weighted images from this dataset, we use the image labels to evaluate the registration performance in terms of the volume weighted average Dice score $D_{vw}$ .",
"Out of the five $T_1$ images, we select Template27 as the reference image $m_1$ and register the other four images to $m_1$ .",
"For the registration, we use the parameter search scheme (see §REF ) to identify best regularization.",
"We use linear interpolation and $n_t=8$ time steps in the semi-Lagrangian solver.",
"For the Jacobian bound, we select $J_{min}=0.1$ .",
"In the parameter search, for each trial $\\beta _v$ and $\\beta _w$ , we drive the relative gradient norm $\\Vert g\\Vert _{2,rel}=\\Vert g\\Vert _2/\\Vert g_0\\Vert _2$ to $1\\text{e-}02$ .",
"Once we have found adequate $\\beta _v$ and $\\beta _w$ for each image pair, we rerun the image registrations using only parameter continuation.",
"For a baseline performance comparison, we also perform registration on the same image pairs using the SyN tool in ANTs [5].",
"For ANTs, we use the “MeanSquares” (i.e., squared $L_2$ -) distance measure.",
"We run CLAIRE on a single NVIDIA V100 GPU with 16GB of memory on TACC's Longhorn supercomputer.",
"We run ANTs on a single node of the TACC Frontera supercomputer (system specs: Intel Xeon Platinum 8280 (“Cascade Lake”) processor with 56 cores on 2 sockets (base clock rate: 2.7GHz)).",
"We use all 56 cores.",
"We report the parameters used for ANTs in §.",
"Table: Experiment 1: Performance of the parameter search scheme implemented in CLAIRE.",
"We report results for the registration of four template images to the reference image Template27.",
"We consider the squared L 2 L_2-distance measure as image similarity metric.",
"We restrict the Jacobian determinant J∈[0.1,10]J\\in [0.1,10] for these registrations.",
"We report the following quantities of interest: (i) optimal regularization parameters β v ☆ \\beta _v^\\star and β w ☆ \\beta _w^\\star , (ii) minimum J min J_{\\text{min}} and maximum J max J_{\\text{max}} Jacobian determinant achieved, (iii) solver wall clock time in seconds, and (iv) label volume weighted Dice average D vw D_{vw} pre and post registration.Table: Experiment 1: Performance of ANTs.",
"We report results for registration of four template images to the reference image Template27 using a squared L 2 L_2-distance metric.",
"We report the following quantities of interest (i) minimum (J min J_{\\text{min}}) and maximum (J max J_{\\text{max}}) determinant of the deformation gradient obtained, (ii) label volume weighted Dice average D vw D_{vw} pre and post registration, and (iii) solver wall clock time in seconds.Figure: Experiment 1: Comparison of Dice scores for CLAIRE and ANTs.",
"The box plots show Dice scores of the individual labels for the registration results reported for CLAIRE in Table and ANTs in Table .Figure: Experiment 1: Exemplary registration results using the parameter search scheme implemented in CLAIRE.",
"We consider the datasets Template16 (template image) and Template27 (reference image).",
"We refer to Table and the text for details about the setup.",
"We show (from left to right) the template, reference, deformed template image (top row) and their corresponding labels (bottom row).",
"We also visualize the residual before and after the registration along with the determinant of the deformation gradient and an orientation map for the velocity field.We report the obtained estimates for $\\beta _v$ and $\\beta _w$ as well as results for registration quality in Table REF .",
"In Figure REF , we provide a representative illustration of the obtained registration results.",
"We report baseline registration performance using ANTs in Table REF .",
"We compare the Dice scores obtained for CLAIRE and ANTs in Figure REF .",
"CLAIRE allows us to precisely control the properties of the deformation without having to tune any parameters manually.",
"The only free parameters are the Jacobian bounds, which depend on the overall workflow related to the dataset.",
"The volume weighted Dice scores $D_{vw}$ obtained for CLAIRE (see Table REF ) are competitive to those produced by ANTs (see Table REF ).",
"The average runtime for ANTs for all the registrations reported in Table REF is 201 seconds ($\\approx 3$ minutes).",
"For CLAIRE, the average wall clock time of CLAIRE in the parameter search mode is 9.8 minutes ($3\\times $ slower than ANTs; we search for adequate regularization parameters), while, in the continuation mode, the runtime of CLAIRE is 64 seconds ($3\\times $ faster than ANTs; we apply the optimal regularization parameter and do not search for them)."
],
[
"Experiment 2A: High Resolution Synthetic Data Registration",
"In this experiment, we answer Q1.",
"We attempt this by executing our registration algorithm on synthetic imaging data.",
"The advantages of using such images over real datasets are as follows: They are noise-free, high contrast, and sharp, unlike real-world images.",
"There is a scarcity of high resolution real image data because it is expensive and time-consuming to acquire.",
"We can control the resolution of synthetic data because the images are created using analytically known functions.",
"We can control the number of discrete image intensity levels, i.e., labels.",
"Because these labels are available as ground truth, we can use them to precisely quantify registration accuracy through the Dice coefficient, avoiding inter- and intra-observer variabilities and other issues associated with establishing ground truth labels in real imaging data.",
"By performing image registration at different resolutions (and applying the resulting velocity to transform the high resolution original images), we want to check whether the registration at higher resolutions is more accurate than performing the registration at a lower resolution."
],
[
"Dataset",
"We use the SYN dataset (see §) for this experiment.",
"We execute registration at different resolutions for the original resolution images and quantify the accuracy using the Dice coefficient for labels before and after the registration.",
"We compare the Dice statistics for different resolutions.",
"More specifically, we take the following steps: We register the template image $m_0$ to the reference image $m_1$ at the base resolution $n$ to get the velocity field $v_{n}$ .",
"We transport $m_0$ using the velocity $v_{n}$ to get the deformed template image $m(t=1)$ by solving ().",
"Then, we compute the Dice score between $l^i(t=1) $ and $ l_1^i $ , $i \\in {1,\\ldots ,10}$ which are discrete labels for $m(t=1)$ and $m_1$ , respectively, using (REF ).",
"We downsample $m_0$ and $m_1$ using nearest neighbor interpolation to half the base resolution (for example, $n/2 = (512,512,512)$We treat $n_x = (N_1,N_2,N_3) $ as a tuple.",
"When we say $ n_x/2 $ , we mean $ n_x/2 = (N_1/2 , N_2/2 , N_3/2)$ .",
"and register the downsampled images to get the velocity $\\hat{v}_{n/2}$ .",
"We upsample $\\hat{v}_{n/2}$ to the base resolution $n$ using spectral prolongation and call it $v_{n/2}$ .",
"We transport $m_0$ using $v_{n/2}$ by solving () to get the deformed template image $m(t=1)$ and then compute the Dice score for this new deformed template image.",
"We repeat the procedure in step 2 for resolutions $n/4$ and $n/8$ and compute the corresponding Dice scores.",
"For the registration, we fix the determinant $J$ of the deformation gradient to be within $[5\\text{e-}02, 20]$ and search for the regularization parameters using the proposed parameter search scheme as described above in §.",
"Note that we perform a search for an optimal regularization parameter for each individual dataset because we want to obtain the best result for each pair of images.",
"In practical applications, this is not necessary (see comments below; we also refer to [69] for a discussion).",
"We fix the tolerance for the reduction of the gradient to $5\\text{e-}02$ , which we have found to be sufficiently accurate for most image registration problems (see [69]).",
"We use linear interpolation in the semi-Lagrangian scheme.",
"Another hyperparameter in our registration solver is the number of time steps $n_t$ for the semi-Lagrangian (SL) scheme.",
"We consider two cases for selecting $n_t$ : $n_t$ changes with resolution: We use $n_t=4$ time steps for the coarsest resolution $n_x=n/8$ and double $n_t$ when we double the resolution in order to keep the CFL number fixed.",
"All other solver parameters, except for the regularization parameters, are the same at each resolution.",
"$n_t$ fixed with resolution: In order to study the effect of $n_t$ on the Dice score we keep $n_t$ fixed for each $n_x$ , instead of increasing $n_t$ proportionately to $n_x$ .",
"Table: Experiment 2A: Registration performance for CLAIRE for case 1 (n t n_t changes proportionally to the image resolution, see §).",
"Comparison of registration accuracy based on the Dice score at different resolutions for the synthetic dataset SYN.",
"KK denotes the frequency of the synthetic velocity field in ().",
"n=(1024,1024,1024)n=(1024,1024,1024) is the base image resolution.",
"We fix the tolerance for the reduction of the gradient to 5e-025\\text{e-}02 and use linear interpolation.",
"The Jacobian bounds for the parameter search are [0.05,20][0.05,20].",
"We report β v ☆ \\beta ^\\star _v and β w ☆ \\beta ^\\star _w (the optimal regularization parameters obtained with the proposed parameter search scheme), and J min J_\\text{min} and J max J_\\text{max} (the minimum and maximum values for the determinant of the deformation gradient).",
"For the Dice score, we report average Dice (D a D_a), the volume weighted average Dice (D vw D_{vw}), and the inverse volume weighted average Dice (D ivw D_{ivw}), pre and post registration.",
"We also report the wall clock time for the parameter search.Table: Experiment 2A: Registration performance for CLAIRE for case 2 (n t n_t independent of the image resolution).",
"Comparison of registration accuracy using Dice at different resolutions for the synthetic dataset SYN.",
"KK denotes the frequency of the synthetic velocity field in ().",
"n=(1024,1024,1024)n=(1024,1024,1024) is the base image resolution.",
"We fix the tolerance for the reduction of the gradient to 5e-025\\text{e-}02 and use linear interpolation.",
"The Jacobian bounds for parameter search is [0.05,20][0.05,20].",
"For each value of n t n_t, we report results for different resolutions.",
"We report β v ☆ \\beta ^\\star _v and β w ☆ \\beta ^\\star _w (the optimal regularization parameters obtained with the proposed parameter search scheme), and J min J_\\text{min} and J max J_\\text{max} (the minimum and maximum values for the determinant of the deformation gradient).",
"For the Dice score, we report average Dice (D a D_a), the volume weighted average Dice (D vw D_{vw}), and the inverse volume weighted average Dice (D ivw D_{ivw}), pre and post the registration.",
"We also report the wall clock time for the parameter search.",
"The missing cases for K=8K=8 failed to finish in a reasonable time frame.",
"We only report a couple of cases for K=16K=16 and expect a behavior similar to K=8K=8 for the rest.Figure: Experiment 2A: Visualization of registration results for case 1.",
"In column 1, from top to bottom, we visualize the template, reference and deformed template images for registrations done at different resolutions.",
"These images correspond to the runs #1-4 in Table .",
"The value in the parentheses in column 1 indicates the resolution at which registration was done.",
"The visualization is done at the original resolution n=(1024,1024,1024)n = (1024,1024,1024).",
"In column 2 and 3, we visualize cropped portions of the images shown in column 1 for specific label values.",
"In column 2, we show label 1, in column 3, we show the union of labels with intensity value ≥5\\ge 5.",
"Note that higher label values have smaller volumes and more fine-grained features.",
"We plot the label boundaries for the reference image in green to visualize the registration errors.Figure: Experiment 2A: Quantitative results for the registration results corresponding to case 1.",
"We show a plot of the Dice scores against the label volume fraction α\\alpha for each label l i l^i, i=1,...,10i=1,\\ldots ,10 for the registration of the synthetic data set SYN at different resolutions.",
"This figure corresponds to the registration runs #1-4 in Table for K=4K=4.Figure: Experiment 2A: Quantitative results for the registration results corresponding to case 1.",
"We show box plots of the Dice scores for the individual labels before and after registration for different resolutions.",
"We consider the synthetic test problem SYN.",
"This figure corresponds to the registration results reported in Table .In Figure REF , we visualize the template, reference and deformed template images for the synthetic problem constructed with $K=4$ .",
"We report quantitative results for CLAIRE in Table REF and Table REF , respectively.",
"In Figure REF , we compare the Dice score for individual labels as a function of their volume fraction $\\alpha $ .",
"In Figure REF , we visualize box plots of the Dice score for the registrations reported in Table REF .",
"The most important observations are: (i) The Dice score averages are better for registrations performed at the base resolution $n$ with progressively worse Dice scores for registrations done at coarser resolutions.",
"(ii) The difference between Dice scores for registrations done at successively coarser resolutions for $K=16$ (rougher velocity field) is higher than at $K=4$ (smoother velocity field).",
"(iii) Keeping $n_t$ fixed for the base and coarser resolutions does not affect the Dice score trend, i.e., the Dice decreases as $n_x$ is decreased.",
"In the following, we give more details for these general observations.",
"Regarding Dice score averages in Table REF , we observe that $D_a$ , the arithmetic mean of the Dice scores of individual labels, drops by as much as 7% between run #13 and #14.",
"However, the percentage drop in volume weighted Dice average $D_{vw}$ is smaller than in $D_a$ .",
"This indicates that labels with higher volume are still easier to register at coarser resolutions.",
"The inverse weighted Dice average $D_{ivw}$ , which gives more weight to smaller labels, features a more pronounced decrease because smaller regions contribute to the high frequency content in the image; this information is lost when the images are downsampled.",
"We observe a 21.8% difference in $D_{ivw}$ for the high frequency images in run #13 and #14 for $K=16$ .",
"As we increase the frequency $K$ of the synthetic ground truth velocity, we see that the difference in all Dice score averages between successive resolutions increases.",
"As $K$ increases, we get increasingly rougher velocity fields, which we can not recover by registering the original images at coarser resolutions.",
"In Table REF , the Dice scores behave the same way even when $n_t$ is fixed for different $n_x$ , indicating that the loss in accuracy is primarily because of the reduction in the spatial resolution (and not the temporal resolution).",
"We also observe that for the full resolution of $n_x=n$ , using $n_t < 32$ results in slow solver convergence; the run did not finish in under 2 hrs.",
"We attribute this slow convergence rate to the loss in numerical accuracy in the computation of the reduced gradient in (REF ).",
"If we compare run #1 and #9 in Table REF , we see that the difference in $D_a$ is marginal in comparison to the run time cost overhead for run #9.",
"However, the accuracy difference increases as $K$ is increased, and the images get less smooth (see runs #13 and #14).",
"These quantitative observations are confirmed by the visual analysis in the figures shown: From Figure REF , we observe that at lower resolutions (top to bottom), the alignment of the outlines (green lines; reference image) with the structures (white areas; deformed template image) is less accurate.",
"Figure REF : shows that the Dice score is worse for labels with smaller volume fractions, i.e., fine structures are matched less accurately at coarse resolutions.",
"Looking at Figure REF , we observe that the average registration accuracy decreases as we decrease the resolution.",
"We use 32 GPUs for registration at $n_x = (1024,1024,1024)$ 4 GPUs for $n_x = (512,512,512)$ and a single GPU for $n_x = (256,256,256)$ and $n_x = (128,128,128)$ .",
"Registration for $n_x = (1024,1024,1024)$ takes on average 44 minutes wall clock time.",
"It is important to note that this includes the time spent in the search for optimal regularization parameters (i.e., we solve the inverse problem multiple times using warm starts; see §REF for details regarding the scheme).",
"For the large-scale runs that use multiple GPUs, the overall runtime of the solver is dominated by communication between MPI processes [16].",
"Adding more resources does not necessarily reduce the runtime because of this increase in communication cost.",
"Registrations for $n_x = (512,512,1512)$ and lower resolutions are much quicker and run in the order of 10 min or less.",
"In the present work, we perform the parameter search for each individual case because we want to obtain the best result for each pair of images.",
"However, in practice where a medical imaging pipeline requires registrations for several similar images, we suggest running the parameter search scheme on one pair of images and use the obtained regularization parameters to run the cohort registration for all images, as we have done in our previous work [69].",
"This strategy reduces the computational cost drastically.",
"One downside to this strategy is that some images in the cohort will not be registered as accurately as others.",
"Our experiment with synthetic images suggests that Dice scores are better when registrations are done in the original, high resolution at which the labels were created.",
"Registration accuracy is affected more significantly if high frequency velocity fields are considered.",
"The images used in this experiment are synthetic and free of noise.",
"We use these images for both registration and evaluation of performance using Dice scores.",
"Because the ground truth labels for these images at the highest resolution are known with certainty, we have high confidence in our observations regarding registration accuracy: the Dice scores become worse when registration is conducted at lower resolutions.",
"However, in practical applications, images have noise and low contrast.",
"To evaluate the registration accuracy for real images using Dice scores, we first evaluate their segmentation using external segmentation tools.",
"This segmentation step is prone to errors (not only due to noise and a lack of contrast but also due to inherent limitations in segmentation software themselves).",
"These errors result in a misalignment between the structures present in the original image and its segmentation, which complicates our analysis.",
"Having said this, we conduct experiments on real brain MRIs in the next section to explore if we can provide experimental evidence that at least partially confirms the observations we have made in this section."
],
[
"Experiment 2B: High Resolution Real Data Registration",
"In this experiment, we aim at answering Q1 as well as Q2.",
"We do this by registering real human brain MRI datasets instead of synthetic images.",
"Unlike synthetic images, these images are not noise-free.",
"Moreover, they lack high contrast."
],
[
"Datasets",
"We use the NIREP and the MRI250 image datasets (see §) for this experiment.",
"We designate the MRI250 image as the template image $m_0$ .",
"We generate the reference images $m_1$ from the images na01–na10 from the NIREP dataset since we do not have access to other $T_1$ -weighted MRI from a different subject at the original resolution of 250 $\\mu $ m. The acquired spatial resolution of the NIREP data is 1 mm, which is $4\\times $ larger than 250 $\\mu $ m. Therefore, in order to generate a reference image $m_1$ that are 250 $\\mu $ m in spatial resolution, we take the following steps: Upsample the respective NIREP image from $256\\times 300\\times 256$ to $640\\times 880\\times 880$ using linear interpolation.",
"Register MRI250 to the upsampled NIREP image using CLAIRE and transport $m_0$ (which corresponds to the MRI250 image) using the resulting velocity $v$ and solving () to obtain the deformed template image $m_1 = m(t=1)$ .",
"We set the tolerance for the relative gradient norm to $g_\\text{tol} = 1\\text{e-}02$ .",
"We lower the tolerance compared to other runs to obtain a potentially more accurate registration result.",
"We use the default regularization parameters $\\beta _v=1\\text{e-}02$ and $\\beta _w=1\\text{e-}04$ .",
"Consequently, we do not perform a parameter search to estimate an optimal regularization parameter for this registration.",
"We want to keep the downstream registration performance analysis, where we will use parameter search, oblivious to the process of generating the high resolution reference image.",
"To generate a segmentation that we can use to compute Dice scores (not for the registration itself, which is done on the original unsegmented images), we use the tool fast from FSL [48] both on the template image $m_0$ and on the reference image $m_1$ .",
"We generate labels WM, GM, and CSF.",
"The remaining steps for this experiment are the same as described in experiment 2A in §REF except that here we are registering real $T_1$ -weighted images instead of noise-free synthetic images.",
"The base resolution for this experiment is $n_x = n = (640,880,880)$ .",
"We consider $n_x=n/2$ and $n_x=n/4$ for the downsampled resolutions.",
"We also consider the two sub-cases for selecting $n_t$ as we did in §REF .",
"For the case where $n_t$ changes with resolution, we use $n_t=4$ for $n_x=n/4$ , $n_t=8$ for $n_x=n/2$ and $n_t=16$ for $n_x=n$ .",
"Figure: Experiment 2B: Illustration of registration results for the multi-resolution registration experiment on real brain images.",
"The images shown here correspond to the runs #1, #2, and #3 in Table ).",
"The base resolution is n x =n=(640,880,880)n_x=n=(640,880,880).",
"In row 1, from left to right, we show the T1-weighted MRI250 datasets (template image m 0 m_0), the upsampled na01 dataset (reference image m 1 m_1) from the NIREP data repository, and the deformed template images obtained from registration at resolutions n x n_x, n x /2n_x/2 and n x /4n_x/4, respectively.",
"In row 2, we show a cropped portion of the images from row 1.",
"In rows 3 and 4, we show the label maps consisting of white matter (WM; white), gray matter (GM; light gray) and cerebro-spinal fluid (CSF; dark gray) and their cropped versions, respectively.",
"In row 5, we show the image residuals before and after registration with respect to each resolution level.Table: Experiment 2B: Registration performance for CLAIRE for case 1 (n t n_t changes proportional to the image resolution).",
"Comparison of registration accuracy using Dice and relative residual rr at different resolutions for the registration of the MRI250 brain image to templates generated from ten real MRI scans from the NIREP dataset.",
"We consider three resolution levels n x ={n,n/2,n/4}n_x=\\lbrace n, n/2, n/4\\rbrace where n=(640,880,880)n = (640,880,880).",
"We fix the tolerance for the relative gradient to 5e-025\\text{e-}02.",
"We use linear interpolation in the semi Lagrangian scheme.",
"The bounds for the determinant of the deformation gradient for the parameter search are [0.05,20][0.05,20].",
"We report the regularization parameters β v ☆ \\beta _v^\\star and β w ☆ \\beta _w^\\star obtained through the proposed parameter search scheme, the minimum and maximum determinant of the deformation gradient (J min J_\\text{min} and J max J_\\text{max}), the relative residual (rr), the average Dice (D a D_a), pre and post the registration, as well as the wall clock time for the parameter search.Table: Experiment 2B: Registration performance for CLAIRE for case 2 (n t n_t independent of the image resolution).",
"Comparison of registration accuracy using Dice and relative residual rr for a fixed number of time steps n t n_t at different resolutions for the registration of the real MRI datasets MRI250 and the reference image m 1 m_1 generated from na01 from the NIREP repository.",
"We consider three resolution levels n x ={n,n/2,n/4}n_x=\\lbrace n,n/2,n/4\\rbrace where n=(640,880,880)n=(640,880,880).",
"We fix the tolerance for the relative gradient to 5e-025\\text{e-}02.",
"We use linear interpolation in the semi Lagrangian schreme.",
"The bounds for the determinant of the deformation gradient for the parameter search are [0.05,20][0.05,20].",
"We keep the time step n t n_t fixed.",
"We report the regularization parameters β v ☆ \\beta _v^\\star and β w ☆ \\beta _w^\\star obtained through the proposed parameter search scheme, the minimum and maximum determinant of the deformation gradient (J min J_\\text{min} and J max J_\\text{max}), the relative residual (rr), the average Dice (D a D_a), pre and post the registration, as well as the wall clock time for the parameter search.",
"The case with n x =nn_x=n and n t =4n_t=4 failed to finish in under 4 hrs.We report the solver parameters for our registration with CLAIRE along with the relative residual $r$ and Dice score averages for GM, WM and CSF before and after the registration in Table REF .",
"The relative residual $r$ and the Dice score are always computed at the base resolution $n=(640,880,880)$ .",
"The respective results with $n_t$ fixed independent of the resolution are given in Table REF for na01.",
"We visualize the image registration results for the reference image na01 in Figure REF .",
"The most important observation is that the relative residual $r$ increases and Dice score averages decrease for registrations done at coarser resolutions irrespective of whether we increase $n_t$ proportionally to the resolution, see Table REF or keep $n_t$ fixed for different $n_x$ , see Table REF .",
"This observation is in line with the experiment for the synthetic dataset SYN in §REF .",
"Except for the case of na04 (see runs #10 and #11 in Table REF ), all other cases exhibit increasingly worse registration performance at coarser resolutions.",
"In REF , we used synthetic, noise-free, high-contrast images for assessing the registration accuracy at different resolutions.",
"Here, we repeat the same experiment with real world images—T1-weighted MR images of the human brain.",
"We used an external software to segment these images to provide the necessary labels to be able to quantify registration performance in terms of Dice score.",
"Notice that this additional segmentation step will inevitably introduce additional errors to our analysis.",
"Due to these additional errors at the native resolution, we expect that the improvement in registration performance at high resolution may not be as pronounced as for the synthetic images considered in REF (which did not require this additional segmentation step).",
"This hypothesis is confirmed if we compare the average Dice score $D_a$ across experiments.",
"In particular, if we reduce the resolution from $n$ to $n/4$ in REF (see Table REF ) and REF (see Table REF ), the Dice score drops by 15.25% compared to 9.5%, respectively.",
"In Table REF , the case with $n_t=4$ and $n_x=n$ took very long to converge (>4 hrs).",
"For this case the $CFL$ number is $15.66$ during the inverse solve while for $n_t=16$ , the $CFL$ number is 4.",
"The larger $CFL$ number for $n_t=4$ yields a higher adjoint error in the SL scheme.",
"This leads to higher errors in the computation of the reduced gradient, which results in worse convergence of the inverse solver for $n_t=4$ .",
"The run time overhead associated with using $n_t=16$ against $n_t=4$ is easily compensated by better solver convergence.",
"We refer to [65] for a thorough study on the effect of $n_t$ on the numerical accuracy of the reduced gradient."
],
[
"Experiment 3: Registration of Mouse Brain CLARITY Images",
"This experiment aims to answer both Q1 and Q2 by examining the performance of our scalable registration solver on ultra-high resolution mouse brain images acquired using the CLARITY imaging technique [89], [19].",
"As opposed to the previous datasets, the dataset in this experiment does not provide any real metrics for its assessment other than the relative residual (nor are we aware of any segmentation software that would work on these data)."
],
[
"Dataset",
"We use the CLARITY dataset (see §) for this experiment.",
"Preprocessing: For all unprocessed images, the background intensity is non-zero.",
"We normalize the image intensities such that they lie in the range [0,1] with the background intensity re-scaled to zero.",
"Next, we affinely register all images to Control182 at $8\\times $ downsampled resolution using the SyN tool in ANTs.",
"We report the parameter settings for the affine registration in the appendix.",
"Subsequently, we zero-pad the images to ensure that periodic boundary conditions are satisfied for CLAIRE.",
"After preprocessing, the base image resolution is $n_x=n$ where $n=(2816,3016,1162)$ and $n/8=(328,412,1162$ ), respectively.",
"We only conduct the parameter search for a single pair of images (at both resolutions independently) for these sets of images and then perform the parameter continuation on the entire dataset.",
"We only report wall clock times for the parameter continuation and not for the parameter search.",
"Deformable Registration: We register all images to the reference image Control182 using CLAIRE.",
"We use the proposed parameter continuation scheme.",
"We set $J_\\text{min}$ to $0.05$ .",
"We do this for both resolution levels.",
"To compare the registration accuracy between each resolution level, we follow the same steps from §REF .",
"We compare the registration performance using the relative residual $r$ .",
"We do not have access to image segmentation for this dataset and, therefore, we cannot evaluate accuracy using Dice scores.",
"Figure: Experiment 3: Illustration of the registration performance for CLAIRE for the CLARITY mouse brain imaging data.",
"We report registration results for the Cocaine178 dataset registered to the Control182 dataset.",
"In row 1 (from left to right), we have the template image m 0 m_0 (Cocaine178), the reference image m 1 m_1 (Control182) and the deformed template image.",
"The resolution of the images is n=(2816,3015,1162)n=(2816,3015,1162).",
"In row 2, we show the determinant of the deformation gradient and the image residuals before and after registration.Table: Experiment 3: Registration performance for CLAIRE for the CLARITY imaging data at resolutions n=(2816,3016,1162)n=(2816,3016,1162) and n/8=(328,412,1162)n/8=(328,412,1162).",
"Control182 is the fixed (reference) image.",
"All other images selected from the CLARITY dataset are registered to Control182 using a parameter continuation scheme.",
"We fix the tolerance for the relative gradient to 5e-025\\text{e-}02.",
"We use linear interpolation for the semi Langrangian scheme.",
"The bounds on the determinant JJ of the deformation gradient for the parameter search are [0.05,20][0.05,20].",
"We report the estimated regularization parameters β v ☆ \\beta _v^\\star and β w ☆ \\beta _w^\\star , the minimum and maximum values for the determinant of the deformation gradient (J min J_\\text{min} and J max J_\\text{max}), the relative residual (rr), as well as the wall clock time for the parameter continuation.We report the quantitative results for the registration of the CLARITY data in Table REF .",
"We showcase exemplary registration results in Figure REF .",
"The most important observation is that we can register high resolution real medical images reasonably well in under 2 hrs (see run #1 and #3 in Table REF ).",
"Unlike the previous experiments in §REF and §REF , the reported wall clock time in Table REF is for performing the parameter continuation and not the parameter search.",
"The average time spent for the regularization parameter search for resolution $n_x=n$ is $\\sim $ 2 hrs.",
"Another observation, which is in agreement with the results reported for the experiments carried out in §REF and §REF , is that the registration performed at downsampled resolution (see Table REF ) results in a larger relative residual and, therefore, worse registration accuracy.",
"We had a maximum of 256 GPUs (64 nodes, 4 GPUs per node) available to us at the TACC Longhorn supercomputer.",
"Because of this resource constraint, our solver ran out of memory for certain parameter configurations (for example, for run #1 and #3, we could not use $n_t > 16$ time steps).",
"Moreover, for all the runs in Table REF we used the zero velocity approximation of $H $ as the preconditioner and applied it at full resolution.",
"We did not use the two-level coarse grid approximation to apply the preconditioner because it requires additional memory for the coarse grid spectral operations."
],
[
"Conclusions",
"In this publication, we apply our previously developed multi-node, multi-GPU 3D image registration solver [14] to study and analyze large-scale image registration.",
"This work builds upon our former contributions on constrained large deformation diffeomorphic image registration [63], [69], [67], [65], [66].",
"The main observations are: (i) We are able to register CLARITY mouse brain images of unprecedented ultra-high spatial resolution ($2816\\times 3016\\times 1162$ ) in 23 minutes using parameter continuation.",
"To the best of our knowledge, images of this scale have not been registered in previous work [14], [55], [57].",
"(ii) We conduct detailed experiments to compare image registration performance at full and downsampled resolutions using synthetic and real images.",
"We find that image registration at higher (native) image resolution is more accurate.",
"To quantify the accuracy, we use Dice coefficients wherever image segmentation is available and relative residuals otherwise.",
"We also do a sensitivity analysis for the overall solver accuracy with respect to the number of time steps $n_t$ in the SL scheme.",
"Overall, CLAIRE performed as expected: fully automatic parameter tuning works well, and higher image resolutions result in improved image similarity compared to the registration results in lower resolution.",
"We note that these improvements in registration accuracy are less pronounced for real imaging data compared to synthetic data for the experiments conducted in this study.",
"We attribute these observations to uncertainties and errors introduced during the registration and segmentation steps due to noise and low contrast.",
"We discuss this in more detail in §."
],
[
"Acknowledgements and Funding.",
"This work was partly supported by the National Science Foundation (DMS-1854853, DMS-2012825, CCF-1817048, CCF-1725743), by the NVIDIA Corporation (NVIDIA GPU Grant Program), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC 2075-390740016, by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC0019393, by the U.S. Air Force Office of Scientific Research award FA9550-17-1-0190, by the Portugal Foundation for Science and Technology and the UT Austin-Portugal program, and by NIH award 5R01NS042645-11A1.",
"Any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the DFG, AFOSR, DOE, NIH, and NSF.",
"Computing time on the Texas Advanced Computing Center's (TACC) systems was provided by an allocation from TACC and the NSF.",
"This work was completed in part with resources provided by the Research Computing Data Core at the University of Houston."
],
[
"Deformable registration parameters for ANTs",
"We report the deformable registration parameters for ANTs which were used for comparison with CLAIRE in the parameter search experiment §REF .",
"#!/bin/bash \tantsRegistration --dimensionality 3 \t--float 1 \t--output [$output_directory/,$output_directory/deformed-template.nii.gz] \t--interpolation Linear \t--winsorize-image-intensities [0.005,0.995] \t--use-histogram-matching 1 \t--initial-moving-transform [$moving_image,$template_image,1] \t--transform Rigid[0.1] \t--metric MI[$reference_image,$template_image,1,32,Regular,0.25] \t--convergence [1000x500x250x100,1e-6,10] \t--shrink-factors 8x4x2x1 \t--smoothing-sigmas 3x2x1x0vox \t--transform Affine[0.1] \t--metric MI[$reference_image,$template_image,1,32,Regular,0.25] \t--convergence [1000x500x250x100,1e-6,10] \t--shrink-factors 8x4x2x1 \t--smoothing-sigmas 3x2x1x0vox \t--transform SyN[0.1,3,0] \t--metric MeanSquares[$reference_image,$template_image,1] \t--convergence [100x70x50x20,1e-6,10] \t--shrink-factors 8x4x2x1 \t--smoothing-sigmas 3x2x1x0vox"
],
[
"Determining $\\beta _{w,\\text{max}}$",
"We report the runs for comparison of runtime and Dice scores for different values of $\\beta _{w,\\text{max}}$ for the experiment conducted in §REF .",
"Table: Experiment 1b: effect of β w,max \\beta _{w,\\text{max}} on registration performance for real brain images: Comparison of registration accuracy using Dice and relative residual rr for different values of β w,max \\beta _{w,\\text{max}} at different resolutions for registration of MRI250 brain image to na01 and na02 from NIREP dataset.",
"We fix β w,min =1e-09\\beta _{w,\\text{min}}=1\\text{e-}09.",
"We consider n x ={n,n/4}n_x=\\lbrace n, n/4\\rbrace where n=(640,880,880)n = (640,880,880).",
"We fix the tolerance for the relative gradient to 5e-025\\text{e-}02.",
"We use linear interpolation.",
"The Jacobian bounds for parameter search are [0.05,20][0.05,20].",
"We increase the number of time steps n t n_t proportionately with increase in resolution.",
"We report β v ☆ \\beta ^\\star _v and β w ☆ \\beta ^\\star _w, the regularization parameters from the parameter search scheme, J min J_\\text{min} and J max J_\\text{max}, the minimum and maximum Jacobian determinant the relative residual rr, average Dice D a D_a pre and post the registration and the wall clock time for the parameter search for the registration.",
"We highlight the best Dice scores for each resolution and for each NIREP image."
]
] | 2209.08189 |
[
[
"ASIR: Robust Agent-based Representation Of SIR Model"
],
[
"Abstract Compartmental models (written as $CM$) and agent-based models (written as $AM$) are dominant methods in the field of epidemic simulation.",
"But in the literature there lacks discussion on how to build the \\textbf{quantitative relationship} between them.",
"In this paper, we propose an agent-based $SIR$ model: $ASIR$.",
"$ASIR$ can robustly reproduce the infection curve predicted by a given SIR model (the simplest $CM$.)",
"Notably, one can deduce any parameter of $ASIR$ from parameters of $SIR$ without manual tuning.",
"$ASIR$ offers epidemiologists a method to transform a calibrated $SIR$ model into an agent-based model that inherit $SIR$'s performance without another round of calibration.",
"The design $ASIR$ is inspirational for building a general quantitative relationship between $CM$ and $AM$."
],
[
"Introduction",
"Compartmental models (written as $CM$ ) and agent-based models (written as $AM$ ) are dominant methods in the field of epidemic simulation.. $CM$ capture the population level dynamics by a set of ordinary differential equations.",
"$AM$ capture the individual level dynamics by an agent-based programming environment.",
"$CM$ and $AM$ have complementary nature.",
"$CM$ are easy to calibrate but have less flexible parameter space to apply a priori; $AM$ have a flexible parameter space to apply a priori, but are hard to calibrate.",
"In the current literature, there lacks discussion on developing the quantitative relationship between $CM$ and $AM$ .",
"Driven by this fact, we wish to bridge the gap between $CM$ and $AM$ by finding an $AM$ that robustly reproduces the infection curve predicted by a $CM$ .",
"We start from the $SIR$ model (the simplest $CM$ ): where $P_{sir}$ is the set of all parameters; and functions $S_{sir}(t)$ , $I_{sir}(t)$ , $R_{sir}(t)$ are the population size of being susceptible, infected and recovered with respect to time $t$ .",
"We propose an agent-based SIR model, $ASIR$ , that achieves the following interesting properties: $P_{asir}$ only depends on $P_{sir}$ .",
"Any parameter $p \\in P_{asir}$ can be deduced from $P_{sir}$ , i.e.",
"can be written as a determinate expression of $\\lbrace p_1, p_2, ..., p_k\\rbrace \\subset P_{sir}$ .",
"$ASIR$ robustly reproduces the infection curve predicted by $SIR$ .",
"$ASIR$ is expected to predict the same $S(t), I(t), R(t)$ as $SIR$ , i.e.",
"$\\forall t: \\mathbb {E}\\big [ S_{asir}(t) \\big ] = S_{sir}(t)$ ; $\\mathbb {E}\\big [ I_{asir}(t) \\big ] = I_{sir}(t)$ ; $\\mathbb {E}\\big [ R_{asir}(t)\\big ] = R_{sir}(t) $ We validate $ASIR$ 's properties by giving: 1).",
"a proof of robustness, 2).",
"two implementations in GAMA and Agents.jl."
],
[
"Related Work",
"$SIR$ model is the simplest compartmental model ($CM$ ).",
"In , the authors give us an overview of the design behind the compartmental model.",
"From this work, we learned that the core idea of $CM$ is to use a set of ordinary differential equations to model the population infection and recovery, and use parameters to control their rate.",
"In , the authors give us an overview of the agent-based simulation's application in the field of epidemiology.",
"From this work, we learned that representation of \"space\" is what distinguish agent-based model ($AM$ ) from $CM$ , and drew our attention to the design of agents' Move behavior.",
"In , the authors describe a general method for the conversion of an equation-based model to an agent-based simulation.",
"Their method was not built on solid mathematics, but the discussion about the relationship between population behavior and individual behavior has greatly inspired our idea behind $ASIR$ .",
"Our model can be seen as translating their rough ideology into rigorous proof in mathematics.",
"and are the multiagent programming environments we use to implement $ASIR$ .",
"Their design is the direct source of our perception of what is agent-based simulation.",
"$ASIR$ has been influenced by the concepts of \"Agent,\" \"Step,\" and \"Map\" that were implemented by and .",
"is the book we used as a reference for the necessary condition for the existence of a Markov chain's stationary distribution."
],
[
"Solution",
"In this section, we introduce $ASIR$ in the following order: 1).",
"idea behind, 2).",
"model specification, and 3).",
"proof of robustness."
],
[
"Idea Behind $ASIR$",
"Let us first briefly recap the design of the $SIR$ model.",
"Figure: Diagram of SIR modelThe $SIR$ model consists of two parameters: $\\lbrace \\alpha , \\beta \\rbrace $ and three ordinary differential equations: $\\frac{d S}{d t}&=-\\frac{\\alpha S I}{N} \\\\\\frac{d I}{d t}&=\\frac{\\alpha S I}{N}-\\beta I \\\\\\frac{d R}{d t}&=\\beta I $ $N$ is the total population size.",
"$S$ is the susceptible population size.",
"$I$ is the infected population size.",
"$R$ is the recovered population size.",
"$\\alpha $ controls the transition speed from Susceptible into Infected.",
"$\\beta $ controls the transition speed from Infected into Recovered.",
"Equation REF models the transition speed of Susceptible population size.",
"Equation models the transition speed of Infected population size.",
"Equation models the transition speed of Recovered population size.",
"The intuitions behind $ASIR$ are: Population infection is an integral of individual infection.",
"Population recovery is an integral of individual recovery.",
"To translate these intuitions into mathematics, we adopt the theory of probability by treating population infection/recovery as the joint distribution of individual infection/recovery.",
"The core ideas behind $ASIR$ are: Model individual infection/recovery as mutually independent and identically distributed random events.",
"Use parameters to control the event probability.",
"The transition speed on population-level = the expected value of the integral of event probability on individual-level.",
"Notably, since a Susceptible individual must be infected by an Infected individual, an individual infection at time $t+1$ is conditional on another individual's infection at time $t$ (or ahead of $t$ ).",
"To guarantee independence between each individual's infection, we model the movement of every agent (or individuals, we are using these words interchangeably) using the same transition matrix $T_{\\text{map} }$ .",
"The matrix below shows a simple example where our $\\text{map}$ consists of three locations: $\\mathbf {T_{\\text{map}}} ={\\begin{bordermatrix } & \\text{Store} & \\text{School} & \\text{Stop} \\\\\\text{Store} & 0.5 & 0.3 & 0.2 \\\\\\text{School} & 0.3 & 0.3 & 0.4 \\\\\\text{Stop} & 0.2 & 0.4 & 0.4 \\end{bordermatrix }} \\qquad $ In this example, coordinate $T_{mn}$ is the probability of moving from $ m \\text{ to } n$ .",
"We focus on the period after every agent's trajectory reaches $T_{\\text{map} }$ 's stationary distribution.",
"We discuss why stationary distribution is critical for robustness later in this section.",
"In the following subsection, we introduce $ASIR$ 's detailed specification in the following order: 1).",
"agents' state, 2).",
"agents' behavior, 3).",
"model's parameter setting."
],
[
"Model Specification",
"Each agent's state can be written as a 3-element tuple: $\\Big (\\text{Timestamp}, \\text{Health}, \\text{Position} \\Big )$ .",
"\"An agent $a_1$ has state $\\Big (8, I, \\text{School} \\Big )$ \" translates as: \"at the 8-$th$ timestamp, $a_1$ is being Infected at $\\text{School}$ .\"",
"In this paper, we will use: $\\big (a_k, h_t\\big )$ or $H^{t}_{a_k}$ to denote an agent's health at timestamp $t$ , $\\big (a_k, p_t\\big )$ or $P^{t}_{a_k}$ to denote an agent's position at timestamp $t$ , $\\big (a_k , h_t, p_t \\big )$ or $\\big ( H^{t}_{a_k}, P^{t}_{a_k}\\big )$ to denote an agent's state at timestamp $t$ .",
"Table: Symbol reference for agent state.At timestamp $t$ , each agent $a_k$ has three (potential) behaviors: Move.",
"$a_k$ moves from $p_{t-1}$ to $p_t$ (which can be the same position as $p_{t-1}$ ).",
"Written as: $P_{a_k}^{t-1} \\rightarrow P_{a_k}^{t}$ As mentioned in the introduction, we model the movement of every agent using the same transition matrix $T_{\\text{map} }$ .",
"The trajectory of every agent forms a Markov chain as the example below shows: Figure: Markov chain of the sample T map T_{\\text{map}} We insist on focusing on the period after every agent's position reaches $T_{\\text{map} }$ 's stationary distribution, because the stationary distribution offers us the following critical property to deduce $ASIR$ 's robustness: Property When every agent's position reaches the $T_{\\text{map} }$ 's stationary distribution, the probability that any two agents become neighbor at time $t$ (i.e.",
"stay at the same position $p_t$ ) equals to a constant ${P}(\\text{meetup})$ .",
"${P}(\\text{meetup})$ is completely determined by $T_{\\text{map} }$ .",
"The proof is trivial.",
"An intuition is that agents' locations are mutually independent and identically distributed, therefore $\\forall j,k,m,n, \\ {P}(a_k \\text{ meets } a_j) = {P}(a_m \\text{ meets } a_n)$ Turn infected.",
"When $a_k$ was Susceptible before moving to position $X$ , it has a chance to turn infected when there is an \"Infected neighbor\" at $X$ , or more precisely, $\\exists _{\\ a_j \\ne a_k} H_{a_j}^t = I, P^{a_j}_t = P^{a_k}_t $ .",
"Written as: $\\Bigg ( H_{a_k}^{t-1}\\rightarrow H_{a_k}^{t} = S\\rightarrow I \\Bigg \\vert \\exists _{\\ a_j \\ne a_k} H_{a_j}^t = I, P^{a_j}_t = P^{a_k}_t \\Bigg )$ or simply: $\\Bigg ( H_{a_k}^{t-1}\\rightarrow H_{a_k}^{t} = S\\rightarrow I \\Bigg \\vert a_k \\text{ has an infected neighbor at t} \\Bigg )$ Figure: Conclusion"
]
] | 2209.08214 |
[
[
"Asymmetric Mediator in Scotogenic Model"
],
[
"Abstract The scotogenic model is the Standard Model (SM) with Z_2 symmetry and the addition of Z_2 odd right-handed Majorana neutrinos and SU(2)_L doublet scalar fields.",
"We have extended the original scotogenic model by an additional Z_2 odd singlet scalar field that plays a role in dark matter.",
"In our model, the asymmetries of the lepton and Z_2 odd doublet scalar are simultaneously produced through CP-violating right-handed neutrino decays.",
"While the former is converted into baryon asymmetry through the sphaleron process, the latter is relaid to the DM density through the decay of SU(2)_L doublet scalar that is named \"asymmetric mediator\".",
"In this way, we provide an extended scotogenic model that predicts the energy densities of baryon and dark matter being in the same order of magnitude, and also explains the low-energy neutrino masses and mixing angles."
],
[
"Introduction",
"The existence of dark matter (DM) and the non-zero value of the baryon asymmetry of the universe (BAU) are long-standing unsolved puzzles in the standard theory of cosmology and particle physics.",
"In fact there is no candidate for the DM particle in the standard model (SM), and accordingly, various particles have been suggested and studied intensively for the DM in particle theories beyond the SM (BSM).",
"One of the most promising candidates of DM is the so-called weakly interacting massive particle (WIMP), and the relic abundance of the DM can be calculated by its annihilation cross-section for the thermal freeze-out scenario.",
"The other problem, namely, BAU is positively realized through the mechanism of leptogenesis [1].",
"Lepton asymmetry is generated by the CP-violating decay of the right-handed neutrinos.",
"Then this lepton asymmetry is converted into baryon asymmetry through the sphaleron process.",
"It is obvious that the amount of the produced baryon asymmetry is determined by the masses of the right-handed neutrinos and Yukawa couplings in this thermal leptogenesis scenario.",
"The relic abundance of the DM and baryon are measured by the Planck observations of the cosmic microwave background (CMB), and the current values are given in the following [2]: $\\Omega _{\\rm DM} h^2 &= 0.120 \\pm 0.001~, \\\\\\Omega _{\\rm B} h^2 &= 0.0224 \\pm 0.0001 ~,$ where we express the Hubble constant $h$ in units of 100 km/s/Mpc.",
"Surprisingly, these two abundances are strikingly similar as $\\Omega _{\\rm DM} / \\Omega _{\\rm B} \\approx 5$ , although the DM and BAU are independently produced through different processes in general.",
"This coincidence of these relic abundances implies the existence of mechanisms that link the productions of the DM and BAU together.",
"Asymmetric dark matter (ADM) [3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15],[16],[17] is one of the frameworks where the coincidence between the relic abundances of the DM and baryon is realized.",
"In this framework, an asymmetry of the DM and anti-DM number densities is produced in the early universe.",
"As the universe cools, the symmetric component annihilates into the SM particles, and then the remaining asymmetry component explains the observed relic abundance of the DM.",
"Generations of the DM asymmetry are roughly classified into two types.",
"One is the sharing mechanism that the asymmetry related with BAU in the SM sector is firstly generated, and then the produced asymmetry is shared between the DM and SM sectors through some interactions.",
"The other is the cogenesis mechanism in which the asymmetries of the matter and DM are generated simultaneously.",
"In this article, we focus on the cogenesis mechanism in the scotogenic model.",
"The scotogenic model [18] is one of the seesaw models [19],[20],[21],[22].",
"In scotogenic model, right-handed neutrinos and a neutrino-philic inert SU(2)$_L$ doublet scalar are introduced, and these fields transform odd under an exact $Z_2$ symmetry.",
"The masses of the light neutrinos are generated through an one-loop diagram.",
"Moreover, the lightest $Z_2$ odd field is stable and can be a candidate for the DM.",
"However, this $Z_2$ odd scalar can not be a DM in the context of the ADM.",
"This is because the mass of the DM should be the same order of magnitude as that of proton, and such a light DM which interacts with the weak gauge bosons is strongly constrained by the requirement that neutron stars do not gravitationally collapse into black holes [23],[24],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39].",
"From these reasons we introduce an additional $Z_2$ odd singlet real scalar as the DM in the scotogenic model.",
"In this new model, the SU(2)$_L$ doublet scalar plays a role of the mediator which links the DM and right-handed neutrinos and relays the asymmetry to the DM.",
"Firstly, the CP-violating decays of the right-handed neutrinos generate the same amount of the lepton and mediator asymmetries simultaneously.",
"After annihilation of the symmetric component of the mediators, the asymmetric component decays into the DM, and then the mediator asymmetry converts into the relic abundance of the DM.",
"Thus, the same order of number densities of the baryon and DM are realized.",
"1There is another model which realizes the coincidence between DM abundance and baryon asymmetry in frameworks of the scotogenic model [40],[41].",
"In these papers, the lepton asymmetry is generated through annihilation and coannihilation of dark sector particles.",
"This article organized as follows.",
"Next section we review the scotogenic model and neutrino parameters.",
"In Sec.",
", we discuss the leptogenesis and DM production in the scotogenic model with a real singlet scalar DM.",
"In Sec.",
", we show the parameter region where the model in this paper explains the observed baryon asymmetry, DM density, and neutrino mixing parameters simultaneously.",
"Finally, our conclusions are discussed in Sec.",
"."
],
[
"Scotogenic Model with Singlet Scalar Dark Matter",
"In this section, we introduce a new scotogenic model by adding a real singlet scalar field.",
"The matter contents of the original scotogenic model are of the SM plus three right-handed neutrinos $N_i~(i = 1,2,3)$ and an inert doublet scalar $\\eta $ .",
"The SM fields are even under a discrete $Z_2$ symmetry but non-SM fields: the right-handed neutrinos $N_i (i = 1,2,3)$ , an inert doublet scalar $\\eta $ , and a single scalar $\\sigma $ , are odd under this symmetry.",
"In Tab.",
"REF , the matter contents of our model are summarized.",
"It is important to note that a singlet scalar $\\sigma $ plays a role of DM in our model 2The scalar field $\\sigma $ may be also a complex scalar field.",
"There is no difference between these choices except for the degree of freedom.",
".",
"As will be mentioned below, this field is the lightest particle among the $Z_2$ -odd fields, and therefore the stability of the dark matter is guaranteed.",
"Table: Matter contents of the extended version of scotogenic model.The SM left-handed lepton doublet, the right-handed charged lepton, and the Higgs doublet scalar are denoted by $L$ , $e_R^{}$ , and $H$ , respectively.",
"Under this setup, the Lagrangian relative to the neutrinos and scalar potential is given by $\\mathcal {L}\\supset &\\,- h_{\\alpha i} \\bar{L}_\\alpha \\tilde{\\eta } N_i + \\frac{1}{2} M_i \\bar{N}_i N_i^c + {\\rm H.c.}~, \\\\V(H,\\eta ,\\sigma ) =&\\,\\mu _H^2 |H|^2 + m_\\eta ^2 |\\eta |^2 + \\frac{1}{2} m_\\sigma ^2 \\sigma ^2 + \\frac{1}{2} \\lambda _1 |H|^4 + \\frac{1}{2} \\lambda _2 |\\eta |^4 + \\frac{1}{2} \\lambda _3 \\sigma ^4 + \\lambda _4 |H|^2 |\\eta |^2 \\nonumber \\\\&\\,+ \\lambda _5 |H^\\dag \\eta |^2 + \\lambda _6 |H|^2 \\sigma ^2 + \\lambda _7 |\\eta |^2 \\sigma ^2 + \\frac{1}{2} \\left[ \\lambda _8 (H^\\dag \\eta )^2 + {\\rm H.c.} \\right] \\nonumber \\\\&\\,+ \\frac{1}{\\sqrt{2}} \\left[ \\mu \\sigma (H^\\dag \\eta ) + {\\rm H.c.} \\right]~,$ where $\\alpha ~ (i)$ denotes the index of the flavor (mass) eigenstates, $M_i$ represents the mass eigenvalue of the heavy neutrino $N_i$ , $\\tilde{\\eta } \\equiv i \\sigma _2 \\eta ^*$ , and $\\mu _H^2$ is negative.",
"All the parameters in the scalar potential can be chosen real without loss of generality.",
"Only the SM Higgs acquires a nonzero vacuum expectation value (VEV), and the other scalars do not.",
"As we will show in the following section, our scenario works under the condition that $m_\\eta $ ($\\mu $ ) is much higher (lower) than the electroweak scale.",
"Therefore the mixing between CP-even neutral components of $\\eta $ and $\\sigma $ is negligible although $\\mu $ plays an important role in the dark matter production and should not be zero.",
"Additionally, we assume $\\lambda _6, \\lambda _7 \\ll 1$ to avoid constraints from direct detection experiments and thermalization of the DM in the early universe.",
"After the SM Higgs acquires a nonzero VEV, the masses of the charged, CP-even, and CP-odd components of the inert doublet scalar, $\\eta = (\\eta ^+, \\eta ^0)^T$ with $\\eta ^0 = (\\eta _R^{} + i \\eta _I^{})/\\sqrt{2}$ , split and are given by $m_{\\eta ^+}^2 =&\\,m_\\eta ^2 + \\frac{1}{2} \\lambda _4 v^2~, \\\\m_{\\eta _R^{}}^2 \\simeq &\\,m_\\eta ^2 + \\frac{1}{2} (\\lambda _4 + \\lambda _5 + \\lambda _8) v^2~, \\\\m_{\\eta _I^{}}^2 \\simeq &\\,m_\\eta ^2 + \\frac{1}{2} (\\lambda _4 + \\lambda _5 - \\lambda _8) v^2~,$ where $v$ is the VEV of the SM Higgs field.",
"That of the singlet scalar DM is given by $m_{\\rm DM}^2 \\simeq m_\\sigma ^2 + \\frac{1}{2} \\lambda _6 v^2~.$ The neutrino masses are radiatively generated as shown in Fig.",
"REF .",
"3There is another contribution to the active neutrino mass by $\\sigma $ .",
"Note that effectively $\\lambda _8$ term is induced by the exchange of $\\sigma $ with the coupling $\\mu \\sigma (H^\\dag \\eta )$ .",
"It is negligible because $\\lambda _8 \\gg {({\\mu }/{m_\\sigma })}^2$ in this model, and we do not discuss it here.",
"Figure: One-loop diagram that generates the neutrino masses.The mass matrix of the active neutrinos are obtain as $(\\mathcal {M}_\\nu )_{\\alpha \\beta } =\\sum _i \\frac{h_{\\alpha i}^* h_{\\beta i}^*}{32 \\pi ^2} M_i \\left[ \\frac{m_{\\eta _R^{}}^2}{m_{\\eta _R^{}}^2 - M_i^2} \\ln {\\frac{m_{\\eta _R^{}}^2}{M_i^2}} - \\frac{m_{\\eta _I^{}}^2}{m_{\\eta _I^{}}^2 - M_i^2} \\ln {\\frac{m_{\\eta _I^{}}^2}{M_i^2}} \\right]~.$ In addition to the assumption that $m_\\eta $ is much higher than the electroweak scale, we assume further that the mass of the right-handed neutrinos are much heavier than $m_{\\eta _R^{}}$ and $m_{\\eta _I^{}}$ .",
"Thus, for $m_{\\eta _{R,I}^{}}^{} \\ll M_i$ and $\\lambda _8 v^2 \\ll m_\\eta ^2$ , the mass matrix of the active neutrinos can be approximated as follows: $(\\mathcal {M}_\\nu )_{\\alpha \\beta } \\simeq \\frac{\\lambda _8 v^2}{32 \\pi ^2} \\sum _i \\frac{h_{\\alpha i}^* h_{\\beta i}^*}{M_i} \\left[ \\ln {\\frac{M_i^2}{m_0^2} - 1} \\right]~,$ where $m_0^2 \\equiv (m_{\\eta _R^{}}^2 + m_{\\eta _I^{}}^2) / 2$ .",
"For convenience, we introduce the Casas-Ibarra (CI) parametrization [42], following Ref.",
"[43] in which the leptogenesis scenario in the scotogenic model is discussed.",
"The mass matrix for the light neutrinos is rewritten by the diagonal matrix $\\mathcal {D}_\\Lambda ^{}$ in the following way: $\\mathcal {M}_\\nu &= h^* \\mathcal {D}_\\Lambda ^{-1} h^\\dag ~, \\\\(\\mathcal {D}_\\Lambda ^{})_{ii} &= \\frac{2\\pi ^2}{\\lambda _8} \\xi _i \\frac{2 M_i}{v^2} \\equiv \\Lambda _i~,$ with $\\xi _i \\equiv \\left\\lbrace \\frac{1}{8} \\frac{M_i^2}{m_{\\eta _R^{}}^2 - m_{\\eta _I^{}}^2} \\left( \\frac{m_{\\eta _R^{}}^2}{m_{\\eta _R^{}}^2 - M_i^2} \\ln {\\frac{m_{\\eta _R^{}}^2}{M_i^2}} - \\frac{m_{\\eta _I^{}}^2}{m_{\\eta _I^{}}^2 - M_i^2} \\ln {\\frac{m_{\\eta _I^{}}^2}{M_i^2}} \\right) \\right\\rbrace ^{-1}~.$ For the hierarchical masses structure between the right-handed neutrinos and inert scalars ($m_\\eta \\ll M_i$ ) and the small scalar four-point coupling ($\\lambda _8 v^2 \\ll m_\\eta ^2$ ), the parameters $\\xi _i$ can be approximated as $\\xi _i \\approx \\frac{8}{\\left[ \\ln {(M_i^2/m_0^2)} - 1 \\right]} ~.$ The mass matrix for the light neutrinos can be diagonalized by a unitary matrix called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix $U$ [44],[45],[46],[47] as $U^T \\mathcal {M}_\\nu U = {\\rm diag}(m_1, m_2, m_3) \\equiv D_\\nu $ Note that we follow the convention of the Particle Data Group [48], and it is different from that of Ref. [43].",
"The Yukawa couplings are written as follows: $h_{\\alpha i} = \\left( U\\, D_\\nu ^{\\frac{1}{2}}\\, R^\\dag \\, D_{\\Lambda }^{\\frac{1}{2}} \\right)_{\\alpha i}~,$ where $R$ is an arbitrary complex orthogonal matrix satisfying $R R^T = 1$ ."
],
[
"Cogenesis Mechanism in the Scotogenic Model",
"In this section, we discuss the generation of the baryon asymmetry and DM through the cogenesis mechanism in our extended scotogenic model.",
"Firstly, we summarize the story of the generation of the DM and baryon asymmetries in the extended scotogenic model.",
"Our setup is displayed in Fig.",
"REF .",
"In the early universe, the right-handed neutrinos are thermally produced in the SM thermal plasma.",
"After the temperature gets lower than the mass of the lightest right-handed neutrino $N_1$ 4In this article, we assume that the asymmetries of the baryon and inert doublet scalar are dominantly generated by the decay of the lightest right-handed neutrino $N_1.$ , the decay process of $N_1$ becomes out-of-equilibrium.",
"Then, the asymmetries of the lepton and mediator are generated by the CP-violating decays of $N_1$ as shown in Fig.",
"REF .",
"Figure: Feynman diagrams contributing to the asymmetries of the baryon and inert doublet scalar.We must notice that the interaction $\\eta \\eta \\leftrightarrow H H$ should become out-of-equilibrium to avoid the wash-out of the mediator asymmetry.",
"Therefore the coupling $\\lambda _8$ should be small, but on the other hand, sufficiently too small $\\lambda _8$ spoils the generation of the neutrino masses by the scotogenic seesaw mechanism.",
"After the temperature drops lower than the mediator mass, they quickly annihilate into the SM fields through the SU(2)$_L$ interaction.",
"The mediator asymmetry is generated in the same process as that of the lepton, and the annihilation process does not change the difference of the number density between $\\eta $ and $\\eta ^\\dag $ : $n_{\\Delta \\eta }^{} \\equiv n_\\eta - n_{\\eta ^\\dag }$ with $n_{\\eta } \\equiv n_{\\eta ^0} + n_{\\eta ^+}$ and $n_{\\eta ^\\dag } \\equiv n_{\\eta ^{0*}} + n_{\\eta ^-}$ , as long as $\\lambda _8$ is small enough to neglect the CP violating annihilation, $\\eta \\eta \\rightarrow H H\\, (\\eta ^\\dag \\eta ^\\dag \\rightarrow H^\\dag H^\\dag )$ .",
"During the annihilation, $n_{\\Delta \\eta }^{}$ is, therefore, equal to the lepton asymmetry $n_{\\Delta L}^{}$ if the decay rate of $\\eta $ and $\\eta ^\\dag $ is less than the Hubble parameter $H$ .",
"After falling out of equilibrium of the annihilation process, the density of $\\eta $ becomes much smaller than that of $\\eta ^\\dag $ , and the hierarchy of the number densities is realized as $|n_{\\Delta \\eta }^{}| \\simeq n_{\\eta ^\\dag } \\gg n_{\\eta }$ .",
"Subsequently, the decays of the remaining $\\eta ^\\dag $ start at the temperature $T_{\\rm dec}$ , and then, $n_{\\Delta \\eta }^{}$ is converted into the DM number density $n_{\\rm DM}^{}$ .",
"Thus, the number density of the DM has the same order as those of the SM lepton and baryon.",
"Hereafter, we discuss the details of each period of the early universe."
],
[
"Leptogenesis in the scotogenic model",
"The baryon asymmetry of the universe is provided by the thermal leptogenesis scenario in our model.",
"In leptogeneis scenario, the lepton asymmetry is firstly generated by the decays of the right-handed neutrinos.",
"The generated lepton asymmetry depends on the following asymmetry parameters [49] $\\epsilon _i =&\\,\\frac{\\sum _\\alpha \\left[ \\Gamma (N_i \\rightarrow L_\\alpha \\eta ) - \\Gamma (N_i \\rightarrow \\bar{L}_\\alpha \\eta ^\\dag ) \\right]}{\\sum _\\alpha \\left[ \\Gamma (N_i \\rightarrow L_\\alpha \\eta ) + \\Gamma (N_i \\rightarrow \\bar{L}_\\alpha \\eta ^\\dag ) \\right]} \\nonumber \\\\=&\\, \\frac{1}{8\\pi } \\frac{1}{(h^\\dag h)_{ii}} \\sum _{j \\ne i} {\\rm Im}\\left[ \\left\\lbrace \\left( h^\\dag h \\right)_{ij} \\right\\rbrace ^2 \\right] F(r_{ji},\\eta _i)~, \\\\F(x,y) =&\\,\\sqrt{x} \\left[ 1 + \\frac{1+x-2y}{(1-y)^2} \\ln \\left( \\frac{x-y^2}{1+x-2y} \\right) - \\frac{1}{x-1} (1-y)^2 \\right]~.$ Here we denote that the Dirac Yukawa coupling $h$ shown in Eq.",
"(REF ), $\\eta _i \\equiv m_\\eta ^2/M_i^2$ , and $r_{ji} \\equiv M_j^2/M_i^2$ , and the function $F(x,y)$ comes from both the one-loop vertex contribution and the self-energy contribution.",
"For the case that the masses of the right-handed neutrinos are hierarchical, the lepton asymmetry is produced dominantly by the decays of the lightest right-handed neutrino $N_1$ , and thus the baryon to photon number ratio is approximately given by [50] $\\eta _B \\approx - 0.01 \\epsilon _1 \\kappa _1~,$ where $\\kappa _1$ is the efficiency factor that presents the wash-out of the generated lepton asymmetry.",
"This efficiency factor is calculated by the decay parameter $K_1 \\equiv \\Gamma _1 / H(T=M_1)$ [43] defined as $K_1 &=\\frac{2 \\pi ^2}{\\lambda _8} \\xi _1 \\sqrt{\\frac{45}{64 \\pi ^5 g_*}} \\frac{M_{\\rm Pl}}{v^2} ~\\widetilde{m}_{11}~(1 - \\eta _1)^2 \\nonumber \\\\&\\simeq 15 \\cdot \\frac{10^{-7}}{\\lambda _8} \\left( \\frac{-10}{\\ln {\\left(\\eta _1\\right)}} \\right) \\frac{\\widetilde{m}_{11}}{10^{-10}\\,{\\rm eV}}~,$ where $g_*$ stands for the effective number of relativistic degree of freedom, $M_{\\rm Pl}$ does the Planck mass, and $\\displaystyle \\widetilde{m} \\equiv R D_\\nu R^\\dag $ .",
"As shown in Eq.",
"(REF ), the decay parameter is much larger than 1 for the parameter region where we mainly focus and investigate.",
"Thus in our scenario, the lepton asymmetry is generated via the strong wash-out regime.",
"For the large value of $K_1$ , the efficiency factor can be approximated by [50] $\\kappa _1(K_1) =\\frac{1}{1.2 K_1 [\\ln K_1]^{0.8}}~.$ The asymmetry $n_{\\Delta _\\eta } = n_\\eta - n_{\\eta ^\\dag }$ makes the reaction rate of $\\eta \\eta \\rightarrow H H$ larger than that of $\\eta ^\\dag \\eta ^\\dag \\rightarrow H^\\dag H^\\dag $ .",
"The coupling constant $\\lambda _8$ should be small so as that the mediator asymmetry is relayed to the DM asymmetry.",
"In the non-relativistic regime of the mediator, the number densities $n_\\eta $ and $n_{\\eta ^\\dag }$ exponentially decay with temperature cooling.",
"This implies that, if $(\\sigma v_{\\rm rel})_{\\eta \\eta \\rightarrow H H}^{}\\, n_{\\eta ^0} < H (T)$ in the relativistic regime, it holds in all of the regimes.",
"Here $H(T)$ denotes the Hubble parameter.",
"The annihilation cross section is roughly estimated in the relativistic regime by $(\\sigma v_{\\rm rel})_{\\eta \\eta \\rightarrow H H} =(\\sigma v_{\\rm rel})_{\\eta ^\\dag \\eta ^\\dag \\rightarrow H^\\dag H^\\dag } =\\frac{3 \\lambda _8^2}{128 \\pi T^2}~.$ The requirement $(\\sigma v_{\\rm rel})_{\\eta \\eta \\rightarrow H H}^{}\\, n_{\\eta ^0} < H (T)$ finds the following constraint; $\\lambda _8 < 3.9 \\times 10^{-8} \\sqrt{ \\frac{T}{{\\rm GeV}} }~.$ The relativistic cross section in Eq.",
"(REF ) is available at $T \\gtrsim m_\\eta $ , and takes maximal for $T=m_\\eta $ .",
"Most conservative constraint is found for $T=m_\\eta $ , e.g., $\\lambda _8 < 3.9 \\times 10^{-6}$ for $m_\\eta = 10\\,\\text{TeV}$ ."
],
[
"Mediator Annihilation and Dark Matter Production",
"The asymmetry of the mediator is produced through the CP-violating decays of the right-handed neutrinos.",
"However, the mediator stays in thermal equilibrium in the early universe, and the asymmetry of $\\eta $ and $\\eta ^\\dag $ is much smaller than their number density, thus $n_{\\Delta \\eta }^{} \\ll n_\\eta , n_{\\eta ^\\dag }$ .",
"As the temperature of the universe falls below the mediator mass, the mediators annihilate into the weak and hypercharge gauge bosons.",
"After the annihilation of the mediators and satisfying $n_\\eta \\ll n_{\\eta ^\\dag } \\approx n_{\\Delta \\eta }$ , the asymmetric component of the mediators decays into the DM, and the mediator asymmetry converts into the DM density.",
"For verifying whether the symmetric component of the mediator sufficiently annihilates, we evaluate the relic abundance of the mediator, assuming that the mediator has no asymmetry.",
"The relic density of the mediator after the annihilation process gets out of equilibrium can be calculated by [51] $Y_{\\eta ,\\infty }\\equiv \\frac{n_{\\eta ,\\infty }}{s}= 2 \\times \\frac{3.80\\,x_{\\mathrm {f}}}{\\left(g_{* s} / g_{*}^{1 / 2}\\right) M_{\\rm Pl} m_{\\eta } \\mathinner {\\langle {\\sigma _{\\rm g} v_{\\rm rel}}\\rangle }}~,$ where $s$ stands for the entropy density, $g_{*s}$ does the total relativistic degrees of freedom for entropy, and $\\mathinner {\\langle {\\sigma _{\\rm g} v_{\\rm rel}}\\rangle }$ does the thermally averaged cross section of the mediator annihilation through the gauge interaction.",
"The factor of 2 comes from the sum of the densities of $\\eta ^0$ and $\\eta ^+$ .",
"In Eq.",
"(REF ), the ratio of the freeze-out temperature to the mediator mass, $x_{\\rm f}$ , is evaluated by $x_{\\rm f} \\equiv \\frac{m_\\eta }{T_{\\rm f}} =&\\ln \\left[ 0.038 \\left( g / g_{*}^{1/2} \\right) M_{\\rm Pl} m_{\\eta } \\mathinner {\\langle {\\sigma _{\\rm g} v_{\\rm rel}}\\rangle } \\right] \\\\&- \\frac{1}{2}\\ln \\left\\lbrace \\ln \\left[0.038 \\left(g / g_{*}^{1 / 2}\\right) M_{\\rm Pl} m_{\\eta } \\mathinner {\\langle {\\sigma _{\\rm g} v_{\\rm rel}}\\rangle } \\right]\\right\\rbrace ~,$ where $g$ is the internal degrees of freedom.",
"Here we approximate the thermally averaged annihilation cross section by its non-relativistic limit as follows: $\\mathinner {\\langle {\\sigma _{\\rm g} v_{\\rm rel}}\\rangle } \\simeq \\frac{ (g_1)^4 + 6 \\cdot (g_1 g_2)^2 + 3 \\cdot (g_2)^4}{256 \\pi {m_\\eta }^2 }~,$ where $g_1$ and $g_2$ stand for the gauge couplings of the hypercharge and weak gauge bosons, respectively.",
"For the sufficient annihilation of the symmetric component, $Y_{\\eta ,\\infty }$ should be smaller than the ratio of the mediator asymmetric component to the entropy density as $\\displaystyle Y_{\\eta ,\\infty } < Y_{\\Delta \\eta }$ .",
"Note that we emphasize that our estimation above is conservative.",
"Assuming the existence of the asymmetric component, the number density of the annihilation partner is larger than that in no asymmetry case from the viewpoint of $\\eta $ .",
"Therefore the annihilation of the mediator and decrease of the density of $\\eta $ proceed more effectively, and the relation, $n_{\\eta } \\ll n_{\\eta ^\\dag } \\approx n_{\\Delta \\eta }$ , can be realized more easily.",
"After the pair annihilation of the mediator, the asymmetric component decays into the DM and SM Higgs boson as $\\eta ^\\dag \\rightarrow H^\\dag \\sigma $ .",
"The decay width of the mediator is given by $\\Gamma _{\\rm decay} =\\frac{\\mu ^2}{16 \\pi m_\\eta }~,$ and the temperature when the mediator decay gets active, $T_{\\rm decay}$ , is obtained by $\\Gamma _{\\rm decay} = H(T_{\\rm decay})$ .",
"For the successful coincidence, the mediator should not decay before the annihilation of the symmetric component, and $T_{\\rm decay}$ should satisfy $T_{\\rm f} > T_{\\rm decay}$ .",
"From this condition, there is an upper limit on the scalar three-point coupling $\\mu $ .",
"On the other hand, the mediator decay during or after the BBN is cosmologically dangerous, and thus we request that the mediator decay starts by the BBN era, and the scalar three-point coupling satisfies $\\Gamma _{\\rm decay} > H(T_{\\rm BBN})$ with $T_{\\rm BBN} \\simeq 1\\,{\\rm MeV}$ being the temperature when the BBN begins.",
"These two conditions give the following upper and lower bounds on the scalar three-point coupling: $8.4 \\times 10^{-12}\\, \\frac{T_{\\rm BBN}}{1\\,{\\rm MeV}} \\sqrt{\\frac{m_\\eta }{\\rm GeV}}<\\frac{\\mu }{\\rm GeV}<8.4 \\times 10^{-9}\\, \\frac{T_{\\rm decay}}{\\rm GeV} \\sqrt{\\frac{m_\\eta }{\\rm GeV}}~.$"
],
[
"Cosmological Constraints on Asymmetric Mediator",
"In the era of the BBN, a part of the DMs, which are produced by late time decays of the asymmetric mediators, can be relativistic.",
"5For $(m_\\eta , m_\\sigma ) = (10^4\\,{\\rm GeV}, 5\\,{\\rm GeV})$ and $T_{\\rm decay} = 1\\,{\\rm GeV}$ , the momentum of the dark matter is given by $|\\mathbf {p}_\\sigma (t_{\\rm BBN})| = m_\\eta a(t_{\\rm decay}) / 2 a(t_{\\rm BBN}) = 0.5\\,{\\rm GeV}$ .",
"Therefore, there is a possibility that a part of the DMs which decay for $T < T_{\\rm decay}$ is relativistic.",
"Such a relativistic DM contributes to the expansion of the universe in the BBN and alter the BBN prediction.",
"In this subsection, we estimate this contribution by the DM and confirm that the DM in our model avoids the constraint from the BBN.",
"If the DM is relativistic in the BBN era, the energy density of the DM is given by $\\rho _\\sigma (t_{\\rm BBN}) &=E_\\sigma (t_{\\rm BBN}) n_\\sigma (t_{\\rm BBN}) \\nonumber \\\\&\\simeq \\frac{m_\\eta }{2} \\left( \\frac{a(t_{\\rm decay})}{a(t_{\\rm BBN})} \\right) \\cdot n_\\sigma (t_0) \\left( \\frac{a(t_0)}{a(t_{\\rm BBN})} \\right)^3 \\nonumber \\\\&= 10^{25}\\,{\\rm GeV/cm^3} \\cdot \\frac{m_\\eta }{10\\,{\\rm TeV}} \\frac{a(t_{\\rm decay})/a(t_{\\rm BBN})}{10^{-3}} \\nonumber \\\\&\\qquad \\quad \\times \\left( \\frac{1~{\\rm GeV}}{m_\\sigma } \\frac{\\rho _\\sigma (t_0)}{10^3~{\\rm eV/cm^3}} \\right) \\left( \\frac{a(t_0)/a(t_{\\rm BBN})}{10^{10}} \\right)^3~,$ where $t_{\\rm BBN}$ , $t_{\\rm decay}$ , and $t_0$ are the times at the BBN, decays of the mediator, and present, respectively.",
"In Eq.",
"(REF ), $\\rho _\\sigma $ and $n_\\sigma $ stand for the energy and number densities of $\\sigma $ , respectively, and $E_\\sigma $ does the energy of $\\sigma $ .",
"The deviation of the effective number of neutrino species, $\\Delta N_{\\rm eff} \\equiv N_{\\rm eff} - N_{\\rm eff}^{\\rm SM}$ , is given by $\\Delta N_{\\rm eff} |_{\\rm BBN} &=\\frac{8}{7} \\left( \\frac{11}{4} \\right)^{\\frac{4}{3}} \\frac{\\rho _\\sigma (t_{\\rm BBN})}{\\rho _\\gamma (t_{\\rm BBN})} \\nonumber \\\\&= 2.7 \\times 10^{-4} \\cdot \\frac{\\rho _\\sigma (t_{\\rm BBN})}{10^{25}\\,{\\rm GeV/cm^3}} \\frac{10^{28}\\,{\\rm GeV/cm^3}}{\\rho _\\gamma (t_{\\rm BBN})}~,$ with $\\rho _\\gamma $ being the energy density of the photon.",
"The contribution to the effective number of neutrino species by the DM in this model is much smaller than the SM prediction: $N_{\\rm eff}^{SM} \\simeq 3.044$ [52],[53],[54],[55], and therefore the constraint on the DM from $N_{\\rm eff}$ in the BBN era can be negligible."
],
[
"Results",
"So far we have discussed the leptogenesis, annihilation of the symmetric component of the mediator, and DM production in the scotogenic model.",
"In this section, we discuss the parameter region where our model realizes the coincidence between the observed baryon asymmetry and DM relic density without conflicting with the results of neutrino oscillation measurements.",
"In the following results, the neutrino oscillation parameters, such as the three mixing angles and two squared mass differences, and Dirac CP phase are fixed to be their best-fit values [56].",
"The two Majorana CP phases are fixed to be zero as a reference.",
"The three complex angles in the complex orthogonal matrix $R$ are varied as $10^{-10} < \\left|\\omega _i\\right| < 1$ and $- \\pi < {\\rm arg} \\left(\\omega _i \\right)< \\pi $ ($i=1,2,3$ ).",
"The masses of the lightest active neutrino and mediator field are fixed to be $m_1 = 10^{-10}\\,{\\rm eV}$ and $\\eta _1 \\equiv m_\\eta ^2/M_1^2 = 10^{-6}$ , respectively.",
"The mass hierarchy of the right-handed neutrinos are assumed to be $M_2 / M_1 = M_3 / M_2 = 1.5$ .",
"As a reference, the value of $\\lambda _8$ is fixed to be $\\lambda _8 = 10^{-7}$ and $10^{-5}$ .",
"Figure: The baryon-to-photon ratio as a function of the mass of the lightest right-handed neutrino.All the scattered points avoid the constraint on the sum of the active neutrino masses and triviality bounds for the neutrino Yukawa couplings.Moreover, we focus on the strong washout regime, and all the points satisfy K 1 >10K_1 > 10.The black dotted line shows the observed baryon-to-photon ratio: η B obs =6.1×10 -10 \\eta _B^{\\rm obs} = 6.1 \\times 10^{-10}.In the gray shaded region, the mediator is lighter than 1 TeV, and we do not consider such a region to avoid the severe collider bounds.The brown shaded region conflicts with the requirement in Eq.",
"().In Fig.",
"REF , we show the scatter plots of the baryon-to-photon ratio versus the mass of the lightest right-handed neutrino.",
"All the scattered points in Figs.",
"REF avoid the constraint on the sum of the active neutrino masses and triviality bounds for the neutrino Yukawa couplings, that is, $\\sum _{i=1}^3 m_i < 0.16$ eV (95% C.L.)",
"[57] and $\\displaystyle \\left|h_{\\alpha i}\\right|< 1$ for all $\\alpha $ and $i$ .",
"Moreover, we focus on the strong wash-out regime, and all the points satisfy $K_1 > 10$ .",
"Therefore, the approximate formula of the efficiency factor in Eq.",
"(REF ) is valid.",
"The black dotted line shows the observed baryon-to-photon ratio: $\\eta _B^{\\rm obs} = 6.1 \\times 10^{-10}$ .",
"In the gray shaded region, the mediator is lighter than 1 TeV, and we do not consider such a region to avoid the severe collider bounds.",
"The brown shaded region conflicts with the requirement in Eq.",
"(REF ).",
"As shown in Fig.",
"REF , the observed baryon asymmetry can be generated in broad range of the lightest right-handed neutrino mass.",
"The large-$\\eta _B$ boundary of the scattered region is determined by the active neutrino masses.",
"For a fixed $M_1$ (and $m_\\eta $ ), too large Yukawa couplings cannot realize the light masses of the active neutrinos.",
"There is, therefore, a maximal value of the baryon-to-photon ratio.",
"The heavy-$M_1$ boundary is determined by the triviality bound of the neutrino Yukawa couplings.",
"From Eq.",
"(REF ), the masses of the active neutrinos are obtained as $\\mathcal {M}_\\nu \\simeq 0.05\\,{\\rm eV} \\cdot \\frac{\\lambda _8}{10^{-7}}\\cdot \\frac{h_{\\alpha i} h_{\\beta i}}{1} \\cdot \\frac{5 \\times 10^{6}\\,{\\rm GeV}}{M_1}~,$ for $m_\\eta / M_1 = 10^{-3}$ .",
"Thus, for $\\left|h_{\\alpha i}\\right| < 1$ and $\\lambda _8 = 10^{-7}$ , the lightest right-handed neutrino with a mass heavier than $5 \\times 10^6$ GeV conflicts with the observed neutrino oscillation and measured mass squared difference.",
"Figure: The relic abundance of η\\eta after freeze-out, Y η,∞ Y_{\\eta ,\\infty }, as a function of the mediator mass.The black dashed line corresponds to the observed baryon asymmetry, Y B =8.66×10 -11 Y_B = 8.66 \\times 10^{-11}.In Fig.",
"REF , we show the relic abundance of $\\eta $ after freeze-out as a function of the mediator mass.",
"The black dashed line corresponds to the observed baryon asymmetry, $Y_B = 8.66 \\times 10^{-11}$ .",
"From Fig.",
"REF , we find that the mediator sufficiently annihilates, and hence the relation, $Y_{\\eta ,\\infty } < Y_B$ , is satisfied for $m_\\eta \\lesssim 10^{5}\\,{\\rm GeV}$ .",
"In addition, if the mediator is heavier than the electroweak scale as $m_\\eta \\gg 10^2$ GeV, our model can escape from severe collider bounds.",
"From Eq.",
"(REF ), the freeze-out temperature is roughly obtained as $T_{\\rm f} \\sim m_\\eta /22$ .",
"The viable parameter region of the scalar three-point coupling is obtained as follows: $8.4 \\times 10^{-12}\\, \\sqrt{\\frac{m_\\eta }{\\rm Gev}}<\\frac{\\mu }{\\rm GeV}<3.8 \\times 10^{-10}\\, \\frac{T_{\\rm decay}}{m_\\eta /22} {\\left(\\frac{m_\\eta }{\\rm GeV}\\right)}^{\\frac{3}{2}}~.$ According to the above results, our model can realizes the coincidence between the number densities of baryon and DM is realized as $Y_B \\sim Y_L \\simeq Y_{\\Delta \\eta } \\simeq Y_{\\rm DM}$ for the TeV scale mediator ($1\\,{\\rm TeV} \\lesssim m_\\eta \\lesssim 10^2\\,{\\rm TeV}$ ) and the small scalar four and three-point couplings ($\\lambda _8 \\lesssim 10^{-8} \\sqrt{m_\\eta /{\\rm GeV}},~10^{-11}\\,{\\rm GeV}\\sqrt{m_\\eta /{\\rm GeV}} \\lesssim \\mu \\lesssim 10^{-10}\\,{\\rm GeV} (m_\\eta /{\\rm GeV})^{3/2}$ )."
],
[
"Summary and discussion",
"The Scotogenic model is an excellent extension of the standard model to explain not only the light neutrino masses, mixing, and also DM.",
"The purpose of this article is to combine the idea of ADM with the Scotogenic model to construct a model that simultaneously explains the relationship between the long-standing problems of the SM, namely, the neutrino mass and mixing, DM, and BAU.",
"Particularly, we focus on the coincidence between the observed energy densities of the DM and baryon as $\\Omega _{\\rm DM} / \\Omega _B \\approx 5$ .",
"In this paper, we consider an extended model of the scotogenic model with a $Z_2$ odd singlet scalar field $\\sigma $ that plays a role of DM.",
"The $Z_2$ odd SU(2)$_L$ doublet scalar field $\\eta $ is the mediator field, and it decays into $\\sigma $ .",
"Since the mediator $\\eta $ is simultaneously produced with the SM lepton $L_\\alpha $ by the decay of the right-handed neutrino $N_i$ , the asymmetry of the mediator, $n_{\\Delta \\eta }$ , is the exactly same as that of the lepton, $n_{\\Delta L}$ .",
"After the annihilation of the symmetric component of the mediators preserving the asymmetric component $n_{\\Delta \\eta }$ , $n_\\eta $ becomes much smaller than $n_{\\eta ^\\dag }$ and $n_\\eta \\ll n_{\\eta ^\\dag } \\approx n_{\\Delta \\eta }$ .",
"Subsequently, the mediator decays into the DM, and the mediator asymmetry converts into the DM number density.",
"In this way, the DM number density can be related to the lepton asymmetry and baryon asymmetry of the universe generated by the leptogenesis scenario.",
"For the successful coincidence between the number densities of the lepton and mediator asymmetries, the CP violating annihilation, the interaction rates of $\\eta \\eta \\rightarrow H H$ and $\\eta ^\\dag \\eta ^\\dag \\rightarrow H^\\dag H^\\dag $ are required to be inactive, $\\Gamma _{ \\eta \\eta \\rightarrow H H (\\eta ^\\dag \\eta ^\\dag \\rightarrow H^\\dag H^\\dag ) } < H(T)$ , because the unbalance rate between $\\eta \\eta \\rightarrow H H$ and $\\eta ^\\dag \\eta ^\\dag \\rightarrow H^\\dag H^\\dag $ distort the mediator asymmetry.",
"It requires that the scalar four-point coupling $\\lambda _8$ should be small, though too small $\\lambda _8$ spoils the radiative generation of the neutrino masses.",
"From this condition requirement, the scalar four-point coupling should satisfy $\\lambda _8 < 3.9 \\times 10^{-8} \\times \\sqrt{{m_\\eta }/{\\rm GeV}}$ .",
"We calculate the baryon to photon number ratio $\\eta _B$ taking $ \\lambda _8 = 10^{-7}$ and $10^{-5}$ as reference values.",
"Then we find that there exist successful parameters in range $10^{3}~{\\rm GeV} < m_\\eta < 10^{6}~{\\rm GeV}$ that satisfies the observed value $\\eta _B \\simeq 6.1 \\times 10^{-10}$ .",
"Moreover, in order to match the number densities of baryons and DM, we require $|Y_{\\eta +\\eta ^\\dag }| \\simeq |Y_{\\Delta \\eta }|$ , as DM is generated from $\\eta $ and $\\eta ^\\dagger $ decay.",
"It means the symmetric component between $\\eta $ and $\\eta ^\\dagger $ annihilate completely.",
"From this condition, we found that the mediator mass should be lighter than $10^5$ GeV.",
"In conclusion, the successful coincidence can be realized for $10^{3}\\,{\\rm GeV} \\lesssim m_\\eta \\lesssim 10^{5}\\,{\\rm GeV}$ , $\\lambda _8 \\lesssim 10^{-8} \\sqrt{m_\\eta /{\\rm GeV}}$ , and $~10^{-11}\\,{\\rm GeV}\\sqrt{m_\\eta /{\\rm GeV}} \\lesssim \\mu \\lesssim 10^{-10}\\,{\\rm GeV} (m_\\eta /{\\rm GeV})^{3/2}$ .",
"There is a possibility to observe the mediator in this model in accelerator experiments in the near future.",
"This study will be shown elsewhere."
],
[
"Acknowledgments",
"The authors would like to thank Hiroaki Sugiyama for useful comments.",
"This work is supported in part by the Grant-in-Aid for Research Activity Start-up (No.21K20365 [KA]) and Scientific Research (No.18H01210 [KA, JS, and YT], 22K03638, 22K03602, JP20H05852 [MY]) and MEXT KAKENHI Grant (No.18H05543 [KA, JS, and YT]).",
"This work was partly supported by MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849 [MY]."
]
] | 2209.08257 |
[
[
"The kinematics and excitation of infrared water vapor emission from\n planet-forming disks: results from spectrally-resolved surveys and guidelines\n for JWST spectra"
],
[
"Abstract This work presents ground-based spectrally-resolved water emission at R = 30000-100000 over infrared wavelengths covered by JWST (2.9-12.8 $\\mu$m).",
"Two new surveys with iSHELL and VISIR are combined with previous spectra from CRIRES and TEXES to cover parts of multiple ro-vibrational and rotational bands observable within telluric transmission bands, for a total of $\\approx160$ spectra and 85 disks (30 of which are JWST targets in Cycle 1).",
"The general expectation of a range of regions and excitation conditions traced by infrared water spectra is for the first time supported by the combined kinematics and excitation as spectrally resolved at multiple wavelengths.",
"The main findings from this analysis are: 1) water lines are progressively narrower from the ro-vibrational bands at 2-9 $\\mu$m to the rotational lines at 12 $\\mu$m, and partly match a broad (BC) and narrow (NC) emission components, respectively, as extracted from ro-vibrational CO spectra; 2) rotation diagrams of resolved water lines from upper level energies of 4000-9500 K show vertical spread and curvatures indicative of optically thick emission ($\\approx 10^{18}$ cm$^{-2}$) from a range of excitation temperatures ($\\approx 800$-1100 K); 3) the new 5 $\\mu$m spectra demonstrate that slab model fits to the rotational lines at $> 10$ $\\mu$m strongly over-predict the ro-vibrational emission bands at $< 9$ $\\mu$m, implying non-LTE vibrational excitation.",
"We discuss these findings in the context of emission from a disk surface and a molecular inner disk wind, and provide a list of guidelines to support the analysis of spectrally-unresolved JWST spectra."
],
[
"Introduction",
"The inner regions of young protoplanetary disks, within $\\approx $ 5–10 au from the central stars, present the right conditions for molecular chemistry to thrive in a warm disk layer irradiated by stellar and accretion shock radiations [50], [86], [38], [22], [134], [4].",
"The warm molecular layer is then observed through a forest of emission lines at infrared wavelengths from a number of species, especially from CO, H2O, OH, HCN, C2H2, CO2 [31], [99], [112], [109], [47], [82], [92], [14], [17].",
"These molecular spectra have been found to reflect the irradiation, physical, and chemical structure of inner disks through their excitation and kinematics.",
"Line fluxes from different molecules present trends with stellar temperature [99], [92], with stellar and accretion luminosity [113], [23], [14], with the formation of an inner disk dust cavity [113], [14], and with the dust disk mass and radius as observed at millimeter wavelengths [87], [16].",
"The line kinematics, when spectrally-resolved in high-resolution spectra, reveal inner regions of gas depletion [25], [108], [12], a different gas excitation in dust-free vs dust-rich regions [26], [58], [9], variability in line kinematics [24], [17], and kinematic profiles tracing both a disk surface in Keplerian rotation and a slow wind launched from inner disks [97], [28], [17].",
"Infrared molecular spectra are therefore fundamental diagnostics of the physics and chemistry of inner disks at the time of disk dissipation and planet formation.",
"Figure: Overview of near- and mid-infrared water emission bands as observed from space and the ground (see Section ).",
"Top: Earth's atmospheric transmission (in green), with the principal observing bands labelled in orange.",
"Middle: water emission model with a temperature of 700 K and column density of 10 18 10^{18} cm -2 ^{-2}.",
"The coverage of different spectrographs is shown with black boxes to illustrate which parts of the spectrum can be observed: high-resolution (R ∼\\sim 30,000–95,000) spectrographs from the ground can spectrally-resolve relatively weak high-energy lines in small spectral windows, while JWST from space can observe a wider spectral range but only at moderate resolution (R ∼\\sim 1,500–3,700).",
"Bottom: zoomed-in spectral regions to illustrate the difference in resolution between JWST and ground-based data, and including models for OH and CO lines for identification , , .Among the multiple molecules observed at infrared wavelengths, water presents unique opportunities and challenges.",
"Water can be observed from molecular clouds through the phases of star and planet formation all the way to disks and exoplanets [129].",
"Water is also expected to play a major role in planet formation from the dynamics [34], to the accretion of solids [107], to habitability [69], and it is a major driver of exoplanet science today.",
"However, the molecular structure of water, an asymmetric top molecule with three vibrational ($\\nu _1 ~ \\nu _2 ~ \\nu _3$ )for symmetric stretching, bending, and asymmetric stretching modes respectively.",
"and three rotational ($J ~ K_a ~ K_c$ ) quantum numbers, produces a complex rotational and ro-vibrational spectrum that spreads across infrared wavelengths, posing multiple technical challenges in observations (see Section REF ) and in the analysis and interpretation of spectra [84], [66].",
"Observing water and its evolution in disks at the time of planet formation is one of the major drivers of science in the field of planet formation with the James Webb Space Telescope (JWST), which is uniquely suited for water observations thanks to the wide spectral coverage, high sensitivity, and the location outside of Earth's atmosphere.",
"Despite the amount of water spectra from disks collected to date [14], a number of fundamental questions remain: 1) which inner disk region(s) do the infrared water lines trace, and are there multiple water reservoirs (with different temperature and density); 2) what is the water abundance in inner disks and what determines it (chemistry vs dynamics); 3) what is the relative role of different excitation processes, is water emission in LTE; 4) is water present in a molecular inner disk wind; and 5) how can we correctly interpret the complex spectra observed across infrared wavelengths within a unified picture of inner disks?",
"In this work we offer a contribution to this field at a time when the first water spectra are being observed with JWST.",
"We present and analyze spectrally-resolved water emission lines from ro-vibrational and rotational bands at multiple wavelengths between 2.9 and 12.9 $\\mu $ m tracing upper level energies between 4000 and 9500 K, and compare them to the velocity and excitation of ro-vibrational CO spectra observed at 4.5–5.25 $\\mu $ m tracing the same range in upper level energies.",
"The goal of this work is to present a roadmap to support the analysis of both individual and large samples of spectrally-unresolved water spectra from JWST.",
"c c c c c c c c c 0pt Summary of high-resolution water spectra observed in disks with four ground-based instruments.",
"$\\lambda $ Instrument R # lines $E_{u}$ $n_{\\rm {crit}}$ Sample Detections Reference ($\\mu $ m) (K) (cm$^{-3}$ ) (# disks) (# disks) 9 2.9–2.98 CRIRES 95,000 $\\approx 25$ 8,000–9,500 $10^{15}$ –$10^{16}$ 46 16 1 4.52–5.24 iSHELL 60,000 $\\approx 40$ 4,500–9,500 $\\approx 10^{13}$ 60 10 this work, 2 12.23–12.87 VISIR 30,000 2–11 3,200–7,000 $\\approx 10^{11}$ 64 10 this work 12.37–12.46 TEXES 100,000 3–7 3,600–6,000 $\\approx 10^{11}$ 9 7 3 Critical densities $n_{\\rm {crit}}$ = $A_{ul}$ /$C_{ul}$ are estimated using molecular data adopted in RADEX (see Section REF ).",
"As collision temperature, we use the excitation temperatures from Table REF .",
"Since only $E_{u}$ up to 7200 K are included in [45], as collisional rates $C_{ul}$ for the 2.9–2.98 $\\mu $ m lines we take those of lines near 2.7 $\\mu $ m with same range in $A_{ul}$ but $E_{u}$ of 6000–7200 K. $^{1}$ [14], [13], [82], [47], [114]; $^{2}$ [17]; $^{3}$ [88], [111].",
"A few additional high-resolution water spectra have been published using CSHELL and NIRSPEC [33], [112], [37].",
"Water spectra in protoplanetary disks have so far been observed with instruments that fall under two main categories: space-based low-resolution spectrographs (e.g.",
"IRS on Spitzer with maximum resolving power R $\\approx 700$ ), and ground-based high-resolution echelle spectrographs (e.g.",
"CRIRES on the VLT with R $\\approx 95000$ ).",
"Each type of instrument presents specific advantages and disadvantages, which we illustrate in Figure REF .",
"Space-based spectrographs provide wide spectral coverage (e.g.",
"10–37 $\\mu $ m with Spitzer-IRS, and now 4.9–28 $\\mu $ m with JWST-MIRI) but lose kinematic information on the observed emission, and blend together lines from multiple emission/absorption components and different chemical species.",
"Conversely, ground-based spectrographs provide high resolving power that separates the emission from different lines and molecules and reveals the gas kinematics in high detail, but they can only observe in limited spectral windows of high telluric transmission within the $L$ , $M$ , and $N$ bands (Figure REF and Table ).",
"In Figure REF we report a water emission model for quick guidance on the structure of the spectrum across wavelengths.",
"We visually scale the model to approximately match the observed emission in DR Tau just for line identification (in Section REF we will present actual model fits to the data).",
"Purely rotational lines populate a large part of the spectrum at wavelengths longer than 10 $\\mu $ m, while ro-vibrational transitions in the bending and stretching modes present more compact bands at 4–9 $\\mu $ m and 2.5–3.5 $\\mu $ m respectively.",
"Comparison of Earth's transmission to the water emission spectrum in the top two panels in Figure REF demonstrates that ground-based instruments enable observations of the water spectrum only where it is relatively weak (the tails of ro-vibrational bands and a few rotational lines near 12.4 $\\mu $ m).",
"Using observations from space or from the ground, previous work has therefore focused on datasets or samples that were biased one way or the other, resulting in an incomplete view of the distribution and excitation of water in inner disks.",
"Some studies analyzed large samples (up to $\\approx 100$ objects) of spectrally-unresolved rotational water emission observed with Spitzer-IRS at 10–37 $\\mu $ m or Herschel-PACS at 50–200 $\\mu $ m; due to the low resolving power and blending of emission lines, Spitzer spectra only provided degenerate model solutions for the properties (temperature and column density) of multiple molecules and no direct kinematic information on the radial location of emitting regions [32], [113], [92], while Herschel observations provided very low detection rates of a few lines only [106].",
"Other studies analyzed small samples of high-resolution spectra from the ground, mostly biased towards highly accreting disks that had strong water emission in Spitzer spectra; these spectra provided first kinematic information on the water emission but were very limited in sensitivity and by strong telluric absorption [100], [111] and, in the case of the 2.9 $\\mu $ m water band, by absorption in stellar photospheres [14].",
"The properties of water emission as observed in different samples and different wavelengths have ranged from high temperatures in very optically thick regions [37], [114] in the $L$ band, to moderate temperatures and opacity [88], [111] in a narrow window in the $N$ band, down to cooler temperatures at mid- and far-infrared wavelengths [32], [113], [106], [78].",
"Apart from those analyzing Spitzer spectra, these previous results were always based on small samples of 1–10 bright sources.",
"Supporting the scenario of a range of water emitting regions, the combined analysis of spectrally-unresolved mid- and far-infrared water spectra has suggested that water emission should scan disk surfaces from the dust sublimation radius out to 10–100 au [6], [20], [89], where different emission lines probe a range of disk regions and layers depending on their upper level energy and Einstein-$A$ coefficient.",
"However, a unified picture of water in inner disks across different emission bands, including the ro-vibrational lines in the near-infrared, and calibrated on spectrally-resolved water line kinematics is still lacking.",
"JWST now covers both the rotational lines at $>10$ $\\mu $ m ($E_{u} \\approx $ 1000–6000 K) and the ro-vibrational lines from the bending mode ($E_{u} \\approx $ 4000–10000 K) simultaneously and at similar resolving power (R = 1500–3700 across MIRI wavelengths), providing unprecedented leverage on the excitation of water across inner disk radii.",
"The main goal of this paper is to support the analysis of JWST spectra by providing spectrally-resolved line kinematics at multiple wavelengths from a suite of high-resolution ground-based spectrographs (Table ).",
"A comprehensive view of water in disks will require the combination of multiple instruments and multi-wavelength spectra at least until a space-based infrared telescope with high resolving power may be built [101], [67].",
"c l c c c c c 0pt List of prominent H$_2$ O lines covered in this work.",
"Wavelength Transition (upper-lower levels) $A_{ul}$ $E_{u}$ ($\\mu $ m) (level format: $\\nu _1 \\nu _2 \\nu _3~~J _{K_a ~ K_c}$ ) (s$^{-1}$ ) (K) 4 2.90813 001-000 $14_{\\:0\\:14} - 15_{\\:0\\:15}$ 48.4 8341 2.90829 001-000 $11_{\\:4\\:8} - 12_{\\:4\\:9}$ 42.5 8004 2.90911 001-000 $10_{\\:6\\:4} - 11_{\\:6\\:5}$ 31.8 8031 2.90998 001-000 $12_{\\:2\\:10} - 13_{\\:2\\:11}$ 48.8 8177 2.91012 001-000 $13_{\\:2\\:12} - 14_{\\:2\\:13}$ 48.8 8293 2.92726 001-000 $15_{\\:1\\:15} - 16_{\\:1\\:16}$ 49.5 8744 2.92726 001-000 $15_{\\:0\\:15} - 16_{\\:0\\:16}$ 48.9 8744 2.92780 001-000 $12_{\\:3\\:9} - 13_{\\:3\\:10}$ 51.6 8388 2.92912 001-000 $13_{\\:3\\:11} - 14_{\\:3\\:12}$ 48.3 8583 2.92920 001-000 $14_{\\:1\\:13} - 15_{\\:1\\:14}$ 48.7 8698 2.93096 001-000 $11_{\\:6\\:6} - 12_{\\:6\\:7}$ 33.6 8411 2.94665 001-000 $16_{\\:0\\:16} - 17_{\\:0\\:17}$ 48.9 9172 2.94861 001-000 $14_{\\:2\\:12} - 15_{\\:2\\:13}$ 48.9 9012 4.72810 010-000 $9_{\\:7\\:2} - 8_{\\:6\\:3}$ 2.4 5074 4.72942 010-000 $11_{\\:6\\:5} - 10_{\\:5\\:6}$ 2.1 5515 4.78446 010-000 $8_{\\:7\\:2} - 7_{\\:6\\:1}$ 2.7 4757 4.79063 001-000 $10_{\\:6\\:5} - 9_{\\:5\\:4}$ 2.6 5129 4.84063 010-000 $9_{\\:6\\:3} - 8_{\\:5\\:4}$ 2.6 4778 4.88205 010-000 $20_{\\:3\\:18} - 19_{\\:2\\:17}$ 10.6 9179 4.91324 010-000 $22_{\\:2\\:21} - 21_{\\:1\\:20}$ 14.3 9864 4.95058 020-010 $9_{\\:5\\:4} - 8_{\\:4\\:5}$ 4.7 6884 4.96548 010-000 $16_{\\:4\\:13} - 15_{\\:3\\:12}$ 6.7 7329 4.99774 020-010 $6_{\\:6\\:1} - 5_{\\:5\\:0}$ 19.0 6341 5.00798 010-000 $15_{\\:4\\:12} - 14_{\\:3\\:11}$ 6.2 6814 5.06240 010-000 $16_{\\:3\\:14} - 15_{\\:2\\:13}$ 9.6 6975 5.06429 010-000 $16_{\\:2\\:14} - 15_{\\:3\\:13}$ 9.5 6974 5.07548 020-010 $7_{\\:5\\:2} - 6_{\\:4\\:3}$ 5.9 6286 5.10815 010-000 $12_{\\:4\\:9} - 11_{\\:3\\:8}$ 4.2 5425 5.13018 010-000 $11_{\\:4\\:8} - 10_{\\:3\\:7}$ 3.7 5018 5.16471 010-000 $20_{\\:1\\:20} - 19_{\\:0\\:19}$ 18.2 8074 5.16709 010-000 $14_{\\:2\\:12} - 13_{\\:3\\:11}$ 8.6 6019 5.22140 010-000 $15_{\\:1\\:14} - 14_{\\:2\\:13}$ 12.4 6105 5.22307 010-000 $13_{\\:2\\:11} - 12_{\\:3\\:10}$ 8.0 5578 12.2481 000-000 $17_{\\:5\\:13} - 16_{\\:2\\:14}$ 7.8 5795 12.2654 000-000 $18_{\\:7\\:12} - 17_{\\:4\\:13}$ 12.3 6954 12.2772 000-000 $12_{\\:8\\:5} - 11_{\\:5\\:6}$ 0.5 4048 12.2876 000-000 $9_{\\:9\\:1} - 8_{\\:6\\:2}$ 0.02 3202 12.3757 000-000 $16_{\\:4\\:13} - 15_{\\:1\\:14}$ 4.2 4948 12.3962 000-000 $17_{\\:4\\:13} - 16_{\\:3\\:14}$ 7.7 5781 12.4070 000-000 $16_{\\:3\\:13} - 15_{\\:2\\:14}$ 4.2 4945 12.4448 000-000 $11_{\\:8\\:3} - 10_{\\:5\\:6}$ 0.3 3629 12.4535 000-000 $13_{\\:7\\:6} - 12_{\\:4\\:9}$ 1.2 4213 12.8319 000-000 $10_{\\:8\\:2} - 9_{\\:5\\:5}$ 0.2 3243 12.8702 000-000 $10_{\\:8\\:3} - 9_{\\:5\\:4}$ 0.2 3243 Line properties are from HITRAN [52]."
],
[
"Spectra included in this work",
"Table reports the samples of water spectra available from four high-resolution ground-based instruments and some properties of the water emission they cover.",
"Each dataset is described in the next sections instrument by instrument; the data newly presented in this work come from two separate surveys performed with VISIR and iSHELL, while the CRIRES water spectra [14] and TEXES water spectra [88], [111] were published before.",
"All the reduced, science-ready spectra from CRIRES, VISIR, and iSHELL are available on www.spexodisks.com [95].",
"A list of the most prominent water lines covered in these spectra is reported in Table REF .",
"A gallery of portions of the water spectra at multiple wavelengths object by object is included in Appendix .",
"Figure: Example of a CRIRES LL-band spectrum, for the disk of DR Tau.",
"A water emission spectrum is shown in light blue, using the LTE slab model fit described in Section .",
"The prominent feature near 2.935 μ\\mu m is from OH."
],
[
"VLT-CRIRES spectra at 2.9 $\\mu $ m",
"Water ro-vibrational emission spectra at 2.9–2.98 $\\mu $ m are included as observed with the Cryogenic Infrared Echelle Spectrometer [65] on the Very Large Telescope (VLT) as part of a survey from 2007-2008 [97], [28].",
"The water spectra were obtained for $\\approx 50\\%$ of the sample in that survey (35 out of 69 disks, mostly T Tauri stars), and were presented and analyzed in [14], apart for the very different emission with a much larger number of high-excitation lines observed in VV CrA S that is published in [114].",
"Additional spectra for 11 Herbig Ae/Be stars were presented in [47].",
"The spectra were taken with a resolving power of $\\approx 95000$ or $\\approx 3$ km/s, and most of them required correction for stellar photospheric absorption on top of telluric correction.",
"The observed water emission lines were found to be generally as broad as the broad component of CO emission (FWHM up to $\\approx 100$ km/s in some disks), and in a few cases to include also weak narrower central emission similar to the kinematic structure of CO [14].",
"An example $L$ -band spectrum from CRIRES is shown in Figure REF , showing $\\approx $ 15–20 prominent water lines that in some cases are blends of two nearby transitions [14]."
],
[
"IRTF-iSHELL spectra at 5 $\\mu $ m",
"Water ro-vibrational emission spectra at 4.5–5.25 $\\mu $ m are included as observed with iSHELL [103], [104] at the NASA Infrared Telescope Facility (IRTF), as part of an ongoing $M$ -band survey of protoplanetary disks [17].",
"Most spectra were taken with the 0.75\" slit and a resolving power of $\\approx 60000$ or $\\approx 5$ km/s, while the narrower slit with $\\approx 92000$ or $\\approx 3.3$ km/s was used only for the brightest disks (mostly around Herbig Ae/Be stars, but where water is not detected).",
"The iSHELL spectra included here were reduced with Spextool v5.0.3 [35] and have been presented in [17] or obtained in January-July 2022, for a total current sample of 60 disks.",
"The water lines have been discovered in the disk of AS 205 N and reported for the first time in [17], and in this new work we present and analyze all the current detections.",
"An example is presented in Figure REF .",
"In the currently available $M$ -band spectra, a total of $\\approx 40$ ro-vibrational water lines are observed (mostly from $\\nu =1$ and some from $\\nu =2$ , see Table REF ), some of which are blended with CO or HI emission.",
"In this work, we stack up to nine water lines that are not blended with CO to increase S/N in the observed velocity profile; these lines are listed in Table REF .",
"Figure: Example of part of an iSHELL MM-band spectrum, for the disk of FZ Tau.",
"A water emission spectrum is shown in light blue, using the LTE slab model fit described in Section .",
"12 ^{12}CO and 13 ^{13}CO lines are marked in other colors and labelled for reference, but the plot zooms-in on the water lines for better visualization of these and cuts off the peaks of ν=1-0\\nu = 1-0 CO lines.",
"HI emission from the Hu δ\\delta line blended with water near 5.13 μ\\mu m is marked in light green."
],
[
"VLT-VISIR spectra at 12.4 $\\mu $ m",
"Water rotational emission spectra at 12.2–12.9 $\\mu $ m were obtained within a Large Program with the European Southern Observatory (ESO) Very Large Telescope (VLT) Imager and Spectrometer for the Mid-Infrared [74], and are presented and analyzed in this work for the first time.",
"This survey used VISIR at the VLT right after its upgrade [68] and over semesters between August 2016 and August 2018 (program IDs 095.C-0203(A) and 198.C-0104(A,B,C,D,E,F), PI: K. Pontoppidan).",
"A spectrum is shown in Figure REF as an example.",
"The survey included three settings: two echelle settings centered at 12.27 $\\mu $ m and 12.41 $\\mu $ m to cover emission from H$_2$ O lines and the 0-0 S(2) H$_2$ line at 12.2786 $\\mu $ m (and possibly OH), and one long-slit setting centered at 12.84 $\\mu $ m to include the [NeII] line at 12.8136 $\\mu $ m and two water lines with energy 3200 K. The central wavelengths of all settings were set to maximize the number of H$_2$ O emission lines to be observed, a total of eleven (Table REF ).",
"We used the 0.75\" slit providing a resolving power R $\\approx 30,000$ (or 10 km/s).",
"The data reduction is based on a custom pipeline initially made for VISIR 1 data [10] and adapted to the upgraded VISIR.",
"Appendix reports more details on the observations and reduction.",
"The total numbers of spectra obtained in each setting, including multiple epochs and/or slit orientations for a given target, are: 60 spectra in the H$_2$ O setting, 16 spectra in the H$_2$ setting, and 45 spectra in the [NeII] setting.",
"Out of 64 targets, 22 were observed in at least two settings, and 4 targets were observed in all three settings (CrA-IRS2, SCrA, TCrA, VVCrA; see Appendix ).",
"To increase S/N, we have combined spectra from multiple epochs and/or slit orientations obtaining one spectrum per object per setting.",
"The [NeII] detections obtained in this survey have been presented and analyzed in [94]; H2O and [NeII] detections overlap in two disks only, RU Lup and VW Cha, but the lines have very different kinematic structure where [NeII] only traces gas at high blue-shifted velocities of -50 to -200 km/s in these two systems.",
"No H2 detections are reported from this survey, consistent with previous low detection rates [30], [19].",
"Figure: Example of all three spectral settings from the VISIR survey, for the disk of CrA-IRS2 (CHLT 1).",
"Up to 11 water lines are covered, and up to 7 detected in this survey.",
"Water lines near 12.85 μ\\mu m are from levels with upper level energies 3200 K, and could not be corrected from the deep and broad telluric lines (see Appendix ).",
"A water emission spectrum is shown in light blue, using the LTE slab model fit described in Section .",
"The positions of [NeII] and H 2 _2 lines are marked with dotted lines (neither is detected in this spectrum)."
],
[
"Gemini-TEXES spectra at 12.4 $\\mu $ m",
"Additional water rotational emission spectra at 12.2–12.4 $\\mu $ m are included as obtained with TEXES on Gemini [73] with a resolving power of 100,000 or 3 km/s and previously presented in [88] and [111].",
"In this work we only analyze the highest S/N among these spectra, obtained in 4 disks (see Section REF )."
],
[
"Sample",
"The sample included in this work is the combination of samples from the different programs described above as obtained with each instrument.",
"The total is 85 protoplanetary disks around pre-main sequence stars with temperatures of 3000–20000 K and masses of 0.4–5 M$_{\\odot }$ (i.e.",
"including both T Tauri and Herbig Ae/Be stars) in nearby star-forming regions (mostly within $\\approx 200$ pc).",
"Accretion luminosities are in the range of 0.01–200 L$_{\\odot }$ and millimeter disk radii of 10–200 au, including a number of disks with inner dust cavities [29], [49], [16] and some disks in wide multiple systems (See Section REF ).",
"The sample is listed in full in the Appendix and sample properties are visualized in Section REF ."
],
[
"Wide multiple systems",
"There are some wide multiple systems in the sample [102], [81], [90]: AS 205 (separation 1.3\"), DoAr 24E (GSS 31, 2.2\"), S CrA (1.3\"), VV CrA (1.9\"), DK Tau (2.4\"), UY Aur (0.9\"), T Tau (0.7\"), KK Oph (1.6\").",
"In good seeing conditions, the slit was aligned along the system axes so that the main binary components could be extracted.",
"AS 205 A is the brighter component in the N, AS 205 B the fainter component in the S. DoAr 24E A is the brighter component in the N, but it becomes fainter than the B component in the S in the $L$ band and at longer wavelengths [102], [28].",
"Therefore, in our data DoAr 24E B (S) is the brighter component.",
"SCrA A is the brighter component in the NW, SCrA B is the fainter component in the SE; VVCrA A is the brighter component in the S, VVCrA B is the fainter component in the N [118].",
"DK Tau A, UY Aur A, T Tau A, and KK Oph A are the brighter components in the N in each system [81], [90].",
"The first point we address in this work is about the line kinematics and emitting regions of water as spectrally-resolved at multiple wavelengths, addressing questions 1) and 4) listed in Section .",
"Previous work found that 2.9 $\\mu $ m lines include or are dominated by a broad emission component that matches the width of the broad component in CO emission lines in the same systems [14].",
"At 12.4 $\\mu $ m, instead, line velocities were found to be narrower and match the narrow component in CO emission in a few disks where data were available [14], [111].",
"With the new surveys and larger sample included in this work, it is now possible to expand the comparison between CO emission components and water emission lines on a larger number of detections and, for the first time, including water emission in the $M$ band around 5 $\\mu $ m. Figures REF and REF show a comparative analysis of the H2O line profiles using the CO line profiles observed in each object as reference.",
"Line profiles from the fundamental $v=1-0$ lines have been observed with VLT-CRIRES and IRTF-iSHELL [97], [28], [17], and have been decomposed into broad and narrow velocity components as described in [12], [17].",
"The 10 iSHELL spectra where H2O is detected show that the kinematic profile of H2O emission at 5 $\\mu $ m is broader than the NC but narrower than the BC in at least 6/10 cases.",
"In the other 4 cases (DO Tau, DF Tau, RNO 90, and V1331 Cyg), water lines are as broad as the BC.",
"In DF Tau, and tentatively in RNO 90 too, water shows a double-peak shape that matches the width of the BC.",
"In V1331 Cyg, water seems to be as broad as the BC and possibly have a narrower blue-shifted absorption; its near-infrared spectrum is overall more complex due to an uncommonly high excitation of both CO and water [37], and will be analyzed in detail in a future work.",
"Additional tentative detections are found in three more disks (UY Aur A, Elias 24, and HD 35929), which are all included in the plots in the Appendix.",
"The 10 H2O lines detected in the VISIR spectra show that the kinematic profile of H2O emission at 12.4 $\\mu $ m is rather well matched by the narrow CO component in 6/10 cases, within the uncertainty of the much lower pixel sampling and resolution of the VISIR spectra.",
"The comparison between CO and H2O lines is ambiguous in two cases (RU Lup and Elias 20), due to the low S/N of the water spectra.",
"The case of VW Cha seems to show a broader water line with potentially a blue-shifted absorption, which will be discussed below.",
"The case of IRAS 04303 shows a very broad water line, similarly broad as its CO line; in this case, Figure REF shows the complete CO line profile as observed, rather than decomposed into BC and NC as in the other objects, to illustrate the presence of a broad, deep, and blue-shifted absorption component in the CO line [17].",
"Given the higher resolution of TEXES spectra, which matches or exceeds that of CRIRES and iSHELL CO spectra, in Figure REF we show fits to the three water emission lines observed in the four highest quality spectra from [88], [111]Water was detected in three more disks in [111], DoAr 44, HL Tau, and RW Aur, but with too low S/N to apply this analysis., here attempting to reproduce the full line shape with a combination of BC and NC as done for the 2.9 $\\mu $ m lines in [14].",
"The analysis of this small sample confirms that water rotational lines at 12.4 $\\mu $ m match well the shape of the NC, but it also shows that lines include weak broad wings from a broader velocity component similar to the BC.",
"All together, the picture emerging from the kinematics of spectrally-resolved water emission observed between 2.9 and 12.4 $\\mu $ m is that water lines are broader at shorter wavelengths in the ro-vibrational bands and narrower at longer wavelengths in the purely rotational transitions.",
"This suggests that ro-vibrational lines are excited predominantly (or exclusively) in an inner, hotter region as compared to the rotational lines, a region that may match that of the BC in CO.",
"The rotational lines, instead, are dominated by an emission component that matches the NC in CO, at least down to upper level energies of $\\sim 4000$ K as covered in the VISIR and TEXES spectra; lower energy levels at longer wavelengths covered by JWST-MIRI remain to be spectrally-resolved, and might be even narrower by being excited in outer disk regions (See Section ).",
"The fact that water lines at different wavelengths should have different FWHM reflecting excitation from a wide range of emitting regions (broader to narrower from higher to lower level energies) has been the fundamental expectation of model predictions for some time [20], [135], and this multi-wavelength high-resolution dataset directly confirms it for the first time.",
"We will combine the picture emerging from line kinematics to that emerging from line excitation in Section REF .",
"Figure: Emission and absorption components in VW Cha, showing the two water lines at 12.3962 and 12.4070 μ\\mu m (with velocity axis centered on the stronger line).",
"The stellar RV is marked with a dashed grey line.",
"Relative to the stellar RV, the Gaussian component in emission peaks at -7.3 km/s, and the absorption at -14.5 km/s.",
"The absorption line width is consistent with being unresolved (10 km/s).One peculiar case to note in the context of water line profiles and their kinematic structure is that of VW Cha (Figure REF ).",
"The water spectrum observed with VISIR in this object shows a potential narrow absorption line blue-shifted from the peak of the emission line.",
"This emission+absorption structure is observed similarly in both water lines that are covered and detected in this object, and the absorption line is much narrower than the telluric lines (Appendix ); we cannot attribute this feature to any artifact possibly present in the VISIR spectra at the time of observation, and we present it here as a potential first detection of a blue-shifted absorption feature in mid-infrared water spectra from disks.",
"It should be noted that highly blue-shifted [NeII] emission at -150 and -40 km/s has been observed in VW Cha in this same VISIR survey [94], demonstrating the presence of an outflow that could also be linked to the absorption feature in the water lines (although if that is the case, [NeII] would trace an atomic part of the outflow at higher velocities).",
"A unified scenario for emission and absorption in the context of an inner disk wind has been proposed to explain blue-shifted narrow absorption lines that are observed on top of CO emission lines in other disks [97], [17], conditions that are found since earlier stages of disk evolution with additional complexity due to envelopes and multiple absorption components [60].",
"A similar blue-shifted absorption is possibly detected in the $M$ -band water spectrum of V1331 Cyg (Figure REF ), a known source of a large-scale jet and molecular outflow [85], [137].",
"More high-resolution and high-sensitivity observations are required to further investigate these potential absorption components in infrared water spectra.",
"Figure: Overview of line luminosity for CO and water lines between 2.9 and 30 μ\\mu m, as a function of stellar and disk parameters (Section ).",
"Datapoints are classified according to the kinematic shape of the CO line using a line shape parameter SS : “triangular\" lines with S>2.5S > 2.5 associated to disk+wind emission (marked with blue triangles), and double-peak lines with S<2.5S < 2.5 associated to purely Keplerian motion (marked with orange squares).",
"Upper limits are marked with thin symbols."
],
[
"H2O line detections and trends",
"A second point that we investigate in this work is the excitation of water transitions from different energy levels as observed at different wavelengths, addressing questions 1) and 3) from Section .",
"Previous work identified the following trends from spectrally-unresolved water emission observed at 12–17 $\\mu $ m: 1) water detections are much higher in K and M stars as compared to earlier types [99] and lower in M-dwarf disks [92], 2) water detections are lower in disks with inner dust cavities [113], [14], 3) water line fluxes correlate with stellar luminosity in T Tauri stars [113] and, more strongly, with their accretion luminosity [16], 4) water line fluxes anti-correlate with the disk dust radius as spatially-resolved with mm interferometry [16].",
"We provide in Figure REF an overview of these and other trends by considering spectrally-resolved water lines at 2.9, 5, and 12.4 $\\mu $ m, and spectrally-unresolved water lines at 17 and 30 $\\mu $ m (from Spitzer-IRS), including the $M$ -band spectrally-resolved CO lines for reference.",
"The line luminosities in the figure are respectively as measured in: the low-$J$ $\\nu =1-0$ CO lines [17], the three water lines around 2.9285 $\\mu $ m [14], the 5 $\\mu $ m stacked line (Figure REF ), and the few prominent water features around 17.25 $\\mu $ m and 30.7 $\\mu $ m included in the plots in Appendix .",
"In this work we do not re-analyze trends that have already been presented and discussed in previous work, but we find it useful to include the full grid of plots in Figure REF to combine in one place tracers, samples, and trends that have been previously considered only separately, for future reference.",
"In this section, we briefly describe the general trends and any notable differences between different tracers, and remark that a comprehensive view of this kind will be increasingly valuable in future work to correctly interpret individual spectra and small samples from JWST observations within the broader context of the global structure and evolution of molecular gas in inner disks (see guidelines in Section REF ).",
"CO lines provide the highest detection rates in molecular lines tracing inner disks [109], [28], [127], and a fundamental reference to interpret other molecular tracers.",
"Therefore, each datapoint in Figure REF is classified according to the kinematic shape of its CO line using a line shape parameter $S$ = FW10%/FW75% (the ratio of the line width at 10% and 75% of the peak flux): “triangular\" lines with $S > 2.5$ (with broad wings and a much narrower line center, associated to disk+wind emission) and double-peak lines with $S < 2.5$ [17]."
],
[
"Trends in CO luminosity",
"The CO line luminosity $L_{CO}$ measured in this sample shows trends that are sometimes different or even opposite in the two types of lines (top of Figure REF ): in particular, in triangular lines $L_{CO}$ presents a large scatter without an obvious trend as a function of $M_{\\star }$ , while in double-peak lines there is a strong positive correlation above 1 M$_{\\odot }$ .",
"Other notable trends in $L_{CO}$ are with disk inclination, infrared index $n_{13-30}$ , and millimeter dust disk radius [16].",
"The trends observed in $L_{CO}$ are analyzed and discussed in Perez Chavez et al., in prep."
],
[
"Trends in 2.9 $\\mu $ m water luminosity",
"The 2.9 $\\mu $ m water sample was biased toward disks around T Tauri stars with moderate to high accretion rates [14], resulting in perceived high detection rates ($\\sim 45\\%$ , Table ).",
"The relatively small sample in this case does not allow to report strong trends, apart from a general agreement with those observed in CO."
],
[
"Trends in 5 $\\mu $ m and 12.4 {{formula:fd82472e-399a-43e9-98e8-1edb8ae0bdc6}} m water luminosity",
"The samples at 5 $\\mu $ m and 12.4 $\\mu $ m are larger but provided detection rates of only $\\sim 20\\%$ ; despite that, some differences from the trends observed in CO emerge clearly.",
"Water detections are exclusively found in objects that have a triangular CO line shape.",
"These objects have in common a moderate accretion luminosity (0.1-1 L$_{\\odot }$ ), T$_{\\rm {eff}} < 6000$ K, detection of jets and disk winds, and CO emitting from within the dust sublimation radius [17].",
"The upper limits measured in the 5 $\\mu $ m water lines in some disks around intermediate-mass stars appear in stark contrast with their high CO luminosity, suggesting a different C/O ratio in these disks."
],
[
"Trends in 17 $\\mu $ m and 30 {{formula:8bc70792-5dfe-43c6-993b-4346a7d65728}} m water luminosity",
"Water line detections and luminosity at 17 $\\mu $ m and 30 $\\mu $ m, tracing lower-excitation transitions at 2400–3300 K and 1800 K respectively, are still predominantly detected in disks with a triangular line shape.",
"An important difference with the higher-excitation water lines at shorter wavelengths is the detection in a Herbig disk [59], [17] and in a few disks that have an inner dust cavity (DoAr 44, TW Hya, SR 9, SU Aur).",
"Detection of water in Spitzer-IRS spectra of these disks was previously reported in [46], [113], [14].",
"These cases demonstrate that rotationally-cold water emission can still be observed in some disks with dust cavities and disks around Herbig Ae stars (plots of multi-wavelength water spectra are included for reference in Appendix )."
],
[
"General trends in water luminosity",
"Across different energy levels (and wavelengths), the water luminosity is in general higher in T Tauri disks than in Herbig disks.",
"In T Tauri disks, the water luminosity increases with accretion luminosity and with line shape parameter $S$ , and decreases with disk inclination, infrared index $n_{13-30}$ , millimeter disk radius $R_{disk}$ , and radius of CO emission $R_{CO}$ (estimated from the line width at 10%, as a tracer of the inner emitting radius for warm molecular gas) [17].",
"The higher detection rates of water in spectral lines with larger $S$ values (e.g.",
"AS 205 N and DR Tau) as observed at high-resolution, therefore, is not simply due to the higher line-to-continuum contrast.",
"Broader lines with lower $S$ (e.g.",
"in RNO 90) are more blended with nearby lines (e.g.",
"CO, in $M$ -band spectra) and do require a higher S/N per pixel to be detected, implying that their detection rate will be lower.",
"However, two trends suggest an intrinsically decreasing water (and CO) luminosity due to 1) the $S$ parameter itself, where triangular lines with $S > 5$ have a median luminosity that is $\\approx 5$ times higher than triangular lines with $S < 5$ , and 2) the viewing angle, where disks with $incl < 25$ deg have a median luminosity a few times higher than triangular lines with $incl \\approx $ 30–50 deg.",
"Both trends suggest that the geometry of emitting regions as viewed under different inclinations determines the observed properties of molecular spectra [17].",
"Figure: Rotation diagrams of infrared water emission lines, covering mid-infrared rotational lines.",
"Three different models are used to illustrate the effects of different excitation temperatures, line opacity, and LTE/non-LTE excitation: a slab model in LTE , a slab model in non-LTE using RADEX , and a two-dimensional disk structure in LTE using RADLite .",
"See Section for details."
],
[
"Water rotation diagrams",
"The third point we address in this work is the excitation of water spectra at multiple wavelengths, addressing questions 1) and 3) from Section .",
"To do so, we describe the use of rotation diagrams of infrared water emission, which for the first time can be analyzed from spectrally-resolved emission covering upper level energies between 4000 and 9500 K. The rotation diagram technique has been thoroughly described for space applications in [51], but has never been systematically applied to water emission from protoplanetary disks due to the lack of spectrally-resolved line fluxes over a broad range in Einstein-$A$ coefficients $A_{ul}$ and in upper level energies $E_{u}$ .",
"While this is not necessary with linear molecules in the simplest excitation conditions, covering a large range in $E_{u}$ and $A_{ul}$ is essential in the case of an asymmetric top polyatomic molecule like water, especially in circumstellar disks where the emission is not optically thin and non-LTE conditions may sub-thermally excite lines with high $A_{ul}$ [84], [62], [66].",
"Below, we summarize previous explorations by [8] to describe how different conditions (line opacity, LTE/non-LTE excitation) produce distinct features in water rotation diagrams.",
"Rotation diagrams are defined such that, in conditions of optically thin emission at a single excitation temperature and in local thermal equilibrium (LTE), the fluxes from lines at frequency $\\nu $ observed from an unresolved point source form a straight line as [75]: $ln\\left( \\frac{4 \\pi F}{h \\nu g_{u} A_{ul}}\\right) = ln\\left( \\frac{N \\Omega }{Q(T)}\\right) - \\frac{E_{u}}{T} \\, ,$ where $F$ is the integrated line flux, $\\Omega $ is the solid angle $A/d^2$ ($A$ the emitting area in the sky and $d$ the distance to the source), $g_{u}$ is the statistical weight of the upper level, $Q(T)$ is the partition function.",
"The excitation temperature $T$ and the column density $N$ can be easily derived from the slope and intercept of a linear fit using Equation REF .",
"This technique is useful, however, even when the emission becomes optically thick and/or the excitation deviates from LTE.",
"In fact, in these cases the observed molecular line fluxes would produce specific curvatures and/or a large spread of lines departing from the linear behavior given by the LTE optically thin case.",
"Figure: Similar to Figure , but including two more properties: the line intensity proportional to dot sizes, and the line opacity τ\\tau (proportional to A ul A_{ul}) in blue-color scale.",
"Moderate to high column densities produce optically thick emission and increase the spread of lines in the diagram.",
"The model to the right assumes an extreme high column density, to better illustrate the effect of the spread in line opacities.The basic properties of rotation diagrams of infrared water emission are displayed in Figure REF using a slab model in LTE [15], where the free parameters are the temperature $T$ , the column density $N$ , and the emitting area $A$ (which we vary in this figure to align different models on top of each other and highlight the change in shape rather than the offsets between them).",
"In the optically thin case ($N=10^{15}$ cm$^{-2}$ , top left in the figure), water lines form a straight line in the diagram, with slope set by the temperature.",
"When line opacities increase by increasing the column density, lines spread over the diagram following a specific pattern.",
"When a line gets optically thick ($\\tau \\gg 1$ ) it “freezes\" on the diagram, i.e.",
"its intensity is only weakly dependent on the column density.",
"This happens first to those lines with large $A_{ul}$ (the low-energy, bottom left corner of the diagram, see also Figure REF ), as the line opacity is proportional to $A_{ul}$ .",
"The optically thin lines, instead, still follow Equation REF and can rise in the diagram together with $N$ .",
"Therefore, unlike the case of a linear molecule like CO where a curve is produced, an increase in optical depth for the non-linear water molecule spreads lines vertically in the diagram.",
"The most optically thin lines (those with the lowest $A_{ul}$ at any given bin in $E_{u}$ ) define the upper edge of the spread, while the most optically thick ones progressively set the middle and lower edge (Figure REF ).",
"In these conditions, the vertical spread in the diagram is primarily sensitive to the column density.",
"However, as Figure REF shows, line intensities are larger at the center of the diagram, indicating that low-sensitivity spectra may only detect part of the full rotation diagram (as it typically happens with ground-based instruments, see next section).",
"To explore non-LTE conditions, we simulate the emission from a slab of gas using the RADEX code [128], which accounts for the volume density of the collisional partner H$_{2}$ .",
"RADEX uses molecular data from the LAMDA database [116] complemented with data from [121], [45], [39].",
"To illustrate the effects of non-LTE excitation, we adopt $n$ (H$_{2}$ ) = $10^{\\rm 6}$ cm$^{-3}$ , where water emission is not thermalized by collisions [84].",
"We remark that this is not meant to be an exhaustive exploration but rather a quick reference to the LTE case; a more extensive discussion of non-LTE excitation can be found in [84].",
"Overall, the rotation diagram shape produced in non-LTE is different from the LTE case (see Figure REF ): it displays larger spread and/or curvatures, and it mimics steeper slopes, i.e.",
"lower temperatures in a “quasi-thermal\" behavior [62].",
"Figure: Rotation diagram of water emission at multiple wavelengths in a selection of disks with water detections.",
"The last two plots in the bottom are examples of rotationally-cold water detected in Spitzer-IRS spectra.To explore a more realistic disk structure, we adopt the two-dimensional RADLite code [98].",
"RADLite is a raytracer for infrared molecular emission from circumstellar disks, based on the dust temperature and density structure calculated self-consistently using the RADMC code [40].",
"RADLite accounts for dust and gas opacities, and here we set the gas temperature equal to the dust temperature and LTE excitation for simplicity.",
"In modeling the water emission, RADLite takes into account the effects of a snow line set by the temperature and density structure of the disk [20].",
"We set the water abundance to solar oxygen values ($\\approx 10^{-4}$ relative to hydrogen) inward of the snow line (where $T>170$ K), and a low, constant value of $10^{-9}$ per hydrogen outside of it to simulate diffusion and freeze-out onto grains.",
"For illustration, as a reference model we assume a 0.01 $M_{\\odot }$ flared disk around a 1 $M_{\\odot }$ star and a gas-to-dust ratio (GTD) of $10^2$ , to mimic ISM conditions, and as high as $10^4$ to mimic dust settling as in [84].",
"The most distinct feature that can be observed in rotation diagrams of water emission from RADLite, as compared to LTE slab models, is a curvature due to the contribution of a range in temperature and density from different disk radii (Figure REF , plots to the right).",
"Models also show that the gas-to-dust ratio determines the spread of lines in the rotational diagram, by regulating the column of water observable above the dust continuum.",
"Even in this case, the vertical spread in the rotation diagram is linked to the spread in observed line opacities, similarly to slab models, and indicative of the observed water column density.",
"The shape and spread of lines in the rotational diagram of water vapor emission can therefore be useful to generally evaluate the contributions from different effects, even in the case of non-LTE excitation, provided that the observations cover a large enough range in $E_{u}$ and $A_{ul}$ .",
"l c c c c c c c c c c c 0pt Emission properties from LTE slab model fits at different wavelengths.",
"Object 3|c|H$_{2}$ O 2.9 $\\mu $ m 3|c|H$_{2}$ O 5 $\\mu $ m 3|c|H$_{2}$ O 12.4 $\\mu $ m 1|c|CO 5 $\\mu $ m - BC 1|c|CO 5 $\\mu $ m - NC FWHM $T$ log $N$ FWHM $T$ log $N$ FWHM $T$ log $N$ FWHM FWHM (km/s) (K) (cm$^{-2}$ ) (km/s) (K) (cm$^{-2}$ ) (km/s) (K) (cm$^{-2}$ ) (km/s) (km/s) 12 AS 205 N 28 1200 8.2e17 40 1140 3.4e18 23 780 2.0e18 58 16 CrA-IRS 2 – – – 45 970 2.5e18 33 790 2.0e18 60 18 DF Tau (80) – – 100 1880 2.1e18 – – – 113 49 DO Tau (70) – – (70) nc nc – – – 70 19 DR Tau 27 970 1.7e18 22 1160 1.2e18 16 nc nc 33 12 FZ Tau – – – 40 1060 2.5e18 24 730 6.0e16 51 19 IRAS 04303 – – – – – – 100 585 2.3e18 (200) (90) ISO-Oph204 – – – 40 1160 1.1e18 – – – 54 22 RNO 90 – – – (100) nc nc 83 (810) (7.4e18) 110 62 RU Lup (65) – – (50) nc nc (56) nc nc 77 26 VV CrA S – – – – – – 25 840 2.2e18 76 19 VW Cha (80) – – – – – 44 695 6.3e17 72 26 Best-fit slab models are shown for a few examples in Figures REF , REF , and REF .",
"“nc\": model fit not converged, typically for the scarcity of line flux detections.",
"Values in parentheses have an uncertainty larger than 20%.",
"Typical uncertainties for the best-fit results are of the order of 100 K in T and a factor of a few in N. Measurements of CO line widths are given in the last 4 columns for reference to water emission properties (see Figure REF ) and are obtained from iSHELL spectra, apart for VV CrA S and VW Cha that are from CRIRES spectra.",
"The range of CO emitting radii for BC and NC, assuming a purely Keplerian interpretation of the measured FWHM, can be found in [17]."
],
[
"Observed rotation diagrams and LTE slab fits",
"After briefly describing general properties of the technique in the previous section, we now illustrate in Figure REF for the first time rotation diagrams of spectrally-resolved infrared water emission at energies of 4000–9500 K, for the small sample of disks where the emission is detected in this energy range (see Section ).",
"Previous attempts to retrieve line fluxes over a similarly large range were done by de-blending Spitzer-IRS spectra in [8], but were strongly limited by the low resolution of the data and blending between lines from different levels and molecules.",
"Here, we simply measure and include in Figure REF the total flux from $\\sim 20$ lines as observed in Spitzer-IRS spectra to extend the observed rotation diagram down to 1000 K, but we warn the reader that this is done for illustration only and that all the measured fluxes are blends of multiple transitions.",
"This problem will be solved with JWST-MIRI spectra, as discussed below (Section REF ).",
"The largest portion of the observed rotation diagrams is provided by $M$ -band spectra, which cover the largest number of water lines from a single high-resolution spectrum (Section ) and energies between 4000 K and 9500 K. In this work, these lines are observed in the iSHELL spectra, which are flux calibrated using WISE W2 photometry [36].",
"The high-energy end of these lines overlaps with lines extracted from $L$ -band spectra previously observed with CRIRES (flux calibrated using WISE W1 photometry), while the lower-energy end overlaps with the rotational lines observed with VISIR and TEXES (flux calibrated using Spitzer-IRS, where available, or WISE W3 photometry).",
"Overall, all lines overlap well in the rotation diagram apart from the $L$ -band lines, which look under-excited as compared to the $M$ -band lines in all disks; this suggests some difference in excitation between the different ro-vibrational bands (see Section ).",
"Considering line fluxes from different instruments, the observed rotation diagrams show an overall curvature that is reminiscent of what is found from the 2D disk model in Figure REF as due to emission from a range of temperatures at different disk radii.",
"These curvatures could not be seen before in previous work that only covered $< 10$ lines in a narrow portion of the diagram at 4000–6000 K [100], [111], and are now most evident in the $M$ -band spectra alone, as well as in combination with the Spitzer-IRS spectra.",
"In addition to a curvature, lines from the Spitzer-IRS spectra spread vertically as expected in case of large water column densities (Figure REF ).",
"The vertical spread is more visible than in the case of spectra from ground-based instruments, because these have more stringent sensitivity limits and only detect stronger lines close to the upper edge of the rotational spread (Figure REF ).",
"By better de-blending line fluxes at mid-infrared wavelengths and detecting weaker lines with lower $A_{ul}$ , the analysis of JWST spectra will reveal the shape and spread in rotation diagrams in high detail (see Section ).",
"To investigate the excitation of water in different bands and energies, we fit the measured line fluxes with the same LTE slab model adopted above in Section REF .",
"The fit is done on the line fluxes directly by simulating spectra with the resolving power and observed line FWHM from each instrument, and minimizing the chisquare between measured and model line fluxes over the same individual spectral window around each line (the flux included within twice the FWHM of each line).",
"The model includes line opacity and accounts for line blending as observed at the resolution of each instrument, rather than assuming optically thin emission to perform a linear regression fit in the rotation diagram.",
"Figure: Overview of excitation and kinematics of water emission as spectrally-resolved at multiple wavelengths.",
"Top left and bottom: representative LTE slab models from results in Table ; the overall curvature in the observed rotation diagrams (Figure ) can be approximated by a series of slab models fitted to the different wavelengths covered by different instruments (i.e.",
"different ranges of level energies).",
"Non-LTE excitation causes the suppression of ro-vibrational bands as compared to slab model fits to the rotational lines (see Section ).",
"Top right: observed gradient in line widths as a function of level energy for water and CO (see Section for details); grey datapoints at 100–2400 K are model predictions for three water lines in RNO 90 using RADLite, from .",
"Critical densities n crit n_{\\rm {crit}} are reported for reference (see Table ).Table REF reports the fit results as obtained from spectra at different wavelengths, which are illustrated above in a few examples in Figures REF , REF , and REF .",
"We report results in terms of T and N, which are most sensitive to the measured flux ratios between lines of different energy and Einstein-$A$ coefficients (especially in the moderately optically thick conditions found in the results); in terms of emitting area A, the parameter that most relies on the overall flux calibration, we find similar values of $\\approx 0.1$ au$^2$ for the 2.9 $\\mu $ m and 5 $\\mu $ m ro-vibrational lines, and $> 1$ au$^2$ for the 12.4 $\\mu $ m and 16–33 $\\mu $ m rotational lines, indicating widely different emitting areas for different transition bands.",
"The $L$ -band spectrum can be fitted in two disks only, due to the narrow spectral range available, the multiple spectral gaps introduced by telluric and photospheric lines, and severe blending of lines that are typically broad at these wavelengths [14].",
"The best-fit excitation temperature is around 1000 K [82] and the column density about $10^{18}$ cm$^{-2}$ , indicating moderately optically thick emission.",
"Here we use the full line profile as dominated by a broad component, even though in a few cases there could be a weak contribution from a narrow line peak [14].",
"The $M$ -band spectrum can be fitted in most disks where water is detected, but does not converge in case of very broad and very faint emission (DO Tau, RNO 90, RU Lup).",
"The best-fit excitation temperature is around 1100 K and the column density about a few $10^{18}$ cm$^{-2}$ , apart from the case of DF Tau where T is possibly as high as 1900 K. The similar results found from the $L$ -band and $M$ -band spectra suggest that the ro-vibrational bands share a similar excitation, with the stretching mode slightly sub-thermally excited.",
"At 12.4 $\\mu $ m, the fit converges only in spectra where at least 3 lines are detected, while it does not converge in RNO 90 and RU Lup with only 2 lines.",
"The best-fit excitation temperature is overall around 700–800 K and consistent with what has been found in previous work fitting the 12.4 $\\mu $ m lines [88], [111], and the column density about a few $10^{18}$ cm$^{-2}$ , again indicating moderate optical depth.",
"At longer wavelengths observed with Spitzer-IRS, temperatures are lower in the range of $\\approx 450$ K, as found in previous work [113].",
"As a comparison test, we have measured line fluxes at 16–33 $\\mu $ m from the IRS spectrum of DR Tau and find a best fit with T = 430 K and N = 2.6e18 cm-2, a result that is consistent with that found previously in [113]."
],
[
"The emerging picture from spectrally-resolved data",
"The overall picture emerging from this analysis is summarized in Figure REF , by combining results from Section in terms of line excitation and kinematics as spectrally-resolved at multiple wavelengths.",
"LTE slab model fits show that the infrared water spectrum probes moderately optically thick emission from hotter inner regions exciting the higher energy levels to colder outer regions populating the lower energy levels, qualitatively supporting general expectations from previous models (Section ).",
"However, the wide range of observable/observed inner disk water column densities ($10^{17}$ –$10^{20}$ cm$^{-2}$ ) produced by previous modeling works [86], [38], [130], [2], [134] or estimated from spectrally-unresolved observations [32], [113], [78] demonstrates that this parameter, and more fundamentally the water vapor abundance as a function of disk radius, are still critical unknowns to study in future work, especially with MIRI (see guidelines below).",
"The similar column density found in slab fits to the spectra at different wavelengths (a few times $10^{18}$ cm$^{-2}$ ) across the sample in Table REF is therefore particularly interesting: recent models that include chemical heating and UV-shielding propose that this is the column density that ends up being probed in the disk surface due to the combination of the most prominent lines becoming optically thick, and the temperature steeply falling off at higher densities [21].",
"The results presented above suggest that water spectra across infrared wavelengths may trace gas where the right density for excitation is met at any given disk radius, resulting in a similar column density across radii but different excitation temperatures.",
"The overall curvature observed in rotation diagrams (Figure REF ) can be approximated by a series of slab models fitted to the different wavelengths covered by different instruments (i.e.",
"different ranges of energy levels), as reported in Table REF .",
"In the rotation diagram in Figure REF , we show with small dots each model including all transitions between 2.5 and 33 $\\mu $ m, and then highlight with larger dots the stronger transitions that can be observed in the settings covered by each instrument (the color code follows that of Figure REF ).",
"This way, it can be seen that while all models present a large spread of lines due to the moderately large column density (Section REF ), each instrument ends up observing only a fraction of stronger lines close to the upper edge of the rotational diagram (as observed in Figure REF ).",
"The only exception is the rotational lines at $>16$ $\\mu $ m, which can trace a larger part of the vertical spread in the rotation diagram thanks to the wider spectral coverage and higher sensitivity obtained from space.",
"The same slab models are illustrated in the lower part of Figure REF , marking which spectral range each model is actually sensitive to due to the coverage of each instrument.",
"For guidance, we report representative values for $T$ , $N$ , and $A$ for each model from the fits in Section REF .",
"The data demonstrate that no single slab model can reproduce the whole spectrum, an issue that was noticed early on from fits to the rotational lines in Spitzer spectra [32], [113], [78] and that this analysis extends to observations of ro-vibrational bands at $< 10$ $\\mu $ m for the first time."
],
[
"Non-LTE vibrational excitation",
"Further, it now becomes clear that the vibrational excitation is not in LTE: the fits to rotational lines at $> 12$ $\\mu $ m (from VISIR, TEXES, and Spitzer) strongly over-predict the ro-vibrational bands at 2.7 $\\mu $ m and 6.5 $\\mu $ m, that are instead observed to be much weaker (using iSHELL and CRIRES).",
"A similar effect is possibly seen also between the two vibrational bands, where the fit to the bending mode over-predicts the emission in the stretching modes.",
"This effect has already been seen in the excitation of $M$ -band CO emission: while an LTE model can reproduce quite well the CO spectrum in both BC and NC in the $\\nu = 1-0$ lines, it over-predicts the emission observed from the higher vibrational bands, suggesting sub-thermal vibrational excitation [17].",
"Water seems to have a similar excitation: slab models can reproduce quite well the emission as observed over different limited spectral ranges and in different bands (Section REF ), suggesting that the rotational excitation is close to LTE [134], [21], but globally they cannot reproduce the vibrational excitation between the the ground state and higher states, implying non-LTE vibrational excitation.",
"The same conclusion has been reached recently in thermo-chemical modeling of water in inner disks in [21], proposing that observing weaker emission in the 6.5 $\\mu $ m band as compared to the excitation of the rotational lines would be a signature of non-LTE excitation in low-density gas.",
"We should here specify that it is the emitting layer of the rotational lines to be at gas densities lower than the critical density of ro-vibrational bands ($\\approx 10^{13}$ cm$^{-3}$ ), because it is the slab model fit to the rotational lines that strongly over-predicts ro-vibrational emission at $<9$ $\\mu $ m. This is once again consistent with what is observed in CO: it is the NC to show evident sub-thermal excitation, while the BC looks much closer to LTE [17].",
"The higher excitation temperature, larger FWHM, and smaller emitting area found for the ro-vibrational bands suggest that these must be excited in an inner, denser disk region at higher temperature.",
"The excitation of different energy levels and bands, therefore, globally corresponds to the gradient in critical densities: from $10^{13} - 10^{15}$ cm$^{-3}$ for the water ro-vibrational bands and the high-$J$ ($E_{u} = 6000$ –9000 K) $\\nu = 1-0$ CO lines detected in BC, down to $10^{10} - 10^{11}$ cm$^{-3}$ for the water rotational lines near 12 $\\mu $ m and the CO NC at $E_{u} = 3000 - 6000$ K [123], [135].",
"Water lines at longer wavelengths have lower critical densities of $10^{8} - 10^{10}$ cm$^{-3}$ [84].",
"These conditions are in accord with detailed laboratory pump-probe experiments that have determined the excitation/de-excitation of vibrational modes in water (and CO) to be orders of magnitude less efficient than the collisional processes that determine the equilibration of the rotational and translational degrees of freedom [48]."
],
[
"A gradient in FWHM and emitting regions",
"The top right of Figure REF summarizes the picture emerging from the spectrally-resolved line kinematics, using values reported in Table REF .",
"We make this figure in a way to generalize the results to any other disk (see Section below), assuming that the results from this small sample of $\\approx 10$ disks may reflect properties of Class II disks in general.",
"For each line tracer, we report the median value of the sample distribution of the observed FWHM as normalized to the FWHM of the CO BC in each disk.",
"In this way, FWHM differences due to different viewing angles across the sample are removed, putting the figure in the general reference frame of the innermost CO gas observed in any disk.",
"Each tracer in the figure has an horizontal bar that covers the range in energy levels covered at different wavelengths with the different instruments, and a vertical error-bar that is the median absolute deviation of the distribution of values across the sample from Table REF .",
"For reference to the H2O lines, we report also the CO BC (at a value of 1 by definition, by being normalized to itself in each disk), which is typically detected up to $E_{u} \\approx 9000$ K in this sample, and the CO NC, which instead gets weaker more rapidly than BC at higher energy levels and is more representative of levels up to $E_{u} < 6000$ K [17].",
"Figure: Example of a portion of a Spitzer-IRS spectrum (black, upper plot), compared to an LTE slab model of water emission (in light blue) from .",
"The lower plot shows how the same water model would be observed at the resolution of JWST-MIRI, assuming S/N=100; representative models for other molecules previously identified in Spitzer spectra are included for visualization of the improved de-blending between different lines and molecular spectra.As reported above in Section REF , the spectrally-resolved H2O line kinematics approach the FWHM of the BC in the ro-vibrational lines at 2.9 $\\mu $ m and 5 $\\mu $ m and are instead much narrower and approaching the NC in the rotational lines at lower $E_{u}$ , defining the global trend visible in Figure REF .",
"While the rotational lines at $> 13$ $\\mu $ m remain to be spectrally-resolved, we report the expected FWHM values for three lines (again divided by the FWHM of the BC) from levels down to 100 K as reported in [20] using RADLite to model mid- and far-infrared water emission in RNO 90.",
"These previous model predictions, which assumed a radially decreasing temperature profile in a Keplerian disk, show a line FWHM increasing with $E_{u}$ due to higher energy lines tracing progressively hotter inner regions, consistent with the global trend emerging from spectrally-resolved water emission at higher $E_{u}$ in this work.",
"In the rest of this section, we discuss some important implications of this global picture for the analysis of spectrally-unresolved JWST spectra.",
"Figure: Gallery of representative CO line shapes from MM-band spectra of inner disks observed with iSHELL , illustrating the loss of kinematic information and blending of different components when observed at the resolution of MIRI (in black, note how absorption and emission will cancel each other out in MIRI spectra)."
],
[
"JWST-MIRI spectra: opportunities and challenges",
"A total of $\\sim 70$ Class II protoplanetary disks are going to be observed with JWST spectrographs in Cycle 1, most of them with the Mid-Infrared Instrument [105] providing R $\\approx $ 1500–3700 (80–200 km/s) between 4.9 and 28 $\\mu $ m [72].",
"The wide spectral coverage of MIRI includes the ro-vibrational bending mode and rotational lines of water vapor emission in disks for the first time at a similar resolution [115].",
"The higher resolving power of MIRI will de-blend many of the emission features previously observed with Spitzer (see example in Figure REF ).",
"Fundamental expectations from MIRI spectra are therefore a much improved characterization of the water (and OH) spectrum and its excitation across near- and mid-infrared wavelengths, a better characterization of the emission from some organic molecules (although their ro-vibrational bands near 14 $\\mu $ m will still be severely unresolved, see Figure REF ), and the discovery of features from additional molecules previously unidentified in Spitzer spectra.",
"MIRI will not provide, instead, measurements of the gas kinematics and the distinction between double-peak (Keplerian) and triangular line shapes, between BC and NC (where present), and between emission and absorption components observed in inner disks (Figure REF ).",
"At most, MIRI is expected to partially resolve the wings of emission lines that are broader than $\\approx 100$ km/s.",
"By combining advantages and disadvantages, we describe in the following a number of guidelines for future work to support the analysis of space data and propose how combination to ground data may be used to obtain a comprehensive understanding of molecular spectra from inner disks."
],
[
"The emitting regions and excitation of H$_2$ O",
"From the data and analysis included in this and previous work, it is clear that infrared water spectra observed from inner disks trace a range of emitting regions producing different line widths and different excitation (Figure REF ).",
"This picture is naturally expected by disk models (Section REF ) and provided good fits with slab models that include a distribution of temperatures [78], yet the absence of spectrally-resolved observations, the several uncertainties in the physical and thermal structure of inner disks (both radially and vertically), and the possibility of non-LTE excitation made previous thermo-chemical modeling of infrared water spectra challenging and degenerate [84], [66], [6], [134], [21].",
"Without direct support from spectrally-resolved line kinematics, the modeling of MIRI spectra will still have to essentially rely on fitting line fluxes from different $E_{u}$ and $A_{ul}$ to reconstruct the radial and vertical distribution of water vapor across disk radii.",
"The analysis included in this work identifies some useful guidelines: Rotation diagrams of observed water spectra will reveal large spread and curvature across energy levels (Figures REF and REF ), indicative of optically thick emission (current results suggest a column density of a few $\\times 10^{18}$ cm$^{-2}$ ) from a range of excitation temperatures ($\\approx $ 300–1100 K); MIRI spectra will for the first time reveal the full spread of observed rotation diagrams and better measure the observable column density by detecting weak lines with low $A_{ul}$ .",
"Single-temperature slab models in LTE can be expected to provide good fits to JWST spectra over narrow ranges of $E_{u}$ and within the same band (e.g.",
"the ro-vibrational bending mode around 6.5 $\\mu $ m), but the rotational spectrum at $> 10$ $\\mu $ m, which overall spans a very large range in $E_{u}$ from a larger emitting area, will require accounting for gradients in excitation conditions (at least a temperature gradient as a function of disk radius).",
"Using the spectrally-resolved line shape and FWHM from ro-vibrational CO spectra as observed from the ground, which are available or being observed for all JWST targets in Cycle 1, Figure REF can provide general guidance on the expected FWHM for water lines as a function of $E_{u}$ , which models can use to determine the emitting regions in disks.",
"The ro-vibrational bands at 2.7 and 6.5 $\\mu $ m show different temperature and non-LTE excitation as compared to the rotational lines at $> 10$ $\\mu $ m, and trace different disk regions as shown by their different FWHM (Section ); modeling of JWST spectra will need to account for these differences in excitation conditions.",
"The infrared water spectrum can be expected to first approximation to be rotationally in LTE, but not vibrationally, similarly to what observed in CO spectra (Section REF ); detailed analysis of MIRI spectra might reveal more subtle non-LTE effects in the rotational excitation too [84].",
"Obtaining a good model fit to the observed water spectra will be important for the correct analysis of emission from other molecules too, e.g.",
"the organics at 13–15 $\\mu $ m that are blended with water (Figure REF ); future analyses should therefore use models that reproduce the curvature in rotation diagrams, i.e.",
"either a series of slab models or a temperature profile $T(r)$ [78]."
],
[
"CO- and water-rich inner disk winds?",
"The kinematic structure of $M$ -band CO lines observed at high-resolution from most T Tauri disks can be decomposed into two velocity components, BC and NC, that show very different FWHM and vibrational excitation [18], [12], [17].",
"Of these components, NC has been associated to a disk wind due to a number of observed properties, most notably the common presence of blue-shifted asymmetries [28], [17] and an asymmetric spectro-astrometric signal detected in a few disks [97].",
"The similar properties of NC across T Tauri disks, once projection effects at different viewing angles are considered, and the presence of blue-shifted absorption components suggest that NC may trace a slow molecular inner disk wind in T Tauri disks in general [17].",
"The similarity between the profile of NC in CO lines and that of water lines at 12.4 $\\mu $ m (Figures REF and REF ) therefore raises the question whether water may trace the same molecular inner disk wind as CO. MHD disk wind models expect the survival of water and CO in the upper disk layers launching winds from within 10 au from the central star [91], [138], [131], suggesting that part of the molecular spectra observed from inner disks could indeed trace a wind.",
"The narrow absorption tentatively detected in water lines in two disks (Section REF ) may also suggest water-rich outflowing gas, similarly to the absorption observed in CO.",
"The current dataset does not show a correspondence between CO absorption and water absorption in lines at $E_{u} \\approx 4000$ –6000 K (Figure REF and Appendix ), but emission at lower $E_{u}$ is yet to be spectrally-resolved.",
"The quality and number of high-resolution water spectra do not allow a definitive answer yet on water, but a few useful guidelines can be formulated for future work to study molecular inner disk winds at infrared wavelengths: Absorption components, where present, will be blended to emission at the resolution of MIRI (Figure REF ); MIRI spectra will observe, in these situations, weaker emission lines that will be particularly affected in case of deep absorption, as observed in some CO spectra, at the shortest MIRI wavelengths (4.9–5.3 $\\mu $ m).",
"These situations can be expected to be more frequent at high disk inclinations [17]; high-resolution CO spectra from the ground will allow to identify disks with deep absorption, but it is still unclear how much that may affect the water spectra and over which energy levels.",
"In absence of direct measurements of gas kinematics from spectrally-resolved water emission lines in individual disks, models may have to explore the distinction between disk emission and any emission from a wind from the excitation of water and CO spectra, e.g.",
"the inferred density and temperature.",
"Advancements in understanding the properties of molecular inner disk winds in protoplanetary disks [91], [131], [120] may also provide guidance in the future."
],
[
"Measuring H2O/CO ratios in inner disks",
"Another fundamental expectation from the analysis of JWST spectra is the possibility to characterize inner disk chemistry and its connections to planet formation.",
"Models expect that the excitation of infrared molecular lines and bands will reflect chemical signatures (e.g.",
"different C/O ratios) as well as dynamical processes (e.g.",
"inner disk oxygen enrichment from icy pebble drift) in individual as well as ratioed molecular tracers [86], [22], [134].",
"Some trends possibly related to different C/O ratios due to the drift of icy solids have been measured from water and HCN spectra observed with Spitzer [87], [16], but their interpretation is still tentative.",
"A fundamental problem is how to reliably estimate column densities from the observed spectra using disk models, and in turn the elemental C/O ratio [21].",
"MIRI covers both CO and water in a single spectrum (e.g.",
"Figure REF ), avoiding issues with flux calibration and variability that currently affect data obtained from the ground with different instruments (Section ).",
"However, MIRI only covers the high-$J$ levels of CO lines at $>4.9$ $\\mu $ m (P25 and up, or $E_{u} > 4700$ K) and of $^{13}$ CO (P15 and up, or $E_{u} > 3500$ K), in a regime where the rotation diagram is flat and could be misinterpreted as optically thin emission by missing the strong curvature at lower $J$ levels [17].",
"Measuring H2O/CO abundance ratios in inner disks as a function of disk radius will therefore be in principle possible from MIRI spectra, but it requires the identification of CO and H2O lines that trace the same disk region and layer (Figure REF ); gaining a more global view of C/O will however imply the need to reconstruct a radial density profile in CO and H2O as observed at multiple wavelengths.",
"Even here, the available spectrally-resolved data analyzed in this work suggest some guidelines: While ro-vibrational CO lines, when observed at high resolution, can generally be separated into two discrete components that show different kinematics and excitation (BC and NC), infrared water lines seem to show a more continuous range of line shapes and widths that can only partly be matched by BC and NC or their combination (Section REF ), making it hard to identify specific lines from different molecules that trace the same disk region.",
"Nonetheless, by considering the measured FWHM as a function of $E_{u}$ (Figure REF ), rotational water lines with $E_{u} \\approx 4000$ K, close to that of the 12.4 $\\mu $ m lines that almost entirely match the NC line shape (Figure REF ), could be used to empirically measure H2O/CO(NC) ratios and their dependence on different stellar and disk properties and their evolution (Figure REF ).",
"The question remains of which region in a disk (or wind) layer these specific lines would trace.",
"When observing CO in MIRI spectra, it should be kept in mind that only the high-$J$ lines at $E_{u} > 4700$ K are covered, and that these are typically dominated by BC and are not a good match to the rotational water lines at $>10$ $\\mu $ m. To measure empirical H2O/CO ratios from MIRI spectra alone, water lines from the ro-vibrational band at 6.5 $\\mu $ m and similar $E_{u} \\approx 5000$ K may need to be used."
],
[
"Summary & conclusions",
"In this work, we presented the largest dataset of spectrally-resolved water emission from Class II protoplanetary disks collected to date, as observed with multiple high-resolution (R $\\sim $ 30,000–95,000) spectrographs at wavelengths covered by JWST (from 2.9 $\\mu $ m with NIRSpec to 5 $\\mu $ m and 12.4 $\\mu $ m with MIRI), for a total of 85 disks and 160 spectra.",
"We presented an analysis of the kinematics and excitation of water emission from ro-vibrational and rotational lines in different bands observable from the ground, to support the analysis of spectrally-unresolved JWST spectra.",
"A correct interpretation of water spectra from disks requires observations across IR wavelengths, supporting the key role that JWST is going to play in this field.",
"This analysis shows that MIRI spectra will observe significant curvature in rotation diagrams of water emission, by tracing a range of emitting regions in a moderately optically thick upper disk layer with excitation temperatures of 300–1100 K. We provided in Section a list of guidelines to support the analysis of the large number of disk spectra to be observed with JWST, and describe how ground-based high-resolution data will be fundamental to provide the missing kinematic information.",
"The main findings of this work are: ro-vibrational water lines at 2.9 $\\mu $ m and 5 $\\mu $ m ($E_{u}$ of 4500–10000 K) have larger FWHM and higher excitation temperature (1000–1100 K), while rotational lines near 12.4 $\\mu $ m ($E_{u}$ of 4000–6000 K) have narrower FWHM and lower excitation temperature (800 K); the column density that is observed is very similar across all bands (a few $\\times 10^{18}$ cm$^{-2}$ ); the new 5 $\\mu $ m spectra for the first time directly demonstrate that slab model fits to the rotational lines at $> 10$ $\\mu $ m strongly over-predict the ro-vibrational emission bands at $< 9$ $\\mu $ m; this is interpreted as evidence for non-LTE excitation, as discussed in [21]; the picture emerging from combining line kinematics and excitation is in good agreement with the range in critical density for the different bands: 1) an inner, hotter disk region exciting the ro-vibrational water bands and the broad CO component, and 2) a progressively cooler disk surface exciting the rotational lines (with density $< 10^{13}$ cm$^{-3}$ so as to sub-thermally excite the ro-vibrational lines); the relative contribution of disk vs wind emission across the different radial regions remains to be clarified.",
"These results are valid for a small sample of objects where water is observed at high S/N from the ground (Table REF ), which is composed of Class II disk with high accretion and winds.",
"Water spectra observed with JWST, supported with line kinematics from higher-resolution ground-based observations, will enable a comprehensive characterization of excitation conditions and of the density and distribution of water (and other molecules) in inner disks on a larger population.",
"Analyses of large samples of de-blended (or fully spectrally-resolved) molecular spectra will be fundamental to study the properties of molecules in inner disks and translate global trends currently observed in line luminosities (Figure REF ) into trends in physical and chemical conditions that are evolving in inner disks, to inform models of planet formation and disk dispersal.",
"We therefore invite the community to make use of the large database of infrared spectra available on www.spexodisks.com, currently $> 1000$ spectra from 6 spectrographs, to support and complement the analysis of spectrally-unresolved JWST spectra in the coming years.",
"IRTF, VLT, Spitzer mpfit [83] , matplotlib [64] , seaborn [132]"
],
[
"Multi-wavelength spectral plots",
"Figures REF to REF illustrate multi-wavelength water spectra in disks that have CRIRES, iSHELL, and/or VISIR spectra of water emission, as included in this work.",
"Figure: Multi-wavelength water spectra included in this work.",
"H2 and OH emission near 17 μ\\mu m and 30.5 μ\\mu m are marked in pink and grey in the Spitzer spectra.Figure: Same as Figure .Figure: Same as Figure .Figure: Same as Figure , but for disks with inner dust cavities.Figure: Same as Figure , but for disks around Herbig Ae/Be stars."
],
[
"Data reduction",
"With VISIR in echelle mode, the telescope chops off the slit due to the short slit of 4.1\"; the nod B positions are therefore used only for background subtraction.",
"Mechanical oscillations of the grating between sequential chops and nods, and sometimes a difficult guiding on weak sources, introduce small shifts (between fractions of a pixel to a few pixels) in both the spatial and spectral directions on the detector.",
"These are corrected by cross-correlation of the position of one echelle order (for the spatial direction) and telluric lines (for the spectral direction), during combination of individual chops and nods.",
"This procedure ensures the correct combination of the PSF as observed in different nods, and avoids producing an artificial PSF broadening and residuals from the shift of telluric lines.",
"A similar procedure for the spectral direction was previously implemented for VISIR 1 data [30], [10].",
"Observing $N$ -band water spectra from the ground is exceptionally challenging due the variability of Earth's atmospheric conditions, especially in the precipitable water vapor (PWV).",
"Apart from rare exceptional conditions, the sky typically changes significantly between sequential nods and causes residuals from variable telluric lines in the spectra.",
"In this VISIR survey, we carefully monitored the observing conditions nod by nod, and reject individual nods where the sky varied by more than five sigma from the mean sky variation in each observing block.",
"Any remaining sky residuals were removed by averaging the signal over 10–15 pixels at the two sides of the PSF pixel by pixel in the spectral direction, fitting this average with a Savitzky-Golay polynomial smoothing filter, and then subtracting the fit from the PSF before extracting the 1D spectrum.",
"We extensively tested this correction, which in previous work was performed using a simple median subtraction [100], [10], and found that it gives the best results in removing telluric residuals while minimizing noise addition to the spectral PSF.",
"The 1D spectrum is then extracted using optimal extraction.",
"After the detector upgrade in 2016, strong high-frequency fringing was found to affect VISIR high-resolution spectra.",
"In connection to that, the sensitivity was also found to be a factor $\\approx 10$ lower.",
"In April 2017, the entrance window of VISIR was replaced to solve this issue.",
"This replacement removed completely the high-frequency fringing in the echelle mode, and suppressed it in the long-slit mode; however, the low-frequency fringing that was affecting VISIR 1 was still present in all modes (see e.g.",
"Figure REF ).",
"Since VISIR observations do not include flats, but fringing was found to not be stable enough to be removed simply by division with the standard, we fitted and removed fringes in each spectrum after extraction of the 1D spectrum and prior to telluric correction.",
"Some fringe residuals remained when fringes varied between a science target spectrum and its telluric standard spectrum and in the spectra of binary systems, where the two components were observed in a different position on the detector as compared to the telluric standard.",
"Telluric correction of the extracted 1D spectra was performed using observations of several bright asteroids and a few bright stars (Alpha Cen, Alpha Car, Alpha CMa = Sirius, Theta Sco = HR6553).",
"Telluric observations were interspersed during science observations in order to ensure a good match (difference $< 0.1$ ) in airmass and PWV with the science targets.",
"This was achieved in most but not all cases.",
"Differences in both airmass and PWV between science targets and telluric standards are compensated during reduction using telluric water and CO$_2$ lines.",
"We have tested using the airmass and PWV values recorded in fits headers, but these average values do not fully capture real differences and variability happening during observations.",
"We therefore scaled telluric water lines until they match those observed in the science targets to find the best PWV correction, and do the same with a CO$_2$ line to find the best airmass correction.",
"These correction factors were applied to the telluric spectra as an exponential correction of the form $f_{corr} = f_{obs}^{(A_{science}/A_{telluric})}$ as in previous work [100], [10].",
"The science/teluric spectra were then normalized to their continua, aligned by cross-correlation of telluric lines, and then divided after airmass and PWV correction applied to the telluric spectrum.",
"Wavelength calibration of individual spectra was done by cross-correlation in each setting with an atmospheric model set on typical Paranal conditions.",
"We fine-tuned the wavelength solution on low-PWV spectra that allowed us to use all the observed telluric lines out to the detector edges (Figure REF ), and then we applied that solution to all other spectra after shifting it by cross-correlation of sky lines in each spectrum."
],
[
"VISIR 2 performance",
"We measured the spectral resolution of VISIR from unresolved O$_3$ lines detected in telluric absorption spectra.",
"We used several asteroid observations obtained during the survey and consistently measure a FWHM of 10 km/s (corresponding to R $\\sim 30,000$ ) in four O$_3$ lines detected in the 12.84 $\\mu $ m setting.",
"In the other two settings, O$_3$ lines are much weaker and possibly detected only in a few spectra; these measurements are less consistent and provide FWHM in the range 5–13.6 km/s, suggesting a similar resolution of R $\\approx 30,000$ also in echelle mode at 12.2–12.5 $\\mu $ m. In terms of sensitivity, we find that VISIR 2 observations in the echelle and long-slit settings utilized here delivered a similar S/N per pixel (Figure REF ).",
"Using as reference spectra of DR Tau, T Cha, and Sz 73 taken in 2008–2012 [93], [10], the new sensitivity is $\\approx 60\\%$ higher than VISIR 1 observations, after the different pixel sampling has been accounted for (VISIR 2 has a pixel/wavelength larger by 60% than VISIR 1).",
"Figure: VISIR 2 sensitivity estimated from this survey, as compared to previous data from VISIR 1 , .",
"Linear fits to the echelle and long-slit modes are shown with dashed lines.",
"A lower sensitivity is recorded in 2016 before the entrance window was replaced, as well as in brighter sources.Figure: Spectral settings covered in the VISIR survey, shown as extracted from the combined 2D spectra (but before any correction/calibration is applied).",
"The target is CrA-IRS2, which is one of the four targets observed in all three settings.",
"Telluric water absorption is marked with dashed lines, telluric CO 2 _2 absorption is marked with dotted lines.",
"The echelle settings are affected by lower-frequency fringes, the long slit setting by higher-frequency fringes.Figure: Venn diagram showing spectral settings and sample obtained in the VISIR survey included in this work.",
"Detections are shown in boldface for water, and underlined for [NeII].",
"No detections are reported for H2."
],
[
"Sample and line flux measurements",
"Table reports the full sample of 85 disks and the line flux measurements used in Figure REF .",
"Line fluxes are measured from the CRIRES, iSHELL, and VISIR spectra as available from the different surveys (Section ), and as explained in Section REF .",
"Sample properties used in this work have been collected and reported in [17] for most of the sample.",
"We include here references used in that paper, which are also used for the rest of the sample included in this work: distances are from [7]; stellar and accretion properties are from [56], [133], [61], [110], [5], [44], [117]; the infrared index $n_{13-30}$ is measured from Spitzer-IRS spectra as in [16]; WISE fluxes are taken from the AllWISE Data Release [36]; references for $R_{\\rm {NIR}}$ , the stellar RV, and the disk inclination are: [76]; [53], [122], [54], [55]; [43], [42]; [71]; [3], [41]; [44], [11]; [77], [126], [136]; [57]; [119]; [1], [26]; [79], [80]; [63], [70]; [71]; [96], [124]; [125]; [97]; [27].",
"l c c c c c c c 0pt Water line flux measurements.",
"Name $S$ $F$ (2.9 $\\mu $ m) err(2.9 $\\mu $ m) $F$ (5 $\\mu $ m) err(5 $\\mu $ m) $F$ (12.4 $\\mu $ m) err(12.4 $\\mu $ m) 8 51Oph – – – – – -3.02e-16 5.66e-17 AATau 1.8 1.42e-15 1.42e-15 – – – – ABAur 3.6 – – 3.73e-17 2.31e-16 – – AS205N 6.8 7.33e-14 2.71e-14 8.73e-15 1.12e-16 6.35e-15 6.25e-16 AS209 6.4 – – 1.58e-15 3.46e-15 9.06e-16 1.16e-15 CITau 4.8 – – 3.77e-15 9.91e-15 – – CQTau 3.0 – – 3.21e-16 6.20e-16 3.85e-16 9.74e-16 CrA-IRS2 6.9 – – 7.17e-15 9.23e-17 2.65e-15 4.10e-16 CWTau 2.7 2.30e-15 2.30e-15 6.95e-16 3.73e-15 1.55e-15 6.51e-15 DFTau 2.8 4.53e-14 5.89e-15 1.24e-14 1.90e-16 – – DKTauA 7.6 – – 1.32e-15 4.41e-15 – – DKTauB 15.1 – – -3.44e-16 2.86e-14 – – DoAr24EN – – – – – -9.42e-16 1.10e-15 DoAr24ES 6.4 5.68e-15 5.68e-15 8.12e-16 4.34e-15 5.74e-16 1.05e-15 DoAr44 3.5 3.82e-15 1.01e-14 5.66e-18 5.72e-15 – – DOTau 13.4 2.32e-14 1.86e-15 8.17e-15 2.74e-16 – – DRTau 5.1 2.82e-14 4.24e-15 4.65e-15 1.01e-16 8.43e-15 4.36e-16 EC82 – 1.39e-15 2.99e-15 – – -2.20e-16 4.90e-15 Elias20 5.6 – – 2.50e-16 1.08e-14 8.27e-15 7.24e-15 Elias24 4.6 – – 3.03e-15 5.00e-15 – – EXLup 11.4 3.26e-15 1.54e-16 – – 3.85e-15 3.08e-14 FNTau 4.0 2.27e-17 4.54e-17 – – – – FZTau 5.4 – – 1.45e-14 1.87e-16 1.43e-14 1.02e-15 GQLup 2.4 – – 1.32e-15 8.90e-15 -9.79e-16 5.34e-15 HD100546 – – – – – 1.70e-16 1.06e-15 HD101412 – – – – – 6.08e-16 2.79e-15 HD135344B 5.2 2.43e-15 9.60e-16 6.28e-17 1.12e-15 – – HD141569 2.0 – – 7.34e-17 1.13e-15 – – HD142666 1.3 – – 2.35e-17 4.98e-16 – – HD143006 1.6 – – -1.34e-18 7.55e-16 – – HD144432S 2.8 – – – – -2.73e-16 1.08e-15 HD144668 2.1 – – 1.69e-16 6.36e-16 2.76e-16 1.16e-15 HD150193 1.4 – – 3.02e-17 5.18e-16 3.86e-16 4.40e-16 HD163296 3.3 – – 6.83e-16 4.32e-17 3.60e-17 7.95e-16 HD169142 2.5 – – 3.99e-18 6.69e-16 – – HD179218 1.7 – – 7.92e-18 2.25e-16 – – HD190073 4.3 – – 1.93e-17 1.23e-16 – – HD259431 2.4 – – 5.25e-17 1.09e-15 – – HD35929 1.9 – – 1.72e-15 1.79e-15 -3.32e-15 7.06e-15 HD36917 1.8 – – 1.14e-16 1.76e-15 – – HD37806 1.4 – – 1.92e-17 5.75e-16 – – HD97048 – – – – – 3.48e-16 1.17e-14 HTLup – 3.60e-15 3.60e-15 – – – – IMLup 2.9 9.00e-17 9.00e-17 – – – – IRAS04303 2.5 – – 1.55e-15 4.00e-15 1.38e-14 8.42e-16 IRS48 1.5 – – 2.33e-16 1.38e-15 – – ISO-Oph204 5.3 – – 1.33e-14 1.76e-16 – – KKOphA 7.5 – – -7.50e-18 1.55e-15 – – KKOphB – – – -3.27e-15 9.50e-15 – – LkHa330 5.2 – – 1.85e-18 8.74e-16 – – MWC297 – – – 8.41e-17 1.07e-16 – – MWC480 4.1 – – 9.96e-18 9.02e-16 – – MWC758 5.8 – – 5.18e-17 7.54e-16 – – PDS156 3.6 – – 1.46e-17 2.38e-15 – – PDS520 9.0 – – 1.59e-16 5.04e-15 – – RCrA – – – – – 4.05e-18 2.10e-16 RNO90 3.2 2.61e-14 3.40e-15 4.88e-15 1.09e-16 6.74e-15 5.96e-16 RULup 6.3 4.27e-14 5.99e-15 8.35e-15 2.27e-16 2.62e-15 3.44e-16 RWAur – – – – – -6.09e-15 4.82e-15 RYLup 8.0 – – 5.66e-18 5.20e-15 – – RYTau 3.8 – – 5.13e-16 1.77e-15 – – SCrAN 7.3 8.68e-14 2.69e-14 – – 4.09e-16 7.65e-16 SCrAS – 6.46e-14 2.00e-14 – – 9.37e-16 3.90e-15 SR21 2.3 1.75e-16 1.80e-16 2.79e-16 8.88e-16 – – SR4 3.9 – – 8.38e-16 5.72e-15 – – SR9 3.9 – – 6.07e-16 6.87e-15 – – SUAur 3.8 – – -1.20e-18 1.10e-15 – – TCrA – – – -8.77e-17 9.01e-16 3.08e-16 8.11e-16 TTauN 6.1 1.27e-13 1.02e-13 – – – – TTauS – 9.78e-14 7.83e-14 – – – – TWCha 4.6 4.91e-15 3.80e-17 – – – – TWHya 2.8 2.07e-16 2.10e-16 2.42e-20 1.46e-15 – – UYAurA 9.7 – – 7.44e-15 2.29e-16 – – UYAurB 3.7 – – 6.73e-16 6.73e-15 – – V1331Cyg 10.6 – – 3.49e-15 1.75e-16 – – V1647Ori – – – – – 8.21e-16 1.77e-15 V853Oph – – – – – 4.65e-15 1.50e-14 V892Tau 3.8 – – 9.67e-17 4.02e-16 – – VVCrAN – – – – – 9.65e-16 8.90e-16 VVCrAS 11.0 – – – – 1.56e-15 1.14e-16 VVSer 1.8 – – 4.17e-17 5.46e-16 -1.17e-15 1.09e-15 VWCha 5.9 2.07e-14 1.09e-15 – – 1.80e-14 1.25e-15 VZCha 4.0 7.36e-15 8.48e-17 – – – – WaOph6 3.7 2.67e-14 1.87e-15 7.77e-16 7.30e-15 5.34e-16 2.89e-15 WXCha 2.7 4.60e-15 1.53e-16 – – 1.92e-14 2.41e-14 Line fluxes are in units of erg s$^{-1}$ cm$^{-2}$ .",
"The CO line shape parameter $S$ is reported in the second column for reference [17]."
]
] | 2209.08216 |
[
[
"Solutions of the Variational Equation for an nth Order Boundary Value\n Problem with an Integral Boundary Condition"
],
[
"Abstract In this paper, we discuss differentiation of solutions to the boundary value problem $y^{(n)} = f(x, y, y^{'}, y^{''}, \\ldots, y^{(n-1)}), \\; a<x<b,\\; y^{(i)}(x_j) = y_{ij},\\; 0\\leq i \\leq m_j, \\; 1 \\leq j \\leq k-1$, and $y^{(i)}(x_k) + \\int_c^d p y(x)\\;dx = y_{ik}, \\;0 \\leq i \\leq m_k,\\;\\sum_{i=1}^km_i=n$ with respect to the boundary data.",
"We show that under certain conditions, partial derivatives of the solution $y(x)$ of the boundary value problem with respect to the various boundary data exist and solve the associated variational equation along $y(x)$."
],
[
"Introduction",
"Our concern is characterizing partial derivatives with respect to the boundary data of solutions to the $n$ th order nonlocal boundary value problem $y^{(n)} = f\\left(x, y, y^{^{\\prime }}, y^{^{\\prime \\prime }}, \\ldots , y^{(n-1)}\\right), \\; a<x<b$ satisfying $\\begin{array}{c}y^{(i)}\\left(x_j\\right) = y_{ij},\\; 0\\le i \\le m_j, \\; 1 \\le j \\le k-1, \\\\y^{(i)}\\left(x_k\\right) + \\displaystyle \\int _c^d p y(x)\\;dx = y_{ik}, \\;0 \\le i \\le m_k\\end{array}$ where and throughout $k,n\\in \\mathbb {N}$ with $2 \\le k \\le n,\\; m_1, \\ldots , m_k\\in \\mathbb {Z}^+$ such that $\\sum _{i=1}^km_i=n,$ and $a < x_1 < x_2 < \\cdots < x_k <c<d <b,p\\in \\mathbb {R}$ .",
"Differentiation of solutions of initial value problems with respect to initial conditions has been a well-known result in the field of differential equations for a long time.",
"In his book [10], Hartman attributes the theorem and proof to Peano.",
"Hence, the result is commonly referred to as a theorem of Peano.",
"These derivatives solve the associated variational equation to the differential equation.",
"Subsequently, similar results were obtained for boundary value problems and relied heavily upon the continuous dependence of solutions of boundary value problems on boundary conditions.",
"The continuous dependence result utilizes a map of initial conditions to boundary conditions and the Brouwer Invariance of Domain Theorem.",
"Results for boundary value problems on differential equations with standard boundary conditions may be found in [11], [26], [27], [28], [29].",
"Direct analogues also exist for difference equations [17] and dynamic equations on times scales [1].",
"The mathematics community has added a parameter to the nonlinearity [14], [15], [25].",
"Researchers have also produced results for various types of boundary conditions including nonlocal [2], [9], [18], [13], [19], [21], [23], [22], functional [4], [8], [5], [6], [7], and integral [3], [24].",
"In this paper, we extend the results of [20] to an $n$ th order differential equation using the procedure outlined in [12].",
"The general idea is to use continuous dependence to write the solution of the boundary value problem as the solution to an initial value problem.",
"After multiple applications of the Mean Value Theorem, we can apply Peano's theorem directly to the problem at hand.",
"The remainder of this paper is organized as follows.",
"In section two, we present the boundary value problem and define its associated variational equation.",
"We also introduce five hypotheses that are imposed upon the differential equation along with Peano's Theorem and the continuous dependence result.",
"Our boundary value problem with integral condition analogue is found in section three."
],
[
"Assumptions and Background Theorems",
"We establish a few conditions that are imposed upon (REF ): $f\\left(x,y_1, \\ldots ,y_n\\right): (a,b) \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ is continuous, $\\frac{\\partial f}{\\partial y_i} \\left(x,y_1, \\ldots ,y_n\\right): (a,b) \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ is continuous, $i = 1,\\ldots ,n,$ solutions of initial value problems for (REF ) extend to $(a,b).$ Remark 2.1 Note that (iii) is not a necessary condition but lets us avoid continually making statements about maximal intervals of existence inside $(a,b)$ .",
"Next, the results discussed rely upon the definition of the variational equation which we present here.",
"Definition 2.1 Given a solution $y(x)$ of (REF ) and for $i=1,2,\\ldots ,n,$ we define the variational equation along $y(x)$ by $z^{(n)} = \\sum _{i=1}^n\\frac{\\partial f}{\\partial y_i} \\left(x,y,y^{\\prime },\\ldots ,y^{(n-1)}\\right) z^{(i-1)}.$ Our aim is an analogue of the following theorem that Hartman [10] attributes to Peano for (REF ), (REF ).",
"Theorem 2.1 [A Peano Theorem] Assume that, with respect to (REF ), conditions (i)-(iii) are satisfied.",
"Let $x_0 \\in (a,b)$ and $y(x):= y\\left(x, x_0, c_0 ,c_1,\\ldots ,c_{n-1}\\right)$ denote the solution of (REF ) satisfying the initial conditions $y^{(i)}\\left(x_0\\right) = c_i,\\; 0\\le i\\le n-1.$ Then, for each $0\\le j\\le n-1$ , $\\alpha _j(x) := \\frac{\\partial y}{\\partial c_j}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $y(x)$ satisfying the initial conditions $\\alpha _j^{(i)}\\left(x_0\\right) = \\delta _{ij}, \\; 0\\le i \\le n-1.$ $\\beta (x):= \\frac{\\partial y}{\\partial x_0}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $y(x)$ satisfying the initial conditions $\\beta ^{(i)}\\left(x_0\\right) = -y^{(i)}\\left(x_0\\right), \\; 0\\le i \\le n-1.$ $\\frac{\\partial y}{\\partial x_0} (x) = -\\sum _{i=0}^{n-1} y^{(i)}(x_0) \\frac{\\partial y}{\\partial c_i} (x).$ The next condition guarantees uniqueness of solutions of (REF ), (REF ) and is a nonlocal analogue of $(m_1,\\ldots ,m_k)$ -disconjugacy.",
"If, for $0\\le i\\le m_j-1, \\; 1\\le j\\le k-1,$ $y^{(i)}\\left(x_j\\right) = z^{(i)}\\left(x_j\\right),$ and, for $0\\le i\\le m_k-1,$ $y^{(i)}\\left(x_k\\right) + \\int _c^d p y(x)\\;dx = z^{(i)}\\left(x_k\\right) + \\int _c^d pz(x)\\;dx,$ where $y(x)$ and $z(x)$ are solutions of (REF ), then, on $(a,b),$ $y(x) \\equiv z(x).$ The last condition provides uniqueness of solutions of (REF ) along all solutions of (REF ) and again is a nonlocal analogue of $(m_1,\\ldots ,m_k)$ -disconjugacy.",
"Given a solution $y(x)$ of (REF ), if, for $0\\le i\\le m_j-1, \\; 1\\le j\\le k-1,$ $u^{(i)}\\left(x_j\\right)=0,$ and, for $0\\le i\\le m_k-1,$ $u^{(i)}\\left(x_k\\right) + \\int _c^d p u(x)\\;dx = 0,$ where $u(x)$ is a solution of (REF ) along $y(x),$ then, on $(a,b)$ , $u(x) \\equiv 0.$ We also make use of the following continuous dependence result for boundary value problems.",
"A typical proof may be found in [16].",
"Theorem 2.2 [Continuous Dependence on Boundary Conditions] Assume (i)-(iv) are satisfied with respect to (REF ).",
"Let $y(x)$ be a solution of (REF ) on $(a,b)$ .",
"Then, there exists a $\\delta > 0$ such that, for $\\left|x_j - t_j\\right| < \\delta ,\\; 1\\le j\\le k,$ $\\left|c-\\xi \\right|<\\delta ,\\; \\left|d-\\Delta \\right|<\\delta ,\\;\\left|p-\\rho \\right|<\\delta ,$ $\\left|y^{(i)}\\left(x_j\\right) - y_{ij} \\right| < \\delta ,\\; 0\\le i\\le m_j-1, \\; 1\\le j\\le k-1,$ and $\\left|y^{(i)}\\left(x_k\\right)+ \\int _c^d p y(x)\\;dx - y_{ik}\\right| < \\delta , \\; 0\\le i\\le m_k-1,$ there exists a unique solution $y_{\\delta } (x)$ of (REF ) such that $y^{(i)}_{\\delta }\\left(t_j\\right) = y_{ij},\\; 0\\le i\\le m_j-1, \\; 1\\le j\\le k-1,$ $y_{\\delta }^{(i)}\\left(t_k\\right) + \\int _{\\xi }^{\\Delta } \\rho y_{\\delta }(x)\\;dx = y_{ik},\\; 0\\le i\\le m_k-1,$ and, for $0\\le i\\le n-1,$ $\\lbrace y_\\delta ^{(i)}(x)\\rbrace $ converges uniformly to $y^{(i)}(x)$ as $\\delta \\rightarrow 0$ on $[\\alpha ,\\beta ]\\subset (a,b)$ ."
],
[
"Analogue of Peano's Theorem",
"In this section, we present our analogue to Theorem REF stated in five parts.",
"Theorem 3.1 Assume conditions (i)-(v) are satisfied.",
"Let $u(x)=\\\\u(x, x_1,\\ldots ,x_k,y_{01},\\ldots ,y_{m_k-1,k},p,c,d)$ be the solution of (REF ) on $(a,b)$ satisfying $u^{(i)}\\left(x_j\\right)=y_{ij}, \\; 0\\le i\\le m_j-1, \\; 1\\le j\\le k-1,$ and $u^{(i)}\\left(x_k\\right)+\\int _c^d pu(x)dx=y_{ik},\\; 0\\le i\\le m_k-1.$ Then, for each $1\\le l\\le k-1$ and $0\\le r\\le m_l-1,\\; Y_{rl}(x):=\\frac{\\partial u}{\\partial y_{rl}}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $u(x)$ satisfying the boundary conditions $&Y^{(i)}_{rl}\\left(x_j\\right) = 0, \\; 0 \\le i \\le m_j -1,\\; 1 \\le j \\le k-1,\\; j \\ne l \\\\&Y^{(i)}_{rl}\\left(x_l\\right) = 0,\\;0 \\le i \\le m_l - 1,\\; i \\ne r \\\\&Y^{(r)}_{rl}\\left(x_l\\right) = 1 \\\\&Y^{(i)}_{rl}\\left(x_k\\right) + \\int _c^dpY_{rl}(x)dx = 0,\\; 0 \\le i \\le m_k - 1,$ and for $0\\le r \\le m_k-1,\\; Y_{rk}:=\\frac{\\partial u}{\\partial y_{rk}}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $u(x)$ satisfying the boundary conditions $&Y^{(i)}_{rk}\\left(x_j\\right) = 0,\\; 0 \\le i \\le m_j -1,\\; 1 \\le j \\le k-1, \\\\&Y^{(i)}_{rk}\\left(x_k\\right) + \\int _c^dpY_{rk}(x)dx = 0,\\; 0 \\le i \\le m_k - 1, \\; i\\ne r,\\\\&Y^{(r)}_{rk}\\left(x_k\\right) + \\int _c^dpY_{rk}(x)dx = 1,$ for each $1\\le l\\le k-1,\\; X_{l}(x):=\\frac{\\partial u}{\\partial x_{l}}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $u(x)$ satisfying the boundary conditions $&X^{(i)}_{l}\\left(x_j\\right) = 0,\\;0\\le i\\le m_j-1,\\; 1 \\le j \\le k-1,\\; j \\ne l \\\\&X^{(i)}_{l}\\left(x_l\\right) = -u^{(i+1)}(x_l),\\; 0 \\le i \\le m_l - 1, \\\\&X^{(i)}_{l}\\left(x_k\\right) + \\int _c^dpX_{l}(x)dx = 0,\\; 0 \\le i \\le m_k - 1,$ and $X_k:=\\frac{\\partial u}{\\partial x_k}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $u(x)$ satisfying the boundary conditions $&X^{(i)}_{k}\\left(x_j\\right) = 0,\\; 0\\le i\\le m_j-1,\\;1 \\le j \\le k-1, \\\\&X^{(i)}_{k}\\left(x_k\\right) + \\int _c^dpX_{k}(x)dx = -u^{(i+1)}\\left(x_k\\right),\\; 0 \\le i \\le m_k - 1.$ $C(x):=\\frac{\\partial u}{\\partial c}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $u(x)$ satisfying the boundary conditions $&C^{(i)}\\left(x_j\\right) = 0,\\; 0\\le i\\le m_j-1,\\;1 \\le j \\le k-1, \\\\&C^{(i)}\\left(x_k\\right) + \\int _c^dpC(x)dx = -pu(c),\\; 0 \\le i \\le m_k - 1.$ $D(x):=\\frac{\\partial u}{\\partial d}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $u(x)$ satisfying the boundary conditions $&D^{(i)}\\left(x_j\\right) = 0,\\; 0\\le i\\le m_j-1,\\;1 \\le j \\le k-1, \\\\&D^{(i)}\\left(x_k\\right) + \\int _c^dpD(x)dx = pu(d),\\; 0 \\le i \\le m_k - 1.$ $P(x):=\\frac{\\partial u}{\\partial p}(x)$ exists on $(a,b)$ and is the solution of the variational equation (REF ) along $u(x)$ satisfying the boundary conditions $&P^{(i)}\\left(x_j\\right) = 0,\\; 0\\le i\\le m_j-1,\\;1 \\le j \\le k-1, \\\\&P^{(i)}\\left(x_k\\right) + \\int _c^dpP(x)dx = -\\int _c^du(x)dx,\\; 0 \\le i \\le m_k - 1.$ We only prove part (a) as the proofs of (b)-(e) follow similarly.",
"Fix integers $1 \\le l\\le k-1$ and $0\\le r\\le m_l-1.$ We consider $Y_{rl}(x)=\\frac{\\partial u}{\\partial y_{rl}}(x).$ Since the argument for the case of $Y_{ik}(x)=\\frac{\\partial u}{\\partial y_{ik}},\\;\\;0\\le i\\le m_k-1,$ is similar, we omit its proof.",
"To ease the burdensome notation and realizing that all boundary data are fixed except $y_{rl}$ , we denote $u(x, x_1,\\ldots , x_k,\\\\ y_{01}, \\ldots , y_{rl}, \\ldots , y_{m_k -1, k},p,c,d)$ by $u(x, y_{rl})$ .",
"Let $\\delta > 0$ be as in Theorem REF with $0 \\le |h| \\le \\delta $ , and define the difference quotient for $y_{rl}$ by $Y_{rlh}(x) = \\frac{1}{h}\\left[u\\left(x, y_{rl} + h\\right) - u\\left(x, y_{rl}\\right)\\right].$ First, we inspect the boundary conditions for $Y_{rlh}$ .",
"Note that for every $h \\ne 0$ and $0 \\le i \\le m_j - 1,\\; $ $1 \\le j \\le k-1,\\; j\\ne l,$ $Y_{rlh}^{(i)}\\left(x_j\\right) &= \\frac{1}{h}\\left[u^{(i)}\\left(x_j, y_{rl} + h\\right) - u^{(i)}\\left(x_j, y_{rl}\\right)\\right] \\\\&= \\frac{1}{h}\\left[y_{ij} - y_{ij}\\right] \\\\&= 0,$ for every $0 \\le i \\le m_l - 1,\\;i \\ne r$ $Y_{rlh}^{(i)}\\left(x_l\\right) &= \\frac{1}{h}\\left[u^{(i)}\\left(x_l, y_{rl} + h\\right) - u^{(i)}\\left(x_l, y_{rl}\\right)\\right]\\\\&= \\frac{1}{h}\\left[y_{il} - y_{il}\\right]\\\\&= 0,$ and $Y_{rlh}^{(r)}\\left(x_l\\right) &= \\frac{1}{h}\\left[u^{(r)}\\left(x_l, y_{rl} + h\\right) - u^{(r)}\\left(x_l, y_{rl}\\right)\\right]\\\\&= \\frac{1}{h}\\left[y_{rl} + h - y_{rl}\\right]\\\\&= 1.$ Finally, for every $0 \\le i \\le m_k -1$ $Y_{rlh}^{(i)}\\left(x_k\\right) + \\int _c^d pY_{rlh}(x)dx &= \\frac{1}{h}\\left[u^{(i)}\\left(x_k, y_{rl} + h\\right) - u^{(i)}\\left(x_k, y_{rl}\\right)\\right.\\\\&+\\left.\\int _c^d p\\left(u(x,y_{rl}+h)- u(x,y_{rl})\\right)dx\\right]\\\\&= \\frac{1}{h}\\left[y_{ik} - y_{ik}\\right] \\\\&= 0.$ Next, we show that $Y_{rlh}(x)$ is a solution of the variational equation.",
"To that end, for $m_l \\le i \\le n-1$ , let $\\mu _i = u^{(i)}\\left(x_l, y_{rl}\\right)$ and $\\nu _i = \\nu _i(h) = y^{(i)}\\left(x_l, y_{rl} + h\\right) - \\mu _i$ Note by Theorem REF , for $m_l\\le i\\le n-1, \\;\\nu _i = \\nu _i (h) \\rightarrow 0$ as $h\\rightarrow 0.$ Using the notation of Theorem REF for solutions of initial value problems for (REF ), viewing $u(x)$ as the solution of an initial value problem at $x_l,$ and denoting this solution as an IVP, i.e.",
"$u(x) = y\\left(x,x_l,y_{0l},\\ldots ,y_{m_l-1,l},\\mu _{m_l},\\ldots ,\\mu _{n-1}\\right)$ , we have $Y_{rlh}(x) = \\frac{1}{h}[ &y(x, x_l, y_{0l}, \\ldots , y_{rl} + h, \\ldots , y_{m_l -1,l}, \\mu _{m_l} + \\nu _{m_l}, \\mu _{m_l + 1} + \\nu _{m_l + 1}, \\ldots , \\mu _{n-1} + \\nu _{n-1}) \\\\&- y(x, x_l, y_{0l}, \\ldots , y_{rl}, \\ldots , y_{m_l -1,l}, \\mu _{m_l}, \\mu _{m_l + 1}, \\ldots , \\mu _{n-1})].$ Next, by utilizing telescoping sums to vary only one component at a time, we have $Y_{rlh}(x) =& \\frac{1}{h}[y(x, x_l, y_{0l},\\dots ,y_{rl} + h, \\ldots ,\\mu _{m_l} + \\nu _{m_l}, \\mu _{m_l + 1} + \\nu _{m_l + 1}, \\ldots ,\\mu _{n-1} + \\nu _{n-1})\\\\&- y(x, x_l, y_{0l},\\ldots ,y_{rl},\\ldots , \\mu _{m_l} + \\nu _{m_l}, \\mu _{m_l + 1} + \\nu _{m_l + 1}, \\ldots , \\mu _{n-1} + \\nu _{n-1})\\\\&+ y(x, x_l, y_{0l},\\ldots ,y_{rl},\\ldots , \\mu _{m_l} + \\nu _{m_l}, \\mu _{m_l + 1} + \\nu _{m_l + 1}, \\ldots , \\mu _{n-1} + \\nu _{n-1}) \\\\&- y(x, x_l, y_{0l},\\ldots ,y_{rl}, \\ldots , \\mu _{m_l}, \\mu _{m_l + 1} + \\nu _{m_l + 1}, \\ldots , \\mu _{n-1} + \\nu _{n-1}) \\\\ &+ - \\cdots \\\\ &+ y(x, x_l, y_{0l},\\ldots ,y_{rl},\\ldots , \\mu _{m_l}, \\mu _{m_l + 1}, \\, \\mu _{n-1} + \\nu _{n-1}) \\\\ &+ y(x, x_l, y_{0l},\\ldots , y_{rl},\\ldots , \\mu _{m_l}, \\ldots , \\mu _{n- 1})].$ By Theorem REF and the Mean Value Theorem, we obtain $Y_{rlh}(x) &= \\alpha _r(x; y(x, x_l, y_{0l},\\ldots ,y_{rl} + \\bar{h}, \\ldots ,\\mu _{ml} + \\nu _{ml}, \\ldots , \\mu _{n - 1} + \\nu _{n-1}))\\\\ &+ \\frac{\\nu _{m_l}}{h}\\alpha _{ml}(x;y(x; x_l, y_{0l},\\ldots ,y_{rl},\\ldots , \\mu _{m_l} + \\bar{\\nu }_{m_l},\\mu _{m_l + 1} + \\nu _{m_l + 1}, \\ldots , \\mu _{n-1} + \\nu _{n-1}))\\\\&+ \\cdots \\\\&+ \\frac{\\nu _{n-1}}{h}\\alpha _{n-1}(x;y(x, x_l, y_{0l}, \\ldots ,\\mu _{m_l}, \\mu _{m_l + 1}, \\ldots , \\mu _{n-1} + \\bar{\\nu }_{n-1})),$ where for $0 \\le j \\le n-1,$ $\\alpha _j(x; y(\\cdot ))$ is the solution of the variational equation (REF ) along $y(\\cdot )$ satisfying $\\alpha ^{(i)}_j\\left(x_l\\right) = \\delta _{ij}, \\; 0 \\le i \\le n-1.$ Furthermore, $y_{rl} + \\bar{h}$ is between $y_{rl}$ and $y_{rl} + h$ , and for each $m_l\\le i\\le n-1$ , $\\mu _i + \\bar{\\nu _i}$ is between $\\mu _i$ and $\\mu _i + \\nu _i$ .",
"Note that we use $y(\\cdot )$ to simplify the notation.",
"Thus, to show $\\displaystyle \\lim _{h \\rightarrow 0} Y_{rlh}$ exists, it suffices to show, for $m_l \\le i \\le n-1,$ $\\displaystyle \\lim _{h \\rightarrow 0}$ $\\frac{\\nu _i}{h}$ exists.",
"Recall that $&Y_{rlh}^{(i)}\\left(x_j\\right) = 0, \\; 0 \\le i \\le m_j - 1,\\; 1 \\le j \\le k-1, \\;j \\ne l, \\\\&Y_{rlh}^{(i)}\\left(x_k\\right) + \\int _c^d pY_{rlh}(x)dx = 0, \\;0 \\le i \\le m_k - 1.$ Hence, by substituting into the equations above and solving each for $\\alpha _r,$ we create a system of $n-m_l$ equations with $n-m_l$ unknowns $-\\alpha _r^{(i)}\\left(x_j; y(\\cdot )\\right) = \\frac{\\nu _{m_l}}{h}\\alpha ^{(i)}_{m_l}\\left(x_j; y(\\cdot )\\right)& + \\cdots + \\frac{\\nu _{n-1}}{h}\\alpha ^{(i)}_{n-1}\\left(x_j; y(\\cdot )\\right),\\\\& 0\\le i\\le m_j-1,\\;1\\le j\\le k-1,\\;j\\ne l$ and $-\\alpha _r^{(i)}\\left(x_k; y(\\cdot )\\right) &- \\int _c^d p\\alpha _r\\left(x;y(\\cdot )\\right)dx= \\frac{\\nu _{m_l}}{h}\\alpha ^{(i)}_{m_l}\\left(x_k; y(\\cdot )\\right)+ \\int _c^d p\\alpha _{m_l}(x;y(\\cdot ))dx \\\\&+ \\cdots + \\frac{\\nu _{n-1}}{h}\\alpha ^{(i)}_{n-1}\\left(x_k; y(\\cdot )\\right) + \\int _c^d p\\alpha _{n-1}(x;y(\\cdot ))dx,\\;0\\le i\\le m_k-1.$ In the system of equations above, we notice that $y(\\cdot )$ is not always the same.",
"Therefore, we consider the matrix along $y(x).$ $ M := \\begin{pmatrix}\\alpha _{m_l}(x_1; y(x)) & \\alpha _{m_l+1}(x_1;y(x)) & \\cdots & \\alpha _{n-1}(x_1; y(x))\\\\\\alpha ^{^{\\prime }}_{m_l}(x_1; y(x)) & \\alpha ^{^{\\prime }}_{m_l+1}(x_1; y(x)) & \\cdots & \\alpha ^{^{\\prime }}_{n-1}(x_1; y(x))\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\alpha _{m_l}^{(m_1 -1)}(x_1; y(x)) & \\alpha ^{(m_1 -1)}_{m_l+1}(x_1; y(x)) & \\cdots & \\alpha ^{(m_1 -1)}_{n-1}(x_1; y(x))\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\alpha _{m_l}^{(m_{l-1} -1)}(x_{l-1}; y(x)) & \\alpha ^{(m_{l-1} -1)}_{m_l+1}(x_{l-1}; y(x)) & \\cdots & \\alpha ^{(m_{l-1} -1)}_{n-1}(x_{l-1}; y(x))\\\\\\alpha _{m_l}(x_{l+1}; y(x)) & \\alpha _{m_l + 1}(x_{l+1}; y(x)) & \\cdots & \\alpha _{n-1}(x_{l+1}; y(x))\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\end{pmatrix}\\alpha _{m_l}(x_k; y(x)) & \\alpha _{m_l+1}(x_k; y(x)) & \\cdots & \\alpha _{n-1}(x_k; y(x))\\\\+ \\int _c^d p\\alpha _{m_l}(x;y(x))dx & +\\int _c^d p\\alpha _{m_l + 1}(x;y(x))dx & \\cdots & +\\int _c^d p\\alpha _{n-1}(x;y(x))dx\\\\$ $\\vdots $ $\\vdots $ $\\vdots $ ml(mk -1)(xk; y(x)) ml + 1(mk -1)(xk; y(x)) n-1(mk -1)(xk; y(x)) + cd pml (x;y(x))dx + cd pml + 1(x;y(x))dx + cd pn-1(x;y(x))dx $$ We claim that $\\det (M) \\ne 0$ .",
"Suppose to the contrary that $\\det (M) = 0$ .",
"Then, there exists a linear combination of the column vectors with scalars $p_i\\in \\mathbb {R},\\;m_l\\le i\\le n-1$ such that at least one $p_i$ is nonzero $p_{m_l} \\begin{pmatrix}\\alpha _{m_l}(x_1; y(x))\\\\\\alpha ^{^{\\prime }}_{ml}(x_1; y(x))\\\\\\vdots \\\\\\alpha _{m_l}^{(m_{l-1} -1)}(x_{l-1}; y(x))\\\\\\alpha _{m_l}(x_{l+1}; y(x))\\\\\\vdots \\\\\\alpha _{m_l}^{(m_k -1)}(x_k; y(x) \\\\ + \\int _c^d p\\alpha _{m_l}(x;y(x))dx\\end{pmatrix} + \\cdots + p_{n-1} \\begin{pmatrix}\\alpha _{n-1}(x_1; y(x))\\\\\\alpha ^{^{\\prime }}_{n-1}(x_1; y(x))\\\\\\vdots \\\\\\alpha _{n-1}^{(m_{l-1} -1)}(x_{l-1}; y(x))\\\\\\alpha _{n-1}(x_{l+1}; y(x))\\\\\\vdots \\\\\\alpha _{n-1}^{(m_k -1)}(x_k; y(x)) \\\\ + \\int _c^d p\\alpha _{n-1}(x;y(x))dx\\end{pmatrix} = \\begin{pmatrix}0 \\\\0\\\\\\vdots \\\\0\\\\0\\\\\\vdots \\\\0\\\\\\end{pmatrix}.$ Set $w(x; y(x)) := p_{m_l}\\alpha _{m_l}(x; y(x)) + \\cdots + p_{n-1}\\alpha _{n-1}(x;y(x)).$ Then by Theorem REF , $w(x; y(x))$ is a nontrivial solution of (REF ), but $w^{(i)}(x_j; y(x)) = 0,\\; 0 \\le i \\le m_j -1, \\;1 \\le j \\le k-1,\\; j\\ne l$ and $w^{(i)}(x_k; y(x)) + \\int _c^d pw(x; y(x))dx= 0,\\;0\\le i\\le m_k-1.$ When coupled with hypothesis (v), we have $w(x; y(x)) \\equiv 0$ .",
"Since each alpha function is not identically zero, $p_{m_l} = p_{m_{l+1}} = \\cdots = p_{n-1} = 0$ which is a contradiction to the choice of $p_i$ 's.",
"Hence, $\\det (M) \\ne 0$ implying $M$ and, subsequently by Theorem REF , $M(h)$ have inverses.",
"Here, $M(h)$ is the appropriately defined matrix from the system of equations using the correct $y(\\cdot )$ .",
"Therefore, for each $m_l \\le i \\le n-1$ , we can solve for $\\frac{\\nu _i}{h}$ by using Cramer's Rule.",
"and suppressing the arguments of each $\\alpha $ : $\\frac{\\nu _i}{h} &=\\frac{1}{M(h)} \\times \\\\ &\\begin{vmatrix}\\alpha _{m_l} & \\cdots & \\alpha _{i-1} & -\\alpha _r & \\alpha _{i+1} & \\cdots & \\alpha _{n-1} \\\\\\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\\alpha _{m_l} + \\int p\\alpha _{m_l}& \\cdots & \\alpha _{i-1} + \\int p\\alpha _{i - 1} & -\\alpha _r - \\int p\\alpha _{r}& \\alpha _{i+1} + \\int p\\alpha _{i + 1} & \\cdots & \\alpha _{n-1} + \\int p\\alpha _{n-1}\\end{vmatrix}$ Note as $h \\rightarrow 0$ , $\\det (M(h)) \\rightarrow \\det (M),$ and so, for $1 \\le i \\le n-1 $ , $\\nu _i(h)/h \\rightarrow \\det (M_i) / \\det (M) := B_i$ as $h \\rightarrow 0$ , where $M_i$ is the $n- m_l \\times n- m_l$ matrix found by replacing the appropriate column of the matrix $M$ by $\\textup {col}\\Big [&-\\alpha _r(x_1;y(x)),\\ldots ,-\\alpha _r^{(m_1-1)}(x_1;y(x)),\\ldots ,-\\alpha _r(x_{l-1};y(x)),\\ldots ,-\\alpha _r^{(m_{l-1}-1)}(x_{l-1};y(x)),\\\\&-\\alpha _r(x_{l+1};y(x)),\\ldots ,-\\alpha _r^{(m_{l+1}-1)}(x_{l+1};y(x)),\\ldots ,-\\alpha _r(x_k;y(x))-\\int _c^d p\\alpha _r(x;y(x))dx,\\\\&\\ldots ,-\\alpha _r^{(m_k-1)}(x_k;y(x))-\\int _c^d p\\alpha _r(x;y(x))dx\\Big ].$ Now, let $Y_{rl}(x) = \\displaystyle \\lim _{h \\rightarrow 0}Y_{rlh}(x)$ , and note by construction $Y_{rl}(x) = \\frac{\\partial y}{\\partial y_{rl}}(x)=\\frac{\\partial u}{\\partial y_{rl}}(x).$ Futhermore, $Y_{rl}(x) = \\lim _{h \\rightarrow 0} Y_{rlh}(x) =\\alpha _r\\left(x;u(x)\\right)+\\sum _{i=m_l}^{n-1}B_i\\alpha _i\\left(x;u(x)\\right)$ which is a solution of the variational equation (REF ) along $u(x)$ .",
"In addition, $&Y_{rl}^{(i)}\\left(x_j\\right) = \\lim _{h \\rightarrow 0}Y^{(i)}_{rlh}\\left(x_j\\right) = 0, \\; 0 \\le i \\le m_j-1, \\; 1 \\le j \\le k-1,j\\ne l,\\\\&Y_{rl}^{(i)}\\left(x_l\\right) = \\lim _{h \\rightarrow 0}Y_{rlh}^{(i)}\\left(x_l\\right) = 0, \\; 0 \\le i \\le m_j-1, i\\ne r,\\\\&Y_{rl}^{(r)}\\left(x_l\\right) = \\lim _{h \\rightarrow 0}Y^{(i)}_{rlh}\\left(x_l\\right) = 1,\\\\&Y_{rl}^{(i)}\\left(x_k\\right) + \\int _c^d p Y_{rl}(x)\\;dx = \\lim _{h \\rightarrow 0} \\left[Y_{rlh}^{(i)}\\left(x_k\\right) + \\int _c^d Y_{rlh}(x)\\;dx\\right] = 0,\\;0 \\le i \\le m_k -1.$ Finally, we note that similar to part (c) of Peano's theorem, the solutions found in (a)-(e) of the main result may be written as various combinations of one another due to the dimensionality of the solution space.",
"We refer the reader to Corollary 4.1 in [23] for an example."
]
] | 2209.08164 |
[
[
"Cell Attention Networks"
],
[
"Abstract Since their introduction, graph attention networks achieved outstanding results in graph representation learning tasks.",
"However, these networks consider only pairwise relationships among nodes and then they are not able to fully exploit higher-order interactions present in many real world data-sets.",
"In this paper, we introduce Cell Attention Networks (CANs), a neural architecture operating on data defined over the vertices of a graph, representing the graph as the 1-skeleton of a cell complex introduced to capture higher order interactions.",
"In particular, we exploit the lower and upper neighborhoods, as encoded in the cell complex, to design two independent masked self-attention mechanisms, thus generalizing the conventional graph attention strategy.",
"The approach used in CANs is hierarchical and it incorporates the following steps: i) a lifting algorithm that learns {\\it edge features} from {\\it node features}; ii) a cell attention mechanism to find the optimal combination of edge features over both lower and upper neighbors; iii) a hierarchical {\\it edge pooling} mechanism to extract a compact meaningful set of features.",
"The experimental results show that CAN is a low complexity strategy that compares favorably with state of the art results on graph-based learning tasks."
],
[
"Introduction",
"Graph Neural Networks (GNNs) find applications in a plethora of fields, like computational chemistry [1], social networks [2] and physics simulations [3].",
"Since their introduction [4], [5], GNNs have shown remarkable results in learning tasks when data are defined over a graph domain, where the flexibility of neural networks is coupled with prior knowledge about data relationships, expressed in terms of the underlying topology.",
"The literature on GNNs is large and numerous techniques have been studied, usually categorized in spectral [6], [7] and non-spectral methods [8], [9], [10].",
"Generally speaking, the idea is learning representations of node attributes using local aggregation, where the neighborhood is formally represented by the graph topology.",
"By leveraging this basic but powerful idea, outstanding performance has been achieved in many traditional tasks such as classification for nodes or entire graphs [7], [11], [8] or link prediction [12] as well as more specialized ones such as protein folding [13] and neural algorithmic reasoning [14], [15].",
"At the same time, a major performance boost to deep learning algorithms has been offered by the inclusion of attention mechanisms, introduced to handle sequence-based tasks [16], enabling for variable sized inputs, to concentrate on their most important features.",
"Then, pioneering works introduced graph attention networks [11], [17], [18] achieving state-of-the-art results in most of the aforementioned tasks.",
"Graphs can be also seen as a simple instance of a topological space, able to capture pairwise interactions through the presence of an edge between any pair of directly interacting nodes.",
"However, despite their overwhelming popularity, graph-based representations are unable to consistently model higher order relations, which play a crucial role in many practical applications.",
"Examples where multiway relations cannot be reduced to an ensemble of pairwise relations are gene regulatory networks [19], [20], where some reactions occur only when a set of multiple (not only two) genes interact, or applications in network neuroscience [21], [22].",
"To overcome this problem, many architectures defined on general hypergraphs have been proposed [23], [24], [25], [26], [27], and some works incorporating attention mechanisms for hypergraphs neural networks have been published [28], [29].",
"However, in the meanwhile, pioneering works on Topological Signal Processing [30], [31] demonstrated the benefit of processing signals defined on simplicial and cell complexes, which are specific examples of hyper-graphs with a rich algebraic description, and can easily encode multi-way relationships hidden in the data in a low-complexity fashion.",
"Consequently, there was a natural interest in the design of (deep) neural networks architectures able to learn from data defined on simplicial and cell complexes, as summarized in the following state of the art."
],
[
"Related Works",
"Despite Topological Deep Learning is an emerging research area that has been introduced quite recently, numerous pioneering works appeared in this field.",
"In [32], message passing neural networks (MPNNs) [1] are adapted to simplicial complexes (SCs), and a Simplicial Weisfeiler-Lehman (SWL) colouring procedure for distinguishing non-isomorphic SCs is introduced.",
"The aggregation and updating functions are able to process the data exploiting lower and upper neighbourhoods.",
"The architectures in [32], namely Message Passing Simplicial Networks (MPSNs), are a generalization of Graph Isomorphism Network (GIN) [33].",
"In [34], message passing neural networks able to handle data defined over regular cell complexes are introduced under the name of CW Networks (CWNs), and they are proven to be not less powerful than the 3-WL test.",
"The work in ( [35]) introduced Neural Sheaf Diffusion Models, neural architectures grounded in the theory of cellular sheaves able to improve learning performance on graph-based tasks especially in heterophilic settings.",
"In [36], a novel attention neural architecture that operates on data defined on simplicial complexes leveraging masked self-attention layers is introduced, taking into account lower and upper neighbourhoods and introducing a proper technique to process the harmonic component of the data based on the design of a sparse projection operator.",
"A similiar architecture is proposed in [37], which re-weights the interactions between neighbouring simplices through an orientation equivariant attention mechanism.",
"However, none of these works considered masked self-attention mechanisms for architectures designed to handle data defined over cell complexes.",
"Finally, the work in [38] introduced a broad class of attentional architectures operating on generalized higher-order domains called Combinatorial Complexes."
],
[
"Contribution",
"The aim of this paper is to introduce cell attention networks, i.e., a fully attentional low-complexity architecture able to learn from data defined over the nodes of a graph by incorporating the graph within a cell complex and working at the edge-level in order to extract higher order interactions.",
"The idea is to implement an attention mechanism that handles relations among nearby edges, where the neighborhood is formally represented by a cell complex.",
"To this aim, we exploit a hierarchical approach that lifts up node features to derive edge features, and then it computes the attention coefficients between nearby edges to find the optimal combination of edge features.",
"In particular, we devise an attentional lift mechanism that learns the data over the edges of the complex leveraging a self-attention mechanism over the features of the vertices that are on their boundaries.",
"Consequently, our architecture is also equipped with a cellular lifting map algorithm that embeds the graph domain into a regular cell complex (CW).",
"Other graph lifting techniques have been used in previous works, such as clique complexes [39], [40], or more sophisticated structures based on incidence tensors [41].Finally, please notice that the proposed architecture presents nice features of explainability due to its fully attention-driven design: by simply inspecting the attention coefficients, it is possible to understand the contribution of the single cells for the learning task.",
"For instance, in a computational chemistry task, the upper attention coefficients may represent the importance of the interaction between two atoms inside the rings of a molecule.",
"Also, considering a network traffic problem, the pooling attention coefficients may tell us which links can be considered obsolete, or even detect communities in social networks."
],
[
"Cell Complexes",
"In this section we recall the basics of regular cell complexes, which are topological spaces provided with a rich algebraic structure that enables an efficient representation of high-order interaction systems, and we explain their relation with usual graphs $\\mathcal {G}=(\\mathcal {V},\\mathcal {E)}$ .",
"In particular, we will first introduce the definition of a regular cell complex, then we will describe few additional properties enabling the representation of cell complexes via boundary operators.",
"Definition 1 (Regular cell complex) [42], [34].",
"A regular cell complex is a topological space $\\mathcal {C}$ together with a partition $\\lbrace \\mathcal {X}_{\\sigma }\\rbrace _{\\sigma \\in \\mathcal {P}_{\\mathcal {C}}}$ of subspaces $\\mathcal {X}_{\\sigma }$ of $\\mathcal {C}$ called $\\mathbf {cells}$ , where $\\mathcal {P}_{\\mathcal {C}}$ is the indexing set of $\\mathcal {C}$ , such that For each $c$ $\\in $ $\\mathcal {C}$ , every sufficient small neighborhood of $c$ intersects finitely many $\\mathcal {X}_{\\sigma }$ ; For all $\\mathcal {\\tau }$ ,$\\mathcal {\\sigma }$ we have that $\\mathcal {X}_{\\tau }$ $\\cap $ $\\overline{\\mathcal {X}}_{\\sigma }$ $\\ne $ $\\varnothing $ iff $\\mathcal {X}_{\\tau }$ $\\subseteq $ $\\overline{\\mathcal {X}}_{\\sigma }$ , where $\\overline{\\mathcal {X}}_{\\sigma }$ is the closure of the cell; Every $\\mathcal {X}_{\\sigma }$ is homeomorphic to $\\mathbb {R}^{n_{\\sigma }}$ for some $n_{\\sigma }$ ; For every $\\sigma $ $\\in $ $\\mathcal {P}_{\\mathcal {C}}$ there is a homeomorphism $\\phi $ of a closed ball in $\\mathbb {R}^{{n}_{\\sigma }}$ to $\\overline{\\mathcal {X}}_{\\sigma }$ such that the restriction of $\\phi $ to the interior of the ball is a homeomorphism onto $\\mathcal {X}_{\\sigma }$ .",
"Condition (2) implies that the indexing set $P_{\\mathcal {C}}$ has a poset structure, given by $\\tau $ $\\le $ $\\sigma $ iff $\\mathcal {X}_{\\tau }$ $\\subseteq $ $\\overline{\\mathcal {X}_\\sigma }$ .",
"This is known as the face poset of $\\mathcal {C}$ .",
"The regularity condition (4) implies that all topological information about $\\mathcal {C}$ is encoded in the poset structure of $P_{\\mathcal {C}}$ .",
"Then, a regular cell complex can be identified with its face poset.",
"Definition 2 (k-skeleton).",
"A k-skeleton of a cell complex $\\mathcal {C}$ , denoted $\\mathcal {C}^{(k)}$ , is the subcomplex of $\\mathcal {C}$ consisting of cells of dimension at most k. From Definition 1 and Definition 2, it is trivial to check the 0-skeleton of a cell complex is the set of vertices $\\mathcal {C}^{(0)} = \\mathcal {V}$ and the 1-skeleton is the underlying graph $\\mathcal {C}^{(1)}=\\mathcal {G}(\\mathcal {V},\\mathcal {E)}$ ; we refer to 2-cells as polygons and, in general, there is little interest with dimensions above two.",
"Regular cell complexes can be described via an incidence relation (boundary relation) with a reflexive and transitive closure that is consistent with the partial order introduced in Definition 1.",
"The boundary relation describes which cells are on the boundary of other cells.",
"Definition 3 (Boundary relation).We have the boundary relation $\\sigma $ $\\prec $ $\\tau $ iff $\\dim ({\\sigma })$ $\\le $ $\\dim ({\\tau })$ and there is no cell $\\delta $ such that $\\sigma $ $\\le $ $\\delta $ $\\le $ $\\tau $ .",
"We can use the previous definitions to define the four types of (local) adjacencies present in cell complexes, following the approach from [34]: Definition 4 (Cell complex adjacencies) [34].",
"For a cell complex $\\mathcal {C}$ and a cell $\\sigma \\in \\mathcal {P}_{\\mathcal {C}}$ , we define: The boundary adjacent cells $\\mathcal {B}(\\sigma )$ $=$ $\\lbrace \\tau $ $|$ $\\tau \\prec \\sigma \\rbrace $ , are the lower-dimensional cells on the boundary of $\\sigma $ .",
"For instance, the boundary cells of an edge are its vertices and the boundary cells of a polygon are its edges.",
"The co-boundary adjacent cell $\\mathcal {CB}(\\sigma )$ $=$ $\\lbrace \\tau $ $|$ $ \\sigma \\prec \\tau \\rbrace $ , are the higher-dimensional cells with $\\sigma $ on their boundary.",
"For instance, the co-boundary cells of a vertex are the edges having that vertex as an endpoint and the co-boundary of an edge are the polygons having that edge as one of its sides.",
"The lower adjacent cells $\\mathcal {N}_{d}(\\sigma )$ $=$ $\\lbrace \\tau $ $|$ $ \\exists \\delta $ such that $\\delta \\prec \\sigma $ and $\\delta \\prec \\tau \\rbrace $ , are the cells of the same dimension as $\\sigma $ that share a lower dimensional cell on their boundary.",
"The line graph adjacencies between the edges are a classic example of this.",
"The upper adjacent cells $\\mathcal {N}_{u}(\\sigma )$ $=$ $\\lbrace \\tau $ $|$ $ \\exists \\delta $ such that $\\sigma \\prec \\delta $ and $\\tau \\prec \\delta \\rbrace $ .",
"These are the cells of the same dimension as $\\sigma $ that are on the boundary of the same higher-dimensional cell as $\\sigma $ .",
"Figure: Illustration of a geometric cell complex 𝒞=(𝒱,ℰ,𝒫)\\mathcal {C} = (\\mathcal {V}, \\mathcal {E}, \\mathcal {P}).",
"In this figure, it is possible to distinguish the topology of 𝒫\\mathcal {P} by the color attached to its elements,i.e., the polygons.",
"Two polygons share the same color if the have the same number of boundary elements, i.e., triangles are orange, squares are green, and so on.Finally, it is possible to incorporate a graph $\\mathcal {G}$ in a higher order cell complex $\\mathcal {C}_\\mathcal {G}$ by attaching polygons to closed paths of edges having no internal chords."
],
[
"Algebraic Representation",
"Let us now introduce an algebraic representation of the cell complexes based on the incidence relations among cells.",
"We need first to introduce the orientation of a cell complex by generalizing the concept of orientation of a simplex [32], [43].",
"To define the orientation of a $k$ -cell, we may apply a simplicial decomposition [43], which consists in subdividing the cell into a set of internal $k$ -simplices, so that, by orienting a single internal simplex, the orientation propagates to the entire cell.",
"Defining the transposition as the permutation of two elements, two orientations are equivalent if each of them can be recovered from the other through an even number of transpositions.",
"Given an oriented cell complex ${\\cal C}$ , let ${S}_{k}$ denote the number of cells in dimension $k$ , and $\\tau _i$ and $\\sigma _j$ denote two cells of the complex with dimension $\\dim (\\sigma _j) = k$ and $\\dim (\\tau _i) = k-1$ respectively.",
"The $k-th$ signed boundary matrix $\\mathbf {B}_{k} \\in \\mathbb {R}^{S_{k-1} \\times S_{k}}$ of ${\\mathcal {C}}$ is: $[\\mathbf {B}_k]_{i,j}=\\left\\lbrace \\begin{array}{rll}1, & \\text{if} \\; \\tau _i \\prec _{+} \\sigma _j \\\\-1,& \\text{if} \\; \\tau _i \\prec _{-} \\sigma _j\\\\0,& \\; \\text{otherwise}\\\\\\end{array}\\right.$ where we use the notation $\\prec _{+}$ to indicate coherent orientation between two cells and $\\prec _{-}$ to indicate a opposite orientation between two cells.",
"As a particular case, let us consider a cell complex of order two $\\mathcal {C}=\\lbrace \\mathcal {V},\\mathcal {E},\\mathcal {P}\\rbrace $ , where $\\mathcal {V}$ , $\\mathcal {E}$ , $\\mathcal {P}$ denote the set of 0, 1 and 2-cells, i.e., vertices, edges and polygons, respectively.",
"We denote their cardinality by $|\\mathcal {V}|=V$ , $|\\mathcal {E}|=E$ and $|\\mathcal {P}|=P$ .",
"Then, the two incidence matrices describing the connectivity of the complex are $\\mbox{$\\mathbf {B}$}_1 \\in \\mathbb {R}^{V\\times E}$ and $\\mbox{$\\mathbf {B}$}_2 \\in \\mathbb {R}^{E\\times P}$ , where $\\mbox{$\\mathbf {B}$}_2$ can be written as in [31]: $\\mbox{$\\mathbf {B}$}_2=[\\mbox{$\\mathbf {B}$}_{T},\\mbox{$\\mathbf {B}$}_{Q},\\ldots ,\\mbox{$\\mathbf {B}$}_{P_{S}} ]$ , with $\\mbox{$\\mathbf {B}$}_{T}$ , $\\mbox{$\\mathbf {B}$}_{Q}$ and $\\mbox{$\\mathbf {B}$}_{P_{S}}$ denoting the incidences between edges and, respectively, triangles, quadrilaterals, up to polygons with $P_{S}$ sides (where each polygon does not include any internal chord between any pair of its vertices).",
"An interesting property of the incidence matrices is that $\\mbox{$\\mathbf {B}$}_k \\mbox{$\\mathbf {B}$}_{k+1}=\\mathbf {0}$ , for all $k$ .",
"Finally, the structure of a $K$ -cell complex can be described through the higher-order combinatorial Laplacians defined as [44], [45]: $&\\mbox{$\\mathbf {L}$}_0=\\mbox{$\\mathbf {B}$}_1\\mbox{$\\mathbf {B}$}_1^T, \\nonumber \\\\&\\mbox{$\\mathbf {L}$}_k= \\mbox{$\\mathbf {L}$}_{k}^{d}+\\mbox{$\\mathbf {L}$}_{k}^{u} ,\\;\\;\\qquad \\hbox{$\\;\\;$ for $k=1,\\ldots ,K-1$,} \\nonumber \\\\&\\mbox{$\\mathbf {L}$}_K=\\mbox{$\\mathbf {B}$}_K^T\\mbox{$\\mathbf {B}$}_K,$ where $\\mbox{$\\mathbf {L}$}_{k}^{d} =\\mbox{$\\mathbf {B}$}_{k}^T \\mbox{$\\mathbf {B}$}_{k}$ and $\\mbox{$\\mathbf {L}$}_{k}^{u} =\\mbox{$\\mathbf {B}$}_{k+1} \\mbox{$\\mathbf {B}$}_{k+1}^T$ are respectively the lower and upper Laplacians, encoding the lower and upper adjacencies of the $k$ -order cells.",
"Note that $\\mathbf {L}_{0}$ corresponds to the combinatorial Laplacian used for graph representations."
],
[
"Data over Cell Complexes",
"Let $\\mathcal {C}_k$ be the set of k-th order cells in a cell complex $\\mathcal {C}$ .",
"In most of the cases the focus is on complex $\\mathcal {C}^{(2)}$ of order up to two, thus a set of vertices $\\mathcal {V}$ with $|\\mathcal {V}| = V$ , a set of edges $\\mathcal {E}$ with $|\\mathcal {E}|=E$ and a set of polygons $\\mathcal {P}$ with $|\\mathcal {P}| = P$ are considered, resulting in ${\\cal C}_{0}={\\cal V}$ (cells of order 0), ${\\cal C}_{1}={\\cal E}$ (cells of order 1) and ${\\cal C}_{2}={\\cal P}$ (cells of order 2).",
"In Fig.",
"REF we sketch an example of a cell complex of order 2.",
"A $k$ -cell signal is defined as a mapping from the set of all $k$ -cells contained in the complex to real numbers: $ \\mathbf {x}_{k}: {\\cal C}_{k} \\rightarrow \\mathbb {R}, \\,\\,\\quad k=0, 1, \\ldots K.$ The order of the signal is one less the cardinality of the elements of ${\\cal C}_{k}$ .",
"Therefore, for a complex $\\mathcal {C}^{(2)}$ , the $k$ -cell signals are defined as the following mappings: $\\mathbf {x}_{0}: {\\cal V} \\rightarrow \\mathbb {R} , \\qquad \\mathbf {x}_{1}: {\\cal E} \\rightarrow \\mathbb {R} , \\qquad \\mathbf {x}_{2}: {\\cal P} \\rightarrow \\mathbb {R} ,$ representing vertex, edge and polygon signals, respectively.",
"In this work, we will consider only vertex and edge signals.",
"In particular, we will refer to an instance of the former as $\\mathbf {x}_{i}$ while the instance of the latter will be referred as $\\mathbf {x}_{e}$ ."
],
[
"Cell Attention Networks",
"The aim of this work is to extend Graph Attention Networks introduced in [11] to account for multi-way relationships, i.e., performing a masked self-attention mechanisms at the edge level.",
"The proposed hierarchical architecture, which we refer to as Cell Attention Network (CAN) (REF ), starts with the embedding of the input graphs in regular cell complexes via a skeleton-preserving cellular lifting map and an attentional lift procedure enabling the derivation of edge features from node features.",
"Then, we introduce a novel edge-level attentional message passing scheme.",
"After each round of message passing, we perform a novel edge pooling operation and a local readout to reduce complexity; finally, after the last message-passing round, a global readout is applied.",
"As in the previous sections, we denote the input graph(s) with $\\mathcal {G}=(\\mathcal {V},\\mathcal {E)}$ and the input node features of node $i \\in \\mathcal {V}$ with $\\mathbf {x}_i \\in \\mathbb {R}^{F_n}$ ."
],
[
"Cellular Lifting Map",
"We first need to incorporate input graphs in regular cell complexes.",
"To address this challenge, we exploit the notion of skeleton-preserving cellular lifting map presented in [34] and defined as: Definition 5 (Skeleton-Preserving Cellular Lifting Map) [34].",
"A cellular lifting map $s: \\mathcal {G} \\rightarrow \\mathcal {C}_{\\mathcal {G}}$ is a skeleton preserving function that incorporates a graph $\\mathcal {G}$ into a regular cell complex $\\mathcal {C}_{\\mathcal {G}}$ , such that, for any graph $\\mathcal {G}$ , the 1-skeleton (i.e., the underlying graph) of $s(\\mathcal {G})$ and $\\mathcal {G}$ are isomorphic .",
"Informally, Definition 5 just requires that the lifting map keeps the underlying graph structure unchanged.",
"Several cellular lifting map can be exploited, in this work we opted for a lifting map that attach cells to all the induced (or chordless) cycles, where $k$ can be considered a hyperparameter to be chosen arbitrarily, and which controls the maximum size of the polygons (2-cells) of the complex."
],
[
"Attentional lift",
"After the Cellular Lifting Map, we need to learn edge features, thus performing a lift operation on the node features that we refer to as attentional lift.",
"To this aim, we exploit a masked multi-head self-attention mechanism [11].",
"The procedure is based on the computation of $F^0$ attention heads such that, for each pair of nodes $i$ ,$j \\in \\mathcal {V}$ connected by an edge $e \\in \\mathcal {E}$ , the corresponding edge features $\\mathbf {x}_{e} \\in \\mathbb {R}^{F^0}$ are given by the concatenation of the resulting attention scores.",
"Definition 6 (Attentional Lift).",
"An Attentional Lift is a learnable function $g: \\mathbb {R}^{F_n} \\times \\mathbb {R}^{F_n} \\rightarrow \\mathbb {R}^{F^0}$ , of the form: $\\mathbf {x}_{e} =g(\\mathbf {x}_i,\\mathbf {x}_j)= \\overset{F^0}{\\underset{k=1}{||}}a_n^k(\\mathbf {x}_i,\\mathbf {x}_j), \\quad \\forall e \\in \\mathcal {E}.$ where $a_n^k: \\mathbb {R}^{F_n} \\times \\mathbb {R}^{F_n} \\rightarrow \\mathbb {R}$ is the $k$ -th (shared across nodes) learnable attention function, and $||$ is the concatenation operator.",
"Since the order of the nodes connected by an edge should not change the corresponding lifted edge features, we assume that the functions $a_n^k$ are symmetric."
],
[
"Cell Attention",
"In this section we introduce Cellular Attentional Message-passing, an attentional message passing scheme operating at edges level on the learned edges features of Eq.",
"(REF ) exploiting the connectivity given by the regular cell complex $\\mathcal {C}_{\\mathcal {G}}$ computed via the cellular lifting map of Definition 5.",
"Before describing the proposed scheme, please notice that, as previously introduced, we will perform an edge pooling operation after the message-passing round at each layer $l \\in \\lbrace 1,...,L\\rbrace $ , meaning that the architecture will produce a sequence of cell complexes $\\lbrace \\mathcal {C}^l\\rbrace _l$ such that $\\mathcal {C}^{l+1} \\subseteq \\mathcal {C}^{l}$ (due to the fact that the corresponding edge sets are such that $\\mathcal {E}^{l+1} \\subseteq \\mathcal {E}^{l}$ ); we will describe the edge pooling in details in the next section.",
"As already introduced in Section 2, there are two types of adjacencies that can be exploited when dealing with cell complexes.",
"In particular, since the message exchange happen at the edges level, in each layer $l$ , our message-passing scheme exploits upper and lower edge adjacencies $\\mathcal {N}^l_{d}(e)$ and $\\mathcal {N}^l_{u}(e)$ , associated with the cell complex $\\mathcal {C}_{l}$ , $l=1,\\ldots ,L.$ At each layer $l$ , we introduce a learnable upper attention function $a^l_{u}: \\mathbb {R}^{F^l} \\times \\mathbb {R}^{F^l} \\rightarrow \\mathbb {R}$ , responsible to evaluate the reciprocal importance of two edges that are part of the same polygon, and a lower attention function $a^l_{d}: \\mathbb {R}^{F^l} \\times \\mathbb {R}^{F^l} \\rightarrow \\mathbb {R}$ , responsible to evaluate the reciprocal importance of two edges that share a common node.",
"Therefore, edges embedding are updated in the $l-th$ message passing round as: $\\widetilde{\\mathbf {h}}_{e}^{l} = \\phi ^l\\Bigg (\\mathbf {h}_{e}^{l}, \\, \\bigoplus _{k \\in \\mathcal {N}^l_d(e)} a^l_d(\\mathbf {h}_{e}^l,\\mathbf {h}_{k}^l)\\, \\psi _d^l(\\mathbf {h}_{k}^l), \\, \\bigoplus _{k \\in \\mathcal {N}^l_u(e)} a^l_u(\\mathbf {h}_{e}^l,\\mathbf {h}_{k}^l)\\, \\psi _u^l(\\mathbf {h}_{k}^l) \\Bigg ) \\in \\mathbb {R}^{F^{l+1}}, \\; \\forall e \\in \\mathcal {E}^l,$ where $\\bigoplus $ is any permutation invariant (aggregation) operator (e.g., sum, mean, max, ...), $\\phi ^l$ is a (possibly) learnable function, $\\psi _u^l$ and $\\psi _d^l$ are learnable functions sharing the weights with $a^l_u$ and $a^l_d$ , respectively (as usual in attentional settings), $\\mathbf {h}_{e}^{0} = \\mathbf {x}_{e}$ , $\\mathcal {C}^0 = \\mathcal {C}_{\\mathcal {G}}$ (thus $\\mathcal {E}^0 = \\mathcal {E}$ ).",
"Obviously, multi-head attention can be trivially injected following the usual concatenation or averaging approach [36], [11].",
"A pictorial example of how upper and lower attention work is depicted in Figure REF ."
],
[
"Edge Pooling",
"In this section, we present a self-attention edge pooling technique, adopting a variation of the method used in [46].",
"Let $\\widetilde{\\mathbf {h}}_{e}^{l} \\in \\mathbb {R}^{F^{l+1}}$ be the hidden feature vector associated to edge $e$ obtained via attentional message-passing after the $l$ -th message-passing round.",
"The edge attention pooling operation consists in computing a self-attention score $\\gamma _{e}^{l} \\in \\mathbb {R}$ for each edge of the complex via a pooling learnable attention function $a^l_{p}: \\mathbb {R}^{F^{l+1}} \\rightarrow \\mathbb {R}$ : $\\gamma _{e}^{l} = a^l_p \\left(\\widetilde{\\mathbf {h}}_{e}^{l} \\right) \\qquad \\forall e \\in \\mathcal {E}^l.$ Let $k \\in (0, 1]$ be the pooling ratio, i.e., the fraction of the edges that will be retained over the number of edges in input to the self-attention edge pooling layer.",
"At this point, we keep the $\\lceil k |\\mathcal {E}^l| \\rceil $ edges belonging to the set $\\mathcal {E}^{l+1} = \\lbrace e : e \\in \\mathcal {E}^l \\textrm { and } \\gamma _e^{l} \\in \\text{top-k}(\\lbrace \\gamma _{e}^{l}\\rbrace _{e \\in \\mathcal {E}^l}, \\lceil k |\\mathcal {E}^l| \\rceil ) \\rbrace \\subseteq \\mathcal {E}^l$ where $\\text{top-k}(\\lbrace \\gamma ^l_e\\rbrace _{e \\in \\mathcal {E}^l}, \\lceil k |\\mathcal {E}^l| \\rceil )$ is the set of the highest $\\lceil k |\\mathcal {E}^l| \\rceil $ self-attention scores.",
"Finally, the feature vectors that will be kept after the pooling stage are scaled as: $\\mathbf {h}_{e}^{l+1} = \\gamma _{e}^{l} \\widetilde{\\mathbf {h}}^{l}_{e} , \\;\\; \\forall e \\in \\mathcal {E}^{l+1}.$ After the edge pooling, we consequently need to adjust the structure of the cell complex $\\mathcal {C}^l$ to obtain a consistent updated complex $\\mathcal {C}^{l+1}$ .",
"To this aim, we apply the procedure depicted in Fig.",
"REF : If an edge $e$ belongs to $\\mathcal {E}^{l}$ but is not contained in $ \\mathcal {E}^{l+1}$ , the lower connectivity is updated by disconnecting the nodes that are on the boundary of $e$ , while the upper connectivity is updated by removing the polygons that have $e$ on their boundaries.",
"Figure: Illustration of the proposed edge pooling procedureFinally, we considered also a hierarchical version of the aforementioned self-attention edge pooling operation as in [47].",
"To this aim we employ a (by-layer) readout operation on the hidden feature $\\lbrace \\mathbf {h}_e^{l+1}\\rbrace _{e \\in \\mathcal {E}^{l+1}}$ to obtain an aggregate embedding of the whole complex $\\mathcal {C}^{l+1}$ as: $\\mathbf {h}_{\\mathcal {C}^{l+1}} = \\bigoplus _{e \\in \\mathcal {E}^{l+1}} \\mathbf {h}_{e}^{l+1}.$ Then, after the last hidden layer, a final (global) readout operation is performed, e.g., by aggregating all the previously computed complexes embeddings: $\\mathbf {h}_{\\mathcal {C}} = \\bigoplus _{l} \\mathbf {h}_{\\mathcal {C}^{l}}.$ Finally, the result of the final aggregation is fed to a multi-layer perceptron (MLP) if needed for the learning task."
],
[
"CAN Architecture and Symmetries",
"In summary, a Cell Attention Network with input graph(s) $\\mathcal {G}=(\\mathcal {V},\\mathcal {E})$ and input node features $\\lbrace \\mathbf {x}_i\\rbrace _{i \\in \\mathcal {V}}$ is defined as the stack of: (i) a skeleton-preserving cellular lifting map to obtain a regular cell complex $\\mathcal {C}_{\\mathcal {G}}$ from a graph $\\mathcal {G}$ ; (ii) a multi-head attentional lift to obtain edge features $\\lbrace \\mathbf {x}_e\\rbrace _{e \\in \\mathcal {E}}$ from node features $\\lbrace \\mathbf {x}_i\\rbrace _{i \\in \\mathcal {V}}$ ; (iii) a stack of $L$ cell attention layers, each of them composed by a message passing round as in Eq.",
"(REF ), an edge pooling stage as in Eq.",
"(REF ) and Eq.",
"(REF ), and a (by-layer) readout function as in Eq.",
"(REF ); (v) a (global) readout function, e.g.",
"as in Eq.",
"(REF ).",
"A schematic view of the whole architecture is illustrated in Figure REF .",
"Finally, we present the following result about equivariance properties of the proposed architecture: Theorem 1.",
"Cell Attention Networks are permutation invariant.",
"In literature, a GNN is permutation invariant if a permutation of nodes produces the same output without the permutation.",
"More formally, a GNN $f(\\cdot )$ taking an input graph $\\mathcal {G}$ with adjacency matrix $\\mathbf {A}$ and input node features matrix $\\mathbf {X} = \\lbrace \\mathbf {x}_i\\rbrace _{i \\in \\mathcal {V}}$ is (node) permutation invariant if $ f(\\mathbf {P}\\mathbf {A}\\mathbf {P}^T,\\mathbf {P}\\mathbf {X}) = f(\\mathbf {A},\\mathbf {X})$ for any permutation matrix $\\mathbf {P}$ .",
"In the same way, Cell Attention Networks are permutation invariant w.r.t.",
"permutations of nodes, edges and polygons.",
"Proof.",
"We can assert, w.l.o.g., that Attentional Lift is permutation equivariant by construction.",
"The operation $g(\\cdot )$ and $a(\\cdot )$ are both learnable functions acting on edges, and since $a(\\cdot )$ is symmetric by definition, both $a(\\cdot )$ and $g(\\cdot )$ are symmetric w.r.t.",
"to the vertices that are endpoint of the edges we are considering.",
"This leads to have the whole function permutation equivariant.",
"The scheme followed in Edge Pooling is the selecting $top-k$ element of edge set referring to self-attentional coefficients $\\gamma _{e}$ .",
"In order to select the $top-k$ elements of $\\mathcal {E}$ , the vector $\\gamma _{e}$ must be sorted.",
"So no matter what is the permutation on the set, after sorting we obtain always the same result.",
"For this reason Edge Pooling is permutation invariant.",
"Finally, since the proposed CAN architecture is the composition of a permutation equivariant function (i.e., the attentional lift) and a permutation invariant function (i.e., the edge pooling), it readily follows that CAN are permutation equivariant.",
"Please notice that the proposed architecture without the pooling stage is clearly permutation equivariant.",
"Figure: Illustration of a cell attention network"
],
[
"Experimental Results",
"In this section we asses the performance of the proposed architecture when solving several real-world graph classification problems, focusing on well known molecular benchmarks on TUDataset [48].",
"In every experiment, if the dataset is equipped with edge features, we concatenate them to the result of the lift layer (Eq.",
"(REF )).",
"We included small molecules with class labels such as MUTAG [49] and PTC [50].",
"In the former dataset, the task is to identify mutagenic molecular compounds for potentially commercial drugs, while in the latter the goal is to identify chemical compounds based on their carcinogenicity in rodents.",
"The PROTEINS dataset [51] is composed mainly by macromolecules.",
"Here, nodes represent secondary structure elements and are annotated by their type.",
"Nodes are connected by an edge if the two nodes are neighbours on the amino acid sequence or one of three nearest neighbors in space; the task is to understand if a protein is an enzyme or not.",
"Using these type of data in a Cell Complex based architecture has an underlying importance since molecules have polyadic structures.",
"Finally, NCI1 and NCI109 are two datasets aimed at identifying chemical compounds against the activity of non-small lung cancer and ovarian cancer cells [52].",
"Considering the aforementioned datasets, we compare CAN with other state of the art techniques in graph representation learning.",
"Since there are no official splits for the training and test sets, to validate the proposed architecture, we followed the method used in [34]: we run a 10-fold cross-validation reporting the maximum of the average validation accuracy across folds.",
"Table: Experimental results on TUDatasets.",
"The first part shows the accuracy of graph kernel methods, while the second assess graph neural networks.",
"Cell attention networks scores top on four out of five experiments.The performance of CAN is reported in Table REF , along with those of graph kernel methods: Random Walk Kernel (RWK, [53]), Graph Kernel (GK, [54]), Propagation Kernels (PK, [55]), Weisfeiler-Lehman graph kernels (WLK, [56]); other GNNs: Diffusion-Convolutional Neural Networks (DCNN, [10]), Deep Graph Convolutional Neural Network (DGCNN, [57]), Invariant and Equivariant Graph Networks (IGN, [58]), Graph Isomorphism Networks (GIN, [33]), Provably Powerful Graph Networks (PPGNs, [59]), Natural Graph Networks (NGN, [60]), Graph Substructure Network (GSN [61]) and topological networks: Simplicial Isomorphism Network (SIN, [32], Cell Isomorphism Network (CIN, [34]).",
"As we can see from Table REF , CAN achieves the best performance on four out of five benchmarks, while performing very similarly to CIN in the last experiment (i.e., NCI109).",
"Since CAN has a much lower computational complexity than CIN (cf.",
"Appendix C), these results support the validity and the performance obtained of the proposed architecture.",
"The tested models have been implemented using PyTorch [62].",
"The datasets have been taken from the PyTorch Geometric library [63].",
"The operations involved during cellular lifting maps use the code provided by [34] under MIT license.",
"PyTorch, NumPy, SciPy and are made available under the BSD license, Matplotlib under the PSF license, graph-tool under the GNU LGPL v3 license.",
"PyTorch Geometric is made available under the MIT license.",
"All the experimental results have been made on NVIDIA® GeForce RTX 3090 GPUs with 10,496 CUDA cores and 24GB GPU memory.",
"The operative system used for the experiment was Ubuntu 22.04 LTS 64-bit.",
"See Appendix for an extensive description of the tested architectures and an ablation study The code implementation for the proposed architecture is available at: https://github.com/lrnzgiusti/can."
],
[
"Conclusion and Discussion",
"In this work we presented Cell Attention Networks (CANs), novel neural architectures operating on data defined over the nodes of a graph incorporated into a a regular cell complex, exploiting generalized masked self-attention mechanisms.",
"It builds on skeleton-preserving cellular lifting maps, a novel attentional features lift and a novel edge-level attentional message-passing scheme with two attention functions that operate on the upper and lower connectivities induced by the cell complex.",
"The proposed architecture is also equipped with a novel hierarchical edge pooling technique that leverage a self-attention mechanism to downsample the data in the network's hidden layers while extracting significant features for the learning task.",
"The Cell Attention Network architecture proposed and tested in the previous sections shows promising results and it is grounded in the theory of regular cell complexes; however, some directions can be explored to enrich the proposed formulation.",
"In particular, a signal processing perspective [31] can be exploited to reinterpret and modify the proposed architecture following a similar approach to [36]; an expressivity analysis can be carried out based on the renewed Weisfeiler-Lehman approach [33], on its generalization to cell complexes [34], [38], or based on spectral approaches [64].",
"We leave these problems to be addressed in future works."
],
[
"Model Implementation",
"In our experiments, we employ cell attention networks to regular cell complexes of order two obtained by applying the structural lifting map to the original graphs, i.e.",
"we consider nodes as 0-cells and edges as 1-cells, and the chordless cycles of size up to $R=6$ as 2-cells.",
"In our case, each node of the original graphs is always equipped with an input feature vector.",
"Throughout all experiments, we employ cell attention networks with the following structure.",
"The attentional lift mechanism in Eq.",
"(REF ) is given by: $&\\mathbf {h}_{e}^0 = \\mathbf {x}_e = \\overset{F^0}{\\underset{k=1}{||}}\\phi _n \\underbrace{\\left( \\left( \\left(\\mathbf {a}_{n}^{k} \\right)^T [\\mathbf {x}_i \\vert \\vert \\mathbf {x}_j] \\right) \\, \\big \\vert \\big \\vert \\, \\tilde{\\mathbf {x}}_e \\right)}_{a_n^k}, \\quad i,j \\in \\mathcal {B}(e),$ where $\\tilde{\\mathbf {x}}_e$ is the input feature vector of the edge $e$ .",
"If not provided by the specific benchmark, $\\tilde{\\mathbf {x}}_e$ can be considered as an empty vector.",
"Also, $\\mathbf {a}_n^k \\in \\mathbb {R}^{2F_n}$ is the vector of attention coefficients associated to the k-th feature of the input edge feature vector, and $\\phi _n$ is the non-linear activation function for the lift layer.",
"Please notice that the employed functions $a_n^k$ are not symmetric, but they give the best learning performance on the proposed tasks.",
"The lower and upper attentional functions $a_d(\\mathbf {h}_{e},\\mathbf {h}_{k})$ and $a_u(\\mathbf {h}_{e},\\mathbf {h}_{k})$ in Eq.",
"(REF ) are chosen as two independent masked self-attention schemes.",
"They can be chosen following any of the known approaches from graphs [11], [65].",
"In this paper we follow the approach from [11]: formally, let $&\\omega _{e,k}^{l,d} = \\phi _a \\left( (\\mathbf {a}_{d}^l)^T \\left[ \\mathbf {W}_{d}^{l} \\mathbf {h}_{e}^{l} \\vert \\vert \\mathbf {W}_{d}^{l} \\mathbf {h}_{k}^{l} \\right] \\right) \\\\&\\omega _{e,k}^{l,u} = \\phi _a \\left( (\\mathbf {a}_{u}^l)^T \\left[ \\mathbf {W}_{u}^{l} \\mathbf {h}_{e}^{l} \\vert \\vert \\mathbf {W}_{u}^{l} \\mathbf {h}_{k}^{l} \\right] \\right), $ where $\\mathbf {a}_{d}^l, \\mathbf {a}_{u}^l \\in \\mathbb {R}^{F^{l}}$ are two independent vectors of attention coefficients, $\\mathbf {W}_{d}^l, \\mathbf {W}_{u}^l \\in \\mathbb {R}^{F^{l+1} \\times F^{l}}$ are two learnable linear transformations shared by the lower and upper neighbourhoods of the complex, respectively, and $\\phi _a$ is a pointwise non-linear activation.",
"The coefficients $\\omega ^u$ and $\\omega ^d$ in Eq.",
"(REF ,) represent the importance of the features of edge k when exchanging messages with edge e over lower and upper neighborhoods, respectively.",
"It worth to emphasize that since the attention schemes are decoupled, these importance coefficients will be different over the upper and lower neighborhoods.",
"In line with the approach of [11], we make coefficients easily comparable across different edge by normalizing them across all choices of $k$ using the softmax function: $&\\alpha _{e,k}^{l,d} = \\frac{\\exp \\left( \\omega _{e,k}^{l,d} \\right) }{\\sum _{\\iota \\in \\mathcal {N}_{d}^{l}(e)} \\exp \\left( \\omega _{e,\\iota }^{l,d} \\right)} \\\\[5pt]&\\alpha _{e,k}^{l,u} = \\frac{\\exp \\left( \\omega _{e,k}^{l,u} \\right) }{\\sum _{\\iota \\in \\mathcal {N}_{u}^{l}(e)} \\exp \\left( \\omega _{e,\\iota }^{l,u} \\right)}$ Thus, for layer $l$ we have that $a^l_d(\\mathbf {h}_{e}^l,\\mathbf {h}_{k}^l) = \\alpha _{e,k}^{l,d}$ and $a^l_u(\\mathbf {h}_{e}^l,\\mathbf {h}_{k}^l) = \\alpha _{e,k}^{l,u}$ .",
"Once the attention coefficients have been normalized, to update the representation of an edge $e$ , a linear combination of the edge features and the normalized attention coefficients corresponding to them is computed for both the lower and upper neighbourhoods and the results are aggregated alongside with the current edge representation.",
"Formally: $&\\widetilde{\\mathbf {h}}_{e}^{l} = \\phi \\Bigg ((1+\\varepsilon ) \\, \\mathbf {W}_{s}^{l} \\mathbf {h}_{e}^{l} + \\sum _{k \\in \\mathcal {N}^l_d(e)} \\alpha _{e,k}^{l,d} \\, \\mathbf {W}_{d}^{l} \\mathbf {h}_{k}^l + \\sum _{k \\in \\mathcal {N}^l_u(e)} \\alpha _{e,k}^{l,u} \\, \\mathbf {W}_{u}^{l} \\mathbf {h}_{k}^l)\\Bigg ),$ here $\\mathbf {W}_{s}^l \\in \\mathbb {R}^{F^{l+1} \\times F^{l}}$ is a shared linear transformation applied to the current hidden representation of the edges of the complex.",
"Notice that the functions $\\psi _d(\\mathbf {h}_{k}^l)$ and $\\psi _u(\\mathbf {h}_{k}^l)$ in Eq.",
"(REF ) are implemented respectively as: $\\mathbf {W}_{d}^{l} \\mathbf {h}_{k}^l$ and $\\mathbf {W}_{u}^{l} \\mathbf {h}_{k}^l$ .",
"In the pooling layer, the hidden feature vectors are updated using Eq.",
"(REF ) by scaling the features $\\widetilde{\\mathbf {h}}_{e}^{l}$ with the corresponding score $\\gamma _e^l$ : $&\\mathbf {h}_{e}^{l+1} = \\underbrace{\\phi _p \\left( ( \\mathbf {a}_p^l )^T \\tilde{\\mathbf {h}}_e^l \\right)}_{\\gamma _e^l} \\tilde{\\mathbf {h}}_e^l, \\quad \\forall e \\in \\mathcal {E}^{l+1},$ where the vector $\\mathbf {a}_p^l$ plays the role of a collection of attention coefficient that weight the features of $\\tilde{\\mathbf {h}}_e^l$ to compute the corresponding score $\\gamma _e^l$ , which that represents the importance of edge $e$ in the learning task.",
"Following the approach of [46], the weight $( \\mathbf {a}_p^l )^T \\tilde{\\mathbf {h}}_e^l$ of the edge $e$ is also forwarded into a non-linear activation function $\\phi _p$ to produce the score $\\gamma _e^l \\in \\mathbb {R}$ , which is then multiplied to $\\tilde{\\mathbf {h}}_e^l$ to obtain $\\mathbf {h}_{e}^{l+1}$ .",
"Readout operations are performed as follows: If the pooling approach is hierarchical, the readout is performed layer-wise.",
"In particular, we choose the sum as the permutation equivariant aggregation function of Eq.",
"(REF which results in a hierarchical representation of the complex i.e.",
"a collection $\\lbrace \\mathbf {h}_{\\mathcal {C}^l}\\rbrace _{l=0}^{L-1}$ of hidden representations.",
"Then, Eq.",
"(REF ) is computed as the sum over the collection defined previously.",
"In the case of a global pooling, the readout is computed only in the last layer, which constitute the overall representation of the complex: $\\mathbf {h}_{\\mathcal {C}}= \\mathbf {h}_{\\mathcal {C}^{L-1}} = \\sum _{e \\in \\mathcal {E}^{L-1}} \\mathbf {h}_{e}^{L-1}.$ Once $\\mathbf {h}_{\\mathcal {C}}$ is obtained, it is forwarded into a 2-Layer MLP with $\\phi $ as activation function to perform the prediction.",
"In all layers, we adopt a Batch Normalization technique [66] and all training operations are performed with the AdamW optimization algorithm [67].",
"In Table REF we report the hyper-parameters used in our experiments for each dataset.",
"Table: Hyperparameter used for the experiments on TUDatasets."
],
[
"Complexity Analysis",
"In this section, we review the computational complexity of each operation involved the proposed architecture referring to the model implementation define above: Cellular Lifting Map: Although this operation can be precomputed for the entire dataset and the connectivity results stored for a later usage, it worth to elicit its complexity noticing that for some applications the storage of the upper and lower connectivity for the entire dataset might be not possible.",
"We considered Cellular Lifting Maps that assign 2-cells to all the chordless cycles of a graph with a maximum number of nodes in the cycles up to $R$ as maximum cycle size.",
"The chord-less cycles in a graph can thus be enumerated in $\\mathcal {O}((|E| + |V| R) \\, \\textrm {polylog} |V|)$ time [68].",
"Similar to [34], in our experimental setup we have that $R$ can upper bounded by a small constant.",
"Thus, the complexity of this operation can be approximated to be linear in the size of the complex.",
"Attentional Lift: The complexity of this operation consists of a multi-head attention message passing scheme over the entire graph [11].",
"For a single node pair $i,j \\in \\mathcal {V}$ connected by an edge $e \\in \\mathcal {E}$ , the attentional lift defined in Eq.",
"(REF ) can be decomposed into $F^0$ independent self-attention schemes.",
"Each attention scheme requires $\\mathcal {O}(F_n)$ computations, where $F_n$ is the number of input node features.",
"Thus, for the pair $i,j$ , the attentional lift is performed in $\\mathcal {O}(F^0 F_n)$ , where $F^0$ is a parameter to be chosen as the number of input edge features.",
"Accounting all the edges of the complex yields an amount of $\\mathcal {O}(\\vert \\mathcal {E} \\vert F^0 F_n))$ operations to lift the given node features into edge ones.",
"Cell Attention Layer: This operation consists in two independent masked self-attention message passing schemes over the upper and lower neighbourhoods of the complex, namely cell attention, an inner linear transformation of the edges' features and an outer point-wise nonlinear activation (Eq.",
"(REF )).",
"For the lower neighbourhood, a single edge $e$ receives at most 2 messages for each cell in its coboundary, $\\mathcal {CB}(e)$ (cf Fig.",
"REF .1 ); thus, for a single edge the attention over the lower neighbourhood of the complex is $\\mathcal {O}(\\vert \\mathcal {CB}(e) \\vert )$ , where $\\vert \\mathcal {CB}(e) \\vert $ is the number of cells that are co-faces of the edge $e$ .",
"Regarding the upper neighbourhood, if $R$ is the maximum ring size we have that a single edge $e$ receives $\\mathcal {O}(R \\cdot \\vert \\mathcal {CB}(e) \\vert )$ messages (cf.",
"Fig.",
"REF .2.",
"Recalling that $R$ is upper bounded by a small constant [34], cell attention is an $\\mathcal {O}(\\vert \\mathcal {CB}(e) \\vert )$ operation for both neighbourhoods of an arbitrary edge $e$ i.e.",
"linear in the size of the complex.",
"The inner linear transformation that propagates the information contained in $\\mathbf {h}^{l}_e$ is upper bounded by $\\mathcal {O}(F^{l})$ .",
"Extending this to all edges of the complex, we have that the complexity of a cell attention layer can be rewritten as $\\mathcal {O}(|\\mathcal {E}^l|\\, F^{l})$ .",
"In the case of a multi-head cell attention, the complexity receives an overhead induced by the number of attention heads involved within the layer, i.e., a multiplication by a factor $H_c$ , the number of cell attention heads.",
"Attentional Pooling: The operations involved in the pooling layer can be decomposed in: (i) computing the self-attention scores for each edge of the complex ($\\gamma _e^l$ in Eq.",
"(REF )); (ii) select the highest $\\lceil k \\, \\vert \\mathcal {E}^{l} \\vert \\, \\rceil $ values from a collection of self-attention scores ($\\text{top-k}(\\lbrace \\gamma ^l_e\\rbrace _{e \\in \\mathcal {E}^l}, \\lceil k |\\mathcal {E}^l| \\rceil )$ ); and (iii) adjust the connectivity of the complex (see Fig.",
"REF ).",
"To compute the computational complexity of this layer it is convenient to see the selection operation as a combination of a sorting algorithm over a collection of self-attention scores and a selection of the first $\\lceil k \\, \\vert \\mathcal {E}^{l} \\vert \\, \\rceil $ elements from the sorted collection.",
"Since the computations involved in (i) and (iii) are linear in the dimension of the complex, the overall complexity of this layer in can be upper bounded by the sorting algorithm, i.e., $\\mathcal {O}(|\\mathcal {E}^{l}| \\; log (|\\mathcal {E}^{l}|))$ .",
"In practice, all the computations involved in a cell attention network are local formulations completely disjoint from each other.",
"Thus, using an efficient GPU-based implementation, we can rewrite all the analysis in terms of the longest sequential chain of operations in a concurrent execution over the edges of the input domain, i.e., $\\mathcal {O}(log(|E|))$ .",
"For an in-depth concurrency analysis we refer readers to [69], where the authors report a complete taxonomy of parallelism in GNNs."
],
[
"Learnable Parameters",
"The total number of learnable parameters of a CAN can be decomposed into: Cellular Lifting Map: Lifting the input graph $\\mathcal {G}$ to a cell complex $\\mathcal {C}$ is an operation that assign a cell $\\sigma $ to all the chord-less cycles of $\\mathcal {G}$ up to a maximum cycle size $R$ .",
"Intuitively this operation does not involve any parameter to be learned during the network's training phase.",
"Thus, the number of learnable parameters for the lift is $\\mathcal {O}(1)$ .",
"Attentional Lift: In the context of lifting a pair of graph node features $\\mathbf {x}_i, \\mathbf {x}_j \\in \\mathbb {R}^{F_n}$ to a signal defined over the edge of the complex, $\\mathbf {x}_{e} \\in \\mathbb {R}^{F^0}$ , we have to learn a vector of attention coefficients $\\mathbf {a}_{n} \\in \\mathbb {R}^{2 F_n}$ for each input edge feature.",
"The vector $\\mathbf {a}_{n}$ has a number of learnable parameters in $\\Theta \\left( F_n \\right)$ .",
"Accounting multiple input edge features, the overall number of parameters for the attention lift operation is $\\Theta (F^0 F_n)$ , where $F^0$ is the number of input edge features computed as multiple independent attention heads.",
"Cell Attention: In terms of learnable parameters, a single cell attention layer is composed of: two independent vectors of attention coefficients $\\mathbf {a}_d^l, \\mathbf {a}_u^l \\in \\mathbb {R}^{F^{l}}$ for properly weighting the lower and upper neighbourhoods, respectively.",
"Moreover, the layer is equipped with three linear transformations, $\\mathbf {W}_s^l, \\mathbf {W}_d^l, \\mathbf {W}_u^l \\in \\mathbb {R}^{F^{l} \\times F^{l+1}}$ acting respectively on: $\\mathbf {h}_e^l$ , the hidden feature vector of edge $e$ at layer $l$ and the hidden feature vectors $\\mathbf {h}_k^l$ in the lower and upper neighbourhoods of the edge $e$ .",
"Thus, the number of learnable parameters of a cell attention layer is $\\mathcal {O}(F^{l} F^{l+1})$ .",
"Pooling: For this layer, learnable parameters are employed only in computing the self-attention scores ($\\gamma _e^l$ , Eq.",
"(REF )).",
"The shared vector of attentional scores' coefficients $\\mathbf {a}_p^l \\in \\mathbb {R}^{F^{l}}$ , similarly to the lift layer, is known to have a number of learnable parameters in $\\Theta \\left( F^{l} \\right)$ ."
],
[
"Ablation Study",
"In this section we take a detailed look at the performance of each operation involved in cell attention networks by performing different ablation studies and show their individual importance and contributions.",
"Figure: TUDataset: Results of the ablation of different CAN features with respect to Table (g.t.).",
"The ablation study shows the benefits of incorporating all the proposed operations into the message passing procedure when operating on data defined over cell complexes.In particular, we followed the same experimental setup used in section by fixing the hyper-parameters as in Table REF and removing one-by-one the cell attention network operations: (i) removing the lift refers to assign a feature $\\mathbf {x}_e$ to an edge $e$ using a linear function that takes the feature vectors $\\mathbf {x}_i, \\mathbf {x}_j$ from the vertices $i,j \\in \\mathcal {B}(e)$ , i.e.",
"a simple scalar product between $\\mathbf {x}_i$ and $\\mathbf {x}_j$ ($\\mathbf {x}_e = \\langle \\mathbf {x}_i, \\mathbf {x}_j \\rangle $ ); (ii) removing the lower attention can be intended as initializing the lower attention coefficients as: $\\mathbf {a}_{d}^l = 1$ and left them unmodified during the update step in the training phase; (iii) similarly, for the upper attention we replicated the idea of (ii) but now only the upper attention coefficients are involved, i.e.",
"$\\mathbf {a}_{u}^l = 1$ and kept fixed for the entire optimization stage; (iv) removing the attention means to remove both the upper and lower attention simultaneously as explained in (ii) and (iii); removing the pooling means to detach the pooling layer from the network and remove eventual intermediate readout computations involved in the hierarchical pooling setup.",
"As shown in Figure REF and in Table REF , we observe a decrease in the overall performance when removing parts of the cell attention network architecture as expected.",
"Of particular interest is the ablation study on NCI1, which shows a slightly higher accuracy in every case we kept the attention coefficients fixed and without the pooling but a drastic drop in the performance when the edge features are no longer learned.",
"Moreover we see that there are no evident \"patters\" inside the ablation study with the except that for NCI109 we observe the same behaviour of NCI1 when removing the lift layer.",
"This fact can be explained by noticing that the aforementioned datasets experience, on average, a very similar topology (Table REF ).",
"Table: Analysis of the impact of the operations involved in cell attention networks."
]
] | 2209.08179 |
[
[
"Linear Network Coding Based Fast Data Synchronization for Wireless Ad\n Hoc Networks with Controlled Topology"
],
[
"Abstract Fast data synchronization in wireless ad hoc networks is a challenging and critical problem.",
"It is fundamental for efficient information fusion, control and decision in distributed systems.",
"Previously, distributed data synchronization was mainly studied in the latency-tolerant distributed databases, or assuming the general model of wireless ad hoc networks.",
"In this paper, we propose a pair of linear network coding (NC) and all-to-all broadcast based fast data synchronization algorithms for wireless ad hoc networks whose topology is under operator's control.",
"We consider both data block selection and transmitting node selection for exploiting the benefits of NC.",
"Instead of using the store-and-forward protocol as in the conventional uncoded approach, a compute-and-forward protocol is used in our scheme, which improves the transmission efficiency.",
"The performance of the proposed algorithms is studied under different values of network size, network connection degree, and per-hop packet error rate.",
"Simulation results demonstrate that our algorithms significantly reduce the times slots used for data synchronization compared with the baseline that does not use NC."
],
[
"Introduction",
"Wireless ad hoc networks, such as the networked sensors, robots and unmanned aerial vehicles (UAVs), constitute a distributed, flexible and cooperative information-sharing system [1], [2], [3].",
"Fast data synchronization among network nodes is important for wireless ad hoc networks, since it is expected to provide essential information reliably for high-layer real-time decision and control algorithms.",
"Unfortunately, in general this is a great challenge, because in many cases the network topologies and the wireless channels between nodes are highly dynamic and random [4], [5].",
"Owing to the openness of wireless channels, all-to-all broadcast has the potential to serve as a highly efficient approach for achieving fast data synchronization in wireless ad hoc networks.",
"However, the transmission efficiency of all-to-all broadcast is still constrained by the conventional store-and-forward protocol, which does not take advantage of more sophisticated data processing and thus limits the distributed system's efficiency of sensing and reacting to the environment.",
"The network coding (NC) technique [6], [7] offers an attractive approach for data synchronization in distributed systems, since it is capable of reducing the total time cost of data synchronization by exploiting both the broadcast property of wireless channels and the XOR operation in NC, thus increasing the effective data transmission rate [8], [9].",
"Regarding the related work, in [10] the authors proposed a proactive NC scheme for all-to-all broadcast while using the random access schemes of IEEE 802.11 and considering several different network topologies.",
"In [11] an all-to-all broadcast protocol was designed for wireless ad hoc networks that use directional antennas, but NC was not considered.",
"In [12] the reliability of a random neighbor NC based all-to-all broadcast approach was analyzed.",
"In [13] an adaptive transmission protocol suite integrated with the random linear NC was proposed, where the modulation and channel coding parameters are adaptively selected for each packet to improve the packet loss performance.",
"However, all the above contributions are designed for general ad hoc networks, while ignoring the unique properties of specific systems.",
"There indeed exist some particular scenarios, where the original distributed data synchronization problem can become less challenging or enjoy more benefits.",
"For instance, compared with the data integrity and security, the latency requirement is less stringent for the data synchronization in distributed databases [14], [15].",
"Additionally, more benefits can be gleaned in the scenario where the network topology is under the operator's control.",
"Topology control is important and practical for wireless ad hoc networks, as it is beneficial for reducing energy consumption (thus extending the network lifetime) and radio interference (thus increasing the network communication capacity) [16], [17], [18].",
"By using topology control, a more stable and convenient graph representation of the network can be obtained.",
"Wireless ad hoc networks with controlled topology have found applications in many areas, such as the flight formation in military operations, the truck platooning in Internet of Vehicles (IoV) as highlighted by 5G, and the low earth orbit (LEO) satellite constellations etc.",
"In this paper we propose a pair of linear NC and all-to-all broadcast based fast data synchronization algorithms for wireless ad hoc networks with controlled topology.",
"We study the performance of the proposed algorithms with a large number of randomly generated network topology samples.",
"It is shown that the average gain in terms of the time-slot usage reduction upon applying the proposed algorithms can be over five times compared with the time-division multiple-access (TDMA) based all-to-all broadcast algorithm that does not use NC.",
"Furthermore, this substantial gain is achievable in a wide range of network topologies, in particular for the topologies that have a low or medium degree of network connectivity."
],
[
"System Model",
"As shown in Figure REF , we consider an ad-hoc network that consists of $N$ nodes connected in any arbitrary topology.",
"We assume that for each node that carries out linear NC, the corresponding decoding is performed by its immediate neighbouring nodes.",
"For each hop, a single action of transmitting-and-receiving occupies one time slot.",
"The data synchronization throughout the network is achieved as follows.",
"The node $n$ broadcasts its own data block $p_n$ , whose original or processed copy is then disseminated to the other $N-1$ nodes by relaying through their respective neighbouring nodes, where $n=1, 2, \\cdots , N$ .",
"The number of neighbouring nodes of node $n$ and the number of data blocks stored at node $n$ are denoted as $m_n$ and $d_n$ , respectively.",
"It is required that all the nodes in the network obtain the set of data blocks ${\\mathbb {A}} = \\left\\lbrace {{p_1},{p_2},\\cdots ,{p_N}} \\right\\rbrace $ as fast as possible, which means that ${d_n}$ must equal $N$ after the data synchronization is achieved throughout the network, i.e., $\\max {\\lbrace d_n\\rbrace } = N$ , $\\forall n= 1, 2, \\cdots , N$ .",
"In a given time slot the node $n$ shares its data blocks with its neighbouring nodes $n_k$ , where $1 \\le k \\le m_n$ , $1 \\le n_k \\le N$ and $n_k \\ne n$ .",
"To this end, we define the following sets: ${{\\mathbb {A}}_{n}}{\\rm { = }}\\left\\lbrace {{p_{1}},{p_{2}},\\cdots ,{p_{{d_n}}}} \\right\\rbrace $ : The data blocks stored at node $n$ .",
"$\\bar{\\mathbb {A}}_{n_k} = \\lbrace \\left.",
"{{p_j}} \\right|{p_j} \\in {\\mathbb {A}}, {p_j} \\notin {{\\mathbb {A}}_{n_k}}, j = 1, 2, \\cdots , N\\rbrace $ : The data blocks that node ${n_k}$ has not obtained.",
"Then $\\bar{\\mathbb {A}}_{n_k} \\cap {\\mathbb {A}}_n$ represents the data blocks that node $n_k$ can obtain from node $n$ .",
"Figure: The system model of the wireless ad hoc network considered, where the shaded rectangle p n p_n denotes the data block stored at node nn in the initial stage."
],
[
"The Proposed Data Synchronization Algorithms",
"Below we describe the proposed data synchronization algorithms in detail, whose flowchart is shown in Figure REF .",
"Step 0: This is the initialization stage, where each node carries out data acquisition by monitoring its own state or the state of the environment.",
"Note that it is possible for a single node to have multiple data blocks.",
"Step 1: Each node broadcasts the data blocks, which are stored on it and encapsulated into packets, to its neighbours in a TDMA manner.",
"Step 2: If a packet is received correctly and network-coded, it is fed into the network-decoding module deployed on the node that receives the packet, and the data blocks extracted from the decoder are stored on the node.",
"If the packet is not received correctly, each node updates its data block storage status.",
"Otherwise, the packet is directly stored on the node in the format of data blocks, and each node updates its data block storage status accordingly.",
"Step 3: Check whether all the data blocks are synchronized in all nodes.",
"This is achieved at each node by examining if $d_n = N$ .",
"If yes, the algorithms terminate.",
"Otherwise, select an appropriate transmitting node that contributes on average the maximum innovative data to each neighbouring node according to (REF ) (optional, used in Alg.",
"2, by examining the feedback from the subsequent data block selection (DBS), as elaborated in Sec.",
"REF ), and an appropriate set of data blocks that are innovative and decodable to the maximum number of the neighbouring nodes of the selected node according to (REF ) (mandatory, used in both Alg.",
"1 and Alg.",
"2, by exploiting the output of the decoder in Step 2, as elaborated in Sec.",
"REF ).",
"These selected data blocks are stored in their host node to participate in the subsequent network-encoding operation.",
"Step 4: Carry out network-encoding operation with respect to the data blocks selected, and jump to Step 1, where the above selected node broadcasts the network-coded data blocks to their respective neighbours.",
"This process is repeated until all the data blocks on all the nodes are synchronized.",
"Note that the DBS results are fed back to serve as an input for the node selection (NS) module, according to the outputs of the decoding module introduced in Sec.",
"REF .",
"Figure: The flowchart of the proposed data synchronization algorithms, where the node selection is only invoked in Alg.",
"2."
],
[
"Decoding",
"For each TX-RX pair, the receiver is anticipated to decode the packets sent by the transmitter and then keep the data blocks that are absent from this receiver previously.",
"In other words, each receiver is only interested in the data blocks that are innovative to itself.",
"To clarify our design philosophy, the following theorem is given.",
"For packets generated by linear NC, the receiver is capable of decoding the packet if at most one component data block is unknown.",
"Let us use the mathematical induction method to prove the theorem.",
"Firstly, we have the following proposition: the component data block $x$ can be extracted from packet $P = x \\oplus {a_1} \\oplus {a_2} \\oplus \\cdots {a_l} \\cdots \\oplus {a_L}$ provided that all the component data blocks $a_l, l = 1, 2, \\cdots , L$ are known, where $L \\in {\\mathbb {Z}_+} $ is a positive integer.",
"1) When $L=1$ , the proposition is obviously true.",
"2) Assume that the above proposition is true when $L = r$ , with $r \\in {\\mathbb {Z}_+}$ being an arbitrarily given positive integer.",
"Then $x$ can be solved from $P = x \\oplus a_1 \\oplus a_2 \\oplus \\cdots {a_l} \\cdots \\oplus a_r$ if and only if $a_l$ is known, $\\forall l \\in \\lbrace 1,2, \\cdots , r\\rbrace $ .",
"3) When $L=r+1$ , the packet $P$ is written as $P=x \\oplus a_1 \\oplus a_2 \\oplus \\cdots {a_l} \\cdots \\oplus a_r \\oplus a_{r+1}.$ Then if $\\forall l \\in \\lbrace 1,2, \\cdots , r+1\\rbrace $ , where $a_l$ is unknown, $x$ is obviously unsolvable.",
"Let us assume that at least one $a_l$ is known.",
"For convenience, assume that $a_{r+1}$ is known.",
"Then Eq.",
"(REF ) is rewritten as $P=P^{\\prime } \\oplus a_{r+1},$ where $P^{\\prime }=x \\oplus a_1 \\oplus a_2 \\oplus ... \\oplus a_r$ and it is decodable.",
"Furthermore, to decode the data block $x$ , $\\forall a_l, l=1,2,...,r$ must be known because of the above statement 2).",
"Thus the proof has been established.",
"Therefore, upon receiving the packet $P$ , the node $n_{k}$ is able to obtain innovative data block from node $n$ if $|{\\mathbb {A}_n} \\cap {\\bar{\\mathbb {A}}_{n_k}}| = 1.$"
],
[
"The DBS operation and encoding",
"To maximize the gain of all-to-all broadcast, the network encoding operation is expected to select the particular data blocks that enable as many neighbouring receivers as possible to decode the packet transmitted and each of these receivers must obtain innovative data block from the decoding.",
"In other words, the network encoding should operate on the data blocks that are the solutions to the following optimization problem $\\max _{{\\tilde{\\mathbb {A}}_n} \\subseteq {\\mathbb {A}_n}} {\\beta _{n}},$ where $\\tilde{\\mathbb {A}}_n$ denotes the set of data blocks selected on the node $n$ and $ {\\beta _n} $ is defined as the particular number of the neighbouring nodes that can decode and obtain innovative data from node $n$ : ${\\beta _n} = \\sum \\limits _j^{m_n} {{\\varepsilon _{nj}}}.$ More specifically, ${\\varepsilon _{nj}}{\\rm { = }}\\left\\lbrace {\\begin{array}{*{20}{c}}1&{ |\\tilde{{\\mathbb {A}}}_n \\cap {\\bar{\\mathbb {A}}_{n_j}}| = 1 }\\\\0&{\\textrm {otherwise}}\\end{array}} \\right.$ , and $j$ represents the index of receiving nodes.",
"(REF ) is a combinatorial optimization problem, which can be solved by any established search method $f(\\cdot )$ over all the subsets of ${\\mathbb {A}_n}$ .",
"To be more efficient, the node $n$ can select data blocks to transmit from the set $\\mathbb {B}=(\\bar{\\mathbb {A}}_{n_1} \\cap {\\mathbb {A}}_n) \\cup (\\bar{\\mathbb {A}}_{n_2} \\cap {\\mathbb {A}}_n) \\cup \\cdots \\cup (\\bar{\\mathbb {A}}_{n_{m_n}} \\cap {\\mathbb {A}}_n)$ with the specific approach $f(\\cdot )$ introduced aboveIn other words, $\\tilde{\\mathbb {A}}_n \\subseteq \\mathbb {A}_n$ in (REF ) can be replaced with $\\tilde{\\mathbb {A}}_n \\subseteq \\mathbb {B}$ , resulting in a smaller search space., and encodes these component data blocks into packet $P$ with the aid of the bitwise XOR operation, i.e., $P = {p_{d_1}} \\oplus {p_{d_2}} \\oplus \\cdots \\oplus {p_{d_j}}$ , where $ \\oplus $ stands for bitwise XOR, $1 \\le d_j \\le N$ , $1 \\le j \\le | \\tilde{{\\mathbb {A}}}_n | $ and the selected data blocks ${p_{d_j}} \\in \\tilde{\\mathbb {A}}_n = f (\\mathbb {B})$ ."
],
[
"The NS operation",
"For a given network topology, it is important to determine which node should broadcast its data to the neighbours at a particular time instant.",
"In other words, it is necessary to carry out NS, in addition to DBS.",
"Thus, based on $\\beta _n$ we further consider the ratio of the neighbouring nodes that can be helped among all the neighbouring nodes of the selected node per broadcast or time slot.",
"Then, the node is selected according to $\\max _i {\\beta _i}/{m_i},$ where $i = 1, 2, \\cdots , N$ .",
"By solving (REF ), the node that enables the largest proportion of neighbour nodes to be capable of obtaining innovative data blocks is selected, thus the transmission efficiency is improved."
],
[
"Simulation Results and Discussions",
"In this section, the performance of the proposed all-to-all broadcast based distributed data synchronization algorithms are demonstrated with Monte Carlo simulations.",
"We consider a wireless ad hoc network, where the number of nodes is from 5 to 11.",
"For each network size, $10^5$ network samples are randomly generated, and in each network sample the positions of the nodes are generated with uniform distributions, thus guaranteeing a diverse range of network topologies to be examined.",
"Additionally, we only keep the network samples that can be represented by connected graph, in order to ensure that it is feasible to achieve data synchronization throughout the network.",
"The comparisons are made among three data synchronization schemes: Uncoded+DBS (baseline, abbreviated as U-DBS): The data synchronization is performed without using NC.",
"In a single transmission cycle, each node broadcasts its individual data in turn following a given order, and each node uses a single time slot.",
"DBS is also invoked by each node in this scheme, using the criterion similar to (REF ), where $\\tilde{\\mathbb {A}}_n$ changes from a multi-element set to a single-element set.",
"In other words, each node broadcasts the data block that is innovative to the largest number of receivers.",
"This process is repeated in the next cycle, until the data synchronization across the network is achieved using the conventional store-and-forward protocol.",
"Coded+DBS (Alg.",
"1, abbreviated as C-DBS): The data synchronization is achieved by employing the DBS based linear NC, as presented in Sec.",
"REF , while NS is not invoked.",
"The transmission protocol can now be termed as compute-and-forward.",
"The remaining operations are the same as the scheme of U-DBS.",
"Coded+DBS+NS (Alg.",
"2, abbreviated as C-DBS-NS): In this scheme, not only the DBS based linear NC, but also the NS as introduced in Sec.",
"REF , is invoked.",
"Thus, in each time slot, only the node selected broadcasts its data.",
"This process is repeated until the data synchronization across the network is achieved.",
"As such, the concept of transmission cycle as used in the schemes of U-DBS and C-DBS becomes invalid, and the transmission protocol is also the compute-and-forward.",
"Firstly, by using the concept of average degree as defined in graph theory for characterizing the nodes' and network's connectivityAs the average degree increases, the network connection state is improved.",
", in Figure REF we evaluate the average number of time slots used by the three schemes considered, under different values of network degree, network size and packet error rate $P_e$ of a single hop.",
"The average number of time slots $S$ under a particular average degree $d$ is defined as $S{}_{d} = \\frac{{\\sum \\limits _i^{{N_d}} {s{_{id}}} }}{N_d},$ where $s{_{id}}$ denotes the number of time slots used in the data synchronization process for the $i$ th network sample under the average degree $d$ , and $N_d$ denotes the number of randomly generated network samples, which essentially also represent different network topologies.",
"Figure: Average gain versus average degreeWe can observe that the proposed C-DBS-NS scheme (the red curves) and C-DBS scheme (the blue curves) both enjoy a significant time slot usage reduction, compared to the U-DBS scheme (the green curves) in the weakly or moderately connected networks.",
"Additionally, the C-DBS-NS scheme performs best among the three schemes, especially under the moderately connected network topology, but the advantage of the C-DBS-NS scheme over the C-DBS scheme is limited.",
"Furthermore, the time slot usage increases when we have a larger $P_e$ and a larger network size.",
"Secondly, the average relative processing gain (RPG) of the proposed two schemes to the baseline U-DBS scheme under different values of network degree, network size and packet error rate $P_e$ of a single hop is depicted in Figure REF .",
"Here the average RPG is defined as ${G_d} = \\frac{{\\sum \\limits _i^{{N_d}} {{\\hat{s}_{id}}/{{\\tilde{s}}_{id}}} }}{{{N_d}}},$ where ${\\tilde{s}}_{id}$ and ${\\hat{s}}_{id}$ respectively denote the number of time slots used in the data synchronization process of the U-DBS scheme and of either the proposed C-DBS or C-DBS-NS schemes for the $i$ th network sample under the average degree $d$ .",
"We observe that for either large or small average degree, the average RPG of the C-DBS scheme is similar to that of the C-DBS-NS scheme, which is a common phenomenon under different values of network size.",
"For medium average degree, the average RPG of the C-DBS-NS scheme is significantly higher than that of the C-DBS scheme.",
"In other words, for poorly connected networks, the C-DBS scheme has a similar performance compared to the most complicated C-DBS-NS scheme; for well connected networks, the data synchronization performance (i.e., synchronization speed) of all the three schemes, including the simplest U-DBS scheme, tends to be identical; while for moderately connected networks, the C-DBS-NS scheme is shown to have the highest synchronization speed.",
"Finally, in Figure REF we compare the computational complexity of the three schemes in terms of the average number of FLOPSDue to the randomness of the topology and the network size, it is difficult to give analytic results of the computational complexity of the three schemes.",
"under different numbers of nodes.",
"We see that as the number of nodes rises, the average number of FLOPS of the C-DBS-NS scheme grows much faster than that of the U-DBS scheme, while the C-DBS scheme has a moderately higher computational complexity than the U-DBS scheme.",
"This implies that when choosing appropriate data synchronization algorithm for large-scale wireless ad hoc networks, both the time slot usage and the computational complexity of the algorithm should be considered.",
"It is advised to use the divide-and-conquer approach to reduce the network size to a moderate value.",
"Figure: Computational complexity versus the network size"
],
[
"Conclusion",
"In this paper, we have proposed a pair of linear NC and all-to-all broadcast based fast data synchronization algorithms for wireless ad hoc networks that have controlled topology.",
"For better exploiting the benefits of NC, the first algorithm C-DBS exploits data block selection, while the second algorithm C-DBS-NS exploits both the transmitting node selection and data block selection.",
"Thus, compared with the conventional uncoded approach that uses store-and-forward protocol, a more efficient compute-and-forward protocol is used in our algorithms.",
"We show that C-DBS-NS performs best among the three schemes in the moderately connected networks, while in the weakly connected networks, C-DBS achieves similarly good performance at a lower computational complexity compared to C-DBS-NS.",
"For well connected networks, however, the three schemes have almost identical performance, hence there is no need to perform NC and the simplest U-DBS approach suffices."
],
[
"ACKNOWLEDGEMENT",
"This work is financially supported by Beijing Municipal Natural Science Foundation (No.",
"L202012), the Open Research Project of the State Key Laboratory of Media Convergence and Communication, Communication University of China (No.",
"SKLMCC2020KF008), and the Fundamental Research Funds for the Central Universities (No.",
"2020RC05).",
"hudie.epsDie Hureceived the B.Eng degree in communication engineering from Wuhan University, China, in Jul.",
"2018, and the M.Eng degree in system design of flight vehicle from China Academy of Launch Vehicle Technology in Jun.",
"2021.",
"She was also a visiting student at Beijing University of Posts and Telecommunications from Jan. 2020 to Apr.",
"2021.",
"She is currently a research engineer at China Academy of Launch Vehicle Technology.",
"Her research interests include distributed information system, flying ad hoc networks and routing protocols.",
"zhuxuejun.epsXuejun Zhureceived the B.Eng degree in automatic control from National University of Defense Technology, China, in 1984, and the M.Eng degree in aircraft navigation and control from China Academy of Launch Vehicle Technology in 1987.",
"She is currently a chief designer and professorial research fellow with China Academy of Launch Vehicle Technology.",
"In 2019, she was elected to the Academician of Chinese Academy of Sciences.",
"Her research interests include networking technologies, flight vehicle design and electrical system design.",
"gongmin.epsMin Gong received the B.Eng degree in information engineering from Beijing Institute of Technology, China, in 2005, and the PhD degree in information and communication engineering from Tsinghua University, China, in 2010.",
"He is currently a chief designer with China Academy of Launch Vehicle Technology.",
"His research interests include mobile ad hoc networks, distributed information processing, flight vehicle design and game theory.",
"ShaoshiYang.epsShaoshi Yangreceived the B.Eng degree in information engineering from Beijing University of Posts and Telecommunications (BUPT), China, in 2006, and the PhD degree in electronics and electrical engineering from University of Southampton, U.K., in 2013.",
"From 2008 to 2009, he was a researcher of WiMAX standardization with Intel Labs China.",
"From 2013 to 2016, he was a Research Fellow with the School of Electronics and Computer Science, University of Southampton.",
"From 2016 to 2018, he was a Principal Engineer with Huawei Technologies Co. Ltd., where he made significant contributions to the company’s products and solutions associated with 5G base stations, wideband IoT, and cloud gaming/VR.",
"He is currently a Full Professor with BUPT.",
"His research expertise includes 5G wireless networks, massive MIMO, iterative detection and decoding, mobile ad hoc networks, distributed artificial intelligence, and cloud gaming/VR.",
"He is a member of the Isaac Newton Institute for Mathematical Sciences, Cambridge University, and a Senior Member of IEEE.",
"He received the Dean’s Award for Early Career Research Excellence from University of Southampton in 2015, the Huawei President Award of Wireless Innovations in 2018, the IEEE Technical Committee on Green Communications & Computing (TCGCC) Best Journal Paper Award in 2019, and the IEEE Communications Society Best Survey Paper Award in 2020.",
"He is an Editor for IEEE Systems Journal, IEEE Wireless Communications Letters, and Signal Processing (Elsevier).",
"He was also an invited international reviewer of the Austrian Science Fund (FWF)."
]
] | 2209.08255 |
[
[
"Necessity Specifications for Robustness"
],
[
"Abstract Robust modules guarantee to do only what they are supposed to do - even in the presence of untrusted, malicious clients, and considering not just the direct behaviour of individual methods, but also the emergent behaviour from calls to more than one method.",
"Necessity is a language for specifying robustness, based on novel necessity operators capturing temporal implication, and a proof logic that derives explicit robustness specifications from functional specifications.",
"Soundness and an exemplar proof are mechanised in Coq."
],
[
"More Bank Specifications",
"NecessityBankSpec$_a$ $\\triangleq$ from a:Account $\\wedge$ a.balance==bal next a.balance < bal onlyIf $\\exists$ o.",
"[$\\external{\\prg{o}}$ $\\wedge$ $\\access{\\prg{o}}{\\prg{a.pwd}}$] NecessityBankSpec$_b$ $\\triangleq$ from a:Account $\\wedge$ a.balance==bal next a.balance < bal onlyIf $\\exists$ o.",
"[$\\external{\\prg{o}}$ $\\wedge$ $\\calls{\\prg{o}}{\\prg{a}}{\\prg{transfer}}{\\prg{\\_, \\_, \\_}}$] NecessityBankSpec$_c$ $\\triangleq$ from a:Account $\\wedge$ a.balance==bal to a.balance < bal onlyIf $\\exists$ o.",
"[$\\external{\\prg{o}}$ $\\wedge$ $\\calls{\\prg{o}}{\\prg{a}}{\\prg{transfer}}{\\prg{\\_, \\_, \\_}}$] NecessityBankSpec$_d$ $\\triangleq$ from a:Account $\\wedge$ a.balance==bal to a.balance < bal onlyThrough $\\exists$ o.",
"[$\\external{\\prg{o}}$ $\\wedge$ $\\calls{\\prg{o}}{\\prg{a}}{\\prg{transfer}}{\\prg{\\_, \\_, \\_}}$] Prove Encapsulation (Part 1): We start by proving that a.balance is encapsulated, and subsequently that a.balance = bal is encapsulated (since bal is an integer and thus also encapsulated).",
"Per-Method Specifications (Part 2): We use a Hoare logic to prove the following classcial spec for the transfer method { a : Account $\\wedge$ a.balance == bal $\\wedge$ $\\neg$ (a' == a $\\wedge$ pass == a.pwd) } a'.transfer(pass, _, _) {$\\neg$ a.balance < bal} We then use If1-Classical and the above specification to arrive at from a : Account $\\wedge$ a.balance == bal $\\wedge$ _ calls a'.transfer(pass, _, _) next balance < bal onlyIf a' == a $\\wedge$ pass == a.pwd Then we use the Hoare logic to prove each other method in Mdl satisfies the following classical spec: { a : Account $\\wedge$ a.balance == bal } x.m($\\overline{\\prg{y}}$) {$\\neg$ a.balance < bal} where x is some object of class ${C}\\ \\in \\ {Mdl}$ .",
"We subsequently use If1-Classical to prove from a : Account $\\wedge$ a.balance == bal $\\wedge$ _ calls x.m($\\overline{\\prg{y}}$) next a.balance < bal onlyIf false Per-Step Specification (Part 3): Next we use the result from Parts 1 and 2 and Nxt-Internal to prove the following specification: from a:Account $\\wedge$ a.balance == bal next a.balance < bal onlyIf _ calls a.transfer(a.pwd, _, _) NecessityBankSpec$_a$ follows easily from above."
],
[
"NecessityBankSpec$_b$",
"The proof of NecssitiyBankSpec$_b$ is an intermediate result of NecessityBankSpec$_a$ ."
],
[
"NecessityBankSpec$_c$",
"This is not true!!!!",
"!"
],
[
"Proof Outline of The DOM",
"DOMSpec $\\triangleq$ from nd : Node $\\wedge$ nd.property = p to nd.property != p onlyIf $\\exists$ o.",
"[ $\\external {\\prg{o}}$ $\\wedge$ $( \\ \\exists$ nd':Node.",
"[ $\\access{\\prg{o}}{\\prg{nd'}}$ ] $\\vee$ $\\exists$ pr:Proxy,k:$\\mathbb{N}$.",
"[$\\, \\access{\\prg{o}}{\\prg{pr}}$ $\\wedge$ nd.parent$^{\\prg{k}}$=pr.node.parent$^{\\prg{pr.height}}$ ] $\\,$ ) $\\,$ ] Part 1: Part 2: Part 3: Part 4:"
],
[
"ERC20",
"ERC20Spec1 $\\triangleq$ from e : ERC20 $\\wedge$ e.balance(p) = m + m' $\\wedge$ m > 0 next e.balance(p) = m' onlyIf $\\exists$ p' p''.",
"[$\\calls{\\prg{p'}}{\\prg{e}}{\\prg{transfer}}{\\prg{p, m}}$ $\\vee$ e.allowed(p, p'') $\\geq$ m $\\wedge$ $\\calls{\\prg{p''}}{\\prg{e}}{\\prg{transferFrom}}{\\prg{p', m}}$] The proof of ERC20Spec1 follows much the same format as that of NecessityBankSpec, except adding a case where an authorized participant may transfer money from another's account.",
"ERC20Spec2 $\\triangleq$ from e : ERC20 $\\wedge$ p : Object $\\wedge$ p' : Object $\\wedge$ m : Nat next e.allowed(p, p') = m onlyIf $\\calls{\\prg{p}}{\\prg{e}}{\\prg{approve}}{\\prg{p', m}}$ $\\vee$ (e.allowed(p, p') = m $\\wedge$ $\\neg$ ($\\calls{\\prg{p'}}{\\prg{e}}{\\prg{transferFrom}}{\\prg{p, \\_}}$ $\\vee$ $\\calls{\\prg{p}}{\\prg{e}}{\\prg{allowed}}{\\prg{p, \\_}}$)) $\\vee$ $\\exists$ p''.",
"[e.allowed(p, p') = m + m' $\\wedge$ $\\calls{\\prg{p'}}{\\prg{e}}{\\prg{transferFrom}}{\\prg{p'', m'}}$] As with ERC20Spec1, the proof of ERC20Spec2 is similar to that of NecessityBankSpec, where we first show that $\\lnot $ e.allowed(p, p') = m is encapsulated by the internal ERC module."
],
[
"DAO",
"DAOSpec1 $\\triangleq$ from d : DAO $\\wedge$ p : Object to d.balance(p) > d.ether onlyIf false DAOSpec2 $\\triangleq$ from d : DAO $\\wedge$ p : Object next d.balance(p) = m onlyIf $\\calls{\\prg{p}}{\\prg{d}}{\\prg{repay}}{\\prg{\\_}}$ $\\wedge$ m = 0 $\\vee$ $\\calls{\\prg{p}}{\\prg{d}}{\\prg{join}}{\\prg{m}}$ $\\vee$ d.balance(p) = m The proof of DAOSpec2 starts by proving that d.balance(p) is encapsulated.",
"Using a Hoare logic and If1-Classical, we prove that if d.balance(p) changes, then either d.repay(_) or d.join(m) was called, and in both cases p being the caller was a necessary precondition.",
"For all other methods in the DAO module, we prove that false is a necessary precondition to changing d.balance(p).",
"We then use If1-Classical to raise these per-method specifications to per-step specifications.",
"We then use Excluded-Middle and logic rules to prove DAOSpec2."
],
[
"The Safe",
"SafeSpec $\\triangleq$ from s : Safe $\\wedge$ s.treasure != null to s.treasure == null onlyIf $\\neg$ inside(s.secret) The module SafeModule described below satisfies SafeSpec.",
"module SafeModule class Secret{} class Treasure{} class Safe{ field treasure : Treasure field secret : Secret method take(scr : Secret){ if (this.secret==scr) then { t=treasure this.treasure = null return t } } } The proof of SafeModule's satisfaction of SafeSpec starts by proving that $inside (s.secret)$ is encapsulated.",
"Then using Hoare logic and If1-Inside, we show that take can only expose s.treasure if s is the reciever of the call, and s.secret is the argument.",
"We then raise that per-method specification to a per-step specification using If1-Internal to prove that exposing s.treasure in a single step requires a call s.take(s.secret).",
"Then using Changes we raise the per-step specification to a general specification for all executions.",
"Finally it is enough to show that it is not possible to expose s.secret to get the desired result."
],
[
"Crowdsale",
"lacks some of the features of solidity (in particular the ability to refer to the caller of a method), and thus it is not possible to encode the Crowdsale example in .",
"It is however possible to construct proofs of parts of the specifications in Crowdsale in if we imagine a version of built upon Solidity.",
"R0 $\\triangleq$ {e : Escrow $\\wedge$ e.deposit(p) = m $\\wedge$ e.balance = m + n} e.claimRefund(p) {e.deposit(p) = 0 $\\wedge$ e.balance = n} R0 can be encoded as a traditional Hoare specification.",
"R1 $\\triangleq$ e : Escrow $\\wedge$ e.State != SUCCESS $\\longrightarrow$ sum(e.deposits) $\\leq$ e.balance R1 is encoded as a property that is true in all program states, and as such falls under our assumed logic.",
"R2_1 $\\triangleq$ from e : Escrow $\\wedge$ _ calls e.withdraw() to _ calls e.claimRefund(_) onlyIf false 1.",
"Prove that for all e : Escrow and c : Crowdsale, e.State, c.closedTime, c.raised, and c.goal are enclosed by the Crowdsale module.",
"2.",
"Prove for all methods m in Crowdsale and Escrow, the goal is never changed: {c : Crowdsale $\\wedge$ c.goal = x } c.m() {c.goal = x} {e : Escrow $\\wedge$ c : Crowdsale $\\wedge$ c.goal = x } e.m() {c.goal = x} 3.",
"Similarly to 2, prove that closedTime is never changed, and raised is never decreased.",
"4.",
"From 2, 3, and If1-Internal we get that goal, and closedTime are never changed and raised is never decreased, due to any single step computation.",
"We subsequently extend this property to an arbitrary number of steps.",
"5.",
"Following similar reasoning as in the previous steps, we prove: from e : Escrow $\\wedge$ c : Crowdsale $\\wedge$ e.State == SUCCESS $\\wedge$ e.owner == c next e.State == REFUND onlyIf now > c.closeTime $\\wedge$ c.raised < c.goal $\\wedge$ _ calls c.close() 6.",
"We then need to prove a slightly different property to show that if the escrow state is changed to REFUND, then c.raised can never be equal to or greater than c.goal.",
"from e : Escrow $\\wedge$ c : Crowdsale $\\wedge$ e.State == SUCCESS $\\wedge$ e.owner == c to e.State == REFUND $\\wedge$ c.raised >= c.goal onlyIf false 7.",
"Prove from e : Escrow $\\wedge$ _ calls e.claimRefund(_) next true onlyIf e.State == REFUND 8.",
"Prove from e : Escrow $\\wedge$ _ calls e.withdraw() next true onlyIf e.State == SUCCESS 9.",
"Prove from e : Escrow $\\wedge$ e.State == REFUND next e.State != REFUND onlyIf false 10. extend 7 to get from e : Escrow $\\wedge$ _ calls e.claimRefund(_) to true onlyIf e.State == REFUND magentaJulian: magentaJulian: This is interesting.",
"It is quite an involved proof.",
"If we can prove it, assuming some solidity language features, then it will show we can construct fairly complex proofs.",
"R2_2 $\\triangleq$ from e : Escrow $\\wedge$ _ calls e.claimRefund(_) to _ calls e.withdraw() onlyIf false Proof follows the similarly to R2_1 R3 $\\triangleq$ from e : Escrow to e.claimRefund(_) onlyIf sum(e.deposits) < goal This material is based upon work supported by the GS100000001National Science Foundationhttp://dx.doi.org/10.13039/100000001 under Grant No.",
"GS100000001nnnnnnn and Grant No. GS100000001mmmmmmm.",
"Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation."
]
] | 2209.08205 |
[
[
"Few-Shot Classification with Contrastive Learning"
],
[
"Abstract A two-stage training paradigm consisting of sequential pre-training and meta-training stages has been widely used in current few-shot learning (FSL) research.",
"Many of these methods use self-supervised learning and contrastive learning to achieve new state-of-the-art results.",
"However, the potential of contrastive learning in both stages of FSL training paradigm is still not fully exploited.",
"In this paper, we propose a novel contrastive learning-based framework that seamlessly integrates contrastive learning into both stages to improve the performance of few-shot classification.",
"In the pre-training stage, we propose a self-supervised contrastive loss in the forms of feature vector vs. feature map and feature map vs. feature map, which uses global and local information to learn good initial representations.",
"In the meta-training stage, we propose a cross-view episodic training mechanism to perform the nearest centroid classification on two different views of the same episode and adopt a distance-scaled contrastive loss based on them.",
"These two strategies force the model to overcome the bias between views and promote the transferability of representations.",
"Extensive experiments on three benchmark datasets demonstrate that our method achieves competitive results."
],
[
"Additional Ablation Study",
"We conduct ablation studies in the pre-training stage on the miniImageNet.",
"Here, except for the model trained with only $L_{CE}$ (see the first line in REF ), we adopt cross-view episodic training (CVET) mechanism and distance-scaled contrastive loss in the meta-training stage for all experiments in REF and REF .",
"Table: Ablation experiments in the pre-training stage on miniImageNet.",
"Table: Effectiveness of vector-map and map-map modules on miniImageNet.As shown in REF , compared to training with $L_{CE}$ alone, each contrastive loss used in the pre-training stage plays an important role, with $L_{local}^{ss}$ contributing the most.",
"The results from the methods introducing $L_{local}^{ss}$ indicate that contrastive learning based on extra local information can learn more generalizable representations.",
"Meanwhile, we can observe that the results obtained by training with supervision only are much lower than the results obtained by using both supervision and self-supervision.",
"The results in REF validate the effectiveness of our proposed contrastive losses, and we obtain the best results when employing all proposed contrastive losses in the pre-training stage.",
"Based on $L_{CE}$ and the other two contrastive losses using global information, we verify the effectiveness of vector-map and map-map modules by conducting experiments on them separately, as shown in REF .",
"Compared to using only global information, contrastive learning that leverages local information either in the form of vector-map or map-map can improve the transferability of the representations.",
"The best results are achieved when both forms work together."
],
[
"Experiments on Hyperparameters in the Pre-training",
"We investigate the effect of hyperparameters in the pre-training stage on classification performance.",
"Note that we use the inverse temperature parameters in our implementation.",
"To make it easy to understand, we draw the graph according to the inverse temperature parameters.",
"That is, the horizontal axis of REF actually represents the inverse of $\\tau _{1,2,3,4}$ .",
"As shown in REF , we empirically find that the best results can be achieved by setting the four temperature parameters $\\tau _{1,2,3,4}$ to 0.1 and the three balance scalars $\\alpha _{1,2,3}$ to 1.0 at the same time.",
"The results indicate that our pre-training approach is less sensitive to the two kinds of hyperparameters (within a certain range), which means it is robust.",
"Thus it is effortless to apply our approach to other methods.",
"Figure: Effect of hyperparameters τ 1,2,3,4 \\tau _{1,2,3,4} and α 1,2,3 \\alpha _{1,2,3} on miniImageNet."
],
[
"Quantitative Analysis of Model Parameters",
"In the case of using the same backbone network, we compare the number of parameters of our proposed method with the two-stage methods ProtoNet and FEAT .",
"The results are shown in REF .",
"In the pre-training stage, the number of parameters of our method is 1.9M more than ProtoNet and baseline (FEAT) due to the introduction of projection heads and a fully connected layer for our contrastive losses.",
"In the meta-training stage, a single projection head used in distance-scaled contrastive loss results in 0.4M more parameters for our method than baseline.",
"Furthermore, during meta-testing, the number of parameters of our method is the same as in baseline.",
"Therefore, the small number of additional parameters we introduce in the pre-training and meta-training stages is acceptable considering the improved classification performance of our method.",
"Table: Comparison of the number of parameters for several methods."
],
[
"Qualitative Ablation Study",
"In this section, we give qualitative analysis to verify the effectiveness of each component proposed in our method.",
"The classification results of our components are shown in REF , Thanks to the successive use of each component, the model can better adapt to novel tasks with more pictures in which the background is dominant.",
"Moreover, the accurate recognition of small objects reflects that our framework enables the representations to learn meta-knowledge that is useful for few-shot classification.",
"Therefore, it can be considered that each of our proposed components is effective.",
"Figure: 5-way 1-shot classification results on miniImageNet.",
"The green box indicates the correct classification result, while the red box indicates the incorrect result."
],
[
"Visualization and Quantitative Analysis of Feature Embeddings",
"In this section, we present the complete visualization results of Sect.",
"4.4 in the paper.",
"The results from REF illustrate that our proposed framework generates more transferable and discriminative representations on novel classes.",
"Figure: Visualization of 100 randomly sampled images for each of the 5 meta-test classes from miniImageNet by t-SNE.",
"Table: Quantitative analysis of feature embeddings using SVM.We further give quantitative analysis below.",
"We use the same sampling strategy as in the visualization experiments above and divide the training and test sets in a 1:4 ratio.",
"Then the features extracted by the four methods ProtoNet, ProtoNet+Ours, baseline and Ours are classified with SVM.",
"The mean accuracies of these methods are shown in REF .",
"The results indicate that the two-stage methods combined with our framework make feature embeddings more separable."
]
] | 2209.08224 |
[
[
"Thompson Sampling with Virtual Helping Agents"
],
[
"Abstract We address the problem of online sequential decision making, i.e., balancing the trade-off between exploiting the current knowledge to maximize immediate performance and exploring the new information to gain long-term benefits using the multi-armed bandit framework.",
"Thompson sampling is one of the heuristics for choosing actions that address this exploration-exploitation dilemma.",
"We first propose a general framework that helps heuristically tune the exploration versus exploitation trade-off in Thompson sampling using multiple samples from the posterior distribution.",
"Utilizing this framework, we propose two algorithms for the multi-armed bandit problem and provide theoretical bounds on the cumulative regret.",
"Next, we demonstrate the empirical improvement in the cumulative regret performance of the proposed algorithm over Thompson Sampling.",
"We also show the effectiveness of the proposed algorithm on real-world datasets.",
"Contrary to the existing methods, our framework provides a mechanism to vary the amount of exploration/ exploitation based on the task at hand.",
"Towards this end, we extend our framework for two additional problems, i.e., best arm identification and time-sensitive learning in bandits and compare our algorithm with existing methods."
],
[
"Introduction",
"In a stochastic multi-armed bandit (MAB) setting, an agent faces the problem of sequential decision making in the face of uncertainty.",
"At each time step, the agent takes an action from a set of actions and each action produces a reward drawn from an underlying, fixed but unknown, distribution associated with that action.",
"As the agent observes the reward at each time step, she learns about the underlying reward distributions and tries to optimize her long-term performance.",
"The agent faces the dilemma of exploiting the already acquired knowledge to maximize her immediate rewards or exploring actions from which few/no observations have been made to acquire more knowledge for potential future gains while facing the risk of immediate loss.",
"Various algorithms have been proposed to solve the exploitation-exploration dilemma in the stochastic MAB problem.",
"They include simple heuristics such as greedy and $\\epsilon $ -greedy algorithms [1], computationally intensive approaches such as Gittins indices [2], and the Upper Confidence Bound (UCB) family of algorithms which offer low computational cost and strong theoretical guarantees on the performance [3], [4], [5], [6], [7], [8].",
"Thompson proposed a simple heuristic for the stochastic MAB with Bernoulli rewards [9].",
"Starting with a prior distribution over the unknown parameters of the reward distribution of each action, the algorithm updates the posterior distributions as the actions are played.",
"At each time step, an action is chosen according to its posterior probability of being the best action.",
"This algorithm is known as Thompson sampling (TS) (and also as posterior sampling, probability matching) and has attracted a lot of attention in recent times.",
"While [10], [11], [12], [13], [14] presented empirical studies showing excellent performance of TS in comparison with other state-of-the-art algorithms along with some weak theoretical guarantees of TS, [15], [16], [17], [18], [19] and [20] have presented rigorous theoretical analysis establishing tight bounds on the regret performance of TS.",
"In this paper, we present a modified TS algorithm, referred to as Thompson Sampling with Virtual Helping Agents and Combining (TS-VHA-$\\mathsf {C}$ ).",
"The real (or, primary) agent playing the MAB game is assisted by $N-1 > 0$ virtual helping agents, with each agent generating an independent sample from the posterior distribution of each arm; All the $N$ samples ($N-1$ samples generated by the $N-1$ virtual helping agents and the one generated by the primary agent), corresponding to each arm, are processed using a combiner and which arm to play next is decided based on the values of the combined samples.",
"Here, we propose two linear combiners $\\mathsf {C1}$ and $\\mathsf {C2}$ .",
"Compared to the (conventional) TS, $\\mathsf {C1}$ increases the exploitation at the expense of exploration and $\\mathsf {C2}$ increases the exploration at the expense of exploitation.",
"Importantly, our work may be considered as a framework for varying exploration vs. exploitation for Thompson sampling, by choosing the number of virtual helping agents and the type of combiner, enabling us to achieve a better regret performance (compared to TS) for some of the MAB problems.",
"It is to be noted that one can design other combiners that achieve a different exploitation-exploration tradeoff.",
"Rest of the paper is organized as follows.",
"After introducing the details of stochastic MAB problem and the Thompson sampling in Section , we present the TS-VHA algorithm in Section .",
"Section states the main theoretical results that we present and the corresponding proofs.",
"In Section , we present simulation results to substantiate our theoretical results and Section concludes the paper."
],
[
"The Stochastic Multi-armed Bandit Problem",
"Consider an agent faced with a stochastic MAB problem.",
"Given a slot machine with $K$ arms, the agent has to choose an arm to play at each time step $t\\in \\mathbb {Z}_{>0}$ .",
"The real-valued reward produced by each arm, when played, is a random variable whose distribution is fixed but unknown with a finite support over $[0,1]$ .",
"The rewards obtained by playing an arm repeatedly are independent and identically distributed (i.i.d) and are independent of the plays of the other arms.",
"The agent has to decide which arm to play at each time $t$ , based on its observations of the past $t-1$ plays and their outcomes, to maximize the expected total reward at time $T$ , a widely used performance metric in the stochastic MAB setting.",
"The set of arms can also be referred to as the set of actions and playing arm $i$ is equivalent to choosing action $i$ .",
"Denoting the (unknown) expected reward of arm $i$ with $\\mu _i$ and the index of the arm played at time $t$ with $i(t)$ , the expected total reward at time $T$ is given by $\\mathbb {E}\\left[\\sum _{t=1}^T \\mu _{i(t)}\\right]$ .",
"An equivalent (and convenient) metric to work with is the expected total regret, given by $\\mathbb {E}[R(T)] = \\mathbb {E}\\left[\\sum _{t=1}^{T}\\mu ^{*}-\\mu _{i(t)}\\right],$ where $\\mu ^{\\ast } \\max _i \\mu _i$ and the expectation is over the random choices of arms played by the algorithm."
],
[
"Thompson Sampling",
"As stated before, Thompson sampling takes a Bayesian approach.",
"It starts by assuming an independent prior belief $P(\\tilde{\\mu }_i)$ over the expected reward of each arm $i$ and a likelihood function $P(r \\mid \\tilde{\\mu }_i)$ representing the probability of observing reward $r$ upon playing arm $i$ .",
"When an arm $i$ is played, its posterior is updated based on the observed reward $r$ using the Bayes rule: $P(\\tilde{\\mu }_i \\mid r) \\propto P(r \\mid \\tilde{\\mu }_i) P(\\tilde{\\mu }_i)$ .",
"At each time $t$ , an arm is played according to its posterior probability of having the highest mean reward; In practice, this is done by simply drawing a sample from the posterior distribution of each arm and playing the arm that produces the largest sample.",
"Algorithm REF presents the Thompson sampling.",
"[t] Thompson Sampling (TS) $K$ , priors $P(\\hat{\\mu }_i)$ , likelihood $P(r\\mid \\hat{\\mu }_i),~i=1, \\ldots , K$ .",
"$t = 1, 2, \\ldots $ Sample: Draw $\\theta _i(t) \\sim P(\\hat{\\mu }_i), i=1,\\ldots ,K$ Select action: Play arm $i(t) = \\arg \\max _i \\theta _i(t)$ and observe its reward $r(t)$ Update distribution: $P(\\hat{\\mu }_{i(t)}) \\leftarrow P(\\hat{\\mu }_{i(t)} \\mid r(t))$ , where $P(\\hat{\\mu }_{i(t)} \\mid r(t)) \\propto P(r(t) \\mid \\hat{\\mu }_{i(t)}) P(\\hat{\\mu }_{i(t)})$"
],
[
"Thompson Sampling with Virtual Helping Agents",
"Thompson sampling has three essential steps.",
"First, the agent draws a sample from the posterior distribution of the expected reward of each arm, which acts as an estimate of the arm's expected reward.",
"Next, the agent selects the arm with the largest sample and observes a reward.",
"Finally, the agent updates the posterior distribution of the expected reward of the selected arm based on the observed reward.",
"Our proposed algorithm introduces two significant changes to the Thompson sampling.",
"First, we modify the sampling step of TS by employing $N-1$ virtual helping agents.",
"Let $\\mathcal {K}=\\lbrace 1,2,\\ldots ,K\\rbrace $ represent the set of arms and let $\\mathcal {A}=\\lbrace 1\\rbrace \\cup \\lbrace 2,\\ldots ,N\\rbrace = \\lbrace 1, 2,\\ldots ,N\\rbrace $ denote the set of all agents, containing the real agent and the $N-1$ virtual helping agents.",
"At every time step $t$ , all the $N$ agents perform the sampling activity; i.e., every agent $n\\in \\mathcal {A}$ draws a sample, independently, from $P(\\hat{\\mu }_i)$ , the posterior distribution of the expected reward of $i^{\\text{th}}$ arm, $\\forall i \\in \\mathcal {K}$ .",
"At the end of the sampling step, each agent $n\\in \\mathcal {A}$ has $K$ samples $\\theta _{i,n}(t), i = 1, \\ldots , K$ , where $\\theta _{i,n}(t)$ is the sample drawn by agent $n$ from the posterior distribution of arm $i$ at time $t$ .",
"Note that the sampling activity (for generating the samples $\\theta _{i,n}(t), i = 1, \\ldots , K, n = 1, \\ldots , N$ ) is independent across the agents and across the arms.",
"Next, for each arm $i\\in \\mathcal {K}$ , we combine the samples $\\theta _{i,n}(t), n=1, \\ldots , N$ , using a combiner $f:\\mathbb {R}^{1 \\times N} \\rightarrow \\mathbb {R}$ to arrive at the final combined estimate of the expected reward $\\theta _{i}(t)$ of the $i^{\\text{th}}$ arm.",
"After the combining step, like in TS, we select the arm with the largest combined sample, observe the reward and update the posterior of the selected arm based on the observed reward.",
"Note that the posterior update is same as that of the TS.",
"[t] TS with Virtual Helping Agents (TS-VHA) $K$ , $N$ , priors $P(\\hat{\\mu }_i)$ , likelihood $P(r\\mid \\hat{\\mu }_i),~i=1, \\ldots , K$ , Combiner $f$ .",
"$t = 1, 2, \\ldots $ Sample: $n = 1, 2, \\ldots ,N$ Draw $\\theta _{i,n}(t) \\sim P(\\hat{\\mu }_i), \\forall i \\in \\mathcal {K}$ Combine: $\\theta _{i}(t) = f(\\theta _{i,1}(t), \\ldots , \\theta _{i,N}(t)), \\forall i \\in \\mathcal {K}$ Select action: Play arm $i(t) = \\arg \\max _i \\theta _i(t)$ and observe its reward $r(t)$ Update distribution: $P(\\hat{\\mu }_{i(t)}) \\leftarrow P(\\hat{\\mu }_{i(t)} \\mid r(t))$ , where $P(\\hat{\\mu }_{i(t)} \\mid r(t)) \\propto P(r(t) \\mid \\hat{\\mu }_{i(t)}) P(\\hat{\\mu }_{i(t)})$ Algorithm details the proposed Thompson sampling with virtual helping agents (TS-VHA).",
"With no virtual agents (i.e., $N-1=0$ ) and identity function as the combiner (i.e., $f(\\theta _i)=\\theta _i)$ , TS-VHA reduces to TS.",
"In other words, TS-VHA can be interpreted as TS with $N-1$ virtual helping agents and a combiner.",
"The $N-1$ virtual agents provide the real agent (who is actually trying to solve the MAB problem) with additional samples from the posterior to help her in manipulating the exploitation vs. exploration and deciding which arm to play at each time step.The agents are virtual as they do not really play the arms.",
"Compared to TS, the additional cost of TS-VHA is in generating $N-1$ additional samples and processing them through the combiner function.",
"In this work, we propose two combiners, $\\mathsf {C1}$ and $\\mathsf {C2}$ , which enable us to increase the exploitation (at the cost of exploration) and exploration (at the cost of exploitation), respectively.",
"Both the combiners are linear, having the generic form given below.",
"$f(\\theta _{i,1}(t), \\ldots , \\theta _{i,N}(t)) = \\sum _{n=1}^{N} c_n \\theta _{i,n}, \\forall i \\in \\mathcal {K}.$ Combiners $\\mathsf {C1}$ and $\\mathsf {C2}$ differ only in the choice of the coefficients $c_n$ as described in the following."
],
[
"Combiner $\\mathsf {C1}$ : Increasing Exploitation",
"Combiner $\\mathsf {C1}$ is given by $c_n = \\frac{1}{N}, \\forall n \\in \\mathcal {A}$ Employing combiner $\\mathsf {C1}$ in TS-VHA (which will be referred to as TS-VHA-$\\mathsf {C1}$ now onward) leads to higher exploitation as compared to TS.",
"Observe that, for any arm $i$ , the variance of the distribution of $\\theta _{i}(t)$ is $\\frac{1}{N}$ times that of the distribution of $\\theta _{i,n}(t), n=1,\\ldots ,N$ , whereas the mean remains the same.",
"Thus, for each arm, the variance of the distribution of the combined sample is lower compared to the variance of the posterior distribution of that particular arm.",
"Thus, TS-VHA-$\\mathsf {C1}$ places more confidence on its empirical estimates $\\hat{\\mu }_i(t)$ , resulting in increased exploitation and lower exploration when compared with TS.",
"As $N\\rightarrow \\infty $ , $\\theta _i(t) \\rightarrow \\hat{\\mu }_i(t)$ and TS-VHA-$\\mathsf {C1}$ emulates greedy decision making."
],
[
"Combiner $\\mathsf {C2}$ : Increasing Exploration",
"Combiner $\\mathsf {C2}$ is for increasing the variance of the distribution of $\\theta _i(t)$ , the combined sample for arm $i$ , compared to the posterior distribution of arm $i$ and is given by the following set of coefficients $c_n$ : $& When~N~is~an~even~integer, \\\\&c_n = \\frac{1}{N}+\\left(\\sqrt{\\frac{N^2+1}{N}}\\right)^{n+1}, \\ n=1, \\ldots , N. \\\\& When~N~is~an~odd~integer, \\\\&c_n = \\left\\lbrace \\begin{array}{l l} \\frac{1}{N}+\\left(\\sqrt{\\frac{N+1}{N}}\\right)^{n+1}, & n=1, \\ldots , N-1, \\\\\\frac{1}{n}, & n=N.",
"\\end{array} \\right.$ Observe that, for any arm $i$ , the variance of the distribution of $\\theta _{i}(t)$ is $N$ times that of the distribution of $\\theta _{i,n}(t), n=1,\\ldots ,N$ , whereas the mean remains the same.",
"Due to this increase in the variance, TS-VHA with $\\mathsf {C2}$ as its combiner (which will be referred to as TS-VHA-$\\mathsf {C2}$ hereafter) places less confidence on its empirical estimates $\\hat{\\mu }_i(t)$ and leads to higher exploration as compared to TS.",
"Increasing the number of agents in this case makes TS-VHA-$\\mathsf {C2}$ to over-explore.",
"Though our focus in this work is on the two linear combiners $\\mathsf {C1}$ and $\\mathsf {C2}$ (given by (REF ), (REF ), (REF ), ()), note that TS-VHA provides a generic framework to manipulate exploitation vs. exploration.",
"One can design other forms of combiners with desired exploitation-exploration tradeoff for a wide-range of MAB problems for which TS can be applied.",
"To highlight this point, we present $\\mathsf {C3}$ , a third combiner."
],
[
"Combiner $\\mathsf {C3}$ : Dynamic Exploitation",
"$\\mathsf {C3}$ is a non-linear combiner that computes the combined sample $\\theta _i(t)$ for each arm $i\\in \\mathcal {K}$ as $\\theta _i(t) = \\max \\left(\\sum _{n=1}^{N(t)} \\frac{1}{N(t)}\\theta _{i,n}(t),\\min _{j \\in \\mathcal {K}}(\\hat{\\mu }_j(t))\\right), \\forall i \\in \\mathcal {K}$ where $\\hat{\\mu }_j(t)$ is the observed empirical mean reward of arm $i$ at time $t$ , $\\theta _{i,n}(t)$ is the sample generated by agent $n$ for arm $i$ at time $t$ and $N(t)$ is the total number of agents (among which $N-1$ are the virtual helping agents) which is dynamically determined at each time step $t$ as, $N(t) = \\left\\lfloor {\\max (1, t\\cdot \\tilde{\\Delta })}\\right\\rfloor $ where $\\tilde{\\Delta }=\\hat{\\mu }^{(1)}(t)-\\hat{\\mu }^{(2)}(t)$ , and $\\hat{\\mu }^{(1)}(t)$ , $\\hat{\\mu }^{(2)}(t)$ are the largest and the second-largest values, respectively, in the set $\\lbrace \\hat{\\mu }_1(t), \\ldots , \\hat{\\mu }_K(t)\\rbrace $ .",
"In (REF ), the term $\\sum _{n=1}^{N(t)} \\frac{1}{N(t)}\\theta _{i,n}(t)$ is similar to combiner $\\mathsf {C1}$ with $N(t)$ number of agents.",
"Note that, here, the number of agents $N(t)$ is a function of time unlike in $\\mathsf {C1}$ .",
"As discussed previously, the value of $N(t)$ commands the exploitation-exploration trade-off.",
"The expression for $N(t)$ in (REF ) is based on the following intuition: With time, we expect our best empirical arm to be the optimal arm with increasing confidence.",
"Thus, tuning TS to increase exploitation with time could reduce the regret incurred at later time steps and improve its performance.",
"If the difference in the empirical means of the top two candidate arms $\\tilde{\\Delta }$ is high, it may suggest that the best empirical arm is indeed the optimal arm.",
"Increasing exploitation in this case may help reduce the cumulative regret on average.",
"On the other hand, if $\\tilde{\\Delta }$ is low, we may have to explore more to better discern the optimal arm.",
"Hence, we set $N(t) \\propto \\tilde{\\Delta }$ .",
"Based on the above intuitionWe would like to emphasize that these points are only our intuition and we do not have a mathematical justification, we hypothesize that the number of agents to be deployed should be dependent on time $t$ and $\\hat{\\mu }^1(t) - \\hat{\\mu }^2(t)$ .",
"Next, inspired from [14] we apply a max operation (REF ).",
"Note that, unlike OBS, we take the maximum of the averaged estimate and the minimum among the empirical means of all arms.",
"We observe that this step yields superior empirical results.",
"In Section , we provide simulation studies that prove the effectiveness of TS-VHA, for all the three combiners $\\mathsf {C1}$ , $\\mathsf {C2}$ and $\\mathsf {C3}$ , on Bernoulli bandits and Gaussian bandits.",
"However, due to the analytical tractability of Gaussian distribution, we focus on Gaussian bandits while analyzing the regret performance of TS-VHA.",
"We provide the mathematical regret analysis only for combiners $\\mathsf {C1}$ and $\\mathsf {C2}$ and leave the analysis of the non-linear combiner $\\mathsf {C3}$ for future work."
],
[
"Gaussian Bandits",
"In the rest of the paper, we mainly focus on stochastic multi-armed bandits where the likelihood of the reward distributions are Gaussian; To be precise, the likelihood of the reward of arm $i\\in \\mathcal {K}$ is Gaussian distributed with mean $\\mu _i$ (unknown a priori) and unit varianceAs in [16], we consider the single-parameter model where only the mean of the reward distribution is unknown.",
"We do not consider the two-parameter model where both mean and variance of the reward distribution are unknown.. Equivalently, the likelihood of $r_i(t)$ , reward from arm $i$ at time $t$ , given parameter $\\mu _i(t)$ , is given by $\\mathcal {N}(\\mu _i(t),1)$ .",
"Denote the arm played at time $t$ with $i(t)$ and the number of plays of arm $i$ until (and including) $t-1$ with $k_i(t)$ .",
"Define $\\hat{\\mu }_i(t) \\frac{\\sum _{\\tau =1:i(\\tau )=i}^{t-1}r_i(\\tau )}{k_i(t)+1}$ , and $\\hat{\\mu }_i(1) 0$ .",
"With the Gaussian likelihood, it is convenient to use Gaussian priors.",
"Consider $\\mathcal {N}\\left(\\hat{\\mu }_i(t), \\frac{1}{k_i(t)+1}\\right)$ as the prior for $\\mu _i$ at time $t$ .",
"When arm $i$ is played at time $t$ , the posterior distribution for $\\mu _i$ , by applying Bayes rule, turns out to be $\\mathcal {N}\\left(\\hat{\\mu }_i(t+1),\\frac{1}{k_i(t+1)+1}\\right)$ .",
"By using the Gaussian priors and likelihoods in Algorithm REF , TS can be employed for Gaussian bandits and is referred to as TS using Gaussian priors."
],
[
"TS-VHA using Gaussian Priors",
"For a Gaussian bandit, we can apply TS-VHA by using Gaussian priors and Gaussian likelihoods in Algorithm resulting in Algorithm REF .",
"The additional step of generating multiple samples and combining them in TS-VHA may alter the distribution of $\\theta _i(t)$ in Algorithm .",
"However, with Gaussian distributions and a linear combiner $f$ (such as the one given by (REF )), the distribution of $\\theta _i(t)$ remains Gaussian.",
"[t] TS-VHA using Gaussian Priors $\\mathcal {K}, N, Set \\ \\mu _i(1) = 0 \\: \\forall \\: i \\in $ priors $P(\\hat{\\mu }_i)$ , likelihood $P(r\\mid \\hat{\\mu }_i),~i=1, \\ldots , K$ , Combiner $f$ .",
"$t = 1, 2, \\ldots $ Sample: $n = 1, 2, \\ldots ,N$ Draw $\\theta _{i,n}(t) \\sim P(\\hat{\\mu }_i), \\forall i \\in \\mathcal {K}$ Combine: $\\theta _{i}(t) = f(\\theta _{i,1}(t), \\ldots , \\theta _{i,N}(t)), \\forall i \\in \\mathcal {K}$ Select action: Play arm $i(t) = \\arg \\max _i \\theta _i(t)$ and observe its reward $r(t)$ Update distribution: $P(\\hat{\\mu }_{i(t)}) \\leftarrow P(\\hat{\\mu }_{i(t)} \\mid r(t))$ , where $P(\\hat{\\mu }_{i(t)} \\mid r(t)) \\propto P(r(t) \\mid \\hat{\\mu }_{i(t)}) P(\\hat{\\mu }_{i(t)})$ In the next Section, we bound the finite time expected regret of TS-VHA-$\\mathsf {C1}$ and TS-VHA-$\\mathsf {C2}$ for Gaussian bandits.",
"Note that we have investigated the performance of TS-VHA-$\\mathsf {C3}$ only through simulation experiments, presented in Section .",
"Understanding the theoretical implications of $\\mathsf {C3}$ would be interesting and we will consider it in our future work."
],
[
"Regret Analysis",
"For the finite time regret analysis presented in this section, we consider employing TS-VHA with $N-1 > 0$ virtual helping agents for Gaussian bandits with reward distribution being finite support over [0, 1].",
"When we chose $\\mathsf {C1}$ as the combiner, the variance of the combined sample $\\theta _i(t), \\forall i \\in \\mathcal {K}$ , gets scaled by $1/N$ compared to the variance of the posterior distribution of arm $i, \\forall i \\in \\mathcal {K}$ .",
"With combiner $\\mathsf {C2}$ , variance of $\\theta _i(t), \\forall i \\in \\mathcal {K}$ , gets scaled by $N$ compared to the variance of the posterior distribution of arm $i, \\forall i \\in \\mathcal {K}$ .",
"In both the cases, the mean of the combined sample for each arm is equal to the mean of the posterior distribution of the corresponding arm.",
"Equivalently, at each time step $t$ , TS-VHA-$\\mathsf {C1}$ results in scaling the variance of $\\theta _i(t), \\forall i \\in \\mathcal {K}$ by a factor $1/N$ and TS-VHA-$\\mathsf {C2}$ scales the variance of $\\theta _i(t), \\forall i \\in \\mathcal {K}$ by $N$ , when compared to Thompson sampling.",
"To unify the regret analysis of TS-VHA-$\\mathsf {C1}$ and TS-VHA-$\\mathsf {C2}$ , we introduce $1/\\gamma $ as the factor that determines the variance scaling.",
"Thus, $\\gamma > 1$ corresponds to TS-VHA-$\\mathsf {C1}$ , $\\gamma \\in (0,1)$ corresponds to TS-VHA-$\\mathsf {C2}$ .",
"Theorem 1 For the $K$ -armed stochastic bandit problem, Thompson sampling with virtual helping agents using Gaussian priors and with variance scaling factor $\\gamma $ has expected regret at time $T \\ge K$ .",
"$For\\ \\gamma \\ \\in (0,4),$ $\\mathbb {E}[R(T)]\\le \\sum _{i=2}^{K} \\left(c^1_i \\ln {T\\Delta _i^2} + f_i^1(\\beta ,\\gamma ,\\epsilon )\\Delta _i+ \\frac{9.5}{\\Delta _i}\\right)$ $For\\ \\gamma \\ \\ge 4,$ $\\mathbb {E}[R(T)]\\le \\sum _{i=2}^{K}(c^1_i \\ln {T\\Delta _i^2} +\\\\c^{\\prime }\\left(\\frac{T^{1+\\epsilon -\\frac{2\\beta }{\\gamma }}-1 }{1+\\epsilon -\\frac{2\\beta }{\\gamma }} + g(\\epsilon )+1\\right)\\Delta _i+ \\frac{9.5}{\\Delta _i})$ where $c_i^1 = \\frac{2(H(\\beta )+1)\\Delta _i}{\\gamma (y_i-x_i )^2}, f_i^1(\\beta ,\\gamma ,\\epsilon ) = c^{\\prime }(g(e)+\\zeta (\\frac{2\\beta }{\\gamma }-\\epsilon ) )+1$ and $\\beta , \\epsilon , y_i, x_i, \\Delta _i, c^{\\prime }$ are all constants at time $T \\ge K$ ."
],
[
"Proof of Theorem ",
"We adopt the notation and definitions from [20] and follow the same methodology as that of [20] in analyzing the finite cumulative regret achieved by TS-VHA using Gaussian priors when employed over a $K$ armed Gaussian bandit.",
"Without loss of generality, we assume that $\\mu ^{\\ast }=\\mu _1> \\arg \\max _{i\\ne 1} \\mu _i$ .",
"Definition 1 $i(t)$ denotes the arm played at time $t$ , $k_{i}(t)$ denotes the number of plays of arm $i$ until, and including, time $t-1$ .",
"$\\hat{\\mu }_i(t)$ denotes the empirical mean, given by $\\hat{\\mu }_i(t)=\\frac{\\sum _{\\tau =1:i(\\tau )=i}^{t-1}r_i(\\tau )}{k_i(t)+1}$ , where $r_i(t)$ denotes the reward observed from arm $i$ at time $t$ and $\\hat{\\mu }_i(t)=0$ when $k_i(t)=0$ .",
"Definition 2 $\\theta _{i,n}(t)$ denotes the $n^{\\text{th}}$ sample generated, independently, from $\\mathcal {N}\\left( \\hat{\\mu }_i(t), \\frac{1}{k_i(t)+1}\\right)$ , the posterior distribution of arm $i$ at time $t$ and $\\theta _{i}(t)=f(\\theta _{i,1}, \\ldots , \\theta _{i,N})$ .",
"Definition 3 For arm $i=2, \\ldots , K$ , $x_i$ and $y_i$ denote thresholds such that $\\mu _i<x_i<y_i<\\mu _1$ .",
"Definition 4 For $i=2,\\ldots , K$ , $E_i^\\mu (t)$ is the event $\\hat{\\mu }_i(t)\\le x_i$ and $E_i^\\theta (t)$ is the event $\\theta _i(t)\\le y_i$ .",
"Definition 5 $\\mathcal {F}_t = \\lbrace i(\\tau ),r_{i(\\tau )}(\\tau ),\\tau =1,2,\\ldots , t\\rbrace $ is the history of arm play until time $t$ , where $i(\\tau )$ is the arm played at time $\\tau $ and $r_{i(\\tau )}(\\tau )$ is the reward observed from arm $i(\\tau )$ at time $\\tau $ .",
"Define $\\mathcal {F}_0=\\emptyset $ .",
"By definition, $\\mathcal {F}_0 \\subseteq \\mathcal {F}_1 \\subseteq \\ldots \\mathcal {F}_{T-1}$ .",
"Definition 6 Define $p_{i,t}$ as the probability $p_{i,t} = \\text{Pr}(\\theta _1(t)>y_i\\mid \\mathcal {F}_{t-1}).$ The expected total regret in time $T$ is given by $\\mathbb {E}[R(T)] = \\mathbb {E}\\left[\\sum _{t=1}^{T}(\\mu ^{*}-\\mu _{i(t)})\\right]=\\sum _{i}\\Delta _{i}\\mathbb {E}[k_{i}(T)], $ where $\\Delta _i=\\mu ^{\\ast }-\\mu _i$ and $\\mu ^{\\ast } \\max _i \\mu _i$ .",
"In order to bound the expected regret, we need to bound $\\mathbb {E}[k_{i}(T)]$ for $i \\ne 1$ , which can be decomposed into three terms as follows: $\\mathbb {E}[k_{i}(T)]&=\\sum _{t=1}^{T}\\text{Pr}(i(t)=i) \\\\&=\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t), E_{i}^{\\theta }(t)\\right) \\\\&\\quad +\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t), \\overline{E_{i}^{\\theta }(t)}\\right)\\\\ &\\qquad +\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,\\overline{E_{i}^{\\mu }(t)}\\right)$ We will now consider the terms (), () and () individually.",
"In the following, we consider $x_i=\\mu _i+\\frac{\\Delta _i}{3}$ , $y_i=\\mu _1-\\frac{\\Delta _i}{3}$ and let $L_i(T) = \\frac{2 \\ln {T\\Delta _i^2}}{\\gamma (y_i-x_i)^2}$ ."
],
[
"Term (",
"For $k\\ge 1$ , let $\\tau _{k}$ be the time step at which the first arm is played for the $k^{\\text{th}}$ time, and let $\\tau _{0} = 0$ .",
"Then, as shown in [20] (cf.",
"Eqn.",
"(4), Section 2.1 in [20]), for $i \\ne 1$ , $\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t), E_{i}^{\\theta }(t)\\right) \\le \\sum _{k=0}^{T-1}\\mathbb {E}\\left[\\frac{(1-p_{i,\\tau _{k}+1})}{p_{i,\\tau _{k}+1}}\\right].$ It is easy to verify that (REF ) does not get affected by the distribution of the sample $\\theta _i(t), \\forall i \\in \\mathcal {K}$ .",
"Hence, (REF ) holds good for TS as well as TS-VHA.",
"We utilize (REF ) to prove the following bound on term ().",
"Lemma 1 For $\\gamma \\in (0,4)$ , $\\sum _{t=1}^{T} \\text{Pr} ( i(t) = i, E_{i}^{\\mu }(t), E_{i}^{\\theta }(t))\\le H(\\beta )L_i(T) + \\\\ c^{\\prime } \\left( g(\\epsilon ) + \\zeta (\\frac{2\\beta }{\\gamma }-\\epsilon ) \\right) + \\frac{4}{\\Delta _i^2},$ For $\\gamma \\ge 4$ , $\\sum _{t=1}^{T}\\text{Pr}( i(t) = i, E_{i}^{\\mu }(t), E_{i}^{\\theta }(t))\\le H(\\beta )L_i(T) + \\\\ c^{\\prime } \\left( g(\\epsilon ) + \\frac{T^{1+\\epsilon -\\frac{2\\beta }{\\gamma }}-1 }{1+\\epsilon -\\frac{2\\beta }{\\gamma }}\\right) + \\frac{4}{\\Delta _i^2},$ where $\\beta \\in [1,2)$ , $\\epsilon > 0$ and $\\zeta $ is the Riemann zeta function.",
"Please refer to Appendix ."
],
[
"Term (",
"Lemma 2 For $i \\ne 1$ , $\\sum _{t=1}^{T}Pr\\left(i(t)=i,E_{i}^{\\mu }(t), \\overline{E_{i}^{\\theta }(t)}\\right) \\le L_i(T)+\\frac{1}{\\Delta _i^2}$ $\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t), \\overline{E_{i}^{\\theta }(t)}\\right)$ can be subdivided into two parts based on the values of $k_i(T)$ .",
"$\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t), \\overline{E_{i}^{\\theta }(t)}\\right) = \\\\ \\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t),k_i(T)\\le L_i(T), \\overline{E_{i}^{\\theta }(t)}\\right) \\\\ + \\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t),k_i(T)> L_i(T), \\overline{E_{i}^{\\theta }(t)}\\right)$ The first term on the RHS of (REF ) is bounded by $\\mathbb {E}\\left[\\sum _{t=1}^T I\\left(i(t)=i,k_i(t)<L_i(T)\\right) \\right]$ which is upper bounded by $L_i(T)$ .",
"We now bound the second term on the RHS.",
"$\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,E_{i}^{\\mu }(t),k_i(T)> L_i(T), \\overline{E_{i}^{\\theta }(t)}\\right) \\\\\\le \\mathbb {E}\\left[\\sum _{t=1}^T \\text{Pr}\\left(i(t)=i,\\overline{E_{i}^{\\theta }(t)}\\;|\\; k_i(t)>L_i(T),E_{i}^{\\mu }(t),\\mathcal {F}_{t-1}\\right)\\right] \\\\\\le \\mathbb {E}\\left[\\sum _{t=1}^T \\text{Pr}\\left( \\theta _i(t)>y_i\\;|\\; k_i(t)>L_i(T),\\hat{\\mu }_i(t)\\le x_i,\\mathcal {F}_{t-1} \\right)\\right]$ Note that, $\\theta _i(t) \\sim \\mathcal {N}\\left(\\hat{\\mu }_i(t),\\frac{1}{\\gamma (k_i(t)+1)}\\right)$ .",
"Let $\\nu _i(t) \\sim \\mathcal {N}\\left(x_i,\\frac{1}{\\gamma (k_i(t)+1)}\\right)$ .",
"Then, as $\\hat{\\mu }_i(t)\\le x_i$ $\\text{Pr}\\left(\\theta _i(t)>y_i|k_i(t)>L_i(T),\\hat{\\mu }_i(t)\\le x_i,\\mathcal {F}_{t-1}\\right) \\\\\\le \\text{Pr}\\left(\\nu _i(t)>y_i|k_i(t)>L_i(T),\\hat{\\mu }_i(t)\\le x_i,\\mathcal {F}_{t-1}\\right)$ Using [ineq3]Inequality 3, for any fixed $k_i(t)>L_i(T)=\\frac{2\\ln (T\\Delta _i^2)}{\\gamma (y_i-x_i)^2}$ , $\\text{Pr}(\\nu _i(t)>y_i) & \\le \\frac{1}{2} e^{-\\frac{\\gamma (k_i(t)+1)(y_i-x_i)^2}{2}} \\\\ & \\le \\frac{1}{2} e^{-\\frac{\\gamma L_i(T)(y_i-x_i)^2}{2}}\\\\ & \\le \\frac{1}{T \\Delta _i^2}$ This results in, $\\sum _{t=1}^T \\text{Pr}\\left( \\theta _i(t)>y_i\\;|\\; k_i(t)>L_i(T),\\hat{\\mu }_i(t)\\le x_i,\\mathcal {F}_{t-1} \\right) \\le \\frac{1}{\\Delta _i^2}, $ bounding the second term on the RHS of (REF ) with $\\frac{1}{\\Delta _i^2}$ ."
],
[
"Term (",
"Term () denotes the probability of pulling the sub-optimal arm $i$ when it is neither well estimated nor well sampled.",
"Lemma 3 For $i \\ne 1$ , $\\sum _{t=1}^{T}\\text{Pr}\\left(i(t)=i,\\overline{E_{i}^{\\mu }(t)}\\right)\\le \\frac{1}{d_i(x_i,\\mu _i)} \\le \\frac{9}{2\\Delta _i^2}+1.$ The proof for Lemma REF follows from [20].",
"Since, the proof for the Lemma REF doesn't depend on the posterior distribution of the arms, the proof provided for Lemma 2.15 in[20] holds valid as a proof for our Lemma REF .",
"$\\mathbb {E}[k_{i}(T)]$ can be bounded by substituting Lemma REF , REF and REF in (REF ) and using this bound on $\\mathbb {E}[k_{i}(T)]$ in (REF ) completes the proof of Theorem REF ."
],
[
"Simulation Experiments",
"In this section, we present computational experiments that illustrate the potential benefits of TS-VHA.",
"In the following, TS-VHA-$\\mathsf {C1}$ -VA$n$ and TS-VHA-$\\mathsf {C2}$ -VA$n$ denote TS-VHA with $n$ virtual helping agents, with combiner $\\mathsf {C1}$ and $\\mathsf {C2}$ , respectively.",
"Note that, as per the notation introduced in Section , $N-1=n$ and TS corresponds to $N=1$ with identity function as the combiner."
],
[
"Gaussian Bandits",
"We evaluate the performance of TS-VHA-$\\mathsf {C1}$ and TS-VHA-$\\mathsf {C2}$ and compare it with TS for Gaussian bandits.",
"First, we consider a 20 armed bandit problem with reward from arm $i$ modeled as $\\mathcal {N}(\\mu _i,1)$ , where the mean reward $\\mu _i$ is independently sampled from $\\mathcal {U}[0,1]$ .",
"Fig.",
"REF shows the cumulative regret over 10000 time steps, averaged over 1000 independently sampled problem instances.",
"Fig.",
"REF corresponds to a second Gaussian bandit problem with 200 arms, keeping all the other details same as that of the 20 armed bandit.",
"For TS-VHA-$\\mathsf {C1}$ -VA$n$ (TS-VHA-$\\mathsf {C2}$ -VA$n$ ), exploitation (exploration) increases with $n$ , as compared to TS.",
"As can be observed from the plots, increasing exploitation through TS-VHA-$\\mathsf {C1}$ improves the regret performance.",
"It should be noted that having more exploitation might turn out to be counter-productive.",
"As discussed in Section REF , as $n$ grows to a higher value, TS-VHA-$\\mathsf {C1}$ -VA$n$ starts behaving like the greedy algorithm.",
"Observe that, in Fig.",
"REF , TS-VHA-$\\mathsf {C1}$ -VA4 accumulates more regret and performs poorly relative to TS-VHA-$\\mathsf {C1}$ -VA$n$ , $n=1,2,3$ .",
"Figure: Variation in the cumulative regret with TS, TS-VHA-𝖢1\\mathsf {C1} and TS-VHA-𝖢2\\mathsf {C2} for Gaussian bandits.Fig.",
"REF shows the distribution of final cumulative regret at the end of 10000 time steps, over 1000 runs, for a Gaussian bandit with 20 arms.",
"Reward from arm $i$ is distributed as $\\mathcal {N}(\\mu _i,1)$ , where $\\mu _i, i=1,\\ldots ,20$ , is chosen by sampling independently from $\\mathcal {U}[0,1]$ once at the beginning of the experiment and kept constant throughout the 1000 runs.",
"TS-VHA-$\\mathsf {C1}$ has a higher variance in its final cumulative regret and is thus not suitable for risk-sensitive scenarios.",
"On the other hand, TS-VHA-$\\mathsf {C2}$ results in a lower variance in its final cumulative regret and may be preferred in risk-averse applications."
],
[
"Bernoulli Bandits",
"We now evaluate the performance of TS-VHA over Bernoulli bandits, i.e., bandit problems with Bernoulli distributed rewards and Beta distribution as the prior."
],
[
"Bernoulli Bandit with Randomized Mean Rewards",
"Similar to the Gaussian bandits discussed above, we consider two Bernoulli bandit problems, one with 20 arms and the other with 200 arms, with mean reward of each arm is independently sampled from $\\mathcal {U}[0,1]$ .",
"Fig.",
"REF and Fig.",
"REF shows the cumulative regret over 100000 time steps, averaged over 1000 independently sampled problem instances, for the 20 armed bandit and the 200 armed bandit, respectively."
],
[
"Real World Datasets",
"Here, we show the effectiveness of TS-VHA-$\\mathsf {C1}$ on the real-world data sets Coupon-Purchase [25] and edX-Courses [26].",
"Figure: Considering coupon purchase rate multiplied by the normalized selling price as the mean reward of each arm.The Coupon-Purchase dataset contains discount coupons applied to online purchases.",
"From the dataset, we have considered only 142 coupons that correspond to products priced less than or equal to 200 price units and purchased by at least one customer (as in [27]).",
"For these 142 coupons, we have extracted the purchase rate that lies within $[0, 0.3]$ and the final selling price normalized by 200 price units, which lies within $(0, 1]$ .",
"With each coupon as an independent arm that (when played) generates a binary valued reward according to a Bernoulli distribution, we formulate two bandit problems.",
"In the first one, the mean reward of an arm is equal to the corresponding coupon purchase rate and, in the second problem, the mean reward of each arm is equal to the coupon purchase rate multiplied by the corresponding selling price normalized by 200.",
"By modeling the mean reward of each arm using a Beta distribution, we present the performance of TS and TS-VHA-$\\mathsf {C1}$ -VA$n$ , $n=1,2,3$ , in Fig.",
"REF and Fig.",
"REF , corresponding to the first and the second problem, respectively.",
"As can be seen, TS-VHA with combiner $\\mathsf {C1}$ helps achieving a lower cumulative regret for both the problems.",
"Figure: Considering the course certification rate multiplied by the course participation rate as the mean reward of each arm.The edX-Courses dataset contains information regarding 290 Harvard and MIT courses and, as in [28], we compute the normalized course participation rates (that lie within unit interval) through min-max normalization of the number of participants in each course and obtain the course certification rates by dividing the number of certified participants in each course by the number of course participants.",
"We formulate two bandit problems by considering each course as an independent arm that returns a Bernoulli distributed reward.",
"In the first problem, the mean reward of each arm is given by the course certification rate and in the second, course certification rate multiplied by the course participation rate is the mean reward.",
"With Beta distribution as the prior for the mean reward of each arm, Fig.",
"REF and Fig.",
"REF compare the cumulative regret performance of TS and and TS-VHA-$\\mathsf {C1}$ -VA$n$ , $n=1,2,3$ , for the first and the second problem, respectively.",
"Figure: Linear GaussianFigure: Bernoulli bandit with two arms.",
"Mean rewards: 0.51, 0.5.Figure: Gaussian bandit, 20 arms"
],
[
"Time-Sensitive Bandit Learning",
"Most of the bandit algorithms focus on learning the optimal arm (or, action).",
"Often, especially for bandit problems having a very large set of arms, convergence to optimality may take a long time rendering them not useful in some practical applications.",
"For example, in the case of a recommender system, the learning agent may be required to impress upon the users through its near optimal recommendations during the early interactions; Or, the learning agent may not have enough number of interactions with each user to converge onto perfect recommendations.",
"In [29], the authors have addressed the problem of learning near-optimal satisficing actions considering situations where the near term performance is more important than the performance over an asymptotically long time horizon, or, the optimal action is costly to learn relative to near-optimal actions.",
"Satisficing Thompson Sampling (STS), proposed in [29], performs time-sensitive learning by modifying the Select action step of TS.",
"Recall that, in TS (i.e., Algorithm REF ), $\\theta _i(\\tau )$ is the sample drawn from posterior of arm $i$ at time $\\tau $ , $i(\\tau )$ is the index of arm played at time $\\tau $ and $\\theta _{i(\\tau )}$ is the expected reward of the arm played at time $\\tau $ .",
"At each time step $t$ , STS identifies an $\\epsilon $ -optimal action through the following Select action step.",
"Select action (in STS): Let $i(t) = \\arg \\max _i \\theta _i(t)$ .",
"Let $ {\\hat{\\tau }} = \\min \\lbrace \\tau \\in \\lbrace 1, \\ldots , t-1\\rbrace : \\theta _{i(\\tau )} + \\epsilon \\ge \\theta _{i(t)}\\rbrace $ .",
"If $\\hat{\\tau }$ is not null, then $i(t)=i(\\hat{\\tau })$ .",
"Essentially, at each time step $t$ , STS chooses to play an arm $k$ that has already been played in the past, as long as the estimate of the expected reward from arm $k$ is not lower than the estimate of the expected reward from an optimal arm (optimal at time $t$ as per the TS) by $\\epsilon $ units.",
"Thus, STS exploits more by re-using near-optimal satisficing arms rather than exploring un-used arms.",
"With per period regret as the performance metric (that captures the time preference), the simulation experiments reported in [29] show that STS can significantly outperform TS when the optimal action is costly to learn relative to satisficing near-optimal actions.",
"We consider four simulation experiments that are same as those considered in [29] and compare the performance of TS-VHA-$\\mathsf {C1}$ with that of STS and TS in Fig.",
"REF .",
"For all the four experiments, we compute the per period regret over 500 time steps, averaged over 5000 runs.",
"Fig.",
"REF considers a deterministic bandit with 250 arms with mean reward of each arm sampled independently from $\\mathcal {U}[0,1]$ .",
"As every arm, when played, returns the reward equal to its mean reward, it is referred to as a deterministic bandit.",
"Performance of TS-VHA-$\\mathsf {C1}$ -VA2 and TS-VHA-$\\mathsf {C}1$ -VA3 is very close to that of STS.",
"Fig.",
"REF corresponds to a bandit that differs from that of Fig.",
"REF as follows: Whenever an arm is played, the observed reward is a Bernoulli random variable with success probability equal to the mean reward.",
"Note that, in Fig.",
"REF and Fig.",
"REF , we consider $\\epsilon =0.05$ for the STS.",
"It can be observed that TS-VHA-$\\mathsf {C1}$ -VA2 and TS-VHA-$\\mathsf {C1}$ -VA3 perform better than STS for time step values above (approximately) 150 and 50, respectively.",
"Fig.",
"REF corresponds to a 250 armed Gaussian bandit with mean reward of each arm sampled independently from $\\mathcal {N}(0,1)$ ; When an arm is played, the realized reward is the sum of the arm's mean reward and an independent sample from $\\mathcal {N}(0,1)$ .",
"Here, $\\epsilon =0.5$ for the STS.",
"Finally, we consider linear Gaussian bandit with 250 arms in Fig.",
"REF .",
"The mean rewards are given by the vector $\\mathbf {L}\\theta \\in \\mathbb {R}^{250\\times 1}$ , where $\\theta \\in \\mathbb {R}^{250\\times 1}$ is sampled from $\\mathcal {N}(0,\\mathbf {I})$ and $\\mathbf {L}\\in \\mathbb {R}^{250\\times 250}$ is a random matrix with each row drawn independently and uniformly from the unit sphere.",
"While $\\theta $ is unknown a priori, $\\mathbf {L}$ is known before hand.",
"When an arm is played the observed reward is the sum of the mean reward and an independent sample from $\\mathcal {N}(0,2)$ .",
"As can be observed from Fig.",
"REF and Fig.",
"REF , TS-VHA-$\\mathsf {C1}$ outperforms TS and STS for both independent Gaussian and Linear Gaussian bandits.",
"We think, the above simulation experiments only indicate that it might be interesting to investigate (and analyze) the TS-VHA-$\\mathsf {C1}$ from the aspect of time-sensitive learning."
],
[
"Best Arm Identification",
"Next, we consider the fixed budget setting of the Best Arm Identification problem as discussed in [30].",
"The idea is to identify the best arm amongst all the bandit arms by playing them intelligently for a fixed number of time steps $t$ .",
"The metric used to compare algorithms is the probability of error in identifying the best arm after the fixed time step $t$ .",
"TS can be utilized to solve this problem by designating the arm with the highest empirical mean after $t$ time steps as the best arm.",
"However, TS performs poorly for this pure-exploration problem because of its high exploitative nature.",
"Therefore, with the intention to increase the exploration in TS, we evaluate the applicability of TS-VHA-$\\mathsf {C2}$ in this scenario.",
"In Fig.",
"REF , we consider the Bernoulli bandit as well as Gaussian bandit, each having two arms.",
"The plots on the left and right have arms with mean rewards equal to $(0.5, 0.25)$ and $(0.51, 0.5)$ , respectively.",
"For both the scenarios, we observe that TS-VHA-$\\mathsf {C2}$ outperforms TS empirically."
],
[
"Combiner $\\mathsf {C3}$",
"Finally, we evaluate the cumulative regret performance of Combiner $\\mathsf {C3}$ , through simulations, for Bernoulli bandits and Gaussian bandits and compare its performance with TS and TS-VHA-$\\mathsf {C1}$ .",
"Similar to section -A and -B, we first evaluate the performance of $\\mathsf {C3}$ on the randomized 20 arms case.",
"As shown in Fig.",
"REF and Fig.",
"REF , for both Gaussian and Bernoulli bandits, TS-VHA-$\\mathsf {C3}$ outperforms TS.",
"Next, we choose the same randomized scenario but with two arms in Fig.",
"REF and Fig.",
"REF .",
"In this case, $\\mathsf {C3}$ outperforms both TS and TS-VHA-$\\mathsf {C1}$ significantly.",
"Interestingly, for the Gaussian bandits, TS-VHA-$\\mathsf {C1}$ performs inferior to TS, suggesting that, in some cases, increasing exploitation from the beginning does not help in optimizing the cumulative regret.",
"But, dynamically adjusting the amount of exploitation over time by $\\mathsf {C3}$ provides superior performance.",
"A mathematical analysis of the regret bound for $\\mathsf {C3}$ would help gaining more insight into it."
],
[
"Conclusion",
"We have proposed a general framework, Thompson Sampling with Virtual Helping Agents (TS-VHA), that combines samples drawn by the virtual agents to maneuver the exploration vs exploitation tradeoff in Thompson Sampling.",
"Based on this framework, we developed two linear combiners (TS-VHA-$\\mathsf {C1}$ and TS-VHA-$\\mathsf {C2}$ ) and analysed theoretically their cumulative regret performance on Gaussian Bandits.",
"Moreover, we showed their empirical efficacy on both Gaussian and Bernoulli bandits for multiple metrics: cumulative regret, best-arm identification and time-sensitive learning.",
"We defer the analysis of the regret bounds on these metrics for our future work.",
"To exhibit the broad scope of the framework, we also put forth a nonlinear combiner TS-VHA-$\\mathsf {C3}$ that dynamically tunes the amount of exploration/exploitation and offers superior empirical performance.",
"It would be interesting to experiment and devise more sophisticated combiners.",
"TS-VHA can be applied wherever Thompson Sampling can be applied and we leave extending TS-VHA (along with designing combiners) for contextual bandits, non-stationary bandits and restless bandits for future work.",
"Finally, exploring the usage of neural networks in developing combiners would be an exciting avenue for future work."
],
[
"Inequalities used in the regret analysis",
"Inequality 1 (Chernoff-Hoeffding Bound) Let $X_1, \\ldots , X_n$ be independent 0 - 1 r.v.s with $E[X_i]= p_i$ (not necessarily equal).",
"Let $X = \\frac{1}{n}\\sum _i X_i$ , $\\mu = E[X]=\\frac{1}{n}\\sum _{i=1}^{n}p_i$ .",
"Then, for any $0<\\lambda <1-\\mu $ , $\\text{Pr}(X \\ge \\mu + \\lambda ) \\le e^{-nd(\\mu +\\lambda ,\\mu )},$ and for any $0<\\lambda <\\mu $ , $\\text{Pr}(X \\ge \\mu - \\lambda ) \\le e^{-nd(\\mu -\\lambda ,\\mu )},$ where $d(a,b) = a\\ln {\\frac{a}{b}+(1-a)\\ln {\\frac{1-a}{1-b}}}$ Inequality 2 (Chernoff-Hoeffding Bound) Let $X_1, \\ldots , X_n$ be random variables with common range $[0,1]$ and such that $\\mathbb {E}[X_t | X_1, \\ldots , X_{t-1}]=\\mu $ .",
"Let $S_n = \\sum _{i=1}^{n} X_i$ .",
"Then, for all $a \\ge 0$ , $\\text{Pr}(S_n \\ge n\\mu + a) \\le e^{-2a^2/n},$ and $\\text{Pr}(S_n \\le n\\mu - a) \\le e^{-2a^2/n}.$ The following inequalities can be derived for a Gaussian random variable from Formula $7.1.13$ in [24].",
"Inequality 3 For a Gaussian distributed random variable $Z$ with mean $m$ and variance $\\sigma ^2$ , $\\text{Pr}(Z > m+x\\sigma ) \\ge \\frac{x}{\\sqrt{2\\pi } (x^2+1)}e^{-x^2/2}.$ Inequality 4 For a Gaussian distributed random variable $Z$ with mean $m$ and variance $\\sigma ^2$ , for any $z$ , $\\frac{1}{4\\sqrt{\\pi }} e^{-7z^2/2} < Pr (|Z-m| > z\\sigma ) \\le \\frac{1}{2} e^{-z^2/2}.$ Inequality 5 Let $S_n = \\sum _{i=1}^{n} \\frac{1}{i^p}$ .",
"Then for $0 < p < 1$ from [31], $S_n < 1+\\frac{(n+1)^{1-p}-1}{1-p}$"
],
[
"Proof of Lemma 1",
"Recall that $\\theta _i(t)\\sim \\mathcal {N}\\left(\\hat{\\mu }_{i}(t), \\frac{1}{\\gamma (k_i(t)+1)}\\right)$ , $x_i=\\mu _i+\\frac{\\Delta _i}{3}$ , $y_i=\\mu _1-\\frac{\\Delta _i}{3}$ and $L_i(T) = \\frac{2 \\ln {T\\Delta _i^2}}{\\gamma (y_i-x_i)^2}$ .",
"Given $\\mathcal {F}_{\\tau _{k}}$ , let $\\Theta _{k}$ denote a Gaussian random variable distributed as $\\mathcal {N}\\left(\\hat{\\mu }_{1}(\\tau _{k}+1), \\frac{1}{\\gamma (k+1)}\\right)$ .",
"For convenience, we denote $\\hat{\\mu }_1(\\tau _{k}+1)$ with $\\hat{\\mu }_{1}$ in the following.",
"Let $G_k$ be the geometric random variable representing the number of consecutive independent trials until a sample of $\\Theta _{k}$ becomes greater than $y_{i}$ .",
"Using $\\Theta _k$ and [defpit]Definition 6, we can write $p_{i,\\tau _k+1}=\\text{Pr}(\\Theta _{k}>y_{i}|\\mathcal {F}_{\\tau _k})$ , and $\\mathbb {E}\\left[\\frac{(1-p_{i,\\tau _{k}+1})}{p_{i,\\tau _{k}+1}}\\right] = \\mathbb {E}[\\mathbb {E}[G_k|\\ \\mathcal {F}_{\\tau _{k}}]] = \\mathbb {E}[G_k].$ Therefore, $\\sum _{k=0}^{T-1}\\mathbb {E}\\left[\\frac{(1-p_{i,\\tau _{k}+1})}{p_{i,\\tau _{k}+1}}\\right]= \\underbrace{\\sum _{k=0}^{4L_{i}(T)-1} \\mathbb {E}[G_k]}_{\\text{Sum (\\ref {eq:sums}a)}}+\\underbrace{\\sum _{k=4L_{i}(T)}^{T-1} \\mathbb {E}[G_k]}_{\\text{Sum (\\ref {eq:sums}b)}} $ We will now bound Sum (REF a), first term on the RHS of (REF ).",
"Let $z=\\sqrt{\\ln {r^{\\beta }}}$ , where $r\\ge 1$ is an integer, $\\beta \\in [1,2)$ , and let $M_{r}$ denote the maximum of $r$ independent samples of $\\Theta _{k}$ .",
"$\\text{Pr}(G_k < r) & \\ge \\text{Pr}(M_{r} > y_{i}) \\\\& \\ge \\text{Pr}\\left(M_{r} > \\hat{\\mu }_{1} + \\frac{z}{\\sqrt{\\gamma (k+1)}} > y_{i}\\right)\\\\& = \\mathbb {E}\\left[\\mathbb {E}\\left[M_{r} >\\eta > y_{i} \\mid \\mathcal {F}_{\\tau _{k}}\\right]\\right]\\\\& = \\mathbb {E}\\left[I\\left(\\eta > y_{i}\\right)\\text{Pr}\\left(M_{r} > \\eta \\mid \\mathcal {F}_{\\tau _{k}}\\right)\\right], $ where $\\eta =\\hat{\\mu }_1 + \\frac{z}{\\sqrt{\\gamma (k+1)}}$ .",
"Since $\\Theta _{k} \\sim \\mathcal {N}\\left(\\hat{\\mu }_{1}, \\frac{1}{\\gamma (k+1)}\\right)$ , using [ineq3]Inequality 3, we can write $\\text{Pr}\\left(M_{r} > \\eta \\; | \\; \\mathcal {F}_{\\tau _{k}} = F_{\\tau _{k}} \\right) & \\ge 1 - \\left(1-\\frac{1}{\\sqrt{2\\pi }}\\frac{z}{z^{2}+1}e^{\\frac{-z^2}{2}}\\right)^{r} \\\\& \\ge 1 - e^{-\\frac{r^{1-\\frac{\\beta }{2}}}{\\sqrt{2\\beta \\pi \\ln {r}}}}.$ Note that $F_{\\tau _{k}}$ is any realization of $\\mathcal {F}_{\\tau _{k}}$ .",
"As $\\beta \\in [1,2)$ , there exists a number $h(\\beta ) \\in \\mathbb {R}_{>0}$ such that $e^{-\\frac{r^{1-\\frac{\\beta }{2}}}{\\sqrt{2\\beta \\pi \\ln {r}}}}\\le \\frac{1}{r^2}$ for $r\\ge h(\\beta )$ .",
"Hence, for any $r\\ge h(\\beta )$ and any $\\gamma >0$ , $\\text{Pr}\\left( M_{r} > \\eta \\; | \\; \\mathcal {F}_{\\tau _{k}} = F_{\\tau _{k}} \\right) \\ge 1 - \\frac{1}{r^2}$ On substituting (REF ) in (REF ) we get, for any $r\\ge h(\\beta )$ , $\\text{Pr}(G_k < r) &\\ge \\mathbb {E}\\left[I\\left( \\eta \\ge y_{i} \\right)\\left(1-\\frac{1}{r^2}\\right) \\right]\\\\&= \\left(1-\\frac{1}{r^2}\\right)\\text{Pr}( \\eta \\ge y_{i}) $ We will now find a lower bound on $\\text{Pr}( \\eta \\ge y_{i})$ .",
"$\\text{Pr}(\\eta \\ge y_{i}) = \\text{Pr}\\left( \\hat{\\mu }_{1}+\\frac{z}{\\sqrt{\\gamma (k+1)}} \\ge \\mu _{1}-\\frac{\\Delta _{i}}{3}\\right)=\\\\\\text{Pr}\\left( \\hat{\\mu }_{1}+\\frac{1}{k+1}\\ge \\mu _{1}-\\left(\\frac{\\Delta _{i}}{3}+\\frac{z}{\\sqrt{\\gamma (k+1)}}-\\frac{1}{k+1}\\right)\\right) $ $\\frac{1}{k + 1}$ was added to $\\hat{\\mu }_{1}$ to account for the fact that $\\hat{\\mu }_1$ is not the average of the past $k$ observations, but it is the sum of the past $k$ observations divided by $(k+1)$ .",
"Applying [ineq2]Inequality 2 to (REF ), $\\text{Pr}(\\eta \\ge y_{i}) \\ge 1 - e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{(k+1)}}{3}+\\frac{z}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k+1}}\\right)^2}.",
"$ Substituting (REF ) back into (REF ), for any $r\\ge h(\\beta )$ , $\\text{Pr}(G_k < r) &\\ge \\left(1-\\frac{1}{r^2}\\right)\\left(1 - e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{(k+1)}}{3}+\\frac{z}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k+1}}\\right)^2}\\right) \\\\& \\ge 1 - \\frac{1}{r^2} - e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{(k+1)}}{3}+\\frac{z}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k+1}}\\right)^2}.$ This leads us to $\\sum _{k=0}^{4L_{i}(T)-1} \\mathbb {E}[G_k] = \\sum _{k=0}^{4L_{i}(T)-1} \\sum _{r=0}^{T} \\text{Pr}(G_k \\ge r) \\\\ \\le \\sum _{k=0}^{\\small {4L_{i}(T)-1}}\\sum _{r=0}^{T} \\left( \\frac{1}{r^2} \\right.",
"\\\\ + \\left.",
"e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{(k+1)}}{3}+\\frac{z}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k+1}}\\right)^2}\\right) $ First term on the RHS in (REF ) can be upper bounded as follows.",
"$\\sum _{k=0}^{4L_{i}(T)-1}\\left(\\sum _{r=0}^{T} \\frac{1}{r^2} \\right) &\\le \\sum _{k=0}^{4L_{i}(T)-1} \\left( h(\\beta ) + \\sum _{r\\ge h(\\beta )} \\frac{1}{r^2}\\right) \\\\& \\le \\sum _{k=0}^{4L_{i}(T)-1} \\left( h(\\beta ) + \\zeta (2)\\right) \\\\& \\le (h(\\beta ) + \\zeta (2))4L_{i}(T) ,$ where $\\zeta $ is the Riemann zeta function.",
"Next, we consider the second term on the RHS of (REF ) and use $k^{\\prime }=k+1$ for convenience.",
"$&\\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}}\\sum _{r=0}^{T} e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{k^{\\prime }}}{3}+\\frac{\\sqrt{\\beta \\ln {r}}}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k^{\\prime }}}\\right)^2} \\\\&=\\sum _{r=0}^{T}\\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}} e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{k^{\\prime }}}{3}+\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma }}\\right)^2}e^{\\frac{-2}{k^{\\prime }}}e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{k^{\\prime }\\gamma }}\\right)}e^{\\frac{4\\Delta _i}{3}} \\\\&\\le \\sum _{r=0}^{T}\\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}} e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{k^{\\prime }}}{3}+\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma }}\\right)^2}e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{k^{\\prime }\\gamma }}\\right)}e^{\\frac{4\\Delta _i}{3}} \\\\&= \\sum _{r=0}^{T}\\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}} e^{\\frac{-2\\beta \\ln {r}}{\\gamma }}e^{\\frac{-2\\Delta _i^2k^{\\prime }}{9}}e^{\\frac{-4\\Delta _i}{3}\\sqrt{\\frac{\\beta k^{\\prime }\\ln {r}}{\\gamma }}}e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma k^{\\prime }}}\\right)}e^{\\frac{4\\Delta _i}{3}} \\\\& \\overset{(a)}{\\le }\\sum _{r=0}^{T}e^{\\frac{-2\\beta \\ln {r}}{\\gamma }}e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma }}\\left(1-\\frac{\\Delta _i}{3}\\right)\\right)}e^{\\frac{4\\Delta _i}{3}} \\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}} e^{\\frac{-2\\Delta _i^2 k^{\\prime }}{9}} \\\\&\\overset{}{\\le }\\sum _{r=0}^{T} e^{\\frac{-2\\beta \\ln {r}}{\\gamma }}e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma }}\\left(1-\\frac{\\Delta _i}{3}\\right)\\right)}e^{\\frac{2\\Delta _i}{3}}\\frac{1}{e^{(\\frac{2\\Delta _i^2}{9}-1)}} \\\\&\\overset{(b)}{=} \\sum _{r=0}^{T} \\frac{c^{\\prime }}{r^{\\frac{2\\beta }{\\gamma }}}e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma }}\\left(1-\\frac{\\Delta _i}{3}\\right)\\right)}$ $(a)$ is due to the fact that $\\max \\left(e^{\\frac{-4\\Delta _i}{3}\\left(\\sqrt{\\frac{\\beta k^{\\prime }\\ln {r}}{\\gamma }}\\right)}e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma k^{\\prime }}}\\right)}\\right) = e^{4\\left(\\sqrt{\\frac{\\beta \\ln {r}}{\\gamma }}\\left(1-\\frac{\\Delta _i}{3}\\right)\\right)}$ at $k^{\\prime }=1$ .",
"In $(b)$ , $c^{\\prime }=e^{\\frac{4\\Delta _i}{3}}/(e^{\\frac{2\\Delta _i^2}{9}-1}) $ .",
"For any $\\epsilon > 0$ , there exists a number $g(\\epsilon $ ) such that $\\frac{e^{4\\sqrt{\\frac{\\ln {r}}{\\gamma }}\\left(1-\\frac{\\Delta _i}{3}\\right)}}{r^{\\frac{2\\beta }{\\gamma }}} \\le \\frac{1}{r^{\\frac{2\\beta }{\\gamma }-\\epsilon }}$ for $r\\ge g(\\epsilon )$ .",
"Hence, for $\\beta \\in [1,2)$ , $\\gamma >0$ , $\\epsilon >0$ and $r\\ge g(\\epsilon )$ , $ \\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}}\\sum _{r=0}^{T} e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{k^{\\prime }}}{3}+\\frac{\\sqrt{\\beta \\ln {r}}}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k^{\\prime }}}\\right)^2} \\le \\sum _{r=0}^{T}\\frac{c^{\\prime }}{r^{\\frac{2\\beta }{\\gamma }-\\epsilon }}$ We will analyze (REF ) separately for $\\gamma \\in (0,4)$ and $\\gamma \\ge 4$ .",
"For any value of $\\gamma \\in (0,4)$ , we choose $\\beta \\in [1,2)$ such that $\\gamma <2\\beta $ .",
"Then, we select $\\epsilon >0$ to have $\\frac{2\\beta }{\\gamma }-\\epsilon >1$ .",
"Thus, for $\\gamma \\in (0,4)$ , (REF ) can be further simplified as, $\\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}}\\sum _{r=0}^{T} e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{k^{\\prime }}}{3}+\\frac{\\sqrt{\\beta \\ln {r}}}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k^{\\prime }}}\\right)^2}\\le \\sum _{r=0}^{T}\\frac{c^{\\prime }}{r^{\\frac{2\\beta }{\\gamma }-\\epsilon }} \\\\\\le c^{\\prime }g(\\epsilon )+\\sum _{r\\ge c^{\\prime }g(\\epsilon )}\\frac{c^{\\prime }}{r^{\\frac{2\\beta }{\\gamma }-\\epsilon }} \\\\\\le c^{\\prime } \\left(g(\\epsilon ) + \\zeta \\left(\\frac{2\\beta }{\\gamma }-\\epsilon \\right)\\right)$ Since $\\frac{2\\beta }{\\gamma }-\\epsilon >1$ and $\\zeta $ is the Riemann zeta function, $\\zeta \\left(\\frac{2\\beta }{\\gamma }-\\epsilon \\right)$ is a finite number.",
"On the other hand, for $\\gamma \\ge 4$ , $\\frac{2\\beta }{\\gamma }-\\epsilon <1$ for any choice of $\\beta $ and $\\epsilon $ .",
"If we fix $\\beta \\in [1,2)$ and $\\epsilon >0$ such that $\\frac{2\\beta }{\\gamma }-\\epsilon > 0$ (REF ) results in, $\\sum _{k^{\\prime }=1}^{\\small {4L_{i}(T)}}\\sum _{r=0}^{T}& e^{-2\\left(\\frac{\\Delta _{i}\\sqrt{k^{\\prime }}}{3}+\\frac{\\sqrt{\\beta \\ln {r}}}{\\sqrt{\\gamma }}-\\frac{1}{\\sqrt{k^{\\prime }}}\\right)^2} \\\\ & \\le \\sum _{r=0}^{T}\\frac{c^{\\prime }}{r^{\\frac{2\\beta }{\\gamma }-\\epsilon }} \\\\& \\le c^{\\prime }\\left( g(\\epsilon )+\\sum _{r\\ge 1}\\frac{c^{\\prime }}{r^{\\frac{2\\beta }{\\gamma }-\\epsilon }} \\right)\\\\& \\overset{(a)}{\\le } c^{\\prime }\\left( g(\\epsilon ) + 1 + \\frac{T^{1+\\epsilon -\\frac{2\\beta }{\\gamma }}-1 }{1+\\epsilon -\\frac{2\\beta }{\\gamma }}\\right)$ The inequality $(a)$ in (REF ) follows from [ineq5]Inequality 5.",
"On substituting (REF ), (REF ) and (REF ) back into (REF ) gives us the bound for Sum (REF a), the first term on the RHS of (REF ).",
"$\\sum _{k=0}^{4L_{i}(T)-1} \\mathbb {E}[G_k] \\le \\\\ \\left\\lbrace \\begin{array}{l}H(\\beta ) L_i(T)+ c^{\\prime } \\left(g(\\epsilon )+\\zeta (\\frac{2\\beta }{\\gamma }-\\epsilon )\\right)~ {\\text{for}}~\\gamma \\in (0,4),\\\\H(\\beta )L_i(T)+c^{\\prime } \\left( g(\\epsilon ) + \\frac{T^{1+\\epsilon -\\frac{2\\beta }{\\gamma }}-1 }{1+\\epsilon -\\frac{2\\beta }{\\gamma }}\\right)~{\\text{for}}~\\gamma \\ge 4,\\end{array}\\right.$ where, $H(\\beta )=4(h(\\beta ) + \\zeta (2))$ .",
"Next, we bound Sum (REF b), second term on the RHS of (REF )) where the index of summation $k\\ge 4L_i(T)$ .",
"We will start by defining $A_{t-1}$ as the event in which $\\hat{\\mu }_1(t)-\\frac{\\Delta _i}{6}>y_i$ and use the notation $\\mathcal {F}_{t-1}|_{A_{t-1}}$ to indicate random variable $\\mathcal {F}_{t-1}$ conditioned on $A_{t-1}$ being true.",
"Then, $\\mathbb {E}\\left[\\frac{1}{p_{i,\\tau _{k}+1}}\\right] &= \\mathbb {E}\\left[\\frac{1}{\\text{Pr}(\\Theta _k>y_i|\\mathcal {F}_{\\tau _k})}\\right] \\\\& \\le \\mathbb {E}\\left[\\frac{1}{\\text{Pr}\\left(\\Theta _k>y_i\\Big |\\mathcal {F}_{\\tau _k}|_{A_{\\tau _k}}\\right)\\text{Pr}(A_{\\tau _k})}\\right] $ We now bound $\\text{Pr}(\\Theta _k>y_i|\\mathcal {F}_{\\tau _k}|_{A_{\\tau _k}})$ and $\\text{Pr}(A_{\\tau _k})$ .",
"$\\text{Pr}\\left(\\Theta _k>y_i \\; | \\;\\mathcal {F}_{\\tau _k}|_{A_{\\tau _k}}\\right) &\\ge \\text{Pr}\\left(\\Theta _k>\\hat{\\mu }_1-\\frac{\\Delta _i}{6} \\; |\\; \\mathcal {F}_{\\tau _k}|_{A_{\\tau _k}} \\right) \\\\&\\overset{(a)}{\\ge } 1-e^{-\\gamma (k+1)\\Delta _i^2/72} \\\\&\\overset{(b)}{\\ge } 1-e^{-\\gamma (4L_i(T))\\Delta _i^2/72} \\\\&\\ge 1- \\frac{1}{T\\Delta _i^2} $ In the above, $(a)$ follows from [ineq2]Inequality 2 with $z=\\sqrt{\\gamma (k+1)}\\Delta _i/6$ and $(b)$ is due to the fact that $k\\ge 4L_i(T)$ .",
"Note that we can use [ineq2]Inequality 2 here because we assume that the reward distribution has a finite support over $[0,1]$ .",
"Observe that for any $t\\ge \\tau _k+1$ , we have $k_1(t)\\ge k\\ge 4L_i(T)$ , and, using [ineq2]Inequality 2, we obtain, $\\text{Pr}(A_{\\tau _k}) = \\text{Pr}\\left(\\hat{\\mu }_1(t) > \\mu _1 - \\frac{\\Delta _i}{6}\\right) &\\ge 1- e^{-2\\gamma k_1(t)\\Delta _i^2/36} \\\\&\\ge 1 - \\frac{1}{T\\Delta _i^2} $ Substituting (REF ) and (REF ) into (REF ), for $k\\ge 4L_i(T)$ , $\\mathbb {E}\\left[\\frac{1}{p_{i,\\tau _{k}+1}}\\right] -1 &\\le \\frac{1}{\\left(1- \\frac{1}{T\\Delta _i^2}\\right)^2}-1 \\\\&\\le \\frac{4}{T\\Delta _i^2} $ For any $\\gamma >0$ , using (REF ), we get the following bound on Sum (REF b).",
"$\\sum _{\\small {4L_{i}(T)}}^{T-1} \\mathbb {E}[G_k] &\\le \\sum _{\\small {4L_{i}(T)}}^{T-1} \\frac{4}{T\\Delta _i^2} \\\\& \\le \\frac{4}{\\Delta _i^2} $ Combining the results from (REF ), (REF ), (REF ) and (REF ) completes the proof of Lemma REF .",
"[Figure: NO_CAPTION He is currently a Master's student and a Graduate Research Assistant in Aeronautical and Astronautical Engineering at Purdue University.",
"His research interests include Robotics and Control theory.",
"He is a recipient of the Chintakindi Amba Rao Fellowship at Purdue University in 2021.",
"[Figure: NO_CAPTION He is currently a Master's student in computational social science at Stanford University.",
"His research interests include optimization techniques/theory and machine learning.",
"He was a recipient of the Director’s Gold Medal at IIT BHU Varanasi, for outstanding all-round performance in the graduating class [Figure: NO_CAPTION He was a Postdoctoral Fellow at the Department of Electrical and Computer Engineering, University of Toronto, from March 2009 to October 2011.",
"From January 2015 to December 2018, he was a Faculty member at the Indian Institute of Technology (BHU), Varanasi, India.",
"His past industry experience includes working at the Indian Space Research Organisation, Samsung Electronics and Nokia Networks and Ericsson.",
"In 2022 he joined Motorola Mobility/Lenovo as a researcher working on 5G NR standardization activities.",
"His research interests include wireless communications, with emphasis on physical and MAC layer algorithms, and machine learning."
]
] | 2209.08197 |
[
[
"Quantum Computing Methods for Supply Chain Management"
],
[
"Abstract Quantum computing is expected to have transformative influences on many domains, but its practical deployments on industry problems are underexplored.",
"We focus on applying quantum computing to operations management problems in industry, and in particular, supply chain management.",
"Many problems in supply chain management involve large state and action spaces and pose computational challenges on classic computers.",
"We develop a quantized policy iteration algorithm to solve an inventory control problem and demonstrative its effectiveness.",
"We also discuss in-depth the hardware requirements and potential challenges on implementing this quantum algorithm in the near term.",
"Our simulations and experiments are powered by \\texttt{IBM Qiskit} and the \\texttt{qBraid} system."
],
[
"Introduction",
"Recent surveys indicate promising prospects of quantum computing in many fields, for example, in finance [1], [2] and quantum chemistry simulation [3].",
"In this work, we explore what advantages quantum computing may provide for addressing important problems in the field of operations management.",
"Supply chain management is a central problem in the field of operations management.",
"Supply chain management is a discipline studying the flow of goods or services through all the different stages of the process.",
"From the early days when inventory decisions were manually documented with pen and paper, supply chains have undergone a major transformation to be more automated and efficient, thanks to the advances of new technologies such as digitalization, software development, and more computing power.",
"Therefore, it is natural to wonder what the technology of quantum computing may offer to supply chain management.",
"In particular, it is well known that the large decision spaces of inventory control problems brought major computational challenges to supply chain management, and some recent studies have explored using deep learning methods for supply chain management [4].",
"In this work, we conduct a modest inquiry of the prospect of quantum computing on supply chain management by focusing on a classic inventory control problem.",
"We describe special features of this inventory control problem that make quantum computing a suitable tool.",
"We also discuss the practical limitations of certain quantum approaches in terms of hardware requirements and error correction.",
"We run numerical experiments on IBM Qiskit [5] and the qBraid system [6] to establish the validity."
],
[
"Inventory Control Problem",
"We consider a classic periodic-review inventory control problem in supply chain management [7].",
"At each time period, the manager of a warehouse needs to decide on the number of product units to order.",
"We assume that there is no lead time and the ordering cost is proportional to the number of units ordered.",
"At each time period, the demand is stochastic following some probability distribution known to the manager.",
"When there is no sufficient inventory, excess demand is lost.",
"The manager needs to make ordering decisions to balance the ordering cost and lost sales cost in the long run.",
"This inventory control problem can be naturally modeled by the Markov Decision Process.",
"Let $s_t$ represent the inventory level at the beginning of the $t$ -th time period, and $a_t$ represent the number of units ordered by the inventory manager in the beginning of the period $t$ .",
"Correspondingly, the state space of $s_t$ and action space of $a_t$ are denoted by $\\mathcal {S} = \\lbrace 1,\\dots , |\\mathcal {S}|\\rbrace $ and $\\mathcal {A} = \\lbrace 1,\\dots ,|\\mathcal {A}|\\rbrace $ , respectively.",
"Let $D_t$ represent the stochastic demand of this time period.",
"We assume that the probability distribution of demand is stationary over time and can be specified by values $p_j = P(D_t = d)$ for $d = 1,2,\\dots , D$ for some integer $D>0$ .",
"The inventory level in the beginning of $t+1$ , denoted by $s_{t+1}$ , is captured by the following system transition function, $s_{t+1} = [s_t +a_t - D_t]^+,$ where $x^+:= \\max \\lbrace x,0\\rbrace $ .",
"The sequence of events is illustrated in Fig.",
"REF .",
"Figure: Sequence of events in the inventory control problem.The reward of period $t$ , denoted by $r(s_t,a_t)$ , is given by $r(s_t,a_t) = - h \\cdot [s_t+a_t-D_t]^+- l \\cdot [D_t-s_t-a_t]^+ ,$ where $h$ is the unit holding cost, $l$ is the unit lost sales cost, and the negative signs are added because minimizing negative costs is equivalent to maximizing reward.",
"Further, we use $r\\in \\mathbb {R}^{|\\mathcal {S}||\\mathcal {A}|}$ to denote the reward vector of a state-action pair, where $r(i,j)$ is given in (REF ).",
"We consider an infinite horizon discounted setting with some given discount factor $\\gamma \\in (0,1)$ , and it is well-known that (see, e.g., [8]) in this setting there exists a deterministic and stationary optimal policy $\\pi :\\mathcal {S} \\rightarrow \\mathcal {A}$ that maximizes the total discounted reward over an infinite horizon.",
"We follow the Q-function formulation [9], and the goal is to find policy $\\pi $ that maximizes the Q-value function $Q^\\pi (i,j)\\in \\mathbb {R}^{|\\mathcal {S}||\\mathcal {A}|}$ defined as $Q^\\pi (i,j) = \\mathbb {E}\\left[\\sum _{t=0}^\\infty \\gamma ^t r(s_t,a_t)|\\pi ,s_0 = i, a_0 = j \\right].$ We use $P^{\\pi } \\in \\mathbb {R}^{|\\mathcal {S}||\\mathcal {A}| \\times |\\mathcal {S}||\\mathcal {A}|}$ to denote the transition matrix on state-action pairs under a stationary policy $\\pi $ , where the elements of $P^{\\pi }$ are defined as $P^{\\pi }_{(i,j),(i^{\\prime },j^{\\prime })} = P(i^{\\prime }|i,j)\\cdot \\pi (j^{\\prime }|i^{\\prime }), \\text{ for all } i,i^{\\prime } \\in \\mathcal {S}, j,j^{\\prime }\\in \\mathcal {A}.$ Under the notation of $Q^\\pi $ and $P^\\pi $ , the Q-value function satisfies the following equation, $Q^\\pi = r + \\gamma P^{\\pi } Q^\\pi .$"
],
[
"Preliminaries on Policy Iteration",
"A classic algorithm for solving the Markov Decision Process is the policy iteration algorithm [10], which generates a sequence of policies that gradually converge to the optimal solution over iterations.",
"The policy iteration algorithm enjoys nice convergence properties and actually terminates within finite iterations.",
"Specifically, it is proved in [11] that the number of iterations of policy iteration can be bounded in polynomial time $\\tilde{\\mathcal {O}}(|\\mathcal {S}||\\mathcal {A}|-|\\mathcal {S}|)/(1-\\gamma ))$ , where $|\\mathcal {S}|$ and $|\\mathcal {A}|$ stand for the size of state space and action space, respectively.",
"However, each iteration of the policy iteration algorithm is time-consuming due to the sizes of the state space and the action space.",
"The per iteration complexity of the policy iteration algorithm has a time complexity of $|\\mathcal {S}|^3+ |\\mathcal {S}|^2|\\mathcal {A}|$ .",
"It leaves space to leverage quantum computing to implement the policy iteration algorithm.",
"[!ht] Policy Iteration Algorithm for Inventory Control [1] demand distribution, cost parameters, $K$ optimal inventory reorder policy Initialization : initial policy $\\pi _0$ $k = 0,1,\\dots ,K$ Calculate $Q^{\\pi _k} \\in \\mathbb {R}^{|\\mathcal {S}|}$ by a solving a linear system $Q^{\\pi ^k} = (I - \\gamma P^{\\pi _k})^{-1} r$ Obtain new policy $\\pi _{k+1}$ by greedily solving the following maximization with respect to $v_{\\pi _k}$ : for all $i \\in \\mathcal {S}$ update $\\pi _{k+1}(i) \\in \\arg \\,\\max _{j\\in \\mathcal {A}} Q^{\\pi _k}(i,j)$ $\\pi _{K}$ We have chosen the termination condition of Algorithm REF by setting the total number of iterations as some large enough number $K$ because it is known that policy iteration algorithm can converge in finite iterations [11].",
"In Step REF of Algorithm REF , we use quantum linear system solver to provide quantum states to approximate the Q function $Q^{\\pi _k}$ .",
"Let $B_k$ denote $B_k:=I - \\gamma P^{\\pi _k}$ , then the policy iteration step is finding the value of $B_k^{-1} r$ , where the dimension of the matrix $B_k$ is $|\\mathcal {S}||\\mathcal {A}|$ .",
"In Step REF , the new policy $\\pi _{k+1}(i)$ is obtained by finding the position of the largest element in $Q^{\\pi _k}(i,j)$ for the fixed $i$ .",
"A key observation is that the large matrices $B_k$ and $P^\\pi _k$ have some sparsity properties for any deterministic policy $\\pi $ under a very practical sparsity assumption on the inventory demand.",
"Based on the definition in (REF ), for every given $(i,j)$ , $P^{\\pi }_{(i,j),(i^{\\prime },j^{\\prime })}\\ne 0$ only when $j^{\\prime } = \\pi (i^{\\prime })$ and $P(i^{\\prime }|i,j)>0$ .",
"Assuming that the demand only takes $r\\le |\\mathcal {S}|$ different values, then the number of non-zeros in $P^\\pi _k$ is bounded by $r \\times |\\mathcal {S}||\\mathcal {A}|$ , and consequently the number of non-zeros in $P^\\pi _k$ is bounded by $(r+1) \\times |\\mathcal {S}||\\mathcal {A}|$ , which is much smaller than the total number of elements $|\\mathcal {S}|^2|\\mathcal {A}|^2$ ."
],
[
"HHL Algorithm for Policy Evaluation",
"The Harrow-Hassidim-Lloyd (HHL) quantum algorithm [12] is a fundamental quantum computing algorithm for solving linear systems with exponential speed-up in theory.",
"For any matrix $B$ , HHL algorithm approximates the solution of $B^{-1} r$ in $\\mathcal {O}((\\log (N) ) ^2)$ quantum steps where $N$ is the dimension of the matrix $B$ and $N = |\\mathcal {S}||\\mathcal {A}|$ in our case.",
"Additionally, the time complexity also depends linearly on the sparsity $s$ defined as the number of non-zeros in $B$ , and $\\kappa $ denoting the condition number of $B$ .",
"For notational simplicity, we consider the case of solving $Bq = r$ , and the HHL algorithm works as follows.",
"We first assume that $B$ is Hermitian, because otherwise we can consider the following equivalent formulation involved with a Hermitian matrix, $\\begin{pmatrix}0 & B \\\\B^\\dagger & 0\\end{pmatrix}\\begin{pmatrix}0\\\\q\\end{pmatrix}= \\begin{pmatrix}r\\\\0\\end{pmatrix}.$ The vector $r$ is represented by a quantum state ${r}$ with $\\log _2N$ qubits, and the solution vector $q$ is also considered as a quantum state ${q}$ .",
"Suppose ${r} = \\sum _l r_l {E_l}$ , where ${E_l}$ 's are the eigenvectors of $B$ , and the corresponding eigenvalue of ${E_l}$ is $\\lambda _l$ .",
"Therefore, the equation $q = B^{-1}r$ can be encoded as ${q} = \\sum _l \\lambda _l^{-1} r_l {E_l}.$ Equation (REF ) is achieved by first conducting quantum phase estimation to compute the eigenvalues $\\lambda _l$ , and then rotating an ancillary qubit of angle $\\lambda _{\\min }/\\lambda _l$ , where $\\lambda _{\\min }$ is the smallest eigenvalue, and lastly uncomputing the quantum phase estimation to obtain $\\sum _l r_l {E_l} \\left(\\frac{\\lambda _{\\min }}{\\lambda _l} {1} + \\sqrt{1 - \\frac{\\lambda _{\\min }^2}{\\lambda _l^2}} {0} \\right).$ While the $\\mathcal {O}((\\log N)^2)$ complexity is much faster than $\\mathcal {O}(N\\log N)$ of the classic algorithms and in particular it is an exponential speed-up, retrieving $x$ from ${x}$ requires $\\mathcal {O}(N)$ repetitions to get all $N$ components, and the cost of HHL in practice still remains prohibitive [13]."
],
[
"Variational Algorithms",
"The HHL algorithm is expected to be useful in the long term with fault-tolerant quantum computing technologies.",
"However, in the near-term with the Near-term Intermediate Scale Quantum (NISQ) regime [14], people are interested in finding variational analogs of HHL with similar capabilities in practice [15], [16].",
"Similar to the spirit of deep learning [17], those algorithms do not have a theoretically guaranteed computational advantage against their classic counterparts, but they might practically perform well in the near-term quantum devices [18].",
"More precisely, general variational quantum algorithms are specified by the following unitary operators, $U = \\prod \\nolimits _{\\ell = 1}^L {{W_\\ell }{U_\\ell }} = \\prod \\nolimits _{\\ell = 1}^L {{W_\\ell }\\exp (i{X_\\ell }{\\theta _\\ell })}~,$ where $W_\\ell $ 's are fixed unitary operators (constant gates in the quantum circuits), $X_\\ell $ 's are Hermitian operators (usually elements from the Pauli group), and $\\theta _\\ell $ 's are variational classic parameters.",
"The above unitary operations could be used to make loss functions based on the quantum measurements.",
"Say that we initialize the quantum circuit by the state ${\\Psi _0}$ , we could construct the following loss function, $\\mathcal {L} = {\\Psi _0} U^\\dagger (\\theta )O U {\\Psi _0}~.$ The task now is to find the minimal eigenvalue of the Hermitian operator $O$ , which might be obtained by traditional gradient descent algorithms, $\\theta _\\ell (t+1)-\\theta _\\ell (t) = -\\eta \\frac{\\partial \\mathcal {L}}{\\partial \\theta _\\ell }~,$ with the learning rate $\\eta $ and the number of iterations $t$ .",
"In supervised learning, we could encode supervised data in ${\\Psi _0}$ .",
"Thus, supervised learning could be performed with quantum machines similarly.",
"One could give a variational version of the HHL algorithm by simply preparing variational states ${x(\\theta )} = U(\\theta ) {\\Psi _0}$ such that one could minimize the difference between $A {x(\\theta )} $ and $b$ , in order to solve the matrix inversion of $A$ .",
"We adapt the variational ansatz in [15], where people find a comparable performance against HHL with shorter circuit depth.",
"We will primarily study the implementation of it in the near-term simulation and benchmarks."
],
[
"QRAM Hardware Requirements",
"When applying HHL-like algorithm, one of the primary challenges is to inform quantum computers the matrix elements of $A$ we want to invert.",
"This problem could be solved, in principle, using so-called Quantum Random Access Memory [19] (QRAM).",
"With the data size $N$ , QRAM could use $\\mathcal {O}(N)$ qubits, and $\\mathcal {O}(\\log N)$ time to implement the following unitary operator using a parallel way, $\\sum _i \\alpha _i {i} {0} \\rightarrow \\sum _i \\alpha _i {i} {x_i}$ where $\\alpha _i$ 's are arbitrary coefficients and $x_i$ 's are the data.",
"In practice, large-scale, fault-tolerant QRAMs are challenging to build [20], [21], [22].",
"However, it will be visionary to estimate how hard it is in practice to implement QRAM and HHL for given physical devices and algorithm requirements.",
"In Figure REF and REF , we study how many physical parameters are needed for given requirements from current Supple Chain Management.",
"Here, we make use of the hardware models of QRAM [20], [21].",
"Assuming that the size of the matrix is $10^3$ , with the precision $10^{-3}$ , which could typically happen in the current Supple Chain Management technology, we bound the required decoherence rate $\\kappa +\\gamma $ in the hybrid Circuit quantum electrodynamics system for realizing QRAM.",
"A general conclusion is that it might be challenging to realize those requirements with the current physical devices, it will be helpful to study possibilities of realizing those experiments in the long term.",
"Finally, we comment that the classic-quantum interfaces are essential for HHL-based algorithms.",
"QRAM architectures solve the uploading problem, and downloading the quantum data towards classic devices could be done using classic shadow [23], which is exponentially efficient for sparse quantum state tomography.",
"Figure: Constraints on the error rate ε\\varepsilon from the Markov Decision Process.In Figure REF , we calculate the error rate from the precision of the problem $1-F$ and the size of the data $N$ , with the assumption that$T = \\log N$ .",
"The region bounded by the red box indicates the current constraints used in the Supply Chain Optimization community.",
"We use the infidelity formula proven from [21], $1-F \\sim \\frac{1}{4} \\varepsilon T \\log N \\sim \\frac{1}{4} \\varepsilon \\log ^2 N$ for QRAM architectures.",
"The color density is the error rate $\\varepsilon $ .",
"Figure: Constraints on the precision 1-F1-F from the Markov Decision Process problem, with the decoherence rate κ+γ\\kappa +\\gamma .In Figure REF , we assume $g_d=1 \\text{ kHz} \\times 2\\pi $ , $\\nu = 10 \\text{ MHz} \\times 2\\pi $ , and $c_d =4.5$ which is the average of the CZ and SWAP gates inside the QRAM circuit as an estimate, in the setup of [20] with the formula $\\varepsilon \\approx (\\kappa +\\gamma )\\frac{{{c}_{d}}\\pi }{2{{g}_{d}}}+{{\\left( \\frac{{{g}_{d}}}{\\nu } \\right)}^{2}}$ .",
"Here, $\\kappa $ and $\\gamma $ are the phonon and transmon decoherence rates, $\\nu $ is the free spectral range, and $g_d$ is the direct coupling.",
"We give an emphasis on the size of the modern Supply Chain Management problem in the red line.",
"The color density is the error rate $\\varepsilon $ ."
],
[
"Simulation Experiments",
"In this section, we give precise simulation details about how to solve our Markov Decision Process using quantum computing.",
"Precisely speaking, we implement the policy iteration step (Step REF ) of Algorithm REF in quantum computers.",
"Step REF is essentially solving a large linear system in (REF ), and we use the variational quantum algorithms introduced in Section REF .",
"Under the simplified notation, $B := (I - \\gamma P^\\pi )$ and $q:= Q^{\\pi }$ , we need to solve the following linear system $Bq = r,$ where the dimension of $B$ is $|\\mathcal {S}||\\mathcal {A}|$ ."
],
[
"LCU Coefficients",
"We use the oracle model, so-called the LCU (Linear Combination of Unitaries) decomposition to upload the data of the matrix $B$ we need to invert in each iteration of the Markov Decision Process.",
"The LCU decomposition is defined as $B = \\sum _{i=1}^L a_i P_i~,$ where $a_i$ 's are real coefficients (since $B$ is Hermitian) and $P_i$ s are unitary operations.",
"One could choose $P_i$ as elements of the Pauli group ($4^N$ elements in total with $N$ qubits if we do not count for redundancies from signs), one could compute the coefficients $a_i$ according to $a_i =\\frac{1}{2^N} \\text{Tr}(B P_i).$ Figure REF shows the distribution of the LCU coefficients of a single matrix instance used in the Markov Decision Process.",
"Figure: The distribution of the LCU coefficients of a single matrix instance used in the Markov Decision Process we study.",
"Here we use Pauli group elements as the basis in the LCU oracle.",
"The vertical axis is the value of the LCU coefficients, and the horizontal axis is different terms in LCU."
],
[
"HHL Benchmarks",
"We use the IBM Qiskit system and the qBraid system to decompose the HHL unitaries towards fundamental gates.",
"In Table REF , we give precise gate counting by truncating and scaling the number of LCU terms before implementing them into HHL.",
"With our truncation and approximation scheme, we verify that the gate counting scaling is roughly polynomial and efficient, but the numbers of fundamental gates are generally high.",
"Those results indicate that HHL is more suitable to be implemented with the help of fault-tolerant quantum computing.",
"Table: Gate counting used in the HHL algorithm from real Supple Chain Optimization problems up to 6 qubits.In Table REF , we compute the number of fundamental gates used in IBM Qiskit for the HHL oracle for the matrix inversion task in the Markov Decision Process.",
"We use ✕ to denote the disallowed situation where the dimension of the Hilbert space cannot hold so much independent $L$ .",
"Moreover, in order to scale, we choose the leading submatrix with the corresponding size by given number of qubits $N$ and the number of the LCU terms $L$ ."
],
[
"Variational Circuits",
"We use the IBM Qiskit system and the qBraid system to simulate the variational quantum algorithms for linear system solving, with the Markov Decision Process.",
"With the qBraid environment, we compare the simulations between noiseless and noisy environments provided by IBM Qiskit with the real hardware noise models, with first five truncated LCU decomposition coefficients and 6 qubits, in Figure REF .",
"We find that the variational simulation could converge in a decent amount of time even in the noisy environment.",
"Figure: Solving the matrix inversion problem by variational quantum linear solver.In Figure REF , we use the variational ansatz provided in [15] with 2 layers and 6 qubits.",
"The noise calculation is from the real quantum device model in IBM Qiskit."
],
[
"Conclusion",
"Despite the promising future of quantum computing, more efforts are needed to make it practical in solving real-world problems.",
"This work is the first in exploring the usage of quantum computing in the field of supply chain management by focusing on a canonical inventory control problem.",
"We discuss in-depth a classic inventory control problem and propose to solve it with a quantized policy iteration algorithm.",
"Our experiments on IBM Qiskit and the qBraid system demonstrate the practicality of variational algorithms for solving small-sized inventory control problems.",
"We believe this is a well open area that will be interesting for both academia and industry to explore further."
]
] | 2209.08246 |
[
[
"Discovery of a new Local Group Dwarf Galaxy Candidate in UNIONS:\n Bo\\\"otes V"
],
[
"Abstract We present the discovery of Bo\\\"otes V, a new ultra-faint dwarf galaxy candidate.",
"This satellite is detected as a resolved overdensity of stars during an ongoing search for new Local Group dwarf galaxy candidates in the UNIONS photometric dataset.",
"It has a physical half-light radius of 26.9$^{+7.5}_{-5.4}$ pc, a $V$-band magnitude of $-$4.5 $\\pm$ 0.4 mag, and resides at a heliocentric distance of approximately 100 kpc.",
"We use Gaia DR3 astrometry to identify member stars, characterize the systemic proper motion, and confirm the reality of this faint stellar system.",
"The brightest star in this system was followed up using Gemini GMOS-N long-slit spectroscopy and is measured to have a metallicity of [Fe/H] $=$ $-$2.85 $\\pm$ 0.10 dex and a heliocentric radial velocity of $v_r$ = 5.1 $\\pm$ 13.4 km s$^{-1}$.",
"Bo\\\"otes V is larger (in terms of scale radius), more distant, and more metal-poor than the vast majority of globular clusters.",
"It is likely that Bo\\\"otes V is an ultra-faint dwarf galaxy, though future spectroscopic studies will be necessary to definitively classify this object."
],
[
"Introduction",
"The discovery and study of dwarf galaxies in the Local Group has prospered due to the great advances in wide field imaging surveys over the last two decades [85], [86], [9], [10], [11], [42], [43], [39], [77], [78], [79], [53], [16], [17], [18], [67].",
"The Sloan Digital Sky Survey [88], [1] led the early charge, with the Pan-STARRS consortium [19], the Dark Energy Survey [2], and the DECam Local Volume Exploration Survey [25] contributing mightily to the discovery of Milky Way satellites, while the Pan-Andromeda Archaeological Survey [58] has been prolific in finding dwarf galaxies around M31 [57], [64], [49].",
"The dwarf galaxy population around M31 has also been bolstered by more recent work with the DESI Legacy Imaging Survey [23], [21], [52].",
"Dwarf galaxies play several key roles in testing and developing new models for answering some of the most fundamental questions in astronomy.",
"Dwarf galaxies fainter than around $M_V = -7.7$ have been dubbed “ultra-faint dwarf galaxies” (UFDs; e.g., see review by [72]).",
"They appear to be the smallest, least massive, and most metal-poor galaxies yet observed, and so represent the extreme end of the galaxy luminosity function.",
"They appear to reside in the shallowest gravitational potential wells that have been able to retain gas and stars throughout cosmic time [12], [72], and their existence tests current theories of galaxy formation, particularly the interplay between stellar feedback and gas retention [14], the evolution of faint galaxies in the Milky Way environment [83], and the quenching effects of reionization [15].",
"Critically, UFDs are powerful probes of theories of structure evolution, as they provide parsec-scale tests of dark matter models that were initially developed to explain the largest scale observations of the Universe [47], [14].",
"Discoveries of new Local Group dwarf galaxies continue to provide both unique individual examples and an ever-growing statistical sample of faint systems that are testing these aforementioned fundamental theories of structure and galactic evolution.",
"In this paper, we detail the discovery and characterization of a new ultra-faint dwarf galaxy candidate, which we call Boötes V. In Section we summarise the dataset in which it was detected and the search algorithm that was used.",
"In Section , we characterize the structural parameters of the dwarf galaxy candidate, as well as its distance and luminosity.",
"In Section , we identify member stars observed with Gaia, measure the proper motion of the system, and present first estimates of its metallicity and dynamics.",
"Finally, in Section , we discuss the classification of Boötes V and summarise our results.",
"We draw the attention of the reader to “Six More Ultra-Faint Milky Way Companions Discovered in the DECam Local Volume Exploration Survey” by Cerny et al.",
"(2022), which presents an independent discovery of this new satellite.",
"Boötes V was identified as an overdensity of resolved stars in the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS).",
"UNIONS is a consortium of northern wide field imaging surveys, and consists of the Canada-France Imaging Survey (CFIS) collaboration that uses the Canada-France-Hawaii Telescope (CFHT), team members from Pan-STARRS, and the Wide Imaging with Subaru HyperSuprime-Cam of the Euclid Sky (WISHES).",
"Each group is currently collecting imaging at their respective telescopes: CFHT/CFIS is targeting deep $u$ and $r$ band photometry, Pan-STARRS is obtaining deep $i$ and moderate/deep $z$ bands, and Subaru/WISHES is acquiring deep $z$ band.",
"These independent efforts are directed, in part, to securing optical imaging to complement the Euclid space mission, although UNIONS is a separate consortium aimed at maximizing the science return of these large and deep ground-based surveys of the northern skies.",
"When completed, the combined $ugriz$ survey will cover approximately 5000 deg$^2$ at declinations of $\\delta $ $>$ 30° and Galactic latitudes of $|b|$ $>$ 30° (the northern sky, excluding the Milky Way disk) and will be approximately as deep as one year of the Legacy Survey of Space and Time (LSST) at the Vera C. Rubin Observatory.",
"Our work uses the CFIS-r and Pan-STARRS-i combined dataset.",
"For CFIS-r, the 5-sigma point source depth is 24.9 with a 2 arcsecond () aperture.",
"For Pan-STARRS-i, the final 5-sigma point source depth will be 24.3.",
"However, the Pan-STARRS survey strategy of mapping the entire sky repeatedly means that the i-band is currently, on average, at about 75% of the final depth, although there is significant variation across the survey region.",
"The area in common between both bands at this stage amounts to $\\sim $ 3500 deg$^2$ total across both the North and South Galactic Caps (NGC/SGC).",
"These catalogs were crossed matched using a 0.5 matching tolerance (although typically the sources match to better than 0.13).",
"Star-galaxy separation was performed using morphological criteria in CFIS-r, which has a median image quality of better than 0.7.",
"We correct for the Galactic foreground extinction using the extinction values, $E(B-V)$ , from [69] assuming the conversion factors given by [68] for a reddening parameter of $R_{V}$ = 3.1.",
"For the CFIS-r, we adopt the conversion factor for the DES $r$ -band: the full-width at half-maximum (FHWM) of the DES-r [26] and CFIS-r are identical, and the DES $r$ -band is shifted redwards with respect to the CFIS filter by only 2 nm."
],
[
"GMOS-N Spectroscopy",
"We received Directors Discretionary Time at Gemini North to use the Gemini Multi-Object Spectrograph (GMOS-N) to obtain the spectrum of a single star, the brightest in Boötes V, through the program GN-2022B-DD-201 (PI: S. Smith).",
"Our long-slit spectroscopic observations used the R831 grating and the RG610 spectroscopic blocking filter, with a 1.0 wide slit.",
"We had a 1 $\\times $ 2 CCD binning configuration, resulting in a $\\sim $ 0.38 Å per-pixel spectral resolution.",
"Our observations were comprised of 3 $\\times $ 900 s exposures centered at 8500 Å and 3 $\\times $ 900 s exposures centered at 8600 Å during the nights of August 7th and August 8th, bringing our total exposure time to 5400 s. The science observations were reduced and extracted using standard routines and procedures for GMOS data in the Gemini IRAF package.",
"Bias corrections and flat-fielding were performed, along with the calibration of the wavelength solution using CuAr emission lamp exposures that were taken alongside the science observations.",
"The three frames obtained at each central wavelength were stacked using GemCombine, and a one-dimensional spectrum was extracted from each stack.",
"The flux was then calibrated using a 60 s exposure of a spectrophotometric standard (G191B2B) using an identical instrument set-up with the observations centered at a wavelength of 8500 Å.",
"The 8500 Å-centered stack and 8600 Å-centered stack were observed on consecutive nights, so their extracted spectra were individually corrected for Earth's orbital motion about the Sun before they were added together, producing the final spectrum of the target star.",
"The exposure times, instrument set-up, and wavelength range were selected with the goal of measuring a signal-to-noise ratio (S/N) of $\\sim $ 50 per-pixel in the region of the Calcium ii infrared triplet (CaT) absorption feature.",
"This has been demonstrated as sufficient to measure the heliocentric radial velocity with an uncertainty of $\\sim $ 10 km s$^{-1}$ [60], and also sufficient to measure the CaT Equivalent Width (EW) with adequate precision for reaching an uncertainty of less than 0.2 dex in the metallicity estimate through the CaT EW - [Fe/H] relation of [74].",
"Details of the measured properties of this spectrum are found in Section REF ."
],
[
"Detection",
"Boötes V was identified as one of the most promising candidates in a new and ongoing search for dwarf galaxies in the UNIONS sky.",
"The method of detection is based on a matched-filter approach, a tried and tested methodology that has yielded many previous dwarf galaxy discoveries [48], [8], [24].",
"We start by selecting all stars that are consistent with a 12 Gyr, [Fe/H] = $-$ 2 dex isochrone shifted to a heliocentric distance $d$ .",
"The algorithm filters the sky systematically in logarithmic steps for $d$ in the range [10, 1000] kpc and the isochrones used in this work were obtained from the PARSEC isochrone database [13] for the CFHT and Pan-STARRS 1 (PS1) photometric systems.",
"We adopt a broad color cut around the isochrone that is more than sufficient to account for the empirical photometric errors in both the CFIS-r and Pan-STARRS-i, the intrinsic color spread of any putative dwarf galaxy, and the unknown distance of the dwarf.",
"Formally: $\\big |(r-i)_{\\text{star}} - (r-i)_{\\text{iso}}\\big | \\le \\sqrt{0.1^2 + \\sigma _r^2 + \\sigma _i^2},$ where $(r-i)_{\\text{star}}$ is the color of a given star, $(r-i)_{\\text{iso}}$ is the color of the isochrone at the same magnitude as the given star, and $\\sigma _{r,i}$ are the photometric uncertainties in each passband at the same magnitude as the given star.",
"The algorithm does not consider stars fainter than CFIS-r = 24.6 and Pan-STARRS-i = 24, since below these limits completeness becomes an issue, although all detected stars are used when visually inspecting candidates.",
"Stars meeting this color criteria are selected and projected onto the tangent plane, where they are spatially binned into 0.5 $\\times $ 0.5 pixels.",
"This pixel size was selected to be slightly smaller than the typical angular size of the smallest known dwarfs so that a single galaxy will appear as several pixels in the two dimensional (2D) stellar density map.",
"The next step concerns teasing out local overdensities of stars.",
"We find the mean local stellar density for each pixel by convolving the field with a 2D top hat filter with a width of 20, where the large kernel was chosen to sufficiently smooth the stellar density around a putative dwarf galaxy.",
"We find the standard deviation in the local stellar density by taking the square root of the variance of these convolved maps.",
"By definition, this is the square of the mean local density minus the mean of the squared local density, i.e., $\\sigma _{\\text{loc}} = \\sqrt{\\langle \\rho _{\\text{loc}} \\rangle ^2 - \\langle \\rho _{\\text{loc}}^2 \\rangle }~.$ $\\rho _{\\text{loc}}$ is the density of the local field and $\\sigma _{\\text{loc}}$ is the standard deviation in the local field.",
"Returning to the unsmoothed distributions, we convolve these maps with 2D Gaussian kernels of different sizes, corresponding to dwarf galaxies with different projected sizes.",
"Specifically, we adopt kernels with a FWHM of 1.2, 2.4, and 4.8.",
"We refer to these “smoothed densities” as $\\rho _{\\text{sm}}$ .",
"Finally, for the entire UNIONS sky, we compute a 2D map of the statistical overdensities with respect to the local stellar density as $s = \\frac{\\rho _{\\text{sm}} - \\rho _{\\text{loc}}}{\\sigma _{\\text{loc}}},$ where $s$ is the significance of the overdensity in units of $\\sigma _{\\text{loc}}$ .",
"This process is repeated for a range of distances, where each distance produces three maps, one for each of the smoothing kernels.",
"During initial tests of this method, nearly all known dwarf galaxies in the UNIONS footprint have been detected with high significance.",
"As a reference point, Canes Venatici II is detected with high significance at several distances.",
"At a distance of 100 kpc, with the stellar density map smoothed by a 1.2 kernel, Canes Venatici II (true distance of 160 kpc) has $s = 2.8$ , while Boötes V has a significance value of $s = 4.9$ .",
"Boötes V is one of the most prominent overdensities of match-filtered stars not previously detected, and it is detected strongly at many distances less than 150 kpc.",
"In support of the match-filter detection, we present the CFIS-r imaging of the putative dwarf galaxy (left panel of Figure REF ) wherein a central clustering of sources is seen prominently.",
"As mentioned in Section REF , we separate sources in the imaging into stars and galaxies, and in the right panels of Figure REF , we show the surface density of both of these.",
"Clearly, the clustering is not due to background galaxies.",
"We show the spatial density of the match-filter-selected stars alongside the color-magnitude diagram (CMD) of sources in the vicinity of Boötes V in Figure REF .",
"The positions of all these sources on the CMD suggest that many of the stars in this field of view are consistent with the same old, metal-poor stellar population given a putative distance to the system of 100 kpc."
],
[
"Structure",
"Given the probable identification of a new Milky Way satellite as shown in Figure REF , we determine the most likely values of its structural parameters using a Markov-Chain Monte Carlo (MCMC) approach under the assumption that the system is well described by an exponential model.",
"Our model and approach are based on the work in [48] & [50], and we used emcee [27] to sample the posterior.",
"The radial surface density profile $\\rho _{\\text{dwarf}}(r)$ for a dwarf galaxy can be described with an elliptical, exponential model as a function of $r$ .",
"This profile is defined by the centroid of the profile ($x_0, y_0$ ), an ellipticity $\\epsilon $ (defined as $\\epsilon = 1 - b/a$ where $b/a$ is the minor-to-major-axis ratio of the model), the position angle of the major axis $\\theta $ , (defined East of North), the half-light radius (which is the length of the semi-major axis $r_{\\text{h}}$ ), and the number of stars $N^*$ in the system.",
"The model is written as $\\rho _{\\text{dwarf}}(r) = \\frac{1.68^2}{2\\pi r_h^2 (1-\\epsilon )}N^* \\exp {\\bigg (\\frac{-1.68r}{r_{\\text{h}}}\\bigg )},$ where $r$ , the elliptical radius, is related to the projected sky coordinates ($x$ , $y$ ) by $\\nonumber r = \\Bigg \\lbrace \\bigg [\\frac{1}{1-\\epsilon } \\Big ((x-x_0)\\cos \\theta - (y-y_0)\\sin \\theta \\Big ) \\bigg ]^2 \\\\ + \\bigg [ \\Big ((x-x_0)\\sin \\theta - (y-y_0)\\cos \\theta \\Big )^2 \\bigg ]^2\\Bigg \\rbrace ^{\\frac{1}{2}}.$ We assume that the background stellar density is constant, which is reasonable on the scale of arcminutes up to a degree or so.",
"As such, we model the density of stars in our entire field of view as $\\rho _{\\text{model}}(r) = \\rho _{\\text{dwarf}}(r) + \\Sigma _{\\text{b}},$ where $\\Sigma _{\\text{b}}$ is the constant background density term.",
"This term is defined to be $\\Sigma _{\\text{b}} = \\frac{n - N^*}{A},$ where $n$ is the total number of stars in the field of view and $A$ is the total area, normalizing the background density with respect to the selected region.",
"We use all matched-filter selected sources within a circle of radius 9 surrounding the satellite's initial centroid, projected onto the tangent plane.",
"We mildly constrain the model parameters to be physically consistent with the selection region, but assume flat priors within the constraints (shown in Table REF ).",
"The emcee program used 64 walkers, each going through 15,000 iterations and the first 7500 iterations were thrown out to account for burn-in.",
"Through this analysis, we find that Boötes V is located at a Right Ascension (RA) of 14h 15m 38.6s and a Declination (Dec) of +32° 54 40.",
"We present the results of the MCMC analysis as a corner plot in Figure REF and we include the final parameter estimates alongside all other measured and derived properties of Boötes V in Table REF .",
"Table: Flat priors for each parameter in the MCMC analysis.Table: Measured and Derived Properties for Boötes VTable: Identifier, Magnitudes, and Distance information for three potential BHB stars"
],
[
"Distance Determination",
"Boötes V was initially identified as an overdensity of stars at around 100 kpc.",
"Given this initial distance estimate, comparisons to isochrones suggested that there may be a potential member star near the tip of the red giant branch (RGB), as well as three possible member stars consistent with being blue horizontal branch (BHB) stars.",
"These are shown as a red triangle and blue squares respectively, in Figure REF .",
"Note that the two brightest BHB candidates occupy an almost identical position in this figure.",
"Under the assumption that the bright star is indeed an RGB star, we can estimate the distance to Boötes V by assuming that this star is at the tip of the RGB (TRGB).",
"This is the upper bound on the distance estimate as the lone, bright star may not truly be at the very tip, and there are no other similarly bright stars in the system with which to compare.",
"The TRGB in the $i$ -band has been shown to act as a standard candle and can therefore be used for distance estimation [44], [66].",
"The corrected $i$ -band luminosity of this star is $i_0 = 17.05 \\pm 0.02$ and the absolute magnitude of the TRGB in the PanSTARRS-i band is $M_{i, TRGB} = -3.56$ mag.",
"We adopt an uncertainty on this value of order 0.05 mags from the globular cluster calibration by [29].",
"The gives $(m - M)_i = 20.61 \\pm 0.05$ , yielding a heliocentric distance upper bound of $d_{\\text{upper}} = 132.3 \\pm 3.2$ kpc.",
"For determining the distance to the potential BHB stars, we follow the calibration given by [22] for the absolute $g$ -band magnitude of BHB stars using SDSS $g$ - and $r$ -band magnitudes: $M_g = 0.434 - 0.169(g_0 - r_0) + 2.319(g_0 - r_0)^2 + \\\\\\nonumber 20.449(g_0 - r_0)^3 + 94.517(g_0 - r_0)^4.$ All three potential BHB stars are in SDSS DR17 and their extinction-corrected SDSS $g$ and $r$ magnitudes are presented in Table REF .",
"The faintest BHB star has $(g_0 - r_0)$ = $-$ 0.34, which falls outside the color range for which the above equation is valid ($-0.25$ $<$ $(g_0 - r_0)$ $<$ 0, see [22] for details), so we cannot use it for a distance estimate.",
"Table REF presents derived absolute $g$ magnitudes and distances for the two bright BHBs.",
"They have heliocentric distances of 97.4 $\\pm $ 2.8 and 97.4 $\\pm $ 3.2 kpc, where the uncertainties are only those obtained by propagating the uncertainties in magnitude A third estimate of the distance may be obtained by estimating the distance an isochrone should be shifted to in order to best match the CMD.",
"As such, we determined the best-match isochrone through visual inspection by adopting a 13 Gyr stellar population, and considering [Fe/H] in the range [$-2.9$ , $-1.9$ ] dex in intervals of 0.2 dex, and distance shifts in the range [80, 175] kpc in intervals of 5 kpc.",
"We found that an [Fe/H] $= -2.3$ dex isochrone with a distance of 100 $\\pm $ 20 kpc describes the data reasonably well.",
"Figure REF shows the CMD of all stars within 3 elliptical $r_{\\text{h}}$ , with the isochrone overlayed at distances of 80, 100, and 120 kpc, to demonstrate the range we estimate as reasonable: much closer or much more distant, and the isochrone (both the hydrogen and helium burning branches) does not align well with the bulk of the stars, especially, but not only, the BHB and RGB candidates discussed earlier.",
"While each of these methods of estimating the distance to this satellite are not precise, they all point to a distance of approximately 100 kpc.",
"Consequently, for the remainder of our analysis, we adopt a distance to the satellite of 100 $\\pm $ 20 kpc.",
"This estimate from the by-eye isochrone analysis is fully consistent with estimates from both BHB stars and the upper limit provided by the candidate RGB star.",
"We searched the PS1 RR Lyrae [70] and Gaia variability [30] catalogs, but could not find evidence that any RR Lyrae stars have been detected, so we unfortunately cannot use the properties of these variable stars to constrain the distance to Boötes V. Figure: Yellow sources are those selected as high confidence (probability >> 90%) members by our maximum-likelihood membership selection algorithm.",
"Grey sources/shading are all 5-parameter Gaia detections in a 2° circle around Boötes V.Left: Sky positions of stars in a 12 ×\\times 12 region about Boötes V, projected onto the tangent plane centered at RA, Dec of (14h 15m 38.6s, +32° 54 42).Center: Color-magnitude diagram with a 13 Gyr, [Fe/H] = --2.3 isochrone overlaid.Right: Proper motions of members overlaid on Milky Way foreground stars."
],
[
"Luminosity",
"To estimate the luminosity of Boötes V, we use an approach that is based on the work of [50], which aims to create a suite of synthetic stellar populations that represent the full stellar content of the dwarf galaxy.",
"The construction of each individual synthetic stellar population is performed as follows.",
"We assume that the stellar population in question is well represented by a 13 Gyr, [Fe/H] $= -2.3$ PARSEC isochrone [13], with a standard Kroupa initial mass function [41], that has been corrected for Galactic foreground extinction (same as description in Section REF ).",
"We then draw a random distance from a normal distribution centered at 100 kpc, with a standard deviation of 20 kpc, and shift the isochrone to that distance.",
"We then construct an $i$ -band luminosity function (ignoring the horizontal branch) down to 0.1 $\\textup {M}_\\odot $ , and normalize it so that it behaves as a probability distribution function (PDF).",
"Crucially, these mock stellar populations are characterized by the number of stars above the magnitude limit of our survey, emulating the actual stellar population that was observed.",
"We reran the MCMC analysis using only stars consistent with the main sequence (MS) and red giant branch (RGB) that have Pan-STARRS-i $<$ 24, and we found that Boötes V is estimated to have 34$^{+7.3}_{-6.5}$ stars that fit these criteria.",
"So, we also sample a normal distribution centered at 34, with a standard deviation of 6.9 (average of empirical 16th and 84th quantiles), and set this value to be $N$ , the target number of stars in the mock stellar population brighter than the limiting magnitude of 24 mag in Pan-STARRS-i.",
"Randomly drawing values for both the distance and the number of stars above the magnitude limits for each rendition of the stellar population allows for the propagation of parameter estimate uncertainties into the final derived magnitude of the system.",
"Finally, to create the synthetic stellar population, we sample the PDF using the acceptance-rejection method, recording the $r$ -band, $i$ -band, and $V$ -band magnitudes of each accepted star, and flaging each star that has $m_i < 24$ mag.",
"When we accrue $N$ flagged stars (where $N$ is the target number of stars in that realization of Boötes V), we stop sampling.",
"Finally, we shift apparent magnitudes to absolute magnitudes (using the distance corresponding to that specific realization of the stellar population), convert to fluxes, sum them, and convert to the total absolute magnitude.",
"We generate 1000 instances of the stellar population and find the systemic magnitude of Boötes V to be $M_V = -4.5$ $\\pm $ $0.4$ mag.",
"When converted into total luminosity, we get $5.4^{+2.2}_{-1.6}$ $\\times $ $10^3$ $\\textup {L}_\\odot $ .",
"We calculate the effective surface brightness by dividing half the total flux by the area enclosed by one elliptical half-light radius and converting to mag arcsec$^{-2}$ .",
"We calculate it to be 25.7 $\\pm $ 0.7 mag arcsec$^{-2}$ with all errors from the magnitude, half-light radius, and ellipticity propagated through.",
"These values, including the absolute magnitudes in the $r$ - and $i$ -bands, are included in Table REF ."
],
[
"Proper Motion, Membership using Gaia",
"The Third Data Release from Gaia [31], [32] has a limiting magnitude of about $G = 21$ mag.",
"The majority of stars we selected as members of this system based on UNIONS photometry using our matched-filter method are fainter than the Gaia limits, but several of the brighter stars are successfully detected by Gaia, allowing for the use of Gaia's powerful astrometry [45] in characterizing this system, as well as providing clear confirmation of its reality.",
"We follow the methodology developed by [55] to estimate the systemic proper motion of the system and to clearly identify member stars.",
"The reader is referred to this paper for details.",
"Briefly, the algorithm estimates the most likely proper motion for a putative satellite under the assumption that a field of stars consists of only a Milky Way satellite and Milky Way foreground, by considering the CMD distribution of stars, the spatial distribution of stars, and the proper motion distribution of stars.",
"The foreground/background density is assumed constant in the region of the putative satellite, and the CMD and proper motion distributions of the foreground/background are derived empirically.",
"The CMD distribution of the putative satellite assumes an old, metal-poor system at some distance, and the spatial distribution of the putative satellite assumes a 2D exponential profile described by a centroid, ellipticity, position angle and half-light radius.",
"As such, the distance and structural parameter estimates derived earlier (and their uncertainties) are used in this analysis.",
"The CMD distribution is modelled as an old (13 Gyr), metal poor ([Fe/H] = $-$ 2.3) isochrone in the Gaia passbands [65].",
"Additionally, we incorporated extra constraints for robustly selecting horizontal branch stars (see Jensen et al.",
"2022, in prep, for further details).",
"This method assumes that the proper motions of all stars within the putative satellite share the same intrinsic proper motion, with any variance coming directly from measurement uncertainties, so that the proper motion PDF is modeled as a bivariate Gaussian function.",
"For Boötes V, we follow [56], and select well-measured sources in a circular region with a radius of 2° around the previously determined centroid.",
"We identify six high confidence members (membership probability $>$ 90%) within 3 elliptical $r_{\\text{h}}$ and display these stars in Figure REF .",
"This figure shows the spatial positions, $G_0$ versus $(B_P-R_P)_0$ CMD, and proper motions of the 6 members compared to the field (grey points).",
"We find the systemic proper motion of Boötes V to be $(\\mu _\\alpha $ cos$\\delta , \\mu _\\delta ) = (-0.23$ $\\pm $ 0.04 (stat) + 0.033 (sys), $-0.28$ $\\pm $ 0.07 (stat) + 0.033 (sys)) mas year$^{-1}$ .",
"We note that such a tight clustering in all these spaces, especially proper motion space, provides a robust confirmation that this is a real physical system.",
"Systematic errors are taken from [45].",
"Encouragingly, all of these high confidence members are also found in the original matched-filter selection that identified the stellar overdensity in UNIONS (Section REF ).",
"Of these six likely member stars, two are the brightest stars identified as potential BHB stars in Section REF .",
"The third BHB star, which was not used for a distance estimate, does not have well measured astrometric parameters, and so its membership cannot be classified by this method.",
"Another of the six likely member stars is the bright RGB candidate discussed earlier.",
"The Gaia member stars are cross-matched to the UNIONS data set and are plotted as yellow markers on Figure REF .",
"We note that in Figure REF , several of the faintest stars appear to be HB stars that lie below the isochrone track.",
"While this might suggest we are underestimating the distance, shifting the isochrone to larger distances only produces a reasonable fit to the data up to about 120 kpc, which is within the uncertainty we adopt when only using UNIONS data.",
"A deeper CMD than obtained with Gaia or UNIONS is necessary to allow a more robust distance measurement."
],
[
"Spectroscopic Follow-up Analysis",
"In Section REF we assumed that the brightest star consistent with the isochrone was a red giant star near the TRGB.",
"Our membership analysis has also identified this star as a highly likely member (probability $> 99.9\\%$ ).",
"Fortuitously, this star was observed as part of the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST), and has low-resolution spectroscopy available as part of its data release 5 (DR5).",
"Its LAMOST stellar identifier is LAMOST HD141746N331518M01.",
"LAMOST DR5 [87] provides an estimation for both the metallicity and heliocentric radial velocity of this RGB star, with [Fe/H] = $-$ 2.25 $\\pm $ 0.60 dex $v_r$ = $-5.5 \\pm 22.7$ km s$^{-1}$ .",
"However, the uncertainties on both measurements are rather large, which prompted us to obtain our own low-resolution spectrum of this RGB star using GMOS-N as described in Section REF .",
"The analysis of the reduced spectrum is now presented.",
"We infer the metallicity of the star adopting the method from [74].",
"Their method needs as input the equivalent width (EW) of the second and third components of the Ca ii Triplet ($\\lambda \\lambda 8498.02, 8542.09, 8 662.14$ Å) and the absolute magnitude of the star $\\rm {M_V}$ [74].",
"The EW is measured using the splot routine in IRAF [75], [76] fitting with multiple line profiles.",
"The median and the standard deviation has been adopted as final values for the EW and its uncertainty.",
"$\\rm {M_V}$ is derived converting the Gaia DR3 magnitudes to the Johnson-Cousin filter following the relation from [65] and adopting a heliocentric distance of 100 $\\pm $ 20 kpc.",
"We perform a Monte Carlo analysis with $10^6$ randomizations on the heliocentric distance, the EW, and the de-reddened magnitudes assuming Gaussian distributions for all parameters.",
"The final [Fe/H] and its uncertainty are the median and the standard deviation of the randomizations, respectively.",
"The star is measured to be a very metal-poor star, with [Fe/H] = $-$ 2.85 $\\pm $ 0.10 dex.",
"To calculate the radial velocity, we create a synthetic spectrum using the synth option in MOOG https://www.as.utexas.edu/~chris/moog.html [73] with the list of spectral lines generated by linemakehttps://github.com/vmplacco/linemake [61].",
"We adopted a model atmosphere from the MARCS1 models [33], [62].",
"The synthetic spectrum has been created at the same resolution of the GMOS spectrograph with the stellar parameters $\\rm {T_{eff}}$ , logg, and [Fe/H], where each parameter is derived with the following methods.",
"The effective temperature is derived using the calibration from [59], which takes as input the knowledge on the nature of the star (i.e., dwarf or giant), the metallicity, and the de-reddened Gaia EDR3 photometry.",
"Then, the surface gravity is inferred adopting the Stefan-Boltzmann equation.",
"This needs the distance, the de-reddened G magnitude, the bolometric corrections on the flux [3], an estimate on the effective temperature, and the stellar mass ($0.5-0.8$$\\textup {M}_\\odot $ ).",
"A Monte Carlo randomization has been performed to infer the $\\rm {T_{eff}}$ , logg, and their uncertainties.",
"These methods have proved to yield robust stellar parameters compatible with high-resolution spectroscopic values in the very metal-poor regime [38], [71], [81].",
"We derive $\\rm {T_{eff}}= 4481$ $\\pm $ 77 K and logg $=0.87$ $\\pm $ $0.11$ , and incorporate these parameters into the synthetic spectrum.",
"Then, we added a Poissonian noise to match the observations.",
"The radial velocity is measured by cross-correlating the combined observed spectrum with the downgraded synthetic spectrum using the fxcor routine in IRAF, and is found to be 5.1 $\\pm $ 13.4 km s$^{-1}$ .",
"In the absence of any other stars in Boötes V with measured radial velocities, we adopt this radial velocity as representative of the systemic velocity of the candidate dwarf."
],
[
"Orbital analysis",
"We now use a simple dynamical model to examine the orbit of Boötes V, with the aim of understanding its orbital history and interaction with the Milky Way.",
"We approximate Boötes V as a point-mass in a Milky Way potential, implemented with the python-wrapped package gala [63].",
"The Milky Way potential used for this analysis is identical to that which is described in Section 5 of [36], so we direct the reader to that work, and references therein, for full details.",
"We perform 1000 realizations of the point-mass orbit, where the initial conditions (i.e.",
"input parameters { $\\alpha _{J2000}$ , $\\delta _{J2000}$ , $d$ , $\\mu _{\\alpha }$ cos$\\delta $ , $\\mu _{\\delta }$ , $v_r$ }) for each realization are drawn from normal distributions with means and standard deviations defined by the values and associated errorbars presented in Table REF .",
"Each point-mass is integrated both forwards and backwards in time by 1 Gyr in time steps of 10$^{-3}$ Gyr.",
"For each orbit, the pericenter (closest approach to Milky Way), apocenter (furthest point from Milky Way), Z max (maximum height above the disk), time between pericenters (orbital time), time since last pericenter, and orbital eccentricity are recorded and presented in 1D distributions in Figure REF .",
"Figure: Absolute V-band magnitude (M V _V) vs. half-light radius (r h r_{\\text{h}}) plane showing both the dwarf galaxy (black markers) and globular cluster (blue Xs) populations.",
"We show both candidate and spectroscopically confirmed dwarf galaxies within 300 kpc of the Milky Way.",
"Boötes V is shown as a yellow triangle."
],
[
"Classification and Conclusions",
"In this paper we have presented the discovery of the Boötes V satellite of the Milky Way in UNIONS.",
"We detected it using a matched-filter method on the combined CFIS-r and Pan-STARRS-i photometric catalog, from which we estimated the distance, luminosity and structural parameters of the system.",
"Examination of the Gaia DR3 data allows us to confirm this as a real system, identify its brightest members in the Gaia data, and estimate its proper motion.",
"GMOS-N observations also allow us to provide a first estimate of its spectroscopic metallicity and its radial velocity, which in turn allow us to characterize the orbit of the system.",
"Based on the results of the orbital estimation, Boötes V is approaching the apocenter of its orbit, having been at pericenter $\\sim $ 0.5 Gyr ago.",
"Interestingly, the pericenter of the orbit is 16.6$^{+6.8}_{-7.6}$ kpc from the center of the Milky Way.",
"With an orbital period of $1.4$ Gyr, Boötes V may have had time to complete several orbits.",
"Several such close passes with the inner part of the Milky Way potential may mean that tidal forces have affected this satellite, though there does not appear to be significant visual evidence in support of tidal disruption.",
"Additionally, the orbit is highly eccentric and the motion of the satellite is primarily occurring perpendicular to the disk.",
"We note, however, that the measure of a single radial velocity is a poor proxy for the systemic radial velocity of the satellite, as both dwarf galaxies and globular clusters are measured to have velocity dispersions on the scale of a few to tens of km s$^{-1}$ .",
"We also note that the observed star for which we obtained spectroscopy is a visual double, with a companion that does not have fully measured astrometric parameters in Gaia DR3.",
"However, the double is separated on the sky by 1.72 , which, at a distance of 100 kpc, corresponds to a minimum physical separation of 0.83 pc.",
"It is very unlikely that these two stars are bound, and if they are, a negligible portion of the radial velocity on the measured star would be due to binary orbital motion.",
"Regardless, the radial velocity of this single, bright RGB star is still our best available tool for providing an initial estimate of the orbit for Boötes V. Boötes V is faint ($M_V$ = $-$ 4.6 $\\pm $ 0.4 mag), small ($r_{\\text{h}}$ = 27.1$^{+7.2}_{-5.6}$ pc), and distant ($d$ = 100 $\\pm $ 20 kpc).",
"A question remains as to whether this system is a dwarf galaxy or a globular cluster.",
"Figure REF illustrates the absolute V-band magnitude (M$_V$ ) vs. half-light radius ($r_{\\text{h}}$ ) plane, a commonly used metric for sorting dwarf galaxies and globular clusters.",
"Here, we use only dwarf galaxies within 300 kpc of the Milky Way.",
"The data and separation are taken from the updated dataset of [54]https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/community/nearby/.",
"For the globular clusters, we used data from [7]https://people.smp.uq.edu.au/HolgerBaumgardt/globular/parameter.html.",
"Boötes V resides in a region of this plane that has largely contained satellites whose classification has been contested.",
"However, it is worth noting that, in terms of scale-radius ($r_{\\text{h}}$ ), Boötes V would be the third largest globular cluster while only being the seventh smallest dwarf.",
"Additionally, at a heliocentric distance of 100 kpc, Boötes V would also be the fourth most distant globular cluster, but at a distance typical of dwarfs.",
"The metallicity of this system is very notable.",
"The GMOS-N spectrum was measured to be very metal-poor, with an [Fe/H] = $-$ 2.85 dex, indicating that Boötes V may be among the most metal-poor dwarf galaxies known.",
"In terms of overall mean metallicities, Tucana II is the only dwarf with a mean metallicity more metal poor than this, at $-$ 2.90 dex [20].",
"Four additional dwarf galaxies have $\\langle $ [Fe/H]$\\rangle $ $< -2.7$ [37], [46], [40], [28].",
"Each of these five very metal-poor dwarfs contain individual stars that are measured to have [Fe/H] $\\lesssim $ -3 dex.",
"While this single metallicity measurement certainly suggests Boötes V must be quite metal poor, we do not yet know how representative this star is of the overall mean metallicity of the system, and more spectroscopic observations are needed.",
"Globular clusters do not appear to extend to such low mean metallicities as dwarf galaxies.",
"Indeed, if Boötes V were a globular cluster, we would expect this single measurement to be more representative of the whole system, and it would be nearly 0.5 dex more metal-poor than then most extreme, intact globular cluster [35].",
"However, it is worth noting that two remarkably metal-poor stellar streams have been discovered whose progenitors are thought to be ancient globular clusters, the Phoenix stellar stream [82], and the C-19 stellar stream [51], which have [Fe/H] = $-$ 2.70 and [Fe/H] = $-$ 3.38, respectively.",
"The rather large caveat throughout this entire discussion is that we only have a metallicity measurement for a single star.",
"However, it does immediately make Boötes V an interesting object of future spectroscopic studies as results from the observations of several more stars will be two-fold: measurements of its stellar velocity dispersion will provide a direct estimation of the mass (and by extension, the dark matter content), and measurements of the metallicity dispersion have more recently been used as a proxy for mass, insofar as systems with a large metallicity dispersion likely need to reside, or have resided, in a massive dark matter halo [84].",
"Both measurements will likely provide clarity relating to the classification of this newly discovered system.",
"While it is not inconceivable that this system could be a globular cluster, we consider the balance of evidence to currently suggest that Boötes V is more likely a member of the ever-growing class of ultra-faint dwarf galaxies.",
"Boötes V is the first Milky Way satellite to be discovered in the UNIONS dataset, which, when complete, will provide coverage of approximately 5000 square degrees of the northern extragalactic sky at a depth comparable to the first year of the Legacy Survey of Space and Time by the Vera C. Rubin observatory.",
"The discovery of a single new candidate dwarf galaxy is, in some sense, another drop in the bucket, as the list of known Local Group galaxies continues to grow quickly.",
"Even though each new object is worthy of study in its own right, there is a more comprehensive goal underlying all of this work.",
"The true power of broad searches for new dwarf galaxies is that they build towards amassing a statistically significant population of faint stellar systems whose chemical, dynamical, and structural properties will test theories of dark matter and of galaxy formation on the smallest scales."
],
[
"Ackownledgements",
"We acknowledge and respect the lək̍$^{\\rm w}$ əŋən peoples on whose traditional territory the University of Victoria stands and the Songhees, Esquimalt and $\\underaccent{\\bar{}}{\\rm W}$ SÁNEĆ peoples whose historical relationships with the land continue to this day.",
"It was a pleasure to coordinate the submission of this paper with an independent discovery paper led by William Cerny and the DELVE team, and we thank William, Alex Drlica-Wagner, and the whole DELVE team for the very positive, collaborative interactions that we had with them.",
"This work is based on data obtained as part of the Canada-France Imaging Survey, a CFHT large program of the National Research Council of Canada and the French Centre National de la Recherche Scientifique.",
"Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA Saclay, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers (INSU) of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.",
"This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency.",
"This research is based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.",
"We are honored and grateful for the opportunity of observing the Universe from Maunakea, which has the cultural, historical and natural significance in Hawaii.",
"Pan-STARRS is a project of the Institute for Astronomy of the University of Hawaii, and is supported by the NASA SSO Near Earth Observation Program under grants 80NSSC18K0971, NNX14AM74G, NNX12AR65G, NNX13AQ47G, NNX08AR22G, YORPD20_2-0014 and by the State of Hawaii.",
"This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.",
"Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences.",
"Funding for the project has been provided by the National Development and Reform Commission.",
"LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.",
"AWM acknowledges support from the NSERC Discovery Grant program.",
"NFM gratefully acknowledges support from the French National Research Agency (ANR) funded project “Pristine” (ANR-18-CE31-0017) along with funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No.",
"834148).",
"FS thanks the Dr. Margaret \"Marmie\" Perkins Hess postdoctoral fellowship for funding his work at the University of Victoria.",
"CFHT, PS1, Gemini:Gillett, Gaia astropy [4], [5], [6], emcee [27], numpy [34], scipy [80]"
]
] | 2209.08242 |
[
[
"How to Define the Propagation Environment Semantics and Its Application\n in Scatterer-Based Beam Prediction"
],
[
"Abstract In view of the propagation environment directly determining the channel fading, the application tasks can also be solved with the aid of the environment information.",
"Inspired by task-oriented semantic communication and machine learning (ML) powered environment-channel mapping methods, this work aims to provide a new view of the environment from the semantic level, which defines the propagation environment semantics (PES) as a limited set of propagation environment semantic symbols (PESS) for diverse application tasks.",
"The PESS is extracted oriented to the tasks with channel properties as a foundation.",
"For method validation, the PES-aided beam prediction (PESaBP) is presented in non-line-of-sight (NLOS).",
"The PESS of environment features and graphs are given for the semantic actions of channel quality evaluation and target scatterer detection of maximum power, which can obtain 0.92 and 0.9 precision, respectively, and save over 87% of time cost."
],
[
"Introduction",
"With the ever-increasing diverse scenes and communication requirements, predictive 6G Network with environment sensing enhancement is becoming promising [1][2].",
"Powered by advanced sensing techniques, environment reconstruction can be deployed, and the detailed environment information [3][4] enables the applications with precision improvement.",
"However, the demanded new technologies with increasingly-high data rates require online predictions for dynamic environments, especially in non-line-of-sight (NLOS) scenarios.",
"Powered by natural language processing (NLP) and computer vision (CV) techniques that have lots of potential in processing intelligent tasks, semantic communication [5]-[7] has drawn significant attention, which mainly relies on semantic-based information conversion between different content to achieve efficient, intelligent interaction.",
"Semantic communication focuses on the content between the transmitter (Tx) and receiver (Rx), which considers the difference between the meaning of the transmitted messages and that of recovered ones for different semantic tasks.",
"Similar to the semantic pipeline of semantic communication, the propagation environment and application task also need semantic (one-to-many) mapping that carries the meaning rather than the object (one-to-one) mapping without understandable information because online applications do not always perform with every environment changing.",
"In [8][9], the cluster nuclei is proposed by directly mapping the physical environment to the channel.",
"For representing the propagation environment from different perspectives, the environment features and graph representations are proposed in [10] and [12], which can assist the efficient channel prediction in dynamic environments.",
"Hence, we believe that defining the propagation environment semantics (PES) by considering the different environment representations, i.e., propagation environment semantic symbols (PESS), is crucial for highly efficient prediction by leveraging the environment information directly.",
"In this paper, PES is defined as a PESS set, where the set is limited because of considering the channel with limited properties as the basis.",
"The PESS are deconstructed environment representations at a semantic level for prediction applications.",
"Therefore, semantic mapping can be built between the PES and applications, and the specific task can be implemented by the related PES.",
"For beam prediction implementation in NLOS scenarios, the environment features and graph representations are considered the PESS for the semantic actions: channel quality evaluation and target scatterer detection.",
"Compared with the method that predicts the beam indices and requires the extra process of beam searching [4], the proposed method can provide the scatterer with maximum power directly and further empower other advanced techniques."
],
[
"Problem Formulation",
"The wireless channel is the essential intermediate bridge for the semantic mapping between the propagation environment and the application tasks.",
"Therefore, the task-oriented PES can be defined by the semantic deconstruction of the environment, which considers the channel properties.",
"Then, the tasks can be employed based on the machine learning (ML) method, as shown in Fig.",
"1.",
"Figure: Task-oriented PES construction process."
],
[
"Task-Oriented Environment Deconstruction",
"According to the channel-task prior knowledge, the concerned channel properties are impacted by different environmental information.",
"Diverse properties or property groups are required to meet different application tasks, where the properties include large-scale parameters (LSP): path loss, delay spread (DS), azimuth angle spread of arrival (ASA), azimuth angle spread of departure (ASD), zenith angle spread of arrival (ZSA), zenith angle spread of departure (ZSD), small-scale parameters (SSP): power, delay, azimuth angle of arrival (AOA), azimuth angle of departure (AOD), zenith angle of arrival (ZOA), zenith angle of departure (ZOD), and characteristic: line-of-sight (LOS) blockage.",
"The environment is deconstructed to meet the applications directly at the semantic level.",
"Following the environment-channel prior knowledge, the environment can be represented at large-scale and small-scale levels to meet the environment information requirement of different channel properties.",
"The large-scale level includes the layout and global environment representations for LSP, and the small-scale level consists of local and target representations for SSP and LOS blockage."
],
[
"PES Construction with PESS Extraction",
"The incurred semantics-based action of the application task depends on diverse background information, which decides how to interpret the intermediate information.",
"As for the target task, environment information is not equally important to the specific semantics needs, so the task-related information should be abstracted instead of retaining all of it.",
"As a result, the PES can be defined as the semantic variable that reflects the semantic changes.",
"Unlike the physical environment without specific interactions between objects, the propagation environment should be described with the preset Tx, Rx, and propagation mechanism.",
"The radio waves encountered with the scatterers can produce diverse propagation paths caused by various significant propagation mechanisms, such as LOS transmission, reflection, and diffraction.",
"Therefore, the geometry relationship-correlated propagation mechanisms can be regarded as the considerable environment-channel prior knowledge for PESS extraction.",
"According to the propagation mechanisms, the essential geometric attributes that affect the paths include position, dimensions, and layout.",
"Thereby, the fundamental PESS can be extracted as features $PESS_{feature}$ , linear vectors $PESS_{vector}$ , and non-linear graphs $PESS_{graph}$ to represent the environment characteristics or global structure at small-scale or large-scale level, as shown in Fig.",
"2.",
"Its expandable when extra data form are raised and the basis PES can be presented as the set of the PESS, i.e., $PESS = \\lbrace PESS_{feature}, PESS_{vector}, PESS_{graph}\\rbrace .$"
],
[
"PES for Beam Prediction",
"A beam prediction case is given for method verification to show how the PES works on the prediction tasks."
],
[
"Task-oriented Channel Properties",
"Beam prediction generally aims to improve the communication quantity by switching to a better target.",
"Thus, two task actions should be considered for PES-based beam prediction: channel quality evaluation and target scatterer prediction, as shown in Fig.",
"2.",
"Therefore, the respective channel properties can be analyzed according to the requirements.",
"Figure: The process of action decomposition and corresponding PESS extraction.The LSP can reflect channel quality evaluation related to the performance [10], where the parameter types should not be considered in detail for coarse semantic mapping.",
"In addition to the LSP, the blockage characteristic of LOS also has a significant impact, where the blockage would attenuate the performance.",
"Therefore, the critical channel properties of quality evaluation action $A_{evaluation}$ can be expressed as $A_{evaluation} = \\lbrace LSP, Blockage\\rbrace .$ Once it is determined that the current channel is unqualified, the target scatterer detection should be employed according to the maximum power, which is an SSP-related prediction problem.",
"Hence, the channel properties of target scatterer prediction $A_{detection}$ can be denoted by $A_{detection} = \\lbrace P_j, j \\in J\\rbrace ,$ where $P_j$ is the power of the $j$ -th path and total $J$ paths."
],
[
"Task-Oriented PES Construction",
"According to the channel properties, the environment information influencing the concerned characteristics and parameters should be represented from global and structural aspects for two different semantic actions implementation.",
"However, for practice applications, environment changes might cause propagation path changes but not significant statistics changes.",
"Hence, the environment is represented for all the channel properties instead of each parameter or characteristic.",
"The primary information of Tx, Rx, and internal scatterers are utilized as original data, which are environment-isolated information with no scene constraints.",
"PESS of Features: For the simultaneous description of the blockage characteristics and statistical LSP, the blockage and global features are extracted and constructed into an exclusive PESS representation, as shown in Fig.",
"2.",
"The degree of LOS occlusion is leveraged as a PESS for blockage description.",
"In practice, diverse methods can be utilized for PESS feature calculation according to the geometric relationship between the Tx, Rx, and the scatterer.",
"There, we use the method mentioned in [10], which obtains the blockage feature for each scatterer by describing the extent to which LOS and scatterers intersect.",
"Specifically, the distance $d_i$ between the center point of $i$ -th scatterer and the LOS is calculated according to the position coordinates.",
"Then $b_i$ can be defined as the ratio of the $d_i$ and the width of the $i$ -th scatterer $w_i$ .",
"Then the maximum blockage feature is selected as the environment-level PESS, which can be denoted as $PESS_{blockage} = max \\lbrace b_1,b_2,\\dots ,b_n\\rbrace ,$ where there are $n$ internal scatterers.",
"As for the LSP representation, the PESS that contains the general environment information should be extracted.",
"Because of the uncertain parameter requirement, the embedding feature should be utilized rather than the certain calculated feature.",
"The matrix of original global representation $PESS_{global}$ is constructed by the information of the Tx, Rx, and internal scatterers.",
"In view of the 3-dimensional coordinates vector of Tx and Rx while the 6-dimensional vector for scatterers, 0 paddings are utilized to fill the Tx and Rx row to deal with the inconsistency of dimensions.",
"In which $PESS_{global} \\in \\mathbb {R}^{(n+2)\\times 6}$ for the environment sample with $n$ scatterers.",
"Therefore, the final PESS can be obtained by combining the matrix $PESS_{global}$ and the blockage feature value.",
"To concatenate the two features of different dimensional, $PESS_{global}$ should be first converted into a 1-dimensional vector.",
"The commonly used unsupervised dimensionality reduction algorithm: principal component analysis (PCA), is utilized for compression [11].",
"Hence, the compressed feature $PESS_{global}^{^{\\prime }} \\in \\mathbb {R}^{1\\times 6}$ is obtained.",
"Finally, the blockage and global feature can be concatenated as a whole environment feature, i.e., $PESS_{evaluation} \\in \\mathbb {R}^{1\\times 7}$ .",
"PESS of Environment Graph: Unlike the overall channel properties, which the linear feature can represent, the environment layout needs a nonlinear representation.",
"The graph structure data in non-Euclidean space is utilized to describe the structure information of the propagation environment.",
"Specifically, the environment graph is constructed as the PESS for each scatterer that is to be classified.",
"The graph is constructed by building edges of the pending scatterer node and other nodes to mark the pending scatterer [12].",
"Therefore, let $PESS_{graph} = (V, E)$ denotes the graph with nodes $V$ , edges $E$ , and node feature vectors $X$ .",
"Where $V$ consist of Tx node $v_t$ , Rx node $v_r$ , and $n$ scatterer nodes for the graph of $n$ scatterers, i.e., $V=\\lbrace v_t, v_r, v_1, v_2, \\dots , v_n\\rbrace $ , as shown in Fig.",
"3.",
"Meanwhile, $E$ can be expressed as $E = \\lbrace (v_t, v_p), (v_r, v_p),\\dots ,(v_n, v_p), n\\ne p\\rbrace $ .",
"For Tx and Rx node, the position coordinates are used as feature vectors, that is, $X_t=(x_t, y_t, z_t)$ and $X_r=(x_r, y_r, z_r)$ .",
"The center coordinates $(x_i, y_i, z_i)$ , long $l_i$ , width $w_i$ , and height $h_i$ formed the feature vector of $i$ -th scatterer node, which can be expressed as $X_i = \\lbrace (x_i, y_i, z_i,l_i,w_i,h_i), i \\in n\\rbrace $ .",
"Figure: The PESS graph PESS_{graph} of three scatterers and pending scatterer v p v_p."
],
[
"PESaBP",
"After obtaining the essential PES of the beam prediction, the PESaBP can be implemented, i.e., channel quality evaluation and target scatterer detection can be achieved by leveraging the corresponding PES directly.",
"The proposed PESS of environment features can predict the channel quality, and the constructed PESS of environment graphs can detect the target scatterer for beam prediction."
],
[
"PES-Based Channel Quality Evaluation",
"As for dividing the quality into qualified and unqualified, the quality evaluation can be solved as a binary classification problem, differentiating the unqualified as class 0 and the qualified quality as class 1.",
"In which the quality threshold can be set for diverse requirements according to the cumulative probability of the received power.",
"According to the low-dimensional features for PES representation, the support vector machine (SVM) is built as the classification model rather than the neural networks requiring more learning cost with no significant accuracy gain.",
"The SVM [13] is a classic ML algorithm to maximize a particular mathematical function for a given collection of data that performs classification by constructing the hyperplane.",
"The kernel function is the crucial calculation that enables the SVM to map the data from a low-dimensional space to a higher-dimensional space, which can be denoted by $\\left\\langle a_1 \\cdot a_2 \\right\\rangle \\leftarrow K(a_i, a_j) = \\left\\langle \\Phi (a_i) \\cdot \\Phi (a_j) \\right\\rangle ,$ where $\\Phi $ is a nonlinear function that maps the input space into the feature space and $K$ is the kernel function.",
"Four classical kernel functions are used for nonlinear model learning, including linear, polynomial, sigmoid, and radial basis kernels.",
"In which the linear and polynomial kernel function can be described as $K(a_i, a_j) = \\left\\langle a_i, a_j \\right\\rangle , K(a_i, a_j) = (1 + \\left\\langle a_1, a_2 \\right\\rangle )^d,$ where $d$ is the degree of the kernel function.",
"The radial basis kernel can map the primitive features to infinite dimensions, which can be expressed as $K(a_i, a_j) = exp(- \\frac{\\left\\Vert a_i-a_j \\right\\Vert }{2 \\sigma ^2}).$ While the sigmoid kernel function comes from the neural network, which is generally denoted by $K(a_i, a_j) = tanh(\\gamma \\left\\langle a_1, a_2 \\right\\rangle +r),$ where the $\\gamma $ and $r$ are the kernel parameters."
],
[
"PES-Based Target Scatterer Detection",
"In the case of an unqualified channel, the beam should be switched to the better direction, i.e., the target scatterer with maximum power should be detected.",
"Hence, the issue can be considered a scatterer classification mission by classifying the scatterers into two classes, i.e., scatterer with maximum power $S_{max}$ (class: 1) and other scatterers (class: 0).",
"The graph neural network (GNN) is constructed by utilizing the net-architecture in [14].",
"The GNN is the graph learning method for the graph data process.",
"In which the $\\mathbf {Aggregate}(\\cdot )$ and $\\mathbf {Combine}(\\cdot )$ are the critical operators for modeling, where the former serves as the aggregation function of the neighborhood information, and the latter passes the aggregated node feature to a learnable layer to generate node embedding for the GNN layer.",
"Let $a_v^{(p)}$ stand for the nodes representing the structural information captured within the $p$ -hop network neighborhood in $k$ iterations of aggregation.",
"Hence, the $p$ -th layer can be denoted by $a_v^{(p)}= \\mathbf {Aggregate}^{(p)}(\\lbrace h_u^{(p-1)}:u \\in \\mathcal {N}(v)\\rbrace ).$ $h_v^{(p)}= \\mathbf {Combine}^{(p)}(h_v^{(p-1)},a_v^{(p)}),$ where $h_v^{(p)}$ is the feature vector of node $v$ at the $p$ -th iteration, for $p=1,2,\\cdots , P$ and $\\mathcal {N}(v)$ is a set of nodes adjacent to $v$ .",
"The $h_v^{(0)}$ is initialized with $X_v$ .",
"As for the model, 8 MLPs are constructed.",
"For each MLP, 6 hidden layers are deployed, where 1024 neurons are set for each layer.",
"Finally, scores of two classes can be obtained by a fully-connected network.",
"However, the classification is independently deployed for each internal scatterer.",
"For the unique detected scatterer of one environment, the scatterer classification results of a specific environment are ranked by the classification probability.",
"In practice, the scatterer with the top class 1 probability is selected, or the scatterer with the minimum class 0 probability when all classified 0 is considered the final prediction result."
],
[
"Simulation Settings",
"The environment samples with random scatterer changing are considered.",
"The 3D modeling software Blender and the ray-tracing tool WirelessInSite are utilized for the traceable environment, and channel generation [10] as shown in Fig.",
"4.",
"The propagation area's length, width, and height are set at 15, 10, and 3 m, and the Tx and Rx are set on the two sides of the diagonal.",
"Regular scatterers with random numbers, positions, and dimensions are generated.",
"The dataset includes samples with $J\\in [3,12]$ numbers of internal scatterers.",
"Figure: A simulated sample with a random scatterer layout.The training and testing data are a random selection of samples with different numbers of scatterers for generalization verification of the prediction method, in which the samples of the testing data consist of 4, 8, and 12 scatterers, and the rest samples are training data.",
"The corresponding channels at 28 GHz are produced using the omnidirectional antenna, and six-order reflections and one-order diffraction are set.",
"After selecting the NLOS samples, there are 1475 and 265 samples in training and testing data."
],
[
"Performance Metrics",
"As for the binary classification problem, the precision and receiver operating characteristic curve (ROC) are given for performance analysis with the device of one NVIDIA GeForce RTX 2080.",
"The precision $pre$ can be calculated as $pre = \\rm {\\frac{TP}{TP+TN}}$ where true-positive (TP) and true-negative (TN) are the true samples classified as positive and negative.",
"The ROC is plotted with a false-positive rate (FPR) and true-positive rate (TPR), which can be expressed as $\\rm FPR = \\rm {\\frac{FP}{FP+TN}, TPR = \\frac{TP}{TP+FN}},$ where the false-positive (FP) and false-negative (FN) denote the false samples that be predicted with positive and negative.",
"The crucial feature of ROC is the area under curve (AUC), where the closer the AUC is to 1, the better the performance."
],
[
"Quality and Target Scatterer Classification Results",
"The quality threshold is set of 60% cumulative distribution function (CDF) of received power.",
"The SVM models with linear, polynomial, radial basis, and sigmoid kernel functions get 0.88, 0.92, 0.89, and 0.53 precisions, respectively.",
"In which the polynomial kernel offers the best result.",
"Then, according to the precision calculation, we can obtain the scatterer classification's accuracy is 0.89.",
"The ROC and AUC of the SVM and GNN-based model are shown in Fig.",
"5, indicating the identity ability.",
"Figure: The ROC curve and AUC of quality and scatterer classification.Moreover, based on the scatterer classification results, the target scatterer can be selected for an environment sample by the rank of classification scores.",
"Moreover, the beam indices prediction in [4] is tested by utilizing the code at [15] for comparison.",
"In which the top view images are generated by converting the coordinates of the dataset and setting the scatterers with diverse grayscale according to the different heights.",
"The digital architecture system is employed with 8 antennas corresponding to 8 classes.",
"The results are indicated in TABLE I.",
"Table: The Precision ComparisionsThe precision of the proposed method is around 0.9, while the top-3 precision of the method in [4] is around 0.84, which can hardly adapt to changing environments using few training data.",
"Moreover, the testing time is compared for cost evaluation in TABLE II.",
"The results illustrate that the proposed PESaBP method can save over 87% time cost, which can support the online prediction for changing environments.",
"Table: The Comparisions of Testing Time"
],
[
"Conclusion and Future Work",
"This paper is interested in the PES definition, in which The PES is considered the task-oriented environment representation set according to the concerned channel properties.",
"Therefore, the semantic mapping between the propagation environment and applications can be built directly.",
"Simulation results of the PESaBP method indicate the efficiency and precision in NLOS scenarios, which have the potential to support online prediction in the ever-changing environment."
]
] | 2209.08245 |
[
[
"Unsupervised Lexical Substitution with Decontextualised Embeddings"
],
[
"Abstract We propose a new unsupervised method for lexical substitution using pre-trained language models.",
"Compared to previous approaches that use the generative capability of language models to predict substitutes, our method retrieves substitutes based on the similarity of contextualised and decontextualised word embeddings, i.e.",
"the average contextual representation of a word in multiple contexts.",
"We conduct experiments in English and Italian, and show that our method substantially outperforms strong baselines and establishes a new state-of-the-art without any explicit supervision or fine-tuning.",
"We further show that our method performs particularly well at predicting low-frequency substitutes, and also generates a diverse list of substitute candidates, reducing morphophonetic or morphosyntactic biases induced by article-noun agreement."
],
[
"Introduction",
"There has been growing interest in developing automatic writing support systems to assist humans to write documents.",
"One relevant task to this research goal is lexical substitution, where given a target word and its surrounding context, a system suggests a list of word substitutions that can replace the target word without changing its core meaning.",
"For instance, given the target word great and the context He is a great artist, the model might suggest alternative words such as outstanding, terrific, or distinguished.",
"Writers can use such suggestions to improve the fluency of their writing, reduce lexical repetition, or search for better expressions that represent their ideas more creatively.",
"As with other NLP tasks, recent studies have shown that masked language models such as BERT [7] perform very well on lexical substitution, even without any task-specific supervision.",
"A common approach is to employ language models as generative models and predict substitutes based on their generative capability.",
"However, this approach has some limitations.",
"First, it is extremely difficult for language models to predict rare words — especially those that are segmented into multiple subword tokens — since the models inevitably assign them very low probabilities.",
"Second, word prediction is highly affected by morphosyntactic constraints from the surrounding context, which overshadows the (arguably more important) question of semantic fit.",
"For instance, if the target word is increase in the context ... with an increase in ..., language models tend to suggest words that also start with a vowel sound due to the presence of an before the target word, missing other possible substitutes such as hike or boost.",
"In fact, this problem is even more pronounced in languages where words have grammatical gender (e.g.",
"Italian nouns) or a high degree of inflection (e.g.",
"Japanese verbs).",
"In this paper, we propose a new approach that explicitly deals with these limitations.",
"Instead of generating words based on language model prediction, we propose to find synonymous words based on the similarity of contextualised and decontextualised word embeddings, where the latter refers to the “average” contextual representation of a word in multiple contexts.",
"Experiments on English and Italian lexical substitution show that our fully unsupervised method outperforms previous models by a large margin.",
"Furthermore, we show that our model performs particularly well at predicting low-frequency words, and also generates more diverse substitutes with less morphophonetic or morphosyntactic bias, e.g.",
"as a result of article–noun agreement in English and Italian."
],
[
"Our Approach",
"Given a sentence that contains a target word $x$ and its surrounding context $c$ , we first feed the sentence into a pre-trained transformer model [37] such as BERT and generate the contextualised representations of $x$ : ${f^{\\ell }(x, c)} \\in \\mathbb {R}^{d}$ , where $\\ell ~(\\le L)$ denotes the layer of the model.",
"We propose to predict substitutes of $x$ by retrieving words that have similar representations to ${f^{\\ell }(x, c)}$ .",
"To this end, we calculate $S(y|x,c)$ : the score of $y$ being a substitute of $x$ in context $c$ , as follows: S(y|x,c)= cos(f(y),f(x, c)), f(x,c) =Zf(x,c), f(y) = 1NiNZf(y, c'i), where $f(y) \\in \\mathbb {R}^{d}$ denotes the decontextualised word embedding of $y$ ; $Z$ is a set of selected layers; and $ \\mathrm {cos}(a, b)$ denotes the cosine similarity between $a$ and $b$ .",
"To obtain $f(y)$ , we randomly sample $N$ sentences ($c^{\\prime }_{1},c^{\\prime }_{2} ...,c^{\\prime }_{N}$ ) that contain $y$ from a monolingual corpus, and take the average of the contextualised representations of $y$ given $c^{\\prime }_{i}$ : $f^{\\ell }(y, c^{\\prime }_i)$ .",
"We pre-compute $f(y)$ for each word $y$ in our pre-defined vocabulary $\\tilde{V}$ , which consists of lexical items (i.e.",
"no subwords) and contains different words from the pretrained model's original vocabulary $V$ .",
"If $y$ is segmented into multiple subwords (using the pretrained model's tokeniser), we average its subword representations — this way we can include low-frequency words in $\\tilde{V}$ and generate diverse substitutes.",
"We obtain $f(x,c)$ and $f(y)$ by summing representations across different layers $\\ell \\in Z$ to capture various lexical information.We also tried taking the weighted sum of the different-layer embeddings, but we did not see noticeable improvement."
],
[
"Multi-Sense Embeddings",
"Representing $f(y)$ in Eqn.",
"(REF ) as a simple average of the contextualised representations of $y$ is clearly limited when $y$ has multiple meanings, since the representations will likely vary depending on its usage.",
"For instance, [38] show that BERT representations of polysemous words such as bank create distinguishable clusters based on their usages.",
"To address this issue, we first group the $N$ sentences into $K$ clusters based on the usages of $y$ , and for each cluster $k$ , we obtain the decontextualised embedding $f^{k}(y)$ by averaging the contextualised representations, i.e., $f^{k}(y) = \\dfrac{1}{|{C}^k|}\\sum _{ c^{\\prime } \\in {C}^k}\\sum _{ \\ell \\in Z}{f^{\\ell }(y,{c^{\\prime }})},$ where $C^k$ denotes the set of the sentences that belong to the cluster $k$ .",
"To obtain clusters, we apply $K$ -means [22], [2] to the L2-normalised representations of $y$ in $N$ sentences.We concatenate $f^{\\ell }(y,{c})$ across multiple layers $\\ell \\in Z$ .",
"We expect that if $y$ has multiple senses, $f^k{(y)}$ will to some degree capture the different meanings.Note that the number of clusters $K$ is fixed across all words, forcing the model to “split” and “lump” senses [10] to varying degrees.",
"This methodology has been shown to be effective by [5] on context-independent word similarity tasks.",
"With $f^{k}(y)$ , we can refine the similarity score $S(y|x,c)$ in Eqn.",
"(REF ) as follows: ${S}(y|x,c)=\\underset{k}{\\mathrm {max~}}{\\mathrm {cos}(f^{k}(y),f(x, c))}.$ In this way, we can compare $x$ with $y$ based on the sense that is most relevant to $x$ .",
"Furthermore, we capture global similarity between $x$ and $y$ as: $\\begin{aligned}S(y|x,c) = & \\underset{k}{\\mathrm {~max~}}\\lambda \\mathrm {cos}({f^{k}(y),f(x, c)})\\\\&+(1-\\lambda )\\mathrm {cos}(f^{k}(y),f^{j_c}(x)),\\\\\\end{aligned}\\\\\\begin{aligned}j_c = \\underset{j}{\\mathrm {~argmax~}} \\mathrm {cos}(f^{j}(x),f(x, c)),\\end{aligned}$ where the second term in Eqn.",
"(REF ) corresponds to the global similarity, which compares $x$ and $y$ outside of context $c$ .To obtain $f^{j}(x)$ , we compute the decontextualised embedding of $x$ and apply $K$ -means, as we do to compute $f^{k}(y)$ .",
"When $x$ is not included in our pre-defined vocabulary $\\tilde{V}$ , we set $\\lambda $ to 1 and ignore the second term in Eqn.",
"(REF ).",
"However, it still considers $c$ in Eqn.",
"() to retrieve the cluster that best represents the meaning of $x$ given $c$ .",
"While Eqn.",
"(REF ) generally generates high-quality substitutes, we found that it sometimes retrieves words that share the same root word as $x$ and yet do not make good substitutes (e.g.",
"pay and payer).",
"This is mainly due to the fact that the vocabulary $\\tilde{V}$ contains a large number of derivationally-related words, some of which are out-of-vocabulary (OOV) in the original vocabulary $V$ (e.g.",
"pay ##er).",
"To address this problem, we add a simple heuristic where $y$ is discarded if the normalised edit distanceThe distance normalised by the maximum string length.",
"between $x$ and $y$ is less than a threshold (0.5 for our English and Italian experiments).We tuned this threshold based on English development data (i.e.",
"the development split of SWORDS)."
],
[
"Reranking",
"In Eqn.",
"(REF ), the context $c$ affects the representation of $x$ but not $y$ .",
"Ideally, however, we want to consider the context $c$ on both sides to find the words that best fit the context.",
"Therefore, we first generate top-$M$ candidates based on Eqn.",
"(REF ), and rerank them using the following score: $\\mathrm {S(y|x, c)} = \\frac{1}{|Z|}\\sum _{\\ell \\in Z}{\\mathrm {cos}(f^{\\ell }(y, c),f^{\\ell }(x, c))},$ where $f^{\\ell }(y, c)$ denotes the contextualised representation of $y$ given $c$ , which can be obtained by replacing $x$ in $c$ with $y$ and feeding it into the model.",
"In Eqn.",
"(REF ), we calculate the similarity at each layer $\\ell \\in Z$ and take the average, which yields small yet consistent improvements over averaging the embeddings first and then calculating the similarity.In Eqn.",
"(REF ), we obtained similar results by averaging the embeddings or cosine similarities across layers.",
"We limit the use of this scoring method to the $M$ candidates only, since it is computationally expensive to calculate $f^{\\ell }(y, c)$ for every single word $y$ in $\\tilde{V}$ .",
"Previously, a similar method was employed by [19] but they used the last layer only (i.e.",
"$Z = \\lbrace L\\rbrace $ ).",
"We show that using multiple layers substantially improves the performance.",
"Following [19], we set $M$ to 50."
],
[
"Comparison to Previous Approaches",
"Our approach contrasts with previous approaches [42], [19], [40] that employ BERT as a generative model and predict lexical substitutes based on the generation probability $ P(y|x,c)$ : $P(y|x,c) = \\dfrac{\\exp (E_y{f^{\\hat{L}}(x, c)}+b_y)}{\\sum _{{y} \\in V}\\exp (E_{{y}}{f^{\\hat{L}}(x, c)}+b_{{y}})}, $ where $E_{y} \\in \\mathbb {R}^{d}$ denotes the output embedding of $y$ , which is usually tied with the input word embedding; ${f^{\\hat{L}}(x, c)}$ is the representation at the very last layer of the model;Note that this does not always correspond to the last layer of transformer: ${f^{L}(x, c)}$ .",
"E.g., BERT calculates ${f^{\\hat{L}}(x, c)}$ by applying a feed forward network and layer normalisation to ${f^{L}(x, c)}$ , whereas for XLNET, ${f^{\\hat{L}}(x, c)}$ = ${f^{L}(x, c)}$ .When $x$ consists of multiple subwords, the representation of the first or longest token is usually used.",
"and $b_y$ is a scalar bias.",
"While this approach is straightforward and well motivated, its predictions are highly influenced by morphosyntax, as discussed in intro.",
"Moreover, Eqn.",
"(REF ) shows three additional limitations compared to our approach: (1) the prediction is conditioned on the last layer only, despite previous studies showing that different layers capture different information, with the last layer usually containing less semantic information than the lower or middle layers [3], [35]; (2) $y$ is represented by the single vector $E_y$ , which may not work well when $y$ has multiple meanings — we alleviate this by clustering the embeddings (multi-sense); and (3) the model is not capable of generating OOV words, unless we force the model to decode multiple subwords, e.g.",
"by using multiple mask tokens or duplicating $x$ .",
"Our approach, in contrast, can include rare words in the pre-defined vocabulary $\\tilde{V}$ and generate diverse substitutes (vocabeffect)."
],
[
"Data and Evaluation",
"We conduct experiments in two evaluation settings: generation and ranking.",
"In the generation setting, systems produce lexical substitutes given target words and sentences, while in the ranking setting, they are also given substitute candidates and rank them based on their appropriateness.",
"For the generation task, we base our experiments on SWORDS [19], the largest English lexical substitution dataset, which extends and improves CoInCo [16] by introducing a new annotation scheme: in CoInCo, the annotators were asked to come up with substitutes by themselves, whereas in SWORDS, the annotators were given substitute candidates pre-retrieved from a thesaurus, and only had to made binary judgements (“good” or “bad”).The annotators were asked if they would consider using the substitute candidate to replace the target word as the author of the context.",
"A word is regarded as acceptable if it is judged to be good by more than five out of ten annotators, and conceivable if selected by at least one annotator.",
"In this way, SWORDS provides more comprehensive lists of substitutes, including many low-frequency words that are good substitutes and yet difficult for humans to suggest — these words are of particular interest to us.",
"For the evaluation metrics, the authors use the harmonic mean of the precision and recall given the gold and top-10 system-generated substitutes.More precisely, their evaluation script lemmatises the top-50 substitutes first and then extracts the top-10 distinct words.",
"As gold substitutes, they use either the acceptable or conceivable words, and calculate the corresponding scores $F_a$ and $F_c$ , respectively.",
"They also propose to measure those scores in both strict and lenient settings, which differ in that in the lenient setting, candidate words that are not scored under SWORDS are filtered out and discarded.",
"In the ranking task, we evaluate models on the traditional SemEval-2007 Task 10 (“SemEval-07”) data set (trial+test) [23], as well as SWORDS.",
"For the evaluation metric, we follow previous work in using Generalized Average Precision (GAP; [15]): $GAP = \\dfrac{\\sum _{i=1}^{N}I(\\alpha _i)p_i}{\\sum _{i=1}^{R}I(\\beta _i)\\bar{\\beta }_i},~~~ p_i = \\dfrac{\\sum _{k=1}^{i}\\alpha _k}{i}, $ where $\\alpha _i$ and $\\beta _i$ denote the gold weight of the $i$ -th item in the predicted and gold ranked lists respectively, with $N$ and $R$ indicating their sizes; $I(\\alpha _i)$ is a binary function that returns 1 if $\\alpha _i>0$ , and 0 otherwise; and $\\bar{\\beta }_i$ is the average weight of the gold ranked list from the 1st to the $i$ -th items.",
"In our task, the weight corresponds to the aptness of the substitute, which is set to zero if it is not in the gold substitutes.",
"Following previous work [24], [1], we ignore multiword expressions in SemEval-07.We run the evaluation code at https://github.com/orenmel/lexsub with the no-mwe option."
],
[
"Models",
"As shown in Eqn.",
"(REF ), our approach requires only the vector representations of words and hence is applicable to any text encoder model.",
"Therefore, we test our method with various pre-trained models, including five masked language models: BERT [7], mBERT, SpanBERT [14], XLNET [41], and MPNet [30]; one encoder-decoder model: BART [20]; and two discriminative models: ELECTRA [6] and DeBERTa-V3 [11].See modelsource for the details of all models.",
"We also evaluate two sentence-embedding models: MPNet-based sentence transformer [28] and SimCSE [9], both of which are fine-tuned on semantic downstream tasks such as MNLI and achieve good performance on sentence-level tasks.",
"Finally, we also evaluate the encoder of the fine-tuned mBART on English-to-Many translation [34].",
"Note that the discriminative models and embedding models (e.g.",
"NMT-encoder, SimCSE) cannot generate words and hence are incompatible with the previous approach described in previousapproach.",
"Table: The results for the generation task in the lenient and strict settings.",
"The best scores are boldfaced.We use the same vocabulary $\\tilde{V}$ for all models, which consists of the 30,000 most common wordsWe discard tokens that contain numerals, punctuation, or capital letters.",
"As such, $\\tilde{V}$ includes more lexical items (with less noise and no subwords) than the original vocabulary $V$ .",
"in the OSCAR corpus [26].",
"We set the number of sentences we sample from OSCAR to calculate the decontextualised embeddings, i.e.",
"$N$ , to 300; the clustering size $K$ to 4; and $\\lambda $ in Eqn.",
"(REF ) to 0.7.",
"For the set of transformer layers, $Z$ , we employ all layers except for the first and last two, i.e.",
"{3, 4, ..., $L-2$ }.",
"We tune all hyper-parameters on the development split of SWORDS."
],
[
"Results",
"resultswords shows the results on SWORDS, along with baseline scores from previous work (with some bug fixes, as noted).",
"The first row, HUMANS, indicates the agreement of two independent sets of annotators on (a subset of) SWORDS, and approximates the upper bound for this task.",
"The second row, CoInCo, shows the accuracy of the gold standard substitutes in CoInCo, which are suggested by human annotators without access to substitution candidatesSince all of these words are in the substitute candidates of SWORDS, it cannot be evaluated under the strict setting.",
"— this approximates how well humans perform when asked to elicit candidates themselves.",
"The remainder of the rows above OURS denote baseline systems, all of which employ generative approaches.",
"The first baseline uses GPT-3 [4], and achieves the state-of-the-art in the strict setting.",
"It generates substitutes based on “in-context learning”, where the model first reads several triplets of target sentences, queries, and gold-standard substitutes retrieved from the development set, and then performs on-the-fly inference on the test set.",
"As such, it is not exactly comparable to the other fully unsupervised models.",
"BERT-K generates substitutes based on Eqn.",
"(REF ) by feeding the target sentence into BERT, and BERT-M works the same except that the target word is replaced by [MASK].",
"Both models further rerank the candidates based on Eqn.",
"(REF ), using the last layer only; we show the performance without reranking as “w/o rerank” in resultswords.",
"[40] and [42] also use BERT to generate substitutes, and rerank them using their own method.",
"The rows below OURS indicate the performance of our approach using various off-the-shelf models.",
"Our method with BERT substantially outperforms all the BERT-based baselines, even without the edit-distance heuristic (multi-sense) or reranking method (Eqn.",
"(REF )).",
"The best performing models are DeBERTa-V3, XLNet, and BART (Enc-Dec), all of which outperform the weakly-supervised GPT-3 model by a large margin in the strict setting; and XLNet even outperforms CoInCo in the lenient setting.",
"The last three rows show the performance of the top-3 models when they are given the candidate words and rank them based on Eqn.",
"(REF ), which emulates how the SWORDS annotators judged the words.",
"The result shows that all the models still lag behind HUMANS, suggesting there is still substantial room for improvement.",
"Interestingly, BART performs best when we average the scores obtained by its encoder and decoder, suggesting each layer captures complementary information.",
"It is also intriguing to see that the discriminative models (DeBERTa-V3 and ELECTRA) perform much better than BERT, albeit they are not trained to generate words and not compatible with the previous generative approach.",
"The sentence embedding models (SBERT, SimCSE) perform no better than the original models, which contrasts with their strong performance in sentence-level tasks.",
"The multilingual models (mBERT, NMT) perform very poorly, even though the NMT model was fine-tuned on large English-X parallel corpora.",
"Table: GAP scores on SemEval-07 and SWORDS.",
"“LMs+WN” employs multiple language models and WordNet, and “BERT, sup” is a supervised model.resultgap shows the results for the ranking task on SemEval-07 and SWORDS.",
"[25] harness WordNet [8] to obtain synsets of the target word and also their glosses, and employ BERT and RoBERTa [21] to rank candidates.",
"The models proposed by [17] are different from the others in that they fine-tune BERT on lexical substitution data sets.",
"They propose unsupervised (BERT) and supervised (BERT, sup) models, which are fine-tuned on automatically-generated or manually-annotated data.",
"resultgap shows that our method with BERT performs comparably with the unsupervised model of [17] without any fine-tuning, and outperforms [42] and [19] (except for the score of [42] on SemEval-07, which couldn't be reproduced in previous work).",
"Just like the generation task, DeBERTa-V3 achieves the best performance on both data sets and establishes a new state-of-the-art.",
"Other models also follow a similar trend to the generation results, e.g.",
"BART performs best by combining its encoder and decoder, and the multilingual models perform very poorly.",
"We hypothesise that their poor performance is mainly caused by suboptimal segmentation of English words.",
"This hypothesis is also supported by the fact that DeBERTa-V3 has by far the largest vocabulary $V$ of all models.Note that $V$ differs from $\\tilde{V}$ , the pre-defined vocabulary we used for all models.",
"modelsource compares the size of the model's original vocabulary $V$ across different models."
],
[
"Results on Italian Lexical Substitution",
"We further conduct an additional experiment on Italian, based on the data set from the EVALITA 2009 workshop [36].",
"We report $F$ scores given top-10 predictions as in the English generation task, plus two traditional metrics used in the workshop, namely oot and best, which compare the top-10 and top-1 predictions against the gold substitutes.We report precision only, as it is the same as recall under those metrics when predictions are made for every sentence.",
"We lemmatise all the generated words to make them match the gold substitutes, following the SWORDS evaluation script.We used the Italian lemmatiser (it_core_news_sm 3.2.0) in spaCy (ver.",
"3.2.2) [13].",
"We use the same hyper-parameters as for the English experiments, and Italian shows the results.",
"[12] is a strong baseline that retrieves substitute candidates from MultiWordNet [27] and ranks them using a supervised ranker model.",
"We also implement BERT-K using an Italian BERT model [29], with and without the reranking method.",
"The results show that our approach substantially outperforms the baselines, confirming its effectiveness.",
"However, our reranking method is not as effective as in English, which we attribute to the influence of grammatical gender in Italian (which we return to in analysissyntax).",
"The heuristic improves best-P but harms $F$ and oot-P, meaning it removes good candidates as well as bad ones, possibly because we used the threshold tuned on English.",
"Table: The result of Italian lexical substitution.Table: Ablation studies of our method.",
"The scores denote F c F_c in the strict setting on SWORDS."
],
[
"Ablation Studies",
"We perform ablation studies on SWORDS to see the effect of $\\lambda $ and the $K$ -clustered embeddings, and also the heuristic based on edit distance.",
"ablation shows the results.",
"Overall, our method with $\\lambda = 0.7$ performs better than $\\lambda = 1$ or $\\lambda = 0$ , confirming the benefit of considering both in-context and out-of-context similarities.",
"One interesting observation is that while BERT and DeBERTa perform better with $\\lambda = 1$ than with $\\lambda = 0$ , the opposite trend is observed for XLNet and especially BART (and hence the optimal value for $\\lambda $ is smaller than 0.7).",
"This suggests that BART representations are highly influenced by context, containing much information that is not relevant to the semantics of the target word; we further confirm this in the next section.",
"When we set the cluster size $K$ to 1, the performance of all the models drops sharply, indicating the effectiveness of the clustered embeddings.",
"When we retrieve the cluster of $y$ at random instead of the closest one to $x$ in Eqn.",
"(REF ), the performance decreases substantially, suggesting each cluster captures different semantics.",
"The heuristic consistently improves the performance, filtering out derivationally-related yet semantically-dissimilar words to the target word.",
"Lastly, our reranking method substantially improves the performance of all the models, demonstrating that it is important to incorporate the target context $c$ into both the target and candidate word representations."
],
[
"Effects of Morphosyntactic Agreement",
"Compared to previous generative approaches, our method does not depend on the generation probabilities of language models, and hence we expect it to be less sensitive to morphosyntactic agreement effects.",
"To investigate this, we analyse the performance on noun target words which immediately follow one of the following articles: a or an in English, and una, la, le, un, il or i in Italian.",
"The first three Italian articles are used with feminine nouns, and the rest with masculine ones.",
"Our hypothesis is that generative methods will be highly biased by these articles, despite the gold standard being semantically annotated, and thus largely oblivious to local morphosyntactic agreement effects.",
"resultarticle shows the percentage of top-10 predicted candidates that agree with the article.We retrieve Italian gender information using a dictionary API (https://github.com/sphoneix22/italian_dictionary), and English phonetic information using CMUdict (https://github.com/cmusphinx/cmudict) accessed via NLTK [32].",
"It demonstrates that the prediction of BERT-K is highly affected by the proceeding article as expected, resulting in substitutes which don't satisfy this constraint being assigned low probabilities.",
"In contrast, the results of our method are more balanced and close to the gold standard.Note that the big jump in results for an is based on a small number of instances (5 sentences).",
"Conversely, our reranking method actually increases the bias greatly, suggesting that the contextualised embeddings $f^{\\ell }(x,c)$ and $f^{\\ell }(y,c)$ in Eqn.",
"(REF ) become similar when $x$ and $y$ collocate similarly with the words in the context $c$ — overall, this leads to better results, but actually hurts in cases of local agreement effects biasing the results.",
"This is one reason why reranking was not as effective in Italian as in English, as agreement effects are stronger in Italian.",
"Table: The percentage of substitutes whose initial sound agrees with the corresponding English articles.Table: Top-5 predictions when the article an comes before the target word (increase or accord).",
"Gold shows a list of “conceivable” words sorted by their annotated scores (with “acceptable” words shown in italic, and multiword expressions omitted from the table).",
"Words included in Gold are boldfaced.syntacticeffecteng shows the result when we use different pre-trained models in English.",
"First, it shows that BERT-M is more sensitive to the articles than BERT-K, indicating the strong morphophonetic agreement effect on the masked word prediction.",
"Among the pre-trained language models used by our method, SpanBERT and BART are the most sensitive to the article a and an, respectively.",
"This suggests that the embeddings $f(x,c)$ obtained from these models are highly sensitive to the context $c$ , partly explaining why BART performs very poorly with $\\lambda $ = 1, as shown in ablationsec.",
"Lastly, anexamples shows examples of predicted substitutes when the article an comes before the target word.",
"It shows that BERT-K and OURS with BART tend to retrieve words that start with a vowel sound, as quantitatively described in syntacticeffecteng.",
"Figure: Layer-wise performance (F c F_c) on SWORDS."
],
[
"Layer-Wise Performance",
"We analyse the performance on SWORDS using different layers in layerfig (w/o rerank).We perform qualitative analysis in layerexamplessection.",
"First, we clearly see that middle layers perform better than the first or last ones, for all models.We see a similar trend in the ranking task (layerwise-ranking-sec).",
"The performance of BERT peaks at layer 16, in contrast with previous findings that the first quarter of layers perform best on context-independent word similarity tasks [3], likely because lexical substitution critically relies on context.In fact, [35] show that high-level semantic information is encoded in higher layers.",
"Our method using multiple layers performs mostly as well as using the best layer without the need to perform model-wise layer selection (see layer-wise-table in multi-layer-sec).",
"The last layer performs very poorly for all models, highlighting the limitation of the previous approach which uses the last layer only (Eqn.",
"(REF )).",
"The downward trend is particularly evident for MPNet, BART (dec), and SpanBERT; for BART and SpanBERT, we attribute this to the fact that their last-layer representations of the word at position $t$ are used to predict the next word $w_{t+1}$ , or $u$ neighbouring words {$w_{t-u}$ ..,$w_{t-1}$ } or {$w_{t+1}$ ..,$w_{t+u}$ }.SpanBERT does this for Span Boundary Objective.",
"This training objective may also lead to their sensitivity to articles before the target word, as shown in analysissyntax.",
"Interestingly, the sentence-embedding models (SBERT and SimCSE) are no exception to the downward trend, which is somewhat counter-intuitive given that their last layer representations are fine-tuned (and used during inference) to perform semantic downstream tasks.",
"Importantly, they do not perform better than the original models (MPNet and BERT), although in the ranking task, both models benefit moderately from fine-tuning (see layerwise-ranking-sec).",
"Table: The number of correctly predicted substitutes, grouped by their frequency in monolingual data.",
"The number in brackets shows the size of the vocabulary V ˜\\tilde{V}.Table: Results with different numbers of clusters."
],
[
"Analysis of Word Frequency",
"One of the strengths of our approach is that it can generate low-frequency substitutes that are OOV words in the original vocabulary.",
"To confirm this, we analyse how well our method can generate low-frequency words from different vocabulary sizes $\\tilde{V}$ .",
"analysisfreq shows the results, in which we experiment with our BERT-based model with the vocabulary sizes of 5k, 10k, 20k, and 30k.",
"The columns under “# Matched Words” show the numbers of correctly-predicted words, grouped by frequency range: low, med, and high denote words with frequency $<$ 50k, 50k–100k, and $>$ 100k in a large web corpus.",
"The table shows that our method with 30k words generates nearly twice as many low-frequency substitutes as the baseline.",
"Our method with 10k words still outperforms BERT-K in $F_c$ , demonstrating its effectiveness.",
"The last three rows show the performance of our method using other models, further demonstrating its ability to predict low-frequency words."
],
[
"Effects of Cluster Size",
"Finally, we analyse the effect of the cluster size $K$ for ELECTRA, as shown in clusteranalysis.",
"While a larger cluster size yields better performance, the improvement is marginal.",
"Rather than using a fixed $K$ , in future work we are interested in dynamically selecting the number of clusters per word."
],
[
"Related Work",
"In the pre-BERT era, most lexical substitution methods employed linguistic resources such as WordNet [8] to obtain substitute candidates [33], [12].",
"However, recent studies have shown that pre-trained language models such as BERT outperform these models without any external lexical resources.",
"For instance, [42] feed a target sentence into BERT while partially masking the target word using dropout [31], and generate substitutes based on the probability distribution at the target word position.",
"The masking strategy was shown to be effective on SemEval-07 but not on SWORDS.",
"Similarly, [40] feed two sentences into BERT, concatenating the target sentence with itself but with the target word replaced by [MASK], and predict words based on the mask-filling probability.",
"[25] augment pre-trained language models with WordNet and outperform [42].",
"[17] fine-tune BERT on lexical substitution data sets that are automatically generated using BERT.",
"They show that this approach is effective at ranking, and that adding manually-annotated data further boosts performance.",
"[18] fine-tune BART on human-annotated data, and make it generate a list of substitutes given a target sentence in an end-to-end manner.",
"They show that this generative approach rivals [42].",
"Note that all of these recent models are evaluated on English only."
],
[
"Conclusion",
"We present a new unsupervised approach to lexical substitution using pre-trained language models.",
"We showed that our method substantially outperforms previous methods on English and Italian data sets, establishing a new state-of-the-art.",
"By comparing performance on lexical substitution using different layers, we found that middle layers perform better than first or last layers.",
"We also compared the substitutes predicted by the previous generative approach and our method, and showed that our approach works better at predicting low-frequency substitutes and reduces morphophonetic or morphosyntactic biases induced by article–noun agreement in English and Italian."
],
[
"Details of Pre-trained Models",
"mdoeldetails describe the details of the pre-trained models used in our experiments.",
"We sourced these models from the Transformers library [39] except for SpanBERT, which we obtained from the original GitHub repository (https://github.com/facebookresearch/SpanBERT)."
],
[
"Layer-Wise Ranking Performance",
"layerwiserankfig shows the layer-wise performance in the ranking task.",
"Similar to the generation results (layerfig), middle layers perform better than the first or last layers.",
"It also shows that sentence-embedding models (SimCSE/SBERT) outperform their original models (BERT/MPNet) for several layers, different from the generation results where they perform similarly.",
"This suggests that fine-tuning on semantic downstream tasks improves the capacity of the model to differentiate subtle semantic differences between synonymous words, but not their ability to retrieve relevant words from a large pool of words; it also suggests that optimal representations for these objectives might differ."
],
[
"Effectiveness of Using Multiple Layers",
"layer-wise-table shows the generation and ranking performance of our model on SWORDS using different layers.",
"It shows that our method using multiple layers $\\ell \\in Z$ performs comparably or even better than selecting the best layer tuned on the test set for each model.",
"It also shows that the best-performing layer differs across models, suggesting they capture lexical information in a different manner.",
"Figure: Layer-wise performance (GAP) on SWORDS.Table: Details of the pre-trained models used in this paper.",
"|V||V| denotes the original vocabulary size of each model.",
"*The vocabularies of mBERT and mBART contain a great number of non-English words.Table: Generation and ranking performance of our approach on SWORDS using the first, middle (L 2\\frac{L}{2}th), last (LLth), or best layer tuned on the test set (the corresponding layer denoted in brackets); or using multiple layers ℓ∈Z\\ell \\in Z: {3, 4, ..., L-2L-2} (LL = 12 for MPNet and BART, and 24 otherwise).",
"Generation and ranking performance across all layers is illustrated in layerfig and layerwiserankfig.Table: Examples of substitutes predicted by our method (w/o rerank) using different layers.",
"Gold shows a list of “conceivable” words sorted by their annotated scores (with “acceptable” words shown in italic, and multiword expressions omitted from the table).",
"Words included in Gold are boldfaced."
],
[
"Examples of Generated Substitutes",
"layerexamples shows examples of substitutes generated by our method using different layers (without reranking).",
"It shows that the words retrieved by each layer are very different, indicating that each layer encodes very different information about the input word.",
"For instance, given the target word care, the first layer of BERT and BART-Enc/Dec retrieves a large number of words that contain the target word as a sub-morpheme (e.g.",
"aftercare, carefree).Since the edit distances between these words and the target word care are not greater than the threshold (0.5), they weren't filtered out by our heuristic.",
"This is presumably because the first-layer representations are highly affected by the input word embedding, and hence result in retrieving words that share the same subword token (e.g.",
"care ##free) regardless of the semantic similarity.",
"The last layer also performs poorly (as previously shown in layerfig), e.g.",
"BART-DEC (L12) retrieves participation as the closest word to the target word interest.",
"This is because the last-layer representations of BART-decoder are used to directly predict the next word in after interest in the target sentence, and in fact, most of the retrieved words (e.g.",
"uptick, faith, surge) are those that often collocate with in.",
"Oddly, BART-Enc predicts a large number of substitutes that consist of multiple words (segmented by the tokeniser), none of which are relevant to the target word, e.g.",
"aswell, todo, and inbetween as substitutes for interest.",
"In fact, the number of such words increases (and the performance decreases) as the hyper-parameter $\\lambda $ gets bigger (which increases the influence of $f(x,c)$ on the predictions).",
"One possible interpretation is that the last layer representations of the BART encoder may contain vague contextual information rather than the lexical information of the input word, since they are used by the decoder to predict various words (esp.",
"masked words) during pre-training.",
"Lastly, another interesting observation is that, for the target word interest, the last layer representations of BERT and BART-enc retrieve a lot of words that start from a vowel sound, despite the absence of the article an before interest, suggesting that the embeddings contain some morphophonetic information."
]
] | 2209.08236 |
[
[
"Better Hardness Results for the Minimum Spanning Tree Congestion Problem"
],
[
"Abstract In the spanning tree congestion problem, given a connected graph $G$, the objective is to compute a spanning tree $T$ in $G$ for which the maximum edge congestion is minimized, where the congestion of an edge $e$ of $T$ is the number of vertex pairs adjacent in $G$ for which the path connecting them in $T$ traverses $e$.",
"The problem is known to be NP-hard, but its approximability is still poorly understood, and it is not even known whether the optimum can be efficiently approximated with ratio $o(n)$.",
"In the decision version of this problem, denoted STC-$K$, we need to determine if $G$ has a spanning tree with congestion at most $K$.",
"It is known that STC-$K$ is NP-complete for $K\\ge 8$, and this implies a lower bound of $1.125$ on the approximation ratio of minimizing congestion.",
"On the other hand, $3$-STC can be solved in polynomial time, with the complexity status of this problem for $K\\in \\{4,5,6,7\\}$ remaining an open problem.",
"We substantially improve the earlier hardness result by proving that STC-$K$ is NP-complete for $K\\ge 5$.",
"This leaves only the case $K=4$ open, and improves the lower bound on the approximation ratio to $1.2$."
],
[
"Introduction",
"Problems involving constructing a spanning tree that satisfies certain requirements are among the most fundamental tasks in graph theory and algorithmics.",
"One such problem is the spanning tree congestion problem, ${\\textsf {STC}}$ for short, that has been studied extensively for many years.",
"Roughly, in this problem we seek a spanning tree $T$ of a given graph $G$ that approximates the connectivity structure of $G$ in the following sense: Embed $G$ into $T$ by replacing each edge $(u,v)$ of $G$ by the unique $u$ -to-$v$ path in $T$ .",
"Define the congestion of an edge $e$ of $T$ as the number of such paths that traverse $e$ .",
"The objective of ${\\textsf {STC}}$ is to find a spanning tree $T$ that minimizes the maximum edge congestion.",
"The general concept of edge congestion was first introduced in 1986, under the name of load factor, as a measure of quality of an embedding of one graph into another [3] (see also the survey in [20]).",
"The problem of computing trees with low congestion was studied by Khuller et al.",
"[11] in the context of solving commodities network routing problem.",
"The trees considered there were not required to be spanning subtrees, but the variant involving spanning trees was also mentioned.",
"In 2003, Ostrovskii provided independently a formal definition of ${\\textsf {STC}}$ and established various fundamental properties of spanning trees with low congestion [16].",
"Since then, many combinatorial and algorithmic results about this problem have been reported in the literature — we refer the readers to the survey paper by Otachi [18] for complete information, most of which is still up-to-date.",
"As established by Löwenstein [14], ${\\textsf {STC}}$ is ${\\mathbb {NP}}$ -hard.",
"As usual, this is proved by showing ${\\mathbb {NP}}$ -hardness of its decision version, where we are given a graph $G$ and an integer $K$ , and we need to determine if $G$ has a spanning tree with congestion at most $K$ .",
"Otachi et al.",
"[19] strengthened this by proving that the problem remains ${\\mathbb {NP}}$ -hard even for planar graphs.",
"In [15], ${\\textsf {STC}}$ is proven to be ${\\mathbb {NP}}$ -complete for chain graphs and split graphs.",
"On the other hand, computing optimal solutions for ${\\textsf {STC}}$ can be achieved in polynomial-time for some special classes of graphs: complete $k$ -partite graphs, two-dimensional tori [13], outerplanar graphs [17], and two-dimensional Hamming graphs [12].",
"In our paper, we focus on the decision version of ${\\textsf {STC}}$ where the bound $K$ on congestion is a fixed constant.",
"We denote this variant by ${K\\!-\\!\\textsf {STC}}$ .",
"Several results on the complexity of ${K\\!-\\!\\textsf {STC}}$ were reported in [19].",
"For example, the authors show that ${K\\!-\\!\\textsf {STC}}$ is decidable in linear time for planar graphs, for graphs of bounded treewidth (which includes bounded degree graphs), and for all graphs when $K=1,2,3$ .",
"On the other hand, they show that the problem is ${\\mathbb {NP}}$ -complete for any fixed $K \\ge 10$ .",
"In [4], Bodlaender et al.",
"proved that ${K\\!-\\!\\textsf {STC}}$ is linear-time solvable for graphs in apex-minor-free families and chordal graphs.",
"They also show the improved hardness result of ${K\\!-\\!\\textsf {STC}}$ , namely that it is ${\\mathbb {NP}}$ -complete for $K \\ge 8$ , even in the special case of apex graphs that only have one unbounded degree vertex.",
"As stated in [18], the complexity status of ${K\\!-\\!\\textsf {STC}}$ for $K \\in { \\left\\lbrace 4,5,6,7 \\right\\rbrace }$ remains an open problem.",
"Relatively little is known about the approximability of the optimization version of ${\\textsf {STC}}$ .",
"The trivial upper bound for the approximation ratio is $n/2$ [18].",
"As a direct consequence of the ${\\mathbb {NP}}$ -completeness of ${8\\!-\\!\\textsf {STC}}$ , there is no polynomial-time algorithm to approximate the optimum spanning tree congestion with a ratio better than $1.125$ (unless ${\\mathbb {P}}= {\\mathbb {NP}}$ ).",
"Our contribution.",
"Addressing the open question in [18], we provide an improved hardness result for ${K\\!-\\!\\textsf {STC}}$ : For any fixed integer $K \\ge 5$ , ${K\\!-\\!\\textsf {STC}}$ is ${\\mathbb {NP}}$ -complete.",
"The proof of this theorem is given in Section .",
"Combined with the results in [19], Theorem leaves only the status of ${4\\!-\\!\\textsf {STC}}$ open.",
"Furthermore, it also immediately improves the lower bound on the approximation ratio for ${\\textsf {STC}}$ : For $c < 1.2$ there is no polynomial-time $c$ -approximation algorithm for ${\\textsf {STC}}$ , unless ${\\mathbb {P}}={\\mathbb {NP}}$ .",
"We remark that this hardness result remains valid even if an additive constant is allowed in the approximation bound.",
"This follows by an argument in [4].",
"(In essence, the reason is that assigning a positive integer weight $\\beta $ to each edge increases its congestion by a factor $\\beta $ .)",
"Other related work.",
"The spanning tree congestion problem is closely related to the tree spanner problem in which the objective is to find a spanning tree $T$ of $G$ that minimizes the stretch factor, defined as the maximum ratio, over all vertex pairs, between the length of the longest path in $T$ and the length of the shortest path in $G$ between these vertices.",
"In fact, for any planar graph, its spanning tree congestion is equal to its dual's minimum stretch factor plus one [8], [19].",
"This direction of research has been extensively explored, see [5], [6], [7].",
"${\\textsf {STC}}$ is also intimately related to problems involving cycle bases in graphs.",
"As each spanning tree identifies a fundamental cycle basis of a given graph, a spanning tree with low congestion yields a cycle basis for which the edge-cycle incidence matrix is sparse.",
"Sparsity of such matrices is desirable in linear-algebraic approaches to solving some graph optimization problems, for example analyses of distribution networks such as in pipe flow systems [1].",
"${\\textsf {STC}}$ can be considered as an extreme case of the graph sparsification problem, where the objective is, given a graph $G$ , to compute a sparse graph $H$ that captures connectivity properties of $G$ .",
"Such $H$ can be used instead of $G$ for the purpose of various analyses, to improve efficiency.",
"See [2], [9], [23] (and the references therein) for some approaches to graph sparsification."
],
[
"Preliminaries",
"Let $G = (V,E)$ be a simple graph with vertex set $V$ and edge set $E$ .",
"Consider a spanning tree $T\\subseteq E$ of $G$ .",
"If $e = (u,v)\\in T$ , removing $e$ from $T$ splits $T$ into two components.",
"We denote by $T_{u,v}$ the component that contains $u$ and by $T_{v,u}$ the component that contains $v$ .",
"Let the cross-edge set of $e$ , denoted $\\partial _{G,T}(e)$ , be the set of edges in $E$ that have one endpoint in $T_{u,v}$ and the other in $T_{v,u}$ .",
"In other words, $\\partial _{G,T}(e)$ consists of the edges $(u^{\\prime },v^{\\prime }) \\in E$ for which the unique (simple) path in $T$ from $u^{\\prime }$ to $v^{\\prime }$ goes through $e$ .",
"Note that $e \\in \\partial _{G,T}(e)$ .",
"The congestion of $e$, denoted by $\\text{cng}_{G,T}(e)$ , is the cardinality of $\\partial _{G,T}(e)$ .",
"The congestion of tree $T$ is $\\text{cng}_{G}(T) = \\max _{e \\in T} \\text{cng}_{G,T}(e)$ .",
"Finally, the spanning tree congestion of graph $G$, denoted by $\\text{stc}(G)$ , is defined as the minimum value of $\\text{cng}_{G}(T)$ over all spanning trees $T$ of $G$ .",
"The concept of the spanning tree congestion extends naturally to multigraphs.",
"For multigraphs, only one edge between any two given vertices can be in a spanning tree, but all of them belong to the cross-edge set $\\partial _{G,T}(e)$ of any edge $e\\in T$ whose removal separates these vertices in $T$ (and thus all contribute to $\\text{cng}_{G,T}(e)$ ).",
"As observed in [19], edge subdivision does not affect the spanning tree congestion of a graph.",
"Therefore any multigraph can be converted into a simple graph by subdividing all multiple edges, without changing its minimum congestion.",
"We use positive integer weights to represent edge multiplicities: an edge $(u,v)$ with weight $\\omega $ represents a bundle of $\\omega $ edges connecting $u$ to $v$ .",
"While we state our results in terms of simple graphs, we use weighted graphs in our proofs, with the understanding that they actually represent the corresponding simple graphs.",
"As all weights used in the paper are constant, the computational complexity of ${K\\!-\\!\\textsf {STC}}$ is not affected.",
"In fact, it is convenient to generalize this further by introducing edges with double weights.",
"A double weight of an edge $e$ is denoted $\\omega \\!",
":\\!\\omega ^{\\prime }$ , where $\\omega $ and $\\omega ^{\\prime }$ are positive integers such that $\\omega \\le \\omega ^{\\prime }$ , and its interpretation in the context of ${K\\!-\\!\\textsf {STC}}$ is as follows: given a spanning tree $T$ , if $e\\in E\\setminus T$ then $e$ contributes $\\omega $ to the congestion $\\text{cng}_{G,T}(f)$ of any edge $f$ for which $e\\in \\partial _{G,T}(f)$ , and if $e\\in T$ then $e$ contributes $\\omega ^{\\prime }$ to its own congestion, $\\text{cng}_{G,T}(e)$ .",
"The lemma below implies that including edges with double weights that add up to at most $K$ does not affect the computational complexity of ${K\\!-\\!\\textsf {STC}}$ , and therefore we can formulate our proofs in terms of graphs where some edges have double weights.",
"Let $(u,v)$ be an edge in $G$ with double weight $\\omega \\!",
":\\!\\omega ^{\\prime }$ , where $1\\le \\omega \\le \\omega ^{\\prime }$ and $\\omega +\\omega ^{\\prime } \\le K$ for some integer $K$ .",
"Consider another graph $G^{\\prime }$ with vertex set $V^{\\prime } = V \\cup { \\left\\lbrace w \\right\\rbrace }$ and edge set $E^{\\prime } = E \\cup { \\left\\lbrace (u,w), (w,v) \\right\\rbrace } \\setminus { \\left\\lbrace (u,v) \\right\\rbrace }$ , in which the weight of $(u,w)$ is $\\omega $ and the weight of $(w,v)$ is $\\omega ^{\\prime }$ .",
"Then, $\\text{stc}(G) \\le K$ if and only if $\\text{stc}(G^{\\prime }) \\le K$ .",
"($\\Rightarrow $ ) Suppose that $G$ has a spanning tree $T$ with $\\text{cng}_{G}(T) \\le K$ .",
"We will show that there exists a spanning tree $T^{\\prime }$ of $G^{\\prime }$ with $\\text{cng}_{G^{\\prime }}(T^{\\prime }) \\le K$ .",
"We break the proof into two cases, in both cases showing that $\\text{cng}_{G^{\\prime },T^{\\prime }}(e) \\le K$ for each edge $e\\in T^{\\prime }$ .",
"Case 1: $(u,v) \\in T$ .",
"Let $T^{\\prime } = T \\cup { \\left\\lbrace (u,w), (w,v) \\right\\rbrace } \\setminus { \\left\\lbrace (u,v) \\right\\rbrace }$ .",
"$T^{\\prime }$ is clearly a spanning tree of $G^{\\prime }$ .",
"If $(x,y) \\in E^{\\prime }\\setminus { \\left\\lbrace (u,w), (w,v) \\right\\rbrace }$ , the $x$ -to-$y$ paths in $T$ and $T^{\\prime }$ are the same, except that if the $x$ -to-$y$ path in $T$ traverses edge $(u,v)$ then the $x$ -to-$y$ path in $T^{\\prime }$ will traverse $(u,w)$ and $(w,v)$ instead.",
"Therefore, if $e\\in T^{\\prime } \\setminus { \\left\\lbrace (u,w), (w,v) \\right\\rbrace }$ , $\\partial _{G^{\\prime },T^{\\prime }}(e) = \\partial _{G,T}(e)$ , so $\\text{cng}_{G^{\\prime },T^{\\prime }}(e) = \\text{cng}_{G,T}(e) \\le K$ .",
"On the other hand, if $e \\in { \\left\\lbrace (u,w), (w,v) \\right\\rbrace }$ , $\\partial _{G^{\\prime },T^{\\prime }}(e) = \\partial _{G,T}(u,v) \\setminus { \\left\\lbrace (u,v) \\right\\rbrace }\\cup { \\left\\lbrace e \\right\\rbrace }$ .",
"Then, edge $e$ contributes $\\omega $ or $\\omega ^{\\prime }$ to $\\text{cng}_{G^{\\prime },T^{\\prime }}(e)$ , while $(u,v)$ , by the definition of double weights, contributes $\\omega ^{\\prime } \\ge \\omega $ to $\\text{cng}_{G,T}(u,v)$ .",
"Hence, $\\text{cng}_{G^{\\prime },T^{\\prime }}(e) \\le \\text{cng}_{G,T}(u,v) \\le K$ .",
"Case 2: $(u,v)\\notin T$ .",
"Let $T^{\\prime } = T \\cup { \\left\\lbrace (w,v) \\right\\rbrace }$ , which is a spanning tree of $G^{\\prime }$ .",
"If $e \\in T^{\\prime } \\setminus { \\left\\lbrace (w,v) \\right\\rbrace }$ , we have two subcases.",
"If $e$ is not on the $u$ -to-$v$ path in $T^{\\prime }$ , $\\partial _{G^{\\prime },T^{\\prime }}(e) = \\partial _{G,T}(e)$ , so $\\text{cng}_{G^{\\prime },T^{\\prime }}(e) = \\text{cng}_{G,T}(e) \\le K$ .",
"If $e$ is on the $u$ -to-$v$ path in $T^{\\prime }$ , $\\partial _{G^{\\prime },T^{\\prime }}(e) = \\partial _{G,T}(e) \\cup { \\left\\lbrace (u,w) \\right\\rbrace } \\setminus { \\left\\lbrace (u,v) \\right\\rbrace }$ .",
"As $(u,w)$ contributes $\\omega $ to $\\text{cng}_{G^{\\prime },T^{\\prime }}(e)$ and, by the definition of double weights, $(u,v)$ contributes $\\omega $ to $\\text{cng}_{G,T}(e)$ , we obtain that $\\text{cng}_{G^{\\prime },T^{\\prime }}(e) = \\text{cng}_{G,T}(e) \\le K$ .",
"In the remaining case, for $e = (w,v)$ , we have $\\partial _{G^{\\prime },T^{\\prime }}(e) = { \\left\\lbrace (u,w),(w,v) \\right\\rbrace }$ , so $\\text{cng}_{G^{\\prime },T^{\\prime }}(e) = \\omega + \\omega ^{\\prime } \\le K$ .",
"($\\Leftarrow $ ) Let $T^{\\prime }$ be the spanning tree of $G^{\\prime }$ with congestion $\\text{cng}_{G^{\\prime }}(T^{\\prime }) \\le K$ .",
"We will show that there exists a spanning tree $T$ of $G$ with $\\text{cng}_{G}(T) \\le K$ .",
"Note that at least one of edges $(u,w)$ and $(v,w)$ has to be in $T^{\\prime }$ .",
"We now consider three cases, in each case showing that $\\text{cng}_{G,T}(e) \\le K$ for each edge $e\\in T$ .",
"Case 1: $(u,w), (v,w) \\in T^{\\prime }$ .",
"Let $T = T^{\\prime } \\cup { \\left\\lbrace (u,v) \\right\\rbrace } \\setminus { \\left\\lbrace (u,w), (w,v) \\right\\rbrace }$ .",
"$T$ is clearly a spanning tree of $G$ .",
"The argument for this case is similar to Case 1 in the proof for the $(\\Rightarrow )$ implication.",
"For each edge $e\\in T \\setminus { \\left\\lbrace (u,v) \\right\\rbrace }$ , its congestion in $T$ is the same as in $T^{\\prime }$ .",
"The congestion of $(u,v)$ in $T$ is bounded by the congestion of $(w,v)$ in $T^{\\prime }$ , which is at most $K$ .",
"Case 2: $(v,w)\\in T^{\\prime }$ and $(u,w)\\notin T^{\\prime }$ .",
"Let $T = T^{\\prime } \\setminus { \\left\\lbrace (w,v) \\right\\rbrace }$ .",
"$T$ is a spanning tree of $G$ .",
"Here again, the argument is similar to the proof for Case 2 in the $(\\Rightarrow )$ implication.",
"For each edge $e\\in T$ , if $e$ is not on the $u$ -to-$v$ path in $T$ , its congestion in $T$ and $T^{\\prime }$ is the same.",
"If $e$ is on the $u$ -to-$v$ path in $T$ , the contributions of $(u,v)$ and $(u,w)$ to the congestion of $e$ in $T$ and $T^{\\prime }$ are the same.",
"Case 3: $(u,w) \\in T^{\\prime }$ and $(v,w)\\notin T^{\\prime }$ .",
"Consider $T^{\\prime \\prime } = T^{\\prime } \\cup { \\left\\lbrace (v,w) \\right\\rbrace } \\setminus { \\left\\lbrace (u,w) \\right\\rbrace }$ , which is a different spanning tree of $G^{\\prime }$ .",
"It is sufficient to show that $\\text{cng}_{G^{\\prime }}(T^{\\prime \\prime }) \\le \\text{cng}_{G^{\\prime }}(T^{\\prime })$ because it will imply $\\text{cng}_{G^{\\prime }}(T^{\\prime \\prime }) \\le K$ , and then we can apply Case 2 to $T^{\\prime \\prime }$ .",
"We examine the congestion values of each edge $e\\in T^{\\prime \\prime }$ .",
"Suppose first that $e\\ne (u,w)$ .",
"If $e$ is not on the $u$ -to-$v$ path in $T^{\\prime }$ , $\\partial _{G^{\\prime },T^{\\prime \\prime }}(e) = \\partial _{G^{\\prime },T^{\\prime }}(e)$ , so $\\text{cng}_{G^{\\prime },T^{\\prime \\prime }}(e) = \\text{cng}_{G^{\\prime },T^{\\prime }}(e)$ .",
"If $e$ is on the $u$ -to-$v$ path in $T^{\\prime }$ , $\\partial _{G^{\\prime },T^{\\prime \\prime }}(e) = \\partial _{G^{\\prime },T^{\\prime }}(e) \\cup { \\left\\lbrace (u,w) \\right\\rbrace }\\setminus { \\left\\lbrace (v,w) \\right\\rbrace }$ , so $\\text{cng}_{G^{\\prime },T^{\\prime \\prime }}(e) = \\text{cng}_{G^{\\prime },T^{\\prime }}(e) + \\omega - \\omega ^{\\prime } \\le \\text{cng}_{G^{\\prime },T^{\\prime }}(e)$ .",
"In the last case when $e = (v,w)$ , $\\text{cng}_{G^{\\prime },T^{\\prime \\prime }}(e) = \\omega + \\omega ^{\\prime } \\le K$ ."
],
[
"${\\mathbb {NP}}$ -Hardness Proof of {{formula:8368264a-2e16-462b-878f-c9fdc761764c}} for {{formula:8a62ece7-687a-4ef9-85d8-1e876169b2b1}}",
"In this section we prove our main result, the ${\\mathbb {NP}}$ -completeness of ${K\\!-\\!\\textsf {STC}}$ .",
"Our proof uses an ${\\mathbb {NP}}$ -complete variant of the satisfiability problem called (2P1N)-SAT [25].",
"An instance of (2P1N)-SAT is a boolean expression $\\phi $ in conjunctive normal form, where each variable occurs exactly three times, twice positively and once negatively, and each clause contains exactly two or three literals of different variables.",
"The objective is to decide if $\\phi $ is satisfiable, that is if there is a satisfying assignment that makes $\\phi $ true.",
"For each constant $K$ , ${K\\!-\\!\\textsf {STC}}$ is clearly in ${\\mathbb {NP}}$ .",
"To show ${\\mathbb {NP}}$ -hardness we present a polynomial-time reduction from (2P1N)-SAT.",
"In this reduction, given an instance $\\phi $ of (2P1N)-SAT, we construct in polynomial time a graph $G$ with the following property: $(\\ast )$ $\\phi $ has a satisfying truth assignment if and only if $\\text{stc}(G) \\le K$ .",
"Throughout the proof, the three literals of $x_i$ in $\\phi $ will be denoted by $x_i$ , $x^{\\prime }_i$ , and ${\\bar{x}}_i$ , where $x_i$ , $x^{\\prime }_i$ are the two positive occurrences of $x_i$ and ${\\bar{x}}_i$ is the negative occurrence of $x_i$ .",
"We will also use notation ${\\tilde{x}}_i$ to refer to an unspecified literal of $x_i$ , that is ${\\tilde{x}}_i\\in { \\left\\lbrace x_i,x^{\\prime }_i,{\\bar{x}}_i \\right\\rbrace }$ .",
"We now describe the reduction.",
"Set $k_i = K-i$ for $i = 1,2,3,4$ .",
"(In particular, for $K=5$ , we have $k_1 = 4$ , $k_2 = 3$ , $k_3 = 2$ , $k_4 = 1$ .)",
"$G$ will consist of gadgets corresponding to variables, with the gadget corresponding to $x_i$ having three vertices $x_i$ , $x^{\\prime }_i$ , and ${\\bar{x}}_i$ , that represent its three occurrences in the clauses.",
"$G$ will also have vertices representing clauses and edges connecting literals with the clauses where they occur (see Figure REF b for an example).",
"As explained in Section , without any loss of generality we can allow edges in $G$ to have constant-valued weights, single or double.",
"Specifically, starting with $G$ empty, the construction of $G$ proceeds as follows: Add a root vertex $r$ .",
"For each variable $x_i$ , construct the $x_i$ -gadget (see Figure REF a).",
"This gadget has three vertices corresponding to the literals: a negative literal vertex ${\\bar{x}}_{i}$ and two positive literal vertices $x_i, x_i^{\\prime }$ , and two auxiliary vertices $y_i$ and $z_i$ .",
"Its edges and their weights are given in the table below: Table: NO_CAPTION For each clause $c$ , create a clause vertex $c$ .",
"For each literal ${\\tilde{x}}_i$ in $c$ , add the corresponding clause-to-literal edge $(c,{\\tilde{x}}_i)$ of weight $1\\!",
":\\!k_2$ .",
"Importantly, as all literals in $c$ correspond to different variables, these edges will go to different variable gadgets.",
"For each two-literal clause $c$ , add a root-to-clause edge $(r,c)$ of weight $1\\!",
":\\!k_1$ .",
"Figure: (a)The x i x_i-gadget.",
"(b) An example of a partial graph GG for the boolean expression φ=(x ¯ 1 ∨x 4 )∧(x 1 ∨x 2 ∨x ¯ 3 )∧(x 1 ∨x ¯ 2 )∧⋯\\phi = ({\\bar{x}}_1 \\vee x_4) \\wedge (x_1 \\vee x_2 \\vee {\\bar{x}}_3) \\wedge (x_1 \\vee {\\bar{x}}_2) \\wedge \\cdots .Here, c 1 =x ¯ 1 ∨x 4 c_1 = {\\bar{x}}_1 \\vee x_4, c 2 =x 1 ∨x 2 ∨x ¯ 3 c_2 = x_1 \\vee x_2 \\vee {\\bar{x}}_3, and c 3 =x 1 ∨x ¯ 2 c_3 = x_1 \\vee {\\bar{x}}_2.We now show that $G$ has the required property $(\\ast )$ , proving the two implications separately.",
"$(\\Rightarrow )$ Suppose that $\\phi $ has a satisfying assignment.",
"Using this assignment, we construct a spanning tree $T$ of $G$ as follows: For every $x_i$ -gadget, include in $T$ edges $(r, x_i^{\\prime })$ , $(r, y_i)$ , and $(y_i, z_i)$ .",
"If $x_i = 0$ , include in $T$ edges $(\\bar{x}_i, z_i)$ and $(x_i, x_i^{\\prime })$ , otherwise include in $T$ edges $(y_i, \\bar{x}_i)$ and $(z_i, x_i)$ .",
"For each clause $c$ , include in $T$ one clause-to-literal edge that is incident to any literal vertex that satisfies $c$ in our chosen truth assignment for $\\phi $ .",
"By routine inspection, $T$ is indeed a spanning tree: Each $x_i$ -gadget is traversed from $r$ without cycles, and all clause vertices are leaves of $T$ .",
"Figures REF and REF show how $T$ traverses an $x_i$ -gadget in different cases, depending on whether $x_i = 0$ or $x_i = 1$ in the truth assignment for $\\phi $ , and on which literals are chosen to satisfy each clause.",
"Note that the edges with double weights satisfy the assumption of Lemma in Section , that is each such weight $1\\!",
":\\!\\omega ^{\\prime }$ satisfies $1 \\le \\omega ^{\\prime }$ and $1+\\omega ^{\\prime } \\le K$ .",
"We need to verify that each edge in $T$ has congestion at most $K$ .",
"All the clause vertices are leaves in $T$ , thus the congestion of each clause-to-literal edge is $k_2 + 2 = K$ ; this holds for both three-literal and two-literal clauses.",
"To analyze the congestion of the edges inside an $x_i$ -gadget, we consider two cases, depending on the value of $x_i$ in our truth assignment.",
"When $x_i = 0$ , we have two sub-cases as shown in Figure REF .",
"The congestions of the edges in the $x_i$ -gadget are as follows: In both cases, $\\text{cng}_{G, T}(r, x_i^{\\prime })= k_3 + 3$ .",
"In case (a), $\\text{cng}_{G, T}(r,y_i) = k_4 + 3$ .",
"In case (b), it is $k_4 + 2$ .",
"In case (a), $\\text{cng}_{G, T}(y_i,z_i) = k_4 + 4$ .",
"In case (b), it is $k_4 + 3$ .",
"In case (a), $\\text{cng}_{G, T}({\\bar{x}}_i, z_i) = k_3 +3$ .",
"In case (b), it is $k_3 + 2$ .",
"In both cases, $\\text{cng}_{G, T}(x_i, x_i^{\\prime }) = k_2+2$ .",
"Figure: The traversal of the x i x_i-gadget by TT when x i =0x_i = 0.",
"Solid lines are tree edges, dashed lines are non-tree edges.",
"(a) x ¯ i {\\bar{x}}_i is chosen by clause cc.",
"(b) x ¯ i {\\bar{x}}_i is not chosen by clause cc.On the other hand, when $x_i = 1$ , we have four sub-cases as shown in Figure REF .",
"The congestions of the edges in the $x_i$ -gadget are as follows: In cases (a) and (b), $\\text{cng}_{G, T}(r, x_i^{\\prime })=k_3 +3$ .",
"In cases (c) and (d), it is $k_3 + 2$ .",
"In cases (a) and (c), $\\text{cng}_{G, T}(r,y_i) = k_4 + 4$ .",
"In cases (b) and (d), it is $k_4 + 3$ .",
"In cases (a) and (c), $\\text{cng}_{G, T}(y_i, z_i) = k_4 + 4$ .",
"In cases (b) and (d), it is $k_4 + 3$ .",
"In cases (a) and (c), $\\text{cng}_{G, T}(z_i,x_i) = k_3 + 3$ .",
"In cases (b) and (d), it is $k_3 + 2$ .",
"In all cases, $\\text{cng}_{G, T}(y_i, {\\bar{x}}_i) = k_2 +2$ .",
"Figure: The traversal of the x i x_i-gadget by TT when x i =1x_i = 1.By cc, c ' c^{\\prime } and c '' c^{\\prime \\prime } we denote the clauses that contain literals x ¯ i {\\bar{x}}_i, x i x_i and x i ' x^{\\prime }_i, respectively.",
"(a) x i x_i and x i ' x^{\\prime }_i are chosen by clauses c ' c^{\\prime } and c '' c^{\\prime \\prime }.",
"(b) x i ' x^{\\prime }_i is chosen by clause c '' c^{\\prime \\prime }.",
"(c) x i x_i is chosen by clause c ' c^{\\prime }.",
"(d) None of x i ,x i ' x_i, x_i^{\\prime } is chosen.In summary, the congestion of each edge of $T$ is at most $K$ .",
"Thus $\\text{cng}_{G}(T) \\le K$ ; in turn, $\\text{stc}(G) \\le K$ , as claimed.",
"$(\\Leftarrow )$ We now prove the other implication in $(\\ast )$ .",
"We assume that $G$ has a spanning tree $T$ with $\\text{cng}_{G}(T) \\le K$ .",
"We will show how to convert $T$ into a satisfying assignment for $\\phi $ .",
"The proof consists of a sequence of claims showing that $T$ must have a special form that will allow us to define this truth assignment.",
"Each $x_i$ -gadget satisfies the following property: for each literal vertex ${\\tilde{x}}_i$ , if some edge $e$ of $T$ (not necessarily in the $x_i$ -gadget) is on the $r$ -to-${\\tilde{x}}_i$ path in $T$ , then $\\partial _{G,T}(e)$ contains at least two distinct edges from this gadget other than $(y_i, z_i)$ .",
"This claim is straightforward: it follows directly from the fact that there are two edge-disjoint paths from $r$ to any literal vertex ${\\tilde{x}}_i\\in { \\left\\lbrace {\\bar{x}}_i, x_i, x_i^{\\prime } \\right\\rbrace }$ that do not use edge $(y_i, z_i)$ .",
"For each two-literal clause $c$ , edge $(r,c)$ is not in $T$ .",
"To justify Claim , we argue by contradiction.",
"Suppose a root-to-clause edge $e = (r,c)$ is in $T$ .",
"If $c$ is a leaf of $T$ , the congestion of $e$ is $k_1+2 > K$ .",
"Thus at least one clause-to-literal edge of $c$ , say $(c, {\\tilde{x}}_i)$ is in $T$ .",
"By Claim , at least two edges of the $x_i$ -gadget are in $\\partial _{G,T}(e)$ .",
"Hence, the congestion of $e$ is at least $k_1+2>K$ , proving Claim .",
"All clause vertices are leaves in $T$ .",
"To prove Claim , suppose there is a clause $c$ that is not a leaf.",
"Then, by Claim , $c$ has at least two clause-to-literal edges in $T$ , say $(c, {\\tilde{x}}_i)$ and $(c, {\\tilde{x}}_j)$ .",
"We can assume that the last edge on the $r$ -to-$c$ path in $T$ is $e = (c, {\\tilde{x}}_i)$ .",
"Clearly, $r \\in T_{{\\tilde{x}}_i,c}$ and ${\\tilde{x}}_j \\in T_{c,{\\tilde{x}}_i}$ .",
"By Claim , at least two edges of the $x_j$ -gadget are in $\\partial _{G,T}(e)$ , and they contribute at least 2 to $\\text{cng}_{G,T}(e)$ .",
"We now have some cases to consider.",
"If $c$ is a two-literal clause, its root-to-clause edge $(r, c)$ is also in $\\partial _{G,T}(e)$ , by Claim .",
"Thus, $\\text{cng}_{G,T}(e)\\ge k_2+3 > K$ (see Figure REF a).",
"So assume now that $c$ is a three-literal clause, and let ${\\tilde{x}}_l \\ne {\\tilde{x}}_i, {\\tilde{x}}_j$ be the third literal of $c$ .",
"If $T$ contains $(c,{\\tilde{x}}_l)$ , the $x_l$ -gadget would also contribute at least 2 to $\\text{cng}_{G,T}(e)$ , so $\\text{cng}_{G,T}(e) \\ge k_2+4 > K$ (see Figure REF b).",
"Otherwise, $(c,{\\tilde{x}}_l)\\notin T$ , and $(c,{\\tilde{x}}_l)$ itself contributes 1 to $\\text{cng}_{G,T}(e)$ , so $\\text{cng}_{G,T}(e) \\ge k_2+3 > K$ (see Figure REF c).",
"We have shown that if a clause vertex $c$ is not a leaf in $T$ , then in all cases the congestion of $T$ would exceed $K$ , completing the proof of Claim .",
"Figure: Illustration of the proof of Claim .In (a) cc is a two-literal clause; in (b) and (c), cc is a three-literal clause.For each $x_i$ -gadget, edge $(r,x_i^{\\prime })$ is in $T$ .",
"Towards contradiction, suppose that $(r, x_i^{\\prime })$ is not in $T$ .",
"Let $(x_i^{\\prime },c)$ be the clause-to-literal edge of $x_i^{\\prime }$ .",
"If only one of the two edges $(x_i^{\\prime },x_i), (x_i^{\\prime },c)$ is in $T$ , making $x^{\\prime }_i$ a leaf, then the congestion of that edge is $k_3 + k_2 + 1 > K$ .",
"Otherwise, both $(x_i^{\\prime },x_i), (x_i^{\\prime },c)$ are in $T$ .",
"Because $c$ is a leaf in $T$ by Claim , $e= (x_i, x_i^{\\prime })$ is the last edge on the $r$ -to-$x_i^{\\prime }$ path in $T$ .",
"As shown in Figure REF a, $\\text{cng}_{G,T}(e) \\ge k_3+k_2+2 > K$ .",
"This proves Claim .",
"For each $x_i$ -gadget, edge $(r,y_i)$ is in $T$ .",
"To prove this claim, suppose $(r, y_i)$ is not in $T$ .",
"We consider the congestion of the first edge $e$ on the $r$ -to-$y_i$ path in $T$ .",
"By Claims and , we have $e = (r, x_i^{\\prime })$ , all vertices of the $x_i$ -gadget have to be in $T_{x_i^{\\prime }, r}$ , and $T_{x_i^{\\prime }, r}$ does not contain literal vertices of another variable $x_j \\ne x_i$ .",
"For each literal ${\\tilde{x}}_i$ of $x_i$ , if a clause-to-literal edge $(c, {\\tilde{x}}_i)$ is in $T$ , then the two other edges of $c$ contribute 2 to $\\text{cng}_{G,T}(e)$ , otherwise $(c, {\\tilde{x}}_i)$ contributes 1 to $\\text{cng}_{G,T}(e)$ .",
"Then, $\\text{cng}_{G,T}(e) \\ge k_4+k_3+3 > K$ (see Figure REF b), proving Claim .",
"Figure: (a) Illustration of the proof of Claim .",
"(a) Illustration of the proof of Claim .",
"Dot-dashed lines are edges that may or may not be in TT.For each $x_i$ -gadget, exactly one of edges $(z_i, x_i)$ and $(x_i, x_i^{\\prime })$ is in $T$ .",
"By Claims and , edges $(r,y_i)$ and $(r,x_i^{\\prime })$ are in $T$ .",
"Since the clause neighbor $c^{\\prime }$ of $x_i$ is a leaf of $T$ , by Claim , if none of $(z_i, x_i)$ , $(x_i, x_i^{\\prime })$ were in $T$ , $x_i$ would not be reachable from $r$ in $T$ .",
"Thus, at least one of them is in $T$ .",
"Now, assume both $(z_i, x_i)$ and $(x_i, x_i^{\\prime })$ are in $T$ (see Figure REF a).",
"Then, edge $(y_i, z_i)$ is not in $T$ , as otherwise we would create a cycle.",
"Let us consider the congestion of edge $e = (r,x_i^{\\prime })$ .",
"Clearly, $x_i$ and $x_i^{\\prime }$ are in $T_{x_i^{\\prime }, r}$ .",
"The edges of the two clause neighbors $c^{\\prime }$ and $c^{\\prime \\prime }$ of $x_i$ and $x_i^{\\prime }$ contribute at least 2 to $\\text{cng}_{G,T}(e)$ , by Claim .",
"In addition, by Claim , besides $e$ and $(y_i, z_i)$ , $\\partial _{G,T}(e)$ contains another edge of the $x_i$ -gadget which contributes at least another 1 to $\\text{cng}_{G,T}(e)$ .",
"Thus, $\\text{cng}_{G,T}(e) \\ge k_4+k_3+3 > K$ — a contradiction.",
"This proves Claim .",
"For each $x_i$ -gadget, edge $(y_i, z_i)$ is in $T$ .",
"By Claims and , the two edges $(r,x_i^{\\prime })$ and $(r,y_i)$ are in $T$ .",
"Now assume, towards contradiction, that $(y_i, z_i)$ is not in $T$ (see Figure REF b).",
"By Claim , only one of $(z_i, x_i)$ and $(x_i, x_i^{\\prime })$ is in $T$ .",
"Furthermore, the clause neighbor $c^{\\prime }$ of $x_i$ is a leaf of $T$ , by Claim .",
"As a result, $(z_i, x_i)$ cannot be on the $y_i$ -to-$z_i$ path in $T$ .",
"To reach $z_i$ from $y_i$ , the two edges $(y_i, {\\bar{x}}_i), ({\\bar{x}}_i, z_i)$ have to be in $T$ .",
"Let us consider the congestion of $e = (y_i, {\\bar{x}}_i)$ .",
"The edges of the clause neighbor $c$ of ${\\bar{x}}_i$ contribute at least 1 to the congestion of $e$ , by Claim .",
"Also, by Claim , besides $e$ and $(y_i, z_i)$ , $\\partial _{G,T}(e)$ contains another edge of the $x_i$ -gadget which contributes at least 1 to $\\text{cng}_{G,T}(e)$ .",
"In total, $\\text{cng}_{G,T}(e) \\ge k_4+k_2+2 > K$ , reaching a contradiction and completing the proof of Claim .",
"Figure: (a) Illustration of the proof of Claim .",
"(b) Illustration of the proof of Claim .",
"Dot-dashed lines are edges that may or may not be in TT.For each $x_i$ -gadget, if its clause-to-literal edge $({\\bar{x}}_i, c)$ is in $T$ , then its other two clause-to-literal edges $(x_i, c^{\\prime })$ and $(x_i^{\\prime }, c^{\\prime \\prime })$ are not in $T$ .",
"Assume the clause-to-literal edge $({\\bar{x}}_i, c)$ of the $x_i$ -gadget is in $T$ .",
"By Claim , edge $(y_i, z_i)$ is in $T$ .",
"If $(y_i, {\\bar{x}}_i)$ is also in $T$ , edge $({\\bar{x}}_i, z_i)$ cannot be in $T$ , and it contributes 1 to $\\text{cng}_{G,T}(y_i, {\\bar{x}}_i)$ .",
"As shown in Figure REF a, $\\text{cng}_{G,T}(y_i, {\\bar{x}}_i) = k_2 + 3 > K$ .",
"Thus, $(y_i, {\\bar{x}}_i)$ cannot be in $T$ .",
"Since $c$ is a leaf of $T$ , edge $({\\bar{x}}_i, z_i)$ has to be in $T$ , for otherwise ${\\bar{x}}_i$ would not be reachable from $r$ .",
"By Claim , one of edges $(z_i, x_i)$ and $(x_i, x_i^{\\prime })$ is in $T$ .",
"If $(z_i, x_i)$ is in $T$ (see Figure REF b), $\\text{cng}_{G,T}(y_i, z_i) \\ge k_4 + 5 > K$ .",
"Hence, $(z_i, x_i)$ is not in $T$ , which implies that $(x_i, x_i^{\\prime })$ is in $T$ .",
"Figure: Illustration of the proof of Claim .",
"Dot-dashed lines are edges that may or may not be in TT.Now, we proceed by contradiction assuming that at least one other clause-to-literal edge of the $x_i$ -gadget is in $T$ .",
"If edge $(x_i, c^{\\prime })$ is in $T$ , $\\text{cng}_{G,T}(x_i, x_i^{\\prime }) \\ge k_2 + 3 > K$ , as shown in Figure REF c. Similarly, if $(x_i^{\\prime }, c^{\\prime \\prime })$ is in $T$ , $\\text{cng}_{G,T}(r, x_i^{\\prime }) \\ge k_3 + 4 > K$ (see Figure REF d).",
"So we reach a contradiction in both cases, thus proving Claim .",
"We are now ready to complete the proof of the $(\\Leftarrow )$ implication in the equivalence $(\\ast )$ .",
"We use our spanning tree $T$ of congestion at most $K$ to create a truth assignment for $\\phi $ by setting $x_i = 0$ if the clause-to-literal edge of ${\\bar{x}}_i$ is in $T$ , otherwise $x_i = 1$ .",
"By Claim , this truth assignment is well-defined.",
"Each clause has one clause-to-literal edge in $T$ which ensures that all clauses are indeed satisfied."
]
] | 2209.08219 |
[
[
"Three-dimensional non-kinematic simulation of post-emergence evolution\n of bipolar magnetic regions and Babcock-Leighton dynamo of the Sun"
],
[
"Abstract The Babcock-Leighton (BL) flux-transport model is a widely-accepted dynamo model of the Sun.",
"This dynamo model has been extensively studied in a two-dimensional (2D) mean-field framework in both kinematic and non-kinematic regimes.",
"Recent three-dimensional (3D) models have been restricted to the kinematic regime.",
"In these models, the surface poloidal flux is produced by the emergence of bipolar magnetic regions (BMRs) that are tilted according to Joy's law.",
"We investigate the prescription for emergence of a BMR in 3D non-kinematic simulations.",
"We also report initial results of cyclic BL dynamo simulation.",
"We extend a conventional 2D mean-field model of the BL flux-transport dynamo into 3D non-kinematic regime.",
"The large-scale mean flows are driven by the parameterized $\\Lambda$-effect in this model.",
"For the induction equation, we use a BL source term by which the surface BMRs are produced in response to the dynamo-generated toroidal field inside the convection zone.",
"We find that, in the 3D non-kinematic regime, the tilt angle of a newly-emerged BMR is very sensitive to the prescription for the subsurface structure of the BMR.",
"Anti-Joy tilt angles are found unless the BMR is deeply embedded in the convection zone.",
"We also find that the leading spot tends to become stronger than the following spot.",
"The anti-Joy's law trend and the morphological asymmetry of the BMRs can be explained by the Coriolis force acting on the Lorentz-force-driven flows.",
"Furthermore, we demonstrate that the solar-like magnetic cycles can be successfully obtained if the Joy's law is explicitly given in the BL $\\alpha$-effect.",
"In these cyclic dynamo simulation, a strong Lorentz force feedback leads to cycle modulations in the differential rotation and meridional circulation.",
"The non-axisymmetric components of the flows are found to exist as inertial modes such as the equatorial Rossby modes."
],
[
"Introduction",
"The Sun exhibits an 11-year cyclic magnetic activity which is sustained by the dynamo processes in the convection zone [12].",
"The Babcock-Leighton flux-transport model is one of the most promising solar dynamo models at present that can explain many observational features [17].",
"In this model, the equatorward migration of sunspot groups is attributed to the meridional flow near the base of the convection zone [73], [15].",
"This model is supported by the recent helioseismic observations in which the meridional flow is found to be poleward at the surface and equatorward at the base [27].",
"Another characteristic feature of this dynamo model is that the main process generating poloidal fields from toroidal fields is the so-called Babcock-Leighton mechanism, in which the surface poloidal fields are generated by the poleward advection and equatorial cancellation of the bipolar sunspots that are tilted with respect to east-west direction [1], [50].",
"The tendency that the leading spot is located closer to the equator than the following one is called Joy's law [30].",
"The physical origin of the Joy's law is still under debate: Thin flux tube simulations have demonstrated that the Joy's law can be explained by the Coriolis force acting on the buoyantly-rising flux tubes through the convection zone [18], [23], [74].",
"On the other hand, recent observations have shown that the active regions emerge with east-west alignment (with zero tilt) on average and the Joy's law tilts are generated by the north-south separation motions after emergence [65].",
"Numerical investigations of the Babcock-Leighton flux-transport dynamo model have been mostly carried out in a two-dimensional (2D) kinematic mean-field framework [17], [58], [13], [31], [44].",
"In these models, the Babcock-Leighton $\\alpha $ -effect is modeled as the axisymmetric poloidal source term which is localized near the surface.",
"Although there are some non-kinematic studies where the dynamo-generated fields are allowed to give a feedback on the mean flows [62], [39], [40], the longitudinal component of the Lorentz force has been ignored because of the axisymmetry of the system.",
"There are several recent studies that aim to realize the Babcock-Leighton process in a three-dimensional (3D) full-spherical domain.",
"[76] first presented a kinematic model in which the upward velocity perturbation associated with the magnetic buoyancy is explicitly prescribed to produce the tilted bipolar magnetic regions (BMRs) at the surface.",
"This method has also been used in [49] and [75].",
"Furthermore, [55] have developed a different model of the Babcock-Leighton dynamo, in which the BMRs are artificially placed at the surface in response to the toroidal field at the base under the constraint of Joy's law.",
"In fact, this method is regarded as a 3D realization of the so-called “double-ring” algorithm used in 2D mean-field models [19], [58], [57].",
"The same model has also been used to study the long-term cycle variability [45].",
"However, all of these models are kinematic.",
"Therefore, it still remains unclear how the Lorentz-forces of the BMRs affect their post-emergence evolution and the resulting dynamo solution in the non-kinematic regime.",
"The models which include the most physics are provided by magnetohydrodynamic (MHD) convective dynamo simulations in a spherical shell [9], [25], [8], [21], [35], [70].",
"However, they have difficulty in reproducing the large-scale mean flows as we observe when the solar parameters are used [60].",
"Moreover, they still cannot capture the full dynamics of the flux-emergence and the resulting formation of BMRs at the surface comprehensively [59], [21], [14].",
"Therefore, it is still helpful to use mean-fieldIn the solar dynamo community, the “mean-field” models are conventionally regarded as 2D axisymmetric models where the mean is taken over longitudes.",
"In this paper, however, we use the term “mean-field” in a more general sense; the mean should be regarded as an ensemble average or a spatial average over small portions in the convection zone that satisfy the Reynolds' averaging rules.",
"models in which the large-scale mean-flows are largely controlled with parameterizations of small-scale convective angular momentum transport [47] and the flux emergence is modeled via a parametrization.",
"In this paper, we present a new numerical framework to study the Babcock-Leighton dynamo processes of the Sun in a 3D non-kinematic regime, which takes advantage of both the mean-field approach for the solar large-scale mean flows and the 3D realization of the Babcock-Leighton process.",
"Therefore, our model extends both the 2D non-kinematic mean-field models [62] and the 3D kinematic models [56].",
"Although our model is still less complete than 3D MHD convective dynamo models, we can solve the MHD dynamo equations under the constraints of the observed differential rotation and meridional circulation (that are hard to obtain in the 3D MHD convective dynamo models).",
"We believe that our model can potentially provide many future applications such as data assimilation and cycle prediction.",
"Recently, various kinds of inertial modes have been discovered and identified on the Sun [52], [26].",
"Since these inertial modes have different mode properties from those of acoustic (p) modes, they are expected to be useful as an alternative tool to probe the interior of the Sun [26], [3].",
"However, it remains largely uncertain how these modes are affected by the dynamo-generated magnetic fields.",
"We believe that our dynamo model can also be used to study the effects of magnetic fields on various inertial modes in the Sun's convection zone in the nonlinear regime.",
"The organization of this paper is as follows.",
"The numerical model is explained in detail in §.",
"In § , we show how the post-emergence evolution of the BMRs are changed from the previous models.",
"Our initial results of the cyclic dynamo are then presented in §.",
"We close by summarizing our results and discussing the future prospects in §.",
"We numerically solve a set of MHD equations in a spherical coordinate $(r,\\theta ,\\phi )$ : $&& \\frac{\\partial \\rho _{1}}{\\partial t} = -\\nabla \\cdot (\\rho _{0} {\\mbox{$v$}}), \\\\&& \\frac{\\partial {\\mbox{$v$}}}{\\partial t} = -{\\mbox{$v$}}\\cdot \\nabla {\\mbox{$v$}}-\\frac{\\nabla p_{1}}{\\rho _{0}}-\\frac{\\rho _{1}}{\\rho _{0}}g{\\mbox{$e$}}_{r}+2{\\mbox{$v$}}\\times {\\mbox{$\\Omega _{0}$}} \\nonumber \\\\&& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ +\\frac{1}{4\\pi \\rho _{0}} (\\nabla \\times {\\mbox{$B$}})\\times {\\mbox{$B$}}+\\frac{1}{\\rho _{0}}\\nabla \\cdot {\\mbox{$\\Pi $}}, \\\\&& \\frac{\\partial {\\mbox{$B$}}}{\\partial t} = \\nabla \\times ({\\mbox{$v$}}\\times {\\mbox{$B$}}+{\\mbox{$\\mathcal {E}$}}-\\eta \\nabla \\times {\\mbox{$B$}}), \\\\&& \\frac{\\partial s_{1}}{\\partial t} = {\\mbox{$v$}}\\cdot \\nabla s_{1}+c_{\\mathrm {p}}\\delta \\frac{v_{r}}{H_{p}}+\\frac{1}{\\rho _{0}T_{0}}\\nabla \\cdot (\\rho _{0}T_{0}\\kappa \\nabla s_{1}) \\nonumber \\\\&& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ +\\frac{1}{\\rho _{0}T_{0}}\\left[({\\mbox{$\\Pi $}}\\cdot \\nabla )\\cdot {\\mbox{$v$}} +\\frac{\\eta }{4\\pi }|\\nabla \\times {\\mbox{$B$}}|^{2} \\right], $ where $g$ , $\\rho _{0}$ , $p_{0}$ , and $H_{p}$ denote the gravitational acceleration, density, pressure, and pressure scale height of the background state which is in an adiabatically-stratified hydrostatic equilibrium.",
"We use the same radial profiles for the background stratification as the model presented in [61] and [5].",
"The quantities with subscript 1, $\\rho _{1}$ and $p_{1}$ , are the perturbations with respect to the background that are assumed to be sufficiently small, i.e., $|p_{1}/p_{0}| \\approx |\\rho _{1}/\\rho _{0}| \\ll 1$ , so that the equation of state is linearized $&& p_{1}=p_{0}\\left( \\gamma \\frac{\\rho _{1}}{\\rho _{0}} +\\frac{s_{1}}{c_{v}}\\right),$ where $\\gamma =5/3$ is the specific heat ratio and $s_{1}$ is entropy perturbation from the adiabatic background.",
"The rotation rate of the radiative core $\\Omega _{0}/2\\pi =431.3$ nHz is used for a system rotation rate.",
"The tensor ${\\mbox{$\\Pi $}}$ represents the turbulent Reynolds stress associated with small-scale (subgrid-scale) convective motions that are not explicitly resolved in our model.",
"This in principle contains the effects of turbulent diffusion and turbulent momentum transport [47].",
"Therefore, the Reynolds stress is expressed as, $&&\\Pi _{ik}=\\rho _{0} \\nu _{\\mathrm {vis}} \\left( S_{ik}-\\frac{2}{3}\\delta _{ik}\\nabla \\cdot {\\mbox{$v$}} + \\Lambda _{ik}\\Omega _{0} \\right),$ where $S_{ik}$ and $\\delta _{ik}$ denote the velocity deformation tensor and the Kronecker-delta unit tensor.",
"The detailed expression of $S_{ik}$ in a spherical coordinate can be found in [21].",
"In our model, turbulent viscous, thermal, and magnetic diffusivities are all assumed to be isotropic.",
"We use the same radial profiles for the viscous ($\\nu _{\\mathrm {vis}}$ ), thermal ($\\kappa $ ), and magnetic ($\\eta $ ) diffusivities as of [62].",
"The diffusivity values at the top boundary are $\\nu _{\\mathrm {vis}}=\\kappa =5\\times 10^{12}$ cm$^{2}$ s$^{-1}$ , and $\\eta =10^{12}$ cm$^{2}$ s$^{-1}$ .",
"In order to break the Taylor-Proudman's constraint of the differential rotation via the thermal wind balance, a negative (positive) latitudinal entropy gradient in the northern (southern) hemisphere is required.",
"Although there are several proposed mechanisms to generate this latitudinal entropy gradients [47], [54], [33], in this paper, we adopt the idea proposed by [61] that the latitudinal entropy variation is generated by the radial meridional flows when the base of the convection zone is weakly subadiabatic.",
"To this end, we give the superadiabaticity $\\delta =\\nabla -\\nabla _{\\mathrm {ad}}$ , with $\\nabla =d\\ln {T}/d\\ln {p}$ , as $&& \\delta (r,\\theta ) =\\mathcal {T}_{-}(r;r_{\\mathrm {sub}},d_{\\mathrm {sub}}) \\ \\delta _{\\mathrm {sub}}(\\theta ), \\\\&& \\delta _{\\mathrm {sub}}(\\theta ) = \\delta _{\\mathrm {pl}}+(\\delta _{\\mathrm {eq}}-\\delta _{\\mathrm {pl}}) \\sin ^{2}{\\theta }, \\\\&& r_{\\mathrm {sub}}(\\theta )=r_{\\mathrm {pl}}+(r_{\\mathrm {eq}}-r_{\\mathrm {pl}})\\sin ^{2}{\\theta },$ where $\\mathcal {T}$ denotes a transition function defined by $&& \\mathcal {T}_{\\pm }(x;x_{0},d)=\\frac{1}{2}\\left[1\\pm \\tanh {\\left( \\frac{x-x_{0}}{d}\\right)} \\right].$ We set the superadiabaticity at the poles and at the equator as $\\delta _{\\mathrm {pl}}=-1.5\\times 10^{-5}$ and $\\delta _{\\mathrm {eq}}=-2\\times 10^{-5}$ , respectively, at the base of the convection zone.",
"The depths where the stratification changes from subadiabatic to adiabatic are given as $r_{\\mathrm {pl}}=0.725R_{\\odot }$ and $r_{\\mathrm {eq}}=0.735R_{\\odot }$ .",
"The weakly subadiabatic layer near the base is thought to be an outcome of a non-local energy transport of strongly magnetized convection [66], [6] and has been reported in some recent numerical simulations [43], [32], [4].",
"The subadiabaticity is slightly enhanced in the equatorial area owing to the latitudinal variation of the Coriolis force acting on low-entropy downdrafts [46].",
"The dimensionless tensor $\\Lambda _{ik}$ specifies the amplitude and direction of the turbulent momentum transport.",
"In this model, we only consider the turbulent angular momentum transport.",
"Therefore, we parameterize $\\Lambda _{r\\phi }(=\\Lambda _{\\phi r})$ and $\\Lambda _{\\theta \\phi }(=\\Lambda _{\\phi \\theta })$ similarly to the model presented in [61], $&& \\Lambda _{r\\phi }=+\\Lambda _{0}\\tilde{f}_{l}(r,\\theta ) \\cos {(\\theta +\\lambda )} \\left[1+\\zeta _{r}(r,\\theta ,\\phi ) \\right], \\\\&& \\Lambda _{\\theta \\phi }=-\\Lambda _{0}\\tilde{f}_{l}(r,\\theta ) \\sin {(\\theta +\\lambda )} \\left[1+\\zeta _{\\theta }(r,\\theta ,\\phi ) \\right].",
"$ The overall amplitude of the $\\Lambda $ -effect is given by $\\Lambda _{0}=0.85$ .",
"The inclination is set to $\\lambda =+(-)15^{\\circ }$ in the northern (southern) hemisphere.",
"Thus, the associated angular momentum flux becomes largely equatorward and weakly away from the rotational axis.",
"The spatial distribution of the $\\Lambda $ -effect is specified as $&& \\tilde{f}_{l}(r,\\theta )=\\frac{f_{l}(r,\\theta )}{\\mathrm {max}|f_{l}(r,\\theta )|}, \\\\&& f_{l}(r,\\theta )=\\sin ^{2}{\\theta }\\cos {\\theta } \\tanh {\\left( \\frac{r_{\\mathrm {max}}-r}{d_{l}}\\right)}.$ where $d_{l}=0.025R_{\\odot }$ .",
"With this parameterization, the profiles of differential rotation and meridional circulation become similar to observations [38], [27] The quantities $\\zeta _{r}$ and $\\zeta _{\\theta }$ denote random fluctuations due to the unresolved turbulent convection.",
"In our model, the random fields $\\zeta _{r}$ and $\\zeta _{\\theta }$ are separately constructed by simply superposing multiple gaussians as, $\\zeta (r,\\theta ,\\phi )=\\sum _{i=1}^{N} c_{i} \\exp {\\left[ -\\left( \\frac{r-r_{i}}{\\delta r}\\right)^{2}-\\left( \\frac{\\theta -\\theta _{i}}{\\delta \\theta }\\right)^{2}-\\left( \\frac{\\phi -\\phi _{i}}{\\delta \\phi }\\right)^{2} \\right]},$ where the locations of gaussian peaks $(r_{i},\\theta _{i},\\phi _{i})$ are randomly chosen and their amplitudes $c_{i}$ are also randomly determined within the range $-2<c_{i}<2$ .",
"The spatial scale of each gaussian is set as $(\\delta r,\\delta \\theta ,\\delta \\phi )=(0.03R_{\\odot },5^{\\circ },5^{\\circ })$ .",
"In our reference calculation, we set the number of gaussians $N=30$ .",
"We generate the random field ${\\mbox{$\\zeta $}}$ at every time step and therefore it is uncorrelated in time.",
"Note that the non-axisymmetric flows can be partially driven by these random fluctuations of the $\\Lambda $ -effect.",
"In the induction equation (), we add an electro-motive-force ${\\mbox{$\\mathcal {E}$}}$ to model the Babcock-Leighton $\\alpha $ -effect, by which the poloidal field is generated near the surface from the toroidal field near the base of the convection zone.",
"Note that this term is only switched on when the cyclic dynamo is simulated in § .",
"A detail formulation of ${\\mbox{$\\mathcal {E}$}}$ will be given in § REF ."
],
[
"Numerical scheme",
"We numerically solve the Eqs.",
"(REF )–() using the 4th-order centered-difference method for space and 4-step Runge-Kutta scheme for time integration [71].",
"To avoid the severe CFL constraint for time step, we use the reduced speed of sound technique [61] so that the background sound speed is artificially reduced by a factor of $\\xi =200$ , which still ensures that flows remain sufficiently subsonic [34].",
"Moreover, we use the hyperbolic divergence cleaning method (9-wave method) for minimizing the numerical error resulting from the divergence of magnetic field [16].",
"The numerical domain is a full-spherical shell extending from $r_{\\mathrm {min}}=0.65R_{\\odot }$ up to $r_{\\mathrm {max}}=0.985R_{\\odot }$ .",
"The base of the convection zone is located at $r_{\\mathrm {bc}}=0.71R_{\\odot }$ .",
"In order to avoid the singularities in a spherical coordinate at the poles, we use the Yin-Yang grid [42].",
"For more details about the implementation of the Yin-Yang grid, refer [2].",
"The grid resolution is $72(N_{r})\\times 72(N_{\\theta })\\times 216(N_{\\phi })\\times 2$ (Yin and Yang grids).",
"The code is parallelized using message passing interface (MPI).",
"At both radial boundaries, impenetrable and stress-free boundary condition is used for velocity.",
"The magnetic field is assumed to be radial at the top and horizontal at the bottom.",
"Figure: Results of the mean flows from the hydrodynamic calculation.",
"(a) Differential rotation 〈Ω〉=Ω 0 +〈v φ 〉/(rsinθ)\\langle \\Omega \\rangle =\\Omega _{0}+\\langle v_{\\phi } \\rangle /(r\\sin {\\theta }).The black dotted curve shows the location of the base of the convection zone at r=0.71R ⊙ r=0.71R_{\\odot }, below which the stratification is subadiabatic.",
"(b) Meridional circulation 〈v θ 〉\\langle v_{\\theta } \\rangle .The black solid and dashed curves show contours of the streamfunction Ψ\\Psi defined by ρ 0 v m =∇×(Ψe φ )\\rho _{0} {\\mbox{$v_{m}$}}=\\nabla \\times (\\Psi {\\mbox{$e$}}_{\\phi }) where v m =〈v r 〉e r +〈v θ 〉e θ {\\mbox{$v$}}_{m}=\\langle v_{r} \\rangle {\\mbox{$e$}}_{r} +\\langle v_{\\theta } \\rangle {\\mbox{$e$}}_{\\theta }.The meridional circulation is counter-clockwise (clockwise) in the northern (southern) hemisphere, i.e., the flow is poleward (equatorward) at the surface (base).We first carry out a hydrodynamic simulation until the large-scale mean flows become quasi-stationary.",
"Figure REF shows profiles of the differential rotation $\\langle \\Omega \\rangle = \\Omega _{0}+\\langle v_{\\phi } \\rangle /(r\\sin {\\theta })$ and meridional circulation ${\\mbox{$v$}}_{m}=\\langle v_{r}\\rangle {\\mbox{$e$}}_{r}+\\langle v_{\\theta }\\rangle {\\mbox{$e$}}_{\\theta }$ obtained from our hydrodynamic simulation.",
"Here, $\\langle \\ \\rangle $ denotes the longitudinal average.",
"We then add magnetic fields to carry out MHD calculations."
],
[
"Post-emergence evolution of BMRs",
"In this section, we carry out a set of numerical simulations to study how the post-emergence evolution of a BMR is dependent on how it is injected into the simulation.",
"To this end, we solve the MHD equations (REF )–() starting from different initial magnetic field configurations for a newly-emerged single BMR.",
"For simplicity, we only consider the short-term evolution of a BMR and do not discuss the long-term buildup of the polar fields and the resulting dynamo cycles.",
"Hence, we set ${\\mbox{$\\mathcal {E}$}}=0$ .",
"Figure: Temporal evolution of a BMRs from Case 1 at selected temporal points (a) t=0t=0 days, (b) t=2.1t=2.1 days, and (c) t=6.4t=6.4 days.The kinematic simulation with the same initial condition is shown in panel (d) at t=6.4t=6.4 days.Top panels show the radial field B r B_{r} at the surface r=0.985R ⊙ r=0.985R_{\\odot }.Thick black solid curves shows the contour at |B r |=0.3|B_{r}|=0.3 kG.The blue and red cross marks represent the locations of the flux-weighted center for the leading and following spots, and the grey straight lines are drawn to connect these two cross marks.Middle panels show the radial flows v r v_{r} near the surface r=0.98R ⊙ r=0.98R_{\\odot } (color contour) and the horizontal velocities (v θ ,v φ v_{\\theta },v_{\\phi }) at the surface (vector arrows).Bottom panels show cross sections of the magnetic field strength |B||B| at the fixed latitude of 20 ∘ 20^{\\circ } which is denoted by black dashed lines in the top and middle panels.Figure: Schematic illustrations explaining the generation of anti-Joy's law tilt and the morphological asymmetry of the BMRs' field strengths.",
"(a) Cross section at the surface seen from the top.",
"The red and blue arrows show the directions of the Lorentz force and the Coriolis force, respectively.",
"(b) Cross section at the fixed latitude seen from the equator to the north pole.",
"(c) Three-dimensional view of the evolution of a BMRs.Figure: Same as Fig.",
"except from the simulation Case 2.Figure: Same as Fig.",
"except from the simulation Case 3."
],
[
"Initial condition of magnetic fields",
"For simplicity, we simulate the evolution of a BMR with zero initial tilt, i.e., the leading and following polarity spots are perfectly east-west aligned.",
"The initial magnetic field is given as $&& {\\mbox{$B$}} \\propto \\nabla \\times (A_{\\mathrm {ic}} {\\mbox{$e$}}_{\\theta }),$ where the vector potential $A_{\\mathrm {ic}}$ is given by $A_{\\mathrm {ic}}(r,\\theta ,\\phi )=\\left\\lbrace 1-\\tanh {\\left( \\frac{l_{\\mathrm {bmr}}(r,\\phi )-1}{0.5} \\right)}\\right\\rbrace \\exp {\\left[ -\\left( \\frac{\\theta -\\theta ^{*}}{\\Delta \\theta _{\\mathrm {bmr}}}\\right)^{2}\\right]},$ with $&& l_{\\mathrm {bmr}}(r,\\phi )=\\sqrt{\\left( \\frac{r-r_{\\mathrm {max}}}{\\Delta r_{\\mathrm {bmr}}}\\right)^{2}+\\left( \\frac{\\phi -\\phi ^{*}}{\\Delta \\phi _{\\mathrm {bmr}}}\\right)^{2}}.$ Here we set the colatitude $\\theta ^{*}=70^{\\circ }$ and $\\phi ^{*}=0^{\\circ }$ , and therefore a BMR is located at the latitude of $20^{\\circ }$ in the northern hemisphere at the central meridian.",
"The parameters $\\Delta r_{\\mathrm {bmr}}$ , $\\Delta \\theta _{\\mathrm {bmr}}$ , and $\\Delta \\phi _{\\mathrm {bmr}}$ specify the radial, latitudinal, and longitudinal extent of the BMR.",
"We fix $\\Delta \\theta _{\\mathrm {bmr}}=5^{\\circ }$ but vary both $\\Delta r_{\\mathrm {bmr}}$ and $\\Delta \\phi _{\\mathrm {bmr}}$ as free model parameters to change the subsurface shape of the BMR, as sumamrized in Table.",
"REF .",
"In Case 1, $\\Delta r_{\\mathrm {bmr}}$ is relatively small and $\\Delta \\phi _{\\mathrm {bmr}}$ is large, indicating that the subsurface structure of the BMR is very shallow in radius but stretched in longitude.",
"In Case 3, on the other hand, the subsurface field morphology is changed to a vertically-elongated half ellipse.",
"Case 2 is an intermediate case between Case 1 and 3.",
"In all cases, the amplitude of the initial field is determined such that the maximum radial field at the top boundary is 4 kG."
],
[
"Temporal evolution",
"Let us first take a look at Case 1 where the results are most drastically changed from the previous studies.",
"Figure REF shows the evolution of the radial field, flows at the surface, and the subsurface field of the BMR over the first several days after the emergence from Case 1.",
"As soon as the BMR is inserted, there emerge strong upflows at the surface because the mass is expelled from the flux tube due to the pressure imbalance.",
"At the same time, the strong Lorentz force of the BMR drives strong longitudinal converging flows towards the polarity inversion line and latitudinal diverging flows along the polarity inversion line.",
"These are clearly seen in Fig.",
"REF b middle panel.",
"Consequently, the two spots (that are initially separated in longitude) are quickly pulled together and stretched in latitude, as shown in Fig.",
"REF c top panel.",
"For comparison, we show the same snapshot from the corresponding kinematic calculation in as Fig.",
"REF d. The result reveals that the temporal evolution of the BMR is substantially changed in non-kinematic regime where the Lorentz force and the Coriolis force are taken into account.",
"This type of evolution is not observed on the Sun."
],
[
"Tilt angle",
"In order to measure the tilt angle of the BMR, we compute the flux-weighted center locations of the leading and following polarity regions ($\\theta _{L}$ , $\\phi _{L}$ ) and ($\\theta _{F}$ , $\\phi _{F}$ ), respectively.",
"The tilt angle $\\gamma $ is then defined as $\\gamma &=&\\tan ^{-1}{\\left( \\frac{\\sin {\\theta _{L}}-\\sin {\\theta _{F}}}{\\cos {\\theta _{L}}\\sin {\\theta _{L}}-\\cos {\\theta _{F}}\\sin {\\theta _{F}}} \\right)} \\nonumber \\\\& \\approx & \\tan ^{-1}{\\left[ \\frac{\\theta _{L}-\\theta _{F}}{(\\phi _{L}-\\phi _{F}) \\cos {\\theta ^{*}}} \\right]}.$ Since we consider the BMR located in the northern hemisphere, the Joy's law is satisfied when $\\gamma >0$ by definition.",
"It is seen from Fig.",
"REF c (top) that the leading polarity spot is located at slightly higher latitude than the following polarity spot on average, indicating that the associated tilt angle is negative ($\\gamma \\approx -10^{\\circ }<0$ at $t=6.2$ days).",
"This is against the Joy's law.",
"This is because the Coriolis force acts on the longitudinal converging flows (towards the polarity inversion line), as schematically illustrated in Fig.",
"REF a.",
"It should be emphasized that this generation of anti-Joy's law tilt is essentially 3D non-kinematic effect, and thus, cannot be captured neither by the 2D non-kinematic models (which ignores the longitudinal dependence) nor the 3D kinematic models (which ignores the Lorentz force feedback).",
"Needless to say, the generation of negative tilts is contrary to the solar observations.",
"One possible way of reconciling these simulations and observations is that, the Joy's law tilts are generated during the rise of the toroidal flux tubes [18] and thus are already embedded apriori in the emerging BMRs, which overcomes the tendency to generate the anti-Joy's law tilts.",
"In § , we will demonstrate that this is possible.",
"Another possibility is that the BMRs emerge with nearly zero tilt but acquire the positive tilts after the emergence by yet-unknown physical process.",
"For example, [53] proposed that the net positive tilt can be generated from the initial zero-tilt state due to the coupled effects of differential rotation and active region inflows.",
"This effect is not considered in this study since our code does not include the effect of radiative cooling in the BMRs, which geostrophically drives the active region inflows [68]."
],
[
"Morphological asymmetry",
"We also find a significant asymmetry between the leading and following spots: The leading polarity region tends to retain its compact shape and its strong field strength, whereas the following spot tends to gradually expand and the field becomes substantially weaker as time passes.",
"To qualitatively assess this field strength asymmetry, we measure the maximum field strengths in the leading and following polarity regions $B_{L}$ and $B_{F}$ .",
"In our simulation Case 1, the leading spot has about twice stronger field than the following spot, $|B_{L}/B_{F}|\\approx 2$ at $t=6.2$ days.",
"This morphological asymmetry of the BMRs has been a well-known feature observed on the Sun [7], [24], and often explained by the differential stretching of the rising $\\Omega $ -loop due to the Coriolis force [22].",
"See [20] for a more comprehensive review on the observational and theoretical studies on the morphological asymmetry.",
"Here, we provide a different but related explanation for this asymmetry.",
"As illustrated in Fig.",
"REF b, the two spots are attracted with each other by the Lorentz force, and the Coriolis force acting on these longitudinal converging flows drives an downflow (upflow) inside the leading (following) polarity regions of the flux tube.",
"This can be confirmed by the contour map of the radial motion in Fig.",
"REF c (middle).",
"Owing to the mass conservation, these downflow and upflow are accompanied by the horizontal converging and diverging motions, respectively.",
"Therefore, the leading polarity region becomes compressed and the field gets stronger, whereas the following one become broader and the field gets weaker.",
"This is schematically illustrated in Fig.",
"REF c. Figure: Temporal evolution of (a) the tilt angle γ\\gamma and (b) the ratio of the maximum field strengths between the leading and following magnetic regions |B L /B F ||B_{\\mathrm {L}}/B_{\\mathrm {F}}|.The Red, green, and blue curves correspond to the Cases 1, 2, and 3."
],
[
"Dependence on the subsurface structure of a BMR",
"The results from the Case 2 and Case 3 are shown in Fig.",
"REF and Fig.",
"REF , respectively.",
"We can clearly see that the temporal evolution of a BMR is sensitively dependent upon the initial field structure of the BMR.",
"From Case 1 (where the BMR is localized in a shallow surface and the two spots are largely separated in longitude) to Case 3 (where the BMR extends deeper in the convection zone and the longitudinal separation of the spots is small), the dominant component of the Lorentz force is changed from horizontal (longitudinally converging at the surface) to vertical (upward at the bottom apex of the flux tube).",
"Consequently, the generation of a negative tilt is significantly suppressed from Case 1 to Case 3, as shown in Fig.",
"REF a.",
"On the other hand, the field strength asymmetry between the leading and following spots can be found in all cases.",
"This is because, in the Case 3, the Lorentz-force-driven rising motion of the bottom apex of the flux tube drives the counter-rotating flow inside the flux tube, which can enhance the converging (diverging) motion in the leading (following) polarity regions at the surface.",
"Therefore, regardless of the subsurface shape of the BMR, the observed morphological asymmetry can be reproduced.",
"In fact, the asymmetry is most significant in Case 2 where the Coriolis force can act both on the longitudinally converging motions at the surface and on the radially upward motion of the deep flux tube.",
"This is schematically illustrated in Figs.",
"REF b and c. Although the simulations reported in this section are based on a very simplified model of the half-torus-shaped BMR with zero initial tilt, we find that its temporal evolution is extremely sensitive to its subsurface structure in the 3D non-kinematic regime.",
"In particular, this study warns that the shallow BMRs model (which is conventionally used in many 2D non-kinematic models and 3D kinematic models) will lead to drastically different dynamo results when all these realistic 3D non-kinematic effects are included, especially due to the tendency to produce the anti-Joy's law tilts."
],
[
"Cyclic dynamo with Babcock-Leighton $\\alpha $ -effect",
"In the previous section, we see that the newly-emerged BMRs with shallow subsurface root have a general tendency to produce the anti-Joy's law tilts in the 3D non-kinematic regime.",
"In this section, we demonstrate that, despite this trend, the cyclic dynamo solution can be obtained if the Joy's law tilt is explicitly imposed for the emerging BMRs."
],
[
"Babcock-Leighton $\\alpha $ -effect source term",
"Now, we switch on the electro-motive-force ${\\mbox{$\\mathcal {E}$}}$ in the Eq.",
"() that represents the source of the Babcock-Leighton $\\alpha $ -effect, by which the surface poloidal field is produced as a result of the north-south tilt of the BMRs [1], [50].",
"In our model, the emergence of BMRs at the surface is assumed to occur in response to the dynamo-generated toroidal field deep inside the convection zone, i.e., the tilted BMRs are instantaneously generated when the toroidal field near the base exceeds a threshold field strength.",
"Our approach differs from the method presented in [76] and [49] where the upward velocity associated with the magnetic buoyancy of the toroidal flux is prescribed.",
"Rather, our method is similar to that used in [55] and [56] where the BMRs are explicitly spotted at the surface.",
"We take the following steps to construct ${\\mbox{$\\mathcal {E}$}}$ .",
"First, the toroidal field near the base of the convection zone is computed at every time step, $&& \\bar{B}_{\\mathrm {tor}}(\\theta ,\\phi )=\\frac{1}{r_{b}-r_{a}}\\int _{r_{a}}^{r_{b}} B_{\\phi }(r,\\theta ,\\phi ) dr,$ where the average is taken over a narrow radial range near the base of the convection zone from $r_{a}=0.71R_{\\odot }$ to $r_{b}=0.735R_{\\odot }$ .",
"Then, we determine the location of the flux emergence in a spherical surface $(\\theta ^{*},\\phi ^{*})$ .",
"In order to suppress the emergence at high latitudes as suggested by observations, we apply a latitudinal mask to $ \\bar{B}_{\\mathrm {tor}}(\\theta ,\\phi )$ such that $&& \\bar{B}^{*}_{\\mathrm {tor}}(\\theta ,\\phi )=\\mathcal {T}_{+}(\\theta ;\\pi /2-\\theta _{\\mathrm {em}},\\Delta \\theta _{\\mathrm {tran}})\\times \\nonumber \\\\&& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathcal {T}_{-}(\\theta ;\\pi /2+\\theta _{\\mathrm {em}},\\Delta \\theta _{\\mathrm {tran}}) \\bar{B}_{\\mathrm {tor}}(\\theta ,\\phi ),$ where $\\theta _{\\mathrm {em}}=17.5^{\\circ }$ and $\\Delta \\theta _{\\mathrm {tran}}=8.5^{\\circ }$ .",
"We impose a necessary condition for the BMRs emergence to occur, that $|\\bar{B}^{*}_{\\mathrm {tor}}(\\theta ,\\phi )|$ exceeds a threshold field strength $B_{\\mathrm {crit}}=750$ G. When the above condition is satisfied on multiple points, the location of emergence $(\\theta ^{*},\\phi ^{*})$ is randomly chosen between points satisfying the criterion.",
"Eventually, ${\\mbox{$\\mathcal {E}$}}$ is expressed as follows being proportional to $\\bar{B}^{*}_{\\mathrm {tor}}(\\theta ^{*},\\phi ^{*})$ , $&& \\left(\\begin{array}{c}\\mathcal {E}_{r} \\\\\\mathcal {E}_{\\theta } \\\\\\mathcal {E}_{\\phi }\\end{array}\\right)=a_{0} \\tilde{f}^{*}_{\\alpha } (r,\\theta ,\\phi ) \\left(\\begin{array}{c}0 \\\\-\\cos {\\psi ^{*}} \\\\\\sin {\\psi ^{*}}\\end{array}\\right) \\bar{B}_{\\mathrm {tor}}^{*}(\\theta ^{*},\\phi ^{*}), $ where $\\tilde{f}_{\\alpha }$ represents the spatial distribution of BMRs, $\\tilde{f}^{*}_{\\alpha }(r,\\theta ,\\phi ) = \\exp {\\left[ -\\left( \\frac{r-r_{\\mathrm {max}}}{\\Delta r_{\\mathrm {bmr}}}\\right)^2-\\left( \\frac{\\theta -\\theta ^{*}}{\\Delta \\theta _{\\mathrm {bmr}}}\\right)^2 -\\left( \\frac{\\phi -\\phi ^{*}}{\\Delta \\phi _{\\mathrm {bmr}}}\\right)^2 \\right]}.$ Here, $\\Delta r_{\\mathrm {bmr}}$ , $\\Delta \\theta _{\\mathrm {bmr}}$ , and $\\Delta \\phi _{\\mathrm {bmr}}$ denote the radial, latitudinal, and longitudinal size of the BMRs.",
"In order to demonstrate that the cyclic dynamo is possible even with the presence of strong non-kinematic effects (discussed in § ), we set $\\Delta r_{\\mathrm {bmr}}=0.04R_{\\odot }$ so that the BMRs are confined in the shallow surface layer.",
"In the reference calculation, we set $\\Delta \\theta _{\\mathrm {bmr}}=\\Delta \\phi _{\\mathrm {bmr}}=6^{\\circ }$ , which is consistent with observations suggesting the typical size of BMRs of $r_{\\mathrm {bmr}}\\approx 5-100$ Mm [67] that implies $\\Delta \\theta _{\\mathrm {bmr}}=\\Delta \\phi _{\\mathrm {bmr}}\\approx 2r_{\\mathrm {bmr}}/R_{\\odot }\\approx 0.4-8^{\\circ }$ .",
"The overall amplitude of the Babcock-Leighton $\\alpha $ -effect is set to $a_{0}=100$ km s$^{-1}$ .",
"This value, in combination with the typical toroidal field strength near the base $ \\bar{B}_{\\mathrm {tor}}^{*}\\approx 5-20$ kG [17], leads to the total magnetic flux of BMRs of $10^{22}-10^{23}$ Mx, which is consistent with observations [64].",
"Figure: Example structure of BMRs per each hemisphere produced from our Babcock-Leighton α\\alpha -effect model.Radial field at the solar surface is shown where red (blue) points represent positive (negative) B r B_{r}.The Solid black arrows denote the direction of the electro-motive-force ℰ\\mathcal {E} defined by the Eq.",
"() with appropriate Joy's law.Positive (negative) toroidal field in the northern (southern) hemisphere is implicitly assumed near the base of the convection zone.The north-south tilt of the BMRs ($\\psi ^{*}$ ) obeys the Joy's law such that $&& \\psi ^{*}=35^{\\circ }\\cos {\\theta ^{*}}+\\psi ^{\\prime }_{\\mathrm {f}},$ where $\\psi ^{\\prime }_{\\mathrm {f}}$ denotes the random fluctuation of the tilt angle [30], [37], [69], [72].",
"For simplicity, we assume that the probability distribution of $\\psi ^{\\prime }_{\\mathrm {f}}$ is roughly given by the following Gaussian distribution, $&& P_{\\mathrm {f}}(\\psi ^{\\prime }_{\\mathrm {f}})=\\frac{1}{\\sigma _{\\mathrm {f}}\\sqrt{2\\pi }}\\exp {\\left[ -\\psi _{\\mathrm {f}}^{\\prime 2}/(2\\sigma _{\\mathrm {f}}^{2}) \\right]},$ with $\\sigma _{\\mathrm {f}}=15^{\\circ }$ .",
"Unlike the kinematic model of [45], a quenching term is not necessary in our model because the saturation of the dynamo occurs self-consistently owing to the Lorentz-force feedback [62], [39].",
"Figure REF shows examples of ${\\mbox{$\\mathcal {E}$}}$ and the resulting tilted BMRs produced by our Babcock-Leighton $\\alpha $ -effect source where we assume sufficiently strong positive (negative) toroidal field near the base of the convection zone.",
"Figure: Cumulative log-normal distribution of the emergence events 𝒞 em (Δ t )\\mathcal {C}_{\\mathrm {em}}(\\Delta _{t}) used in our model during the activity maxima (red) and activity minima (blue).In order to prevent overlapping emergence events on the same location in a very short time span, we introduce the following time delay algorithm as presented in [55], [56], [45]: We use a cumulative log-normal distribution function of the emergence events defined as $&& \\mathcal {C}_{\\mathrm {em}}(\\Delta _{t})=\\int _{t=t_{s}}^{t_{s}+\\Delta _{t}} P_{\\mathrm {em}}(\\Delta _{t}) \\ dt, \\\\&& P_{\\mathrm {em}}(\\Delta _{t})=\\frac{1}{\\sigma _{t}\\Delta _{t}\\sqrt{2\\pi } }\\exp {\\left[ -\\frac{(\\ln {\\Delta _{t}}-\\mu _{t})^{2}}{2\\sigma _{t}^{2}}\\right]},$ where $\\Delta _{t}=t-t_{s}$ is the time lag since the last emergence event at $t_{s}$ .",
"A flux emergence event is allowed only when the cumulative $\\mathcal {C}_{\\mathrm {em}}$ exceeds a number $\\in [0,1]$ randomly chosen at every time step.",
"The standard deviation $\\sigma _{t}$ and the mean $\\mu _{t}$ are specified by $\\tau _{p}$ and $\\tau _{s}$ as follows.",
"$&& \\sigma _{t}^{2}=\\frac{2}{3}\\ln {\\left( \\frac{\\tau _{s}}{\\tau _{p}}\\right)}, \\ \\ \\ \\mu _{t}=\\sigma _{t}^{2}+\\ln {\\tau _{p}}, \\\\&& \\tau _{p}=\\tau _{p,0}+ \\Delta \\tau _{p} \\ e^{-(\\bar{B}^{*}_{\\mathrm {em}}/B_{\\tau })^{2}}, \\\\&& \\tau _{s}=\\tau _{s,0}+ \\Delta \\tau _{s} \\ e^{-(\\bar{B}^{*}_{\\mathrm {em}}/B_{\\tau })^{2}}.$ Here, we set $\\tau _{p,0}=0.8$ days, $\\tau _{s,0}=1.9$ days, $\\Delta \\tau _{p}=0.75$ days, $\\Delta \\tau _{s}=3.0$ days.",
"The quantity $\\bar{B}^{*}_{\\mathrm {em}}$ represents the horizontally-averaged $\\bar{B}^{*}_{\\mathrm {tor}}$ and $B_{\\tau }=1.5$ kG denotes the threshold value of the toroidal field strength for characterizing phase in the activity cycle.",
"Figure REF shows the two examples of the cumulative $\\mathcal {C}_{\\mathrm {em}}(\\Delta _{t})$ each corresponding to the solar activity minima and maxima.",
"Therefore, the flux emergence becomes more frequent during the activity maxima ($\\bar{B}^{*}_{\\mathrm {em}} > B_{\\tau }$ ) and less frequent when during the activity minima ($\\bar{B}^{*}_{\\mathrm {em}} < B_{\\tau }$ ).",
"We must note that our Babcock-Leighton $\\alpha $ -effect model is strongly spatially-localized and temporally intermittent.",
"This is clearly different from the conventional 2D models with spatially-distributed and temporally-continuous source term [15], [17], [62].",
"Taking into account the tilt angle inclination of ${\\mbox{$\\mathcal {E}$}}$ , the localization in longitudes, and the emergence frequency of the BMRs, we can estimate the corresponding $\\alpha _{0}$ value within the 2D mean-field framework as $\\alpha _{0} &\\approx & \\sin {\\psi }\\left(\\frac{\\Delta \\phi _{\\mathrm {bmr}}}{2\\pi } \\right) \\left( \\frac{\\Delta t_{\\mathrm {CFL}}}{\\Delta _{t}} \\right) a_{0} \\nonumber \\\\&\\approx & 1.2 \\ \\mathrm {m}\\ \\mathrm {s}^{-1},$ where we use the typical values $\\psi =17.5^{\\circ }$ , $\\Delta _{t}=5$ days, and $\\Delta t_{\\mathrm {CFL}}=17$ min.",
"This $\\alpha _{0}$ value is consistent with the previous 2D mean-field models.",
"Figure: Temporal evolution of the longitudinally-averaged magnetic fields and horizontal velocities.",
"(a) Azimuthal mean of the radial field 〈B r 〉\\langle {B}_{r} \\rangle at the surface r=0.985R ⊙ r=0.985R_{\\odot } where the bar denotes the azimuthal mean.",
"(b) Azimuthal mean of the longitudinal field 〈B φ 〉\\langle {B}_{\\phi } \\rangle near the base of the convection zone r=0.715R ⊙ r=0.715R_{\\odot }.Black solid lines are the contours of the emerged BMRs at each time.",
"(c) Torsional oscillation pattern δ〈Ω〉=〈Ω〉-〈Ω〉 t \\delta \\langle \\Omega \\rangle = \\langle \\Omega \\rangle -\\langle \\Omega \\rangle _{t} at the surface where 〈〉 t \\langle \\rangle _{t} denotes the azimuthal and temporal average.",
"(d) Azimuthal mean of the latitudinal velocity 〈v θ 〉\\langle {v}_{\\theta } \\rangle at the surface.Red (blue) in the northern hemisphere represents the equatorward (poleward) flow.",
"(e) The same as (d) but near the base of the convection zone.Black dashed lines denote the contours of the toroidal field at the base (8.5 kG).",
"(f) Entropy perturbation δ〈s 1 〉=〈s 1 〉-〈s 1 〉 t \\delta \\langle s_{1} \\rangle = \\langle s_{1} \\rangle -\\langle s_{1}\\rangle _{t} at the surface.Figure: Temporal evolution of the volume-integrated kinetic and magnetic energies.The red, green, and black lines denote the kinetic energies of the differential rotation KE DR \\mathrm {KE}_{\\mathrm {DR}}, the meridional circulation KE MC \\mathrm {KE}_{\\mathrm {MC}}, and the non-axisymmetric flows KE m≠0 \\mathrm {KE}_{m \\ne 0}.The blue, purple, and orange lines denote the magnetic energies of the mean toroidal field KE tor \\mathrm {KE}_{\\mathrm {tor}}, the mean poloidal field ME pol \\mathrm {ME}_{\\mathrm {pol}}, and the non-axisymmetric fields ME m≠0 \\mathrm {ME}_{m \\ne 0}."
],
[
"Initial condition of magnetic fields",
"We add an axisymmetric dipolar field into the fully-developed hydrodynamic calculation shown in Fig.",
"REF .",
"The simulation is evolved until initial transients disappear and the dynamo cycles with quasi-steady amplitudes are obtained.",
"In the following subsections, we analyze the last three cycles of our simulation."
],
[
"Dynamo cycles and the Lorentz force feedback",
"Figures REF a and b show the tim-latitude plots of the longitudinally-averaged radial field $\\langle {B}_{r} \\rangle $ at the surface and the toroidal field $\\langle {B}_{\\phi } \\rangle $ near the base of the convection zone, represented in terms of the well-known magnetic butterfly diagram.",
"We can clearly see the cyclic polarity reversals that occur roughly at every 9 years, which is slightly shorter than the solar cycle yet comparable.",
"In each cycle, there is an equatorward migration of sunspot groups (BMRs) and the build-up of the polar field by poleward advection of the magnetic fluxes associated with the trailing sunspots.",
"These are owing to the single-cell meridional circulation achieved in our model, which has an amplitude of about 15 m s$^{-1}$ at the surface and 2 m s$^{-1}$ near the base of the convection zone.",
"The black solid lines in Fig.",
"REF b denote the range of the emergence latitudes of BMRs at each time (the so-called active region belt).",
"The phase of the equatorward advection of the toroidal field at the base corresponds to that of the emergence of the BMRs at the surface.",
"In our non-kinematic model, the dynamo-generated fields have strong impacts on flows via the Lorentz force feedback.",
"Figure REF c shows the time-latitude plot of the fluctuation of the differential rotation $\\delta \\langle \\Omega \\rangle = \\langle \\Omega \\rangle -\\langle \\Omega \\rangle _{t}$ where $\\langle \\ \\rangle _{t}$ denotes the longitudinal and temporal average.",
"This is commonly known as torsional oscillations [36].",
"We clearly find both poleward and equatorward propagating oscillation patterns with the typical amplitude of about 5 nHz at the surface.",
"Figures REF d and e show the time-latitude plots of the latitudinal velocity $\\langle v_{\\theta }\\rangle $ at the top and bottom of the convection zone, respectively.",
"Although the poleward flow at the surface and the equatorward flow near the base are strongly suppressed and disturbed during the activity maxima, the feedback is not large enough to switch off the advective transport of the magnetic fields [39].",
"Figure REF f shows the time-latitude plot of the entropy perturbation $\\delta \\langle s_{1} \\rangle = \\langle s_{1} \\rangle -\\langle s_{1}\\rangle _{t}$ at the surface with typical variation amplitude of about 250 erg g$^{-1}$ K$^{-1}$ which corresponds to the temperature fluctuation of about $1.4$ K. The positive entropy fluctuation can be seen along with the active region belt, implying that the surface is heated whenever the BMRs emerge due to the strong magnetic diffusion in our model.",
"Note, however, that this is not likely in the real Sun: The surface is expected to be cooled by enhanced radiation in the BMRs, leading to lower temperature due to the radiative loss in the active region belt [68].",
"The theories suggest that, this radiative loss at the surface can produce the low-latitude branches of the torsional oscillation by inducing the geostrophical flows around the BMRs (thermal forcing) [62], [29].",
"This effect is not included in our model.",
"It should be emphasized that, in our simulation reported here, the artificial heating in the active region belt may be responsible for the low-latitude torsional oscillation branches due to the thermal forcing with the opposite sign.",
"Figure REF shows the volume-integrated kinetic and magnetic energies of various components.",
"Their definitions are $&& \\mathrm {KE}_{\\mathrm {DR}}=\\int _{V} \\frac{\\rho _{0}}{2} \\langle v_{\\phi } \\rangle ^{2} \\ dV, \\\\&& \\mathrm {KE}_{\\mathrm {MC}}=\\int _{V} \\frac{\\rho _{0}}{2} (\\langle v_{r} \\rangle ^{2}+\\langle v_{\\theta } \\rangle ^{2}) \\ dV, \\\\&& \\mathrm {KE}_{m \\ne 0}=\\int _{V} \\frac{\\rho _{0}}{2} ({\\mbox{$v$}}-\\langle {\\mbox{$v$}} \\rangle )^{2} \\ dV, \\\\&& \\mathrm {ME}_{\\mathrm {tor}}=\\int _{V} \\frac{1}{8\\pi } \\langle B_{\\phi } \\rangle ^{2} \\ dV, \\\\&& \\mathrm {ME}_{\\mathrm {pol}}=\\int _{V} \\frac{1}{8\\pi } (\\langle B_{r} \\rangle ^{2}+\\langle B_{\\theta } \\rangle ^{2}) \\ dV, \\\\&& \\mathrm {ME}_{m \\ne 0}=\\int _{V} \\frac{1}{8\\pi } ({\\mbox{$B$}}-\\langle {\\mbox{$B$}} \\rangle )^{2} \\ dV,$ where the integrals are taken over the whole volume of the convection zone.",
"The two largest energy reservoirs in our simulation are the differential rotation kinetic energy $\\mathrm {KE}_{\\mathrm {DR}}$ and the toroidal field magnetic energy $\\mathrm {ME}_{\\mathrm {tor}}$ .",
"When the toroidal field is amplified by the $\\Omega $ -effect, $\\mathrm {KE}_{\\mathrm {DR}}$ is converted to $\\mathrm {ME}_{\\mathrm {tor}}$ .",
"The toroidal field eventually becomes superequipartition ($\\mathrm {ME}_{\\mathrm {tor}}>\\mathrm {KE}_{\\mathrm {DR}}$ ) with respect to the differential rotation on average.",
"In our simulation, this Lorentz force feedback on differential rotation leads to a dynamo saturation.",
"The non-axisymmetric magnetic energy $\\mathrm {ME}_{m \\ne 0}$ is greater than the mean poloidal field energy $\\mathrm {ME}_{\\mathrm {pol}}$ because the BMRs are strongly non-axisymmetric.",
"The non-axisymmetric fields drive strong non-axisymmetric flows, whose kinetic energy $\\mathrm {KE}_{m \\ne 0}$ is also greater than $\\mathrm {ME}_{\\mathrm {pol}}$ .",
"This suggests that the non-axisymmetric components of the magnetic fields and the flows are important for the convection zone dynamics.",
"Figure: Time evolution of magnetic field and velocity.Shown are the snapshots at t=12.9t=12.9 yr (from (a) to (e)), t=14.9t=14.9 yr (from (f) to (j)), t=17.9t=17.9 yr (from (k) to (o)), t=19.9t=19.9 yr (from (p) to (t)) in Fig.",
"!.The mollweide projections on the 1st and 2nd columns show the radial field B r B_{r} at the surface r=0.985R ⊙ r=0.985R_{\\odot } and longitudinal field B φ B_{\\phi } near the base of the convection zone r=0.715R ⊙ r=0.715R_{\\odot }, respectively.The meridional plot in the 3rd column represents the azimuthally-mean toroidal field (color scales) and poloidal field (contours).The meridional plots in the 4th and 5th columns represent the azimuthally-mean differential rotation and streamfunction of the meridional circulation, respectively.An animation of this figure is available online.Figure: Snapshots of the radial field B r B_{r} at the surface (top panels) and the radial velocity v r v_{r} [m s -1 ^{-1}] near the surface (bottom panels) at t=10.3t=10.3 yr in Fig.",
".The black arrows represent the horizontal flow (v θ ,v φ v_{\\theta },v_{\\phi }) at the surface.Panels (c) and (d) are the zoom-in of the panels (a) and (b), focusing on the single BMRs denoted by red thick solid lines.Figure: Equatorial power spectrum of latitudinal velocity v θ v_{\\theta } near the surface r=0.95R ⊙ r=0.95R_{\\odot }.The spectra are computed in a frame rotating at Ω 0 /2π=431.3\\Omega _{\\mathrm {0}}/2\\pi =431.3 nHz.The blue solid lines represent the differential rotation rate at the surface ω=m(Ω sf -Ω 0 )\\omega =m (\\Omega _{\\mathrm {sf}}-\\Omega _{0} ) where Ω sf =〈Ω(0.95R ⊙ ,π/2)〉\\Omega _{\\mathrm {sf}}=\\langle \\Omega (0.95R_{\\odot },\\pi /2) \\rangle .The red points denote the theoretical dispersion relation of the sectoral Rossby modes, ω=-2Ω sf /(m+1)+m(Ω sf -Ω 0 )\\omega =-2\\Omega _{\\mathrm {sf}}/(m+1)+m ( \\Omega _{\\mathrm {sf}}-\\Omega _{0}).Figure REF shows snapshots of the magnetic fields and mean flows over the course of a magnetic cycle.",
"The two leftmost panels show the mollweide projections of the radial magnetic field $B_{r}$ at the surface and the toroidal field $B_{\\phi }$ at the bottom convection zone.",
"As prescribed in our Babcock-Leighton source term, BMRs emerge at low latitudes obeying the Hale's and Joy's laws.",
"Therefore, the radial magnetic field at the surface is substantially non-axisymmetric.",
"On the other hand, toroidal field near the base of the convection zone is found to be almost axisymmetric.",
"Meridional plots on the 3rd, 4th, and 5th columns of Fig.",
"REF show the azimuthally-mean profiles of the poloidal and toroidal magnetic fields, differential rotation, and meridional circulation, respectively.",
"When longitudinally averaged, the dynamo solution shows a qualitatively similar time evolution pattern as the previous 2D mean-field models [62], [39], although our model has a stronger torsional oscillation and the meridional circulation modulations which presumably depends on the radial strucuture we have assumed for flux emergence."
],
[
"Non-axisymmetric flows",
"In our model, non-axisymmetric flows are driven largely by the non-axisymmetric Lorentz forces and only partially by the random fluctuations in the $\\Lambda $ -effect.",
"Figure REF shows a snapshot of the radial field at the surface (top rows) and the radial velocity near the surface (bottom rows).",
"Black arrows represent the horizontal flow motions at the surface.",
"Strong horizontal flows exist only in the vicinity of the BMRs: When a BMR emerge at the surface which happens instantaneously in our model, horizontal converging flows are driven towards the polarity inversion line with the typical amplitudes of about 100 m s$^{-1}$ .",
"This is owing to a strong magnetic tension force of the BMRs that pulls the two spots together.",
"This strong converging flow further drives both horizontal outflows and radial downflows along the polarity inversion line, as shown in Fig.",
"REF c and d. Due to these strong horizontal flows at the surface, a newly-emerged BMR that initially consists of two round-shaped sunspots is quickly deformed into an elongated shape along the polarity inversion line, as seen in Fig.",
"REF a.",
"This temporal evolution is similar the simulation reported in § REF with horizontally-elongated BMRs (Case 1).",
"However, it should be noted that we do not include the radiative cooling associated with the active regions in our simulations.",
"Thus, our model currently lacks the physics required to properly produce the observed inflows associated with active regions [28], [68].",
"Other interesting non-axisymmetric flow features are low-frequency inertial modes of oscillation, in particular, the equatorial Rossby modes that have recently been detected on the Sun [52].",
"Figure REF shows the equatorial power spectrum of latitudinal velocity $v_{\\theta }$ near the surface from our non-kinematic dynamo simulation similarly to [2].",
"Note that all the spectra are computed in a frame rotating at $\\Omega _{\\mathrm {ref}}/2\\pi =431.3$ nHz.",
"We can clearly see the existence of the equatorial Rossby modes as represented by a clear power ridge along the expected dispersion relations (red points) in the spectra for $3 \\le m \\le 12$ .",
"In our simulation, these Rossby modes are excited both by the non-axisymmetric random fluctuations in the $\\Lambda $ -effect and by the non-axisymmetric Lorentz-force, unlike the rotating convection simulation of [2] where they are excited by turbulent convective motions alone.",
"It is implied that our code can be used to study the magnetic cycle dependence of the Rossby modes (or inertial modes in general) in the future."
],
[
"Summary and Discussions",
"In this paper, we have developed a new Babcock-Leighton flux-transport dynamo model of the Sun.",
"In our model, we do not solve the small-scale convection and focus on the large-scale flows and magnetic structures in a full spherical shell.",
"The solar-like large-scale mean flows are driven by proper parameterization of the $\\Lambda $ -effect.",
"The model operates in a 3D non-kinematic regime, and therefore, is more realistic than the 2D non-kinematic models [62], [39] and 3D kinematic models [76], [55].",
"To better illustrate the major differences from the conventional 2D non-kinematic models and the 3D kinematic models, we first carry out a set of simulations for a single BMR with different initial subsurface structure.",
"We find that, when the BMR has a shallow subsurface structure and a large longitudinal separation, the post-emergence evolution of the BMR becomes significantly changed from those from the conventional models: Even if the initial BMR is perfectly east-west aligned (zero tilt angle), it begins to acquire a negative tilt angle (which is opposite to the Joy's law).",
"The strength of the negative tilt angle decreases as the model bipole is embedded deeper in the solar convection zone.",
"Furthermore, we find a strong asymmetry in the field strengths between the leading and following polarity regions.",
"The leading polarity field becomes stronger whereas that of the following spot becomes weaker, which is similar to the observations.",
"These results can be explained by the Coriolis force acting on the flows driven by the Lorentz force of the BMR (see Fig.",
"REF ).",
"We also carry out the cyclic solar dynamo simulation using the source term of the Babcock-Leighton $\\alpha $ -effect which is implemented in a 3D manner where the Joy's law tilts are explicitly given.",
"We have successfully demonstrated that many observational features are reproduced in our model such as the activity cycles with decadal periods, the equatorward migration of the sunspot groups (BMRs), and the poleward transport of the surface radial fields.",
"The nonlinear saturation of the dynamo occurs due to a strong Lorentz force feedback: The magnetic energy of the toroidal field amplified by the $\\Omega $ -effect is found to exceed the kinetic energy of the differential rotation.",
"This strong Lorentz force feedback can be seen in the cyclic modulations of the differential rotation (torsional oscillations) and the meridional circulation.",
"Note, however, that our study does not exclude other nonlinear dynamo saturation mechanisms such as variability in the Babcock-Leighton process [74], [45] and magnetic quenching of the turbulent transport processes [48], [11], [77].",
"Since our model is highly sensitive to various model parameters, there are still several disagreements with the solar observations such as a slightly shorter cycle period of 9 year, stronger radial field strengths at the surface of typical amplitudes of about $200-300$ G, and slightly larger torsional oscillations.",
"Obviously, the model parameters associated with the subsurface structure of the newly-emerged BMRs will be highly influential, as expected from the discussion in § .",
"The other important parameters would be $\\Lambda _{0}$ that determines the amplitudes of the differential rotation and meridional circulation, and $a_{0}$ that determines the field strengths of BMRs at the surface.",
"A detailed parameter study is required in the future.",
"It is noteworthy that our code is applicable to examine the impact of magnetic fields on various kinds of inertial modes which we found to exist in our simulation (see Fig.",
"REF ).",
"Recent observations suggest that the amplitudes and frequencies of some of the solar equatorial Rossby modes exhibit a cycle dependence [51].",
"If we properly understand the effects of deep-seated magnetic fields on the mode frequencies and eigenfunctions of the equatorial Rossby modes, observations could potentially be used to infer the location and strength of the magnetic fields hidden in the Sun.",
"An important physical ingredient still missing in the present model is the enhanced radiative cooling associated with the BMRs, which will affect both the short-term post-emergence evolution of the BMRs and the long-term cyclic dynamo behaviors.",
"This radiative loss will substantially affect the surface horizontal motions by geostrophycally inducing inflows around the active regions [28], [29].",
"These active region inflows are expected to affect the tilt angle [53], regulate the poleward transport of the poloidal fluxes and limit the buildup of the polar fields [41], and thus affect the cycle amplitudes in the Babcock-Leighton solar dynamo [10].",
"Furthermore, it is often argued that the low-latitude branches of the torsional oscillation are attributed to the thermally-induced flows due to the enhanced surface cooling of the BMRs (thermal forcing) [68], [62], [63].",
"In the future model, we plan to include this effect to study how the post-emergence of the BMRs and the nonlinear saturation of the dynamo change in the 3D non-kinematic regime.",
"We thank B. Karak for helpful comments on the manuscript.",
"Y.",
"B. was enrolled in the International Max-Planck Research School for Solar System Science at the University of Göttingen (IMPRS).",
"Y .B.",
"also acknowledges a support from a long-term scholarship program for degree-seeking graduate students abroad from the Japan Student Services Organization (JASSO).",
"We acknowledge a support from ERC Synergy Grant WHOLE SUN 810218.",
"All the numerical computations were performed at the Max-Planck supercomputer RZG in Garching."
]
] | 2209.08178 |
[
[
"PPT: token-Pruned Pose Transformer for monocular and multi-view human\n pose estimation"
],
[
"Abstract Recently, the vision transformer and its variants have played an increasingly important role in both monocular and multi-view human pose estimation.",
"Considering image patches as tokens, transformers can model the global dependencies within the entire image or across images from other views.",
"However, global attention is computationally expensive.",
"As a consequence, it is difficult to scale up these transformer-based methods to high-resolution features and many views.",
"In this paper, we propose the token-Pruned Pose Transformer (PPT) for 2D human pose estimation, which can locate a rough human mask and performs self-attention only within selected tokens.",
"Furthermore, we extend our PPT to multi-view human pose estimation.",
"Built upon PPT, we propose a new cross-view fusion strategy, called human area fusion, which considers all human foreground pixels as corresponding candidates.",
"Experimental results on COCO and MPII demonstrate that our PPT can match the accuracy of previous pose transformer methods while reducing the computation.",
"Moreover, experiments on Human 3.6M and Ski-Pose demonstrate that our Multi-view PPT can efficiently fuse cues from multiple views and achieve new state-of-the-art results."
],
[
"Introduction",
"Human pose estimation aims to localize anatomical keypoints from images.",
"It serves as a foundation for many down-stream tasks such as AR/VR, action recognition [21], [65], and medical diagnosis [11].",
"Over the past decades, deep convolutional neural networks (CNNs) play a dominant role in human pose estimation tasks [53], [62], [40], [63], [50], [59], [61].",
"However, cases including occlusions and oblique viewing are still too difficult to be solved from a monocular image.",
"To this end, some works apply a multi-camera setup [48], [60], [22], [6] to boost the performance of 2D pose detection[43], [19], since difficult cases in one view are potentially easier to be resolved in other views.",
"Meanwhile, human body joints are highly correlated, constrained by strong kinetic and physical constraints [52].",
"However,since the reception fields of CNNs are limited, the long-range constraints among joints are often poorly captured [31].",
"Figure: Different types of cross-view fusion.",
"The first row is the current view, and the second row is the reference view.Recently, the ViT [14] demonstrates that the transformers [55] can achieve impressive performance on many vision tasks [54], [2].",
"Compared with CNN, the self-attention module of transformers can easily model the global dependencies among all visual elements.",
"In the field of pose estimation, many tansformer-based works [31], [67], [37], [33], [74] suggest that the global attention is necessary.",
"In single-view 2D human pose estimation, TransPose [67] and TokenPose [31] achieve new state-of-the-art performance and learn the relationship among keypoints with transformers.",
"In multi-view human pose estimation, the TransFusion [36] uses the transformer to fuse cues from both current and reference views.",
"Typically, these works flatten the feature maps into 1D token sequences, which are then fed into the transformer.",
"In multi-view settings, tokens from all views are usually concatenated together to yield a long sequence.",
"However, the dense global attention of transformers is computationally extensive.",
"As a result, it is challenging to scale up these methods to high-resolution feature maps and many views.",
"For example, the TransFusion [36] can only compute global attention between two views due to the large memory cost.",
"Meanwhile, as empirically shown in Fig.REF , the attention map of keypoints is very sparse, which only focuses on the body or the joint area.",
"This is because the constraints among human keypoints tend to be adjacent and symmetric [31].",
"This observation also suggests that the dense attention among all locations in the image is relatively extravagant.",
"In this paper, we propose a compromised and yet efficient alternative to the global attention in pose estimation, named token-Pruned Pose Transformer (PPT).",
"We calculate attention only within the human body area, rather than over the entire input image.",
"Specifically, we select human body tokens and prune background tokens with the help of attention maps.",
"As the human body only takes a small area of the entire image, the majority of input tokens can be pruned.",
"We reveal that pruning these less informative tokens does not hurt the pose estimation accuracy, but can accelerate the entire networks.",
"Interestingly, as a by-product, PPT can also predict a rough human mask without the guidance of ground truth mask annotations.",
"Moreover, we extend PPT to multi-view settings.",
"As in Fig.REF , previous cross-view fusion methods consider all pixels in the reference view (global fusion) or pixels along the epipolar line (epipolar-based fusion) as candidates.",
"The former is computationally extensive and inevitably introduces noise from the background, and the latter requires accurate calibration and lacks semantic information.",
"Built upon PPT, we propose a new fusion strategy, called human area fusion, which considers human foreground pixels as corresponding candidates.",
"Specifically, we firstly use PPT to locate the human body tokens on each view, and then perform the multi-view fusion among these selected tokens with transformers.",
"Thus, our method is an efficient fusion strategy and can easily be extended to many views.",
"Figure: Attention map for TokenPose (monocular view) and TransFusion (multi-view).",
"The attention maps are very sparse and only attend to a small local regions.Our main contributions are summarized as follows: We propose the token-Pruned Pose Transformer (PPT) for efficient 2D human pose estimation, which can locate the human body area and prune background tokens with the help of a Human Token Identification module.",
"We propose the strategy of “Human area fusion\" for multi-view pose estimation.",
"Built upon PPT, the multi-view PPT can efficiently fuse cues from human areas of multiple views.",
"Experimental results on COCO and MPII demonstrate that our PPT can maintain the pose estimation accuracy while significantly reduce the computational cost.",
"Results on Human 3.6M and Ski-Pose show that human area fusion outperforms previous fusion methods on 2D and 3D metrics.",
"Recently, the transformer [55] achieves great progresses on many computer vision tasks, such as classification [14], [54], object detection [2], [76], [15], and semantic segmentation [75], [58], [66], [68].",
"While being promising in accuracy, the vanilla ViT [14] is cumbersome and computationally intensive.",
"Therefore, many algorithms have been proposed to improve the efficiency of vision transformers.",
"Recent works demonstrate that some popular model compression methods such as network pruning [17], [7], [8], [70], knowledge distillation [20], [54], [9], and quantization [46], [51] can be applied to ViTs.",
"Besides, other methods introduce CNN properties such as hierarchy and locality into the transformers to alleviate the burden of computing global attention [35], [5].",
"On the other hand, some works accelerate the model by slimming the input tokens [71], [3], [45], [44], [29], [32], [38].",
"Specifically, the Token-to-tokens [71] aims to reduce the number of tokens by aggregating neighboring tokens into one token.",
"The TokenLearner [45] mines important tokens by learnable attention weights conditioned on the input feature.",
"The DynamicViT [44] prunes less informative tokens with an extra learned token selector.",
"The EViT [32] reduces and reorganizes image tokens based on the classification token.",
"However, all these models have only been designed for classification, where the final prediction only depends on the special classification token.",
"In the past few years, many successful CNNs are proposed in 2D human pose estimation.",
"They usually capture both low-level and high-level representations [62], [12], [40], [13], [63], [50], or use the structural of skeletons to capture the spatial constraints [52], [24], [42], [26], [27], [10], [28].",
"Recently, many works introduce transformers into pose estimation tasks [67], [31], [37], [30], [33], [74].",
"Specifically, TransPose [67] utilizes transformers to explain dependencies of keypoint predictions.",
"TokenPose [31] applies additional keypoint tokens to learn constraint relationships and appearance cues.",
"Both works demonstrate the necessity of global attention in pose estimation."
],
[
"Efficient 2D Pose Estimation",
"Some recent works also explore efficient architecture design for real-time pose estimation [41], [39], [47], [57], [72], [69].",
"For example, EfficientPose [72] designs an efficient backbone with neural architecture search.",
"Lite-HRNet [69] proposes the conditional channel weighting unit to replace the heavy shuffle blocks of HRNet.",
"However, these works all focus on CNN-based networks, and none of them study transformer-based networks."
],
[
"Multi-view Pose Estimation",
"3D pose estimation from multiple views usually takes two steps: predicting 2D joints on each view separately with a 2D pose detector, and lifting 2D joints to 3D space via triangulation.",
"Recently, many methods focus on enabling the 2D pose detector to fuse information from other views [43], [73], [64], [19], [36].",
"They can be categorized into two groups: 1) Epipolar-based fusion.",
"The features of one pixel in one view is augmented by fusing features along the corresponding epipolar line of other views.",
"Specifically, the AdaFuse [73] adds the largest response on the heatmap along the epipolar line.",
"The epipolar transformer [19] applies the non-local module [56] on intermediate features to obtain the fusion weights.",
"However, this fusion strategy requires precise camera calibration and discard information outside the epipolar lines.",
"2) Global fusion.",
"The features of one pixel in one view are augmented by fusing features of all locations in other views.",
"In detail, the Cross-view Fusion [43] learns a fixed attention matrix to fuse heatmaps in all other views.",
"The TransFusion [36] applies the transformers to fuse features of the reference views and demonstrates that global attention is necessary.",
"However, the computation complexity of global fusion is quadratic to the resolution of input images and number of views.",
"Thus, both categories have their limitations.",
"A fusion algorithm that can overcome these drawbacks and maintains their advantages is in need."
],
[
"Overview",
"Fig.REF is an overview of our token-Pruned Pose Transformer.",
"Following [31], the input RGB image $\\mathbf {I}$ first go through a shallow CNN backbone $\\mathcal {B}(\\cdot )$ to obtain the feature map $\\mathbf {F} \\in \\mathbb {R}^{ H \\times W \\times C}$ .",
"Then $\\mathbf {F}$ is decomposed into flattened image patches $\\mathbf {F}_p \\in \\mathbb {R}^{N_v \\times (C \\cdot P_h \\cdot P_w) }$ , where $(P_h, P_w)$ is the resolution of each image patch, and $ N_v = \\frac{H}{P_h} \\cdot \\frac{W}{P_w}$ is the total number of patches [14].",
"Then a linear projection is applied to project $\\mathbf {F}_p$ into $\\mathbf {X}_p \\in \\mathbb {R}^{N_v \\times D}$ , where $D$ is the dimension of hidden embeddings.",
"The 2D positional encodings $\\mathbf {E} \\in \\mathbb {R}^{N_v \\times D}$ are added to make the transformer aware of position information [55], i.e., $\\mathbf {X}_v = \\mathbf {X}_p + \\mathbf {E}$ , namely the visual token.",
"Meanwhile, following TokenPose [31], we have $J$ additional learnable keypoint tokens $\\mathbf {X}_k \\in \\mathbb {R}^{J \\times D}$ to represent $J$ target keypoints.",
"The input sequence to the transformer is $\\mathbf {X}^0 = [\\mathbf {X}_k, \\mathbf {X}_v] \\in \\mathbb {R}^{N \\times D}$ , where $N = N_v + J$ and $[\\ldots ]$ is the concatenation operation.",
"The transformer has $L$ encoder layers in total.",
"At the $L_1^{th}$ layer, the Human Token Identification (HTI) module locates $K$ most informative visual tokens where human body appears and prunes the remaining tokens.",
"We denote $r=\\frac{K}{N_v} (0<r<1)$ as the keep ratio.",
"As a result, the length of the sequence is reduced to $N^{\\prime }=rN_v+J$ for the following transformer layers.",
"The HTI is conducted $e$ times at the $L_1^{th}, L_2^{th}, \\ldots , L_e^{th}$ layers.",
"Thus, PPT can progressively reduce the length of visual tokens.",
"Finally, the total number of tokens is $r^e N_v + J$ .",
"The prediction head projects the keypoint tokens in the last layer $\\mathbf {X}_k^L \\in \\mathbb {R}^{J\\times D}$ into the output heatmaps $\\mathbf {H} \\in \\mathbb {R}^{J\\times (H_h\\cdot W_h)}$ ."
],
[
"Transformer Encoder Layer. ",
"The encoder layer consists of the multi-headed self-attention (MHSA) and multi-layer perceptron (MLP).",
"Operations in one encoder layer is shown in Fig.",
"REF .",
"The self-attention aims to match a query and a set of key-value pairs to an output [55].",
"Given the input $\\mathbf {X}$ , three linear projections are applied to transfer $\\mathbf {X}$ into three matrices of equal size, namely the query $\\mathbf {Q}$ , the key $\\mathbf {K}$ , and the value $\\mathbf {V}$ .",
"The self-attention (SA) operation is calculated by: $\\text{SA}(\\mathbf {X}) = \\text{Softmax}( \\frac{ \\mathbf {Q} \\mathbf {K}^T }{\\sqrt{D}} )\\mathbf {V},$ For MHSA, $H$ self-attention modules are applied to $\\mathbf {X}$ separately, and each of them produces an output sequence."
],
[
"Human Token Identification (HTI). ",
"The TokenPose [31] conducts self-attention among all visual tokens, which is cumbersome and inefficient.",
"From Equation REF , we know that each keypoint token $\\mathbf {X}_k^j$ interacts with all visual tokens $\\mathbf {X}_v$ via the attention mechanism: $\\text{Softmax} (\\frac{ \\mathbf {q}_k^j \\mathbf {K}_v^T }{\\sqrt{D}}) \\mathbf {V}_v = \\mathbf {a}^j \\mathbf {V}_v,$ where $ \\mathbf {q}_k^j$ denotes the query vector of $\\mathbf {X}_k^j$ , $\\mathbf {K}_v$ and $\\mathbf {V}_v$ are the keys and values of visual tokens $\\mathbf {X}_v$ .",
"To this end, each keypoint token is a linear combination of all value vectors of visual tokens.",
"The combination coefficients $\\mathbf {a}^j \\in \\mathbb {R}^{N_v}$ are the attention values from the query vector for that keypoint token with respect to all visual tokens.",
"To put it differently, the attention value determines how much information of each visual token is fused into the output.",
"Thus, it is natural to assume that the attention value $\\mathbf {a}^j$ indicates the importance of each visual token in the keypoint prediction [32].",
"Typically, a large attention value suggests that the target joint is inside or nearby the corresponded visual token.",
"With this assumption, we propose the Human Token Identification module to select informative visual tokens with the help of attention scores of keypoint tokens.",
"However, each keypoint token usually only attends to a few visual tokens around the target keypoint.",
"And some keypoint tokens (such as the eye and the nose) may attend to close-by or even the same visual tokens.",
"Thus, it is difficult to treat the attention values of each keypoint separately.",
"For simplicity, as all human keypoints make up a rough human body area, we use $\\mathbf {a} = \\sum _j \\mathbf {a}^j$ as the criterion to select visual tokens, which is the summation of all joints' attention maps.",
"In detail, we keep visual tokens with the $K$ largest corresponding values in $\\mathbf {a}$ as the human tokens, and prune the remaining tokens.",
"As a result, only $K$ visual tokens and $J$ keypoint tokens are sent to the following layers.",
"Figure: Overall framework of the Multi-view PPT.",
"A share-weight PPT is applied to extract a subset of visual tokens for each view.",
"Then BB transformer layers are applied to the concatenated tokens from each view to perform cross-view fusion.",
"The output head takes keypoint tokens in each view to predict heatmaps."
],
[
"Human Area Fusion. ",
"We propose the concept of Human area fusion for cross-view fusion in multi-view pose estimation, which considers pixels where human appears as corresponding candidates.",
"Suppose there are $m$ cameras, and each view maintains $n$ pixels (tokens) in its feature map.",
"We summarize three typical types of cross-view fusion strategies in Fig.REF .",
"1) For global fusion, each pixel in each view calculates attention with respect to all $n$ pixels in feature maps of other $m-1$ views.",
"Thus the computational complexity is $\\mathcal {O}(m^2n^2)$ .",
"2) For epipolar-based fusion, each pixel in each view calculates attention with $k (k \\ll n)$ pixels along the corresponded epipolar lines of other $m-1$ views.",
"Thus the computational complexity is $\\mathcal {O}(m^2nk)$ .",
"3) For our human area fusion, we firstly select $k^{\\prime }$ human foreground pixels in each view.",
"Then we perform dense attention among these foreground tokens.",
"As we also reduce the number of query pixels, the computational complexity is $\\mathcal {O}(m^2k^{\\prime 2})$ .",
"Typically, $k < k^{\\prime } \\ll n$ .",
"Thus, our method is an efficient way to perform cross-view fusion.",
"Moreover, it also avoids the useless or even disturbing information from the background tokens and thus makes the model focus on the constraints within the human body."
],
[
"Multi-view PPT. ",
"Naturally, we can apply an off-the-shelf segmentation network [18] to obtain human foreground pixels and then perform human area fusion.",
"However, a large amount of densely annotated images are required to train a segmentation model.",
"To this end, we utilize PPT to efficiently locate a rough human foreground area without any mask labels, and further propose the multi-view PPT for multi-view pose estimation.",
"Specifically, we design our network in a two-stage paradigm, as shown in Fig.REF .",
"Given the image $\\mathbf {I}^m$ in each view, the share-weight PPT firstly produces selected human tokens $\\mathbf { \\tilde{X}}^m_v$ and keypoint tokens $\\mathbf {X}^m_k$ .",
"Then we concatenate tokens from all views together and perform the dense attention among them with $B$ transformer encoder layers.",
"To help the network perceive the 3D space information, we also add the 3D positional encodings [36] on all selected visual tokens.",
"Thus, each keypoint token can fuse visual information from all views.",
"Moreover, it can learn correspondence constraints between keypoints both in the same view and among different views.",
"Finally, a share-weight MLP head is placed on top of the keypoint token of each view to predicts keypoint heatmaps."
],
[
"Datasets & Evaluation Metrics. ",
"We firstly evaluate PPT on monocular 2D human pose estimation benchmarks.",
"COCO [34] contains $200K$ images in the wild and $250K$ human instances with 17 keypoints.",
"Following top-down methods [63], [50], [31], we crop human instances with the ground truth bounding boxes for training and with the bounding boxes provided by SimpleBaseline [63] for inference.",
"The evaluation is based on object keypoint similarity, which measures the distance between the detected keypoint and the corresponding ground truth.",
"The standard average precision (AP) and recall (AR) scores are reported.",
"MPII [1] contains about $25K$ images and $40K$ human instances with 16 keypoints.",
"The evaluation is based on the head-normalized probability of correct keypoint (PCKh) score [1].",
"A keypoint is correct if it falls within a predefined threshold to the groundtruth location.",
"We report the [email protected] score by convention."
],
[
"Implementation Details. ",
"For fair comparison, we build our PPT based upon TokenPose-S, TokenPose-B, and TokenPose-L/D6 [31], namely PPT-S, PPT-B, and PPT-L/D6, respectively.",
"For PPT-S and PPT-B, the number of encoder layers $L$ is set to 12, the embedding size $D$ is set to 192, the number of heads $H$ is set to 8.",
"They take the shallow stem-net and the HRNet-W32 as the CNN backbone, respectively.",
"Following [44], [32], the HTI is performed $e=3$ times and is inserted before the 4th, 7th, and 10th encoder layers.",
"The PPT-L/D6 has $L=12$ encoder layers and takes HRNet-W48 as the backbone.",
"the HTI is inserted before the 2th, 4th, and 5th encoder layers.",
"The number of visual tokens $N_v$ is 256 for all networks, and the keep ratio $r$ is set to $0.7$ by default.",
"Thus, only 88 visual tokens are left after three rounds pruning.",
"We follow the same training recipes as [31].",
"In detail, all networks are optimized by Adam optimizer [25] with Mean Square Error (MSE) loss for 300 epochs.",
"The learning rate is initialized with $0.001$ and decays at the 200-th and the 260-th epoch with ratio $0.1$ .",
"As locating human is difficult at early training stages, the keep ratio is gradually reduced from 1 to $r$ with a cosine schedule during the early 100 epochs.",
"Table: Results on COCO validation dataset.",
"The input size is 256×192256\\times 192.",
"GFLOPs T ^T means the GFLOPs for the transformers only following equations from , as our method only focus on accelerating the transformers.Table: Results on the MPII validation set ([email protected]).",
"The input size is 256×256256\\times 256.The results are shown in Table REF and Table REF for COCO and MPII, respectively.",
"Generally, the transformer-based methods [31], [67] maintain less number of parameters.",
"On COCO, compared with the TokenPose, PPT achieves significant acceleration while matching its accuracy.",
"For example, PPT-S reduces 27% total inference FLOPs while only reducing $0.3$ AP.",
"Compared to SimpleBaseline-ResNet152 [63], PPT-S achieves equal performance but only requires $10\\%$ FLOPS.",
"We can also observe consistent conclusion on PPT-B and PPT-L.",
"Note that, for PPT-B and PPT-L, the CNN backbone takes a large portion of computation.",
"Thus, the reduction of total FLOPs is relatively small.",
"Meanwhile, compared with other efficient pose estimation networks [69], [72], the AP of PPT-S is $72.2$ , which is much better than EfficientPose-C [72] with $71.3$ AP at the same FLOPs level.",
"More over, On MPII, our PPT-S can even improve on the PCKh of TokenPose-S by 1.1%.",
"We believe that slimming the number of tokens can also make the attention focus on key elements [76].",
"Thus, our PPT is efficient yet powerful, and it is applicable to any TokenPose variants.",
"All of these results suggest that pruning background tokens does not hurt the overall accuracy and calculating attention among human foreground tokens is sufficient for 2D human pose estimation."
],
[
"Visualizations",
"We visualize the selected tokens from PPT-S in Fig.",
"REF .",
"We present the original images and the selected tokens at different layers.",
"Remarkably, the human areas are gradually refined as the network deepens.",
"The final selected tokens can be considered as a rough human mask.",
"Thus, our HTI can successfully locate human tokens as expected.",
"Moreover, the HTI can handle quite a few complicated situations such as man-object interaction (Fig.REF ), oblique body pose (Fig.",
"REF ), occlusion (Fig.",
"REF ), and multiple persons (Fig.REF REF ).",
"Nevertheless, when only part of human body appears in the image (Fig.",
"REFREF ), the quality of the located human mask could be imperfect.",
"In these cases, we hypothesize that some keypoint tokens such as ankle and knee cannot locate the corresponding joints as they are invisible.",
"Thus, they may just give equal attention score, which leads to inaccurate token selection.",
"Figure: Visualizations of the selected tokens at each HTI module on COCO.",
"The masked regions represent the pruned tokens (We use blue circles to mask out face for privacy issue).",
"For each image group, the first column is the original image, the 2nd, 3rd, and 4th colums are the selected tokens by HTI at the 4th,7th, and 10th layers, respectively."
],
[
"Ablation Studies",
"The keep ratio $r$ controls the trade-off between the acceleration and the accuracy.",
"Meanwhile, reducing tokens also introduces some regularization [76].",
"We take PPT-S and vary $r$ from $0.6$ to $0.8$ on both COCO and MPII.",
"The results are shown in Table REF .",
"The reduction of AP is always less than 1%.",
"When the $r$ is relatively small, PPT can achieve considerable speedup but may not cover the entire human body.",
"As a result, the accuracy of pose estimation is slightly dropped.",
"To maintain the accuracy, we choose $0.7$ as our default keep ratio.",
"Table: Results of PPT-S on COCO and MPII with different keep ratio rr."
],
[
"Datasets & Evaluation Metrics.",
"We evaluate multi-view PPT on two single-person datasets of multi-view 3D human pose estimation, i.e., Human 3.6M [22], [4] and Ski-Pose [49], [16] Only authors from UCI downloaded and accessed these two datasets.",
"Authors from Tencent and Meta don't have access to them.. Human 3.6M contains video frames captured by $M=4$ indoor cameras.",
"It includes many daily activities such as eating and discussion.",
"We follow the same train-test split as in [43], [23], [19], where subjects $1, 5, 6, 7, 8$ are used for training, and $9, 11$ are for testing.",
"We also exclude some scenes of $S9$ from the evaluation as their 3D annotations are damaged [23].",
"Ski-Pose contains video frames captured by outdoor cameras.",
"It is created to help analyze skiers's giant slalom.",
"There are $8,481$ and $1,716$ frames in the training and testing sets, respectively.",
"We use the Joint Detection Rate (JDR) on original images [43] to evaluate the 2D pose accuracy.",
"JDR measures the percentage of successfully detected keypoints within a predefined distance of the ground truth location.",
"The 3D pose is evaluated by Mean Per Joint Position Error (MPJPE) between the ground truth 3D pose in world coordinates and the estimated 3D pose."
],
[
"Implementation Details. ",
"We build multi-view PPT upon PPT-S.",
"The first 9 transformer layers are used to extract human tokens, and the last 3 transformer layers are used for cross-view fusion.",
"Thus, no additional parameters are introduced.",
"Following the settings in [19], [36], we start from a PPT-S pre-trained on COCO and finetune it on multi-view human pose datasets, as it is difficult to train the transformer from scratch with examples in limited scenes.",
"We apply Adam optimizer and train the model for 20 epochs with MSE loss.",
"The learning rate starts with $0.001$ and later on decays at 10-th and 15-th epoch with ratio $0.1$ .",
"The keep ratio $r$ is set to $0.7$ through the entire training process.",
"We resize input images to $256\\times 256$ and follow the same data augmentation in [43], [36].",
"Table: 2D pose estimation on Human3.6M.",
"The metric is JDR on original image.",
"All inputs are resized to 256×256256\\times 256.",
"#V means the number of views used in cross-view fusion step.",
"The FLOPs is the total computation for each view and cros-view fusion.Table: The MPJPE of each pose sequence on Human 3.6M.",
"The 2D results on Human 3.6m is shown in Table REF .",
"The MACs (multiply-add operations) consider both single-view forward MACs of all views and cross-view fusion MACs.",
"Noticeably, our multi-view PPT outperforms all previous cross-view fusion methods on JDR.",
"The JDR can be further improved with the 3D positional encodings (3DPE) [36] on visual tokens.",
"Meanwhile, it can significantly reduce the computation of all 4 view fusion, i.e., the MACs is reduced from $55.1$ G to $9.7$ G. When only fusing 2 views, multi-view PPT still achieves comparable accuracy with other two-view-fusion methods [19], [36], Moreover, we add the baseline that adds transformers on top of TokenPose to perform cross-view fusion, which can be considered as multi-view PPT without token pruning.",
"The JDR is $97.4\\%$ (-$0.7\\%$ with respect to our multi-view PPT), which supports that our human area fusion is better than global attention in both accuracy and efficiency.",
"The MPJPE of estimated 3D pose is reported in Table REF .",
"We can observe that multi-view PPT also achieves the best MPJPE on 3D pose, especially on sophisticated action sequences such as “Phone\" and “Smoke\", as the result of 3D pose is determined by the accuracy of 2D pose.",
"Therefore, our “human area fusion\" strategy is better than previous fusion strategies as it strikes a good balance between efficiency and accuracy.",
"We can also observe consistent conclusion on Ski-Pose from Table REF .",
"Nevertheless, it seems that the performance in this datatset tends to be saturated.",
"The reason might be that there is limited number of training examples, thus the transformer is easy to overfit.",
"Table: 2D and 3D pose estimation accuracy comparison on Ski-Pose."
],
[
"Human Tokens. ",
"Fig.REF presents the selected human tokens in all views.",
"Similar to the conclusion on COCO, our PPT accurately locates all human areas and prunes background areas in all views.",
"Moreover, the tokens used in the cross-view fusion step can be significantly reduced.",
"Figure: Visualizations of the final located tokens on Human 3.6M validation set.",
"For each group, each column is an image from one view.",
"The masked regions represent the pruned tokens.",
"We perform cross-view fusion among these selected tokens."
],
[
"Qualitative results. ",
"We present examples of predicted 2D heatmaps on the image in Fig.REF , and compare our methods with TransFusion [36].",
"It is observed that our method can solve heavy occlusion cases very well, while TransFusion cannot.",
"For two-view-fusion method, occlusion cases in current view may still be occluded in the neighbor view.",
"For example, the heatmap marked with red box is inaccurate in both view 2 and view 4.",
"Thus, fusing this bad quality heatmap cannot improve the final prediction.",
"However, our method can avoid this problem by fusing clues from all views.",
"Figure: Sample heatmaps of our approach.Figure: Attention maps among keypoint tokens."
],
[
"Attentions. ",
"We present an example of the attention map between keypoint tokens in Fig.REF .",
"Given keypoint tokens in one view, they pay attention to keypoints tokens in all views.",
"For example, the left wrist in the first view (blue dot) is occluded, thus its corresponded keypoint token attends to the keypoint token in the second view, where the keypoint is visible.",
"Therefore, the keypoint token in multi-view PPT can learn the dependencies among joints in different views."
],
[
"Conclusion",
" In this paper, we propose the PPT for 2D human pose estimation.",
"Experiments on COCO and MPII show that the PPT achieves similar accuracy compared with previous transformer-based networks but reduces the computation significantly.",
"We also empirically show that PPT can locate a rough human mask as expected.",
"Furthermore, we propose the multi-view PPT to perform the cross-view fusion among human areas.",
"We demonstrate that multi-view PPT efficiently fuses cues from many views and outperforms previous cross-view fusion methods on Human 3.6M and Ski-Pose."
],
[
"Runtime evaluation for Monocular 2D pose estimation",
"Although the GFLOPs reflects the efficiency of networks, it is not equivalent to the real runtime on hardware due to different implementation.",
"We further report the throughput, which measures the maximal number of input instances the network can process in time a unit.",
"Unlike FPS (frame per second), which involves the processing of a single instance, the throughput evaluates the processing of multiple instances in parallel.",
"During the inference time of the top-down method, given one input image, multiple human instances located by an object detector are usually cropped, resized, and combined into a minibatch to accelerate the inference.",
"Then the minibatch of multiple human instances is fed into the pose detector.",
"Thus, we believe throughput is a more reasonable metric to evaluate top-down 2D human pose estimation networks.",
"We set the batch size to 32 for all networks, and compute the throughput on a single 2080 Ti GPU.",
"Both FPS and throughput of PPT and TokenPose [31] are shown on Table REF .",
"Remarkably, pruning tokens cannot significantly improve the time of a single instance (i.e., FPS).",
"We believe the extra time introduced by the pruning operation is not negligible.",
"Nevertheless, PPT significantly improves the throughput from TokenPose, which is consistent with the improvement of GFLOPs in Table REF .",
"We further show the comparison of throughput with other methods in Figure REF .",
"Our PPT consistently improves the throughput at the same AP level.",
"Thus, pruning token does improve the runtime on hardware in practice.",
"Table: FPS and Throughput on COCO validation dataset.Figure: Comparison of throughput on COCO validation dataset."
]
] | 2209.08194 |
[
[
"ANet: Autoencoder-Based Local Field Potential Feature Extractor for\n Evaluating An Antidepressant Effect in Mice after Administering Kratom Leaf\n Extracts"
],
[
"Abstract Kratom (KT) typically exerts antidepressant (AD) effects.",
"However, evaluating which form of KT extracts possesses AD properties similar to the standard AD fluoxetine (flu) remained challenging.",
"Here, we adopted an autoencoder (AE)-based anomaly detector called ANet to measure the similarity of mice's local field potential (LFP) features that responded to KT leave extracts and AD flu.",
"The features that responded to KT syrup had the highest similarity to those that responded to the AD flu at 85.62 $\\pm$ 0.29%.",
"This finding presents the higher feasibility of using KT syrup as an alternative substance for depressant therapy than KT alkaloids and KT aqueous, which are the other candidates in this study.",
"Apart from the similarity measurement, we utilized ANet as a multi-task AE and evaluated the performance in discriminating multi-class LFP responses corresponding to the effect of different KT extracts and AD flu simultaneously.",
"Furthermore, we visualized learned latent features among LFP responses qualitatively and quantitatively as t-SNE projection and maximum mean discrepancy distance, respectively.",
"The classification results reported the accuracy and F1-score of 79.78 $\\pm$ 0.39% and 79.53 $\\pm$ 0.00%.",
"In summary, the outcomes of this research might help therapeutic design devices for an alternative substance profile evaluation, such as Kratom-based form in real-world applications."
],
[
"Introduction",
"Kratom is a local name of Mitragyna speciosa (Korth.)",
"Havil.",
"and has been broadly recognized as herbal medicine for many decades to treat and improve withdrawal symptoms effectively and psychological disorders induced by abused drugs in animal models [1] and humans [2].",
"On a pre-clinical scale, many forms of KT leave extracts have been reported to possess antidepressant (AD)-liked activity on the central nervous system (CNS) [3], [4].",
"As a result, it was still unclear which kind of KT leaf extract might have a profile similar to standard AD.",
"Local field potential (LFP) recording is an invasive measurement to capture brain working, providing high temporal resolution and sensitivity.",
"The spectral power of LFP oscillates dependently on the alteration of the neuro-biomolecules (NeuroBios) and is a pivotal character in drug profile classification in rodents [5].",
"However, the only statistical analysis may face errors when there are substantial medical databases to be distinguished, and each has similar effects on LFP features.",
"Therefore, the machine learning approach is an alternative way for users to recognize LFP features in higher complexity tasks [6].",
"In the literature, only a few experiments attempted extracting LFP essential features from animals with the deep neural network, which is similar to our interest.",
"However, the publication above primarily relied on a simple convolutional neural network [7], which may not be a complete pipeline distinctively.",
"Therefore, according to our motivation, we desire to propose a novel pipeline for LFP processing.",
"A compressed sensing method, autoencoder (AE), is a computational-based deep learning architecture or model [8].",
"AE primarily consists of two components: an encoder and a decoder.",
"Mapping the input signals to latent space is the encoder network's main action; meanwhile, another network's function is reconstructing the latent vector to be the input signals at the end of the model [9].",
"Integrating a supervised classifier network to the AE, so-called multi-task AE, effectively learning to compress data, reconstruct, and classify the brain signals simultaneously [10].",
"In terms of application, although tiny alteration of the brain working in a fraction of a millisecond, such as during performing imaginary motor tasks as well as examining the test to diagnose developmental dyslexia, which is generally difficult to distinguish by visual inspection, the multi-task AE can classify well by having the brain signals or EEG as the inputs to the models.",
"[11], [12].",
"Meanwhile, as far as we know, none of the previous works apply the multi-task AE to process the LFP signals.",
"Thus the study related to mice's LFP response to given substance, as in the rest of the paper, is a novel issue.",
"Figure: An overview of this study.",
"Note: Medial prefrontal cortex (mPFC), Hippocampus (HP), Nucleus accumbens (NAc), control (con), fluoxetine (flu), KT syrup (KTsyr), KT alkaloids (KTalk), and KT aqueous (KTaqu)Here, we proposed the AE-based anomaly detector called ANet as the LFP feature extractor that can measure the similarity of mice's LFP administering KT extracts and AD fluoxetine (flu).",
"fig:Fig1 illustrated ANet (AE) and extended ANet (multi-task AE, which is AE and supervised learning) architecture or model with preprocessed LFP as the input to the model.",
"ANet is the novel mice's LFP feature extractor automatically extracts the discriminative features from LFP in responses to treated substances.",
"These features play significant roles in the LFP similarity measure or comparison in the form of anomaly detection compared to the target LFP, which responds to this study's standard drug, AD flu.",
"We extended ANet to be the multi-task AE in the feasibility study of simultaneous classifying multiple LFP responses induced by the multiple KT extracts and AD flu.",
"Apart from classification performance, we evaluate the feature extraction capability through the latent space qualitatively and quantitatively as t-distributed stochastic neighbor embedding (t-SNE) projection and maximum mean discrepancy (MMD) distance, respectively.",
"Based on mice's brain responses or LFP evaluated using ANet and the extended version (the multi-task AE), we found that KT syrup produced the highest similar LFP features to AD flu.",
"According to the literature and the proposed findings, we are the first to reveal that using KT syrup extract is an appropriate candidate substance for depressant therapy.",
"However, further confirmation of the clinical study on AD effects from KT syrup remains a future challenge.",
"We collected KT leave from the natural resources in Surat Thani province, Thailand.",
"Supplementary materials explain more details about the preparation of KT extract and the quantification of KT major alkaloids, mitragynine (MT), that accumulated in each form of KT extracts."
],
[
"Mice",
"The committee approved all operations involving animals employed in the scientific study from the Prince of Songkla University's Institute of Animal Care and Use, which followed the criteria of the International Committee on Laboratory Animal Science (ICLAS) [project license number: MHESI 6800.11/845 and reference number: 57/2019].",
"We recruited Male Swiss albino ICR mice from the Nomura Siam International Company, Bangkok, Thailand.",
"They were appropriately at rest for one week before the experiment began to minimize the stress.",
"Moreover, mice stayed in separated stainless steel cages (17 x 28.5 x 17 cm) under the standard condition room (12/12 h light/dark cycle, 22°C, and 55.1$\\%$ relative humidity).",
"Water and commercial food pellets were accessed freely.",
"We conducted the studies between eight a.m. and four p.m."
],
[
"LFP Electrode Implantation",
"Male ICR mice (four months of age) had been through an intraperitoneal injection of a mixed solution of 16 mg/kg xylazine hydrochloride (Xylavet, Sigma-Aldrich International GmbH, Switzerland) and 50 mg/kg zoletil (Tiletamine – zolazepam, Vibac Ah, Inc., USA) to be deeply anesthetized.",
"The head of the animals was then fixed with stereotaxic apparatus before the scalp on the dorsal head in the middle line was exposed.",
"According to the mouse brain atlas [13], electrodes were stereotaxically implanted on the left hemisphere of the brain, from the bregma to the medial prefrontal cortex (mPFC) (AP; +2.5 mm, ML; 0.2 mm; DV; 1.5 mm), hippocampus (HP) (AP; -2.5 mm, ML; 2.0 mm; DV; 1.5 mm) and the nucleus accumbens (NAc) (AP; +1.3 mm, ML; 1.0 mm; DV; 4.2 mm).",
"Over the cerebellum (AP; -6.0 mm, ML; 0.0 mm; DV; 1.5 mm), a ground electrode was inserted.",
"Dental acrylic was applied to hold and secure all placed electrodes.",
"The antibiotic ampicillin (100 mg/kg) (General Drug House Co., Ltd., Bangkok, Thailand) and carprofen (10 mg/kg) (Best Equipment Center Co., Ltd., Thailand) were given intramuscularly once a day for three days [14] to prevent infection and relieve pain.",
"It took at least two weeks to fully recuperate from surgery before they could begin the experiment."
],
[
"Data Collecting in Mice",
"As soon as animals recovered from surgery, they went to the experimental period adapted from the previous investigations [15], [5], as shown in supplementary materials.",
"In brief, after animals habituated for three consecutive days, they went to the testing phase.",
"During this phase, we placed mice individually in the recording chamber (25 cm x 18 cm x 25 cm) to familiarize the experimental conditions for one hour.",
"After that, the LFP signals were harvested in the recording session for 30 minutes after the injection of the treated substances (number of mice = 7 per group) for one hour.",
"The concentrations of KT extract in each sample existed in tab:Table1 according to the high-performance liquid chromatography analysis.",
"KT extract calculated doses are from the whole amount of sample containing MT, fixed at 10 mg/kg, a dose that effectively cured the animal model of depression [16].",
"The AD flu at 10 mg/kg was selected as the standard drug since it showed positive results in previous findings [3].",
"Details of instrumental setup for LFP signals recording are with the previous works [14].",
"In summary, the Dual Bio Amp (AD Instruments, Castle Hill.",
"NSW, Australia) and the PowerLab 16/35 system (AD Instruments, Castle Hill.",
"NSW, Australia) with 16-bit A/D at a sampling frequency for two kHz are amplification and digitization for LFP signals, respectively.",
"Also, we filtered the power line noise artifact at 50 Hz.",
"LabChart 7.3.7 Pro software was a tool for recording LFP signals.",
"Table: Summary of MT content in KT extracts used in this study"
],
[
"Experiments",
"This work has designed three experiment settings: Spectral Power with Statistical Analysis, AE-Based Anomaly Detection, and Multi-Task AE.",
"The purpose and detail of each experiment are described below."
],
[
"Experiment I: Spectral Power with Statistical Analysis",
"This experiment's purpose was to compare LFP responses induced by different substances traditionally.",
"The substances used are control, flu, KT syrup, KT alkaloids, or KT aqueous.",
"First, we downsampled raw LFP to 200 Hz using the MNE python package (version 0.23.4) [17].",
"Then we computed the spectral power of the prepared LFP for all channels by applying the Morlet wavelets transform and divided into six frequency bands: Delta, Theta, Alpha, Beta, Gamma I, and Gamma II according to the previous work [14].",
"Finally, we conducted the statistical analysis using a non-parametric Kruskal-Wallis test with Bonferroni multiple comparisons to test group differences across the frequency bands at each brain region.",
"Comparing statistical results to the deep learning approaches, the recorded LFP signals at each brain region were further split into four seconds/segment, followed by spectral power of the frequency band extraction as the input to ANet and the multi-task AE in the following two experiments as shown in fig:Fig2 and fig:Fig3."
],
[
"Experiment II: AE-Based Anomaly Detection (ANet)",
"This experiment aimed to evaluate the similarity of LFP features responded to each KT extract relative to the reference drug (AD flu).",
"We first designed the structure of the auto-encoder or ANet as the following two components."
],
[
"Encoder",
"The encoder network contained two convolutional neural networks (CNN) stacks; each consisted of a 2-D convolutional layer (Conv2D) performed with an exponential linear unit (ELU) activation function, a batch normalization layer, and an averaging pooling layer (AveragePooling2D), respectively.",
"In this module, the input format was (F*B, T, 1), where F, B, and T denote the number of frequency bands, brain areas, and time points, respectively.",
"During the implementation, we set a data format as the channel last option for Conv2D (Keras API).",
"Flatten was the final layer before mapping to the latent representation or vector."
],
[
"Decoder",
"The decoder aligned symmetrically with the encoder component to shape the dimension of reconstructed signals to be the exact size of the original input.",
"Three CNN blocks reconstruct the information from the latent vector.",
"Each CNN block of the decoder network consisted of a Conv2D activated by the ELU function, followed by a UpSampling2D layer to expand the spatial dimension of the data.",
"To build ANet to be the anomaly detector, as illustrated in fig:Fig2, LFP spectral power responded to AD flu was the original input during training as the conventional AE.",
"Therefore, we calculated mean square error (MSE) loss (Equation (REF )) during ANet learning.",
"The mean and standard deviation (SD) of MSE loss defined the threshold for the normal zone of the anomaly detector.",
"The width of the normal zone was the mean ± SD.",
"Then we used the trained ANet to evaluate the similarity of the unseen LFP features that responded to KT extracts (testing sets).",
"If MSE loss of the unseen LFP was fallen into the normal zone, LFP features that responded to the KT extract are deemed similar to standard AD drugs and vice versa.",
"Then we optimized the model according to the following Subsections REF and REF .",
"Figure: The procedure of Experiment II: AE-Based Anomaly Detection (ANet) Note: Number of frequency bands (F), Number of brain areas (B), Number of time points (T), the LFP features in response to each substance: KT syrup (KTsyr), KT aqueous (KTaqu), KT alkaloids (KTalk), and fluoxetine (flu)Figure: The procedure of Experiment III: Multi-Task AE Note: Number of frequency bands (F), Number of brain areas (B), Number of time points (T), the LFP features in response to each substance: control (con), KT syrup (KTsyr), KT aqueous (KTaqu), KT alkaloids (KTalk), and fluoxetine (flu)"
],
[
"Experiment III: Multi-Task AE",
"This study extended ANet by including the supervised learning component to the AE, which turned out to be the multi-task AE.",
"We aimed to perform the feasibility of extracting discriminative features from multiple LFP responses simultaneously.",
"ANet and Supervised Learning: We added the supervised learning component by having the latent vector layer of ANet as the input, which made the proposed architecture form the multi-task AE, as shown in fig:Fig1.",
"At the end of the supervised task, the softmax function classified the latent vector via layer activation in the FC layer.",
"These allowed our model to learn the reconstruction along with the classification.",
"During the experiment, we explored the possibility of the multi-task AE or the model recognizing the multi-class LFP features for classifying LFP responses.",
"We believed that this experiment could benefit in developing computational tools for future studies of the substance that affected mice's brain activities.",
"The experimental protocol followed fig:Fig3.",
"Firstly, we split the inputs for training, validating, and testing sets.",
"Then we optimized the model according to the following Subsections REF and REF .",
"Eventually, the following bullets and equation explain quantitative and qualitative assessments of the model capability in LFP feature extraction.",
"$t$ -SNE projection is for the qualitative assessment of the visualize reduced dimensional data [18].",
"In this experiment, we defined the latent vector as the input of t-SNE projection algorithm to present the probability distribution of unlearned and learned features, representing the LFP datasets before and after feature extraction by the proposed model, respectively.",
"Maximum Mean Discrepancy (MMD) distance is for the quantitative assessment of the distance between the means of the embedding features of two probability distributions in t-SNE projection.",
"They evaluated whether data in set X and Y generated a probability distribution equally.",
"Here, the ideal concept is that if the MMD distance of LFP features between substance X and Y is low.",
"Then, we might imply that those two LFP inputs seem to originate from a very close origination.",
"The standard calculation of MMD is the statistical measure of the samples from the different sets, expressed in Equation (REF ).",
"$\\small M M D\\left(P_{X}, P_{Y}\\right)=\\left\\Vert \\mathbb {E}_{X \\sim P_{X}}[\\varphi (X)]-\\mathrm {E}_{Y \\sim P_{Y}}[\\varphi (Y)]\\right\\Vert \\kappa $ Where $\\varphi $ represents the map of features of $X$ or $Y \\rightarrow \\kappa $ , $\\kappa $ denotes a reproducing kernel Hilbert space.",
"$P_{X}$ and $P_{Y}$ represent the probability distribution of LFP in sets $X$ and $Y$ , respectively.",
"In addition, the multi-task AE performance also exhibited averaged percentage accuracy, F1-score, and confusion matrix."
],
[
"Network Training",
"In both Experiment II and III, we implemented our model using Keras API (TensorFlow version 2.2.0 as backend) under NVIDIA Tesla v100 GPU setup with 32G memory.",
"With the optimization by Adam optimizer, the learning rate was between 10-4 and 10-5.",
"If there were no improvement in validation loss for five consecutive epochs, the learning rate gradually alleviated with a decay rate of 0.5.",
"The batch size was 128 samples.",
"We applied the function of early stopping to stop the training iteration when the validation loss had not reduced for 20 continuous epochs.",
"According to the described experiments earlier, we optimized mean square error (MSE) loss alone for Experiment II or the study of ANet.",
"In contrast, we optimized both MSE and cross-entropy (CE) losses for Experiment III, which studied the multi-task AE.",
"MSE and CE losses correspond to AE and supervised learning components, respectively, as described below: Mean square error (MSE) loss was adopted to monitor and minimize errors between the input signal and reconstructed data.",
"It was expressed in Equation (REF ).",
"$L_{M S E}(x, \\hat{x})=\\frac{1}{C} \\sum _{i=1}^{C}\\left\\Vert x_{i}-\\hat{x}_{i}\\right\\Vert ^{2}$ $C$ represents a number of channels while $x$ and $\\hat{x}$ are input and reconstructed data of the channel.",
"Cross-entropy (CE) loss was used in supervised learning to assess the error between the actual label and classification probability, as shown in Equation (REF ).",
"$L_{C E}(y, \\hat{y})=-\\sum _{j=1}^{|c|} y_{j} \\log \\hat{y}_{j}$ Where $y_{j}$ and $\\hat{y}_{j}$ denote true and predicted labels of data in class, and $c$ is the number of classes.",
"The summation of the losses, as mentioned earlier, finally evaluated the total loss: $L_{M S E}$ and $L_{C E}$ demonstrated in Equation (REF ) and Equation (REF ), respectively, as shown below.",
"$L_{total}(x,\\hat{x},y,\\hat{y})= \\frac{1}{N} \\sum _{k=1}^{N}(L_{MSE}(x_{k},\\hat{x}_{k}) + L_{CE}(y_{k},\\hat{y}_{k}))$ $N$ is the total number of input signals and the weight of loss assigned for 1.0."
],
[
"Network Validation",
"Experiment II validated ANet for anomaly detection with the subject-independent scheme.",
"The stratified 5-folds cross-validation was adopted to separate data for testing, and training sets, both outer and inner loops, using Scikit-Learn (version 1.1.1).",
"Only the spectral power of LFP that responded to the AD flu was assigned as training and validation sets, while unseen data from KT extracts were used as a testing set, as illustrated in fig:Fig2.",
"The multiple LFP responses classification required a larger scale of samples in the network training, so Experiment III validated the multi-task AE with the subject-dependent scheme.",
"The spectral powers of multi-LFP classes responded to each substance: control, AD flu, KT syrup, KT alkaloids, and KT aqueous gathered from seven mice per class, as shown in fig:Fig3."
],
[
"Experiment I: Spectral Power with Statistical Analysis",
"The conventional hand-crafted feature of LFP signals is spectral power.",
"This experiment used statistical testing across the quantitative spectral power analysis to assess the significant difference in LFP responses among the group of mice treated with different substances.",
"As reported in tab:Table3, the results found that LFP responses to KT syrup and KT alkaloids have no significant difference from AD flu in most brain regions and oscillation bands.",
"In summary, the results convinced both KT extracts could be potential candidates for alternative drugs given the similar effects on the brain to AD flu.",
"However, this conventional approach demonstrated the weakness in differentiating LFP responses from KT syrup and KT alkaloids.",
"Thus, we addressed the weakness by proposing ANet and the multi-task AE approaches in identifying the higher similarity of LFP responses to KT syrup and AD flu compared to those to KT alkaloids.",
"Figure: Histograms on anomaly tests were exhibited.",
"The distribution of MSE loss from LFP features responded to the AD flu was illustrated along with each loss distribution of KT syrup (a), KT alkaloids (b), KT aqueous (c), and all forms of KT extracts (d).",
"The lower and upper thresholds were specified for training loss at 0.026 and 0.014, labeled with black and red dash lines on each graph.",
"Any samples dropped between these thresholds were indicated as similar to the LFP of the AD flu.Figure: Training and validation losses were exhibited as the standard error mean (MSE) (a) and cross-entropy (b) loss.",
"These two losses were summed to show as total loss (c).",
"The confusion matrix of multi-LFP class recognition was shown (d).",
"The con, flu, KTsyr, KTaqu, and KTalk represent the LFP features responded to the control, fluoxetine, KT syrup, KT aqueous, and KT alkaloids, respectively.Figure: Visualization of unlearned (raw spectral power data) (a) and learned (latent space) (b) LFP signal features by using 2-dimensional t-SNE projection for the evaluation the LFP features responded to each substance: control (con), KT syrup (KTsyr), KT aqueous (KTaqu), KT alkaloids (KTalk), and fluoxetine (flu) in a subject-dependent settingFigure: Boxplots of log MMD distances quantitatively assessed from t-SNE projection were exhibited.",
"For example, con: flu means the MMD distance measured between the embedding of LFP features responded to con and flu and so on.",
"Asterisks indicate significant differences at ***: pp << 0.001 performed by a non-parametric Kruskal-Wallis test with Bonferroni correction.",
"The con, flu, KTsyr, KTaqu, and KTalk represent the LFP features responded to the control, fluoxetine, KT syrup, KT aqueous, and KT alkaloids, respectively."
],
[
"Experiment II: AE-Based Anomaly Detection (ANet)",
"Experiment II demonstrated the performance of the proposed anomaly detector or ANet in assessing the similarity of the unseen LFP compared to LFP response to AD flu was the reference class or standard antidepressant substance in producing the drugs.",
"The findings revealed that the number of LFP that responded to KT syrup appeared to distribute mainly in the normal zone indicating the high similarity to the standard drug or AD flu fig:Fig4(a), followed by KT alkaloids fig:Fig4(b), and KT aqueous fig:Fig4(c) extract, respectively.",
"Quantitative data found that the normal detecting rate or defined as the similarity rate for this study, was 85.62 ± 0.29%, 83.64 ± 0.32%, and 40.69 ± 0.38% in detecting LFP responses or testing sets from KT syrup, KT alkaloids, and KT aqueous extract, respectively, referred to LFP response of AD flu or the training sets."
],
[
"Experiment III: Multi-Task AE",
"To investigate an insightful optimization process of the multi-task AE, we assessed the alterations of training and validation losses in the multi-LFP class recognition during the training process.",
"Based on the least iteration stopped by early stopping, fig:Fig5(a), (b), and (c) demonstrated training and validation loss while training for 70 epochs for MSE loss, CE loss, and total loss, respectively.",
"The findings demonstrated that MSE loss reached a relatively stable level after data training for ten epochs.",
"Meanwhile, high fluctuation patterns were in the CE and the total losses during the 20th-40th epochs.",
"However, validation loss converged to the training loss, suggesting that the model training had not been overfitted, confirming this study's well-trained model.",
"All measures of the performance were applied together with the early stopping function.",
"In summary, the proposed multi-task AE reached accuracy, and the F1-score of the model was 79.78 ± 0.39% and 79.53 ± 0.00%, respectively.",
"Furthermore, we also illustrated a confusion matrix of the predicted outputs.",
"While the matrix revealed that almost all LFP samples that responded to the treated substances were primarily correctly classified, it was obvious that there were some misclassifications between the AD flu and KT syrup, as shown in fig:Fig5(d).",
"Thus, LFP samples from both classes might have similar spectral power features originating from sharing mechanisms exerted on the CNS and the brain.",
"The finding was consistent with Subsections REF and REF .",
"The 2-dimension of the t-SNE projection visualized the probability distribution of reduced dimensional data at latent space.",
"It was found that t-SNE of baseline, processed from raw spectral power, diffused and mixed among classes of substances evenly to form one cluster, as shown in Fig6(a).",
"Meanwhile, the characteristics of t-SNE resulting from the learned features or represented latent vectors appeared to be formed for correct clusters.",
"For example, consistency with the experiments mentioned earlier and results, among KT extracts, the KT syrup cluster shared the intersect clusters of the AD flu mostly, as depicted in Fig6(b).",
"Although discriminative LFP features of KT extract distributed with the AD flu overlappingly in the t-SNE visualization, this result seems to provide only qualitative estimation.",
"To get a concrete explanation, MMD distance, a value estimated from the t-SNE projection-based-latent vector evaluation, was thus analyzed quantitatively.",
"The results exposed that LFP features of the AD flu and KT syrup were the first groups showing the shortest MMD distance to control mice and were followed by KT alkaloids and KT aqueous, respectively.",
"Moreover, features in LFP responses from the AD flu: KT syrup and AD flu: KT alkaloids were the first two paired classes that produced the shortest MMD distance.",
"In contrast, the longest MMD distance was detected in the LFP features of AD flu to KT aqueous, as shown in fig:Fig7.",
"This finding also brought us to the conclusive similarity of LFP responses obtained from the group of mice treated with KT syrup and AD flu."
],
[
"Discussion",
"Here, we discussed the results of three experiments to identify the gap in the conventional approach to LFP responses comparison and analysis.",
"Then we addressed the contributions of the proposed approaches, which were ANet and the multi-task AE, in bridging the identified gap.",
"Finally, we explained the feasible impact of the finding from the application point of view on the behavioral brain responses to the alternative AD substances for drug formulation; Kratom or KT extracted solutions.",
"Alternation of spectral power amplitudes is directly reflected in the changes in NeuroBios activity [15].",
"In addition, various experiments have presented the feasibility of using spectral power characteristics of LFP as a biomarker for substance profile classification [5].",
"Therefore, as demonstrated in Section REF , we conducted spectral power analysis and statistical testing.",
"LFP spectral features that responded to KT syrup and KT alkaloids have no significant difference in frequency responses overall brain regions when paired with LFP that responded to AD flu or standard substance for AD drug formulation.",
"Therefore, we might infer that mice's brains were influenced by KT syrup and KT alkaloids, closely to AD flu.",
"To utilize and enhance the findings of the conventional analysis, we proposed the AE-based LFP feature extractor or ANet to automatically detect the similarity of LFP responses from the interested substances compared to the reference drug, which was AD flu in this study.",
"According to Section REF , the defined similarity rate of LFP features extracted by ANet corresponding to KT syrup gave us about two percent higher than those from KT alkaloids referred to LFP that responded to AD flu.",
"These findings performed the feasible applications of ANet in pre-screening the alternative substances which affect the mice's brain similar to the standard or well-known substance in the future drug formulated study.",
"Then, we extended ANet to be the multi-task AE for the multi-LFP response recognition, a more sophisticated task than anomaly or similarity detector.",
"The benefit of this task was enhancing, visualizing, and measuring the difference between LFP responses of the candidate substances, which could not be seen during the conventional spectral power analysis.",
"Here, LFP responses of KT syrup and KT alkaloids were great examples.",
"As reported in Section REF , the conventional approach could not give us a convincing conclusion in comparing LFP responses from the group of mice drugged by KT syrup, KT alkaloids, and AD flu.",
"On the other hand, the proposed multi-task AE with qualitative and quantitative measures was quite promising in summarizing that the LFP responses of KT syrup were much closer to AD flu than those of KT alkaloids.",
"Similar to revealing the similarity of LFP responses to KT syrup and AD flu for the first time, our investigation can inspire people interested in applying neural network-based computational modeling as innovative tools to the field of brain behavioral studies responses to the alternative formulation of novel drugs."
],
[
"Conclusion",
"This study proposed a neural network-based approach or ANet for comparing local field potential or LFP responses among the mice drugs with different substances.",
"ANet automatically extracted the features from the unseen LFP responses and predicted the similarity rate to the reference or the standard responses used during the training process.",
"Moreover, the extended ANet in the form of a multi-task AE presented the possibility of classifying multi-class LFP simultaneously.",
"Here, we applied the proposed approaches to study the effect of Kratom, or KT extracted substances, compared to the standard AD drug via the mice's brain activities.",
"Both qualitative and quantitative assessments were used throughout the studies.",
"As far as we concern, It was the first study in the literature.",
"The outcomes convinced KT extracted using syrup induced the high similarity of LFP responses to the standard antidepressant drug or AD fluoxetine.",
"In conclusion, we might infer that KT syrup is the potential candidate substance for an antidepressant drug formulation.",
"However, further confirmation of the clinical and molecular scaled studies remains a future challenge."
]
] | 2209.08210 |
[
[
"Grid-Free MIMO Beam Alignment through Site-Specific Deep Learning"
],
[
"Abstract Beam alignment is a critical bottleneck in millimeter wave (mmWave) communication.",
"An ideal beam alignment technique should achieve high beamforming (BF) gain with low latency, scale well to systems with higher carrier frequencies, larger antenna arrays and multiple user equipments (UEs), and not require hard-to-obtain context information (CI).",
"These qualities are collectively lacking in existing methods.",
"We depart from the conventional codebook-based (CB) approach where the optimal beam is chosen from quantized codebooks and instead propose a grid-free (GF) beam alignment method that directly synthesizes the transmit (Tx) and receive (Rx) beams from the continuous search space using measurements from a few site-specific probing beams that are found via a deep learning (DL) pipeline.",
"In realistic settings, the proposed method achieves a far superior signal-to-noise ratio (SNR)-latency trade-off compared to the CB baselines: it aligns near-optimal beams 100x faster or equivalently finds beams with 10-15 dB better average SNR in the same number of searches, relative to an exhaustive search over a conventional codebook."
],
[
"Introduction",
"mmWave devices require highly directional BF to compensate for the severe isotropic path loss and to achieve viable signal strength.",
"To reduce the cost and power consumption of a fully digital system, practical mmWave systems often adopt analog beams that concentrate energy in a small angular area.",
"On the other hand, these narrow beams are susceptible to changes in the propagation environment such as blockage and reflection.",
"It is critical for mmWave BS and UE to find good BF directions during initial connection and then track these analog beams as the propagation conditions change, such as when a UE moves and rotates.",
"As future cellular systems move to unlock larger bandwidths at higher carrier frequencies extending to the so-called “Terahertz” bands of up to 300 GHz, devices will adopt ever denser antenna arrays and narrower beams.",
"As a result, beam management – the process of discovering and maintaining good analog BF directions – is a critical bottleneck that will only worsen as we move towards 6G and beyond.",
"Existing beam management approaches typically assume a “grid-of-beams” or CB framework, where BS and UE adopt codebooks of quantized BF angles.",
"In order to ensure coverage in any site, the quantized beams usually distribute energy uniformly in the angular space, such as by using DFT codebooks.",
"In this context, the problem of beam alignment becomes selecting the optimal beams from the finite codebooks at the BS and the UE.",
"The most widely adopted method of beam selection is through a search.",
"For instance, 5G NR adopts a beam management framework based on beam sweeping, measurement and reporting [2], [3].",
"In the downlink, the BS exhaustively searches its codebook by periodically sending beamformed RS called SSB.",
"The UE measures and reports the received signal strength.",
"And the BS selects the best beam based on the UE's feedback.",
"The obvious drawback of an exhaustive search is its beam sweeping latency, which grows linearly with the total number of beam pairs.",
"As systems move up the spectrum and adopt narrower beams, the size of the codebook increases accordingly.",
"A Terahertz system may easily adopt tens of thousands of beam pairs, rendering the the exhaustive search infeasible due to the prohibitive beam sweeping latency.",
"A hierarchical search can reduce the latency for a single UE.",
"By sweeping wider beams first and progress to narrower child beams, it iteratively reduces the search space.",
"In the 5G NR framework, the BS can use SSB to sweep the wide beams and use the more flexible CSI-RS for narrow-beam refinement.",
"On the other hand, the radiation patterns of the wide beams need to be carefully designed since the hierarchical search is more prone to search errors caused by noisy measurements and imperfect wide-beam shapes [4], [5].",
"Another way to reduce the beam sweeping latency is to leverage CI.",
"The optimal beam pair of a link is intuitively a function of the topology of the propagation environment and the locations of the BS and the UE.",
"Such spatial properties can be captured through CI such as sub-6 GHz OOB measurements [6], GNSS coordinates of the UE [7], [8], radar measurements [9], [10], images captured by cameras and 3D point clouds of the environment [11], [12].",
"By constructing a mapping from the CI to the index of the optimal beam through a lookup table or ML models, the search space can be substantially reduced.",
"However, these CI-based methods are difficult to be standardized and widely-adopted in practice due to the requirement of additional sensors and the varying hardware capabilities of mmWave devices.",
"They are also not suitable for standalone systems due to the need of a robust feedback link usually occupying a lower frequency band."
],
[
"Site-specific adaptation",
"A promising way to reduce the beam sweeping latency is by site-specific adaptation.",
"In scenarios where the AoA and AoD distributions are highly non-uniform, such as in particular environment topologies and when there are several spatial clusters of UE, some beams in a uniform codebook may never get used.",
"Leveraging this fact, the BS can compute the statistics of the entire codebook after an exhaustive training phase and prioritize the most frequently used beams [13], [14].",
"By training using explicit CSI [15] or through implicit rewards in a RL framework [16], NN can generate smaller codebooks from scratch that adapt to the environment and strategically direct energy towards the UE.",
"The data-driven approach can also help select analog beams from standard codebooks using sensing measurements by optimizing the mapping function [17] or jointly learning the sensing beams and the mapping function [18].",
"Compared to CB beam management methods whose BF gain is limited by the resolution of the codebook, a GF approach extends the search space for the optimal beam to the high-dimensional continuous space and can in theory achieve better gain.",
"For instance, MRT is known to be optimal under the unit power constraint and EGT under the unit modulus constraint.",
"However, they require full CSI to compute the BF weights, which is generally unavailable before beam alignment.",
"Furthermore, even with CSI, finding the optimal EGT and EGC beams requires a grid search over possible weights in the MIMO setting and is computationally prohibitive [19].",
"Recent works have explored directly predicting the hybrid BF weights for the BS without full CSI by learning sensing matrices [20] or a sequence of interactive sensing vectors based on feedback of previous measurements [21].",
"In our previous work [22], we proposed a CB beam alignment method that uses the measurements of a few site-specific probing beams to predict the index of the optimal Tx narrow beam.",
"However, it benefits significantly from trying a few candidate beams for each UE.",
"As a result, the gain in the beam sweeping overhead diminishes with increasing number of UE.",
"This is a common and important limitation of many existing methods: the hierarchical search requires searching all child beams, many CI-based models require trying the most likely candidates, and active learning-based methods require sweeping a different sequence of sensing beams for each UE.",
"A beam sweeping procedure needs to be repeated for all UE in the cell, each of which may require a different set of beams.",
"Hence the total beam sweeping latency increases linearly with the number of UE, which can be large and unknown in cellular systems.",
"In CB approaches, eventually the entire codebook will need to be searched, a major shortcoming of many of the aforementioned methods."
],
[
"Contributions",
"In this work, we propose DL-GF, an one-shot GF beam alignment method based on unsupervised DL.",
"The proposed method learns a small number of probing beams through site-specific training and directly predicts the optimal analog BF vectors without using a quantized codebook.",
"It possesses several qualities of an ideal beam alignment method, many of which are lacking in existing approaches: High BF gain beyond standard codebooks.",
"The proposed method synthesizes the analog beams from the continuous search space instead of selecting from a quantized codebook.",
"With the additional degree of freedom, the synthesized beams can achieve better BF gain compared to choosing optimally from large quantized codebooks.",
"Low beam sweeping latency with optimal scaling.",
"With site-specific training, the proposed method requires just sweeping a few probing beam pairs to capture sufficient channel information.",
"Unlike many existing approaches that require sweeping a few candidate beams or child beams for each UE, the proposed method synthesizes the analog beams in one-shot.",
"As a result, the beam sweeping latency does not increase with the number of UE.",
"In our setting, DL-GF reduces the beam sweeping overhead by over 500$\\times $ vs. the exhaustive CB method regardless of the number of UE.",
"Easy to adopt in cellular standards.",
"The proposed method does not require hard-to-obtain CI such as the location of UE and OOB measurements.",
"Instead, it purely relies on measuring and reporting a few probing beams.",
"The probing beams are also optimized for coverage, allowing the BS to discover new UE.",
"Joint Tx-Rx beam alignment.",
"Unlike many existing methods that only consider beam alignment for the BS, the proposed method synthesizes the analog beam for both the BS and the UE.",
"It does not require an additional beam alignment process for the UE.",
"The synthesized Tx and Rx beam pairs are jointly optimized to achieve high BF gain, even when the UE have random orientations.",
"The rest of this article is organized as follows.",
"The system model is described in Section .",
"The proposed beam alignment approach, the appropriate metrics and the baselines of comparison are explained in Section .",
"The datasets used are described in Section .",
"The simulation results are presented in Section .",
"Finally, we provide the conclusion and final remarks in Section .",
"Notation: The following notations are used in this paper: $|a|$ denotes the magnitude of the scalar $a$ , $||\\mathbf {A}||_{\\mathrm {F}}$ denotes the Frobenius norm, $\\mathbf {A}^{\\mathrm {real}}, \\mathbf {A}^{\\mathrm {imag}}$ denote the real and imaginary parts of a complex matrix $\\mathbf {A}$ , $\\mathbf {A}^{T}$ denotes the transpose, $\\mathbf {A}^{H}$ denotes the conjugate transpose, $[\\mathbf {a}]_i$ denotes the $i$ th element of the vector $\\mathbf {a}$ , $\\mathbf {a} \\otimes \\mathbf {b}$ denotes the Kronecker product, $\\mathbf {A} \\oslash \\mathbf {B}$ denotes the element-wise division and $\\mathbf {A}^{|\\cdot |}$ denotes the element-wise magnitude."
],
[
"System Model",
"A MIMO system is considered where the BS has an array of $N_{\\textnormal {T}}$ antennas.",
"the UE has an array of $N_{\\textnormal {R}}$ antennas and both perform beam alignment.",
"A geometric channel model with $L$ paths is adopted: $\\mathbf {H} = \\sum _{l=1}^{L} \\alpha _{l} e^{j(\\theta _{l}-2\\pi \\tau _{l}B)} \\mathbf {a}_{\\textnormal {R}}(\\omega _{l}^{az},\\omega _{l}^{el}) \\mathbf {a}_{\\textnormal {T}}(\\phi _{l}^{az},\\phi _{l}^{el})^{H},$ where $\\mathbf {a}_{\\textnormal {T}}, \\mathbf {a}_{\\textnormal {R}}$ are the Tx and Rx array response vectors, $\\alpha _l, \\theta _l, \\tau _l$ are the gain, Doppler shift and delay, $\\omega _{l}^{az}, \\omega _{l}^{el}, \\phi _{l}^{az}, \\phi _{l}^{el}$ are the azimuth and elevation AoA and AoD of path $l$ .",
"For a UPA with $N_y$ and $N_z$ antennas in the $y-z$ plane, its array response vector is given by: $\\mathbf {a}(\\phi ^{az},\\phi ^{el}) = \\mathbf {a}_z(\\phi ^{el}) \\otimes \\mathbf {a}_y(\\phi ^{az},\\phi ^{el}),$ where $\\mathbf {a}_y(\\phi ^{az},\\phi ^{el}) =\\begin{bmatrix}1 & e^{j\\frac{2 \\pi }{\\lambda }d\\sin \\phi ^{az} \\sin \\phi ^{el}} & \\cdots & e^{j(N_{y}-1)\\frac{2 \\pi }{\\lambda }d\\sin \\phi ^{az} \\sin \\phi ^{el}}\\end{bmatrix}^{T},$ $\\mathbf {a}_z(\\phi ^{el}) =\\begin{bmatrix}1 & e^{j\\frac{2 \\pi }{\\lambda }d \\cos \\phi ^{el}} & \\cdots & e^{j(N_{z}-1)\\frac{2 \\pi }{\\lambda }d \\cos \\phi ^{el}}\\end{bmatrix}^{T},$ $\\lambda $ is the carrier wavelength and $d$ is the antenna spacing.",
"In the downlink, if the BS adopts a Tx beam $\\mathbf {f} \\in \\mathbb {C}^{N_{\\textnormal {T}} \\times 1}$ and transmit a symbol $s$ and the UE adopts a Rx beam $\\mathbf {w} \\in \\mathbb {C}^{N_{\\textnormal {R}} \\times 1}$ , the received signal can be written as $y = \\sqrt{P_{\\textnormal {T}}}\\mathbf {w}^{H}\\mathbf {H}\\mathbf {f}s + \\mathbf {w}^{H}\\mathbf {n},$ where $P_{\\textnormal {T}}$ is the Tx power and $\\mathbf {n} \\sim \\mathcal {CN}(0,\\sigma ^2\\mathbb {I})$ is a complex AWGN.",
"Assuming unit-power transmitted symbols, the SNR achieved by the beam pair is $\\textnormal {SNR} = \\frac{P_{\\textnormal {T}}|\\mathbf {w}^{H}\\mathbf {H}\\mathbf {f}|^2}{|\\mathbf {w}^{H}\\mathbf {n}|^2}.$ The BS and the UE are assumed to perform analog BF only.",
"Each device has a single RF chain connected to an array of phase shifters.",
"Hence the BF vectors satisfy the unit-power, constant modulus constraint: $|[\\mathbf {f}]_i| = \\frac{1}{\\sqrt{N_{\\textnormal {T}}}}, i=1, \\dots , N_{\\textnormal {T}} \\ $ $|[\\mathbf {w}]_j| = \\frac{1}{\\sqrt{N_{\\textnormal {R}}}}, j=1, \\dots , N_{\\textnormal {R}}.$ In systems that adopts hybrid BF, the BS and the UE may first undergo beam alignment and select good analog beams, then perform digital precoding over the effective channel.",
"The joint optimization of analog and digital BF is left for future work.",
"The orientation of a UE is modeled with random rotations with respect to the local coordinate system which is attached to and rotates with the UE [23].",
"Each UE first rotates by the $z$ axis, then by the new $y$ axis, and finally by the new $x$ axis.",
"The angles of the three elemental rotations are modeled as random variables.",
"The AoA of the channel is then adjusted according to the orientation of the UE.",
"A more sophisticated model may consider the shape of the UE and self–blockage of the device, which is left for future work."
],
[
"The Proposed Method, Metrics and Baselines",
"Our objective is to directly predict optimal Tx and Rx beams without searching large codebooks or candidate beams.",
"In our previous work [22], we demonstrated that a NN classifier can accurately select the optimal beam index from a large codebook using measurements of a few learned site-specific probing beams, i.e., each BS learns unique probing beams that are well-suited to its propagation environment.",
"Inspired by this idea, we propose to learn a small number of probing beams at both the BS and the UE and use their measurements to synthesize the optimal narrow beam pair from the continuous search space.",
"The BS first sweeps $N_{\\mathbf {F}}$ Tx probing beams while the UE measures the received signal power using $N_{\\mathbf {W}}$ Rx probing beams.",
"After the probing-beam sweeping phase, the UE uses the collected measurements as inputs to its beam synthesizer function $f_{\\textnormal {R}}$ to generate its Rx beam.",
"The measurements are also fed back to the BS, which uses them as inputs to its own beam synthesizer function $f_{\\textnormal {T}}$ to generate the Tx beam.",
"Let $\\mathbf {F} \\in \\mathbb {C}^{N_{\\textnormal {T}} \\times N_{\\mathbf {F}}}, \\mathbf {W} \\in \\mathbb {C}^{N_{\\textnormal {R}} \\times N_{\\mathbf {W}}}$ be matrices representing the BS and UE probing beams.",
"The composite received signal $\\mathbf {Y} \\in \\mathbb {C}^{N_{\\mathbf {W}} \\times N_{\\mathbf {F}}}$ consisting of the received signal of all combinations of probing beams can be written as $\\mathbf {Y} = \\sqrt{P_T}\\mathbf {W}^{H}\\mathbf {H}\\mathbf {F}\\mathbf {s}+\\mathbf {W}^{H}\\mathbf {n},$ where $\\mathbf {s}$ is the vector of transmitted signals and $\\mathbf {n} \\sim \\mathcal {CN}(0,\\sigma ^2\\mathbb {I})$ is the measurement noise.",
"We focus on the special case where $N_{\\mathbf {F}}=N_{\\mathbf {W}}=N_{\\mathrm {probe}}$ : the BS sweeps $N_{\\mathrm {probe}}$ probing beams while the UE uses a different Rx beam for each Tx probing beam so that the total number of probing beam pairs is $N_{\\mathrm {probe}}$ .",
"The UE then reports the $N_{\\mathrm {probe}}$ received signal power measurements.",
"As a result, the input feature to the beam synthesizer functions is $\\mathbf {z} =\\begin{bmatrix}|[\\mathrm {diag}(\\mathbf {Y})]_1|^2 & \\cdots & |[\\mathrm {diag}(\\mathbf {Y})]_{N_{\\mathrm {probe}}}|^2\\end{bmatrix}^T.$ In principle, both the measurement and the feedback procedures are flexible and can be design choices.",
"For example, the number of Tx and Rx probing beams can be arbitrary.",
"The UE can measure and feedback any or all combinations of probing beams, and the beam synthesizer functions can utilize the full $\\mathbb {C}^{N_{\\mathbf {W}} \\times N_{\\mathbf {F}}}$ measurement matrix to generate the BF vectors.",
"The UE may also feedback just $N_{\\mathbf {F}}$ measurements corresponding to the strongest Rx probing beam for each Tx probing beam.",
"The specific design considered in this work is motivated by our wish to reduce the measurement and feedback overhead, i.e., $N_{\\mathrm {probe}} \\ll N_{\\mathbf {F}}N_{\\mathbf {W}}$ .",
"Furthermore, it is also more compatible with the RS feedback procedure in 5G NR, where the UE beam is transparent to the BS.",
"However, there is extensive scope for future work exploring other approaches."
],
[
"Problem formulation",
"The parameters of the proposed method include the Tx and Rx probing beams $\\mathbf {F},\\mathbf {W}$ as well as the beam synthesizers $f_{\\textnormal {T}},f_{\\textnormal {R}}$ , which need to be optimized with respect to a utility function.",
"With the unit modulus constraint of the probing and predicted beams in mind, the optimization problem can be written as: $\\begin{array}{rrclcl}\\displaystyle \\max _{\\mathbf {F},\\mathbf {W},f_{\\textnormal {T}},f_{\\textnormal {R}}} & & \\mathcal {U} & & & \\\\\\textrm {s.t.}",
"& \\mathbf {v}_{\\mathrm {T}} & = & f_{\\textnormal {T}}(\\mathbf {z}) & &\\\\& \\mathbf {v}_{\\mathrm {R}} & = & f_{\\textnormal {R}}(\\mathbf {z}) & &\\\\&|[\\mathbf {v}_{\\mathrm {T}}]_{i}| & = & \\frac{1}{\\sqrt{N_{\\textnormal {T}}}}, & \\forall i=1,\\cdots ,N_{\\textnormal {T}} &\\\\&|[\\mathbf {v}_{\\mathrm {R}}]_{i}| & = & \\frac{1}{\\sqrt{N_{\\textnormal {R}}}}, & \\forall i=1,\\cdots ,N_{\\textnormal {R}} &\\\\&|[\\mathbf {F}]_{i,j}| & = & \\frac{1}{\\sqrt{N_{\\textnormal {T}}}}, & \\forall i=1,\\cdots ,N_{\\textnormal {T}}, & \\forall j=1,\\cdots ,N_{\\mathbf {F}} \\\\&|[\\mathbf {W}]_{i,j}| & = & \\frac{1}{\\sqrt{N_{\\textnormal {R}}}}, & \\forall i=1,\\cdots ,N_{\\textnormal {R}}, & \\forall j=1,\\cdots ,N_{\\mathbf {W}}.",
"\\\\\\end{array}$ There are several considerations when designing the utility function.",
"The beam synthesizer functions should generate beams that tend to maximize the BF gain for each channel realization.",
"The probing beams in $\\mathbf {F}$ and $\\mathbf {W}$ serve two important purposes.",
"First, they provide helpful information to the beam synthesizers and thus should capture characteristics of the channel.",
"Second, they should allow the BS to discover new UE during the IA process and thus should satisfy a minimum SNR requirement.",
"Since the optimal BF vectors can be easily computed in closed-form in the simpler MISO or SIMO setting, the synthesized beams can be optimized to resemble the optimal EGT or EGC beams by minimizing a supervised loss function such as the MSE in [15].",
"However, finding the EGT and EGC beam pairs in the MIMO setting requires solving its own non-convex optimization problem and may require a grid-search over possible weights, making a supervised utility function undesirable.",
"To this end, we propose a two-component unsupervised utility function that does not require explicit labels for the probing or synthesized beams: $ \\mathcal {U} &={\\left\\lbrace \\begin{array}{ll}\\mathcal {U}_{\\textnormal {BF}} & \\text{if } \\mathcal {H}_{\\textnormal {IA}} = \\mathcal {H} \\\\\\gamma \\mathcal {U}_{\\textnormal {BF}} + (1-\\gamma )\\mathcal {U}_{\\textnormal {IA}} & \\text{otherwise} \\\\\\end{array}\\right.}",
"\\\\\\mathcal {U}_{\\textnormal {BF}} &= \\mathop {\\mathbb {E}}\\limits _{\\mathbf {H} \\in \\mathcal {H}} \\left[ \\frac{|\\mathbf {v}_\\mathrm {R}^H \\mathbf {H} \\mathbf {v}_\\mathrm {T}|^2}{||\\mathbf {H}||^2_{\\text{F}}} \\right] \\\\\\mathcal {U}_{\\textnormal {IA}} &= \\mathop {\\mathbb {E}}\\limits _{\\mathbf {H} \\in \\mathcal {H} \\setminus \\mathcal {H}_{\\textnormal {IA}}} \\left[ \\max \\limits _{\\mathbf {f} \\in \\mathcal {F}, \\mathbf {w} \\in \\mathcal {W}} \\frac{|\\mathbf {w}^H \\mathbf {H} \\mathbf {f}|^2}{||\\mathbf {H}||^2_{\\text{F}}} \\right] \\\\\\mathcal {H}_{\\textnormal {IA}} &= \\lbrace \\mathbf {H} \\in \\mathcal {H}: \\max \\limits _{\\mathbf {f} \\in \\mathcal {F}, \\mathbf {w} \\in \\mathcal {W}} \\frac{P_T|\\mathbf {w}^H \\mathbf {H} \\mathbf {f}|^2}{|\\mathbf {w}^H \\mathbf {n}|^2} \\ge \\textnormal {SNR}_{\\textnormal {TH}} \\rbrace $ Let $\\mathcal {H}$ denote the set of channel realizations corresponding to possible UE locations in the cell.",
"The set of Tx and Rx probing beams are denoted by $\\mathcal {F}$ and $\\mathcal {W}$ .",
"The first term $\\mathcal {U}_{\\textnormal {BF}}$ focuses on the BF gain of the synthesized beams and is the end-to-end objective.",
"An obvious choice for the utility function is the average SNR or achievable rate of the predicted beam pairs, such as adopted in [20].",
"However, it tends to emphasize UE with good channels more and neglect cell edge UE.",
"In order to provide better coverage to all UE, we maximize the average BF gain normalized by the channel norm to give equal emphasis even for UE with worse channels.",
"The second term $\\mathcal {U}_{\\textnormal {IA}}$ ensures the coverage of the probing beams.",
"It maximizes the average normalized BF gain of the strongest probing beam pairs for those UE that cannot meet the minimum SNR threshold $\\textnormal {SNR}_{\\textnormal {TH}}$ with any of the probing beam pairs.",
"The coefficient $\\gamma \\in [0,1]$ is a design parameter to balance the trade-off between the gain of the synthesized beams and of the probing beams."
],
[
"The proposed NN architecture",
"The optimization problem in REF is difficult to solve since the unit-modulus constraints are non-convex while the functions $f_{\\textnormal {T}}$ and $f_{\\textnormal {R}}$ are generally unknown.",
"We propose to parameterize the probing beams $\\mathbf {F},\\mathbf {W}$ as well as the beam synthesizers $f_{\\textnormal {T}},f_{\\textnormal {R}}$ using NN and optimize them in a data-driven fashion.",
"The overall NN architecture is illustrated in Fig.",
"REF .",
"During the offline training phase, the input to the entire NN model are the channel matrices.",
"The Tx and Rx probing beams are implemented in the complex NN module as two complex matrices.",
"The entire complex NN module can be considered to perform an affine transformation of the channel matrix.",
"To enforce the unit-modulus constraint, the complex matrices are normalized element-wise by the absolute value.",
"We find this implementation to perform better empirically compared to the implementation used in our previous work [22], where the complex matrices are computed using the phase-shift values.",
"The complex NN module computes the composite matrix of received signals of all combinations of probing beams in REF .",
"The power of the $N_{\\textnormal {probe}}$ probing beam pairs corresponding to the diagonal elements are extracted and fed into the Tx and Rx beam synthesizer functions, each parameterized using an MLP.",
"The MLP consists of 2 hidden layers with ReLU activation and a final linear layer outputting the real and imaginary parts of the synthesized BF vector.",
"The final predicted beams are normalized element-wise to enforce the unit-modulus constraint.",
"With each batch of training channel realizations, the utility can be computed using the BF gain of the synthesized beams and that of the best probing beam pairs.",
"The Tx and Rx probing beams as well as the beam synthesizer functions are trained through stochastic gradient descent and backpropagation.",
"Figure: The architecture of the proposed NN, including the probing beam pairs 𝐅\\mathbf {F} and 𝐖\\mathbf {W} and the beam selection functions f T (·)f_{\\textnormal {T}}(\\cdot ) and f R (·)f_{\\textnormal {R}}(\\cdot ).",
"The Tx and Rx beam synthesizers have the same architecture."
],
[
"Practicality of the proposed method",
"The NN model is trained offline prior to deployment.",
"The training data can be obtained through ray-tracing simulations or from measurements.",
"Due to the difficulty of obtaining CSI in real time, online training and adaptation of the NN remain an open research problem.",
"On the other hand, the model proves to be fairly robust against imperfect training data, as we will show in Section REF .",
"After the offline training phase, the probing beam pairs are extracted from the complex NN module and implemented in RF.",
"The beam synthesizers $f_{\\mathrm {T}}, f_{\\mathrm {R}}$ are implemented at the BS and the UE respectively.",
"During deployment, the BS and the UE periodically sweep the probing beam pairs while the UE measures the received signal power.",
"The probing measurements are then reported to the BS.",
"The BS and the UE use the probing measurements as inputs to $f_{\\mathrm {T}}$ and $f_{\\mathrm {R}}$ to synthesize the Tx and Rx beams.",
"Since the probing beams and the beam synthesizers are site-specific, $\\mathbf {W}$ and $f_{\\mathrm {R}}$ need to be transmitted to the UE.",
"In a non-standalone system, this can be done through a lower-frequency side link.",
"A new UE can then complete IA using the strongest probing beam pair.",
"We have previously shown that the Tx probing beams can be used for IA in the MISO setting [1].",
"Therefore in a standalone system, the BS may sweep the Tx probing beams while a new UE receives using a quasi-omnidirectional beam.",
"The UE can complete IA with the strongest Tx probing beam, download the Rx components $\\mathbf {W}$ and $f_{\\mathrm {R}}$ , then perform the full joint Tx-Rx beam alignment."
],
[
"Baselines and Metrics",
"The proposed method optimizes the GF beam synthesizers and the probing beams through DL, hence is referred to as the DL-GF method.",
"It will be compared with five baselines: the DL-CB method, the exhaustive search, the genie with DFT codebooks, the DFT+EGC method, and MRT+MRC.",
"DL-CB The DL-CB method replaces the Tx and Rx beam synthesizers with two classifiers that predict the optimal beam indices in the BS and UE codebooks.",
"It is an extension of our previous work [22] to the MIMO scenario.",
"To improve its accuracy, the DL-CB method can sweep the top few candidate beam pairs predicted by the classifiers.",
"In our experiment, the best beam pair is be selected after trying all combinations of the top-3 predicted Tx and Rx beams.",
"Exhaustive search The exhaustive search sweeps and measures all combinations of beam pairs in the BS and UE codebooks and selects the beam pair with the highest received signal power.",
"Since the measurements are corrupted by noise, the beam pair selected by the exhaustive search may not be optimal.",
"Genie DFT A genie always selects the optimal beam pair in the BS and UE codebooks.",
"It is the same as the exhaustive search when the measurement noise power is zero.",
"DFT+EGC Only the BS has a codebook in the DFT+EGC baseline.",
"The BS exhaustively tries all beams in its codebook while the UE uses the corresponding EGC vector for each Tx beam.",
"The best beam pair is selected assuming no measurement noise.",
"It is expected to perform better than the genie DFT baseline due to the additional degree of freedom at the UE.",
"MRT+MRC Neither the BS nor the UE uses a codebook.",
"The BS uses the optimal MRT beam while the UE uses the MRC beam.",
"The BF gain can be computed through an eigendecomposition of $\\mathbf {H}^H\\mathbf {H}$ .",
"The MRT+MRC baseline is the theoretical upper bound under the unit-power constraint and cannot be achieved under the stricter unit-modulus constraint.",
"One important performance metric for beam alignment is the SNR achieved by the selected beams.",
"We will compare the average SNR as well as the SNR distribution across different baselines.",
"Secondly, new UE need to satisfy a minimum SNR requirement so that they can be discovered during the IA process and connect to the BS.",
"This is generally not a concern for CB approaches that adopt uniform codebooks, but may be problematic when the probing beams are site-specific and have severe coverage holes.",
"Therefore, we will also investigate the misdetection probability of UE, which is the probability that a UE achieves below-threshold SNR with the strongest probing beam pair, formally defined as $\\textnormal {misdetection probability} = \\mathop {\\mathbb {E}}\\limits _{\\mathbf {H} \\in \\mathcal {H}} \\left[ \\mathbb {1}_{\\mathcal {H}_{\\textnormal {IA}}} (\\mathbf {H})\\right].$ The SNR threshold is chosen to be -5 dB [24].",
"Four scenarios from the public DeepMIMO dataset [25] are considered to capture a wide range of propagation environments, including indoor and outdoor environments, 28 GHz and 60 GHz carrier frequencies, as well as LOS and NLOS UE.",
"The channel realizations are computed through ray-tracing with a state-of-the-art commercial-grade software [26], which is one of the most accurate ways of simulating mmWave channels once the environment topology is specified.",
"The BS and UE adopt UPA with half-wavelength spacing.",
"The simulation parameters are summarized in Table REF for the four scenarios described below.",
"Table: Simulation ParametersO1 Scenario.",
"The O1 scenario captures an outdoor urban street environment with LOS UE.",
"An illustration of the scenario is shown in Fig.",
"REF .",
"We select BS 3 and UE from row #800 to row #1200 from the original dataset.",
"The BS is placed on the street side and a total of 72,581 UE are placed on a uniform grid on the street.",
"The carrier frequency is 28 GHz.",
"I3 Scenarios.",
"The I3 scenarios captures an indoor office environment with both LOS and NLOS UE.",
"An illustration of the environment is shown in Fig.",
"REF .",
"There are two BS in this scenario: BS 1 is placed on the inside wall of the conference room and BS 2 is placed on the opposite wall.",
"A grid of LOS UE is placed inside the conference room and a grid of NLOS UE is placed in the corridor outside.",
"There are a total of 118,959 UE and the carrier frequency is 60 GHz.",
"We consider two scenarios based on this environment, each with one of the two BS activated.",
"O1 Blockage Scenario.",
"The O1 blockage scenario is artificially created based on the O1 scenario with a metal screen placed in front of the BS and two reflectors placed on both ends of the street, as illustrated in Fig.",
"REF .",
"The carrier frequency is 28 GHz and there a total of 497,931 UE positions.",
"While it may not be representative of practical mmWave deployments, the extreme topology allows us to gain some intuition on the beam patterns learned in this environment.",
"Figure: Illustration of the DeepMIMO ray-tracing scenarios."
],
[
"Evaluation",
"The datasets discussed in Section are partitioned so that 60% is used to train the NN, 20% is used for validation and hyper-parameter tuning while we report the performance on the remaining 20% testing data.",
"The NN are trained for 4000 epochs with a batch size of 800 using the ADAM optimizer [27].",
"The $\\gamma $ parameter in the utility function is chosen to be 0.3 empirically.",
"The measurement noise is AWGN.",
"For CB baselines, the BS and the UE adopt over-sampled DFT codebooks with 256 and 64 beams respectively.",
"The rest of the simulation parameters are summarized in Table REF ."
],
[
"Can DL-GF Achieve Good SNR?",
"An ideal beam alignment method should quickly select near-optimal beams and achieve good SNR.",
"The average SNR achieved by the proposed DL-GF method and the baselines in the more realistic O1 and I3 scenarios are shown in Fig.",
"REF , REF , REF .",
"It is evident that with the DL-CB approach, the beam classifiers can accurately select the best beam pairs.",
"With increasing numbers of probing beam pairs, the average SNR achieved by the DL-CB baseline approaches that by the genie DFT.",
"Nevertheless, its performance is limited by the resolution of the BS and UE codebooks since there is still a 1 dB gap between the DFT+EGC baseline and the genie DFT.",
"On the other hand, the proposed DL-GF method improves upon DL-CB with its fully synthesized beams and is able to beat the exhaustive search with 20 probing beam pairs in the O1 scenario, 12 in I3 with BS 1 activated and 8 in I3 with BS 2 activated.",
"It is eventually able to outperform the DFT+EGC baseline with as few as 20 probing beam pairs in the I3 scenarios and 32 in O1.",
"A comparison of the 10th, 50th, 90th percentile and the average SNR with 32 probing beam pairs is shown in Table REF .",
"The proposed DL-GF method outperforms the exhaustive search everywhere in the distribution in both the O1 and the I3 scenario with BS 1 activated, and is only slightly worse in the 10th percentile SNR in the I3 scenario with BS 2 activated.",
"Meanwhile, the misdetection probability is just 0.172$\\%$ , 0.008$\\%$ and 2.283$\\%$ in the O1, I3 BS 1 and I3 BS 2 environments, guaranteeing that the vast majority of UE can complete IA with one of the probing beams.",
"The gain of DL-GF is twofold.",
"First, in the probing phase, the site-specific probing beams allows the BS and the UE to capture crucial channel information with fewer measurements.",
"Second, in the beam prediction phase, the beam synthesizer functions can be intuitively viewed as infinitely large codebooks, whose resolution in practice is only limited by the floating-point precision of the NN and the range of probing measurements.",
"By generating analog beams at increased spatial resolutions that are also adapted to the specific environment, DL-GF achieves additional gain over the standard DFT codebooks.",
"Figure: The average SNR vs. number of probing beam pairs in the O1 scenario.Figure: The average SNR vs. number of probing beam pairs in the I3 BS 1 scenario.Figure: The average SNR vs. number of probing beam pairs in the I3 BS 2 scenario.Table: SNR Comparison with 32 Probing Beam Pairs"
],
[
"The SNR vs. Beam Alignment Speed Trade-off",
"Conventional CB beam alignment approaches can always find better beams more frequently by trying more candidates and increasing the resolution of the codebook.",
"Similarly, the proposed DL-GF method can achieve better SNR by sweeping more probing beams.",
"However, such SNR gain usually occurs at the cost of higher latency.",
"In practical cellular systems with multiple UE, the beam alignment procedure may need to be performed for each UE.",
"It is therefore important for operators to consider the trade-off between the SNR and the overall beam alignment speed, which we define as the reciprocal of the total number of beams swept for all UE.",
"The average SNR achieved at each beam alignment speed for DL-GF, DL-CB and the exhaustive search is shown in Fig.",
"REF .",
"Compared to the exhaustive search, the proposed DL-GF method is better by 5 to 10 dB in terms of the average SNR at a given speed or faster by around two orders of magnitude when the achieved average SNR is from 10 to 15 dB.",
"The proposed GF approach is also strictly better than the CB method, achieving 1-2.5 dB higher SNR at each beam alignment speed.",
"With multiple UE, other beam alignment methods that require an additional beam sweeping phase for each UE are expected to achieve a similar SNR-speed trade-off as the DL-CB with top-$k$ search does.",
"Since DL-GF is faster than DL-CB with top-3 search by an order of magnitude with 10 or 20 UE, similar gains can be expected over methods such as the hierarchical search and active learning approaches.",
"Interestingly, increasing the number of probing beams is often more efficient than searching more candidates for the DL-CB method, particularly when a high beam alignment speed is required.",
"Figure: Average SNR vs. the total beam alignment speed in the O1 scenario.A more detailed analysis of the total number of beams swept for different beam alignment methods in a cell with $K$ UE is shown in Table REF .",
"In the hierarchical search, the BS and the UE adopt 2-tier codebooks with $N_{\\mathbf {F}}$ and $N_{\\mathbf {W}}$ wide beams respectively.",
"The BS first performs the Tx hierarchical search while the UE uses an omnidirectional Rx beam, then the UE performs the Rx hierarchical search.",
"The exhaustive search requires sweeping all combinations of the $M_{\\textnormal {T}}$ Tx beams and $M_{\\textnormal {R}}$ Rx beams, which amounts to a total of 16,384 beam pairs in our setting.",
"Methods such as the hierarchical search and DL-CB with top-$k$ search require an additional beam sweeping phase for each UE.",
"As a result, the overall beam sweeping overhead increases linearly with the number of UE.",
"On the other hand, since the proposed DL-GF method predicts the beams in one shot after sweeping a common set of probing beams for all UE, its beam sweeping overhead is constant regardless of the number of UE.",
"In our setting, DL-GF reduces the beam sweeping overhead by over 500$\\times $ while scaling optimally with increasing number of UE.",
"Table: Beam Sweeping Overhead for KK UEsTo better understand how the probing beam pairs and the beam synthesizer functions achieve better performance, we investigate the learned beam patterns in the O1 Blockage scenario.",
"For easier visualization, the BS and the UE are equipped with ULA arrays that only beamform in the azimuth domain.",
"A UE located on the left of the BS is selected from the testing dataset as a case study.",
"With 4 probing beam pairs, the Tx and Rx probing patterns are heavily adapted to the propagation environment by directing most of the energy towards the reflectors on both ends of the street while avoiding the blockage in the broadside direction, as shown in Fig.",
"REF .",
"On the other hand, the synthesizers predict sub-optimal beams.",
"Both the Tx and Rx beams have a main lobe pointing roughly towards the UE.",
"The Rx beam has two strong lobes while the Tx beam has a single narrow main lobe.",
"As the number of probing beam pairs is increased to 20, the learned probing beams have much larger spatial coverage, as shown in Fig.",
"REF .",
"This is particularly noticeable in the Rx probing beams since the range of AoA is much larger compared to that of the AoD with random UE orientations and more variations in the location of UE.",
"Utilizing the increased information gathered by the probing beams, the beam synthesizers also learn to focus energy more accurately and precisely.",
"Both the predicted Tx and Rx beams are narrower, have a single lobe and point more accurately at the UE.",
"Overall, this allows DL-GF to achieve better SNR by adopting more probing beams.",
"Figure: Radiation patterns of learned probing beam pairs and predicted beams in the O1 blockage scenario, N probe =4N_{\\mathrm {probe}}=4.Figure: Radiation patterns of learned probing beam pairs and predicted beams in the O1 blockage scenario, N probe =20N_{\\mathrm {probe}}=20."
],
[
"Robustness to Noisy Measurements",
"The beam synthesizers rely on the probing measurements to generate the BF weights.",
"While measurement noise can cause search errors in exhaustive and hierarchical searches, it can also impact the prediction of the NN models.",
"A comparison of the average SNR with increasing measurement noise PSD is shown in Fig.",
"REF .",
"As expected, the performance of the DL-GF, DL-CB and exhaustive search deteriorates with increasing measurement noise power, which becomes noticeable with noise PSD larger than -167 dBm/Hz.",
"The exhaustive search is the most robust when the noise PSD is less than -143 dBm/Hz, while the one-shot DL-GF and DL-CB methods performs similarly.",
"The DL-CB baseline is trained at each noise PSD level and can improve its robustness to noise by trying the top-3 candidates.",
"There is also significant gain from training the DL-GF models at each noise PSD level compared to training with a fixed measurement noise power and testing with higher noise power.",
"Interestingly, the average SNR of the exhaustive search becomes the worst when the measurement noise PSD reaches -137 dBm/Hz while that of the DL-based methods plateaus at around -143 dBm/Hz.",
"At this point, the exhaustive search is essentially producing randomly guesses since the beam measurements are dominated by noise.",
"On the other hand, the DL-based methods converge to always predict the same beam pair.",
"In the case of DL-CB, the predicted beam pair happens to be the most frequent optimal beam pair.",
"Figure: Comparison of the average SNR with increasing measurement noise PSD in the O1 scenario.",
"The DL-GF and DL-CB methods have 32 probing beam pairs."
],
[
"Robustness to Imperfect Training Data",
"Training DL models typically requires high-quality data.",
"While ray-tracing is one of the most accurate ways to simulate mmWave channels, it is not easily scalable to a city-wide deployment and the generated data may still differ from actual channels due to mismatched environments and imperfect propagation models.",
"If the training data is acquired through measurement campaigns, noisy measurement and channel estimation errors would also affect the quality of the data.",
"It is therefore important to investigate the performance of the proposed method when there is a mismatch between the channels used for training and the actual channels during deployment.",
"To simulate imperfect training data, the training channel matrices are corrupted with AWGN while the model is tested on clean channel data.",
"The average SNR of the proposed DL-GF method, the DL-CB baseline and the exhaustive search with increasing NMSE in the training data is shown in Fig.",
"REF .",
"Both the DL-GF and the DL-CB methods are relatively robust to noisy training data, experiencing little performance degradation when the channel NMSE is smaller than -3 dB.",
"A small amount of noisy in the training data actually benefits the proposed DL-GF method, which is a common phenomenon in ML where noisy training data can sometimes improve the robustness and generalizability of ML models [28].",
"Even with a channel NMSE of 1 dB, the proposed DL-GF method can still outperform the exhaustive search and the DL-CB baseline trained on clean data.",
"The added noise likely does not fundamentally shift the distribution of channels but instead makes it more “fuzzy”.",
"During the unsupervised training procedure, the proposed NN still learn to beamform on these noisy channels and generalize well to the clean actual channels.",
"Figure: Comparison of the average SNR with increasing training channel NMSE in the O1 scenario.",
"The DL-GF and DL-CB methods have 32 probing beam pairs."
],
[
"Reducing the Feedback Overhead",
"The gain of DL-GF can be partially attributed to the much richer uplink feedback: it uses all the measurements as feature vectors to synthesize the BF weights instead of simply selecting the strongest beam.",
"The feedback overhead of different methods are summarized in Table REF .",
"To reduce the feedback overhead, UE can report measurements of a few strongest probing beam pairs.",
"The vector $\\mathbf {z}$ is masked so that only the top-$m$ highest measurements are non-zero and the NN is trained from scratch.",
"Since the shape of the feature vector is roughly maintained, reducing the feedback only results in a small degradation in the average SNR, as shown in Fig.",
"REF .",
"For instance, with 32 probing beam pairs in total, the average SNR drops by 0.4 dB when reporting 4 beam pairs and by only 0.12 dB when reporting the best 8.",
"Note that $f_{\\textnormal {T}}$ and $f_{\\textnormal {R}}$ use the same masked feature vector in our experiments.",
"In practice, the UE may use the full measurement vector to synthesize its beam while the BS uses the reduced feature, which will likely lead to even smaller performance loss.",
"Figure: Average SNR vs. the number of probing beam pairs with different feedback constraints."
],
[
"Impact of Random UE Orientations",
"The orientation of UE has a large impact on the performance of the proposed DL-GF method.",
"A comparison of the average SNR and the misdetection probability with and without random UE rotation in the O1 scenario is shown in Fig.",
"REF .",
"Without random UE rotation, the proposed method can achieve better average SNR and lower misdetection probability with significantly fewer probing beam pairs.",
"The random UE rotation increases the effective range of AoA.",
"As a result, more probing beams are required to capture sufficient channel information on the UE side.",
"The UE's probing beams also need to distribute energy and cover a larger angular space, leading to reduced BF gain and worse misdetection probability.",
"Figure: The average SNR and misdetection probability of the proposed DL-GF method with and without random UE orientations in the O1 scenario."
],
[
"Reducing the Misdetection Probability",
"The ideal beam alignment solution should not only achieve high BF gain for connected UE, but also be able to discover new UE and allow them to complete IA.",
"With the proposed DL-GF method, a new UE can do so with one of the probing beam pairs if it satisfies a minimum SNR requirement.",
"Naturally, adopting more probing beams will allow each to cover a smaller angular space, thus increasing the gain of the probing beams and reducing the misdetection probability.",
"By explicitly incorporating the IA performance in the utility function, we provide another tuning nob to reduce the misdetection probability.",
"The proposed utility function can indeed further reduce the misdetection probability when the number of probing beam pairs is fixed, as shown in Fig.",
"REF .",
"Compared to simply optimizing the BF gain of the synthesized beams ($\\gamma =1.0$ ), incorporating the IA performance ($\\gamma =0.3$ ) reduces the misdetection probability by roughly 5 percentage points with 4 probing beam pairs and by 2 percentage points with 12 probing beam pairs while suffering from little SNR loss.",
"Although the gain diminishes with more probing beams, the two-component utility function still provides a powerful tool to improve the IA coverage when the number of probing beams is limited.",
"Since the $\\mathcal {U}_{\\textnormal {IA}}$ term only covers UE below a specified SNR threshold, the proposed utility function also allows the NN model to adapt to different SNR requirements for IA.",
"As shown in Fig.",
"REF , the misdetection probability is reduced at various SNR threshold values for IA.",
"The gain increases if a higher SNR threshold is adopted.",
"Figure: The misdetection probability and average SNR with different γ\\gamma in the O1 scenario."
],
[
"Conclusion",
"We propose a beam alignment method that directly synthesizes the analog beams in one shot after sweeping a few probing beam pairs and design a NN architecture to optimize the probing beams and beam synthesizer functions through site-specific training.",
"Beating existing methods in terms of the overall SNR-speed trade-off often by orders of magnitude, the proposed method provides a promising new paradigm for mmWave beam alignment.",
"Next-generation cellular systems can reap the benefits without overhauling the existing beam sweeping-based framework: the probing beams can be transmitted using periodic RS while traditional beamforming codebooks are replaced with NN beam synthesizers.",
"Operators can select the number of probing beams based on the SNR and IA coverage requirement for each site.",
"There are many possible directions that this line of research could be extended, many of which have been identified throughout the paper.",
"For example, scaling the site-specific training to city-wide networks and online adaptation to dynamic environments remain open research problems.",
"The beam alignment problem has mostly been investigated to date from the BS point of view.",
"It is important to also consider the unique beam alignment challenges on the UE side arising from multiple antenna panels, precarious dynamic rotations and more demanding power management."
]
] | 2209.08198 |
[
[
"Theoretical predictions of melting behaviors of hcp iron up to 4000 GPa"
],
[
"Abstract The high-pressure melting diagram of iron is a vital ingredient for the geodynamic modeling of planetary interiors.",
"Nonetheless, available data for molten iron show an alarming discrepancy.",
"Herein, we propose an efficient one-phase approach to capture the solid-liquid transition of iron under extreme conditions.",
"Our basic idea is to extend the statistical moment method to determine the density of iron in the TPa region.",
"On that basis, we adapt the work-heat equivalence principle to appropriately link equation-of-state parameters with melting properties.",
"This strategy allows explaining cutting-edge experimental and ab initio results without massive computational workloads.",
"Our theoretical calculations would be helpful to constrain the chemical composition, internal dynamics, and thermal evolution of the Earth and super-Earths."
],
[
"The effectiveness of the SMM-WHEP SCHEME",
"Recently, physicists have succeeded in constraining the melting curve of aluminum, copper, and vanadium by different methods.",
"Hence, in the present Supplemental Material, these metallic systems are utilized to prove the quality of the SMM-WHEP scheme.",
"Table: The Morse parameters for aluminum, copper, and vanadium [S1, S2].First, we extract the Morse parameters from Refs.",
"[S1, S2] to calculate the SMM EOS (see Table S1).",
"Figure S1 shows that the Morse potential can help us capture the atomic rearrangement of aluminum, copper, and vanadium with high accuracy [S3$-$ S5].",
"Based on the Vinet fitting [S6], we acquire the isothermal bulk modulus $K_0$ and its pressure derivative $K_0^{\\prime }$ in Table S2.",
"Table: The input data for WHEP calculations in the case of aluminum, copper, and vanadium.",
"The detailed information about (P 1 ,T 1 )(P_1,T_1) and (P 2 ,T 2 )(P_2,T_2) can be found in Refs.",
"[S7, S8].Next, we enter $K_0$ and $K_0^{\\prime }$ into the WHEP model to locate the melting point of aluminum, copper, and vanadium between 0 and 500 GPa.",
"As presented in Figures S2, S3, and S4, this strategy enables us to redescribe all previous experimental and computational results [S7$-$ S22] at a quantitative level.",
"Besides, it is explicitly observed that our modified WHEP version (Equation (17)) is superior to the original one (Equation (16)).",
"The melting-temperature underestimation is adequately addressed by replacing the Murnaghan EOS with the SMM EOS.",
"The aforementioned evidence verifies the reliability and flexibility of SMM-WHEP analyses in investigating the melting process at intense pressures."
],
[
"Two-phase simulations for iron",
"To strengthen SMM-WHEP predictions for hcp-iron, we perform classical molecular dynamics (CMD) simulations via the open-source LAMMPS package [S23].",
"The two-phase approach (TPA) [S24$-$ S28] is adopted to determine the melting temperature of hcp-iron with the DFT-based Morse pairwise potential [S2] ($D=0.6317$ eV, $\\alpha =1.4107$ Å$^{-1}$ , and $r_0=2.6141$ Å).",
"Physically, this modern method allows us to overcome the overheating problem of conventional one-phase calculations by adding surface effects [S24$-$ S28].",
"Moreover, it is widely demonstrated that the CMD-TPA can give us exact melting information about the chosen interatomic potential [S24$-$ S28].",
"Technical details about the CMD-TPA are summarized in Figure S5.",
"Overall, our computational process at a given pressure can be divided into two principal stages: (i) preparation (Figure S5(a)) and (ii) production (Figure S5(b)).",
"In stage (i), we construct two identical perfect hcp supercells, each containing 2000 atoms.",
"Then, one of these lattices is isochorically heated to an extreme temperature to melt completely.",
"Finally, we put the obtained solid and liquid structures together in our simulated box.",
"A small gap is created between them to prevent atoms from overlapping and lessen the strain at the common interface.",
"In stage (ii), our two-phase system is simulated in the $NPT$ ensemble.",
"Each CMD run is done within 100 ps (100000 timesteps) to find the equilibrium volume.",
"On that basis, the melting transition is detected via an abrupt change in the volume-temperature diagram.",
"Note that the TPA is quite different from the coexistence method [S29], where the solid and liquid parts can coexist under the $NVE$ condition.",
"For the TPA, the system is equilibrated towards a monophase.",
"To be more specific, it is solidified at $T<T^m$ and liquefied at $T>T^m$ .",
"Our CMD-TPA results are compared to SMM-WHEP data in Figure S6.",
"One can readily see an excellent agreement between the above approaches under Earth-core conditions.",
"For instance, at the ICB, CMD-TPA simulations yield $T^m=6050$ K, which is merely lower than the SMM-WHEP output by 3.09 %.",
"This quantitative consistency reaffirms the “steep-melting-line\" scenario for iron as well as the accuracy of SMM-WHEP approximations.",
"Figure: (Color online) The EOS of aluminum, copper, and vanadium given by our SMM calculations, DAC experiments [S3], RW measurements [S4], and DFT simulations [S5].Figure: (Color online) The melting diagram of aluminum obtained from our SMM-WHEP theory and other approaches [S7, S9--S15].Figure: (Color online) Our SMM-WHEP outputs and the prior measured/simulated results [S7, S14, S16--S21] for the melting profile of copper.Figure: (Color online) Compression effects on the melting temperature of vanadium provided by our SMM-WHEP analyses and recent studies [S8, S22].Figure: (Color online) Main stages to derive the high-pressure melting point of hcp-iron from our CMD-TPA simulations.",
"Structural visualization is made by VESTA [S30].Figure: (Color online) Comparison between CMD-TPA and SMM-WHEP results for the melting curve of hcp-iron.",
"Our CMD-TPA data can be well described by the Simon-Glatzel equation as T m (K)=1811P 26.02+1 0.46 T^m(K)=1811\\left(\\cfrac{P}{26.02}+1\\right)^{0.46}, where PP is in GPa."
]
] | 2209.08220 |
[
[
"Breather Solutions to a Two-dimensional Nonlinear Schr\\\"odinger Equation\n with Non-local Derivatives"
],
[
"Abstract We consider the nonlinear Schr\\\"odinger equation with non-local derivatives in a two-dimensional periodic domain.",
"For certain orders of derivatives, we find a new type of breather solution dominating the field evolution at low nonlinearity levels.",
"With the increase of nonlinearity, the breathers break down, giving way to wave turbulence (or Rayleigh-Jeans) spectra.",
"Phase-space trajectories associated with the breather solutions are found to be close to that of the linear system, revealing a connection between the breather solution and Kolmogorov-Arnold-Moser (KAM) theory."
],
[
"Breather solutions in other situations",
"In the main paper, we restrict our focus to a two-dimensional (2D), defocusing Majda-McLaughlin-Tabak (MMT) model without forcing or dissipation.",
"In this section, we show the occurrence of the breather in additional contexts: the MMT model with a focusing nonlinearity, as well as a defocusing forced-dissipated model.",
"Figure: 5T f 5T_f of the time series of HH (dashed) and H 2 H_2 (solid) for a fully-developed breather solution to the (a) focusing MMT equation with β=3\\beta =3 and (b) forced-dissipated MMT equation with β=2\\beta =2.",
"The corresponding H 4 H_4 for (c) the focusing system and (d) the forced-dissipated system are also provided.",
"The focusing system has ε=0.00028\\varepsilon = 0.00028 and the forced-dissipated system has ε=0.0016\\varepsilon = 0.0016.We begin with the focusing MMT model.",
"The focusing case is given by equation (1) and (2) in the main paper with $\\lambda = -1$ .",
"The parameter $\\lambda $ is well-known to control the modulational instability of the Nonlinear Schrödinger Equation (NLS) as well as the MMT model.",
"In the context of the MMT model, the sign of $\\lambda $ has been shown to affect the emergence of coherent structures in a one-dimensional MMT model with dispersion relation $\\omega =k^{1/2}$ [1].",
"In our results, however, we find no significant change in the breather behavior between the focusing/defocusing equations, suggesting (along with the fact that the breather exists only at weak nonlinearity) that modulational instability is not responsible for our breather.",
"In figure REF a/c, we show 5 fundamental periods $T_f$ of $H$ and its components $H_2$ and $H_4$ in a fully-developed breather state for the focusing equation with $\\beta = 2$ .",
"These results were obtained with an identical numerical setup to that of the main paper, and the presented results occur at low nonlinearity.",
"The pattern of the oscillating breather is also similar to that in the defocusing case.",
"Next, we present results obtained for a forced-dissipated system.",
"We again solve the defocusing 2D MMT model (with $\\beta = 2$ ), however with the addition of forcing and dissipation terms.",
"Specifically, we solve the equation $i\\frac{\\partial \\psi }{\\partial t}=\\vert \\partial _{\\mathbf {x}}\\vert ^{2}\\psi +\\lambda \\vert \\partial _{\\mathbf {x}}\\vert ^{\\beta /4}\\left(\\left|\\vert \\partial _{\\mathbf {x}}\\vert ^{\\beta /4}\\psi \\right|^{2}\\vert \\partial _{\\mathbf {x}}\\vert ^{\\beta /4}\\psi \\right) + F + D_1 + D_2,$ where $F$ represents the forcing and $D_1$ and $D_2$ represent dissipation.",
"These terms are explicitly defined in spectral domain, where $F = \\left\\lbrace \\begin{array}{l}F_{r}+iF_{i} \\ \\ \\ 7\\le k\\le 9 \\\\0 \\ \\ \\ \\text{otherwise},\\end{array} \\right.$ with $F_{r}$ and $F_{i}$ sampled from a Gaussian distribution of zero-mean, producing a standard white-noise forcing.",
"The dissipative terms are defined as $D_1 =& \\left\\lbrace \\begin{array}{l}-i\\nu _1 \\hat{\\psi }_{\\mathbf {k}} \\ \\ \\ k\\ge 100 \\\\0 \\ \\ \\ \\text{otherwise},\\end{array} \\right.",
"\\nonumber \\\\D_2 =& \\left\\lbrace \\begin{array}{l}-i\\nu _2 \\hat{\\psi }_{\\mathbf {k}} \\ \\ \\ k\\le 7 \\\\0 \\ \\ \\ \\text{otherwise},\\end{array} \\right.$ where $\\nu _1$ and $\\nu _2$ are dissipative constants.",
"We solve these equations in an identical manner to that of the main paper with nearly identical initial conditions (we now begin with a spectral peak at $k_p = 10$ ) and on a larger domain of $512\\times 512$ modes.",
"In figure REF b, $H$ and its two components are plotted for $5T_f$ in the breather state.",
"In this case, we do not expect the total Hamiltonian to be conserved, but rather to be quasi-steady for the fully developed system.",
"Nevertheless, the signature of the breather is clearly present.",
"Just as in the unforced case, we find that the wave action spectrum of the system is altered when the breather is present.",
"In Figure REF , we provide the fully developed spectra of the forced-dissipated system for several different orders of nonlinearity.",
"When the nonlinearity is low and the breather is present, we again see departure from a power-law spectrum, with a steeper tail region.",
"When nonlinearity is raised, we observe the the spectra of wave turbulence are restored (and an associated forward energy cascade develops) [2].",
"Figure: The fully-developed, angle-averaged wave action spectra at a few nonlinearity levels of the forced-dissipated system, with the Kolmogorov-Zakharov spectral slope of γ=-10/3\\gamma =-10/3 indicated (dashed)."
],
[
"secondary peaks in the breather cycle",
"We include in this section plots of the secondary peaks of $|\\psi |$ in the breather cycle, supplementing figure 2 in the main paper.",
"We choose $|\\psi |$ rather than Re$[\\psi ]$ (as in the main paper) to better resolve the smaller amplitudes of these secondary structures.",
"While the largest peaks in $H_4$ (Fig.",
"REF e) correspond to single peaks in $|\\psi |$ , the secondary peaks in $H_4$ correspond to grids of smaller peaks in $|\\psi |$ (Fig.",
"REF a/b/c/d).",
"Figure: Contour plots of |ψ||\\psi | at β=3\\beta =3 for ε=0.00071\\varepsilon =0.00071 at various stages of the cycle of the breather (a/b/c/d), corresponding chronologically to the times marked by the blue circles in (e) the time series of H 4 H_4."
],
[
"Numerical Validation of the Breather",
"In this section, we provide analyses and numerical tests that rule out the possibility that the breather we discuss is a numerical artifact.",
"In particular, we show that the breather solution is consistent under the change of integration scheme to a symplectic integrator.",
"the change of our dealiasing procedure.",
"an increase in the number of Fourier modes (spatial resolution).",
"We begin with point A. Symplectic integration of a Hamiltonian system preserves the phase space geometry of its solution.",
"Specifically, under Hamiltonian flow, structures such as sinks and limit cycles are forbidden by Liouville's theorem.",
"When using an integrator such as an explicit 4th-order Runge-Kutta scheme (RK4), however, these structures can be erroneously introduced into the solution, which may change the dynamics.",
"To ensure our breather is not an artifact introduced by non-symplectic integration, we implement a simple symplectic integrator, the implicit midpoint method (IMP) [3], to verify that we still obtain (and preserve) the breather solution.",
"In the IMP method, we solve the implicit nonlinear problem via fixed-point iteration.",
"For an identical numerical setup to that of the main paper, we allow the system to freely evolve under the IMP integration scheme.",
"We set $\\beta =3$ and simulate at the low nonlinearity of $\\varepsilon = 0.001$ .",
"Figure: The time series of (a) HH (dashed) and H 2 H_2 (solid) starting from t=0t=0 for the defocusing MMT equation with β=3\\beta =3 under symplectic integration, with the corresponding time series of (c) H 4 H_4.",
"A detailed view beginning at t=1000T f t=1000T_f of (b) HH, H 2 H_2 and (d) H 4 H_4 over 5T f 5T_f.We provide in figure REF a the evolution of $H$ and $H_2$ from $t=0$ , with $H$ very well conserved and $H_2$ indicating that the breather has already formed by $t=1000T_f$ .",
"The corresponding plot of $H_4$ is provided in REF c. Just as in the main paper, these plots of the initial evolution have a low sampling rate, leading to aliasing.",
"To confirm that the breather has the same signature in $H_2$ and $H_4$ as in the case of non-symplectic integration, high-sampling rate plots of $H$ ,$H_2$ (Fig.",
"REF b) and $H_4$ (Fig.",
"REF d) are also provided over $5T_f$ , showing no difference to those results of the main paper.",
"Thus, the breather is not an artifact of non-symplectic integration.",
"Next, we address point B.",
"In order to prevent the aliasing of modes due to the cubic nonlinearity of the MMT model, a standard $1/2$ dealiasing rule is applied after each product during the evaluation of the nonlinear term.",
"The $1/2$ dealiasing rule is typically implemented via zero-padding the truncated wave number domain: if $k_m$ is the maximum resolved wave number in our simulation (that is oriented along the $x$ and $y$ axes), then, in each direction, zero-padding is included such that for a computation domain of size $[-2k_m,2k_m]^2$ , the non-zero (resolved) Fourier modes are only contained in the box $[-k_m,k_m]^2$ .",
"The zero-padding is enforced by setting all modes outside the box $[-k_m,k_m]^2$ to zero after each product is taken.",
"This procedure, however, has the effect of subtly changing the evolution of the system.",
"In order to be assured that the breather is not an artifact of our dealiasing scheme, we first show that our dealiasing leads to a slightly modified Hamiltonian system (analytically), then we show that the breather is preserved in the original system without modification.",
"We start by writing down the truncated Hamiltonian that we aim to numerically simulate: $H = \\underset{\\begin{array}{c}\\mathbf {k} \\\\ |\\mathbf {k}|_\\infty \\in [-k_m ,k_m ]\\end{array}}{\\sum }k^2\\hat{\\psi }_{\\mathbf {k}}\\hat{\\psi }_{\\mathbf {k}}^* + \\frac{1}{2}\\lambda \\underset{\\begin{array}{c}\\mathbf {k}_1, \\mathbf {k}_2, \\mathbf {k}_3, \\mathbf {k} \\\\ \\mathbf {k}_1 + \\mathbf {k}_2 = \\mathbf {k}_3 + \\mathbf {k} \\\\ |\\mathbf {k}_i|_\\infty \\in [-k_m ,k_m ]\\end{array}}{\\sum }(k_1 k_2 k_3 k_4)^{\\beta /4} \\hat{\\psi }_{\\mathbf {k1}}\\hat{\\psi }_{\\mathbf {k2}}\\hat{\\psi }_{\\mathbf {k3}}^*\\hat{\\psi }_{\\mathbf {k}}^*,$ where the summation is over every permutation over the subscript wave numbers.",
"When computing the nonlinear term, we evaluate (via the Fourier transform) $(\\psi _{\\mathbf {x}}\\psi ^*_{\\mathbf {x}})\\psi _{\\mathbf {x}} = \\left(\\underset{\\begin{array}{c}\\mathbf {k}_1, \\mathbf {k}_3 \\\\ |\\mathbf {k}_i|_\\infty \\in [-k_m ,k_m ] \\\\ {\\color {red}|\\mathbf {k}_1 - \\mathbf {k}_3|_\\infty \\in [-k_m ,k_m ]}\\end{array}}{\\sum }\\hat{\\psi }_{\\mathbf {k1}}\\hat{\\psi }^*_{\\mathbf {k3}}e^{i(\\mathbf {k1}-\\mathbf {k3})\\cdot \\mathbf {x}} \\right) \\times \\underset{\\begin{array}{c}\\mathbf {k}_2 \\\\ |\\mathbf {k}_2|_\\infty \\in [-k_m ,k_m ]\\end{array}}{\\sum }\\hat{\\psi }_{\\mathbf {k2}}e^{i\\mathbf {k2}\\cdot \\mathbf {x}}$ where the derivatives have been neglected for clarity ($\\beta =0$ ).",
"The second condition under the first sum (red) is the first dealiasing step, where any product of modes that is mapped outside the bounded computational domain is excluded from the sum.",
"The effect of dealiasing is therefore to remove certain interactions from the original system.",
"It is not hard to show that including this extra condition modifies the Hamiltonian such that $H^{\\prime } = \\underset{\\begin{array}{c}\\mathbf {k} \\\\ |\\mathbf {k}|_\\infty \\in [-k_m ,k_m ]\\end{array}}{\\sum }k^2\\hat{\\psi }_{\\mathbf {k}}\\hat{\\psi }_{\\mathbf {k}}^* + \\frac{1}{2}\\lambda \\underset{\\begin{array}{c}\\mathbf {k}_1, \\mathbf {k}_2, \\mathbf {k}_3, \\mathbf {k} \\\\ \\mathbf {k}_1 + \\mathbf {k}_2 = \\mathbf {k}_3 + \\mathbf {k} \\\\ |\\mathbf {k}_i|_\\infty \\in [-k_m ,k_m ] \\\\ {\\color {red}|\\mathbf {k}_1 - \\mathbf {k}_3|_\\infty \\in [-k_m ,k_m ]}\\end{array}}{\\sum } (k_1 k_2 k_3 k_4)^{\\beta /4} \\hat{\\psi }_{\\mathbf {k1}}\\hat{\\psi }_{\\mathbf {k2}}\\hat{\\psi }_{\\mathbf {k3}}^*\\hat{\\psi }_{\\mathbf {k}}^*,$ where $H^{\\prime }$ represents the effective Hamiltonian when dealiasing is used.",
"While a second dealiasing step is included after the second product is taken in (REF ), no additional interactions are removed from $H^{\\prime }$ by the second dealiasing step: $|\\mathbf {k}_1 + \\mathbf {k}_2 - \\mathbf {k}_3 |_\\infty \\in [-k_m ,k_m ]$ is accounted for by the fact that we already require $\\mathbf {k}_1 + \\mathbf {k}_2 - \\mathbf {k}_3 = \\mathbf {k}$ and $|\\mathbf {k}|_\\infty \\in [-k_m ,k_m ]$ .",
"We remark that it is not $\\emph {a priori}$ clear that the dealiased system is still Hamiltonian, but this fact is discovered when one attempts to write $H^{\\prime }$ .",
"In order to show that the system evolution according to (REF ) also leads to the breather solution, we perform a different dealiasing scheme for the simulation.",
"Specifically, we skip the dealiasing step in the intermediate stage of computing the cubic term, and only dealias once after cubic multiplication is completed.",
"Since this dealiasing step is equivalent to keeping only the Fourier modes up to $k_m$ , this strategy produces evolution consistent with the system given by $H$ (rather than $H^{\\prime }$ ).",
"We use this scheme in an otherwise identical setup to the main paper, with $\\beta =3$ and $\\varepsilon =0.001$ , simulating until a breather emerges.",
"The evolution of the Hamiltonian $H$ and the component $H_2$ from $t=0$ are presented in figure REF a, and the corresponding $H_4$ in figure REF c. For this supplemental test we use a larger time step that leads to larger dissipation, though the energy loss over $1000T_f$ it is still only $0.5\\%$ of the total energy.",
"We see that a clear peak in $H_4$ has formed before $t=1000T_f$ , indicating the breather has formed.",
"Again, due to the low sampling rate, aliasing is present in the figures REF a and REF c. We provide detailed plots over $5T_f$ of $H$ , $H_2$ in REF b and $H_4$ in REF d with sufficient sampling such that no aliasing is present.",
"It is clear that the breather remains unchanged under our second scheme which preserves the truncated Hamiltonian system, indicating that the breather is not an artifact of dealiasing.",
"Figure: The time series of (a) HH (dashed) and H 2 H_2 (solid) starting from t=0t=0 for the defocusing MMT equation with β=3\\beta =3 using a scheme that avoids the dealiasing step, with the corresponding time series of (c) H 4 H_4.",
"A detailed view beginning at t=1000T f t=1000T_f of (b) HH, H 2 H_2 and (d) H 4 H_4 over 5T f 5T_f.Finally, we address point C. The forced-dissipated results shown in §I are computed on a domain with 16 times as many modes, which shows that the breather emerges and persists in simulations with higher spatial resolution."
]
] | 2209.08155 |
[
[
"Can segmentation models be trained with fully synthetically generated\n data?"
],
[
"Abstract In order to achieve good performance and generalisability, medical image segmentation models should be trained on sizeable datasets with sufficient variability.",
"Due to ethics and governance restrictions, and the costs associated with labelling data, scientific development is often stifled, with models trained and tested on limited data.",
"Data augmentation is often used to artificially increase the variability in the data distribution and improve model generalisability.",
"Recent works have explored deep generative models for image synthesis, as such an approach would enable the generation of an effectively infinite amount of varied data, addressing the generalisability and data access problems.",
"However, many proposed solutions limit the user's control over what is generated.",
"In this work, we propose brainSPADE, a model which combines a synthetic diffusion-based label generator with a semantic image generator.",
"Our model can produce fully synthetic brain labels on-demand, with or without pathology of interest, and then generate a corresponding MRI image of an arbitrary guided style.",
"Experiments show that brainSPADE synthetic data can be used to train segmentation models with performance comparable to that of models trained on real data."
],
[
"Training the Label Generator",
"We trained the VAE for 800 epochs using a learning rate of $5\\times 10^{-5}$ , Adam optimizer ($\\beta _1 = 0.99, \\beta _2 = 0.999$ ) and a batch size of 256 in an NVIDIA DGX A100 node.",
"The training time was approximately 8 hours.",
"The LDM was trained for 1500 epochs, with a learning rate of $2.5\\times 10^{-5}$ , Adam Optimizer ($\\beta _1 = 0.99, \\beta _2 = 0.999$ ) and a batch size of 384 in an NVIDIA DGX A100 node.",
"The training time was approximately 15 hours."
],
[
"Training the Image Generator",
"The weights that were used to balance the different losses were: $L{adv}: 1.0$ , $L_{VGG}: 0.25$ , $L{feature-matching}: 0.05$ , $L_{KLD}: 0.001$ , $L_{contrastive}: 1.0$ , $L_{mod-dat}: (0.1 - 2.5; 0.05 - 0.75)$ .An exponentially decaying learning rate starting at $5\\times 10^{-4}$ was used with an Adam Optimizer ($\\beta _1 = 0.5, \\beta _2 = 0.99)$ for 4800 epochs.",
"The training time was approximately 2 weeks, using a batch size of 6 in a NVIDIA Quadro RTX 8000 GPU.",
"For the training process we used the following MONAI augmentations : Random bias field augmentation, with a coefficient range of 0.2-0.6.",
"Random contrast adjustment, with a $\\gamma $ coefficient range of 0.85-1.25.",
"Random gaussian noise addition, with $\\mu $ = 0.0 and $\\sigma $ range of 0.05-0.15.",
"The images were normalised using Z-normalisation.",
"We used nnU-Net to perform all our segmentation experiments.",
"nnU-Net performs automatic hyperparameter selection based on the task and input data; we downloaded the package from Github https://github.com/MIC-DKFZ/nnUNet.git and selected the `2d' training option.",
"We modified the number of epochs to ensure convergence for all models.",
"Figures REF and REF show example segmentations from our experiments.",
"Figure: Example segmentations for experiments 3.1 (a) and 3.2., near-OoD (b) and far-OoD (c); the segmented regions are CSF (red), GM (green) and WM (blue).Figure: Example segmentations for experiment 3.3.",
"From left to right: input T1 and FLAIR images, ground truth, and predictions of tumours made by our models R les R_{les}, S les S_{les} and H les H_{les}, highlighted in red."
]
] | 2209.08256 |
[
[
"Probing the pressure dependence of sound speed and attenuation in bubbly\n media: Experimental observations, a theoretical model and numerical\n calculations"
],
[
"Abstract The problem of attenuation and sound speed of bubbly media has remained partially unsolved.",
"Comprehensive data regarding pressure-dependent changes of the attenuation and sound speed of a bubbly medium are not available.",
"Our theoretical understanding of the problem is limited to linear or semi-linear theoretical models, which are not accurate in the regime of large amplitude bubble oscillations.",
"Here, by controlling the size of the lipid coated bubbles (mean diameter of ~5.4um), we report the first time observation and characterization of the simultaneous pressure dependence of sound speed and attenuation in bubbly water below, at and above MBs resonance (frequency range between 1-3MHz).",
"With increasing acoustic pressure (between 12.5-100kPa), the frequency of the attenuation and sound speed peaks decreases while maximum and minimum amplitudes of the sound speed increase.",
"We propose a nonlinear model for the estimation of the pressure dependent sound speed and attenuation with good agreement with the experiments.",
"The model calculations are validated by comparing with the linear and semi-linear models predictions.",
"One of the major challenges of the previously developed models is the significant overestimation of the attenuation at the bubble resonance at higher void fractions (e.g.",
"0.005).",
"We addressed this problem by incorporating bubble-bubble interactions and comparing the results to experiments.",
"Influence of the bubble-bubble interactions increases with increasing pressure.",
"Within the examined exposure parameters, we numerically show that, even for low void fractions (e.g.",
"5.1*10-6) with increasing pressure the sound speed may become 4 times higher than the sound speed in the non-bubbly medium."
],
[
"Introduction",
"Acoustically excited microbubbles (MBs) are present in a wide range of phenomena; they have applications in sonochemistry [1]; oceanography and underwater acoustics [2], [3], [4]; material science [5], sonoluminescence [6] and in medicine [7], [8], [10], [11], [12], [13], [14].",
"Due to their broad and exciting biomedical applications, MBs have many emerging applications in diagnostic and therapeutic ultrasound[14].",
"MBs are used in ultrasound molecular imaging [7], [8] and recently have been used for the non-invasive imaging of the brain microvasculature [8], [9].",
"MBs are being investigated for site-specific enhanced drug delivery [10], [11], [12], [13] and for the non-invasive treatment of brain pathologies (by transiently opening the impermeable blood-brain barrier (BBB) to deliver macromolecules [11]; with the first in human clinical BBB opening reported in 2016 [10]).",
"However, several factors limit our understanding of MB dynamics which consequently hinder our ability to optimally employ MBs in these applications.",
"The MB dynamics are nonlinear and chaotic [15], [16], [17]; furthermore, the typical lipid shell coating adds to the complexity of the MBs dynamics due to the nonlinear behavior of the shell (e.g., buckling and rupture [18]).",
"Importantly, the presence of MBs changes the sound speed and attenuation of the medium [19], [20], [21], [22], [23].",
"These changes are highly nonlinear and depend on the MB nonlinear oscillations which in turn depend on the ultrasound pressure and frequency, MB size and shell characteristics [19], [20], [21], [22].",
"The increased attenuation due to the presence of MBs in the beam path may limit the pressure at the target location.",
"This phenomenon is called pre-focal shielding (shadowing) [22], [23].",
"Additionally, changes in the sound speed can change the position and dimensions of the focal region; thus, reducing the accuracy of focal placement (e.g., for targeted drug delivery).",
"In imaging applications, MBs can limit imaging in depth due to the shadowing caused by pre-focal MBs [16], [22], [23], [25], [26].",
"In sonochemistry, changes in the attenuation and the sound speed impact the pressure distribution inside the reactors and reduce the procedure efficacy [20], [21].",
"An accurate estimation of the pressure dependent attenuation and sound speed in bubbly media remains one of the unsolved problems in acoustics [27].",
"Most current models are based on linear approximations which are only valid for small amplitude MB oscillations [19], [28].",
"Nonlinear propagation of pressure waves in bubbly media containing coated and uncoated bubbles is theoretically studied using the Korteweg–de Vries–Burgers (KdVB) equation in [29], [30], [31], [32] and the nonlinear coefficients of wave propagation were derived through linearization on the effective equations.",
"Linear approximations, however, are not valid for the typical exposure conditions encountered in the majority of ultrasound MB applications.",
"In an effort to incorporate the nonlinear MB oscillations in the attenuation estimation of bubbly media, a pressure-dependent MB scattering cross-section has been introduced [2], [33].",
"While the models introduce a degree of pressure dependency (e.g.",
"only the pressure dependence of the scattering cross section were considered while the damping factors were estimated using the linear model), they still incorporate linear approximations for the calculation of the other damping factors (e.g.",
"liquid viscous damping, shell viscous damping and thermal damping).",
"Additionally, they neglect the nonlinear changes of the sound speed in their approximations.",
"We have shown in[34], [35], [36], that the changes in liquid and shell viscous damping and thermal dissipation are pressure dependent and significantly deviate from linear predictions even at moderate pressures (e.g.",
"40 kPa).",
"Louisnard [20] and Holt and Roy [37] have derived models based on employing the energy conservation principle.",
"In Louisnards approach [20] the pressure dependent imaginary part of the wave number is calculated by computing the total nonlinear energy loss during bubble oscillations.",
"However, this method still uses the linear approximations to calculate the real part of the wave number; thus, it is unable to predict the changes of the sound speed with pressure.",
"Holt and Roy calculated the energy loss due to MB nonlinear oscillations and then calculated the attenuation by determining the extinction cross-section [37].",
"Both approaches in [20] and [37] use the analytical form of the energy dissipation terms.",
"In the case of coated MBs with nonlinear shell behavior, such calculations are complex and can result in inaccuracies.",
"The existing approaches for sound speed computations based on the Woods model [37], [38] are either limited to bubbles whose expansion is essentially in phase with the rarefaction phase of the local acoustic pressure, or require tedious calculations in nonlinear regimes of oscillations from the spine of the $\\frac{dP}{dV}$ loops (e.g.",
"[39]) where $P$ is pressure and $V$ is the MB volume.",
"Sojahrood et al.",
"[43], [44] introduced a nonlinear model to calculate the pressure dependent attenuation and sound speed of the bubbly media considering full nonlinear bubble oscillations.",
"Later, Trujillo [47] used a similar approach to derive the pressure dependent terms for attenuation and sound speed.",
"However, the models were only validated against the linear model [19].",
"Pressure dependent predictions of the models were not tested against experiments.",
"Experimental investigation of the pressure and frequency dependence of the attenuation of bubbly media has been limited to few studies of coated MBs suspensions [25], [33], [40].",
"Although pressure dependent attenuation measurements have been performed on mono-disperse bubble populations [33], [40], to our best knowledge there is no study that investigated the pressure dependent sound speed in the bubbly media.",
"Application of mono-disperse or narrow sized bubble populations are critical in observing the influence of pressure on the sound speed.",
"In case of poly-disperse solutions, due to the contributions from different resonant sub-populations at each pressure, inference of the sound speed as a function of acoustic pressure and relating it to the bubble behavior is a near impossible task.",
"Moreover, in the absence of a comprehensive model to calculate the pressure dependent sound speed and attenuation, the relationship between the changes in the acoustic pressure and variations in the sound speed and attenuation are not fully understood.",
"The objective of this work is to gain fundamental insight on the simultaneous dependence of sound speed and attenuation on the excitation pressure.",
"To achieve this, we carried out attenuation and sound speed measurements of bubbly water samples at acoustic pressure amplitudes of (12.5 kPa-100 kPa) using monodispersions of stabilized lipid coated MBs.",
"We then derive a simple model describing the relationship between the acoustic pressure and the sound speed and attenuation in bubbly media at resonant low Mach number (maximum bubble wall velocity $\\approx $ 40m/s) regimes of oscillation which treats the MB oscillations with their full nonlinearity.",
"Here, we report the first time controlled observations of the pressure dependence of the sound speed of a bubbly medium.",
"The predictions of the model are in good agreement with experiments.",
"In the appendix we extended the theoretical analysis.",
"First, the model predictions are verified against the linear and semi-linear models for free bubbles, bubbles encapsulated with elastic shells and bubbles immersed in elastic materials.",
"Then, we extended the numerical simulations to higher void fractions and pressure amplitudes.",
"One of the advantages of the introduced model is the ability to take into account bubble-bubble interaction effects.",
"We show that at higher void fractions the interaction reduces the attention and the frequency of the attenuation peak.",
"Thus, one of the well known problems of the linear models [19] which is the significant overestimation of the attenuation near the resonance frequency of bubbles is potentially addressed.",
"We numerically show that with increasing acoustic pressure the attenuation and sound speed can increase significantly (e.g.",
"4 times the medium sound speed) even for small void fractions (e.g.",
"5.1$\\times $ 10$^{-6}$ ).",
"Finally, noting that the main contribution of this work is laying out the theoretical foundations, the potential future applications of the model for the accurate shell characterization of lipid coated MBs and the sound propagation in bubbly media are discussed.",
"To experimentally explore the pressure-dependent changes of the sound speed and attenuation for coated MBs first we need to make monodisperse MBs sizes.",
"Commercially available MBs are polydisperse, and at each pressure a subpoulation can be resonant and thus the unambiguous identification of pressure dependent effects will be challenging.",
"Thus, we aim to first produce monodisperse bubbles.",
"Monodisperse lipid shell MBs were produced using flow-focusing in a microfluidic device as previously described [40], [41].",
"Figure 1 shows the schematic of the procedure.",
"Figure 2 shows the size distribution of the MBs in our experiments.",
"The setup for the attenuation and sound speed measurements is the same as the one used in [41].",
"Figure 3 shows the setup for the measurements of the attenuation and sound speed.",
"A pair of single-element 2.25 MHz unfocused transducers (Olympus, Center Valley,$\\approx $ PA; bandwidth 1-3.0 MHz) were aligned coaxially in a tank of deionized water and oriented facing each other.",
"Monodisperse MBs were injected into a sample chamber that was made with a plastic frame covered with an acoustically transparent thin film.",
"The dimensions of sample chamber were 1.4 x 3.5 x 3.5 cm (1.4 cm acoustic path length), and a stir bar was used to keep the MBs dispersed.",
"The transmit transducer was excited with a pulse generated by a pulser/receiver (5072PR, Panametrics, Waltham, MA) at a pulse repetition frequency (PRF) of 100 Hz.",
"An attenuator controlled the pressure output of the transmit transducer (50BR-008, JFW, Indianapolis, IN), which was calibrated with a 0.2-mm broadband needle hydrophone (Precision Acoustics, Dorset, UK).",
"Electric signals generated by pulses acquired by the receive transducer were sent to the Gagescope (Lockport, IL) and digitized at a sampling frequency of 50 MHz.",
"All received signals were recorded on a desktop computer (Dell, Round Rock, TX) and processed using Matlab software (The MathWorks, Natick, MA).",
"The peak negative pressures of the acoustic pulses that are used in experiments were 12.5, 25, 50, and 100 kPa.",
"Attenuation and sound speed were then calculated by comparing the power and phase spectra of the received signals before and after injection of the MBs in to the chamber: $\\alpha (\\omega )=\\frac{20}{d}\\log _{10}\\left(\\frac{P_l}{P_{MBs}}\\right)$ and $C(\\omega )=\\frac{\\omega d C_l}{\\omega d +C_l(\\phi _{MBs}-\\phi _l)}$ where $\\alpha (\\omega )$ is the frequency dependent attenuation of the bubbly medium, $d$ is acoustic path length, $P_l$ and $P_{MBs}$ are the power spectrum of the received signals in the absence and in the presence of the MBs respectively.",
"$C(\\omega )$ is the frequency dependent sound speed of the bubbly medium, $C_l$ is the sound speed in the liquid in the absence of the MBs, and $\\phi _{MBs}$ and $\\phi _l$ are the phase of the received signal in the presence and absence of the MBs respectively.",
"Figure: Size distribution of the MBs in the experiments measured by Coulter-counter.",
"The volume fraction β 0 \\beta _0 can directly be calculated from the size distribution.Figure: The schematic of the setup for the measurements.",
"A broadband pulse with 2.25 MHz center frequency is transmitted by the transducer on the right hand.",
"After propagation through the chamber, the pulse will be received by the transducer on the left hand side.To drive the terms for pressure dependent attenuation and sound speed in bubbly media, we start with the Caflisch equation [46] for the propagation of the acoustic waves in a bubbly medium.",
"The model treats the bubbly media as as a continuum.",
"This implies that the radial oscillations of all the bubbles inside a small volume of the mixture at point $r$ , can be described by a continuous spatio-temporal radius function $R(r,t)$ .",
"Using the mass and momentum conservation in the mixture we have: $\\frac{1}{\\rho _l {C_l}^2}\\frac{{\\partial {}}P}{\\partial {}t}+\\nabla {}.v=\\sum _{i=1}^N\\frac{\\partial {}\\beta _i}{\\partial {}{t}}$ $\\rho _l \\frac{\\partial {}v}{\\partial {}t}+\\nabla {}p=0$ In these equations, $P(r,t)$ is pressure, $v(r,t)$ is the velocity field, .",
"is the dot product, $\\ C_l$ is the speed of sound in the liquid in the absence of bubbles, ${\\rho {}}_l$ is the liquid density and ${\\beta {}}_i$ is the local volume fraction occupied by the gas at time $t$ of the ith microbubbles (MBs).",
"${\\beta {}}_i$ is given by ${\\beta {}}_i(t)=\\frac{4}{3}\\pi {}{R_i(t)}^3N_{i}$ where $R_i$(t) is the instantaneous radius of the MBs with initial radius of $R{_{0i}}$ and $N_{i}$ is the number of the corresponding MBs per unit volume in the medium.",
"The summation is performed over the whole population of the MBs inside the volume.",
"The velocity field can be eliminated between Eqs.",
"REF and REF to yield an equation involving only the pressure field [19], [20], [21]: ${\\nabla {}}^2\\left(P\\right)=\\frac{1}{C_l^2}\\frac{{\\partial {}}^2P}{\\partial {}t^2}-{\\sum _{i=1}^N\\ \\ \\rho _l{}}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}$ To calculate the attenuation and sound speed we need to determine the wave number and transfer the time averaged form of Eq.",
"REF in the form of the Helmholtz equation given by: ${\\nabla {}}^2\\left(P\\right)+k^2(P)=0$ where $k$ is the wave number (k=k$_{r}$ -i$\\alpha {}$).",
"The sound speed can be calculated from k$_{r }$ which is the real part of the wave number, and the attenuation $\\alpha {}$ from the imaginary part of the wave number.",
"A general solution to Eq.",
"REF is given as: $P(r,t)=\\frac{1}{2}p(r)e^{(-i\\omega t)}+\\frac{1}{2}\\bar{p}(r)e^{(i\\omega t)}$ where $\\omega $ is the angular frequency and $\\bar{p}$ is the complex conjugate of $p$ .",
"Each term on the right side is a particular solution to Eq.",
"REF .",
"Mathematically, the Helmholtz equation (Eq.",
"REF ) is a homogeneous partial differential equation, thus each particular solution is also a solution to the Eq.",
"REF .",
"Thus, if we input the two solutions in Eq.",
"REF we will have: ${\\nabla {}}^2\\left({p}\\right)=-\\frac{\\omega ^2}{C_l^2}{p}-2{\\sum _{i=1}^N\\ \\ \\rho _l{}}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}$ and ${\\nabla {}}^2\\left(\\bar{p}\\right)=-\\frac{\\omega ^2}{C_l^2}\\bar{p}-2{\\sum _{i=1}^N\\ \\ \\rho _l{}}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}$ where $\\bar{p}$ is the complex conjugate of $p$ .",
"Next step is to calculate the wave number in the presence of the bubbles: $k^2=-\\frac{{\\nabla {}}^2\\left(P\\right)}{P}$ .",
"To achieve this, Eq.",
"REF was multiplied by $\\frac{\\ \\bar{p}}{p\\bar{p}}\\ $ and Eq.",
"REF was multiplied by $\\frac{p}{p\\bar{p}}$ .",
"The pressure dependent real and imaginary parts of $k^2$ were derived using the time average of the results of the addition and subtraction of the new equations: $\\langle \\Re (k^2)\\rangle =\\frac{{\\omega {}}^2}{C_l^2}+\\frac{2{\\rho _l{}}}{T{\\left|{}p\\right|{}}^2}\\sum _{i=1}^N\\int _0^T{\\Re (p)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ $\\langle \\Im (k^2)\\rangle =\\frac{2{\\rho _l{}}}{T{\\left|{}p\\right|{}}^2}\\sum _{i=1}^N\\int _0^T {\\Im (p)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ where $\\Re $ and $\\Im $ denote the real and imaginary parts respectively, $<$$>$ denotes the time average, and T is the time averaging interval.",
"The contribution of each MB with ${\\beta {}}_i$ is summed.",
"Using Eqs.",
"REF and REF , we can now calculate the pressure-dependent sound speed and attenuation in a bubbly medium.",
"To do this, the radial oscillations of the MBs in response to an acoustic wave need to be calculated first.",
"Equations REF and REF need to be solved by integrating over the ${\\beta {}}_i$ of each of the MBs in the population.",
"The advantage of this technique is the simultaneous calculation of the pressure dependent sound speed and attenuation in the bubbly medium.",
"This approach is verified in Appendix A against the linear model and in Appendix B against the semi-linear model.",
"Table: Thermal properties of the gases used in simulations(C=5.528×\\times 10 25 ^{25} W/mK 2 ^2)"
],
[
"Lipid coated bubble model including bubble-bubble interaction",
"To numerically simulate the attenuation and sound speed, radial oscillations of the lipid coated MBs with a size distribution given in Fig.",
"2 should be simulated first.",
"Moreover, effect of MB-MB interaction must be included as the pulsation of each MBs generates a pressure at the location of each MBs in its vicinity, which consequently may influence each MB behavior.",
"To model the MB oscillations the Marmottant model [18], which accounts for radial-dependent shell properties of lipid coated MBs, was modified to include MB multiple scattering using the approach introduced in [108]: $\\begin{gathered}R_i\\ddot{R_i}+\\frac{3}{2}\\dot{R_i}^2=\\frac{1}{\\rho }\\left([P_0+\\frac{2\\sigma (R_{0i})}{R_{0i}}](\\frac{R_i}{R_{i0}})^{3k}(1-\\frac{3k}{C_l}\\dot{R_i})-\\frac{2\\sigma (R_{i})}{R_{i}}-\\frac{4\\mu \\dot{R_i}}{R_i}-\\frac{4\\kappa _s\\dot{R_i}}{R_i^2}-P_0-P_{ac}(t)-{\\rho \\sum _{j=1,j\\ne i}^{N}\\frac{R_j}{d_{ij}}(R_j\\ddot{R_j}+2\\dot{R_j}^2)}\\right)\\end{gathered}$ In this equation $R_{i0}$ is the initial radius of the ith bubble, $P_0$ is the atmospheric pressure $C_l$ is the sound speed, $k$ is the polytropic exponent, $\\mu $ is the viscosity of the liquid.",
"The last term in the right hand side of Eq.",
"REF defines the pressure radiated by neighboring MBs with $R_j$ at the location of ith MB and $d_{ij}$ represents the distance between centers of ith and jth bubble $\\sigma (R_{i})$ is the surface tension acting on the ith MB which is a function of bubble radius and is given by : $\\sigma (R_{i})=\\begin{dcases}0 \\hspace{28.45274pt} if \\hspace{14.22636pt} R_i<=R_{bi}\\\\\\chi \\left({(\\frac{R_i}{R_{bi}})}^2-1\\right) \\hspace{28.45274pt} if \\hspace{14.22636pt} R_{bi}<R_i<R_{ri}\\\\\\sigma _{water} \\hspace{28.45274pt} if \\hspace{14.22636pt} R_i>=R_{ri}\\end{dcases}$ where $\\sigma _{water}$ is the water surface tension, $R_{bi}={R_{0i}}/{\\sqrt{1+\\frac{\\sigma (R_{0i})}{\\chi }}}$ is the buckling radius, $\\sigma (R_{0i})$ is the initial surface tension, $\\chi $ is the shell elasticity.",
"$\\kappa _s$ in Eq.",
"REF , is the surface diltational viscosity.",
"$R_{ri}$ is the rupture radius (break up radius in this paper similar to [85], [109]) and is given by $R_r=R_{bi}\\sqrt{1+{\\sigma _{ri}}/{\\chi }}$ where $\\sigma _{ri}$ is the rupture surface tension of the ith MB.",
"Using the approach in [110], [70], [111], Eq.REF can be written in a matrix format as: Figure: Experimentally measured a) attenuation and b) sound speed of the bubbly medium for four different pressures.$\\begin{pmatrix}\\ddot{R}_1\\\\ \\\\\\ddot{R}_2\\\\ \\\\...\\\\ \\\\\\ddot{R}_N\\\\ \\\\\\end{pmatrix}=\\begin{pmatrix}R_1 & \\frac{R_2^2}{d_{12}} & ... & \\frac{R_N^2}{d_{1N}}\\\\ \\\\\\frac{R_1^2}{d_{21}} & R_1 & ... & \\frac{R_N^2}{d_{2N}}\\\\ \\\\... & ... & ... & ... \\\\ \\\\\\frac{R_1^2}{d_{N1}} & \\frac{R_2^2}{d_{N2}} & ... & R_N\\\\ \\\\\\end{pmatrix}^{-1}\\begin{pmatrix}A_1\\\\ \\\\A_2 \\\\ \\\\...\\\\ \\\\A_N\\end{pmatrix}$ where $\\begin{gathered}A_i=\\frac{1}{\\rho }\\left(\\left[P_0+\\frac{2\\sigma (R_{0i})}{R_{0i}}\\right]\\left(\\frac{R_i}{R_{i0}}\\right)^{3k}\\left(1-\\frac{3k}{C_l}\\dot{R_i}\\right)-\\frac{2\\sigma (R_{i})}{R_{i}}\\right.\\\\-\\left.\\frac{4\\mu \\dot{R_i}}{R_i}-\\frac{4\\kappa \\dot{R_i}}{R_i^2}-P_{ac}(t)-\\frac{3\\rho }{2}\\dot{R_i}^2\\right)-\\sum _{j\\ne i}^{}\\frac{2R_j\\dot{R_j}^2}{d_{ij}}\\end{gathered}$ The gas inside the bubble is C$_3$ F$_8$ and the properties are given in Table 1.",
"We have neglected the thermal effects in the numerical simulations.",
"In [35] we have shown that in case of the coated bubbles that enclose C$_3$ F$_8$ gas cores, thermal dissipation has negligible contribution to the overall dissipated power in the ultrasound exposure range that was used here.",
"Thus, thermal effects are neglected to reduce the complexity of the multiple scattering equation that is used in the simulations (Eq.REF )."
],
[
"Simulation procedure",
"For the numerical simulations, the experimentally measured size of the MBs were used as the input for initial MB sizes.",
"Size and concentration of the MBs in the experiments are given in Fig.",
"2.",
"There are 63 size bins in the range between 1.44$\\mu $ m$<R_0<$ 3.3$\\mu $ m. Each $R_{0i}$ has a number density of $N_i$ .",
"These 63 MBs were distributed randomly in a cube of diameter 0.1cm to simulate the total concentration of 6.3$\\times $ 10$^{4}$ MBs/mL in the experiments.",
"The total gas volume in the experiments was $\\approx $ 5.1$\\times $ 10$^{-12}$ m$^3$ /mL.",
"When written in m$^3$ /m$^3$ , this results in $\\beta _0$ of 5.1$\\times $ 10$^{-6}$ .",
"The minimum distance between the MBs was set to be 15$\\mu $ m to avoid MBs collisions [112], [113], [114].",
"At each case (12.5, 25, 50 and 100kPa), the numerical simulations were repeated for 40 frequency values between 1-3MHz.",
"At each of those frequencies, the corresponding pressure was extracted from the driving experimental pressure spectrum.",
"To calculate the attenuation at a given frequency and pressure, then, Eq.",
"REF was solved and the radial oscillations of each MB was recorded for the duration of the experimental pulse.",
"Next the radial oscillations were used as an input to calculate the attenuation and sound speed in Eq.",
"REF and REF .",
"Since the acoustic pressure that excites the MBs in Eq.",
"REF is given by $P$ =$P_a$$\\sin (2\\pi ft)$ , thus $\\Re (p)$ =$P_a$$\\sin (2\\pi ft)$ and $\\Im (p)$ =-$P_a$$\\cos (2\\pi ft)$ .",
"Similar to [47] this is simply achieved by an angle shift ($P_a$ e$^{(i\\omega t-\\pi /2)}$ =$P_a$$\\left(\\sin (2\\pi ft)-i\\cos (2\\pi ft)\\right)$ .",
"Thus, real and imaginary parts of $k^2$ can be calculated from: $\\langle \\Re (k^2)\\rangle =\\frac{{\\omega {}}^2}{C_l^2}+\\frac{2{\\rho _l{}}}{T{P_a}}\\sum _{i=1}^{63}N_i\\int _0^T{\\sin (2\\pi ft)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ $\\langle \\Im (k^2)\\rangle =-\\frac{2{\\rho _l{}}}{T{P_a}}\\sum _{i=1}^{63}N_i\\int _0^T {\\cos (2\\pi ft)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ where T is the pulse length in the experiments.",
"Attenuation and sound speed can be calculated from: $\\alpha (p,f)=\\Im \\left(\\sqrt{\\Re (k^2)+i\\Im (k^2)}\\right)$ $C(p,f)=\\frac{{\\omega }}{\\Re \\left(\\sqrt{\\Re (k^2)+i\\Im (k^2)}\\right)}$ Dimension of the $\\alpha (p,f)$ is in Np/m.",
"To calculate the attenuation in dB/m Eq.",
"REF is multiplied by 8.6858.",
"This procedure was done for different values of the shell elasticity (0.1 N/m$<$$\\chi {}$ $<$ 2.5 N/m), initial surface tension (0 N/m $<$ $\\sigma {}$($R_0)\\ $$<$ 0.072 N/m), shell viscosity (1$\\times $ 10$^{-9}$ kg/s $<$${\\mu {}}_s$$<$ 1$\\times $ 10$^{-7}$ kg/s), and rupture radius $(R_{0i}<R_{ri}$ $<$ 2$R_{0i})$ .",
"The best fit results are presented.",
"These values are within the parameter ranges that were previously reported for lipid coated MBs [33], [77], [78], [79], [80], [85], [86], [95].",
"Figures 4a-b show representative samples of the experimentally measured attenuation and sound speed of the mixture respectively.",
"The attenuation of the bubbly medium increases as the pressure increases from 12.5 kPa to 100 kPa and the frequency of the maximum attenuation decreases from 2.045 MHz to 1.475 MHz.",
"The maximum sound speed of the medium increases with pressure and the corresponding frequency of the maximum sound speed decreases by pressure increases.",
"To our best knowledge this is the first experimental demonstration of the pressure dependence of the sound speed.",
"It is also interesting to note that at the pressure dependent resonance frequencies (measured attenuation peaks in Fig.",
"4a) the sound speed is equal to the sound speed of the water in the absence of bubbles.",
"To compare the predictions of the model with experiments, Figs.",
"5a-h illustrate the results of the experimentally measured (blue) (with standard deviations of the 100 data points at each condition) and numerically simulated (red) sound speed of the medium as a function of frequency, for 4 different pressure exposures of (12.5, 25, 50 and 100 kPa).",
"The shell parameters that were used to compare with the experimental results are $\\chi {}$=1.1 N/m, $\\sigma {}$($R_0)$ = 0.016 N/m, ${\\mu {}}_s$ =7$\\times $ 10$^{-9}$ kg/s and $R_{breakup}$ =1.1$R_0$ .",
"These values are within the estimated values for lipid coated MBs [33], [77], [78], [79], [80], [85], [86], [95].",
"As the pressure increases, the frequency at which the maximum attenuation occurs (which indicates the resonance frequency) decreases (from 2.02 MHz at 12.5 kPa to 1.475 MHz at 100 kPa) and the magnitude of the attenuation peak increases (from 16.5 dB/cm at 2 MHz to 21.8 dB/cm at 1.475 MHz).",
"At 12.5 kPa and for frequencies below $\\sim $ 2 MHz, the speed of sound in the bubbly medium is smaller than the sound speed of water.",
"Above 2 MHz, speed of sound increases and reaches a maximum at 2.25 MHz with a magnitude of $\\sim $ 1.015$C_l$ .",
"At $\\sim $ 2 MHz the sound speed is equal to $C_l$ .",
"This is also the frequency where the attenuation is maximum.",
"According to the linear theory [103] at the resonance frequency the sound speed of the bubbly medium is equal to the sound speed of the medium without the bubbles.",
"As the pressure increases to 25, 50 and 100 kPa the frequency at which the speed of sound in the bubbly sample is equal to $C_l$ decreases to 1.87, 1.65 and 1.48 MHz respectively.",
"The frequency at which the maximum sound speed occurs decreases as the pressure increases and the magnitude of the maximum sound speed increases to $\\sim $ 1.019$C_l$ at 100 kPa.",
"The minimum sound speed decreases from $\\sim $ 0.989$C_l$ at 12.5 kPa (peak at1.6 MHz) to $\\sim $ 0.981$C_l$ at 100 kPa and (1.25 MHz).",
"At each pressure, the frequency at which attenuation is maximized (the pressure dependent resonance) is approximately equal to the frequency where the sound speed becomes equal to $C_l$ .",
"Thus, it can be observed that, even at the pressure dependent resonance frequency ($PDf_r$ ) which is a nonlinear effect [16], the velocity is in phase with the driving force and $\\frac{C}{C_l}=1$ (page 290) [103].",
"This observation is also consistent with the numerical results of the uncoated bubble in Appendix D in Figs.",
"6a-f and Figs.",
"7a-f."
],
[
"Discussion",
"In ultrasound applications, the bubble activity in the target heavily depends on the acoustic pressure that can reach the target.",
"However, there is no comprehensive model that can be used to estimate attenuation and sound speed changes for large amplitude bubble oscillations.",
"The majority of published works employed linear models that are derived using the Comander and Prosperetti approach [19].",
"Since these models are developed under the assumption of very small amplitude bubble oscillations ($R=R_0(1+x)$ where $x\\ll 1$ ), they are not valid for most ultrasound MB applications [20], [51], [52].",
"On the other hand recent semi-linear models [2], [20], [33] still have some inherent linear assumptions.",
"Sojahrood et al, [36] classified the pressure dependent dissipation regimes of uncoated and coated bubbles over a wide range of pressure and for frequencies below, at, and above resonance in tandem with bifurcation analysis of the bubble radial oscillations.",
"They showed that dissipation due to shell viscosity, radiation, thermal and liquid viscosity exhibit strong pressure dependency and linear estimations are inaccurate even at pressures as low as 25kPa[34], [35], [36].",
"The models that include both the pressure dependent real and imaginary part of $k^2$ [43], [44], [47] have not been compared to experiments in pressure dependent regimes.",
"Trujillo[47] validated their model against only the linear model and experiments at very low excitation pressures ($\\approx $ 10Pa).",
"Despite good agreement with the linear model, the model still suffers from significant overestimation of the attenuation near the bubble resonance.",
"Pressure dependent behavior of the attenuation and sound speed thus remained limited to numerical studies [43], [53] and qualitative comparisons [45].",
"Additionally, simultaneous pressure dependent sound speed and attenuation has not been reported experimentally.",
"Due to the polydisersity of MBs in the majority of applications and the absence of a comprehensive model, thus, there is no detailed study on the simultaneous changes of the attenuation and sound speed as a function of acoustic pressure.",
"In case of the polydisperse conventional contrast agents (1-10$\\mu $ m size range), at each exposure parameter only a small sub-population exhibit resonant behavior.",
"Thus, the overall response of the MBs may mask the intricate changes in MB dynamics [25], [104] and the corresponding changes in the attenuation and sound speed, thus observation of such changes may not be possible.",
"Moreover, due to MB-MB interactions and mode locking between different size MBs[111], interpreting the the overall response of the MBs becomes more difficult.",
"The monodisperse agents used in this work, exhibit unifrom acoustic behavior [33] which enhances the nonlinear effects.",
"Due to this higher potency [25], [33], [104], observation of pressure dependent changes are simpler.",
"Here, by confining the MB sizes (using monodisperse MB solutions), we report the first time, controlled, simultaneous experimental observations of the pressure dependent sound speed and attenuation in a bubbly medium.",
"The complex influence of the bubble-bubble interactions [45], [72], [73], [74] were minimized by using lower void fractions to allow for an unambiguous visualization of the simultaneous relationship of sound speed and attenuation with acoustic pressure.",
"We then presented a pressure dependent attenuation-sound speed model for the low mach number regimes of oscillations.",
"Predictions of the model are in good agreement with experimentally measured attenuation and sound speed of mono-dispersions of lipid coated bubbles."
],
[
"Justification for the choice of acoustic exposure parameters to achieve strong and controllable nonlinear oscillations during experiments",
"The choice of the acoustic exposure parameters ($P_a$ between 12.5-100kPa) would initially appear to result in linear oscillations; however the choice of resonant lipid coated MBs results in strong nonlinear oscillations within the exposure parameter ranges that are tested.",
"The main reasons for the chosen pressure ranges are discussed.",
"Lipid coated MBs are interesting highly nonlinear oscillators.",
"The nonlinear behavior of the shell, including buckling and rupture [12], [126] is intermixed with the nonlinearity of the MB oscillations itself.",
"This results in the generation of the nonlinear oscillations at pressures as low as 10kPa [72], [109], [86], [126], [127].",
"The presence of the shell increases the viscosity at the bubble wall, however, despite this, the pressure threshold for nonlinear oscillations of the lipid coated bubbles is significantly lower than the pressure required for the 1/2 order subharmonic (SH) generations for the uncoated bubbles [127].",
"Sojahrood et al [72], [109] compared the nonlinear oscillations of the uncoated and lipid coated bubbles using bifurcation analysis over a wide range of frequency and pressures.",
"They showed that even at pressures below 30kPa, the lipid coated MB exhibits strong nonlinearity including the generation of 1/2 and 1/3 order SHs and higher order superharmonic resonances [72], [109], [86], [126], [127].",
"Attenuation measurements by Segers et al [33] reveals strong nonlinear behavior (significant 62.5$\\%$ reduction of the frequency of the attenuation peak together with the generation of new peaks at frequencies below and above main resonance) in the 10-50kPa range."
],
[
"Interrogation of the bubble oscillations near resonance",
"In our study, experiments were performed using frequencies in the near vicinity of the bubble resonance frequency.",
"We have previously shown that (for uncoated and viscoelastic shell MBs [34], [35], [36]), in the vicinity of linear resonance frequency (0.5$f_r$ -2$f_r$ ) even an increase in pressure from 1kPa to 30-50kPa will result in significant changes in the dissipated powers and 50-100$\\%$ expansion ratio near the bubble resonance.",
"Additionally, the frequency of the main resonance peak shifts and new sub-resonant peaks in the power dissipation curves appear.",
"This behavior is also studied in detail in [105].",
"These studies predict expansion ratio of up to 280$\\%$ for pressures below the Blake threshold and in the vicinity of resonance.",
"This non-linearity is enhanced for lipid coated MBs.",
"Near the linear resonance frequency of the lipid coated MBs [33], [86], [126], [109], even changes as low as 20kPa can result in $\\approx $ 67$\\%$ shift in the resonance frequency of the MBs.",
"The shift in the resonance frequency is highly nonlinear and can not be described by linear bubble models (pure viscoelastic shell models [28] or Rayleigh-Plesset type models for the uncoated bubble[24]).",
"In our experiments we went far above the onset of nonlinear oscillations.",
"The mean initial radius of the MBs was approximately 2.7$\\mu $ m. A lipid coated MB with the shell parameters obtained in our study, experiences 4.5, 9, 25 and 61$\\%$ expansion at 12.5, 25, 50 and 100 kPa respectively.",
"Thus, at pressures even as low as 12.5kPa the oscillations are beyond the linear regime (expansion ratio $<$ 1$\\%$ ).",
"The same MB attained a maximum wall velocity of 30 m/s at 100kPa.",
"For subresonant regimes (well below the resonance frequency of typical clinical MBs of diameters 1-2$\\mu $ m) that are used in drug delivery studies (500kHz and pressures $\\approx $ 500kPa [128]) and for the case of stable and non-destructive oscillations ($R/R_0<2$[131]), a coated MB with $R_0$ =1$\\mu $ m reaches velocities of $\\approx $ 30-40 m/s before possible MB destruction [36].",
"Thus, both radial expansion and wall velocities in our experiments were in the nonlinear regime and of practical relevance.",
"In sonochemistry, frequencies well below the MB resonance are used.",
"Sonochemical procedures typically use a frequency of 20kHz with mean diameters of the MBs of approximately 10$\\mu $ m[54], [129], [130].",
"For a an uncoated bubble this results in a non-damped resonance frequency of $\\approx $ 720kHz.",
"Below the Blake threshold of $\\approx $ 103kPa, the oscillation amplitude grows slowly with increasing pressure ($\\approx $ 45$\\%$ expansion ratio and 0.3m/s wall velocity at 93kPa).",
"In the vicinity of the Blake threshold and higher, the maximum bubble oscillation increases exponentially with pressure increases.",
"At the Blake threshold the bubble expansion ratio is $\\approx $ 74$\\%$ with maximum wall velocity of 1.6m/s.",
"As soon as the pressure increases above the Blake threshold the oscillation amplitude and bubble wall velocity increases rapidly.",
"At 113kPa, the expansion ratio is 300$\\%$ and wall velocity reaches 40m/s.",
"In such oscillation regimes at frequencies well below the MB resonance, the non-linearity sets in more strongly near and above the Blake threshold Since in our study we interrogated the oscillations near resonance, and additionally used lipid coated MBs, the onset of nonlinearity was at much lower pressures than have been predicted for uncoated microbubbles (e.g.",
"12.5-25kPa)."
],
[
"Possible MB destruction at higher pressures",
"Increasing the pressure beyond the 100kPa may result in MB destruction.",
"In our work, 81$\\%$ of the MB population was between 2.5-3.1$\\mu $ m. At 100kPa, the MBs with initial radius of 3.1$\\mu $ m are predicted to have expansion ratio of 76$\\%$ with wall velocities near 40m/s.",
"This is close to the minimum MB destruction threshold of 100$\\%$ expansion ratio ($R/R_0$ =2[131] and for a review of the minimum threshold please see discussion in [16], [102]).",
"Since we interrogated the MBs in the vicinity of their resonance, at slightly higher pressures (e.g.",
"for an uncoated MB with $R_0$ =2$\\mu $ m, the MB is likely to be destroyed at 125kPa and 80kPa at $f=f_r$ and $f=0.9f_r$[36]), a sub-population of the MBs may undergo destruction, thus interpretation of the attenuation and sound speed results may be difficult.",
"Since, the goal of this study was reporting the pressure dependence of sound speed and attenuation and verifying the predictions of the model we choose the pressure in a range that would minimize the unwanted bubble destruction.",
"Nevertheless, based on numerical simulations, we believe that we achieved wall velocities of 30-40 m/s which are in the range of the maximum predicted wall velocities during stable oscillations for MBs with initial radii between 1-10$\\mu $ m[36], [102]."
],
[
"Inaccuracies due to the increased attenuation at higher pressures",
"Another limitation of using higher pressures is the potential increased attenuation from MBs.",
"This creates a steeper pressure gradient within the experimental chamber and thus complicates the interpretation of the results.",
"Therefore, we conclude that the choice of the acoustic parameters and MB sizes were suitable for strong non-linearity while minimizing possible factors attributing to the discrepancy between the model and the experiments."
],
[
"Advantages of the model, new insights and applications",
"The model predictions were in good agreement with experiments.",
"In Appendix D, the predictions of the model were also validated against the linear models at a very small pressure excitations ($P_a$ =1 kPa) and for three different cases of the uncoated bubble, the coated bubble and the bubble in viscoelastic media.",
"At higher pressures, the predictions of the imaginary part of $k^2$ were validated by comparing the predictions of the model with the Louisnard model [20] and for the three different cases mentioned above.",
"The proposed approach provides several advantages and generates new insights."
],
[
"Pressure dependence of the $\\Re \\left(k^2\\right)$ :",
"Advantages of the developed model in this paper are its simplicity and its ability to calculate the pressure dependent real part and imaginary part of the $k^2$ .",
"We showed that pressure dependent effects of the real part of the $k^2$ that are neglected in semi-linear models are important.",
"Attenuation and sound speed estimations using the semi-linear models that does not take into account the pressure dependent effects of the real part of the $k^2$ deviate significantly from the predictions of the approach presented in this paper (section 3 of Appendix D); and these deviations increase with increasing pressure.",
"These deviations may lead to overestimation of the attenuation by a factor of 2 even at moderate acoustic pressures (Fig.",
"8 in section 3 of Appendix D).",
"Moreover, the pressure dependence of the sound speed can be calculated more precisely since the derived model in this paper can calculate the real and imaginary part of the $k^2$ .",
"To our best knowledge, unlike current sound speed models, this model does not have a $\\frac{dP}{dV}$ term (e.g.",
"[35]); thus it does not encounter difficulties addressing the low amplitude nonlinear oscillations."
],
[
"The requirement of the radial oscillations and pressure as inputs only and possible applications:",
"Another advantage of the model is that it uses as input only the radial oscillations of the MBs.",
"There is no need to calculate the energy loss terms, and thus our approach is simpler and faster.",
"This provides important advantages to applications in underwater acoustics and acoustic characterization of contrast agents.",
"In the measurements related to the gassy intertidal marine sediments, pressures in the range of 15-20kPa are used [4].",
"The bubble size range is between 80$\\mu $ m to 1mm [2], [4], [100].",
"The behavior of such big bubbles is very sensitive to the pressure amplitude changes with thermal effects significantly influencing the bubble oscillations [105].",
"The energy dissipation as calculated by the linear model deviates significantly as the pressure increases due to two reasons: 1- The pressure dependent change in the resonance frequency and 2- Inaccuracy of the linear thermal approximation.",
"These are discussed in detail in [92].",
"Thus accurate modeling and estimation of the bubble populations requires a model that encompass the full non-linear bubble oscillations.",
"Since our nonlinear model only uses the radial oscillations and the acoustic pressure as input, it can robustly be used in combination with many different models of MB oscillation.",
"Moreover, attenuation at multiple pressures provides more data that may be used in more accurate estimation of important parameters that characterize bubble populations in these applications.",
"Moreover, in cases of polymer or highly compressible shells(such as in [17], [106] and [107]) the appropriate models can be used.",
"When nonlinear shell behavior occurs (e.g.",
"[18], [76]) it may provide more accurate estimates since there is no need for simplified analytical expressions.",
"When fitting the shell parameters of ultrasound contrast agents [78], [79], [80], [81], pressure dependent attenuation, and sound speed values provide more information for accurate characterizations of the shell parameters, especially in case of microbubbles with more complex rheology [18], [76].",
"There may exist multiple combination of values of the initial surface tension, shell elasticity, and shell viscosity that fits well with measured attenuation curves; however, only one combination provides good fit to both the measured sound speed and attenuation values, and over all the excitation pressure values.",
"A numerical example of a lipid coated MB is shown in Appendix G. Sound speed curves provide more accurate information on the effect of shell parameters on the bulk modulus of the medium (e.g.",
"shell elasticity) while attenuation graphs are more affected by damping parameters; thus the attenuation and sound speed curves can be used in parallel in order to achieve a more accurate characterization of the shell parameters.",
"Several studies used linear models to estimate the shell properties of lipid coated MBs [41], [40], [78], [79], [80], [81], [82], [83] using attenuation measurements.",
"These studies reported shell elasticity of 0.28-1N/m and shell viscosity of 3-60$\\times $ 10$^{-9}$ kg/s for MBs with 0.7-12.6$\\mu m$ over a frequency range between 0.5-29MHz (for a comprehensive review of these studies please refer to [84]).",
"These studies were not able to report the initial surface tension of the MBs.",
"Segers et al [25] estimated a shell elasticity of 2.5N/m, shell viscosity of 2-5$\\times $ 10$^{-9}$ kg/s and initial surface tension of 0.02N/m for monodisperse MBs of $\\approx $ 5$\\mu $ m over a frequency range between 1-6MHz.",
"Attenuation measurements were performed at multiple pressures between 10-100kPa and a semi-linear model was used to fit the shell parameters.",
"By measuring acoustic scattering from single Definity$®$ MBs and an \"acoustic spectroscopy\" approach, Helfield et al [86] estimated shell elasticity values between 0.5-2.5N/m and shell viscosity of 0.1-6$\\times $ 10$^{-9}$ kg/s for MBs with sizes between 1.7-4$\\mu $ m over 4-14MHz.",
"They studied the scattered pressure from single MBs at pressures $<$ 25kPa and used a linear model for fitting the shell parameters.",
"Using optical imaging and flow cytometry, Tu et al[87] directly fit the radial oscillations of the Definity$®$ MBs to the Marmottant model using 1MHz sonications with acoustic pressures between 95-333kPa.",
"They estimated shell elasticity values of 0.46-0.7 and shell viscosity of 0.1-10$\\times $ 10$^{-9}$ kg/s for MBs with sizes between 1.5-6$\\mu $ m. However, the initial surface tension was not reported in their study.",
"To our best knowledge the latter 3 studies [25], [86], [87] are the only studies that fit the shell parameters of the MBs taking into account the acoustic pressure amplitude.",
"In our study we estimated the shell elasticity of 1.1N/m, shell viscosity of 7$\\times $ 10$^{-9}$ kg/s and initial surface tension of 0.016N/m for MBs with a mean size of 5.6$\\mu $ m over a frequency range of 1-3MHz and acoustic pressure of 12.5-100kPa.",
"Our findings are in good agreement with the reported values in [25], [86], [87].",
"The study by Helfield et al [86] used linear assumptions and was limited to very small pressures and bubble oscillation amplitude.",
"The study by Tue et al [87] has sensitivity to the MB oscillation amplitude, and thus, requires higher pressures for large enough MB oscillations (which is limited by optical resolution).",
"Both of the methods are challenging experiments and time consuming.",
"In the first method the initial surface tension and rupture radius of the MB is not reported due to the linear assumptions in the model.",
"In the second method, due to the higher pressures used, MBs spend more of their oscillation time in the ruptured state and behave similar to free bubbles [109].",
"At such high pressures results become less sensitive to the initial surface tension and are affected more by the other parameters due to expansion dominated behavior [2], [5].",
"Thus, estimation of the initial surface tension that is critical in MB behavior at lower pressures (e.g.",
"50kPa) is not possible and was not reported in [87].",
"In addition, during the collapse where significant changes in radius occurs in a short nano-second interval, the optical methods are less accurate in resolving instantaneous bubble behavior.",
"The methods presented in our work and in the semi-linear work of Segers et al [33] are simpler.",
"Additionally the methods have the advantages of providing means to extract the values for the initial surface tension and rupture radius.",
"Taking into account the pressure dependent sound speed changes was shown to significantly influence the estimated attenuation values.",
"This becomes even more important in case of the lipid coated MB that exhibits large variations in resonance with small pressure amplitude changes.",
"Thus, another advantage of our nonlinear method is that in addition to accounting for nonlinear MB radial oscillations, it also provides more accurate predictions of the sound speed in the bubbly medium.",
"To our best knowledge, our study is the first study that uses the pressure dependent sound speed and attenuation curves in tandem to fit the shell parameters of the MBs."
],
[
"Inclusion of the MB-MB interactions:",
"MB-MB interactions have been shown to significantly influence the resonance frequency [74], [70], [71] and the nonlinear behavior of the MBs [74], [70], [108], [111], [110].",
"As the void fraction increases, multiple scatterings by bubbles become stronger[88].",
"However, even at higher void fractions (e.g.",
"5$\\times $ 10$^{-3}$ ) studies based on linear models [19] were not able to incorporate this effect.",
"As such, one of the well-known challenges of these models was the significant overestimation of the attenuation in the vicinity of the bubble resonance frequency even during the linear regime of the oscillations.",
"Investigations at higher acoustic pressure amplitudes and void fractions, also neglected the MB-MB interactions [20], [22], [47], [53], [54], [55], [56].",
"In this study we numerically examined the interaction effects over a wide range of void fractions $\\beta _0=10^{-7}-10^{-4}$ .",
"We show that even in the relatively low void fractions (e.g.",
"$\\beta _0=5.1\\times 10^{-6}$ used in experiments), bubble-bubble interactions are important (Fig.",
"9).",
"The influence of the bubble-bubble interactions on the attenuation and sound speed increases with void fraction.",
"The reason for strong interaction in our experiments can be explained as follows.",
"The $\\beta $ in experiments was estimated to be 5.1$\\times $ 10$^{-6}$ with 63$\\times $ 10$^{10}$ MBs/$m^3$ .",
"The average distance between the MBs can be calculated as $\\approx $ 1/n$^{(1/3)}$ where n is the number density of the MBs.",
"Thus, we can assume that the mean distance between two neighboring bubbles is $\\approx $ 251$\\mu $ m. This results in a relatively strong interaction between MBs.",
"For example at 100kPa, the resonant pressure radiated by a lipid coated MB with $R_0$ of 2.7$\\mu $ m a distance of 251$\\mu $ m is 12.75kPa.",
"When in a solution, each MB receives a sum of the pressures of the multiple neighboring bubbles, plus the incident acoustic pressure field.",
"Since the average scattered pressure by each MB is not negligible, the interaction effects can not be neglected.",
"Moreover, since we interrogated the MBs behavior near their resonance frequency, this effect became more significant.",
"At a high void fraction $\\beta _0=5.1\\times 10^{-3}$ , we compared the predictions of our nonlinear model with the experimental results of Silberman [64].",
"It was shown that by including the bubble-bubble interaction we can significantly improve the fit to experiments and potentially address the near resonance attenuation overestimation by the linear models (Fig.",
"10 in Appendix E).",
"A wide range of pressure amplitudes are employed in diagnostic and therapeutic ultrasound.",
"As examples, in diagnostic ultrasound peak negative pressures of $\\approx $ 100kPa at 2.5MHz [57] and 127kPa at 5MHz[58] have been applied.",
"In therapeutic ultrasound the focal pressure amplitudes of 180kPa-300kPa have been used in blood brain barrier opening [59], [60].",
"In MB enhanced hyperthermia, sonothrombolysis and drug delivery peak negative pressures between 110-975kPa [128], [61], [62], [63] have been applied.",
"In MB enhanced high intensity focused ultrasound heating pressures in the range of 1.5MPa were applied [37].",
"In all of the therapeutic applications, focused transducers employ large pressure gradients in the focal region.",
"Near and at the focus due to the sharp pressure gradients, the pressure increases by more than 10 fold.",
"Thus, it is estimated that the pressure at the pre-focal region is usually below 200kPa for drug delivery and blood brain barrier opening and below 500kPa for thermal ablation applications.",
"The fundamental knowledge of the pressure dependent attenuation in the pre-focal region is of great importance in optimizing the exposure parameters (e.g.",
"shielding reduction).",
"We numerically investigated the pressure dependent attenuation and sound speed at pressures higher than those in the experiments (200kPa-1MPa).",
"We showed that changes in the sound speed and attenuation are strongly non-linear and bubble-bubble interactions become stronger as pressure increases.",
"An interesting numerical observation is that at higher pressures the sound speed of the bubbly medium (assuming monodispersity) becomes 2.5-4 times higher than the sound speed in water, even for the low void fractions used in clinical ultrasound (Fig.",
"10).",
"In sonochemistry and sonoluminescence applications, due to the large increase in the attenuation above the Blake threshold the wave can get attenuated significantly so that the standing wave patterns are lost in sonoreactor geometries[56], [54].",
"Thus, accurate prediction of the attenuation is necessary in the reactor geometry design and optimization of the acoustic exposure parameters [54], [55], [56] Despite the improvement in the estimation of the pressure distribution using the semi-linear Helmholtz model when compared to experiments [54], [52], these methods still suffer from insufficient accuracy in the estimation of the attenuation or pressure dependent wavelengths[54].",
"By taking into account the simultaneous pressured dependent variations of the real and imaginary part of $k^2$ in the presence of bubble-bubble interactions, a more realistic interpretation of the changes of attenuation and sound speed in bubbly media is expected.",
"The experiments and the introduced nonlinear model have inherent limitations.",
"Despite the good agreement between theory and experiment, there were also small discrepancies between the model predictions and experimental measurements.",
"Some of the limitations of the study are discussed here."
],
[
"Neglecting the pressure variation within the chamber and uniform shell properties",
"In modeling the MB behavior, we have not considered pressure variations within the MB chamber due to MB attenuation.",
"The importance of considering the pressure changes within the chamber is discussed in [68].",
"Additionally, we assumed that all the MBs have the identical shell composition and properties, and effects like strain softening or shear thinning of the shell [76], and possible MB destruction, were neglected.",
"As the purpose of the current work was to investigate the simultaneous pressure dependence of the sound speed and attenuation and to develop a model to accurately predict these effects, we have used the simplest model for lipid coated MBs.",
"Investigation of models that incorporate more complex rheological behaviors of the shell which takes into account the effect of shell properties on sound speed and attenuation is the subject of future work."
],
[
"Low void fraction of $\\approx $ 5.1{{formula:edd16576-bb0d-4846-b3db-14eeda7a49d3}} 10{{formula:038328a6-2a8c-4859-b615-f3177732648b}} of micron sized MBs in experiments",
"The changes of the magnitudes of the sound speed peaks in our experiments were small; this is due to the small size and low concentration of the MBs in our experiments.",
"However, it is not only the peak of the sound speed that changed with pressure but also the frequency of the sound speed peaks (frequency of sound speed peak decreased by $\\approx $ 30$\\%$ between 12.5kPa and 100kPa).",
"Bigger bubbles (mm sized [19], [47]) have stronger effects on the magnitude of the sound speed changes of the medium because of their higher compressibility.",
"Moreover, resonance frequency is predicted to change more rapidly with increasing acoustic pressure [16].",
"When volume matched, however, although the sound speed changes of the medium are larger in case of bigger bubbles, however, the attenuation changes of the medium [19], [47], [64] are quite lower than the case of smaller MBs.",
"As an instance maximum attenuation of water with bubbles of $R_0$ =2.07mm and $\\beta _0$ =5.3$\\times $ 10$^{-2}$ was less than 10dB/cm in Silberman et.",
"al[64] while the maximum sound speed reached $\\approx $ 2$\\times $ 10$^4$ m/s.",
"The focus of this paper is on the dynamics of the micron sized ultrasound contrast agents (1-10$\\mu $ m), thus we did not test the cases of mm sized bubbles.",
"Although the changes of the sound speed amplitude are small, we were able to measure these changes in experiments consistent with model predictions indicating the sensitivity of the measurements and the accuracy of the theoretical approach.",
"The void fraction in our study is also relevant to clinical applications of ultrasound.",
"One of the clinically used UCAs is Definity$^{®}$ which is used at lower concentrations for nonlinear imaging and at higher concentrations for clinical therapeutic applications [65], [66], [67].",
"For therapeutic applications, the the dosage of Definity$^{®}$ MBs is $10\\mu $ L per kg weight of the human body (with maximum allowable dose of $20\\mu $ L per kg weight [67]).",
"In novel ultrasound imaging applications, a small total dose of $0.1-0.2 mL$ is used for diagnostic applications [67].",
"Thus, for a 100kg person with approximately 7500mL total blood volume, 1-2 mL of MBs should be injected for clinical therapeutic applications.",
"Each mL of Definity$^{®}$ has $\\approx $ 1.2$\\times $ 10$^{10}$ MBs [65], [66], [67] with gas volume of $\\approx $ 27$\\times $ 10$^{9}$ $\\mu $ m$^3$ /mL.",
"For a 100kg patient, thus, a total gas volume of $\\approx $ 27-54$\\times $$10^{9}$$\\mu $ m$^3$ is injected.",
"Lower and upper limit of $\\beta _0$ (void fraction in m$^3$ /m$^3$ ) thus can be calculated as 27$\\times $$10^{-9}$ m$^3$ /7.5$\\times $$10^{-3}$ m$^3$ and 54$\\times $$10^{-9}$ m$^3$ /7.5$\\times $$10^{-3}$ m$^3$ respectively.",
"This results in $\\beta _0$ of 3.6-7.2$\\times $ 10$^{-6}$ for clinical therapeutic applications.",
"For imaging applications $\\beta _0$ would be 3.6-7.2$\\times 10^{-7}$ inside blood.",
"In our study the concentration of MBs was 6.3$\\times $ 10$^4$ MBs/mL with $\\beta _0$ of $\\approx $ 5.1 $\\times $ 10$^{-6}$ .",
"Thus, the void fraction in our work is relevant to clinical therapeutic applications of the UCAs in blood.",
"When the whole body is considered as the medium, the void fraction would be even lower.",
"$7\\%$ of the body is blood, and assuming homogeneous distribution of blood the actual physical values for $\\beta _0$ in the human body are between 2.5 $\\times 10^{-8}$ -5 $\\times 10^{-7}$ .",
"In many pre-clinical applications (e.g.",
"drug delivery and vascular disruption) high concentrations (e.g 100-200$\\mu $ L per kg) of microbubbles are employed, thus greater changes in the sound speed amplitude are expected (e.g.",
"please refer to Fig.",
"6 and 9 in Appendix D with $\\beta =10^{-5}$ ).",
"However, experimental measurements of the higher void fractions in case of micron sized bubbles (1-10$\\mu $ m) are challenging.",
"As it was discussed and measured in the current work, the attenuation of smaller MBs are very high even when low void fractions are used (e.g.",
"20dB/cm at 100kPa in Fig.",
"5d).",
"Increasing the $\\beta _0$ further increases the attenuation and thus a very large pressure gradient will occur within the sample [68].",
"For example for an attenuation of 60dB/cm just within the first 1mm of the sample holder, the ultrasound amplitude will reduce in half.",
"This causes difficulty in interpretation of the results or numerical implementation of the model.",
"In order to accurately measure the pressure dependent sound speed and attenuation in high void fractions, technically complex experiments are needed where ultrasound only propagates through a very thin sheet of MBs [68], [69].",
"Moreover, bubble-bubble interactions becomes significant (Please refer to Appendix E and F) which further adds to the difficulty of the interpretation of the results.",
"At higher void fractions the steep pressure gradients changes inside the chamber and the bubble-bubble interaction causes large resonance shifts [70], [74] resulting in significant widening of the attenuation curves even in case of monodisperse MBs [45].",
"Thus, experimental detection of a clear relationship between pressure and sound speed and attenuation becomes very difficult.",
"For more accurate detection of this relationship thus, using lower void fractions are crucial.",
"In Appendix E, we presented numerical simulations on the relationship between void fraction and bubble-bubble interaction on the pressure dependent attenuation and sound speed.",
"The experimental detection of the attenuation and sound speed changes at higher void fractions is outside of the scope of this paper and can be the subject of future studies."
],
[
"Assumption of single frequency wave propagation in the model",
"Similar to the approach by Louisnard [20], we only considered the case of mono-frequency propagation in the model.",
"Since the nonlinear ultrasound propagation effects were neglected, the presented model in this work may loose accuracy at higher pressures where the higher order harmonics are generated along the propagating path of the ultrasound waves.",
"Moreover, the simple Helmholz equation can not model the nonlinear propagation of the waves and dissipation terms should take into account the nonlinear waves effects [29], [30], [31], [32], [93], [94].",
"Derivation of the pressure dependent attenuation and sound speed at higher pressures, and Mach numbers, thus, requires changes to the bubble oscillation model in tandem with considering the nonlinear wave propagation effects and is outside of the scope of the current study."
],
[
"Low Mach numbers in experiments",
"To minimize MBs destruction we applied pressures below the Blake threshold of the MBs and we estimate that the maximum MB velocity reached 40m/s in our experiments.",
"We did not experimentally examine the exposure parameters that result in high Mach number regimes.",
"In dealing with high pressure applications, especially at lower frequencies (e.g.",
"20kHz) used in sonochemistry, care must be taken when pressure exceeds that of the Blake threshold.",
"Above the Blake threshold, with increasing pressure, the maximum bubble wall velocity increases very fast.",
"The Keller-Miksis equation takes into account the compressibility effects by only keeping the first order terms.",
"Thus, it is only valid for when $\\frac{|\\dot{R}|}{c}<1$ [91], [92].",
"However, in numerical simulations using the Keller-Miksis equation, this condition is often violated [22], [54], [55] during the violent bubble collapse above the Blake threshold.",
"At higher wall velocities, energy dissipation is dominated by the compressibility effects.",
"Thus a proper representation of the compressibility effects is required for more accurate estimation of the attenuation.",
"When the bubble wall velocity exceeds that of the water sound speed, either corrections to the Keller-Miksis equation should be applied [91], [92] or Glimore equation should be used as it has a higher validity range up to $\\frac{|\\dot{R}|}{c}<2.2$ [123].",
"Zheng et al [89] included the second-order compressibility terms in the simulation of the bubble oscillations.",
"They showed that above approximately the Mach number of 0.59, the predicted energy dissipation deviates from the Keller-Miksis predictions and thus the influence of the liquid compressibility should be fully considered for proper modeling.",
"It is estimated that, at higher Mach numbers ($>$ 0.59-1[89], [92], [91]), higher order liquid compressibility effects dampen further the bubble oscillations, thus the attenuation will be smaller than the value as predicted by the Keller-Miksis equation.",
"When using the nonlinear model, for more accurate predictions, above Mach numbers between 0.59-1, radial oscillations should be calculated using the Glimore equation or models that include the higher order compressibility terms."
],
[
"MB size distribution",
"In our experiments despite using a microfluidic technique, the bubble size distribution was still not fully monodisperse (1.1$\\mu m<R_0<$ 3.3$\\mu m$ ).",
"Thus, the behavior of the MBs were not quite uniform.",
"This contributed to broadening the attenuation width and sound speed changes and reduction in the attenuation and sound speed peaks.",
"Nevertheless, we still were able to observe the pressure dependent effects and the contribution of each population of MBs at a particular size were considered in the modeling.",
"However, using a narrower size range or using MBs that are generated with more complex acoustic size sorting techniques [75] will allow for a more uniform behavior of the MBs and more clear separation of attenuation peaks at each pressure.",
"Additionally, due to uniform and more potent behavior [33], [104], [75], for the same volume fraction a higher attenuation or sound speed peak is expected.",
"Due to the narrow size distribution of the MBs and near resonance sonication, in our experiments we were able to detect strong nonlinearity at relatively low pressure amplitudes.",
"However, it should be noted that commercial contrast agents are polysiperse and subharmonic emission thresholds and onset of nonlinearity could easily extend to few hundred kPa.",
"In summary, we report the first controlled observation of the pressure dependence of sound speed in bubbly media.",
"The relationship between the sound speed and attenuation with pressure was established both theoretically and experimentally.",
"We have presented a model for the calculation of the pressure-dependent attenuation and sound speed in a bubbly medium.",
"The model is free from any linearization in the MB dynamics.",
"The accuracy of the model was verified by comparing it to the linear model [19] at low pressures and the semi-linear Lousinard model [20] at higher pressure amplitudes that cause nonlinear oscillations.",
"The predictions of the model are in good agreement with experimental observations.",
"We showed that to accurately model the changes of the attenuation and sound propagation in a bubbly medium we also need to take into account how the sound speed changes with pressure and frequency.",
"There is a the limited range of experiments in the near resonance regime and with monodisperse bubbles.",
"Future experiments employing polydisperse commercial MBs, sonication of MBs below their resonance and application of higher acoustic pressure are needed to gain better insight on the attenuation and sound speed phenomena as well as testing the model predictions for the entire range of biomedical ultrasound applications.",
"Nevertheless, application of this model will help to shed light on the effect of the nonlinear oscillations on the acoustical properties of bubbly media.",
"This might help to discover new potential exposure parameters to further optimize and improve the current applications.",
"Acknowledgments We like to thank Prof. Yuan Xun, Prof. Pedro Goldman and Prof. David E. Goertz for helpful discussions.",
"The work is supported by the Natural Sciences and Engineering Research Council of Canada (Discovery Grant RGPIN-2017-06496), NSERC and the Canadian Institutes of Health Research ( Collaborative Health Research Projects ) and the Terry Fox New Frontiers Program Project Grant in Ultrasound and MRI for Cancer Therapy (project $\\#$ 1034).",
"A. J. Sojahrood is supported by a CIHR Vanier Scholarship and Qian Li is supported by NSF CBET grant $\\#$ 1134420.",
"Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request.",
"Verification of the predictions of the Eqs.",
"REF and REF in linear oscillation regimes The appendices structure is as follows: 1- We first recall three models for uncoated bubbles (Eq.",
"REF ), bubbles coated with linear viscoealstic shells (Eq.",
"REF ) and bubbles immersed in elastic materials (Eq.",
"REF ).",
"The linear terms for the attenuation and sound speed of the bubbly media are then presented for each model (Appendix ).",
"The attenuation and sound speed calculated by Eqs.",
"REF and REF are compared with the linear models for each case at acoustic pressure amplitude of 1 kPa (linear regime of oscillations) in Appendix REF .",
"We show that the predictions of the model are in good agreement with the linear model in the linear oscillation regimes.",
"2- At higher pressures, predictions of the linear models are no longer valid.",
"Thus predictions of the model are compared with the Louisnard model [20].",
"First, nonlinear dissipation power terms are presented for each model in Appendix .",
"The maginary part of wave number squared ($\\langle \\Im (k^2)\\rangle $ ) is then calculated at different pressures and model predictions are compared with the Louisnard model [20] at different pressure amplitudes (Appendix REF ).",
"We show that that ($\\langle \\Im (k^2)\\rangle $ ) as calculated by our model and the Louisnard model [20] are in good agreement.",
"However, one advantage of our model is the capability to calculate the ($\\langle \\Re (k^2)\\rangle $ ) which results in correction of the overestimation of the attenuation in the semi-linear model.",
"3- At higher void fractions bubble-bubble interactions become significant.",
"We numerically investigate the influence of the increasing void fraction on the changes of the attenuation and sound speed.",
"We compare the cases with and without bubble-bubble interaction.",
"Finally, we compare the predictions of the introduced model with the one of the experimental results of [64].",
"We show that inclusion of the bubble-bubble interaction can partially improve the significant overestimation of the attenuation at linear resonance [19], [47].",
"The bubble models The volume fraction occupied by a bubble with $\\beta _i$ (Eq.",
"REF ) depends on the $R(t)$ of each bubble.",
"The $R(t)$ curve of each bubble is determined by solving the model that describes the bubble oscillations (Eqs.",
"REF (uncoated bubble), REF (coated bubble) and REF (bubble immersed in elastic materials)).",
"The predictions of Eq.",
"REF and REF , will be numerically verified in case of an uncoated free bubble model, a coated bubble model and a model of a bubble immersed in sediment or tissue.",
"In each case the model predictions are validated against the linear regime of oscillations at very low pressures ($1 kPa$ ) by comparing the predictions with the linear models.",
"Linear models are derived using the Commander $\\&$ Porspereti approach in [19].",
"Uncoated free bubble Radial oscillations of an uncoated bubble can be modeled to the first order of Mach number by solving the Keller-Miksis [24] (KM) model: ${\\rho \\left(1-\\frac{\\dot{R}}{C_l}\\right)R\\ddot{R}+\\rho \\frac{3}{2}\\dot{R}\\left(1-\\frac{R}{3C_l}\\right)=\\left(1+\\frac{\\dot{R}}{C_l}\\right)G_g+\\frac{R}{C_l}\\frac{d}{dt}G_g}$ where $G_g=P_g-\\frac{4\\mu _L\\dot{R}}{R}-\\frac{2\\sigma }{R}-P_0-P_a \\sin (2 \\pi f t)$ .",
"In this equation, $R$ is radius at time t, $R_0$ is the initial bubble radius, $\\dot{R}$ is the wall velocity of the bubble, $\\ddot{R}$ is the wall acceleration, $\\rho {}$ is the liquid density (998 kg/m$^3$ ), $C_l$ is the sound speed (1481 m/s), $P_g$ is the gas pressure, $\\sigma {}$ is the surface tension (0.0725 N/m), $\\mu {}$ is the liquid viscosity (0.001 Pa s), $P_0$ is the atmospheric pressure (101.325 kPa), and $P_a$ and f are the amplitude and frequency of the applied acoustic pressure.",
"The values in the parentheses are for pure water at 293 K. In this paper the gas inside the bubble is either air or C$_3$ F$_8$ and water is the host media.",
"Coated bubble The dynamics of the coated bubble can be modeled using the Keller-Miksis-Church-Hoff model (KMCH) [35].",
"We have derived this model by adding the compressibility effects to the first order of Mach number in [35].",
"The model is presented in Eq.",
"6: $\\begin{gathered}\\rho \\left(\\left(1-\\frac{\\dot{R}}{C_l}\\right)R\\ddot{R}+\\frac{3}{2}\\dot{R}^2\\left(1-\\frac{\\dot{R}}{3C_l}\\right)\\right)=\\\\\\left(1+\\frac{\\dot{R}}{C_l}+\\frac{R}{C_l}\\frac{d}{dt}\\right)\\left(P_g-\\frac{4\\mu _L\\dot{R}}{R}-\\frac{12\\mu _{sh}\\epsilon R_0^2\\dot{R}}{R^4}-12G_s\\epsilon R_0^2 \\left(\\frac{1}{R^3}-\\frac{R_0}{R^4}\\right)-P_0-P\\right)\\end{gathered}$ in this equation $\\mu _{sh}$ is the viscosity of the shell (coating), $\\epsilon $ is the thickness of the coating, $G_s$ is the shell shear modulus, $P_g$ is the gas pressure inside the bubble, $P_0$ is the atmospheric pressure (101.325 kPa) and P is the acoustic pressure given by $P=P_a\\sin (2\\pi ft)$ with $P_a$ and $f$ are respectively the excitation pressure and frequency.",
"In this paper for all of the simulations related to coated bubbles $G_s$ is 45 MPa and $\\mu _{sh}$ is given by 1.49$(R_0$ ($\\mu m)$ -0.86)$/\\theta $ (nm) [48] ($sh$ stands for shell (coating)) with $\\theta $ is 4 nm unless otherwise stated.",
"Bubble in sediment or tissue The Yang and Church model [49] describes the radial oscillations of an uncoated bubble in a viscoelastic medium (e.g.",
"marine sediments or tissue): $\\rho \\left(1-\\frac{\\dot{R}}{C_l}\\right)R\\ddot{R}+\\frac{3}{2}\\rho \\dot{R}^2\\left(1-\\frac{\\dot{R}}{3C_l}\\right)=\\\\\\left(1+\\frac{\\dot{R}}{C_l}+\\frac{R}{C_l}\\frac{d}{dt}\\right)\\left(P_g-\\frac{2\\sigma }{R}-\\frac{4\\mu _s\\dot{R}}{R}-\\frac{4G}{3R^3}\\left(R^3-R_0^3\\right)-P_0-P_a\\sin (2\\pi ft)\\right)$ This equation, similar to Eqs.",
"REF and REF accounts for compressibility effects to the first order of Mach number, thus inherits the acoustic radiation losses.",
"Several approaches for the incorporation of such losses into a Rayleigh-Plesset type equation were outlined in [33].",
"The introduced new constant $G$ describes the shear modulus and $\\mu _s$ describes the shear viscosity of the sediment or tissue.",
"In this paper we considered a tissue with $G$ is 0.5 MPa, $\\mu _s$ =0.00287 Pa.s and $\\sigma $ is 0.056 N/m (blood surface tension)[49].",
"Gas pressure and thermal effects 4a-Linear thermal model The linear thermal model [19], [50] is a popular model that has been widely used in studies related to oceanography [5], [6] and the modeling and charecterization of coated bubble oscillations[76], [77], [78], [79], [80].",
"In this model through linearization, thermal damping is approximated by adding an artificial viscosity term to the liquid viscosity.",
"Furthermore, a variable isoentropic index is used instead of the polytropic exponent of the gas.",
"In this model $P_g$ is given by: $P_g=P_{g0}\\left(\\frac{R_0}{R}\\right)^{3k_i}$ Where the polytropic exponent $\\gamma $ is replaced by isoentropic indice ($k_i$ ): $k_i=\\frac{1}{3}\\Re (\\phi )$ The liquid viscosity is artificially increased by adding a thermal viscosity ($\\mu _{th}$ ) to the liquid viscosity.",
"This thermal viscosity ($\\mu _{th}$ ) is given by: $\\mu _{th}=\\frac{P_{g0} \\Im (\\phi )}{4\\omega }$ In the above equations the complex term $\\phi $ is calculated from $\\phi =\\frac{3\\gamma }{1-3\\left(\\gamma -1\\right)i\\chi \\left[\\left(\\frac{i}{\\chi }\\right)^{\\frac{1}{2}}{\\coth }\\left(\\frac{i}{\\chi }\\right)^{\\frac{1}{2}}-1\\right]}$ where $\\gamma $ is the polytropic exponent and $\\chi =D/\\omega R_0^2$ represents the thermal diffusion length where $D$ is the thermal diffusivity of the gas.",
"$D= L/C_p\\rho _g$ where $C_p$ , $\\rho _g$ , and $L$ are specific heat in constant pressure, density and thermal conductivity of the gas inside the bubble.",
"To calculate the radial oscillations of the coated bubble and uncoated bubble while including the linear thermal effects Eqs.",
"REF , REF $\\&$ REF are coupled with Eq.",
"REF and the liquid viscosity is increased by $\\mu _{th}$ (Eq.REF ).",
"The linear thermal model is used to derive the attenuation and sound speed terms in the regime of linear oscillations.",
"In case of the uncoated bubble and the uncoated bubble in viscoelastic medium (Eq.REF $\\&$REF ) $P_g=P_0$$+$ 2$\\sigma $ /$R_0$ .",
"In the case of the coated bubble (Eq.REF ) $P_g=P_0$ [28].",
"4b- Full thermal model If temperature dependent thermal effects are considered, $P_g$ is given by Eq.",
"12 [96]: $P_g=\\frac{N_gKT}{\\frac{4}{3}\\pi R(t)^3-N_g B}$ here $N_g$ is the total number of the gas molecules, $K$ is the Boltzman constant and B is the molecular co-volume.",
"The average temperature inside the gas can be calculated using Eq.",
"13 [96]: $\\dot{T}=\\frac{4\\pi R(t)^2}{C_v} \\left(\\frac{L\\left(T_0-T\\right)}{L_{th}}-\\dot{R}P_g\\right)$ here $C_v$ is the heat capacity at constant volume, $T_0$ = 293 K is the initial gas temperature, $L_{th}$ is the thickness of the thermal boundary layer.",
"$L_{th}$ is given by $L_{th}={\\min }(\\sqrt{\\frac{DR(t)}{|\\dot{R(t)}|}},\\frac{R(t)}{\\pi })$ where $D$ is the thermal diffusivity of the gas.",
"$D$ can be calculated using $D=\\frac{L}{c_p \\rho _g}$ where $L$ is the gas thermal conductivity, $c_p$ is specific heat at constant pressure and $\\rho _g$ is the gas density.",
"Predictions of the full thermal model have been shown to be in good agreement with predictions of the models that incorporate the thermal effects using the PDEs [97].",
"To calculate the radial oscillations of the coated bubble and uncoated bubble while including the full thermal effects Eqs.REF (coated bubble) or Eq.REF (uncoated bubble) or Eq.",
"REF (bubble in sediment or tissue) are coupled with Eq.",
"REF and Eq.",
"REF and then solved using the ode45 solver in Matlab.",
"The relative and absolute tolerances were 1$\\times $ 10$^{-13}$ and 1$\\times $ 10$^{-14}$ .",
"Attenuation and sound speed equations for linear regime of oscillations The linear thermal equations (Eqs.REF ,REF , REF $\\&$ REF ) were coupled to the bubble models (Eqs.REF ,REF $\\&$ REF ) to derive the attenuation and sound speed terms.",
"For the linear regime, radial oscillations can be considered as $R=R_0(1+x)$ where $x$ is a small displacement amplitude [2], [19], [28], [100].",
"Thus, $\\dot{R}=R_0\\dot{x}$ and $R^{-n}=R_0(1-nx)$ (higher order small terms are neglected in the Taylor series expansion of $R^{-n}$ ).",
"A function $g(t)=e^{i\\omega t}$ is also defined [2], [19], [28], [100].",
"The incident pressure can be linearized as [2], [100]: $P_ag(t)=\\frac{\\rho \\ddot{R}R_0}{\\left(1-\\frac{i\\omega R_0}{C_l}\\right)}$ Using these linear approximations a linear analytical solution to Eqs.",
"REF , REF and REF can be provided.",
"These solutions can be written in the following general form of forced damped oscillations: $\\begin{gathered}\\alpha \\ddot{x}+2\\beta \\dot{x}+\\gamma x=-P_Ae^{i\\omega t}\\end{gathered}$ where constants $\\alpha $ , $\\beta $ and $\\gamma $ can be defined by solving the appropriate equations.",
"We can transfer Eq.REF to the frequency domain by setting $x(t)=x(\\omega )e^{i\\omega t}$ : $\\begin{gathered}-\\alpha \\omega ^2x(\\omega )e^{i\\omega t}+2i\\beta \\omega x(\\omega )e^{i\\omega t}+\\gamma x(\\omega )e^{i\\omega t}=-P_Ae^{i\\omega t}\\end{gathered}$ Eq.REF can be simplified by dividing both sides by $\\alpha e^{i\\omega t}$ and inputting $\\omega _0=\\frac{\\gamma }{\\alpha }$ ($\\omega _0$ is resonance angular frequency).",
"Thus: $x(\\omega )\\left[\\omega _0^2-\\omega ^2+\\frac{2i\\beta }{\\alpha }\\omega \\right]=-\\frac{P_A}{\\alpha }$ Free uncoated bubble (KM model) constants In case of the uncoated bubble model Eq.",
"REF , the constants $\\alpha $ , $\\beta $ and $\\gamma $ of Eq.REF can be derived using an approach similar to [2], [100]: $\\begin{dcases}\\alpha =\\rho R_0^2+\\frac{4\\mu R_0}{C_l}\\\\\\beta =2\\mu _{th}-\\frac{\\sigma }{C_l}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)\\\\\\gamma =P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\end{dcases}$ and the angular resonance frequency $\\omega _0=2\\pi f_r$ ($f_r$ is the linear resonance frequency) is given by: $\\omega _0=\\sqrt{\\frac{P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}}{\\rho R_0^2+\\frac{4\\mu \\ R_0}{C_l}}}$ The constant $\\delta _{total}$ is the total damping and is defined as $\\delta _{total}=\\delta _{liquid}+\\delta _{radiation}+\\delta _{thermal}=\\frac{\\beta }{\\alpha }$ (in left hand side of equation REF ).",
"Thus: $\\begin{gathered}\\delta _{total}=\\frac{\\beta }{\\alpha }=\\frac{2\\mu _{th}-\\frac{\\sigma }{C_l}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\end{gathered}$ where $\\delta _{Vis}$ , $\\delta _{th}$ , $\\delta _{Rad}$ $\\&$ $\\delta _{int}$ are damping constants due to liquid viscosity, thermal loss, re-radiation and interfacial effects.",
"$\\begin{dcases}\\delta _{liquid}=\\frac{2\\mu }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{thermal}=\\frac{2\\mu _{th}}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{radiation}=\\frac{\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{int}=\\frac{-\\sigma }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\end{dcases}$ Coated bubble (KMCH) model Linearizing Eq.",
"REF for coated bubbles we can arrive in an analytical solution in the form of Eq.",
"REF where the angular linear resonance frequency is given by $\\begin{gathered}\\omega _0=\\sqrt{\\frac{P_{g0}\\Re (\\phi )+\\frac{12G_s\\epsilon }{R_0}+\\frac{\\omega ^2 \\rho R_0^2}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}}\\end{gathered}$ and $\\begin{dcases}\\alpha =\\rho R_0^2+\\frac{4\\mu R_0}{C_l}\\\\\\beta =2\\mu _{th}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)-2GR_0\\\\\\gamma =P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}+4G\\end{dcases}$ In this case, existence of the shell introduces an extra term for damping due to shell viscosity $\\delta _{shell}$ .",
"Thus, $\\delta _{total}=\\delta _{liquid}+\\delta _{radiation}+\\delta _{shell}+\\delta _{thermal}$ where: $\\begin{gathered}\\begin{dcases}\\delta _{liquid}=\\frac{2\\mu }{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\\\\\delta _{radiation}=\\frac{\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2} \\left(\\rho R_0^2\\right)}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\\\\\delta _{shell}=\\frac{\\frac{6\\mu _{sh}\\epsilon }{R_0}+\\frac{6G_s\\epsilon }{C_l}}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\\\\\delta _{thermal}=\\frac{2\\mu _{th}}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\end{dcases}\\end{gathered}$ Bubble immersed in tissue or sediment The linear analytical solution to Eq.REF for bubbles immersed in tissue or sediments can be written again in the form of Eq.",
"REF .",
"The constants of the equation can be written as follows [2], [100]: $\\omega _0=\\sqrt{\\frac{P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}+4G}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}}$ and $\\begin{dcases}\\alpha =\\rho R_0^2+\\frac{4\\mu R_0}{C_l}\\\\\\beta =2\\mu _{th}-\\frac{\\sigma }{C_l}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)-2GR_0\\\\\\gamma =P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}+4G\\end{dcases}$ The total damping this case has a term related to the elasticity of the sediment or the tissue ($\\delta _{Sed}$ ).",
"Thus, $\\delta _{total}=\\delta _{liquid}+\\delta _{radiation}+\\delta _{Sed}+\\delta _{thermal}$ where: $\\begin{dcases}\\delta _{liquid}=\\frac{2\\mu }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{thernal}=\\frac{2\\mu _{th}}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{radiation}=\\frac{\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{int}=\\frac{-\\sigma }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{Sed}=\\frac{2GR_0}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\end{dcases}$ Derivation of the linear equations of attenuation and sound speed Using the linear formulations for $R$ , ${\\partial {}^2\\beta {}}/{\\partial {}t^2}=4\\pi {R_0}^3\\ddot{x}$ .",
"Moreover $x(\\omega )$ can be calculated from Eq.",
"REF : $x(\\omega )=\\frac{-\\frac{P_A}{\\alpha }}{\\left[\\omega _0^2-\\omega ^2+2i \\delta _{total} \\omega \\right]}$ and $\\dot{x}(\\omega )=-\\omega x(\\omega )$ $\\&$ $\\ddot{x}(\\omega )=\\omega ^2 x(\\omega )$ .",
"Inputting these into Eq.",
"REF and eliminating $e^{i\\omega t}$ yields: ${\\nabla {}}^2\\left(P\\right)+k^2P=0$ where $k$ is the complex wave number ($k={\\omega }/{C_l}-i\\alpha $ ): $k^2=(\\frac{\\omega }{C_l})^2+4\\pi \\omega ^2 \\sum _{j=1}^N \\frac{R_{0j}}{\\omega _{0}^2-\\omega ^2+2i\\delta _{total} \\omega }$ where $R_{0j}$ is the initial radius of the bubble number j. Attenuation and sound speed can easily be obtained from equation REF .",
"Validation of Eqs.",
"REF REF at nonlinear regime of oscillations Louisnard [20] derived the pressure dependent term for the imaginary part of the wave number.",
"Thus, at higher pressures, predictions of the imaginary part of the wave number as calculated by Eqs.",
"REF and REF are verified numerically with predictions of the Louisnard model [20] and for different pressure amplitudes.",
"The Louisnard model was modified by Jamshidi $\\&$ Brenner [21] to include the compressibility effects to the first order of Mach number.",
"Using this approach, they were able to present the nonlinear terms that describe the power loss due to radiation, thermal and viscous effects.",
"In [34] we provided critical corrections to the derived terms in [21] for uncoated bubbles.",
"Using our approach in [34], we have derived the terms describing the nonlinear power loss in the case of the coated bubbles in [35].",
"Here, we will also derive the terms describing the nonlinear power loss for bubbles that are immersed in sediments or tissues using our approach in [34].",
"In Louisnard's approach, firstly, the terms for the nonlinear energy dissipation are derived accounting for large bubble oscillation amplitude.",
"One starts with the mass and momentum conservation equations in a bubbly medium [46]: $\\begin{dcases}\\frac{1}{\\rho C_l^2}\\frac{\\partial P}{\\partial t}+\\nabla .v=\\frac{\\partial \\beta }{\\partial t}\\\\\\rho \\frac{\\partial v}{\\partial t}=-\\nabla P\\end{dcases}$ here $P(r,t)$ $\\&$ $v(r,t)$ are the pressure and velocity field.",
"The above equation can be written as: $\\frac{\\partial }{\\partial t}\\left(\\frac{1}{2}\\frac{P^2}{\\rho C_l^2}+\\frac{1}{2}\\rho v^2\\right)=NP\\frac{\\partial V}{\\partial t}$ Where $V(r,t)$ is the instantaneous volume of the bubbles at time $t$ and $N$ is the number of bubbles per unit volume.",
"$V$ can be calculated by solving the related bubble model (Eqs.REF , REF and REF ).",
"In order to get an energetic interpretation of the bubble radial motion, both sides of the bubble model (e.g.",
"Eq.",
"REF or REF or REF ) can be multiplied by the time derivative of the bubble volume ${\\partial V}/{\\partial t}$ and using the equation of the kinetic energy of the liquid [20] $K_l=2\\pi \\rho R^3 \\dot{R}^2 $ and Eq.REF , one can arrive at: $\\begin{gathered}\\frac{\\partial }{\\partial t}\\left(\\frac{1}{2}\\frac{P^2}{\\rho C_l^2}+\\frac{1}{2}\\rho v^2+NK_l+4N\\pi R^2 \\sigma \\right)+\\nabla .",
"(Pv)=\\\\-N\\left(\\pi _{total}\\right)\\end{gathered}$ where $\\pi _{total}$ is total dissipated energy term.",
"Because Louisnard used the plain Rayleigh-Plesset equation that does not incorporate the compressibility effects of the liquid; he was not able to derive the terms that describe the nonlinear loss due to radiation effects.",
"Jamshidi $\\&$ Brenner [21] used the K-M equation (Eq.",
"REF ) that incorporates the compressibility effects up to the first order of Mach number.",
"Thus, they were able to derive the nonlinear radiation loss terms.",
"In [34] we provided critical corrections to the terms derived in [21].",
"By taking a time average of both sides of Eq.REF and eliminating terms that are zero: $\\nabla .<Pv>=-N\\left(\\Pi _{Total}\\right)$ Where $\\Pi _{Total}$ is the total dissipated power.",
"Eq.",
"REF expresses the conservation of mechanical energy averaged over one or many periods of oscillations.",
"The physical interpretation of this equation is that the the acoustic energy leaving a volume of bubbly liquid is always smaller than the one incident on it [20].",
"This is due to the losses during the bubble oscillations.",
"Each bubble therefore acts as a dissipator of acoustic energy.",
"The physical origin of wave attenuation is thus self-contained in the Caflish model, even for nonlinear oscillations, provided that a correct model is used to describe the losses in the bubble oscillation.",
"In [46], Caflish et.",
"al proposed a conservation equation similar to Eq.",
"35; however, since they disregarded viscosity and assumed isothermal oscillations, mechanical energy was conserved.",
"Eq.",
"REF as derived by Louisnard [20] reverts the same equation solved in 1D by Rozenberg [101] in the case of purely traveling waves, but in the latter work, the dissipated power was fitted from experimental data, rather than being calculated from single bubble dynamics as done by Louisnard [20].",
"Nonlinear dissipation terms of the uncoated free bubble In case of the uncoated bubble model Eq.REF , $\\Pi _{Total}$ is the sum of the following dissipated powers [34]: $\\begin{dcases}\\Pi _{Thermal}=\\frac{-4\\pi }{T}\\int _{0}^{T}R^2\\dot{R}P_g dt\\\\ \\\\\\Pi _{Liquid}=\\frac{16\\pi \\mu _L}{T}\\int _{0}^{T}R\\dot{R}^2dt\\\\ \\\\\\begin{gathered}\\Pi _{Radiation}=\\frac{1}{T}\\int _{0}^{T} \\left[\\frac{4\\pi }{C_l}\\left(R^2\\dot{R}\\left(\\dot{R}P+R\\dot{P}-\\frac{1}{2}\\rho \\dot{R}^3-\\rho R\\dot{R}\\ddot{R}\\right)\\right)\\right.\\\\\\left.-\\left(\\frac{\\dot{R}}{C_l}P_g+\\frac{R}{C_l}\\dot{P}_g\\right)\\frac{\\partial V}{\\partial t}+\\frac{16\\pi \\mu _LR^2\\dot{R}\\ddot{R}}{C_l}\\right]dt\\end{gathered}\\end{dcases}$ where $T$ is the integration time interval.",
"Nonlinear dissipation terms of the coated bubble For the coated bubble Eq.REF , $\\Pi _{Total}$ is the sum of the following dissipated powers [35]: $\\begin{dcases}\\Pi _{Thermal}=\\frac{-4\\pi }{T}\\int _{0}^{T}R^2\\dot{R}P_g dt\\\\ \\\\\\Pi _{Liquid}=\\frac{16\\pi \\mu _L}{T}\\int _{0}^{T}R\\dot{R}^2dt\\\\ \\\\\\Pi _{Shell}=\\frac{48\\pi \\mu _{sh}\\varepsilon R_0^2}{T}\\int _{0}^{T}\\frac{\\dot{R}^2}{R^2}dt\\\\ \\\\\\Pi _{Gs}=\\frac{48\\pi G_s\\varepsilon R_0^2}{T}\\int _{0}^{T}\\left(\\frac{\\dot{R}}{R}-\\frac{R_0\\dot{R}}{R^2}\\right)dt\\\\ \\\\\\begin{gathered}\\Pi _{Radiation}=\\frac{1}{T}\\int _{0}^{t}\\left(4\\pi \\left[\\frac{R^2\\dot{R}^2}{C_l}\\left(P-P_g\\right)+\\frac{R^3\\dot{R}}{C_l}\\left(\\dot{P}-\\dot{P}_g\\right)+\\frac{4\\mu _LR^2\\dot{R}\\ddot{R}}{C_l}\\right.\\right.\\\\\\left.+12\\mu _{sh}\\varepsilon R0^2 \\left(\\frac{\\dot{R}\\ddot{R}}{C_lR}-\\frac{3\\dot{R}^3}{C_l R^2}\\right)+12G_s\\varepsilon R0^2\\left(\\frac{-2\\dot{R}^2}{cR}+\\frac{3R_0\\dot{R}^2}{C_lR^2}\\right) \\right]\\\\\\left.",
"-\\frac{\\rho R^2\\dot{R}^4}{2C_l}-\\frac{\\rho R^3 \\dot{R}^2\\ddot{R}}{C_l}\\right)dt\\end{gathered}\\end{dcases}$ Nonlinear dissipation terms of the bubble in sediment or tissue For the bubbles immersed in sediments or tissue Eq.REF , $\\Pi _{Total}$ is derived using the same approach in [34], [35]: $\\begin{dcases}\\Pi _{Thermal}=\\frac{-4\\pi }{T}\\int _{0}^{T}R^2\\dot{R}P_g dt\\\\ \\\\\\Pi _{Liquid}=\\frac{16\\pi \\mu _L}{T}\\int _{0}^{T}R\\dot{R}^2dt\\\\ \\\\\\begin{gathered}\\Pi _{Radiation}=\\frac{1}{T}\\int _{0}^{T} \\left[\\frac{4\\pi }{C_l}\\left(R^2\\dot{R}\\left(\\dot{R}P+R\\dot{P}-\\frac{1}{2}\\rho \\dot{R}^3-\\rho R\\dot{R}\\ddot{R}+\\frac{4G\\dot{R}}{3C_lR^3}\\left(R^3-R_0^3\\right)\\right)\\right)\\right.\\\\\\left.-\\left(\\frac{\\dot{R}}{C_l}P_g+\\frac{R}{C_l}\\dot{P}_g\\right)\\frac{\\partial V}{\\partial t}+\\frac{16\\pi \\mu _LR^2\\dot{R}\\ddot{R}}{C_l}\\right]dt\\end{gathered}\\\\ \\\\\\Pi _{Sediment}=\\frac{-1}{T}\\int _{0}^{T}\\frac{16G\\pi \\dot{R}}{3R}\\left(R^3-R_0^3\\right)dt\\end{dcases}$ Pressure dependent attenuation and sound speed in Louisnard model Louisnard [20] used the equations of energy dissipation and obtained the imaginary part of the $k^2$ .",
"In this model, the imaginary part of the $k^2$ is pressure dependent and is given by: $\\Im (k^2)=2 \\rho \\omega \\sum _{j=1}^N \\frac{\\Pi (R_{0j})_{Total}}{|P|^2}$ where $\\Pi (R_{0j})_{Total}$ is the total dissipated power due to the oscillations of the $jth$ bubble with initial radius of $R_{0j}$ .",
"The real part of the $k^2$ is still calculated by the linear model (Eq.",
"REF ) and is given by: $\\Re (k^2)=(\\frac{\\omega }{C_l})^2+4\\pi \\omega ^2 \\sum _{j=1}^N \\frac{R_{0j}(\\omega _{0j}^2-\\omega ^2)}{(\\omega _{0j}^2-\\omega ^2)^2-4\\delta _{totalj}^2 \\omega ^2}$ The Louisnard model [20] (Eq.",
"REF ) incorporates the pressure effects in the imaginary part of the wave number.",
"However, because the real part of the wave number is still estimated from the linear approximations it loses accuracy in predicting the sound speed and attenuation, especially in oscillation regimes where the sound speed changes are significant.",
"Validation results Figure: Case of a bubbly medium with MBs with R 0 R_0= 2 μm\\mu m and β 0 \\beta _0=10 -5 ^{-5}.",
"Attenuation calculated using the linear model and nonlinear model (left) and sound speed calculated using the linear model and the nonlinear model at (P= 1kPa) (Right) for: uncoated bubbles in water (a and b), coated bubbles in water (c and d) and uncoated bubbles in tissue (ρ\\rho =1060 kg/m 3 ^3, C l C_l=1540 m/s, μ s \\mu _s= 0.00287 Pa.s, GG=0.5 MPa, σ\\sigma =0.056 N/m ) (e and f).Validation of the model at linear regimes against the linear models The relevant models (Eq.REF (uncoated bubble), Eq.REF (coated bubble) and Eq.",
"REF (bubble in tissue or sediment)) for large amplitude MB oscillations were coupled with the ordinary differential equations describing the thermal damping effects (Eq.REF and Eq.REF ).",
"The new set of differential equations were solved to calculate the MB radial oscillations.",
"Constants of the linear models were calculated from Eq.",
"REF (uncoated bubble), Eq.REF (coated bubble) and Eq.REF (bubble in tissue or sediment).",
"Attenuation and sound speed were then calculated for each case using Eq.REF .",
"Since the linear model is only valid for narrowband pulses with small pressure amplitudes, pulses of 1kPa amplitude with 60 cycles were chosen at each frequency, and the last 20 cycles of the bubble oscillations were used (to eliminate the transient behavior) for the integration using Eqs.REF and REF .",
"For the linearized model, the initial MB radius is 2 $\\mathrm {\\mu }$ m; the gas inside the MB is air and the thermal properties are chosen from [98] (Table 1) and $\\beta _0$ was set to 10$^{-5}$ .",
"Figures REF a-f compare the attenuation and sound speed predictions between the linear model and the non-linear model given by Eqs.REF and REF .",
"Model predictions are in excellent agreement with the linear model for small amplitude radial oscillations ($R_{max}/R_0<1.01$ ) in Fig.",
"REF .",
"The simple model given by Eqs.REF and REF predicted the attenuation and sound speed of the medium for the uncoated bubble, the coated bubble and the bubble in tissue.",
"The model only requires as input the radial oscillations of the bubbles and reduces the complexity of deriving the linear terms in each cases.",
"Fig.",
"REF also shows that the the effect of encapsulating shell (added viscosity and stiffness) reduced the bubble expansion ratio which translated to smaller changes in attenuation and sound speed when compared to the uncoated counterpart.",
"Figure: Case of a bubbly medium with MBs with R 0 R_0= 2 μ\\mu m and β 0 \\beta _0=10 -5 ^{-5} sonicated at various pressures.",
"Left: ℑk 2 \\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {} calculated using the nonlinear model (Eqs.)",
"and Louisnard model (Eq.)",
"and Right: ℜk 2 \\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {} calculated using the nonlinear model (Eqs.)",
"and the Louisnard model (Eq.)",
"(Louisnard model employs the linear model for the real part; thus it is pressure independent) for: uncoated bubbles in water (a and b), coated bubbles in water (c &\\& d) and uncoated bubbles in tissue (ρ\\rho =1060 kg/m 3 ^3, C l C_l=1540 m/s, μ s \\mu _s= 0.00287 Pa.s, GG=0.5 MPa, σ\\sigma =0.056 N/m ) (e and f).",
"Validation of the model at higher pressures against the semi-linear Louisnard model Figure: Comparison between the predictions the Louisnard &\\& the nonlinear model for sound speed and attenuation.",
"Case of a bubbly medium with uncoated MBs with R 0 R_0= 2 μ\\mu m and β 0 \\beta _0=10 -5 ^{-5}.",
"a) attenuation at P a P_a=40 kPa, b) sound speed at P a P_a=40 kPa, c) attenuation at P a P_a=100 kPa d) sound speed at P a P_a=100 kPa, e) attenuation at P a P_a= 150 kPa and f) sound speed at P a P_a=150 kPa.As the pressure increases, assumptions (e.g.",
"small amplitude MB oscillations) on which the linear model is based on are no longer valid.",
"To investigate the effect of pressure, the radial oscillations of the MBs were simulated for exposures of various acoustic pressure amplitudes.",
"For the uncoated bubble $P_a$ = 40, 70, 100, 150 kPa, for the coated bubble $P_a$ = 40, 70, 100, 150, 200 kPa and for the bubble in tissue $P_a$ = 100, 500, 700, 1000 kPa were chosen.",
"The power dissipation expressions for nonlinear damping effects which are given by Eq.REF (for uncoated bubble), Eq.REF for coated bubble and Eq.REF for the bubble in tissue were used to calculate the total dissipated power.",
"The imaginary and real part of the wave number were then calculated using Eqs.REF and REF in case of the nonlinear model and Eq.REF and Eq.REF in case of the Louisnard model [20].",
"The predictions of the two models are illustrated in Fig.",
"REF .",
"The left column of Fig.",
"REF shows that the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ calculated by Eq.",
"REF is in excellent agreement with the Louisnard model (Eq.",
"REF ) for all the acoustic pressures and the bubble models that investigated.",
"The simple approach introduced here, only needs the radial oscillations of the bubble as input and reduces the complexity of the Louisnard model where the equations for different dissipation mechanisms must be derived for each bubble case.",
"Variations of $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ with pressure shows the importance of the considerations of the pressure effects as the linear model fails to predict phenomena like the resonance shift (e.g.",
"[16]), changes in the amplitude of the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ with pressure and the generation of SuH (e.g.",
"[70]) and subharmonic (SH) resonances (e.g.",
"[102], [70]).",
"As an instance in case of the uncoated bubble in Fig.",
"REF a $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 8.5$\\times $ 10$^{8}$ m$^{-2}$ at pressure dependent resonance( $\\frac{f}{f_r}\\unknown.",
"0.98$ ) when $P_a$ =40 kPa.",
"However, as pressure increases to $P_a$ =150 kPa resonance shifts to $f/f_r\\unknown.",
"$ 0.64 and $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 7.3$\\times $ 10$^{8}$ m$^{-2}$ .",
"Moreover, a SuH occurs at ${f}/{f_r}\\unknown.",
"$ 0.34 with $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 2.4$\\times $ 10$^{8}$ m$^{-2}$ .",
"When $P_a=40 kPa$ and at $\\frac{f}{f_r}\\unknown.",
"$ 0.34, $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 3.9$\\times $ 10$^{8}$ m$^{-2}$ .",
"Thus, the pressure increase has a significant influence on the resonances of the system and the magnitude of the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ .",
"The Louisnard model uses the linear assumptions (Eq.REF ) to calculate the $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ .",
"The predictions of the nonlinear model Eq.REF for $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2$ , are compared with the predictions of Eq.REF in the right hand column of Fig.",
"REF and for 3 bubble cases (uncoated, coated and bubble in tissue).",
"We have subtracted the constant $({\\omega }/{C_l})^2$ from $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ to better highlight the pressure dependent changes.",
"In each case, pressure increase leads to significant changes in $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ , and predictions of Eq.",
"REF significantly deviate from the linear values (Eq.",
"REF ).",
"As an instance for the uncoated bubble (Fig.",
"2b) the linear model predicts a maximum for $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ 4.1$\\times $ 10$^{7}$ m$^{-2}$ at ${f/f_r}\\unknown.",
"$ 0.9 and a minimum for $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ -5$\\times $ 10$^{7}$ m$^{-2}$ at $\\frac{f}{f_r}\\unknown.",
"1.12$ .",
"However, when $P_a=100 kPa$ the maximum of $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ 1.9$\\times $ 10$^{7}$ m$^{-2}$ at ${f}/{f_r}\\unknown.",
"$ 0.761 and the minimum is $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ -8.5$\\times $ 10$^{7}$ m$^{-2}$ at ${f}/{f_r}\\unknown.",
"$ 0.773.",
"The nonlinear model incorporates the pressure-dependent changes in $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ and thus can be used to predict the changes of the $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ with pressure.",
"To our best knowledge this is the first time that the frequency-pressure dependence of the $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ in a bubbly medium has been calculated.",
"The ability of the nonlinear model to calculate both the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ and $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ with pressure changes increase the accuracy of the predictions of the medium attenuation and sound speed changes.",
"Figure: Influence of bubble-bubble interaction on the pressure dependent sound speed and attenuation at 100kPa for a coated bubble with R 0 =2μmR_0=2\\mu m in Eq.",
"A2: a-b) β 0 \\beta _0=10 -7 ^{-7}, c-d) β 0 =5.1×10 -6 \\beta _0=5.1\\times 10^{-6} and e-f) β 0 =10 -4 \\beta _0=10^{-4}.",
"In each case 20 bubbles are considered and randomly distributed in a cube.",
"The side lengths of the cube were chosen to replicate the β 0 \\beta _0 in each case.",
"The side length can be calculated as d=(20×4πR 0 3 /3β 0 ) 1/3 d=(20\\times 4\\pi R_0^3/3{}{\\beta _0})^{1/3}.",
"The minimum distance between neighboring MBs was chosen to be 10μ\\mu m. Importance of the accurate calculation of $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ in pressure dependent attenuation and sound speed estimation Fig.",
"REF compares the attenuation and sound speed that are calculated using the nonlinear model and the Louisnard model.",
"The values are calculated for the uncoated bubble in Figs.",
"REF a-b and at $P_a$ =40 kPa, $P_a$ = 100 kPa and $P_a$ = 150 kPa.",
"At 40 kPa (Figs.",
"REF a-b), the Louisnard model fails to capture the sound speed fluctuation around ${f}/{f_r}\\unknown.",
"$ 0.5 due to the occurrence of 2nd order superharmonic (SuH) regime.",
"Moreover, the Louisnard model over-estimates the attenuation at the resonance frequency by about 10 $\\%$ .",
"The deviation in the predicted values between the two models increases with increasing pressure.",
"At $P_a$ = 100 kPa (Figs.",
"REF c-d), Louisnard model overestimates the attenuation by about 40 $\\%$ .",
"Moreover, Louisnard model can not capture the the shift in the maximum sound speed to lower frequencies as well as the $\\approx $ 15$\\%$ increase in its magnitude.",
"At 150 kPa (Figs.",
"REF e-f) the Louisnard model overestimates the attenuation peak by 77 $\\%$ and underestimates the sound speed peak by about 52 $\\%$ .",
"The nonlinear model predicts a shift in the frequency of the sound speed peak by about 42 $\\%$ .",
"Once again, the frequency at which the attenuation peaks (${f}/{f_r}$ =0.65) corresponds to the frequency at which ${C}/{C_l}$ =1.",
"This, shows that pressure dependent effects of $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2$ can not be neglected and must be included in the calculation of sound speed and attenuation.",
"The proposed nonlinear model has the advantage of calculating both of the pressure dependent $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ and $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ .",
"As the pressure increases, the resonance frequency of the bubbles decreases [16], which is observed as the peak of $\\Im {(k^2)}$ in Fig.",
"REF and attenuation curve in Fig.",
"REF shift towards lower frequencies; this corresponds to the frequencies at which the sound speed in the bubbly medium is equal to the sound speed in the absence of the bubbles.",
"This is seen in Fig.",
"REF where the frequency in which attenuation peaks corresponds to the frequency in which ${C}/{C_l}$ =1 in the blue curves that can only be captured by the nonlinear proposed model.",
"At pressure dependent resonances, the oscillations are in phase with the driving acoustic pressure similar to the case of linear resonance(when $f=f_r$ and at $P_a\\unknown.",
"<$ 1 kPa ${C}/{C_l}$ =1 page 290 [103]).",
"As the pressure increases, the maximum sound speed of the bubbly medium increases and occurs at a lower frequency, which depends on the driving acoustic pressure amplitude.",
"The abrupt increases in the sound speed and attenuation at particular frequencies in Figs.",
"REF c-d and in $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ in Figs.",
"REF a, REF c and REF d are due to the pressure dependent resonance frequency which is described in detail in [16].",
"We have previously shown that when MBs are sonicated with their pressure dependent resonance frequency, the radial oscillation amplitude of the MBs undergo a saddle node bifurcation (rapid increase in amplitude) as soon as the pressure increases above a threshold [16] and the maximum stable scattered pressure increases considerably.",
"Figure: Case of a bubbly medium with β 0 =5.3×10 -3 \\beta _0=5.3\\times 10^{-3} and R 0 R_0=2.07 mm (a- attenuation and b-sound speed curves).",
"Blue circles are constructed by solving the nonlinear model (NM) without bubble-bubble interaction.",
"Blue solid line is constructed by the linear Commander and Prosperetti model .",
"The red line-circle is constructed by solving the nonlinear model and incorporating bubble-bubble interaction.",
"Green diamonds are experimentally measured values by Silberman .",
"For the simulations, similar to pressure amplitude of 10Pa is used.",
"Attenuation and sound speed changes at higher void fractions Some applications of MBs (e.g.",
"pre-clinical drug delivery applications) employ high concentration of MBs.",
"At higher void fractions, bubble-bubble interactions become significant.",
"The resonance frequency and maximum oscillation amplitude has been shown to decrease with MB-MB interaction [74], [70], [124], [125].",
"In order to shed light on the pressure dependent changes of the attenuation and sound speed at higher void fractions we have performed numerical simulations on a monodisperse population of MBs with and without considering MB-MB interactions.",
"Figure 9 shows the results of the numerical simulations for a case of MBs with $R_0$ =2$\\mu $ m at 100kPa of driving acoustic pressure at void fractions of 10$^{-7}$ (very dilute suspension), 5.1$\\times 10^{-6}$ (similar to the void fractions in experiments in Figs.",
"4-5) and 10$^{-4}$ (highly concentrated suspension).",
"Radial oscillations in the absence of interaction at each frequency were computed using Eq.",
"A2.",
"In the presence of MB-MB interactions radial oscillations were calculated by considering the pressure radiated by each MB in the location of other MBs by adding the term $-\\rho \\sum _{j\\ne i}^{20}{R_j}/{d_{ij}}(R_j\\ddot{R_j}+2\\dot{R_j}^2)$ to the right side of Eq.",
"A2.",
"An approach similar to Eq.",
"REF and [70] is used for the solution of the large number of the coupled ordinary differential equations.",
"In case of interacting bubbles, for simplicity we considered a case of randomly distributed 20 MBs in a cube with a side length of $d$ .",
"The side length can simply be calculated from the void fraction ($d=(20\\times 4\\pi R_0^3/3{}{\\beta _0})^{1/3}$ )).",
"For a sinusoidal acoustic excitation, the attenuation and sound speed can be calculated using Eqs.",
"REF and REF as follows: $\\langle \\Re (k^2)\\rangle =\\frac{{\\omega {}}^2}{C_l^2}+\\frac{2{\\rho _l{}}}{T{P_a}}\\frac{N}{20}\\sum _{i=1}^{20}\\int _0^T{\\sin (2\\pi ft)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ $\\langle \\Im (k^2)\\rangle =-\\frac{2{\\rho _l{}}}{T{P_a}}\\frac{N}{20}\\sum _{i=1}^{20}\\int _0^T {\\cos (2\\pi ft)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ where $N$ can be calculated from $N$ =$\\beta $ /(4/3$\\pi $$R_0^3$ ).",
"At the lower void fraction of $\\beta _0$ =1$\\times $ 10$^{-7}$ (Fig.",
"9a-b), MB-MB interactions are negligible.",
"As concentration increases, the attenuation and sound speed peak do not linearly scale.",
"At the void fraction of $\\beta _0$ =5.1$\\times $ 10$^{-6}$ , MB-MB interactions lead to a $\\approx $ 5.6$\\%$ and $\\approx $ 8.5$\\%$ decrease in the peak attenuation and the frequency of the peak respectively (Fig.",
"9c).",
"Sound speed peak increases by 3$\\%$ , while the frequency of the sound speed peak decreased by 9.3$\\%$ (Fig.",
"9d).",
"When in a solution, each MB receives a sum of the pressures of the multiple neighboring bubbles, plus the incident acoustic pressure field.",
"Since the scattered pressure by each MB is not negligible, the interaction effects can not be neglected.",
"Moreover, these effects are stronger near the MB resonance frequency.",
"The effective pressure amplitude (pressure of the sound filed plus the pressure radiated by the other bubbles) at the location of one of the bubbles in Fig.",
"9c-d was calculated.",
"Compared to the case in the absence of interaction, the maximum pressure amplitude felt by the bubble increased by 14$\\%$ (114kPa) suggesting a strong MB-MB interaction which can not be neglected.",
"At the higher void fraction of $\\beta _0$ =1$\\times $ 10$^{-4}$ , MB-MB interactions lead to a 13.6$\\%$ and $\\approx $ 16.2$\\%$ decrease in the peak attenuation and the frequency of the peak respectively (Fig.",
"9e).",
"Sound speed peak increases by 10.6$\\%$ , while the frequency of the sound speed peak decreases by 16.3$\\%$ (Fig.",
"9f).",
"The results of the numerical simulations emphasize the influence of the interactions within MBs.",
"Analysis of the experimental results at large void fractions must include these effects.",
"Bubble-bubble interactions are even important in the linear regime of the oscillations.",
"This is emphasized in Commander and Prosperetti [19] as one of the reasons behind the disagreement between the linear model and the experiments.",
"For a large enough void fraction, the distance between bubbles decreases such that the average pressure field exciting a bubble is smaller than, or comparable with, the pressure wave scattered by a neighboring bubble, and thus the linear model fails [19].",
"Since bubble-bubble interactions were neglected, further analysis by Trujillo [47], [53], also did not achieve good agreement with experiential results near resonance (attenuation was over-estimated by at least an order of magnitude).",
"Here, we briefly investigate this important effect, and extend the experimental verification of our nonlinear model to higher void fractions (in case of bigger uncoated bubbles).",
"We considered the experimental results by Silberman [64] (green diamonds) in Figure 10 for air bubbles with $R_0$ =2.07mm and $\\beta _0=5.3\\times 10^{-3}$ .",
"Similar to Trujillo [47] we used an excitation pressure of 10Pa to solve equation A1 at each given frequency in Fig.",
"10.",
"Equation A1, was solved with and without bubble interactions effects.",
"In case of interaction, we considered 20 randomly distributed bubbles in a cube with a side length $d$ .",
"The minimum distance between bubbles was set to be 10mm.",
"Attenuation and sound speed were then calculated from Eq.",
"REF and REF .",
"Figure 10 shows that the linear model and the nonlinear model in the absence of bubble-bubble interactions overestimate the near resonance attenuation by about and order of magnitude (blue line in Fig.",
"10a).",
"However, when bubble-bubble interactions are considered, there is a significant improvement in the the prediction of the attenuation curve (red line in Fig.",
"10a).",
"Similar to the case of the coated MB in Fig.",
"9d, bubble-bubble interaction increases the sound speed peak and reduces the frequency of the peak (Fig.",
"10b).",
"Comparing Fig.",
"9 and Fig.",
"10 we see that for comparable void fractions attenuation of MBs are orders of magnitude higher than mm sized bubbles.",
"Attenuation of a population of MBs with ${}{\\beta _0}=1\\times 10^{-4}$ is about 800 dB/cm (Fig.",
"9c) while this is only about 10dB/cm for bubbles with $R_0=2.07$ mm at ${}{\\beta _0}=5.4\\times 10^{-3}$ (Fig.",
"10a).",
"For the sound speed changes, one can see the exact opposite relationship (Fig.",
"9d and Fig.",
"10b).",
"Attenuation and sound speed changes at higher pressures Figure: Influence of increasing the pressure amplitude on the sound speed and attenuation at β 0 =5.1×10 -6 \\beta _0=5.1\\times 10^{-6} for a coated bubble with R 0 =2μmR_0=2\\mu m in Eq.",
"A2 when P a P_a is: a-b) 200kPa c-d)500kPa and e-f)1000kPa .",
"In each case 20 bubbles are considered and randomly distributed in a cube.",
"The dimensions of the cube were chosen to replicate the β 0 \\beta _0 in each case.",
"The dimension can be calculated as d=(20×4πR 0 3 /3β 0 ) 1/3 d=(20\\times 4\\pi R_0^3/3{\\beta _0})^{1/3}.",
"The minimum distance between neighboring MBs was chosen to be 50μ\\mu m to eliminate the possibility of MBs collisions at higher pressures .",
"Many applications employ pressures above the Blake threshold of the MBs.",
"To investigate the frequency dependent attenuation at higher pressures, in this section we considered a coated MB with $R_0$ =2$\\mu $ m with $\\beta _0$ =5.1$\\times $ 10$^{-6}$ .",
"It is assumed that MB integrity is maintained for all exposures.",
"Figure 11a-b shows the attenuation and sound speed when $P_a$ is 200kPa.",
"The fundamental frequency of the attenuation peak further decreases (30$\\%$ compared to when $P_a$ =100kPa in Fig.",
"9c) when bubble-bubble interaction is considered.",
"The attenuation of the 2nd order superharmonic (SuH) resonance frequency exceeds that of the main resonance ($\\approx $ 27$\\%$ ) in Fig.",
"11a.",
"In the vicinity of the 2nd order SuH frequency (550kHz), the sound speed peak which was below the medium sound speed at 100kPa (0.95$C_{water}$ ) becomes larger ($\\approx $ 1.04$C_{water}$ ).",
"Bubble-bubble interactions reduce the attenuation maxima and frequencies of the attenuation peaks while increasing the maximum sound speed (Figs.11a-f).",
"The attenuation peak increase with pressure in the studied frequency range (0.5-3.5MHz) and the influence of bubble-bubble interactions becomes stronger with increasing pressure.",
"The maximum sound speed increases with increasing pressure and at 500kPa(625kHz) and 1MPa(575kHz) becomes approximately 2.5 and 4.5 times the $C_{water}$ .",
"It is interesting to note that at a $P_a$ of 100kPa, and at the same frequencies, the sound speed was below that of $C_{water}$ .",
"The results indicate strong nonlinear changes in the attenuation and sound speed even at the simulated low void fraction.",
"We will classify these changes during major non-linear regimes of the oscillations (e.g.",
"[36]) in future studies.",
"Importance of pressure dependent measurements in shell characterization of lipid coated MBs undergoing buckling and rupture Figure: Influence of increasing the pressure amplitude on the sound speed and attenuation of a lipid coated MB with R 0 R_0=2.7μ\\mu m, β 0 =5.1×10 -6 \\beta _0=5.1\\times 10^{-6} and different sets of shell parameters A,B,C and D: a-b)12.5kPa c-d)25kPa and e-f)50kPa, g-h)100kPa.",
"For each group, the shell parameters are given in table .",
"Radial oscillations are calculated using Eq.",
"and bubble-bubble interactions and transducer response are neglected for simplicity.",
"The duration of the sonicating pulse is 3μ\\mu s.Here we demonstrate one of the applications of the introduced nonlinear model which is in the accurate characterization of the shell parameters of lipid coated MBs.",
"The nonlinear behavior of the lipid coating including bucking and rupture [18] intensifies the nonlinear changes of the resonance frequency.",
"The linear resonance frequency of the Marmottant model[83] is given by: $f_r=\\frac{1}{2\\pi R_0}\\sqrt{\\frac{1}{\\rho }\\left(3kP_0+(3k-1)\\frac{2\\sigma _0}{R_0}+\\frac{4\\chi }{R_0}\\right)}$ This equation has limited sensitivity to the $\\sigma _0$ value.",
"For a MB with $R_0$ =2.7$\\mu $ m and $\\chi $ of 1N/m, eq.",
"REF predicts only 80kHz changes in the resonance frequency for $\\sigma _0$ between 0 and 0.072N/m.",
"More importantly, the attenuation measurements are often performed using pressures of $\\approx >$ 20kPa [78], [79], [80], [81], [82].",
"Thus, these MBs are already in their nonlinear regime, and since the resonance frequency of the lipid coated MBs shifts with increasing pressure (as small as 5kPa) thus, these methods may underestimate the shell elasticity.",
"Moreover, using the linear model to fit the pressure dependent shell parameters [19] casts doubts on the accuracy of the claims such as stiffness softening and shear thinning with increasing pressure [19].",
"An accurate fit requires interrogation of the frequency dependent attenuation at multiple pressures in increasing steps such as what was done in this study or in [33].",
"At the lower pressures (e.g.",
"$\\approx $$<$ 50kPa) and in the absence of the shell rupture, the changes in the resonance frequency and attenuation peak are majorly affected by the $\\sigma _0$ , $\\chi $ and $k_s$ (Fig.",
"12).",
"At pressures where the rupture occurs, the resonance frequency undergoes a sudden decrease.",
"The magnitude of the shift in the resonance frequency and the attenuation are largely affected by the $R_r$ , $\\chi $ and $k_s$ .",
"At lower pressures (e.g.",
"12.5kPa) several parameters of $\\sigma _0$ , $\\chi $ , $R_r$ and $k_s$ can provide a good fit to the attenuation and sound speed data (Fig.",
"12a-b).",
"However, as the pressure increases (Figs.",
"12c-h), the predictions of each group diverge and only one group provides the best fit to the experimental curves at all excitation pressures.",
"These behaviors can not be captured by the linear model.",
"Table: Shell properties of the lipid coated bubble in Fig.",
"12"
],
[
"Verification of the predictions of the Eqs. ",
"The appendices structure is as follows: 1- We first recall three models for uncoated bubbles (Eq.",
"REF ), bubbles coated with linear viscoealstic shells (Eq.",
"REF ) and bubbles immersed in elastic materials (Eq.",
"REF ).",
"The linear terms for the attenuation and sound speed of the bubbly media are then presented for each model (Appendix ).",
"The attenuation and sound speed calculated by Eqs.",
"REF and REF are compared with the linear models for each case at acoustic pressure amplitude of 1 kPa (linear regime of oscillations) in Appendix REF .",
"We show that the predictions of the model are in good agreement with the linear model in the linear oscillation regimes.",
"2- At higher pressures, predictions of the linear models are no longer valid.",
"Thus predictions of the model are compared with the Louisnard model [20].",
"First, nonlinear dissipation power terms are presented for each model in Appendix .",
"The maginary part of wave number squared ($\\langle \\Im (k^2)\\rangle $ ) is then calculated at different pressures and model predictions are compared with the Louisnard model [20] at different pressure amplitudes (Appendix REF ).",
"We show that that ($\\langle \\Im (k^2)\\rangle $ ) as calculated by our model and the Louisnard model [20] are in good agreement.",
"However, one advantage of our model is the capability to calculate the ($\\langle \\Re (k^2)\\rangle $ ) which results in correction of the overestimation of the attenuation in the semi-linear model.",
"3- At higher void fractions bubble-bubble interactions become significant.",
"We numerically investigate the influence of the increasing void fraction on the changes of the attenuation and sound speed.",
"We compare the cases with and without bubble-bubble interaction.",
"Finally, we compare the predictions of the introduced model with the one of the experimental results of [64].",
"We show that inclusion of the bubble-bubble interaction can partially improve the significant overestimation of the attenuation at linear resonance [19], [47].",
"The bubble models The volume fraction occupied by a bubble with $\\beta _i$ (Eq.",
"REF ) depends on the $R(t)$ of each bubble.",
"The $R(t)$ curve of each bubble is determined by solving the model that describes the bubble oscillations (Eqs.",
"REF (uncoated bubble), REF (coated bubble) and REF (bubble immersed in elastic materials)).",
"The predictions of Eq.",
"REF and REF , will be numerically verified in case of an uncoated free bubble model, a coated bubble model and a model of a bubble immersed in sediment or tissue.",
"In each case the model predictions are validated against the linear regime of oscillations at very low pressures ($1 kPa$ ) by comparing the predictions with the linear models.",
"Linear models are derived using the Commander $\\&$ Porspereti approach in [19]."
],
[
"Uncoated free bubble",
"Radial oscillations of an uncoated bubble can be modeled to the first order of Mach number by solving the Keller-Miksis [24] (KM) model: ${\\rho \\left(1-\\frac{\\dot{R}}{C_l}\\right)R\\ddot{R}+\\rho \\frac{3}{2}\\dot{R}\\left(1-\\frac{R}{3C_l}\\right)=\\left(1+\\frac{\\dot{R}}{C_l}\\right)G_g+\\frac{R}{C_l}\\frac{d}{dt}G_g}$ where $G_g=P_g-\\frac{4\\mu _L\\dot{R}}{R}-\\frac{2\\sigma }{R}-P_0-P_a \\sin (2 \\pi f t)$ .",
"In this equation, $R$ is radius at time t, $R_0$ is the initial bubble radius, $\\dot{R}$ is the wall velocity of the bubble, $\\ddot{R}$ is the wall acceleration, $\\rho {}$ is the liquid density (998 kg/m$^3$ ), $C_l$ is the sound speed (1481 m/s), $P_g$ is the gas pressure, $\\sigma {}$ is the surface tension (0.0725 N/m), $\\mu {}$ is the liquid viscosity (0.001 Pa s), $P_0$ is the atmospheric pressure (101.325 kPa), and $P_a$ and f are the amplitude and frequency of the applied acoustic pressure.",
"The values in the parentheses are for pure water at 293 K. In this paper the gas inside the bubble is either air or C$_3$ F$_8$ and water is the host media."
],
[
"Coated bubble",
"The dynamics of the coated bubble can be modeled using the Keller-Miksis-Church-Hoff model (KMCH) [35].",
"We have derived this model by adding the compressibility effects to the first order of Mach number in [35].",
"The model is presented in Eq.",
"6: $\\begin{gathered}\\rho \\left(\\left(1-\\frac{\\dot{R}}{C_l}\\right)R\\ddot{R}+\\frac{3}{2}\\dot{R}^2\\left(1-\\frac{\\dot{R}}{3C_l}\\right)\\right)=\\\\\\left(1+\\frac{\\dot{R}}{C_l}+\\frac{R}{C_l}\\frac{d}{dt}\\right)\\left(P_g-\\frac{4\\mu _L\\dot{R}}{R}-\\frac{12\\mu _{sh}\\epsilon R_0^2\\dot{R}}{R^4}-12G_s\\epsilon R_0^2 \\left(\\frac{1}{R^3}-\\frac{R_0}{R^4}\\right)-P_0-P\\right)\\end{gathered}$ in this equation $\\mu _{sh}$ is the viscosity of the shell (coating), $\\epsilon $ is the thickness of the coating, $G_s$ is the shell shear modulus, $P_g$ is the gas pressure inside the bubble, $P_0$ is the atmospheric pressure (101.325 kPa) and P is the acoustic pressure given by $P=P_a\\sin (2\\pi ft)$ with $P_a$ and $f$ are respectively the excitation pressure and frequency.",
"In this paper for all of the simulations related to coated bubbles $G_s$ is 45 MPa and $\\mu _{sh}$ is given by 1.49$(R_0$ ($\\mu m)$ -0.86)$/\\theta $ (nm) [48] ($sh$ stands for shell (coating)) with $\\theta $ is 4 nm unless otherwise stated."
],
[
"Bubble in sediment or tissue",
"The Yang and Church model [49] describes the radial oscillations of an uncoated bubble in a viscoelastic medium (e.g.",
"marine sediments or tissue): $\\rho \\left(1-\\frac{\\dot{R}}{C_l}\\right)R\\ddot{R}+\\frac{3}{2}\\rho \\dot{R}^2\\left(1-\\frac{\\dot{R}}{3C_l}\\right)=\\\\\\left(1+\\frac{\\dot{R}}{C_l}+\\frac{R}{C_l}\\frac{d}{dt}\\right)\\left(P_g-\\frac{2\\sigma }{R}-\\frac{4\\mu _s\\dot{R}}{R}-\\frac{4G}{3R^3}\\left(R^3-R_0^3\\right)-P_0-P_a\\sin (2\\pi ft)\\right)$ This equation, similar to Eqs.",
"REF and REF accounts for compressibility effects to the first order of Mach number, thus inherits the acoustic radiation losses.",
"Several approaches for the incorporation of such losses into a Rayleigh-Plesset type equation were outlined in [33].",
"The introduced new constant $G$ describes the shear modulus and $\\mu _s$ describes the shear viscosity of the sediment or tissue.",
"In this paper we considered a tissue with $G$ is 0.5 MPa, $\\mu _s$ =0.00287 Pa.s and $\\sigma $ is 0.056 N/m (blood surface tension)[49]."
],
[
"Gas pressure and thermal effects",
"4a-Linear thermal model The linear thermal model [19], [50] is a popular model that has been widely used in studies related to oceanography [5], [6] and the modeling and charecterization of coated bubble oscillations[76], [77], [78], [79], [80].",
"In this model through linearization, thermal damping is approximated by adding an artificial viscosity term to the liquid viscosity.",
"Furthermore, a variable isoentropic index is used instead of the polytropic exponent of the gas.",
"In this model $P_g$ is given by: $P_g=P_{g0}\\left(\\frac{R_0}{R}\\right)^{3k_i}$ Where the polytropic exponent $\\gamma $ is replaced by isoentropic indice ($k_i$ ): $k_i=\\frac{1}{3}\\Re (\\phi )$ The liquid viscosity is artificially increased by adding a thermal viscosity ($\\mu _{th}$ ) to the liquid viscosity.",
"This thermal viscosity ($\\mu _{th}$ ) is given by: $\\mu _{th}=\\frac{P_{g0} \\Im (\\phi )}{4\\omega }$ In the above equations the complex term $\\phi $ is calculated from $\\phi =\\frac{3\\gamma }{1-3\\left(\\gamma -1\\right)i\\chi \\left[\\left(\\frac{i}{\\chi }\\right)^{\\frac{1}{2}}{\\coth }\\left(\\frac{i}{\\chi }\\right)^{\\frac{1}{2}}-1\\right]}$ where $\\gamma $ is the polytropic exponent and $\\chi =D/\\omega R_0^2$ represents the thermal diffusion length where $D$ is the thermal diffusivity of the gas.",
"$D= L/C_p\\rho _g$ where $C_p$ , $\\rho _g$ , and $L$ are specific heat in constant pressure, density and thermal conductivity of the gas inside the bubble.",
"To calculate the radial oscillations of the coated bubble and uncoated bubble while including the linear thermal effects Eqs.",
"REF , REF $\\&$ REF are coupled with Eq.",
"REF and the liquid viscosity is increased by $\\mu _{th}$ (Eq.REF ).",
"The linear thermal model is used to derive the attenuation and sound speed terms in the regime of linear oscillations.",
"In case of the uncoated bubble and the uncoated bubble in viscoelastic medium (Eq.REF $\\&$REF ) $P_g=P_0$$+$ 2$\\sigma $ /$R_0$ .",
"In the case of the coated bubble (Eq.REF ) $P_g=P_0$ [28].",
"4b- Full thermal model If temperature dependent thermal effects are considered, $P_g$ is given by Eq.",
"12 [96]: $P_g=\\frac{N_gKT}{\\frac{4}{3}\\pi R(t)^3-N_g B}$ here $N_g$ is the total number of the gas molecules, $K$ is the Boltzman constant and B is the molecular co-volume.",
"The average temperature inside the gas can be calculated using Eq.",
"13 [96]: $\\dot{T}=\\frac{4\\pi R(t)^2}{C_v} \\left(\\frac{L\\left(T_0-T\\right)}{L_{th}}-\\dot{R}P_g\\right)$ here $C_v$ is the heat capacity at constant volume, $T_0$ = 293 K is the initial gas temperature, $L_{th}$ is the thickness of the thermal boundary layer.",
"$L_{th}$ is given by $L_{th}={\\min }(\\sqrt{\\frac{DR(t)}{|\\dot{R(t)}|}},\\frac{R(t)}{\\pi })$ where $D$ is the thermal diffusivity of the gas.",
"$D$ can be calculated using $D=\\frac{L}{c_p \\rho _g}$ where $L$ is the gas thermal conductivity, $c_p$ is specific heat at constant pressure and $\\rho _g$ is the gas density.",
"Predictions of the full thermal model have been shown to be in good agreement with predictions of the models that incorporate the thermal effects using the PDEs [97].",
"To calculate the radial oscillations of the coated bubble and uncoated bubble while including the full thermal effects Eqs.REF (coated bubble) or Eq.REF (uncoated bubble) or Eq.",
"REF (bubble in sediment or tissue) are coupled with Eq.",
"REF and Eq.",
"REF and then solved using the ode45 solver in Matlab.",
"The relative and absolute tolerances were 1$\\times $ 10$^{-13}$ and 1$\\times $ 10$^{-14}$ ."
],
[
"Attenuation and sound speed equations for linear regime of oscillations",
"The linear thermal equations (Eqs.REF ,REF , REF $\\&$ REF ) were coupled to the bubble models (Eqs.REF ,REF $\\&$ REF ) to derive the attenuation and sound speed terms.",
"For the linear regime, radial oscillations can be considered as $R=R_0(1+x)$ where $x$ is a small displacement amplitude [2], [19], [28], [100].",
"Thus, $\\dot{R}=R_0\\dot{x}$ and $R^{-n}=R_0(1-nx)$ (higher order small terms are neglected in the Taylor series expansion of $R^{-n}$ ).",
"A function $g(t)=e^{i\\omega t}$ is also defined [2], [19], [28], [100].",
"The incident pressure can be linearized as [2], [100]: $P_ag(t)=\\frac{\\rho \\ddot{R}R_0}{\\left(1-\\frac{i\\omega R_0}{C_l}\\right)}$ Using these linear approximations a linear analytical solution to Eqs.",
"REF , REF and REF can be provided.",
"These solutions can be written in the following general form of forced damped oscillations: $\\begin{gathered}\\alpha \\ddot{x}+2\\beta \\dot{x}+\\gamma x=-P_Ae^{i\\omega t}\\end{gathered}$ where constants $\\alpha $ , $\\beta $ and $\\gamma $ can be defined by solving the appropriate equations.",
"We can transfer Eq.REF to the frequency domain by setting $x(t)=x(\\omega )e^{i\\omega t}$ : $\\begin{gathered}-\\alpha \\omega ^2x(\\omega )e^{i\\omega t}+2i\\beta \\omega x(\\omega )e^{i\\omega t}+\\gamma x(\\omega )e^{i\\omega t}=-P_Ae^{i\\omega t}\\end{gathered}$ Eq.REF can be simplified by dividing both sides by $\\alpha e^{i\\omega t}$ and inputting $\\omega _0=\\frac{\\gamma }{\\alpha }$ ($\\omega _0$ is resonance angular frequency).",
"Thus: $x(\\omega )\\left[\\omega _0^2-\\omega ^2+\\frac{2i\\beta }{\\alpha }\\omega \\right]=-\\frac{P_A}{\\alpha }$"
],
[
"Free uncoated bubble (KM model) constants",
"In case of the uncoated bubble model Eq.",
"REF , the constants $\\alpha $ , $\\beta $ and $\\gamma $ of Eq.REF can be derived using an approach similar to [2], [100]: $\\begin{dcases}\\alpha =\\rho R_0^2+\\frac{4\\mu R_0}{C_l}\\\\\\beta =2\\mu _{th}-\\frac{\\sigma }{C_l}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)\\\\\\gamma =P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\end{dcases}$ and the angular resonance frequency $\\omega _0=2\\pi f_r$ ($f_r$ is the linear resonance frequency) is given by: $\\omega _0=\\sqrt{\\frac{P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}}{\\rho R_0^2+\\frac{4\\mu \\ R_0}{C_l}}}$ The constant $\\delta _{total}$ is the total damping and is defined as $\\delta _{total}=\\delta _{liquid}+\\delta _{radiation}+\\delta _{thermal}=\\frac{\\beta }{\\alpha }$ (in left hand side of equation REF ).",
"Thus: $\\begin{gathered}\\delta _{total}=\\frac{\\beta }{\\alpha }=\\frac{2\\mu _{th}-\\frac{\\sigma }{C_l}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\end{gathered}$ where $\\delta _{Vis}$ , $\\delta _{th}$ , $\\delta _{Rad}$ $\\&$ $\\delta _{int}$ are damping constants due to liquid viscosity, thermal loss, re-radiation and interfacial effects.",
"$\\begin{dcases}\\delta _{liquid}=\\frac{2\\mu }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{thermal}=\\frac{2\\mu _{th}}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{radiation}=\\frac{\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{int}=\\frac{-\\sigma }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\end{dcases}$"
],
[
"Coated bubble (KMCH) model ",
"Linearizing Eq.",
"REF for coated bubbles we can arrive in an analytical solution in the form of Eq.",
"REF where the angular linear resonance frequency is given by $\\begin{gathered}\\omega _0=\\sqrt{\\frac{P_{g0}\\Re (\\phi )+\\frac{12G_s\\epsilon }{R_0}+\\frac{\\omega ^2 \\rho R_0^2}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}}\\end{gathered}$ and $\\begin{dcases}\\alpha =\\rho R_0^2+\\frac{4\\mu R_0}{C_l}\\\\\\beta =2\\mu _{th}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)-2GR_0\\\\\\gamma =P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}+4G\\end{dcases}$ In this case, existence of the shell introduces an extra term for damping due to shell viscosity $\\delta _{shell}$ .",
"Thus, $\\delta _{total}=\\delta _{liquid}+\\delta _{radiation}+\\delta _{shell}+\\delta _{thermal}$ where: $\\begin{gathered}\\begin{dcases}\\delta _{liquid}=\\frac{2\\mu }{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\\\\\delta _{radiation}=\\frac{\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2} \\left(\\rho R_0^2\\right)}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\\\\\delta _{shell}=\\frac{\\frac{6\\mu _{sh}\\epsilon }{R_0}+\\frac{6G_s\\epsilon }{C_l}}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\\\\\delta _{thermal}=\\frac{2\\mu _{th}}{\\rho R_0^2 +\\frac{4\\mu R_0}{C_l}+\\frac{12\\mu _{sh}\\epsilon }{C_l}}\\\\\\end{dcases}\\end{gathered}$"
],
[
"Bubble immersed in tissue or sediment",
"The linear analytical solution to Eq.REF for bubbles immersed in tissue or sediments can be written again in the form of Eq.",
"REF .",
"The constants of the equation can be written as follows [2], [100]: $\\omega _0=\\sqrt{\\frac{P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}+4G}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}}$ and $\\begin{dcases}\\alpha =\\rho R_0^2+\\frac{4\\mu R_0}{C_l}\\\\\\beta =2\\mu _{th}-\\frac{\\sigma }{C_l}+2\\mu +\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)-2GR_0\\\\\\gamma =P_{g0}\\Re (\\phi )-\\frac{2\\sigma }{R_0}+\\frac{\\omega ^2 \\rho R_0^2 }{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}+4G\\end{dcases}$ The total damping this case has a term related to the elasticity of the sediment or the tissue ($\\delta _{Sed}$ ).",
"Thus, $\\delta _{total}=\\delta _{liquid}+\\delta _{radiation}+\\delta _{Sed}+\\delta _{thermal}$ where: $\\begin{dcases}\\delta _{liquid}=\\frac{2\\mu }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{thernal}=\\frac{2\\mu _{th}}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{radiation}=\\frac{\\frac{\\left(\\frac{\\omega R_0}{C_l}\\right)}{1+\\left(\\frac{\\omega R_0}{C_l}\\right)^2}\\frac{\\omega }{2}\\left(\\rho R_0^2\\right)}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{int}=\\frac{-\\sigma }{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\\\\\delta _{Sed}=\\frac{2GR_0}{\\rho R_0^2+\\frac{4\\mu R_0}{C_l}}\\\\\\end{dcases}$"
],
[
"Derivation of the linear equations of attenuation and sound speed",
"Using the linear formulations for $R$ , ${\\partial {}^2\\beta {}}/{\\partial {}t^2}=4\\pi {R_0}^3\\ddot{x}$ .",
"Moreover $x(\\omega )$ can be calculated from Eq.",
"REF : $x(\\omega )=\\frac{-\\frac{P_A}{\\alpha }}{\\left[\\omega _0^2-\\omega ^2+2i \\delta _{total} \\omega \\right]}$ and $\\dot{x}(\\omega )=-\\omega x(\\omega )$ $\\&$ $\\ddot{x}(\\omega )=\\omega ^2 x(\\omega )$ .",
"Inputting these into Eq.",
"REF and eliminating $e^{i\\omega t}$ yields: ${\\nabla {}}^2\\left(P\\right)+k^2P=0$ where $k$ is the complex wave number ($k={\\omega }/{C_l}-i\\alpha $ ): $k^2=(\\frac{\\omega }{C_l})^2+4\\pi \\omega ^2 \\sum _{j=1}^N \\frac{R_{0j}}{\\omega _{0}^2-\\omega ^2+2i\\delta _{total} \\omega }$ where $R_{0j}$ is the initial radius of the bubble number j. Attenuation and sound speed can easily be obtained from equation REF ."
],
[
"Validation of Eqs. ",
"Louisnard [20] derived the pressure dependent term for the imaginary part of the wave number.",
"Thus, at higher pressures, predictions of the imaginary part of the wave number as calculated by Eqs.",
"REF and REF are verified numerically with predictions of the Louisnard model [20] and for different pressure amplitudes.",
"The Louisnard model was modified by Jamshidi $\\&$ Brenner [21] to include the compressibility effects to the first order of Mach number.",
"Using this approach, they were able to present the nonlinear terms that describe the power loss due to radiation, thermal and viscous effects.",
"In [34] we provided critical corrections to the derived terms in [21] for uncoated bubbles.",
"Using our approach in [34], we have derived the terms describing the nonlinear power loss in the case of the coated bubbles in [35].",
"Here, we will also derive the terms describing the nonlinear power loss for bubbles that are immersed in sediments or tissues using our approach in [34].",
"In Louisnard's approach, firstly, the terms for the nonlinear energy dissipation are derived accounting for large bubble oscillation amplitude.",
"One starts with the mass and momentum conservation equations in a bubbly medium [46]: $\\begin{dcases}\\frac{1}{\\rho C_l^2}\\frac{\\partial P}{\\partial t}+\\nabla .v=\\frac{\\partial \\beta }{\\partial t}\\\\\\rho \\frac{\\partial v}{\\partial t}=-\\nabla P\\end{dcases}$ here $P(r,t)$ $\\&$ $v(r,t)$ are the pressure and velocity field.",
"The above equation can be written as: $\\frac{\\partial }{\\partial t}\\left(\\frac{1}{2}\\frac{P^2}{\\rho C_l^2}+\\frac{1}{2}\\rho v^2\\right)=NP\\frac{\\partial V}{\\partial t}$ Where $V(r,t)$ is the instantaneous volume of the bubbles at time $t$ and $N$ is the number of bubbles per unit volume.",
"$V$ can be calculated by solving the related bubble model (Eqs.REF , REF and REF ).",
"In order to get an energetic interpretation of the bubble radial motion, both sides of the bubble model (e.g.",
"Eq.",
"REF or REF or REF ) can be multiplied by the time derivative of the bubble volume ${\\partial V}/{\\partial t}$ and using the equation of the kinetic energy of the liquid [20] $K_l=2\\pi \\rho R^3 \\dot{R}^2 $ and Eq.REF , one can arrive at: $\\begin{gathered}\\frac{\\partial }{\\partial t}\\left(\\frac{1}{2}\\frac{P^2}{\\rho C_l^2}+\\frac{1}{2}\\rho v^2+NK_l+4N\\pi R^2 \\sigma \\right)+\\nabla .",
"(Pv)=\\\\-N\\left(\\pi _{total}\\right)\\end{gathered}$ where $\\pi _{total}$ is total dissipated energy term.",
"Because Louisnard used the plain Rayleigh-Plesset equation that does not incorporate the compressibility effects of the liquid; he was not able to derive the terms that describe the nonlinear loss due to radiation effects.",
"Jamshidi $\\&$ Brenner [21] used the K-M equation (Eq.",
"REF ) that incorporates the compressibility effects up to the first order of Mach number.",
"Thus, they were able to derive the nonlinear radiation loss terms.",
"In [34] we provided critical corrections to the terms derived in [21].",
"By taking a time average of both sides of Eq.REF and eliminating terms that are zero: $\\nabla .<Pv>=-N\\left(\\Pi _{Total}\\right)$ Where $\\Pi _{Total}$ is the total dissipated power.",
"Eq.",
"REF expresses the conservation of mechanical energy averaged over one or many periods of oscillations.",
"The physical interpretation of this equation is that the the acoustic energy leaving a volume of bubbly liquid is always smaller than the one incident on it [20].",
"This is due to the losses during the bubble oscillations.",
"Each bubble therefore acts as a dissipator of acoustic energy.",
"The physical origin of wave attenuation is thus self-contained in the Caflish model, even for nonlinear oscillations, provided that a correct model is used to describe the losses in the bubble oscillation.",
"In [46], Caflish et.",
"al proposed a conservation equation similar to Eq.",
"35; however, since they disregarded viscosity and assumed isothermal oscillations, mechanical energy was conserved.",
"Eq.",
"REF as derived by Louisnard [20] reverts the same equation solved in 1D by Rozenberg [101] in the case of purely traveling waves, but in the latter work, the dissipated power was fitted from experimental data, rather than being calculated from single bubble dynamics as done by Louisnard [20]."
],
[
"Nonlinear dissipation terms of the uncoated free bubble",
"In case of the uncoated bubble model Eq.REF , $\\Pi _{Total}$ is the sum of the following dissipated powers [34]: $\\begin{dcases}\\Pi _{Thermal}=\\frac{-4\\pi }{T}\\int _{0}^{T}R^2\\dot{R}P_g dt\\\\ \\\\\\Pi _{Liquid}=\\frac{16\\pi \\mu _L}{T}\\int _{0}^{T}R\\dot{R}^2dt\\\\ \\\\\\begin{gathered}\\Pi _{Radiation}=\\frac{1}{T}\\int _{0}^{T} \\left[\\frac{4\\pi }{C_l}\\left(R^2\\dot{R}\\left(\\dot{R}P+R\\dot{P}-\\frac{1}{2}\\rho \\dot{R}^3-\\rho R\\dot{R}\\ddot{R}\\right)\\right)\\right.\\\\\\left.-\\left(\\frac{\\dot{R}}{C_l}P_g+\\frac{R}{C_l}\\dot{P}_g\\right)\\frac{\\partial V}{\\partial t}+\\frac{16\\pi \\mu _LR^2\\dot{R}\\ddot{R}}{C_l}\\right]dt\\end{gathered}\\end{dcases}$ where $T$ is the integration time interval."
],
[
"Nonlinear dissipation terms of the coated bubble",
"For the coated bubble Eq.REF , $\\Pi _{Total}$ is the sum of the following dissipated powers [35]: $\\begin{dcases}\\Pi _{Thermal}=\\frac{-4\\pi }{T}\\int _{0}^{T}R^2\\dot{R}P_g dt\\\\ \\\\\\Pi _{Liquid}=\\frac{16\\pi \\mu _L}{T}\\int _{0}^{T}R\\dot{R}^2dt\\\\ \\\\\\Pi _{Shell}=\\frac{48\\pi \\mu _{sh}\\varepsilon R_0^2}{T}\\int _{0}^{T}\\frac{\\dot{R}^2}{R^2}dt\\\\ \\\\\\Pi _{Gs}=\\frac{48\\pi G_s\\varepsilon R_0^2}{T}\\int _{0}^{T}\\left(\\frac{\\dot{R}}{R}-\\frac{R_0\\dot{R}}{R^2}\\right)dt\\\\ \\\\\\begin{gathered}\\Pi _{Radiation}=\\frac{1}{T}\\int _{0}^{t}\\left(4\\pi \\left[\\frac{R^2\\dot{R}^2}{C_l}\\left(P-P_g\\right)+\\frac{R^3\\dot{R}}{C_l}\\left(\\dot{P}-\\dot{P}_g\\right)+\\frac{4\\mu _LR^2\\dot{R}\\ddot{R}}{C_l}\\right.\\right.\\\\\\left.+12\\mu _{sh}\\varepsilon R0^2 \\left(\\frac{\\dot{R}\\ddot{R}}{C_lR}-\\frac{3\\dot{R}^3}{C_l R^2}\\right)+12G_s\\varepsilon R0^2\\left(\\frac{-2\\dot{R}^2}{cR}+\\frac{3R_0\\dot{R}^2}{C_lR^2}\\right) \\right]\\\\\\left.",
"-\\frac{\\rho R^2\\dot{R}^4}{2C_l}-\\frac{\\rho R^3 \\dot{R}^2\\ddot{R}}{C_l}\\right)dt\\end{gathered}\\end{dcases}$"
],
[
"Nonlinear dissipation terms of the bubble in sediment or tissue",
"For the bubbles immersed in sediments or tissue Eq.REF , $\\Pi _{Total}$ is derived using the same approach in [34], [35]: $\\begin{dcases}\\Pi _{Thermal}=\\frac{-4\\pi }{T}\\int _{0}^{T}R^2\\dot{R}P_g dt\\\\ \\\\\\Pi _{Liquid}=\\frac{16\\pi \\mu _L}{T}\\int _{0}^{T}R\\dot{R}^2dt\\\\ \\\\\\begin{gathered}\\Pi _{Radiation}=\\frac{1}{T}\\int _{0}^{T} \\left[\\frac{4\\pi }{C_l}\\left(R^2\\dot{R}\\left(\\dot{R}P+R\\dot{P}-\\frac{1}{2}\\rho \\dot{R}^3-\\rho R\\dot{R}\\ddot{R}+\\frac{4G\\dot{R}}{3C_lR^3}\\left(R^3-R_0^3\\right)\\right)\\right)\\right.\\\\\\left.-\\left(\\frac{\\dot{R}}{C_l}P_g+\\frac{R}{C_l}\\dot{P}_g\\right)\\frac{\\partial V}{\\partial t}+\\frac{16\\pi \\mu _LR^2\\dot{R}\\ddot{R}}{C_l}\\right]dt\\end{gathered}\\\\ \\\\\\Pi _{Sediment}=\\frac{-1}{T}\\int _{0}^{T}\\frac{16G\\pi \\dot{R}}{3R}\\left(R^3-R_0^3\\right)dt\\end{dcases}$"
],
[
"Pressure dependent attenuation and sound speed in Louisnard model",
"Louisnard [20] used the equations of energy dissipation and obtained the imaginary part of the $k^2$ .",
"In this model, the imaginary part of the $k^2$ is pressure dependent and is given by: $\\Im (k^2)=2 \\rho \\omega \\sum _{j=1}^N \\frac{\\Pi (R_{0j})_{Total}}{|P|^2}$ where $\\Pi (R_{0j})_{Total}$ is the total dissipated power due to the oscillations of the $jth$ bubble with initial radius of $R_{0j}$ .",
"The real part of the $k^2$ is still calculated by the linear model (Eq.",
"REF ) and is given by: $\\Re (k^2)=(\\frac{\\omega }{C_l})^2+4\\pi \\omega ^2 \\sum _{j=1}^N \\frac{R_{0j}(\\omega _{0j}^2-\\omega ^2)}{(\\omega _{0j}^2-\\omega ^2)^2-4\\delta _{totalj}^2 \\omega ^2}$ The Louisnard model [20] (Eq.",
"REF ) incorporates the pressure effects in the imaginary part of the wave number.",
"However, because the real part of the wave number is still estimated from the linear approximations it loses accuracy in predicting the sound speed and attenuation, especially in oscillation regimes where the sound speed changes are significant."
],
[
"Validation of the model at linear regimes against the linear models",
"The relevant models (Eq.REF (uncoated bubble), Eq.REF (coated bubble) and Eq.",
"REF (bubble in tissue or sediment)) for large amplitude MB oscillations were coupled with the ordinary differential equations describing the thermal damping effects (Eq.REF and Eq.REF ).",
"The new set of differential equations were solved to calculate the MB radial oscillations.",
"Constants of the linear models were calculated from Eq.",
"REF (uncoated bubble), Eq.REF (coated bubble) and Eq.REF (bubble in tissue or sediment).",
"Attenuation and sound speed were then calculated for each case using Eq.REF .",
"Since the linear model is only valid for narrowband pulses with small pressure amplitudes, pulses of 1kPa amplitude with 60 cycles were chosen at each frequency, and the last 20 cycles of the bubble oscillations were used (to eliminate the transient behavior) for the integration using Eqs.REF and REF .",
"For the linearized model, the initial MB radius is 2 $\\mathrm {\\mu }$ m; the gas inside the MB is air and the thermal properties are chosen from [98] (Table 1) and $\\beta _0$ was set to 10$^{-5}$ .",
"Figures REF a-f compare the attenuation and sound speed predictions between the linear model and the non-linear model given by Eqs.REF and REF .",
"Model predictions are in excellent agreement with the linear model for small amplitude radial oscillations ($R_{max}/R_0<1.01$ ) in Fig.",
"REF .",
"The simple model given by Eqs.REF and REF predicted the attenuation and sound speed of the medium for the uncoated bubble, the coated bubble and the bubble in tissue.",
"The model only requires as input the radial oscillations of the bubbles and reduces the complexity of deriving the linear terms in each cases.",
"Fig.",
"REF also shows that the the effect of encapsulating shell (added viscosity and stiffness) reduced the bubble expansion ratio which translated to smaller changes in attenuation and sound speed when compared to the uncoated counterpart.",
"Figure: Case of a bubbly medium with MBs with R 0 R_0= 2 μ\\mu m and β 0 \\beta _0=10 -5 ^{-5} sonicated at various pressures.",
"Left: ℑk 2 \\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {} calculated using the nonlinear model (Eqs.)",
"and Louisnard model (Eq.)",
"and Right: ℜk 2 \\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {} calculated using the nonlinear model (Eqs.)",
"and the Louisnard model (Eq.)",
"(Louisnard model employs the linear model for the real part; thus it is pressure independent) for: uncoated bubbles in water (a and b), coated bubbles in water (c &\\& d) and uncoated bubbles in tissue (ρ\\rho =1060 kg/m 3 ^3, C l C_l=1540 m/s, μ s \\mu _s= 0.00287 Pa.s, GG=0.5 MPa, σ\\sigma =0.056 N/m ) (e and f)."
],
[
"Validation of the model at higher pressures against the semi-linear Louisnard model",
"As the pressure increases, assumptions (e.g.",
"small amplitude MB oscillations) on which the linear model is based on are no longer valid.",
"To investigate the effect of pressure, the radial oscillations of the MBs were simulated for exposures of various acoustic pressure amplitudes.",
"For the uncoated bubble $P_a$ = 40, 70, 100, 150 kPa, for the coated bubble $P_a$ = 40, 70, 100, 150, 200 kPa and for the bubble in tissue $P_a$ = 100, 500, 700, 1000 kPa were chosen.",
"The power dissipation expressions for nonlinear damping effects which are given by Eq.REF (for uncoated bubble), Eq.REF for coated bubble and Eq.REF for the bubble in tissue were used to calculate the total dissipated power.",
"The imaginary and real part of the wave number were then calculated using Eqs.REF and REF in case of the nonlinear model and Eq.REF and Eq.REF in case of the Louisnard model [20].",
"The predictions of the two models are illustrated in Fig.",
"REF .",
"The left column of Fig.",
"REF shows that the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ calculated by Eq.",
"REF is in excellent agreement with the Louisnard model (Eq.",
"REF ) for all the acoustic pressures and the bubble models that investigated.",
"The simple approach introduced here, only needs the radial oscillations of the bubble as input and reduces the complexity of the Louisnard model where the equations for different dissipation mechanisms must be derived for each bubble case.",
"Variations of $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ with pressure shows the importance of the considerations of the pressure effects as the linear model fails to predict phenomena like the resonance shift (e.g.",
"[16]), changes in the amplitude of the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ with pressure and the generation of SuH (e.g.",
"[70]) and subharmonic (SH) resonances (e.g.",
"[102], [70]).",
"As an instance in case of the uncoated bubble in Fig.",
"REF a $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 8.5$\\times $ 10$^{8}$ m$^{-2}$ at pressure dependent resonance( $\\frac{f}{f_r}\\unknown.",
"0.98$ ) when $P_a$ =40 kPa.",
"However, as pressure increases to $P_a$ =150 kPa resonance shifts to $f/f_r\\unknown.",
"$ 0.64 and $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 7.3$\\times $ 10$^{8}$ m$^{-2}$ .",
"Moreover, a SuH occurs at ${f}/{f_r}\\unknown.",
"$ 0.34 with $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 2.4$\\times $ 10$^{8}$ m$^{-2}$ .",
"When $P_a=40 kPa$ and at $\\frac{f}{f_r}\\unknown.",
"$ 0.34, $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}\\unknown.",
"$ 3.9$\\times $ 10$^{8}$ m$^{-2}$ .",
"Thus, the pressure increase has a significant influence on the resonances of the system and the magnitude of the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ .",
"The Louisnard model uses the linear assumptions (Eq.REF ) to calculate the $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ .",
"The predictions of the nonlinear model Eq.REF for $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2$ , are compared with the predictions of Eq.REF in the right hand column of Fig.",
"REF and for 3 bubble cases (uncoated, coated and bubble in tissue).",
"We have subtracted the constant $({\\omega }/{C_l})^2$ from $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ to better highlight the pressure dependent changes.",
"In each case, pressure increase leads to significant changes in $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ , and predictions of Eq.",
"REF significantly deviate from the linear values (Eq.",
"REF ).",
"As an instance for the uncoated bubble (Fig.",
"2b) the linear model predicts a maximum for $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ 4.1$\\times $ 10$^{7}$ m$^{-2}$ at ${f/f_r}\\unknown.",
"$ 0.9 and a minimum for $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ -5$\\times $ 10$^{7}$ m$^{-2}$ at $\\frac{f}{f_r}\\unknown.",
"1.12$ .",
"However, when $P_a=100 kPa$ the maximum of $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ 1.9$\\times $ 10$^{7}$ m$^{-2}$ at ${f}/{f_r}\\unknown.",
"$ 0.761 and the minimum is $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2\\unknown.",
"$ -8.5$\\times $ 10$^{7}$ m$^{-2}$ at ${f}/{f_r}\\unknown.",
"$ 0.773.",
"The nonlinear model incorporates the pressure-dependent changes in $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ and thus can be used to predict the changes of the $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ with pressure.",
"To our best knowledge this is the first time that the frequency-pressure dependence of the $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ in a bubbly medium has been calculated.",
"The ability of the nonlinear model to calculate both the $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ and $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ with pressure changes increase the accuracy of the predictions of the medium attenuation and sound speed changes.",
"Figure: Influence of bubble-bubble interaction on the pressure dependent sound speed and attenuation at 100kPa for a coated bubble with R 0 =2μmR_0=2\\mu m in Eq.",
"A2: a-b) β 0 \\beta _0=10 -7 ^{-7}, c-d) β 0 =5.1×10 -6 \\beta _0=5.1\\times 10^{-6} and e-f) β 0 =10 -4 \\beta _0=10^{-4}.",
"In each case 20 bubbles are considered and randomly distributed in a cube.",
"The side lengths of the cube were chosen to replicate the β 0 \\beta _0 in each case.",
"The side length can be calculated as d=(20×4πR 0 3 /3β 0 ) 1/3 d=(20\\times 4\\pi R_0^3/3{}{\\beta _0})^{1/3}.",
"The minimum distance between neighboring MBs was chosen to be 10μ\\mu m."
],
[
"Importance of the accurate calculation of $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ in pressure dependent attenuation and sound speed estimation",
"Fig.",
"REF compares the attenuation and sound speed that are calculated using the nonlinear model and the Louisnard model.",
"The values are calculated for the uncoated bubble in Figs.",
"REF a-b and at $P_a$ =40 kPa, $P_a$ = 100 kPa and $P_a$ = 150 kPa.",
"At 40 kPa (Figs.",
"REF a-b), the Louisnard model fails to capture the sound speed fluctuation around ${f}/{f_r}\\unknown.",
"$ 0.5 due to the occurrence of 2nd order superharmonic (SuH) regime.",
"Moreover, the Louisnard model over-estimates the attenuation at the resonance frequency by about 10 $\\%$ .",
"The deviation in the predicted values between the two models increases with increasing pressure.",
"At $P_a$ = 100 kPa (Figs.",
"REF c-d), Louisnard model overestimates the attenuation by about 40 $\\%$ .",
"Moreover, Louisnard model can not capture the the shift in the maximum sound speed to lower frequencies as well as the $\\approx $ 15$\\%$ increase in its magnitude.",
"At 150 kPa (Figs.",
"REF e-f) the Louisnard model overestimates the attenuation peak by 77 $\\%$ and underestimates the sound speed peak by about 52 $\\%$ .",
"The nonlinear model predicts a shift in the frequency of the sound speed peak by about 42 $\\%$ .",
"Once again, the frequency at which the attenuation peaks (${f}/{f_r}$ =0.65) corresponds to the frequency at which ${C}/{C_l}$ =1.",
"This, shows that pressure dependent effects of $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}-({\\omega }/{C_l})^2$ can not be neglected and must be included in the calculation of sound speed and attenuation.",
"The proposed nonlinear model has the advantage of calculating both of the pressure dependent $\\left\\langle {}{\\Re }\\left(k^2\\right)\\right\\rangle {}$ and $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ .",
"As the pressure increases, the resonance frequency of the bubbles decreases [16], which is observed as the peak of $\\Im {(k^2)}$ in Fig.",
"REF and attenuation curve in Fig.",
"REF shift towards lower frequencies; this corresponds to the frequencies at which the sound speed in the bubbly medium is equal to the sound speed in the absence of the bubbles.",
"This is seen in Fig.",
"REF where the frequency in which attenuation peaks corresponds to the frequency in which ${C}/{C_l}$ =1 in the blue curves that can only be captured by the nonlinear proposed model.",
"At pressure dependent resonances, the oscillations are in phase with the driving acoustic pressure similar to the case of linear resonance(when $f=f_r$ and at $P_a\\unknown.",
"<$ 1 kPa ${C}/{C_l}$ =1 page 290 [103]).",
"As the pressure increases, the maximum sound speed of the bubbly medium increases and occurs at a lower frequency, which depends on the driving acoustic pressure amplitude.",
"The abrupt increases in the sound speed and attenuation at particular frequencies in Figs.",
"REF c-d and in $\\left\\langle {}{\\Im }\\left(k^2\\right)\\right\\rangle {}$ in Figs.",
"REF a, REF c and REF d are due to the pressure dependent resonance frequency which is described in detail in [16].",
"We have previously shown that when MBs are sonicated with their pressure dependent resonance frequency, the radial oscillation amplitude of the MBs undergo a saddle node bifurcation (rapid increase in amplitude) as soon as the pressure increases above a threshold [16] and the maximum stable scattered pressure increases considerably.",
"Figure: Case of a bubbly medium with β 0 =5.3×10 -3 \\beta _0=5.3\\times 10^{-3} and R 0 R_0=2.07 mm (a- attenuation and b-sound speed curves).",
"Blue circles are constructed by solving the nonlinear model (NM) without bubble-bubble interaction.",
"Blue solid line is constructed by the linear Commander and Prosperetti model .",
"The red line-circle is constructed by solving the nonlinear model and incorporating bubble-bubble interaction.",
"Green diamonds are experimentally measured values by Silberman .",
"For the simulations, similar to pressure amplitude of 10Pa is used."
],
[
"Attenuation and sound speed changes at higher void fractions",
"Some applications of MBs (e.g.",
"pre-clinical drug delivery applications) employ high concentration of MBs.",
"At higher void fractions, bubble-bubble interactions become significant.",
"The resonance frequency and maximum oscillation amplitude has been shown to decrease with MB-MB interaction [74], [70], [124], [125].",
"In order to shed light on the pressure dependent changes of the attenuation and sound speed at higher void fractions we have performed numerical simulations on a monodisperse population of MBs with and without considering MB-MB interactions.",
"Figure 9 shows the results of the numerical simulations for a case of MBs with $R_0$ =2$\\mu $ m at 100kPa of driving acoustic pressure at void fractions of 10$^{-7}$ (very dilute suspension), 5.1$\\times 10^{-6}$ (similar to the void fractions in experiments in Figs.",
"4-5) and 10$^{-4}$ (highly concentrated suspension).",
"Radial oscillations in the absence of interaction at each frequency were computed using Eq.",
"A2.",
"In the presence of MB-MB interactions radial oscillations were calculated by considering the pressure radiated by each MB in the location of other MBs by adding the term $-\\rho \\sum _{j\\ne i}^{20}{R_j}/{d_{ij}}(R_j\\ddot{R_j}+2\\dot{R_j}^2)$ to the right side of Eq.",
"A2.",
"An approach similar to Eq.",
"REF and [70] is used for the solution of the large number of the coupled ordinary differential equations.",
"In case of interacting bubbles, for simplicity we considered a case of randomly distributed 20 MBs in a cube with a side length of $d$ .",
"The side length can simply be calculated from the void fraction ($d=(20\\times 4\\pi R_0^3/3{}{\\beta _0})^{1/3}$ )).",
"For a sinusoidal acoustic excitation, the attenuation and sound speed can be calculated using Eqs.",
"REF and REF as follows: $\\langle \\Re (k^2)\\rangle =\\frac{{\\omega {}}^2}{C_l^2}+\\frac{2{\\rho _l{}}}{T{P_a}}\\frac{N}{20}\\sum _{i=1}^{20}\\int _0^T{\\sin (2\\pi ft)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ $\\langle \\Im (k^2)\\rangle =-\\frac{2{\\rho _l{}}}{T{P_a}}\\frac{N}{20}\\sum _{i=1}^{20}\\int _0^T {\\cos (2\\pi ft)}\\frac{{\\partial {}}^2{\\beta {}}_i}{\\partial {}t^2}dt$ where $N$ can be calculated from $N$ =$\\beta $ /(4/3$\\pi $$R_0^3$ ).",
"At the lower void fraction of $\\beta _0$ =1$\\times $ 10$^{-7}$ (Fig.",
"9a-b), MB-MB interactions are negligible.",
"As concentration increases, the attenuation and sound speed peak do not linearly scale.",
"At the void fraction of $\\beta _0$ =5.1$\\times $ 10$^{-6}$ , MB-MB interactions lead to a $\\approx $ 5.6$\\%$ and $\\approx $ 8.5$\\%$ decrease in the peak attenuation and the frequency of the peak respectively (Fig.",
"9c).",
"Sound speed peak increases by 3$\\%$ , while the frequency of the sound speed peak decreased by 9.3$\\%$ (Fig.",
"9d).",
"When in a solution, each MB receives a sum of the pressures of the multiple neighboring bubbles, plus the incident acoustic pressure field.",
"Since the scattered pressure by each MB is not negligible, the interaction effects can not be neglected.",
"Moreover, these effects are stronger near the MB resonance frequency.",
"The effective pressure amplitude (pressure of the sound filed plus the pressure radiated by the other bubbles) at the location of one of the bubbles in Fig.",
"9c-d was calculated.",
"Compared to the case in the absence of interaction, the maximum pressure amplitude felt by the bubble increased by 14$\\%$ (114kPa) suggesting a strong MB-MB interaction which can not be neglected.",
"At the higher void fraction of $\\beta _0$ =1$\\times $ 10$^{-4}$ , MB-MB interactions lead to a 13.6$\\%$ and $\\approx $ 16.2$\\%$ decrease in the peak attenuation and the frequency of the peak respectively (Fig.",
"9e).",
"Sound speed peak increases by 10.6$\\%$ , while the frequency of the sound speed peak decreases by 16.3$\\%$ (Fig.",
"9f).",
"The results of the numerical simulations emphasize the influence of the interactions within MBs.",
"Analysis of the experimental results at large void fractions must include these effects.",
"Bubble-bubble interactions are even important in the linear regime of the oscillations.",
"This is emphasized in Commander and Prosperetti [19] as one of the reasons behind the disagreement between the linear model and the experiments.",
"For a large enough void fraction, the distance between bubbles decreases such that the average pressure field exciting a bubble is smaller than, or comparable with, the pressure wave scattered by a neighboring bubble, and thus the linear model fails [19].",
"Since bubble-bubble interactions were neglected, further analysis by Trujillo [47], [53], also did not achieve good agreement with experiential results near resonance (attenuation was over-estimated by at least an order of magnitude).",
"Here, we briefly investigate this important effect, and extend the experimental verification of our nonlinear model to higher void fractions (in case of bigger uncoated bubbles).",
"We considered the experimental results by Silberman [64] (green diamonds) in Figure 10 for air bubbles with $R_0$ =2.07mm and $\\beta _0=5.3\\times 10^{-3}$ .",
"Similar to Trujillo [47] we used an excitation pressure of 10Pa to solve equation A1 at each given frequency in Fig.",
"10.",
"Equation A1, was solved with and without bubble interactions effects.",
"In case of interaction, we considered 20 randomly distributed bubbles in a cube with a side length $d$ .",
"The minimum distance between bubbles was set to be 10mm.",
"Attenuation and sound speed were then calculated from Eq.",
"REF and REF .",
"Figure 10 shows that the linear model and the nonlinear model in the absence of bubble-bubble interactions overestimate the near resonance attenuation by about and order of magnitude (blue line in Fig.",
"10a).",
"However, when bubble-bubble interactions are considered, there is a significant improvement in the the prediction of the attenuation curve (red line in Fig.",
"10a).",
"Similar to the case of the coated MB in Fig.",
"9d, bubble-bubble interaction increases the sound speed peak and reduces the frequency of the peak (Fig.",
"10b).",
"Comparing Fig.",
"9 and Fig.",
"10 we see that for comparable void fractions attenuation of MBs are orders of magnitude higher than mm sized bubbles.",
"Attenuation of a population of MBs with ${}{\\beta _0}=1\\times 10^{-4}$ is about 800 dB/cm (Fig.",
"9c) while this is only about 10dB/cm for bubbles with $R_0=2.07$ mm at ${}{\\beta _0}=5.4\\times 10^{-3}$ (Fig.",
"10a).",
"For the sound speed changes, one can see the exact opposite relationship (Fig.",
"9d and Fig.",
"10b)."
],
[
"Attenuation and sound speed changes at higher pressures",
" Many applications employ pressures above the Blake threshold of the MBs.",
"To investigate the frequency dependent attenuation at higher pressures, in this section we considered a coated MB with $R_0$ =2$\\mu $ m with $\\beta _0$ =5.1$\\times $ 10$^{-6}$ .",
"It is assumed that MB integrity is maintained for all exposures.",
"Figure 11a-b shows the attenuation and sound speed when $P_a$ is 200kPa.",
"The fundamental frequency of the attenuation peak further decreases (30$\\%$ compared to when $P_a$ =100kPa in Fig.",
"9c) when bubble-bubble interaction is considered.",
"The attenuation of the 2nd order superharmonic (SuH) resonance frequency exceeds that of the main resonance ($\\approx $ 27$\\%$ ) in Fig.",
"11a.",
"In the vicinity of the 2nd order SuH frequency (550kHz), the sound speed peak which was below the medium sound speed at 100kPa (0.95$C_{water}$ ) becomes larger ($\\approx $ 1.04$C_{water}$ ).",
"Bubble-bubble interactions reduce the attenuation maxima and frequencies of the attenuation peaks while increasing the maximum sound speed (Figs.11a-f).",
"The attenuation peak increase with pressure in the studied frequency range (0.5-3.5MHz) and the influence of bubble-bubble interactions becomes stronger with increasing pressure.",
"The maximum sound speed increases with increasing pressure and at 500kPa(625kHz) and 1MPa(575kHz) becomes approximately 2.5 and 4.5 times the $C_{water}$ .",
"It is interesting to note that at a $P_a$ of 100kPa, and at the same frequencies, the sound speed was below that of $C_{water}$ .",
"The results indicate strong nonlinear changes in the attenuation and sound speed even at the simulated low void fraction.",
"We will classify these changes during major non-linear regimes of the oscillations (e.g.",
"[36]) in future studies."
],
[
"Importance of pressure dependent measurements in shell characterization of ",
"Here we demonstrate one of the applications of the introduced nonlinear model which is in the accurate characterization of the shell parameters of lipid coated MBs.",
"The nonlinear behavior of the lipid coating including bucking and rupture [18] intensifies the nonlinear changes of the resonance frequency.",
"The linear resonance frequency of the Marmottant model[83] is given by: $f_r=\\frac{1}{2\\pi R_0}\\sqrt{\\frac{1}{\\rho }\\left(3kP_0+(3k-1)\\frac{2\\sigma _0}{R_0}+\\frac{4\\chi }{R_0}\\right)}$ This equation has limited sensitivity to the $\\sigma _0$ value.",
"For a MB with $R_0$ =2.7$\\mu $ m and $\\chi $ of 1N/m, eq.",
"REF predicts only 80kHz changes in the resonance frequency for $\\sigma _0$ between 0 and 0.072N/m.",
"More importantly, the attenuation measurements are often performed using pressures of $\\approx >$ 20kPa [78], [79], [80], [81], [82].",
"Thus, these MBs are already in their nonlinear regime, and since the resonance frequency of the lipid coated MBs shifts with increasing pressure (as small as 5kPa) thus, these methods may underestimate the shell elasticity.",
"Moreover, using the linear model to fit the pressure dependent shell parameters [19] casts doubts on the accuracy of the claims such as stiffness softening and shear thinning with increasing pressure [19].",
"An accurate fit requires interrogation of the frequency dependent attenuation at multiple pressures in increasing steps such as what was done in this study or in [33].",
"At the lower pressures (e.g.",
"$\\approx $$<$ 50kPa) and in the absence of the shell rupture, the changes in the resonance frequency and attenuation peak are majorly affected by the $\\sigma _0$ , $\\chi $ and $k_s$ (Fig.",
"12).",
"At pressures where the rupture occurs, the resonance frequency undergoes a sudden decrease.",
"The magnitude of the shift in the resonance frequency and the attenuation are largely affected by the $R_r$ , $\\chi $ and $k_s$ .",
"At lower pressures (e.g.",
"12.5kPa) several parameters of $\\sigma _0$ , $\\chi $ , $R_r$ and $k_s$ can provide a good fit to the attenuation and sound speed data (Fig.",
"12a-b).",
"However, as the pressure increases (Figs.",
"12c-h), the predictions of each group diverge and only one group provides the best fit to the experimental curves at all excitation pressures.",
"These behaviors can not be captured by the linear model.",
"Table: Shell properties of the lipid coated bubble in Fig.",
"12"
]
] | 2209.08191 |
[
[
"A real-time dynamic obstacle tracking and mapping system for UAV\n navigation and collision avoidance with an RGB-D camera"
],
[
"Abstract The real-time dynamic environment perception has become vital for autonomous robots in crowded spaces.",
"Although the popular voxel-based mapping methods can efficiently represent 3D obstacles with arbitrarily complex shapes, they can hardly distinguish between static and dynamic obstacles, leading to the limited performance of obstacle avoidance.",
"While plenty of sophisticated learning-based dynamic obstacle detection algorithms exist in autonomous driving, the quadcopter's limited computation resources cannot achieve real-time performance using those approaches.",
"To address these issues, we propose a real-time dynamic obstacle tracking and mapping system for quadcopter obstacle avoidance using an RGB-D camera.",
"The proposed system first utilizes a depth image with an occupancy voxel map to generate potential dynamic obstacle regions as proposals.",
"With the obstacle region proposals, the Kalman filter and our continuity filter are applied to track each dynamic obstacle.",
"Finally, the environment-aware trajectory prediction method is proposed based on the Markov chain using the states of tracked dynamic obstacles.",
"We implemented the proposed system with our custom quadcopter and navigation planner.",
"The simulation and physical experiments show that our methods can successfully track and represent obstacles in dynamic environments in real-time and safely avoid obstacles."
],
[
"Introduction",
"Unmanned Aerial Vehicles (UAV) have been widely used in various fields [1][2][3][4], such as construction, agriculture, exploration, and rescue.",
"The UAVs' working environments are usually challenging due to the complex static structures and unpredictable moving obstacles [5].",
"For safe navigation in dynamic environments, an accurate and lightweight perception system is essential for tracking dynamic obstacles and mapping the static environments.",
"There are mainly two categories of approaches for dynamic obstacle detection and tracking in UAV applications.",
"The first category of methods applies vision-based algorithms to detect obstacles [6][7][8][9].",
"Some works [6][7] use geometric information from a depth image to extract obstacles' bounding boxes and achieve real-time tracking performance.",
"However, they cannot classify the static and dynamic obstacles.",
"The others [8][9] adopt the learning-based methods for obstacle avoidance with relatively higher computation resources.",
"The second category of methods utilizes the pointcloud to build a voxel map and classify each voxel into static and dynamic.",
"These methods can represent the dynamic environments efficiently, but their discrete voxel representation cannot accurately predict dynamic obstacles' future trajectories for collision avoidance.",
"Figure: A physical flight experiment using the proposed mapping system.",
"(a) A person walks toward the robot in the top view.",
"(b) The robot generates the avoidance trajectory based on the detected dynamic obstacles and the voxel map.",
"(c) The side view of the experiment.",
"(d) The 2D bounding boxes of obstacles from the depth image during the flight.This paper introduces a novel real-time dynamic obstacle tracking and mapping system with the trajectory prediction module for dynamic environment navigation.",
"The proposed system uses a 3D hybrid map that adopts an occupancy voxel map for static environment representation and tracks dynamic obstacles using their bounding boxes with velocity information.",
"Our method applies the depth map-based detector to obtain 3D region proposals for dynamic obstacles and refine the bounding boxes' sizes by our occupancy map refinement approach.",
"Then, we apply the Kalman filter and our continuity filter to identify and track dynamic obstacles.",
"Next, the proposed dynamic-region cleaning method is applied to free the moving obstacle areas in the static map.",
"Besides, the Markov chain-based trajectory prediction method is developed for dynamic obstacles, considering the interaction between dynamic obstacles and static environments.",
"The contributions of this paper are as follows: Region proposal detector with map refinement: The proposed method applies a lightweight depth image detector to obtain obstacle region proposals and uses the static map to refine the obstacles' bounding boxes.",
"Dynamic obstacle identification and tracking: We apply the Kalman filter and our continuity filter to track and identify dynamic obstacles.",
"The dynamic-region cleaning approach is then applied to clean the remained trails of dynamic obstacles in the static map.",
"Environment-aware trajectory prediction: The proposed Markov chain-based dynamic obstacle trajectory prediction method considers the interaction between dynamic obstacles and the static environment based on the trajectory probability distribution."
],
[
"Related Work",
"Recent years have seen plenty of obstacle detection and tracking algorithms based on different sensors, such as LiDAR [10][11], monocular camera [12][13], event camera [14][15], and depth camera [6][7].",
"The depth camera, one of the most popular sensors for small-size UAV navigation, can provide the robot with images and pointcloud data.",
"Based on the input data representations, there are two categories of obstacle detection and tracking methods: Vision-based method: In [6][7], the u-depth maps are extracted from the depth images to detect obstacles in the environments.",
"The results from [6] show successful obstacle avoidance in dynamic environments.",
"However, static obstacles are represented using the ellipsoid, which can be overly-conservative when the shape of a static obstacle becomes complex.",
"[16] applies a 2D feature-based perception method to achieve obstacle avoidance.",
"[8] adds a learning-based detector to improve the detection accuracy, and [9] applies the neural network to extract features, helping both perception and decision making.",
"However, introducing learning-based methods makes the whole process slower and more computationally demanding [17][18].",
"Pointcloud-based method: This category utilizes the pointcloud data to build a dynamic map for both static and dynamic obstacles [8][19][20][21][22].",
"[8] applies the pointcloud clustering, using a “per-point velocity” voting strategy to identify dynamic obstacles.",
"Inspired by [8], [19] further fuses the dynamic obstacle detection results into a static map generated from the depth pointcloud.",
"[20] represents dynamic and static obstacles in a particle map, capturing the complicated shapes of dynamic obstacles.",
"[21] uses kernel inference in the mapping process to reduce the computation.",
"[22] leverages the voxel map to identify each voxel into static and dynamic and estimate the velocities of dynamic voxels.",
"Most of the works mentioned earlier are either computationally demanding or unable to represent complicated static structures with dynamic obstacles simultaneously.",
"Besides, some methods only apply linear propagation for obstacle trajectory prediction, resulting in the suboptimal performance of obstacle avoidance.",
"To address these issues, we propose a real-time dynamic obstacle tracking and mapping system with the trajectory prediction module for dynamic environment navigation.",
"Similar to [23][24][25], we adopt the Markov chain and introduce environment impact to trajectory prediction, inspired by the idea of social force [26][27].",
"Figure: System framework.",
"The system contains three dynamic modules with one static occupancy map module.",
"It takes the depth images with the robot poses as the inputs and outputs cleaned static maps and dynamic obstacle states with their predicted trajectories."
],
[
"System Framework",
"The whole system can be split into three dynamic modules and one static map module, shown in Fig.",
"REF .",
"First, the region proposal detector detects static and dynamic obstacles and fuses them with the static map to refine the obstacles' region proposals.",
"Then, the dynamic obstacle identification and tracking module tracks the history of obstacles' states and applies the Kalman filter and our continuity filter to identify the dynamic obstacles.",
"Next, the states of dynamic obstacles are fed into the static map to clean the voxel residuals caused by the dynamic obstacles' motions.",
"In the last module, trajectory predictor, the environment-obstacle interaction is introduced into calculating the environment probability for the Markov chain-based trajectory prediction."
],
[
"Obstacle Region Proposal Detection",
"This section describes depth image-based obstacle detection and obstacle region proposal generation with our map refinement method.",
"1) Obstacles detection: This part generates the “raw” 3D bounding box results of obstacles, which contain the rough positions and sizes of static and dynamic obstacles.",
"The raw detection results are generated from the U map inspired by [6][7].",
"A U map is computed with the column depth value histograms of the original depth image [6].",
"Assuming obstacles have continuously changing depth values in the depth images, which differ dramatically from the background, we can get the 2D boxes of the obstacles in the U map, as shown in Fig.",
"REF d, by grouping regions of pixels with values higher than a threshold.",
"This process can provide us with the rough obstacles' lengths and widths.",
"Similarly, the heights on the depth image could be obtained by searching for pixels with depth values close to the corresponding U map 2D box on each column shown in Fig.",
"REF b.",
"So, a box on the image frame can be represented by its center ${P}^{I}_{o} = [{x}^{I}_{o},{y}^{I}_{o}]^{T}$ and the size $S^{I}_{o}=[{w}^{I}_{o},{h}^{I}_{o}]^{T}$ with the depth $d$ , where $I$ denotes the image frame and $o$ denotes the obstacle.",
"The bounding box could then be projected on the camera frame, and further transformed to the map frame with the center of ${P}^{M}_{o} = [{x}^{M}_{o},{y}^{M}_{o}, d_{o}]^{T}$ and size $S^{M}_{o}=[{w}^{M}_{o},{h}^{M}_{o},l^{M}_{o}]$ , where $M$ represents the map frame.",
"Fig.",
"REF c gives a visualization.",
"Figure: Illustration of detecting the raw bounding box of obstacles.",
"(a) The RGB image.",
"(b) The depth map with a 2D detected obstacle.",
"(c) The 3D bounding box in the map frame.",
"(d) The U-map with the 2D bounding box.2) Map refinement: Since the raw detection results from the previous step are not accurate enough due to the noises from the depth images, we refine the detected raw bounding boxes with the static occupancy map to improve the robustness and accuracy.",
"This strategy is named as map refinement.",
"As shown in Fig.",
"REF a, we inflate the above-derived raw bounding boxes in the x, y, and z directions by multiplying a user-defined coefficient factor, $C_{inflate}$ .",
"Then, we search for the occupied voxels in the static occupancy map inside the proposal regions to get the minimal boxes that contain all occupied voxels as the final dynamic obstacle region proposals shown in Fig.",
"REF b."
],
[
"Dynamic Obstacle Identification and Tracking",
"This module tracks the obstacle states with the histories, identifies dynamic obstacles, and cleans the occupied voxels on the trails of dynamic obstacles in the static map.",
"The obstacle states contain the position, velocity and size data.",
"1) Obstacle tracking: For obstacle tracking, all obstacles must be associated with their previous positions in previous time frames.",
"For the detected obstacles set $^{t}O^{C} = \\left\\lbrace ^{t}o^{C}_{0},^{t}o^{C}_{1} ,\\cdots ,^{t}o^{C}_{n}\\right\\rbrace $ in the camera frame at time $t$ , we associate each bounding box $^{t}o^{C}_{i}$ with its best match $^{t-1}o^{C}_{j}$ at the previous time $t-1$ .",
"The match criteria is based on the closest bounding box center distance with the requirement of a high bounding box overlap ratio $r_{i,j}$ : $r_{i,j} = \\frac{A_{i,j}}{A_{i}},$ where $A_{i,j}$ is the top-view overlap area between $^{t}o^{C}_{i}$ and $^{t-1}o^{C}_{j}$ , and $A_{i}$ is the area of $^{t}o^{C}_{i}$ .",
"By associating the previous $k+1$ time step's states, we obtain a tracking history for each obstacle $^{t}H_{i,k}^{C} = \\left\\lbrace ^{t}o_{i}^{C},^{t-1}o_{i}^{C},\\cdots ,^{t-k}o_{i}^{C}\\right\\rbrace $ .",
"To reduce the effects of partially observed bounding boxes, we fix the bounding box sizes once the entire obstacles show up in the camera's field of view (FOV) for $k^{^{\\prime }}$ frames.",
"This strategy is found to be useful for small FOV cameras.",
"Figure: Illustration of the map refinement.",
"a) The red points represent the voxel in the static map, and the blue box is the raw box generated from the detector, which might have a misestimated shape and size.",
"The brown box is inflated by C inflate C_{inflate} b) After searching for occupied voxels, the green box, which is a refined box, gives an improved result.2) Identify dynamic and static obstacles: We apply the Kalman filter in the map frame to estimate the obstacles' velocities.",
"For simplicity, we assume all variables are in the map frame.",
"We denote the obstacle state vector as $X = [x, y, \\dot{x} , \\dot{y}]^{T}$ and the measurement vector as $Z = [o_{x}, o_{y}, v_{x} , v_{y}]^{T}$ , where $v_{x}$ , $v_{y}$ are velocities in the x and y directions calculated by each obstacle's latest two states.",
"The system's model and measurements are described by: $X_{t|t-1} = AX_{t-1} +Q,$ $Z_{t} = HX_{t} + R,$ where $A$ is the state transition model, $Q$ is the covariance of the model noise, $H$ is the measurement model, and $R$ is the covariance of measurement noise.",
"With the estimated velocity from the Kalman filter, we can identify a detected obstacle as dynamic if its velocity exceeds the user-defined threshold.",
"However, some static obstacles might be incorrectly identified as dynamic due to the sensor measurement noises.",
"To solve this issue, we propose the continuity filter to filter out incorrectly identified static obstacles.",
"In Fig.",
"REF top, 6 frames of dynamic obstacle tracking histories are recorded.",
"The brown points and blue arrows are the measured positions and trajectories of a dynamic obstacle.",
"The green points are the ground truth positions.",
"The filter first calculates the displacement vectors (red arrows) using the obstacle pairs from the tracking history defined as: $Pa=\\left\\lbrace (o_{i}^{t-n},o_{i}^{t})_{0},\\cdots , (o_{i}^{t-k},o_{i}^{t-k+n})_{k-n}\\right\\rbrace .$ Then, the displacement vectors between each point in the pairs are defined as: $D=\\left\\lbrace d_{0}^{t-n,t},d_{1}^{t-n-1,t-1},\\cdots ,d_{k-n}^{t-n+k,t-k}\\right\\rbrace ,$ where $d_{0}^{t-n,t}$ means the distance between point $t-n$ and point $t$ .",
"After that, the cosine values of the angles between each pair of vectors with consecutive indices are obtained: $\\Theta = \\left\\lbrace cos(\\theta _{0,1}), cos(\\theta _{1,2}),\\cdots ,cos(\\theta _{k-n-1,k-n})\\right\\rbrace .$ Assuming moving obstacles move with a constant velocity for a short period, the angles between displacement vectors should be close to 0, as the top figure in Fig.",
"REF shows.",
"Conversely, the bottom figure in Fig.",
"REF shows that a static obstacles' measurement generates large angles between displacement vectors.",
"Therefore, we define the continuity coefficient as: $C_{con} = \\frac{\\Sigma \\Theta }{k-n+1}.$ The detected obstacles with $C_{con}$ less than a threshold, $T_{con}$ , will be marked as dynamic temporarily.",
"Then, we keep track of $c$ frames of voting histories, which record the temporary identification result of each obstacle over the past $c$ frame.",
"All the previous result votes for the classification of the current frame, and only those candidates with more than $T_{c}$ vote as dynamic will be finally determined as dynamic.",
"Figure: Illustration of the continuity filter.",
"The k=6 frames' history of positions of a dynamic obstacle and a static obstacle are shown in the top and bottom, respectively.",
"When the obstacle is dynamic, the angles θ 1 \\theta _{1} and θ 2 \\theta _{2} are smaller.3) Dynamic-region cleaning: Since the dynamic obstacles should not be contained in the static map, it is necessary to clean their motion histories in the static map, and this process is named as dynamic-region cleaning.",
"Simply removing the occupancy voxels on the current dynamic obstacles' position is not enough because it may cause later bounding boxes in the next iteration to fail to search for occupied voxels during the map refinement.",
"Therefore, we first clean the occupancy voxels based on the positions of dynamic obstacles in the past $f$ frames.",
"Then, we record the removed voxels into a hash table known as the clean history in each iteration, and the voxels in clean history would be recognized as occupied during map refinement in the next iteration.",
"Figure: Illustration of the trajectory prediction.",
"At the time t-1t-1, all five paths in the path library are safe, so path 3 has the highest probability.",
"However, at time tt, the previously safe paths 2,3 in the path library are not safe anymore.",
"So, the final prediction is path 4."
],
[
"Trajectory Prediction",
"This module applies an environment-aware Markov chain [23][24][25] to predict obstacle trajectories.",
"The whole process is illustrated in Fig.",
"REF .",
"First, we generate a library of $l$ possible paths by collecting human walking data in experiments and fitting them with polynomials.",
"The path library vector starts with the left-turn paths and ends with the right-turn paths with the straight path at the center.",
"Then, the most likely path is chosen by calculating the probability distribution over $l$ paths, $P_{path}$ .",
"To calculate the probability distribution over paths, the initial state in the Markov chain are defined as the probability of each path: $P_{init} =\\begin{bmatrix}p_{init}^{0} & p_{init}^{1} & \\cdots & p_{init}^{l-1}\\end{bmatrix},$ where all values obey a discrete Gaussian distribution, and the probability values are obtained from Gaussian kernels with $p_{init}^{\\frac{l}{2}}$ as the mean because $p_{init}^{\\frac{l}{2}}$ is the path of going straight, and we assume a person always tends to choose paths close to a straight line if possible.",
"In addition, we define the transition matrix in the Markov chain as: $P_{trans} =\\begin{bmatrix}p_{trans}^{0,0} & p_{trans}^{0,1} & \\cdots &p_{trans}^{0,l-1}\\\\p_{trans}^{1,0} & p_{trans}^{1,1} & \\cdots &p_{trans}^{1,l-1}\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\p_{trans}^{l-1,0} & p_{trans}^{l-1,1} & \\cdots & p_{trans}^{l-1,l-1}\\end{bmatrix},$ where each row is a discrete Gaussian distribution from Gaussian kernels with $p_{trans}^{i,i}$ as the mean since people tend to keep their moving tendencies whenever possible.",
"To consider the environment-obstacle interactions, for each path in the library, we calculate the distance from its start to the collision point with the static map as $Dist_{i}$ .",
"Then, we feed the distances into a softmax function to calculate the environment probability, which describes the probability of choosing a specific path when considering the environment-obstacle interactions: $P_{env} = \\textbf {\\emph {Softmax}}(Dist_{0}, Dist_{1}, ..., Dist_{l-1}),$ Finally, the state is predicted by : $P_{path}^{t+1} = P_{path}^{t}P_{trans}*P_{env}^{t+1},$ where $*$ represents the elementwise multiplication."
],
[
"Implementation details",
"To evaluate the performance of the proposed method, we conduct experiments in simulation and physical environments.",
"The simulation experiments are implemented in C++ with ROS/Gazebo running on AMD Ryzen 7 [email protected].",
"For physical experiments, the system runs on our customized quadcopter with an Intel Realsense D435i depth camera, which provides 640 $\\times $ 480 pixels depth images with 87° by 58° field of view.",
"The Nvidia Xavier NX is used for onboard computation, and the PX4-based flight controller controls the robot in flight tests.",
"We apply the visual-inertial odometry (VIO) [28] to estimate the states of the robot."
],
[
"Simulation and Physical Experiments",
"a) Simulation Experiments: To evaluate the performance of the proposed system, we prepare three simulation environments with dynamic obstacles.",
"An example of simulation environment experiments is shown in Fig.",
"REF .",
"The environment consists of static obstacles with 14 pedestrians as the dynamic obstacles.",
"In the bottom of Fig.",
"REF , we can see that the robot builds a static map for the environment, detects two dynamic obstacles marked as blue bounding boxes, and cleans their static map trails.",
"The obstacles' trajectories are then predicted, and the quadcopter plans a path to navigate safely in the environment, shown as the blue curve.",
"b) Physical Experiments: The physical experiments are conducted in five different environments shown in Fig.",
"REF and Fig.",
"REF .",
"We first test the proposed method, carrying the quadcopter by hand, in the four environments (Fig.",
"REF ).",
"One example of dynamic obstacle tracking and mapping results is shown in Fig.",
"REF .",
"We can see that the walking pedestrians are detected and represented by the blue bounding boxes with the green lines indicating the predicted trajectories, while the person in static is not identified as the dynamic obstacle.",
"Meanwhile, the static structures are captured by the static occupancy map with the dynamic obstacles' region cleaned.",
"In addition, the trajectory prediction module can produce collision-free predicted trajectories considering the static obstacles.",
"In Fig.",
"REF c, the person is walking toward the corner of the L-shape corridor, so a left-turn trajectory is predicted, which matches the actual moving direction.",
"An indoor autonomous flight experiment is also performed to demonstrate the system's capability to avoid dynamic obstacles shown in Fig.",
"REF .",
"We can see that the quadcopter is able to detect a person walking toward it and map the static environment.",
"With the trajectory planner mentioned in [29], our system can successfully navigate dynamic environments.",
"Figure: The physical test environments with pedestrians for evaluating our dynamic obstacle tracking and mapping system.Figure: Visualization of the experiment in a L-shape corridor.",
"The RGB images are shown in the upper figure, and the corresponding depth images with 2D obstacle detection results are shown in the middle.",
"The final maps with dynamic obstacles are visualized in the bottom figure."
],
[
"Benchmark Comparison",
"To quantitatively analyze the system performance, we compare our method with the state-of-the-art dynamic obstacle detectors in the UAV platform [6][19].",
"The comparison of dynamic obstacle detection errors in position and velocity is shown in Table REF .",
"In the physical experiment, the ground truth position is measured by the OptiTrack motion capture system.",
"From Table REF , we can see that our proposed method outperforms Method I in both velocity and position estimation accuracy.",
"From our experiment observations, the worse performance of Method I is because it does not identify static and dynamic obstacles, resulting in incorrect estimations for static obstacles.",
"The result also proves that our system has better velocity and comparable position estimation accuracy compared to Method II.",
"Our proposed velocity estimation method treats dynamic obstacles as a whole, thus reducing noises from point cloud of other parts of the object and giving an accurate estimation.",
"Table: The benchmark of the detected dynamic obstacles' position and velocity errors in simulation and physical experiments.The runtime evaluation of the proposed system on the Nvidia Xavier NX onboard computer is shown in Table REF .",
"Overall, the entire system takes less than 40ms in each iteration and can run over 25Hz.",
"The region proposal detection contributes the majority of computation time.",
"Besides, we also compare our method with the learning-based detection method, YOLO, on the onboard computer.",
"The running time of YOLO is 256.4ms when running with the visual-inertial odometry, which is significantly slower than our system and is not able to handle real-time navigation.",
"Table: The runtime of each module of the proposed system.To evaluate the performance of the proposed environment-aware trajectory prediction module, we collect all predicted trajectories in three simulation environments and count the number of failed predictions.",
"The number of the prediction paths in the library is set to 5.",
"The failure prediction denotes either the trajectory is not collision-free or the predictor cannot find a solution.",
"The comparison of the failure ratio between our proposed method and the linear trajectory prediction is shown in Fig.",
"REF .",
"Among the three environments, the Env.",
"A is the most complicated one, while the Env.",
"C is the simplest one.",
"It is shown that our prediction method still has a much lower failure ratio compared to the linear prediction in three environments.",
"Besides, when the environment becomes more complex, the linear prediction's failure ratio goes up dramatically, while our method can always have a low value.",
"For our methods, we can add the number of paths in the library to decrease the failure ratio.",
"Figure: Comparison of the failure ratio between our proposed method and the linear predictor in three different simulation environments."
],
[
"Conclusion and Future Work",
"This paper presents a novel dynamic obstacle tracking and mapping system for autonomous UAV navigation.",
"The proposed method uses a 3D hybrid map, utilizing the occupancy voxel map to represent static obstacles and track dynamic obstacles as bounding boxes.",
"The proposal detector module obtains the region proposals for obstacles from the depth image, which is refined by the proposed map refinement.",
"Then, the proposed identification and filtering methods are applied to track dynamic obstacles.",
"Besides, this work introduces a novel dynamic obstacle trajectory prediction algorithm based on the Markov chain, which considers the trajectory interaction with static environments.",
"The results show that our dynamic obstacle detection method has low estimation errors in position and velocity and can be used to navigate dynamic environments safely.",
"In the future, it is promising to explore camera models with a larger field of view to track dynamic obstacles."
],
[
"Acknowledgement",
"The authors would like to thank TOPRISE CO., LTD and Obayashi Corporation for their financial support in this work."
]
] | 2209.08258 |
[
[
"On R\\'enyi universality formula of charged flat black holes from\n Hawking-Page phase transition"
],
[
"Abstract Probing the thermodynamic properties of the Hawking-Page phase transition of asymptotically-flat black holes in R\\'enyi statistics.",
"We reveal the evidence for the persisting of the dual universal formula associated with the Hawking-Page (HP) and minimum black hole thermodynamical transition points.",
"Our study unveils the universal properties of the black holes in different statistics, in particular beyond the Gibbs-Boltzmann formalism, and motivates a further promising bridge between the nonextensivity R\\'enyi parameter {\\lambda} and the cosmological constant ?",
".",
"Besides, we discover that universal ratios related to horizon radius, R\\'enyi temperature, and entropy can be shown to appear in presence of such a transition."
],
[
"For a long time, numerous global expressions emerge in the formulation of fundamental theories.",
"Such invariant formulas, known as universal ones, are considered independent of any coordinate system.",
"The common type of such fundamental quantities has been of specific attention in the foundation of theories.",
"Their determinations are vital for the prediction of whether a theory is relevant or not, while their simple expressions as concise equations unveil profound significations and an ultimate straightforward verification.",
"Black holes are laboratories for testing fundamental theories that explain how the Universe works on the largest and the smallest scales (e.g., General relativity and Quantum physics) and a possible conciliation between them i.e quantum gravitational theory.",
"In fact, black hole thermodynamics plays a main role in our understanding of the mysterious nature of such compact objects [1], [2].",
"One of the most intriguing enigmas in black hole thermodynamics is the Hawking-Page ($HP$ ) phase transition between the stable large black hole and the thermal gas.",
"It was first investigated for the Schwarzschild black hole in Ref.",
"[2] and then for the charged black hole in Refs.",
"[3], [4], especially in the context of the AdS/CFT correspondence [5], [6], [7], [8], [9].",
"After the birth of the extended phase space notion in the AdS spacetime, this area was reexamined again with prolific outcomes [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].",
"Moreover, it's largely understood that a black hole in an asymptotically flat space has negative heat capacity and is thus thermodynamically unstable.",
"Therefore, when a black hole radiates via the Hawking mechanism [27], it can no longer maintain thermal equilibrium with the environment and will evaporate eventually.",
"In addition, the non-extensive nature of the Bekenstein-Hawking entropy of black holes which is proportional to the surface area of its event horizon rather than the volume, and in the strong gravitational scheme near the black hole, the condition of negligible long-range type interactions in the standard statistical descriptions breaks down, and consequently, the usual definition of mass and other extensive quantities is not possible locally pushes us to go beyond the standard Gibbs-Boltzmann ($GB$ ) statistical proposal.",
"Alfréd Rényi in 1959 introduced a well-defined entropy function, which also obeys both the equilibrium and the zeroth law compatibility exigency of thermodynamics [28].",
"$S_R=\\frac{1}{\\lambda }\\ln \\sum _ip_i^{1-\\lambda }.$ In which $\\lambda $ denotes a real constant parameter called the non-extensivity one.",
"It is obvious that in the limit $\\lambda \\rightarrow 0$ , the standard Boltzmann-Gibbs entropy, $S_{BG}=-\\sum p_i\\ln p_i$ is recovered.",
"Moreover, the Rényi entropy function follows the non-additive composition rule $S_{12}=S_1+S_2+\\lambda S_1S_2,$ where $S_1$ , $S_2$ , and $S_{12}$ are the entropies of the subsystems and the total system, respectively.",
"Recently, such definition of entropy shows exceptionally fascinating features with regards to the black holes thermodynamics and draws in a special accentuation in literature [29], [30], [31], [32], [33], [34], [35], [36].",
"Through the Ref.",
"[37], in the extended phase space of the AdS spacetime, one can show that the Hawking-Page temperature and the minimum black hole one share similar dependence on the pressure, $T\\sim \\sqrt{P}$ with different $d$ -dependent coefficients $T_{HP}=2\\sqrt{\\frac{d-2}{(d-1)\\pi }\\times P},\\quad T_{min}=2\\sqrt{\\frac{d-3}{(d-2)\\pi }\\times P}.$ Therefrom, it's easy to conclude that $ \\lim _{d\\rightarrow \\infty }T_{min}/T_{HP}=1$ , pointed that there will be no metastable large black hole phase when $d\\rightarrow \\infty $ .",
"Moreover, in Ref.",
"[37], the authors discovered a fascinating holographic-like relation between $T_{HP}(d)$ and $T_{min}(d+1)$ , namely $T_{HP}(d)=T_{min}(d+1).$ Such dual relation has been extended to thermodynamics in cavity [26].",
"So far, this dual formula has not been implemented in the non-Boltzmaniann statistics.",
"Therefore, in this paper, we will focus on the elaboration of a such holographic expression within the Rényi formalism for uncharged and charged black hole solution in flat spacetime."
],
[
"The outline of the paper is as follows: In Sect., we investigate the Hawking-Page phase transition for a high-dimensional asymptotically-flat black hole in Rényi statistics and give the essential of thermodynamical quantities.",
"In particular, we deal with the uncharged and charged solutions, and following the strategy of Ref.",
"[37], we elaborate a generalization of the dual formula associated with the anti-de-Sitter black hole in the Gibbs-Boltzmann statistics to an asymptotically-flat black hole in Rényi formalism.",
"Afterward, Sect.",
"is devoted to the conclusion and open questions.",
"Finally, in the appendix, we investigate the thermodynamics of high-dimension charged black holes in asymptotically flat spacetime via Rényi entropy."
],
[
"Uncharged black hole",
"The Hawking-Page phase transition found a deep interpretation as confinement/deconfinement phase transition in the AdS/CFT conjecture context [5], [6].",
"While it can also be expressed as a solid/liquid phase transition within the black chemistry [19], in which the cosmological constant (anti-de Sitter radius $\\ell $ ) can be identified with the thermodynamic pressure in $d$ -dimensional spacetime such as [38], [39] : $P=-\\frac{\\Lambda }{8\\pi }=\\frac{(d-1)(d-2)}{16\\pi \\ell ^2}.$ In the Rényi statistics, a new kind of extended phase space raises by associating now the pressure with the nonextensive parameter $\\lambda $ in four dimensions as [31], [34] : $P_R=\\frac{3\\lambda }{32},$ and it can be put under a more general form for arbitrary spacetime dimension $d$ by following the strategy of [31] as The full derivation of the Rényi pressure can be found in the appendix.. $P_R=\\displaystyle \\frac{ \\left(d - 1\\right)\\left(d - 3\\right) }{16 \\Omega _{d} } \\lambda r_{h}^{d - 4}.$ In order to further introduce new universal formula associated with this phase transition in the Rényi statistics approach, let us recall the behavior of the Gibbs free energy.",
"In the extended phase space within the non-extensive parameter $\\lambda $ , the mass is interpreted as enthalpy, thus the Gibbs free energy can be derived as $G_R&=& M-T_R S_R, \\\\ \\nonumber &=&\\frac{ r_{h}^{d - 3} (d-2)}{\\Omega _{d}}-\\frac{ r_{h}^{d - 5} \\left(d-3\\right)^2(d-1) \\left(1+ \\frac{64 \\pi P_R r_{h}^{2}}{(d-1)(d-2)(d-3) }\\right) }{64\\pi \\Omega _{d} P_R}\\ln {\\left(1+\\frac{ 64 \\pi P_R r_{h}^{2} }{(d-1)(d-2)(d-3)} \\right)}.$ Where, $\\Omega _d$ is related to the volume of a unit $(d-2)$ -sphere, $Vol(S^{d-2})$ by[40]: $\\Omega _{d}=\\frac{16\\pi }{(d-2)Vol(S^{d-2})}=\\frac{8 \\Gamma \\left(\\frac{d - 1}{2}\\right)}{(d - 2) \\pi ^{\\frac{d-3}{2}}}.",
"$ and the Rényi entropy and temperature as functions of the horizon radius $r_h$ stand for $S_R= \\frac{ (d-1)(d-3) \\ln {\\left(1+\\frac{ 64 \\pi P_R r_{h}^{2} }{(d-1)(d-2)(d-3)} \\right)}}{16 \\Omega _{d} P_R}r_{h}^{d - 4}$ and $T_R= \\frac{\\left(d - 3\\right) \\left(\\Omega _{d} \\left(d - 2\\right) + 4 \\pi \\lambda r_{h}^{d - 2}\\right)}{4 \\pi \\Omega _{d} r_{h} \\left(d - 2\\right)},$ respectively.",
"In terms of the Rényi pressure Eq.",
"(REF ), the Rényi temperature becomes $T_R=\\frac{\\left(d-3\\right)}{4 \\pi r_{h} }+\\frac{ 16 P_{R} r_{h}}{ \\left(d-2\\right)\\left(d-1\\right)}.$ The Gibbs free energy behavior in terms of the Rényi temperature is illustrated in the Fig.",
"REF Figure: The G R G_R-T R T_R diagram of a four-dimensional uncharged-flat black hole in Rényi statistics.",
"The arrows designate increasing black hole horizon radius, and T HP T_{HP} and T min T_{min} stand for the Hawking-Page phase transition Rényi temperature and minimum Rényi temperature.",
"The blue dashed curve is associated with the unstable small black hole branch, whereas the metastable and stable large black hole branches are depicted by the orange thin solid and red solid curves, respectively.Fig.REF clearly unveils several features: first, no black hole solution can exist below the minimal temperature there.",
"$T_{min} =\\sqrt{\\frac{16(d-3)}{\\pi (d-2)(d-1)}}\\sqrt{P_R},$ associated with the following minimal horizon radius $r_{min}=\\sqrt{\\frac{(d-3)(d-2)(d-1)}{64\\pi P_{R}}}.$ Second, there exist two branches above $T_{min}$ , the upper one is for small black holes which are thermodynamically unstable, while the lower branch corresponds to a thermodynamically stable large black holes.",
"The positive Gibbs free energy for $T < T_{HP}$ shows that the system is in a radiation-dominated phase.",
"Furthermore, The criterion of the Hawking-Page phase transition is that the Gibbs free energy of the black hole-thermal asymptotically flat system vanishes, $G_R=0.$ We can see from Eq.",
"(REF ) that this point occurs at: $T_{HP}=\\sqrt{\\frac{2(d-1)}{\\pi (d-2)}}\\sqrt{P_R}.$ The radius where the Hawking-Page takes place is directly linked to the minimal radius via the following ratio, $\\large \\gamma _r=\\frac{r_{min}}{r_{HP}}=\\sqrt{\\frac{d-3}{ 2}}.$ Such a ratio shows beautiful features, which are, that $\\gamma _r$ is only dimension dependent and revealing that both Hawking-Page and minimal radii present the same behaviors in pressure $P_R$ .",
"In addition, $\\gamma _r$ is an increasing function of dimension bounded from below by $\\frac{\\sqrt{2}}{2}$ at $d=4$ .",
"The black hole minimum temperature Eq.",
"(REF ) and Hawking-Page phase transition temperature Eq.",
"(REF ) can be put under a more appropriate form as functions of the nonextensivity parameter $\\lambda $ $T_{min}(d)&=&\\displaystyle \\frac{(d-2)}{4} \\left[\\pi ^{\\frac{d-3}{2}}\\Gamma \\left(\\frac{d-3}{2}\\right) \\frac{1}{\\lambda }\\right]^{\\frac{1}{2-d}},\\\\T_{HP}(d)&=&\\frac{ (d - 2)^2}{4(d - 1)} \\left[\\frac{\\pi ^{\\frac{d-3}{2}}}{2}\\Gamma \\left(\\frac{d-3}{2}\\right) \\frac{1}{\\lambda }\\right]^{\\frac{1}{2-d}}.$ Both temperatures depend on the parameter $\\lambda $ in the same manner, $T\\sim \\lambda ^\\frac{1}{d-2}$ and increase with $d$ .",
"In addition, We see from the expressions above, (REF ) and () that $\\lambda $ fixes in $d$ -dimensional spacetime a physical length scale, $l_{\\lambda }=1/\\lambda ^\\frac{1}{d-2}$ in units of Planck's length $l_p$ , where the phase transition from pure thermal radiation to black-holes begins to take place.",
"For $d=4$ the relevant length scale is $l_4=1/\\sqrt{\\lambda }=\\sqrt{\\frac{3}{32 P_R}}$ and it decreases towards a $\\lambda $ -independent value $l_{\\lambda }=l_p$ as the dimension increases.",
"In addition, looking at the dependence of $T_{min}$ and $T_{HP}$ on $\\lambda $ and because the non-extensive parameter is small $0<\\lambda <<1$ , one sees that as dimension $d$ increases it takes a smaller spacetime region filled with relatively hotter thermal radiation to form a black hole.",
"Next, we depict this dependence in Fig.REF , for $d=4$ Figure: T R T_R-P R P_R diagram of a four-dimensional uncharged-flat black hole in Rényi statistics.",
"The black solid and red dashed lines extend indefinitely and represent the minimum and the Hawking-Page transition temperatures respectively at different pressures.Observing Fig.REF , one clearly notice that three system phases-thermal radiation, a stable large black hole, and a metastable large black hole-are persisting.",
"The minimum temperature $T_{min}$ and the Hawking-Page transition temperature $T_{HP}$ are respectively illustrated by the solid and red dashed curves.",
"There is no terminal point in both coexistence lines, and the phase transitions can happen at all pressures, without a critical point [19].",
"Therefore, it is more like solid/supercooled-liquid/liquid phase structure.",
"The size of the region between the curves which represents metastable large black hole states, widens as the pressure increases.",
"A similar behaviour in solid/liquid (thermal radiation/stable LBH) phase transitions is due to a delayed transformation from one phase to another caused by the system occupying more metastable supercooled-liquid (metastable LBH) states.",
"For $d=4$ , we have, $T_{min}(4)&=&\\sqrt{\\frac{8}{3\\pi }\\times P_R},\\\\T_{HP}(4)&=&\\sqrt{\\frac{3}{\\pi }\\times P_R}.$ The same behavior is observed for any higher dimension $d>4$ .",
"$T_{min}(d)&=& \\sqrt{6}\\sqrt{\\frac{(d-3)}{(d-2)(d-1)}}\\;T_{min}(4),\\\\T_{HP}(d)&=&\\sqrt{\\frac{2}{3}}\\sqrt{\\frac{(d-1)}{(d-2)}}\\;T_{HP}(4).$ In addition, comparing these two quantities, we found $\\gamma _T=\\frac{T_{min}(d)}{T_{HP}(d)}=\\frac{4}{d-1}\\sqrt{\\frac{d-3}{ 2}}=\\displaystyle \\frac{2 \\gamma _r}{\\gamma _r^{2} + 1}$ While for the Rényi entropies we have $\\gamma _S = \\frac{ S_{min}(d)}{ S_{HP}(d)} = \\frac{ \\log {\\left(2 \\right)}}{\\log {\\left(\\frac{d - 1}{d - 3} \\right)}}\\left(\\frac{d-3}{2}\\right)^{\\frac{d-4}{2}}=\\displaystyle \\frac{\\gamma _r \\ln {\\left(2 \\right)}}{2\\ln (2)+\\ln {\\left[ \\gamma _r^{2}\\left( \\gamma _r^{2} + 1\\right) \\right]}}.$ Figure: Variation of temperatures and entropies ratios with spacetime dimension in Rényi and Gibbs-Boltzmann statistics (GB).",
"In Rényi formalism, the maximum for the temperatures' ratio occurs at d=5, while for the entropies' ratio the minimum is attained at d=4.",
"At large dimensions, γ S \\gamma _S grows indefinitely but γ T \\gamma _T vanishes at infinity.",
"In Gibbs-Boltzmann formalism, however, the ratios γ T \\gamma _T and γ S \\gamma _S converge to 1 and e -1 e^{-1} respectively, at large dimensions.",
"As seen from Fig.REF , In Rényi formalism, the ratios of the two temperatures (entropies) decreases (increases) with dimension $d$ , acquiring its highest (lowest) value for $d=5$ ($d=4$ ), as 1 ($ \\frac{\\log {\\left(2 \\right)}}{\\log {\\left(3 \\right)}}\\approx 0.6309$ ).",
"This implies that in lower dimensional spacetime the collapse of thermal radiation to form a black-hole is relatively the easiest compared with higher dimensional spacetime.",
"It is remarkable in this respect that in five dimensions once the possibility of a black hole phase exists, $T_R(5)=T_{min}(5)$ , a Hawking-Page phase transition readily happens, $T_{HP}(5)=T_{min}(5)$ and $S_{HP}(5)=S_{min}(5)$ .",
"Such a behavior is exhibited also in $GB$ formalism but only at infinite dimensions.",
"Furthermore, we have the limit as $ \\lim _{d\\rightarrow \\infty }T_{min}/T_{HP}=0$ ($ \\lim _{d\\rightarrow \\infty }S_{min}/S_{HP}=\\infty $ ) in contrast with the anti de-Sitter space in Ref.",
"[37], [26].",
"This dissimilarity stems from the fact that Rényi pressure $P_R$ depends on the horizon radius $r_h$ , Eq.",
"(REF ), While pressure in anti de-Sitter spacetime is independent of $r_h$ , Eq.",
"(REF ).",
"It should be noted that in Rényi statistics, $\\gamma _S$ becomes larger than unity for $d>5$ , which means that the $S_{min}$ is larger than $S_{HP}$ , This observation decouples the possibility of a Hawking-Page phase transition from the usual wisdom adopted within $GB$ statistics that the HP transition indicates that quantum theories of gravity ought to have a small number of states at lower energies, but a huge number of states at higher energies with a sharp transition at $T_{HP}$[17].",
"Since the black hole is not an isolated system, the only criterion for such a transition is minimizing the Gibbs free energy and not maximizing the entropy.",
"Thus it is not necessary to have $S_{min}<S_{HP}$ as predicted within $GB$ formalism.",
"As mentioned above, in Ref.",
"[37], the authors have established a dual relation Eq.",
"(REF ) in Gibbs-Boltzmann statistics associating the Hawking-Page temperature to minimal one of two successive dimensions.",
"Now, we will turn our attention to checking whether it has a similar phase holographic formula in Rényi statistics formalism.",
"Recalling Eqs.",
"(REF ) and (), a straightforward calculation reveals that $T_{min}\\left(d+1,\\xi _d P_R\\right)= T_{hp}(d,P_R).$ Where we have noted $\\xi _d$ as: $\\xi _d=\\displaystyle \\frac{d}{8}\\left(\\frac{d - 1}{ d - 2}\\right)^{2}.$ Therefore, we are in a position to conclude that the holographic-like relation in Eq.",
"(REF ) is a universal feature of the Hawking-Page phase transitions in different statistics and the same interpretation remains, like the duality between the ground and excited states of the black holes [37]."
],
[
"Charged black hole",
"Our holographic universal formula in Rényi statistics can be extended to the charged black hole solution.",
"An easy and similar computation to the previous section reveals that the Rényi temperature calculated in Eq.",
"(REF ) becomes in the grand canonical ensemble when the electric potential $\\phi =\\frac{(d-2)\\Omega _d}{4(d-3)}\\frac{Q}{r^{d-3}}$ remains fixed $T_R=\\frac{(d-3) }{4 \\pi r_{h}}- \\frac{ \\phi ^{2}(d-3)^2}{2 \\pi r_{h}(d-2)}+ \\frac{16 P_R r_{h}}{ (d-1)(d-2)}.$ Where the expression of the Rényi pressure is generalizes to The full derivation of the Rényi pressure can be found in the appendix.",
"as mentioned above.",
"$P_R=\\displaystyle \\left[\\frac{(d-1)(d-3) }{16 \\Omega _{d} }-\\frac{(d-1)(d-3)^2 \\phi ^{2}}{8 \\Omega _{d} \\left(d - 2\\right)}\\right]\\lambda r_{h}^{d - 4}.$ While the free energy is found to be, to first order in the Rényi pressure $P_R$ , $G_R=\\displaystyle \\frac{r_{h}^{d - 3} \\left[ (d-1)(d-2)-2(d-1)(d-3) \\phi ^{2} - 32 \\pi P_R r_{h}^{2} \\right]}{\\Omega _{d}(d-1)(d-2)^2}.$ Using the criterion of Eq.",
"(REF ), we get the expression of the Hawking-Page quantities, namely the horizon radius $r_{HP}=\\displaystyle \\frac{\\sqrt{2(d-1) [d-2- 2\\phi ^{2} \\left(d-3\\right)] }}{8 \\sqrt{\\pi } \\sqrt{P_R}},$ and its associated temperature $T_{HP}=\\displaystyle \\frac{\\sqrt{2(d-1)}\\sqrt{d-2-2\\phi ^2(d-3)} }{\\sqrt{ \\pi }(d-2) }\\sqrt{P_R}.$ From the above equations and like the uncharged case the dependence on $\\sqrt{P_R}$ is held in the charged black hole background.",
"Minimizing the Rényi temperature, we found the minimal horizon radius and its corresponding Rényi temperature as $r_{min}=\\displaystyle \\frac{\\sqrt{ \\left(d - 3\\right) \\left(d - 1\\right) \\left[ d - 2 - 2\\phi ^{2} \\left(d - 3\\right)\\right]}}{8 \\sqrt{\\pi } \\sqrt{P_R}},$ and $T_{min}=\\displaystyle \\frac{\\sqrt{16(d-3)}\\sqrt{d-2-2\\phi ^2(d-3)} }{\\sqrt{ \\pi }\\sqrt{d-1}(d-2) }\\sqrt{P_R}.$ In addition to the same dependence on the $\\sqrt{P_R}$ , the ratio $\\gamma _r=\\frac{r_{min}}{r_{HP}}$ seems to be insensible to black hole charge, for instance, it has been legitimate to consider it as a universal parameter.",
"In four-dimensional spacetime, the Rényi temperatures read as $T_{min}(4)=\\displaystyle \\frac{2 \\sqrt{6} \\sqrt{P_R} \\sqrt{1 - \\phi ^{2}}}{3 \\sqrt{\\pi }},$ and $T_{HP}(4)=\\displaystyle \\frac{\\sqrt{3} \\sqrt{P_R} \\sqrt{1 - \\phi ^{2}}}{\\sqrt{\\pi }}.$ By recalling, Eqs.",
"(REF ) and Eqs.",
"(REF ), the ratios of temperatures and entropies in the charged case are equal to the uncharged ones, namely, we obtain $\\gamma _T= \\frac{T_{min}(d)}{T_{HP}(d)}=\\displaystyle \\frac{2 \\gamma _r}{\\gamma _r^{2} + 1},$ $\\gamma _S= \\frac{ S_{min}(d)}{ S_{HP}(d)} =\\displaystyle \\frac{\\gamma _r \\ln {\\left(2 \\right)}}{2\\ln (2)+\\ln {\\left[ \\gamma _r^{2}\\left( \\gamma _r^{2} + 1\\right) \\right]}}.$ A simple scaling arrangement allows linking $d$ -dimensional temperatures to four-dimensional ones such as, $T_{min}(d, P_R,\\phi )=T_{min}(4,\\alpha _d P_R,\\beta _d\\phi ),$ $T_{HP}(d, P_R,\\phi )=T_{HP}(4, \\tilde{\\alpha }_d P_R,\\beta _d \\phi ).$ Where the scaling factors are found to be $\\alpha _d=\\sqrt{\\frac{6(d-3)}{(d-1)(d-2)}},\\quad \\tilde{\\alpha }_d=\\sqrt{\\frac{2(d-1)}{3(d-2)}}\\quad \\text{and}\\quad \\beta _d=\\sqrt{\\frac{2(d-3)}{d-2}}.$ such quantities depend only on the spacetime dimension, otherwise, they unveil a suspicion about a possible holographic-like equation due to the existence of $(d-1)$ , $(d-2)$ and $(d-3)$ terms.",
"A close inspection unveils that the calculation is quite complicated deserving appropriate simplifications $T_{min}(d+1,\\xi _d P_R,\\zeta _d \\phi )=T_{HP}(d,P_R,\\phi ).$ Herein, we have set $\\xi _d=\\displaystyle \\frac{d(d-1)^2}{8 (d-2)^2},\\quad \\text{ and }\\quad \\zeta _d=\\displaystyle \\frac{\\sqrt{\\left(d - 3\\right) \\left(d - 1\\right)}}{d - 2}.$ At this point, it is clear that we are in presence of a dual universal relation associated with the Hawking-Page phase transition.",
"Besides, we remark also those other universal ratios related to horizon radius, Rényi temperature and entropy can be shown to appear in presence of such a transition.",
"But it is mandatory to discuss some relevant difference aspects of the dual holographic formula between the Gibbs-Boltzmann and Rényi statistics.",
"In particular, the holographic relation Eq.",
"(REF ) is quite simpler than the Eq.",
"(REF ) and it is essentially due to the presence of some quantities associated with the black holes in the definition of the Rényi pressure in Eq.",
"(REF ), namely the event horizon radius and electric potential.",
"The appearance of such quantities in $P_R$ definition is because the existence of nonextensivity length $L_\\lambda $ , as demonstrated in appendix $B$ , and which suggests that there is a specific gravitational energy value beyond which the usual statistical mechanics is no longer valid.",
"In other words, the fact that when $r_h$ is greater than $L_\\lambda $ , the nonextensive effect appears to play a significant part in defining black hole thermodynamics.",
"Therefore all thermodynamical quantities of the black hole become sensitive to these effects which renders the holographic relations established within non-extensive Rényi statistics relatively more complicated than those formulated in extensive conventional statistics."
],
[
"Conclusion",
"Through this work, the dual relation between the black hole minimum temperature and Hawking-Page phase transition temperature in two successive dimensions was extended to Rényi statistics.",
"Concretely, we reveal that both a $d$ -dimensional Schwarzschild-flat and Reissner-Nordstrom-flat black holes exhibit a universal behavior at the Hawking-Page transition point.",
"For an arbitrary dimension, the systems are characterized by two special temperatures: the Hawking-Page phase transition temperature and the black hole minimum one, which are pressure-dependent and are equal for two successive dimensions.",
"Such equality is reminiscent of the AdS/CFT correspondence.",
"Rigorously speaking, if $T_{min}$ is the temperature of a physical quantity in the bulk, thus $T_{HP}$ can be assimilated to the temperature of the dual physical quantity on the boundary.",
"It is worth noting that the ratios $\\gamma _r$ and $\\gamma _T$ can be considered as universal quantities predicted by the charged and uncharged flat black holes.",
"We note also that an attempt to reveal such universality in Kerr-flat black hole shows an absence of such a property, which suggests that the perfect spherical symmetry of the Schwarzschild-flat and Reissner-Nordstrom-flat black holes may be an important requirement of the observed universality.",
"Furthermore, from the present work, this lack of universality for axisymmetric black holes is independent of the statistical formalism adopted[17]."
],
[
"High-dimensional flat black hole thermodynamics in Rényi formalism",
"In the present section, we derive the expression of the Rényi pressure $P_R$ in $d$ -dimensional spacetime through the generalization of the first law of thermodynamics and the Smarr formula of asymptotically flat charged black hole in Rényi formalism.",
"The starting trivial point is the line element of the $d$ -dimensional Reissner-Nordstrom black-hole of mass $M$ and electric charge $Q$ , in asymptotically flat spacetime is given by[40] $ds^2=-f(r)dt^2+\\frac{dr^2}{f(r)}+r^2d\\omega _{d-2}.$ Where, the blackening function stands for $f(r)=1-\\frac{\\Omega _d M}{r^{d-3}} + \\frac{\\Omega _{d}^{2} \\left(d - 2\\right)Q^{2} }{8 \\left(d - 3\\right)r^{2 d-6}},$ and, $\\Omega _{d}=\\frac{16\\pi }{(d-2)Vol(S^{d-2})}=\\frac{8 \\Gamma \\left(\\frac{d - 1}{2}\\right)}{(d - 2) \\pi ^{\\frac{d-3}{2}}}.$ In the $f(r)$ function, $Vol(S^{d-2})$ and $d\\omega _{d-2}$ denote the volume and the line element of the unit $(d-2)$ -sphere respectively.",
"The mass $M$ and the electric charge $Q$ , are related to the electric potential $\\phi $ at the black hole outer horizon of radius $r_h$ , as measured by an observer at infinity by, $M &=\\displaystyle \\frac{r_h^{d-3}}{ \\Omega _{d}}+ \\frac{2 (d - 3)\\phi ^{2}r_h^{d-3} }{\\left(d - 2\\right)\\Omega _{d}},\\\\Q &=\\displaystyle \\frac{4 \\phi r_{h}^{d - 3} \\left(d - 3\\right)}{\\Omega _{d} \\left(d - 2\\right)}.$ It's obvious that thermodynamical quantities at the outer horizon verify the first law of black hole thermodynamics $dM=T_HdS_H+\\phi dQ,$ And the Smarr formula $(d-3)M=(d-2)T_H S_H+\\phi Q.$ Here, $T_H$ and $S_H$ are the Hawking temperature and the Hawking-Bekenstein entropy respectively, and are given by $T_H=\\frac{f^{^{\\prime }}(r)}{4\\pi },\\quad S_H=\\frac{r_h^{d-2}Vol(S^{d-2})}{4}.$ In Rényi formalism, the Hawking-Bekenstein $S_H$ is considered as the Tsallis entropy and Rényi entropy $S_R$ is defined as its formal logarithm such as [31] $S_R=\\frac{1}{\\lambda }\\ln (1+\\lambda S_H),$ with $\\lambda $ is nothing but the non-extensivity parameter.",
"Then, Rényi temperature $T_R$ is defined through the standard thermodynamic relation, as $\\frac{1}{T_R}=\\frac{\\partial S_R}{\\partial M}=\\frac{1}{T_H(1+\\lambda S_H)}.$ Now, we can readily rewrite the first law of thermodynamics Eq.",
"(REF ), and the Smarr formula Eq.",
"(REF ), in Rényi formalism by applying the following substitutions, $T_H=T_R\\exp (-\\lambda S_R),\\quad S_H=\\frac{\\exp (\\lambda S_R)-1}{\\lambda }.$ Where $\\lambda $ is taken as an infinitesimal of order one.",
"We obtain keeping the leading order terms in $\\lambda $ , $dM &= T_RdS_R+VdP_R+\\phi dQ,\\\\(d-3)M &= (d-2)T_R S_R-(d-2)P_RV+\\phi Q.$ In the second line, we have identified the black hole thermodynamical volume $V$ and the Rényi pressure $P_R$ with $V &=\\displaystyle \\frac{16 \\pi r_{h}^{d - 1}}{\\Omega _{d} \\left(d - 2\\right) \\left(d - 1\\right)},\\\\P_R &=\\displaystyle \\lambda r_{h}^{d - 4}\\frac{(d-1)(d-3) }{16 \\Omega _{d} }\\left[\\displaystyle 1-\\frac{2(d-3) \\phi ^{2}}{(d - 2)}\\right].$ Eq.",
"(REF ) permits one to express all thermodynamical quantities where $\\lambda $ appears in term of the more relevant quantity, in fact, Rényi pressure $P_R$ , Which leads to the following formulas used in the main text, $T_R &=\\displaystyle \\frac{(d-3) }{4 \\pi r_{h}}- \\frac{ \\phi ^{2}(d-3)^2}{2 \\pi r_{h}(d-2)}+ \\frac{16 P_R r_{h}}{ (d-1)(d-2)}, \\\\S_R &=\\displaystyle \\frac{r_h^{d-4}(d-1)(d-3) \\left[d-2-2 (d-3) \\phi ^2\\right] \\ln \\left(1+\\frac{64 \\pi P_R r_h^2}{(d-3) (d-1) \\left(d-2-2 (d-3) \\phi ^2\\right)}\\right)}{16 P_R(d-2) \\Omega _d}.\\\\G_R &=\\displaystyle \\frac{r_{h}^{d - 3} \\left[ (d-1)(d-2)-2(d-1)(d-3) \\phi ^{2} - 32 \\pi P_R r_{h}^{2} \\right]}{\\Omega _{d}(d-1)(d-2)^2}.$"
],
[
"Nonextensivity scale and black hole stability requirement\n",
" Over here, we investigate the effect brought up by the non-extensivity on the stability of the black hole.",
"To this end, we consider the microcanonical ensemble of charged black holes together with thermal radiation in asymptotically flat spacetime.",
"The constant total energy density $E$ of the composite system is given as, $E&=&E_{BH}+E_{rad}=const,\\\\0&=& dE_{BH}+dE_{rad},$ where $E_{BH}=M-\\phi Q$ and $E_{rad}=\\sigma T_{rad}^4$ represent the energy density of black holes and thermal radiation, respectively.",
"At thermal equilibrium, the total entropy density of the system is given by, $S=S_{BH}+S_{rad}$ reaches its maximum value.",
"Consequently, the requirements of thermal equilibrium between the black hole and its thermal environment are $\\frac{\\partial S}{\\partial E_{BH}}=0,\\quad \\text{ and } \\quad \\frac{\\partial ^2 S}{\\partial E_{BH}^2}<0.$ These two conditions serve as the stability criteria of the black hole since once the thermal equilibrium is established, the black hole can neither grow in size by absorbing energy from its surroundings nor shrink by giving energy to it.",
"The first condition translates to $\\nonumber dS=0 &=& dS_{BH}+dS_{rad}\\\\0 &=& \\frac{dE_{BH}}{T_{BH}}+\\frac{dE_{rad}}{T_{rad}}\\\\ \\nonumber 0 &=&\\frac{dE_{BH}}{T_{BH}}-\\frac{dE_{BH}}{T_{rad}}$ Where Eq.",
"(REF ) is used.",
"Thus we obtain the equality of the temperatures, $T_{BH}=T_{rad}$ at thermal equilibrium as expected.",
"The second condition for stability gives, $\\nonumber \\frac{\\partial ^2 S}{\\partial E_{BH}^2}&=&\\frac{\\partial ^2 S_{BH}}{\\partial E_{BH}^2}+\\frac{\\partial ^2 S_{rad}}{\\partial E_{BH}^2}<0,\\\\ \\nonumber &=&\\frac{\\partial ^2 S_{BH}}{\\partial E_{BH}^2}+\\frac{\\partial ^2 S_{rad}}{\\partial E_{rad}^2}<0 \\qquad \\text{ since }\\quad (dE_{rad}^2=dE_{BH}^2)\\\\&=&-\\frac{1}{T_{BH}^2}\\frac{\\partial ^2 T_{BH}}{\\partial E_{BH}^2}-\\frac{1}{T_{rad}^2}\\frac{\\partial ^2 T_{rad}}{\\partial E_{rad}^2}<0,\\\\ \\nonumber &=&-\\frac{1}{T_{BH}^2}(\\frac{1}{C_{BH}}+\\frac{1}{C_{rad}})<0.$ We have therefore a condition on the heat capacities of the system $\\left(\\frac{1}{C_{BH}}+\\frac{1}{C_{rad}}\\right)>0$ Considering the case of an infinite heat bath in thermal equilibrium with the charged black hole, the heat bath thus has an infinite number of degrees of freedom and its heat capacity becomes infinite, $C_{rad}\\longrightarrow \\infty $ .",
"Using Eqs.",
"(REF ), (), (REF ) and substituting (REF ) and (REF ), the inequality (REF ) becomes $\\displaystyle \\frac{16 \\pi ^{\\frac{5}{2} - \\frac{d}{2}} r_{h}^{8- d} \\left(d - 2\\right) \\Gamma ^{2}\\left(\\frac{d-3}{2} \\right)\\mathbf {\\left[\\left( d -3 \\right) \\pi ^{\\frac{d}{2} + \\frac{1}{2}}\\lambda r_{h}^{d-2} - 2 \\pi \\Gamma \\left(\\frac{d-1}{2} \\right)\\right]} }{\\left(\\pi ^{\\frac{d}{2}} \\lambda r_{h}^{d} + 2 \\sqrt{\\pi } r_{h}^{2} \\Gamma \\left(\\frac{d-1}{2} \\right)\\right)^{2} \\left(2 d \\phi ^{2} - d - 6 \\phi ^{2} + 2\\right)^{2}}>0.$ Such constraint can be put under more elegant form as $r_h>L_{\\lambda }.$ Where the nonextensivity length scale $L_{\\lambda }$ is defined by, $L_{\\lambda }=\\pi ^{\\frac{1 - d}{2 \\left(d - 2\\right)}} \\left[\\frac{2}{\\lambda (d-3)} \\Gamma \\left(\\frac{d-1}{2} \\right)\\right]^{\\frac{1}{d - 2}}$ This implies that the limit of stability of a charged black hole in asymptotically flat spacetime is fixed by the characteristic length $L_{\\lambda }$ which depends only on the non-extensivity parameter $\\lambda $ .",
"In energy terms, there is a particular value of gravitational energy beyond which the conventional extensive statistical mechanics is no longer valid.",
"This is because the nonextensive effects play an essential role in describing black hole thermodynamics when $r_h$ is larger than $L_{\\lambda }$ ."
]
] | 2209.08195 |
[
[
"Compose & Embellish: Well-Structured Piano Performance Generation via A\n Two-Stage Approach"
],
[
"Abstract Even with strong sequence models like Transformers, generating expressive piano performances with long-range musical structures remains challenging.",
"Meanwhile, methods to compose well-structured melodies or lead sheets (melody + chords), i.e., simpler forms of music, gained more success.",
"Observing the above, we devise a two-stage Transformer-based framework that Composes a lead sheet first, and then Embellishes it with accompaniment and expressive touches.",
"Such a factorization also enables pretraining on non-piano data.",
"Our objective and subjective experiments show that Compose & Embellish shrinks the gap in structureness between a current state of the art and real performances by half, and improves other musical aspects such as richness and coherence as well."
],
[
"Introduction",
"Recent years have witnessed a multitude of research works on leveraging Transformers [1] for symbolic music generation.",
"Generating piano performances emerged as a quintessential arena for such studies, for the rich musical content and texture piano playing can entail without having to deal with the complicated orchestration of instruments.",
"Thanks to Transformers' outstanding capability of modeling long sequences containing complex inter-token relations, generating several minutes-long expressive piano music end-to-end has been made possible [2], [3], [4].",
"Though these works all claimed to have improved upon their predecessors in creating repetitive structures, a central element of music, it has been repeatedly shown that they fail to come up with overarching repetitions and musical development that hold a piece together [5], [6], [7].",
"On the other hand, a line of research that tackles simpler forms of music, e.g., melodies or lead sheets (melody $+$ chords) has seen promising results in composing well-structured pieces [8], [9], [10].",
"A reasonable conjecture then follows: Could it be too demanding for a monolithic model to generate virtuosic performances end-to-end, as it has to process local nuances in texture or emotions, and the high-level musical flow, all at once?",
"Therefore, in this paper, we split piano performance generation into two stages, and propose the Compose & Embellish framework that rests on performant prior works [4], [5].",
"The Compose step writes the lead sheet that sets the overall structure of a song, while the Embellish step conditions on the lead sheet, and adds expressivity to it through accompaniment, dynamics, and timing.",
"Through experiments, we strive to answer the following research questions: [noitemsep, topsep=0pt, leftmargin=*] RQ #1: Can the two-stage framework compose better-structured piano performances than an end-to-end model, without adversely impacting diversity of musical content?",
"RQ #2: Does being able to pretrain the Compose step with larger amounts of non-piano data bring performance gains?",
"RQ #3: How well does the Embellish step follow the structure of music generated by the Compose step?",
"Fig.",
"REF offers a glimpse of our improvement over a state of the art [4].",
"We plan to open source our implementation and trained model weights upon paper publication.",
"Readers are encouraged to listen to samples generated by our framework.Generated samples: https://bit.ly/comp_embel.",
"Figure: Fitness scape plots of pieces randomly drawn from generations by Compose & Embellish, by CP Transformer , and from real data.",
"Darker colors towards top of the triangle indicate more significant long-range repetitive structures.Figure: System overview of Compose & Embellish."
],
[
"Related Work",
"For expressive piano performances, [2] and [3] showed respectively that relative positional encoding and beat-based music representation enhance generation quality.",
"[4] designed a more compact representation and utilized memory-efficient attention to fit entire performances into a Transformer.",
"[6] directly addressed musical structure with a multi-granular Transformer, but weakened expressivity by abstracting timing and dynamics away.",
"Bar-level blueprints [12] and musical themes [13] may help to maintain long-range structure, but neither of these systems are capable of unconditioned generation.",
"On composing melodies or lead sheets, researchers used note-level repeat detection and modeling [8], phrase tokens [5], hierarchical generative pipeline [9], and bar-level similarity relations [10] to induce repetitive structures.",
"Out of these, [5] is the most straightforward one that avoids potential error propagation between multiple components, and is hence adopted by our framework."
],
[
"Input Sequences",
"We represent a polyphonic musical piece or performance with a sequence of tokens $X$ .",
"To encode the expressiveness of a performance, besides chord progression and onset time/pitch/duration of notes, $X$ also contains the velocity (i.e., loudness) of each note, as well as beat-level tempo changes.",
"Any existing method that extracts the monophonic melody line (e.g., the skyline algorithm [14]) may then be applied to $X$ .",
"Using the chord progression in $X$ and the extracted melody, we additionally leverage a structure analysis algorithm [15] based on edit similarity and A$^{*}$ search to capture repetitive phrases (in a form like A$_1$ B$_1$ A$_2$ ...) of the piece.",
"The melody, chords, and structure information would constitute a lead sheet of the piece, denoted by $M$ .",
"Typically, $M$ is much shorter for having less notes, and the expressive aspects being discarded.",
"For both $X$ and $M$ , there exists a mapping $\\mathrm {bar}(t)$ that gives the index of the bar the $t^{\\text{th}}$ token belongs to.",
"With the mappings, we may segment $X$ and $M$ into $\\lbrace X^{(1)},\\dots ,X^{(B)}\\rbrace $ and $\\lbrace M^{(1)},\\dots ,M^{(B)}\\rbrace $ , where $B$ is the piece's number of bars.",
"The segmented sequences will be used in our model."
],
[
"Data Representation",
"Our token vocabulary is designed based on Revamped MIDI-derived Events (REMI) [3].",
"In a full performance $X$ , a [Bar] token appears whenever a new musical bar begins.",
"[SubBeat_*] token indicate timepoints within a bar, in 16th note () resolution.",
"Each note is represented by three tokens: [Pitch_*] (A0 to C8), [Duration_*] ( to ), and [Velocity_*] (32 levels).",
"Moreover, [Tempo_*] (32$\\sim $ 224 bpm) tokens set the pace, and [Chord_*] tokens (12 roots $\\times $ 11 qualities) provide harmonic context.",
"The two above may appear as frequently as every beat.",
"For lead sheets $M$ , we take the mean tempo, and place only one [Tempo_*] token at the very beginning.",
"We also omit [Velocity_*] of each note.",
"To add structure information to $M$ , we refer to [5]: At a phrase's starting bar, we put [Phrase_*] (8 possible letters) and [RepStart_*] (1st to 16th repetition) right after [Bar].",
"At the phrase's ending, [Phrase_*] and [RepEnd_*] close that bar.",
"An [EOS] token ends the entire lead sheet.",
"Other tokens types are the same as in $X$ .",
"By our construction, the vocabulary size for both $X$ and $M$ is about 370."
],
[
"Models and Objectives",
"Fig.",
"REF is a birds-eye view of our Compose & Embellish framework.",
"It is made up of two generative models: the lead sheet model (Compose) $p(M)$ , and the performance model (Embellish) $p(X | M)$ .",
"While being trained independently, the two models work in tandem during inference.",
"For the lead sheet model, we simply factorize $p(M)$ into $\\sum _t p(m_t \\, | \\, M_{<t})$ .",
"It can complete a lead sheet autoregressively given a start token, i.e., [Tempo_*].",
"For $p(X | M)$ , we follow the conditioned generation case in CP Transformer [4], and interleave one-bar segments from $M$ and $X$ as $\\lbrace M^{(1)}, X^{(1)}, M^{(2)}, X^{(2)},\\dots \\rbrace $ .",
"This way, when generating the performance for a bar, the completed lead sheet of that bar is always the closest piece of context the model may refer to, thereby encouraging it to stay faithful to $M$ .",
"Mathematically, $p(X | M)$ can be factorized as $\\sum _t p(x_t \\, | \\, X_{<t}; \\, M^{(\\le \\mathrm {bar}(t))})$ .",
"For the model to distinguish interleaved segments, we place [Track_$M$ ] and [Track_$X$ ] in front of each $M^{(\\cdot )}$ and $X^{(\\cdot )}$ respectively.",
"At inference time, we would move on to the next bar whenever [Track_$M$ ] is generated.",
"Both models minimize the negative log-likelihood ($- \\log p(\\cdot )$ ) of the sequences.",
"One can use any type of sequence decoder for both models.",
"Due to the long sequence length (mostly $>$ 1k) of our data, our choice is Transformers with a causal attention mask.",
"Since the models are trained separately, we may pretrain the lead sheet model on a larger amount of data ($\\mathcal {D}_\\mathrm {p}$ in Fig.",
"REF ) extracted from, e.g., various multitrack pieces.",
"These pieces, though not played only by the piano, or at all, still likely feature a well-structured melody that is either sung or played by another instrument.",
"Then, we just need to finetune it on the piano performance dataset that $p(X | M)$ is trained on ($\\mathcal {D}_\\mathrm {f}$ ) to align their domain."
],
[
"Datasets and Preprocessing",
"We adopt the full Lakh MIDI Dataset (LMD-full) [16] as the pretraining dataset ($\\mathcal {D}_\\mathrm {p}$ ) for our lead sheet model.",
"The LMD-full contains over 100k multitrack MIDIs with various instrument combinations in each of them.",
"The dataset for finetuning our leadsheet model and training our performance model ($\\mathcal {D}_\\mathrm {f}$ ) is Pop1K7 compiled in [4].",
"It features about 1,700 transcribed piano performances of Western, Japanese, and Korean pop songs.",
"We use different algorithms to extract melodies from $\\mathcal {D}_\\mathrm {p}$ and $\\mathcal {D}_\\mathrm {f}$ .",
"For $\\mathcal {D}_\\mathrm {p}$ , we leverage the open-source codehttps://github.com/gulnazaki/lyrics-melody from [17], which searches for the instrument track whose note onset times align best with those of the song's lyrics, and regards that track as the melody.Songs without lyrics annotations are not considered.",
"For $\\mathcal {D}_\\mathrm {f}$ , we employ the skyline algorithm [14], which keeps only the highest-pitched note from each set of simultaneous onsets.",
"This simple heuristic has been shown to have a nearly 80% accuracy on identifying pop songs' melodies [18].",
"We then perform structure analysis [15] on the extracted melodies and discard songs with phrases spanning $>$ 32 bars.",
"The statistics of the processed datasets are displayed in Table REF .",
"CP representation [4] for $\\mathcal {D}_\\mathrm {f}$ is made for baselining purpose.",
"10% of each dataset is reserved for validation.",
"Table: Summary of datasets used in our experiments.",
"The numbers in the last three columns are averages across a dataset.Table: Objective evaluation results.",
"(All metrics are the closer to real data, the better.",
"Stdevs follow ±\\pm .)"
],
[
"Model Implementation Details",
"Our lead sheet model $p(M)$ is parameterized by a 12-layer Transformer [1] (512 hidden state dim., 8 attention heads, 41 million trainable parameters) with relative positional encoding proposed in [19].",
"With a batch size of 2, we can set maximum sequence length to 2,400 (longer than 98% & 90% of songs in $\\mathcal {D}_\\mathrm {p}$ & $\\mathcal {D}_\\mathrm {f}$ , respectively) on an RTX3090-Ti GPU with 24G memory.",
"We use Adam optimizer with 200 steps of learning rate warmup to a peak of 1e$-$ 4, followed by 500k steps of cosine decay.",
"We keep a checkpoint after each epoch, finding that checkpoints with around 0.4 training NLL produces outputs of the best perceived quality.",
"Pretraining on $\\mathcal {D}_\\mathrm {p}$ requires over 5 days, while finetuning on $\\mathcal {D}_\\mathrm {f}$ takes less than half a day.",
"The performance model $p(X|M)$ is a 12-layer linear Transformer with Performer [20] attention approximation.",
"In each epoch, a random 3,072 token-long crop of interleaved $M$ and $X$ segments (see Sec.",
"REF for explanation) of each song is fed to the model in a batch size of 4.",
"This sequence length corresponds to roughly 40 bars of performance.",
"While its possible to feed full performances using batch size$=$ 1, we observe that the increased attention overhead and reduced batch size would render the training slow and unstable.",
"Training the performance model takes around 3 days.Model hyperparameters, hardware, optimizer settings, and checkpoint selection are the same as those for the lead sheet model.",
"Nucleus sampling [21] with temperatured softmax is employed during inference.",
"We discover that the temperature $\\tau $ and probability mass truncation point $p$ greatly affect the intra-sequence repetitiveness and diversity of generated lead sheets.",
"Thus, we follow [22] and search within $\\tau =\\lbrace 1.2, 1.3, 1.4\\rbrace $ and $p=\\lbrace .95, .97, .98, .99\\rbrace $ to find a combination with which our lead sheet model generates outputs with the closest mean perplexity (measured by the model itself) to that of validation real data.",
"Finally, $\\tau = 1.2$ and $p=.97$ are chosen.",
"The effects of these two hyperparameters on the performance model are, however, more subtle.",
"We pick $\\tau =1.1$ and $p=.99$ for they lead to the most pleasant sounding performances to our ears."
],
[
"Baselines and Ablations",
"We select Compound Word (CP) Transformer [4] as our baseline, for it represents the state of the art end-to-end model for unconditional expressive piano performance generation that has access to the full context of previously generated tokens, due to its low memory footprint.",
"However, this advantage does not lead to well-structured generations, as pointed out by [10].",
"For our Compose & Embellish framework, to examine whether (1) pretraining on $\\mathcal {D}_\\mathrm {p}$ , and (2) adding structure (phrase) tokens contribute to its success, we put to test three ablated versions without either, or both, of them.Sampling hyperparameters ($\\tau $ & $p$ ) for the ablated models are chosen in the same way as described in Sec.",
"REF .",
"Additionally, we sample single-bar, and single-phrase, excerpts from the real data and repeat such excerpts to the song's length (in bars), to see how naive repetitions would compare to the models.",
"Table: Difference in structureness indicator scores between lead sheets and full performances.Table: User study results.",
"(Ch: Coherence, Cr: Correctness, S: Structureness, R: Richness, O: Overall.",
"Stdevs follow ±\\pm .)"
],
[
"Objective Evaluation",
"We utilize a set of metrics that can be computed on a song's notes, or its synthesized audio, to evaluate the intra-song structureness, diversity, and general quality of the generated music.",
"[itemsep=0pt, topsep=0pt, leftmargin=*] Structureness Indicators ($\\mathcal {SI}$ ): proposed first by [5], this metric takes the maximum value from a specific timescale range (e.g., all 10$\\sim $ 20 seconds long segments) of the fitness scape plot [11] (see Fig.",
"REF for examples) computed on the audio of a song.",
"This value represents the extent to which the most salient repetitive segment in that timescale range is repeated throughout the entire song.",
"We set the timescale ranges to 4$\\sim $12, 12$\\sim $32, and over 32 seconds to capture the short-, medium-, and long-term structureness (denoted as $\\mathcal {SI}_{\\text{short}}$ , $\\mathcal {SI}_{\\text{mid}}$ , and $\\mathcal {SI}_{\\text{long}}$ ) respectively.",
"Percentage of Distinct Pitch N-grams in Melody ($\\mathcal {DN}$ ): following the popular dist-n [23] metric used in natural language generation to evaluate the diversity of generated content, we compute the percentage of distinct n-grams in the pitch sequence of the skyline extracted from each full performance.not directly on melody lines composed by our lead sheet model.",
"We regard 3$\\sim $ 5, 6$\\sim $ 10, and 11$\\sim $ 20 contiguous notes as short, medium, and long excerpts, and compute $\\mathcal {DN}_{\\text{short}}$ , $\\mathcal {DN}_{\\text{mid}}$ , and $\\mathcal {DN}_{\\text{long}}$ accordingly.",
"Pitch Class Histogram Entropy ($\\mathcal {H}_1$ , $\\mathcal {H}_4$ ): proposed in [5] to see if a model uses primarily a few pitch classes (i.e., C, C#,..., B, B, 12 in total), or more, and more evenly, of them, hence leading to a higher harmonic diversity.",
"The subscripts denote whether histograms are accumulated over 1-, or 4-bar segments.",
"Grooving Similarity ($\\mathcal {GS}$ ): used also in [5].",
"It calculates the pairwise similarity of each bar's groove vector $\\mathbf {g}$ (binary, indicating which sub-beats have note onsets) as $1 - \\mathrm {HammDist}(\\mathbf {g}_a, \\mathbf {g}_b)$ .",
"All bar pairs $(a,b)$ are involved, not just adjacent ones.",
"Higher $\\mathcal {GS}$ suggests more consistent rhythm patterns across the song.",
"We generate 100 songs with each model to compute the metrics.",
"Due to high computation cost of fitness scape plots, we only sample 200 songs from real data ($\\mathcal {D}_\\mathrm {f}$ ) for comparison."
],
[
"User Study",
"We recruit 15 subjects who are willing to spend around half an hour to take part in our listening test.",
"Each test taker is given three independent sets of music.",
"There are three piano performances (full song, about 3$\\sim $ 5 minutes long each) in each set, composed respectively by (1) a human composer, (2) Compose & Embellish, and (3) CP Transformer.",
"To facilitate comparison, the three performances in a set share the same 8-bar prompt drawn from our validation split.",
"Test takers are asked to rate each performance on the 5-point Likert scale, on the following aspects: [itemsep=0pt, topsep=1pt, leftmargin=*] Coherence (Ch): Does the music follow the prompt well, and unfold smoothly throughout the piece?",
"Correctness (Cr): Is the music free of inharmonious notes, and unnatural rhythms and phrasing?",
"Structureness (S): Are recurring motifs / phrases / sections, and reasonable musical development present?",
"Richness (R): Is the music intriguing and full of variations within?",
"Overall (O): Subjectively, how much do you like the music?"
],
[
"Results and Discussion",
"We run the metrics described in Sec.",
"REF on all our model variants and baselines.",
"The results are shown in Table REF .",
"First and foremost, we compare CP Transformer (abbr.",
"as CPT henceforth) to the full Compose & Embellish (C&E).",
"On $\\mathcal {SI}$ metrics, C&E sits right in the middle of CPT and real data, with its advantage over CPT increasing as the timescale goes up.",
"Although CPT scores high on $\\mathcal {DN}$ 's, we should keep in mind that it is not trained to take extra care of the melody, and hence likely does not know some melodic content should be repeated.",
"On the other hand, $\\mathcal {DN}$ scores of C&E are close to those of real data, and a lot better than the two excessively repetitive baselines.",
"An affirmative answer to our RQ #1 may be given: C&E composes better-structured piano performances without sacrificing musical diversity within a piece.",
"Worthwhile to note is that, despite the high $\\mathcal {DN}$ 's, CPT gets considerably lower $\\mathcal {H}_1, \\mathcal {H}_4$ , and slightly higher $\\mathcal {GS}$ (vs. C&E and real data), suggesting that its music may actually sound blander harmonically and rhythmically.",
"Next, we pay attention to the full vs. ablated versions of Compose & Embellish.",
"From Table REF , we may observe that variants not pretrained on $\\mathcal {D}_\\mathrm {p}$ suffer losses on $\\mathcal {SI}_{\\text{mid}}$ and $\\mathcal {SI}_{\\text{long}}$ .",
"Somewhat to our surprise is that w/o pretrain performs worse than w/o struct & pretrain.",
"A possible explanation is that the introduction of structure-related tokens renders a larger amount training data necessary, as the concept of long-range repetition those tokens carry is less explicit than direct interactions between notes.",
"Whether this reasoning holds, however, warrants further study.",
"Despite worse longer-range $\\mathcal {SI}$ scores, we discover that the two w/o pretraining variants are, contrarily, more prone to over-repetition (which we arbitrarily define as: one bar of melody being exactly and consecutively repeated over 6 times, or two neighboring bars of melody repeated over 4 times)—5.5% of these two variants' generations suffer from it, while only 2.5% of pieces by pretrained variants, and 0.5% of real data do.",
"We may now answer another yes to our RQ #2: pretraining the lead sheet model helps with not only the structureness, but also the quality consistency, of the generated music.",
"Table REF displays how much longer-range structureness slip after we feed the lead sheets to our performance model.",
"The performance model is not able to match the structureness of real performances, no matter whose lead sheets it conditions on, while our best lead sheet model already performs similarly with the real data.",
"(The pretrained C&E variants' lead sheet $\\mathcal {SI}$ 's reaffirm our answer to RQ #2.)",
"Lead sheets by the w/o struct variant cause particularly large $\\Delta $ 's ($p < .05$ compared to the full version).",
"To get a better sense our performance model's issues, we check the melody matchness [4] it achieves (i.e., the percentage of notes in a melody that a performance model copies and pastes into the performance; ideally 100%)—it gets $>$ 98% on lead sheets by all four variants.",
"Hence, a reasonable response to our RQ #3 would be: The performance model follows the melody faithfully, but some aspects of repetitive structures come inherently with the performance, which cannot be captured even with effective melody conditioning.",
"The mean opinion scores (MOS) obtained from our user study is listed in Table REF .",
"As expected, C&E holds significant advantage over CPT on all five aspects ($p < .01$ on 45 sets of comparisons).",
"Coherence (Ch) and structureness (S) are what our model does particularly well, gaining $>$ 1 points over CPT.",
"This indicates that explicit modeling of lead sheets helps C&E better glue generated music into one piece, as well as reuse and develop musical content.",
"Nonetheless, the scores also corroborate ($p<.01$ ) that, by every criterion, our model still has a long way to go to rival real performances."
],
[
"Conclusion",
"In this paper, we have introduced a two-stage framework, Compose & Embellish, to generate piano performances with lead sheets as the intermediate output.",
"Promising prior art [4], [5] were chosen and integrated to form our model backbone.",
"We showed via objective and subjective study that our framework composes better-structured and higher-quality piano performances compared to an end-to-end model.",
"Furthermore, pretraining the 1st-stage (i.e., lead sheet) model with extra data contributed to a sizable performance gain.",
"Future endeavors may focus on redesigning the performance model to further close the gap between generated and real performances."
]
] | 2209.08212 |
[
[
"Models for $(\\infty,n)$-categories with discreteness conditions, I"
],
[
"Abstract We establish cartesian model structures for variants of $\\Theta_n$-spaces in which we replace some or all of the completeness conditions by discreteness conditions.",
"We prove that they are all equivalent to each other and to the $\\Theta_n$-space model, and we give a criterion for which combinations of discreteness and completeness give non-overlapping models.",
"These models can be thought of as generalizations of Segal categories in the framework of $\\Theta_n$-diagrams.",
"In the process, we give a characterization of the Dwyer-Kan equivalences in the $\\Theta_n$-space model, generalizing the one given by Rezk for complete Segal spaces."
],
[
"Introduction",
"An $(\\infty ,n)$ -category should be a higher categorical structure with objects and $i$ -morphisms for all $i \\ge 1$ , satisfying weak associativity and unitality, and such that all $i$ -morphisms are weakly invertible for $i>n$ .",
"There have been a number of different approaches to modeling such a structure as a concretely-defined mathematical object, including the Segal $n$ -categories of Hirschowitz and Simpson [19] and Pelissier [28], the $n$ -fold complete Segal spaces of Barwick and Lurie [24], the $\\Theta _n$ -spaces of Rezk [31], and the $n$ -relative categories of Barwick and Kan [1].",
"More recently, Ozornova and Rovelli have developed and compared models [26], [27] that are based off Verity's complicial sets [35], and Campion, Kapulkin, and Yaehara have made progress with cubical models [13].",
"Equivalences between some of these different models have been given by the author and Rezk [10], [11], Haugseng [17], and Doherty, Kapulkin, and Maehara [14], and, using an axiomatic approach, Barwick and Schommer-Pries [3].",
"In the special case when $n=2$ , there are further results by the author with Ozornova and Rovelli [9], and by Gagna, Harpaz, and Lanari [16].",
"Results when $n=1$ are now well-established, and include the comparisons of Barwick and Kan [2], the author [8], Dugger and Spivak [15], Joyal [21], Joyal and Tierney [22], Lurie [23], and the axiomatic approach of Toën [34].",
"Many of these models are given by some kind of diagram of simplicial sets; for example, $n$ -fold complete Segal spaces and Segal $n$ -categories are given by multisimplicial diagrams, and $\\Theta _n$ -spaces are given by functors out of the category $\\Theta _n^{\\operatorname{op}}$ .",
"Such diagrams are required to satisfy $n$ levels of Segal conditions, which essentially encode an up-to-homotopy composition for each of the $n$ levels of (not necessarily invertible) morphisms.",
"However, Segal diagrams without further assumptions do not quite model $(\\infty , n)$ -categories, which should behave like iterated enriched categories and in particular have a discrete space of objects and discrete spaces of $k$ -morphisms for all $1 \\le k <n$ .",
"Without such assumptions, we get structures more reflective of $n$ -categories internal to spaces.",
"In general, there are two ways to impose extra structure on Segal diagrams to get models for $(\\infty ,n)$ -categories.",
"The first is straightforward: simply ask that the desired spaces in the diagram be discrete.",
"This approach was the one taken by Hirschowitz and Simpson in their definition of Segal $n$ -categories [19].",
"This simplicity of definition, however, comes at a cost.",
"Being discrete is a rather unnatural condition from the perspective of homotopy theory, and as such causes many complications in setting up appropriate model structures.",
"Rezk took a different approach in his development of complete Segal spaces as models for $(\\infty ,1)$ -categories, and asked instead for a completeness condition, which essentially asks that the space of objects be weakly equivalent to the subspace of morphisms that behave suitably like homotopy equivalences.",
"Thus, the data of the whole object space is already encoded in the space of morphisms, and does not give substantially new information.",
"This approach was generalized to higher $(\\infty ,n)$ -categories via the $n$ -fold complete Segal space and $\\Theta _n$ -space models, in which $n$ different completeness conditions are assumed.",
"For example, the space of 1-morphisms is required to be weakly equivalent to the space of 2-morphisms that are homotopy equivalences in an appropriate sense, and similarly for higher morphisms.",
"Thus, when we look at existing diagrammatic models for $(\\infty ,n)$ -categories, taking multisimplicial models has an accompanying choice of whether to impose discreteness or completeness conditions, whereas thus far the $\\Theta _n$ -model has only been considered with completeness conditions.",
"A natural question is whether there is a corresponding model given by $\\Theta _n$ -diagrams with discreteness conditions, and answering it is the primary motivation for this paper.",
"However, another question arises: do we need to make a single choice of either discreteness or completeness conditions everywhere, or can these conditions be “mixed and matched\"?",
"In [10] we show that Segal category objects in $\\Theta _{n-1}$ -spaces give a model for $(\\infty ,n)$ -categories; such objects have one discreteness condition and $(n-1)$ -completeness conditions in the setting of $\\Delta \\times \\Theta _{n-1}$ -diagrams.",
"Here we address this question for $\\Theta _n$ -diagrams, and show that discreteness can be imposed “from the bottom up\": we get distinct models for $(\\infty ,n)$ -categories by taking $\\Theta _n$ -diagrams of simplicial sets with discreteness imposed at level $k$ , for some fixed $0 \\le k <n$ , and completeness imposed for any $k < i < n$ .",
"Essentially, if we ask for discreteness at a given level of morphism, the spaces of all lower levels of morphisms are forced to be discrete also.",
"This answer in the $\\Theta _n$ -diagram setting naturally leads us back to ask the analogous question in the context of multisimplicial diagrams, as well as hybrid diagrams, indexed by categories of the form $\\Delta ^i \\times \\Theta _{n-i}$ for some $0<i<n$ , of which Segal category objects in $\\Theta _{n-1}$ -spaces are an example.",
"In a sequel paper [6], we address such models, and in particular give an explicit comparison between $n$ -fold complete Segal spaces and Segal $n$ -categories via these hybrids, a result which has been assumed but which does not seem to be in the literature.",
"The idea that $\\Theta _n$ -diagrams with discreteness assumptions should be a viable model for $(\\infty ,n)$ -categories developed at the early stages of the comparison of $\\Theta _n$ -spaces with categories enriched in $\\Theta _{n-1}$ -spaces, which the author established with Rezk.",
"However, complications with understanding Dwyer-Kan equivalences, and a suitable completion functor generalizing the one for complete Segal spaces in [32] seemed prohibitively difficult.",
"Fortunately, results from that comparison program with Rezk have greatly facilitated the development of this model and its comparison with $\\Theta _n$ -spaces, as we show here.",
"Our motivation for this work is primary aesthetic, in that we believe that establishing all possible options for these kind of models results in a satisfying picture of the choices we have for this flavor of models for $(\\infty ,n)$ -categories.",
"However, it would be worth investigating whether examples originally modeled by Segal $n$ -categories might be more compactly described using $\\Theta _n$ -diagrams rather than multisimplicial ones.",
"After some general background in Section 2, we review the Segal category and complete Segal space models in Section 3.",
"In Section 4, we recall Rezk's $\\Theta _n$ -space model, and in Section 5, we recall the development of Segal and complete Segal objects in $\\Theta _n$ -spaces.",
"We devote Section 6 to understanding Dwyer-Kan equivalences in $\\Theta _n$ -spaces.",
"In Section 7 we give a general treatment of making objects in general diagrams discrete, which we then use in Section 8 to give a model structure on the category of $\\Theta _n$ -diagrams with certain objects discrete.",
"In Section 9 we give our comparison result between these models and Rezk's original $\\Theta _n$ -space model.",
"Finally in Section 10 we give the proof of a deferred technical result."
],
[
"Some background",
"Our goal is to put model structures on certain categories of functors in such a way that the objects that are both fibrant and cofibrant give models for $(\\infty ,n)$ -categories.",
"Here, we give a brief review of some of the model category tools that we need.",
"The models that we consider in this paper are all given by functors from some small category to the category of simplicial sets.",
"Recall that a simplicial set is a functor $\\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {S}ets$ , where $\\Delta ^{\\operatorname{op}}$ is the opposite of the category of finite ordered sets, and $\\mathcal {S}ets$ denotes the category of sets.",
"We denote by $\\mathcal {SS}ets$ the category of simplicial sets.",
"First, we recall the classical model structure on the category of simplicial sets, originally due to Quillen [29].",
"The weak equivalences are the maps whose geometric realizations are weak homotopy equivalences of topological spaces, the cofibrations are the monomorphisms, and the fibrations are the Kan fibrations.",
"Since this model structure is the only one that we consider on the category of simplicial sets, we simply use the notation $\\mathcal {SS}ets$ to refer to it.",
"Given a small category $\\mathcal {C}$ , there are two canonical model structures on the category $\\mathcal {SS}ets^\\mathcal {C}$ of functors $\\mathcal {C} \\rightarrow \\mathcal {SS}ets$ , both of which have weak equivalences given by levelwise weak equivalences of simplicial sets.",
"In the injective model structure, we take the cofibrations to be given levelwise, whereas in the projective model structure, we take the fibrations to be given levelwise.",
"In this paper, we are primarily interested in the injective model structure.",
"Notice in particular that all objects are cofibrant in this model structure.",
"However, the disadvantage of the injective model structure is that we generally do not have explicit descriptions of sets of generating cofibrations and acyclic cofibrations, only arguments for their existence.",
"When $\\mathcal {C}$ has the additional structure of a Reedy category, then there is a third option: the Reedy model structure, which also has levelwise weak equivalences but fibrations and cofibrations defined in terms of matching and latching objects [30], [18].",
"In the particularly nice situation when $\\mathcal {C}$ has the structure of an elegant Reedy category in the sense of [12], the Reedy model structure coincides with the injective structure.",
"Thus, we have both advantages: the cofibrancy of all objects from the injective model structure but also the explicit generating cofibrations and acyclic cofibrations for the Reedy model structure.",
"All of the indexing categories that we consider in this paper, specifically the categories $\\Theta _n^{\\operatorname{op}}$ for all $n \\ge 0$ , are elegant Reedy categories.",
"In nice cases, models for $(\\infty ,n)$ -categories can be obtained by localizing this Reedy model structure with respect to a set of maps in such way that the fibrant objects are the local objects with respect to these maps.",
"Let us recall this process briefly.",
"Recall that in any model category, there is a notion of homotopy mapping space $\\operatorname{Map}^h(X,Y)$ between any two objects $X$ and $Y$ .",
"When the model category in question is simplicial, then the homotopy mapping spaces can be obtained by taking the simplicial mapping spaces of cofibrant-fibrant replacements of $X$ and $Y$ , but they can be defined more generally [18].",
"Now, let $\\mathcal {M}$ be a model category, and $S$ a set of maps in $\\mathcal {M}$ .",
"A fibrant object $Z$ of $\\mathcal {M}$ is $S$ -local if, for any map $A \\rightarrow B$ in $S$ , the induced map $ \\operatorname{Map}^h(B,Z) \\rightarrow \\operatorname{Map}^h(A,Z) $ is a weak equivalence of simplicial sets.",
"An arbitrary map $X \\rightarrow Y$ of $\\mathcal {M}$ is an $S$ -local equivalence if, for any $S$ -local object $Z$ , the induced map $ \\operatorname{Map}^h(Y, Z) \\rightarrow \\operatorname{Map}^h(X,Z) $ is a weak equivalence of simplicial sets.",
"If $\\mathcal {M}$ is a sufficiently nice model category, then there exists a model structure $\\mathcal {L}_S \\mathcal {M}$ on the underlying category of $\\mathcal {M}$ in which the weak equivalences are the $S$ -local equivalences, the cofibrations are the same as those of $\\mathcal {M}$ , and the fibrant objects are the $S$ -local objects [18].",
"However, in many examples we consider in this paper, we do not have a suitable model structure from which we can obtain our desired model structure as a localization.",
"We thus have to prove the existence of such a model structure from scratch.",
"We thus include the following recognition principle for cofibrantly generated model categories, originally due to Kan. Theorem 2.1 [18] Let $\\mathcal {M}$ be a category which has all small limits and colimits.",
"Suppose that $\\mathcal {M}$ has a class of weak equivalences which satisfies the two-out-of-three property and which is closed under retracts.",
"Let $I$ and $J$ be sets of maps in $\\mathcal {M}$ which satisfy the following conditions: Both $I$ and $J$ permit the small object argument [18].",
"Every $J$ -cofibration is an $I$ -cofibration and a weak equivalence.",
"Every $I$ -injective is a $J$ -injective and a weak equivalence.",
"One of the following conditions holds: A map that is an $I$ -cofibration and a weak equivalence is a $J$ -cofibration, or A map that is both a $J$ -injective and a weak equivalence is an $I$ -injective.",
"Then there is a cofibrantly generated model category structure on $\\mathcal {M}$ in which $I$ is a set of generating cofibrations and $J$ is a set of generating acyclic cofibrations.",
"Finally, we consider the additional structure of cartesian model categories.",
"We take the following definition from [32]; the version there includes an additional equivalent formulation of the last condition, but we omit it as we do not need it here.",
"Definition 2.2 A model category $\\mathcal {M}$ is cartesian if the underlying category is cartesian closed, its terminal object is cofibrant, and, if $f \\colon X \\rightarrow X^{\\prime }$ and $g \\colon Y \\rightarrow Y^{\\prime }$ are cofibrations in $\\mathcal {M}$ , then the pushout-corner map $ X \\times Y^{\\prime } \\amalg _{X \\times Y} X^{\\prime } \\times Y \\rightarrow X^{\\prime } \\times Y^{\\prime } $ is a cofibration that is a weak equivalence if either $f$ or $g$ is."
],
[
"Complete Segal spaces and Segal categories",
"In this section, we recall the complete Segal space and Segal category models for $(\\infty ,1)$ -categories.",
"All the models for $(\\infty ,n)$ -categories that we develop in this paper can be regarded as suitable generalizations of one or both of these models.",
"To start, for any $k \\ge 0$ , consider the $k$ -simplex, or representable simplicial set $\\Delta [k] = \\operatorname{Hom}_\\Delta (-, [k])$ .",
"We are interested in the inclusion of the sub-simplicial set $G[k]$ , defined to be the colimit of the diagram $ @1{\\Delta [1] [r]^{d_0} & \\Delta [0] & \\Delta [1] [l]_{d_1} [r] & \\cdots & \\Delta [1] [l]_{d_1}} $ in which there are $k$ copies of $\\Delta [1]$ glued together along copies of $\\Delta [1]$ .",
"We can depict $G[k]$ as a string of $k$ consecutive arrows $ \\bullet \\rightarrow \\bullet \\rightarrow \\cdots \\rightarrow \\bullet $ but with no higher simplices.",
"This simplicial set $G(k)$ is sometimes called the spine of $\\Delta [k]$ .",
"Now we would like to regard these simplicial sets as discrete simplicial spaces, or functors $\\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ ; given a simplicial set $K$ we denote the corresponding discrete simplicial space by $K^t$ .",
"In particular, $(K^t)_k = K_k$ for each $k \\ge 0$ .",
"In contrast, the constant simplicial space, which we also denote by $K$ , is given by the simplicial set $K$ as each level.",
"The two are “transpose\" to one another by reversing the role of the two simplicial directions, but they should be regarded as different from one another.",
"In particular, the simplicial spaces $\\Delta [k]^t$ are representable simplicial spaces, and so for any simplicial space $X$ we have $ \\operatorname{Map}(\\Delta [k]^t, X) \\cong X_k.",
"$ Given any simplicial space $X$ , and any $k \\ge 0$ , consider the Segal map induced by the inclusion $G[k]^t \\rightarrow \\Delta [k]^t$ : $ X_n = \\operatorname{Map}(\\Delta [k]^t, X) \\rightarrow \\operatorname{Map}(G[k]^t, X) \\cong \\underbrace{X_1 \\times _{X_0} \\cdots \\times _{X_0} X_1}_k.",
"$ These maps are isomorphisms for $k=0,1$ , so we restrict our attention to $k \\ge 2$ .",
"Definition 3.1 A Reedy fibrant simplicial space $X$ is a Segal space if, for all $k \\ge 2$ , the Segal maps are all weak equivalences of simplicial sets.",
"The following model structure can be obtained by localizing the Reedy model structure on simplicial spaces with respect to the maps $G[k]^t \\rightarrow \\Delta [k]^t$ for $k \\ge 2$ .",
"Theorem 3.2 [32] There is a cartesian model structure $\\mathcal {S}e\\mathcal {S}p$ on the category of simplicial spaces in which the fibrant objects are precisely the Segal spaces.",
"Remark 3.3 We follow the convention of Rezk and require that a Segal space be Reedy fibrant, so that we have the above concise description of the fibrant objects in the corresponding model structure.",
"Doing so additionally permits us to consider the Segal maps as we have described them, rather than taking homotopy mapping spaces, and hence a homotopy limit on the right-hand side, in the definition of Segal maps.",
"Segal spaces behave like categories up to homotopy, an idea which can be made precise in the following sense.",
"We can define the set of objects of a Segal space to be $X_{0,0}$ .",
"Given two objects $x,y$ of $X$ , the mapping space between $x$ and $y$ in $X$ is defined as the pullback $ {\\operatorname{map}_X(x,y) [r] [d] & X_1 [d]^{(d_1, d_0)} \\\\\\lbrace (x,y)\\rbrace [r] & X_0 \\times X_0.}",
"$ Since $X$ is assumed to be Reedy fibrant, the right-hand vertical map is a fibration, so the mapping space is in fact a homotopy pullback of this diagram.",
"The definition of a Segal space in terms of the Segal maps guarantees the existence of an up-to-homotopy composition on mapping spaces: again because we have assumed that $X$ is Reedy fibrant, the left-hand map in the diagram $ @1{X_1 \\times _{X_0} X_1 & X_2 [l]_-{(d_1, d_0)} [r]^{d_1} & X_1} $ is an acyclic fibration and hence has a section which serves as a homotopy inverse.",
"We can also define the homotopy category $\\operatorname{Ho}(X)$ of a Segal space $X$ , whose objects are those of $X$ and whose morphisms are the path components of the mapping spaces of $X$ .",
"However, in the above definition we have only used the 0-simplices of $X_0$ as the objects, so in some sense the additional simplicial data of $X_0$ is extraneous.",
"Indeed, to get an $(\\infty ,1)$ -category we want to have a discrete space of objects, rather than an arbitrary space as in a general Segal space.",
"There are two approches to remedying this situation.",
"We start with the simplest: requiring that $X_0$ be discrete, so that all its higher simplices are degenerate.",
"Definition 3.4 A Segal category is a Segal space $X$ such that $X_0$ is discrete.",
"Remark 3.5 Note that here we have assumed that a Segal category is Reedy fibrant.",
"This assumption is not made in many other treatments of Segal categories, for example in [8].",
"In some situations, we want to consider Segal categories that are projective fibrant, rather than Reedy fibrant, and hence do not specify one or the other in the basic definition.",
"In this paper, we only consider Segal categories and generalizations that are Reedy fibrant, so to be as streamlined as possible we include this assumption here.",
"For many homotopy-theoretic purposes, however, requiring that a space be discrete is unnatural.",
"An alternative approach is to require $X_0$ to be equivalent to the space of homotopy equivalences in $X_1$ , as described by Rezk [32].",
"Since a Segal space has a notion of up-to-homotopy composition, as well as identity maps defined by those in the image of the degeneracy map $X_0 \\rightarrow X_1$ , there is a natural definition of homotopy equivalences as those maps that have an inverse up to homotopy.",
"They form a subspace $X_{\\operatorname{heq}} \\subseteq X_1$ , and indeed comprise some of the path components of $X_1$ [32].",
"The degeneracy map $s_0 \\colon X_0 \\rightarrow X_1$ factors through $X_{\\operatorname{heq}}$ , allowing for the following definition.",
"Definition 3.6 A Segal space $X$ is complete if the map $s_0 \\colon X_0 \\rightarrow X_{\\operatorname{heq}}$ is a weak equivalence of simplicial sets.",
"The proof that Segal categories and complete Segal spaces are equivalent models for $(\\infty ,1)$ -categories is given by a Quillen equivalence of appropriate model categories.",
"The model structure for complete Segal spaces is obtained as a further localization of the Segal space model structure $\\mathcal {S}e\\mathcal {S}p$ .",
"We localize with respect to the map $E^t \\rightarrow \\Delta [0]^t$ , where $E$ is the simplicial set given by the nerve of the category with two objects and a single isomorphism between them.",
"One can check that $\\operatorname{Map}(E^t, X) \\simeq X_{\\operatorname{heq}}$ , verifying that the local objects with respect to this map are indeed complete.",
"Theorem 3.7 [32] There is a cartesian model structure $\\mathcal {CSS}$ on the category of simplicial spaces such that the fibrant objects are the complete Segal spaces.",
"For Segal categories, the underlying category for the model structure is the category of Segal precategories, or simplicial spaces $X$ with $X_0$ discrete, such that the fibrant objects are the Segal categories.",
"However, since this category does not admit a Reedy-like model structure with levelwise weak equivalences, we cannot obtain the Segal category model structure as a localization; see [8] for more details.",
"Roughly speaking, the problem is that with the discreteness requirement we cannot always obtain the required factorizations to have a model structure with levelwise weak equivalences, so we do not have a good model structure to localize.",
"To establish the desired model structure, we thus need a precise definition of the desired weak equivalences.",
"To this end, let us return to the Segal space model structure for a moment.",
"Given any simplicial space $X$ , we can take a functorial fibrant replacement of it the model structure $\\mathcal {S}e\\mathcal {S}p$ , in which the fibrant objects are the (not necessarily complete) Segal spaces.",
"Denoting this fibrant replacement by $L_{\\operatorname{Se}}$ , we make the following definition.",
"Definition 3.8 A map $f \\colon X \\rightarrow Y$ of simplicial spaces is a Dwyer-Kan equivalence if: for any $x,y \\in \\operatorname{ob}(X)$ , the map $ \\operatorname{map}_{L_{\\operatorname{Se}}X}(x,y) \\rightarrow \\operatorname{map}_{L_{\\operatorname{Se}}Y}(fx,fy) $ is a weak equivalence of simplicial sets; and the induced functor $ \\operatorname{Ho}(L_{\\operatorname{Se}}X) \\rightarrow \\operatorname{Ho}(L_{\\operatorname{Se}}Y) $ is essentially surjective.",
"This definition can be adapted to the setting of Segal precategories, using a suitable modification of the functor $L_{\\operatorname{Se}}$ that retains the necessary discreteness condition [8]; see also the arguments in Proposition REF .",
"The importance of Dwyer-Kan equivalences is illustrated in the following results of Rezk.",
"Theorem 3.9 [32] A map $X \\rightarrow Y$ of Segal spaces is a Dwyer-Kan equivalence if and only if it is a weak equivalence in $\\mathcal {CSS}$ .",
"A map $X \\rightarrow Y$ of complete Segal spaces is a Dwyer-Kan equivalence if and only if it is a levelwise weak equivalence of simplicial sets.",
"In particular, if we want a model structure whose weak equivalences between Segal categories behave like the weak equivalences of $\\mathcal {CSS}$ , the Dwyer-Kan equivalences provide good candidates.",
"Theorem 3.10 [8], [28] There is a cartesian model structure $\\mathcal {S}e\\mathcal {C}at$ on the category of Segal precategories in which the weak equivalences are the Dwyer-Kan equivalences and the fibrant objects are the Segal categories.",
"The common notion of Dwyer-Kan equivalence in the model structures $\\mathcal {CSS}$ and $\\mathcal {S}e\\mathcal {C}at$ is key to the proof of the following theorem.",
"Theorem 3.11 [8] The inclusion functor from the category of Segal precategories into the category of all simplicial spaces has a right adjoint, and this adjunction induces a Quillen equivalence $ \\mathcal {S}e\\mathcal {C}at\\rightleftarrows \\mathcal {CSS}.",
"$ The right adjoint functor $R$ serves as a discretization functor, and can be described on objects as follows.",
"Let $W$ be a simplicial space.",
"Let $U=\\operatorname{cosk}_0(W)$ be the 0-coskeleton of $W$ , and $V=U_{*,0}$ , the discrete simplicial space given by the 0-simplices in each degree of $U$ .",
"Alternatively, $V = \\operatorname{cosk}_0(W_{*,0})$ , where $W_{*,0}$ denotes the discrete simplicial space consisting of the zero simplices in each degree of $W$ .",
"Then $RW$ is defined to be the pullback $ {RW [r] [d] & V [d] \\\\W [r]& U,} $ where $V \\rightarrow U$ is the inclusion and $W \\rightarrow U$ is the canonical map from the coskeleton."
],
[
"$\\Theta _n$ -spaces",
"In this section, we review the definition of $\\Theta _n$ -spaces, which serve as a model for higher-categorical complete Segal spaces, and summarize some of the key constructions that we need here.",
"The categories $\\Theta _n$ were originally described by Joyal, using a direct definition [20].",
"Here we have chosen to use their inductive description via the $\\Theta $ -construction of Berger [4], which is also described by Rezk in [31].",
"Definition 4.1 Let $\\mathcal {C}$ be a small category.",
"Define $\\Theta \\mathcal {C}$ to be the category with objects $[m](c_1, \\ldots , c_m)$ where $[m]$ is an object of $\\Delta $ and each $c_i$ is an object of $\\mathcal {C}$ .",
"A morphism $ [q](c_1, \\ldots ,c_q) \\rightarrow [m](d_1, \\ldots , d_m) $ is given by $(\\delta , \\lbrace f_{ij}\\rbrace )$ where $\\delta \\colon [q] \\rightarrow [m]$ in $\\Delta $ and $f_{ij} \\colon c_i \\rightarrow d_j$ are morphisms in $\\mathcal {C}$ indexed by $1 \\le i \\le q$ and $1 \\le j \\le m$ where $\\delta (i-1) < j \\le \\delta (i)$ .",
"Now let us use this definition to describe our categories of interest here.",
"Definition 4.2 Let $\\Theta _0$ be the terminal category with a single object and no non-identity morphisms.",
"Inductively define $\\Theta _n=\\Theta \\Theta _{n-1}$ .",
"Observe that $\\Theta _1=\\Delta $ .",
"To build some intuition about $\\Theta _n$ for higher $n$ , let us look more closely at $\\Theta _2 = \\Theta \\Delta $ .",
"Its objects are of the form $[q]([c_1], \\ldots , [c_q])$ , where $[q]$ , as well as each $[c_i]$ , is an object of $\\Delta $ .",
"We can think of this object as being a copy of the diagram $[q]$ whose arrows are labeled by $[c_1], \\ldots , [c_q]$ .",
"For example, the object $[4]\\left([2], [3], [0], [1]\\right)$ can be depicted as $ @1{0 [r]^{[2]} & 1 [r]^{[3]} & 2 [r]^{[0]} & 3[r]^{[1]} & 4.}",
"$ Since these labels themselves can be visualized as strings of arrows, we can further illustrate our object as ${ {0} @/^2pc/[r]^{}=\"10\" [r]^{}=\"11\" @/_2pc/[r]^{}=\"12\" @{=>}\"10\";\"11\" @{=>}\"11\";\"12\"& {1} @/^3pc/[r]^{}=\"20\" @/^1pc/[r]^{}=\"21\"@/_1pc/[r]^{}=\"22\" @/_3pc/[r]^{}=\"23\" @{=>}\"20\";\"21\"@{=>}\"21\";\"22\" @{=>}\"22\";\"23\"& {2} [r]& {3} @/^1pc/[r]^{}=\"30\" @/_1pc/[r]^{}=\"31\" @{=>}\"30\";\"31\"& {4.",
"}}$ This diagram can be regarded as generating a strict 2-category by composing 1-cells and 2-cells whenever possible.",
"In other words, the objects of $\\Theta _2$ can be seen as encoding all possible finite composites, whether horizontal or vertical, that can take place in a 2-category, much as the objects of $\\Delta $ can be thought of as listing all the finite composites that can occur in an ordinary category.",
"Example 4.3 Of key importance in this paper are the objects of $\\Theta _n$ given by a single morphism, or free-standing “cell\", of each dimension up to $n$ .",
"For example, we have a single object $\\bullet $ , a single 1-cell $\\bullet \\rightarrow \\bullet $ , and a single 2-cell ${{\\bullet } @/^1pc/[r]^{}=\"30\" @/_1pc/[r]^{}=\"31\" @{=>}\"30\";\"31\"& {\\bullet .}",
"}$ These objects are denoted by $[0]$ , $[1]([0])$ , and $[1]([1])$ as objects of $\\Theta _2$ .",
"More generally, in $\\Theta _n$ we have objects $[1]([1]( \\cdots ([0])) \\cdots )$ where we have up to $n-1$ occurrences of $[1]$ concluding with a $[0]$ , and finally the object $ [1]([1](\\cdots [1]) \\cdots ).",
"$ These objects can be depicted as the free-standing cells of increasing dimension, starting with dimension 0 (an “object\") and going up through dimension $n$ (an “$n$ -morphism\" or “$n$ -cell\").",
"To simplify the notation, we often write $[1]^{(0)}$ for $[0]$ , and then $[1]^{(i)}=[1]([1]^{(i-1)})$ for any $0<i<n$ .",
"Observe that $[0]$ is the terminal object in any $\\Theta _n$ , and likewise that $[1]^{(i)}$ can be considered as an object of $\\Theta _n$ for any $n >i$ .",
"While there is some ambiguity here about what $n$ is, the idea is simply that we have $i$ -morphisms in any $n$ -category for $n>i$ .",
"We include in our notation the free-standing $n$ -cell, which corresponds to the object $[1]([1](\\cdots [1]) \\cdots )$ of $\\Theta _n$ , and denote this object by $[1]^{(n)}$ .",
"This notation is in potential conflict with the above, if we move from $\\Theta _n$ to $\\Theta _k$ for some $k>n$ , but the context should make this distinction, especially since the essential shape remains the same.",
"Indeed, this object is simply given by $ \\underbrace{[1]([1](\\cdots [1]}_n([0])) \\cdots ) $ in $\\Theta _k$ when $k>n$ .",
"Let us consider $\\Theta _n$ -sets, which are functors $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {S}ets$ .",
"For any object $[q](c_1, \\ldots , c_q)$ , let $\\Theta [q](c_1, \\ldots , c_q)$ denote the representable functor $\\operatorname{Hom}_{\\Theta _n}(-,[q](c_1, \\ldots , c_q))$ .",
"Since we want to generalize complete Segal spaces, we are more interested in functors $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ .",
"Observe that any simplicial set can be regarded as a constant functor of this kind, and any functor $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {S}ets$ , in particular the representable functors just described, can be regarded as levelwise discrete functors to $\\mathcal {SS}ets$ .",
"Since $\\Theta _n^{\\operatorname{op}}$ is a Reedy category [4], the functor category $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ can be equipped with the Reedy model structure, which we prove in [12] agrees with the injective model structure.",
"In particular, the cofibrations are the levelwise monomorphisms of simplicial sets, and every object is cofibrant.",
"We recall from [10] that a set of generating cofibrations of this model structure are given by $ \\partial \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q) \\cup \\Delta [m] \\times \\partial \\Theta [q](c_1, \\ldots , c_q) \\rightarrow \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q) $ where $m, q \\ge 0$ and $c_1, \\ldots , c_q \\in \\operatorname{ob}(\\Theta _{n_1})$ , and where $\\partial \\Theta [q](c_1, \\ldots , c_q)$ denotes the boundary of the representable object $\\Theta [q](c_1, \\ldots , c_q)$ , defined by mapping out of objects of strictly lower degree than that of $[q](c_1, \\ldots , c_q)$ .",
"Note that the simplicial sets and $\\Theta _n$ -sets appearing in this definition are appropriately constant in the $\\Theta _n$ and simplicial directions, respectively.",
"A set of generating acyclic cofibrations can be defined similarly, replacing the boundaries $\\partial \\Delta [m]$ with horns $\\Lambda ^k[m]$ for $m \\ge 1$ and $0 \\le k \\le m$ .",
"To obtain models for $(\\infty ,n)$ -categories, we want to ask for appropriate Segal maps to be weak equivalences, and so we generalize the development of Segal spaces as follows.",
"Given $q \\ge 2$ and $c_1, \\ldots , c_q$ objects of $\\Theta _{n-1}$ , define the object $ G[m](c_1, \\ldots , c_m)= \\operatorname{colim}(\\Theta [1](c_1) \\leftarrow \\Theta [0] \\rightarrow \\cdots \\leftarrow \\Theta [0] \\rightarrow \\Theta [1](c_m)).",
"$ There is an inclusion map $\\operatorname{se}^{(c_1, \\ldots , c_q)} \\colon G[q](c_1, \\ldots , c_q) \\rightarrow \\Theta [q](c_1, \\ldots , c_q).",
"$ Now define the set $ \\operatorname{Se}_{\\Theta _n} = \\lbrace \\operatorname{se}^{(c_1, \\ldots , c_q)} \\mid q \\ge 2, c_1, \\ldots c_q \\in \\operatorname{ob}(\\Theta _{n-1})\\rbrace .",
"$ Example 4.4 Referring to the example of the object $ [4]\\left([2], [3], [0], [1]\\right) $ of $\\Theta _2$ above, we have the representable functor $ \\Theta [4]\\left([2], [3], [0], [1]\\right).",
"$ The corresponding functor $ G[4]\\left([2], [3], [0], [1]\\right) $ consists of the union of the representable functors $\\Theta [1]([2])$ , $\\Theta [1]([3])$ , $\\Theta [1]([0])$ , and $\\Theta [1]([1])$ , glued together along the representable functors corresponding to the intersection points, each given by $\\Theta [0]$ .",
"If we localize with respect to the inclusion $ G[4]\\left([2], [3], [0], [1]\\right) \\rightarrow \\Theta [4]\\left([2], [3], [0], [1]\\right), $ then an object $X$ is local if having these vertical composites guarantees the existence of all horizontal composites of 1-cells and 2-cells.",
"Namely, the induced map $ X[4]([2], [3], [0], [1]) \\rightarrow X[1]([2]) \\times _{X[0]} X[1]([3]) \\times _{X[0]} X[1]([0]) \\times _{X[0]} X[1]([1]) $ is a weak equivalence of simplicial sets.",
"However, such a localization only gives us horizontal composition, not vertical composition.",
"For example, we also want the map $ X[1]([2]) \\rightarrow X[1]([1]) \\times _{X[1]([0])} X[1]([1]) $ to be a weak equivalence of simplicial sets.",
"The previous example illustrates that being local with respect to the maps in $Se_{\\Theta _n}$ is not sufficient when $n>1$ , as it only gives an up-to-homotopy composition at level $n$ .",
"Encoding other levels of composition is achieved inductively, making use of the intertwining functor $V[1] \\colon \\mathcal {SS}ets^{\\Theta _{n-1}^{op}} \\rightarrow \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ to translate a set $\\mathcal {S}$ of maps in $\\mathcal {SS}ets^{\\Theta _{n-1}^{\\operatorname{op}}}$ into a set $V[1](\\mathcal {S})$ of maps in $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ .",
"Let us briefly recall this functor; full details can be found in [31].",
"Given a functor $A \\colon \\Theta _{n-1}^{\\operatorname{op}} \\rightarrow \\mathcal {SS}ets$ , define $V[1](A) \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ by $ [q](c_1, \\ldots , c_q) \\mapsto \\coprod _{\\delta \\colon [q] \\rightarrow [1]} \\prod _{i=1}^q A(c_i).",
"$ The idea is that $V[1](A)$ models a category enriched in $\\mathcal {SS}ets^{\\Theta _{n-1}^{\\operatorname{op}}}$ , with two objects $x$ and $y$ and one nontrivial mapping object from $x$ to $y$ given by $A$ .",
"The mapping object from $y$ to $x$ is empty, and the mapping objects at $x$ and $y$ each consist of an identity morphism only.",
"Let $\\mathcal {S}_1=\\operatorname{Se}_{\\bf \\Delta } = \\lbrace G(n)^t \\rightarrow \\Delta [n]^t \\mid n \\ge 2\\rbrace $ , and for $n \\ge 2$ , inductively define $\\mathcal {S}_n=\\operatorname{Se}_{\\Theta _n} \\cup V[1](\\mathcal {S}_{n-1})$ .",
"Thus, in $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ , we have $n$ different Segal conditions, corresponding to the desired composition in each of the $n$ categorical levels.",
"Theorem 4.5 [31] Localizing the Reedy model structure $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ with respect to $\\mathcal {S}_n$ results in a cartesian model category whose fibrant objects are higher-order analogues of Segal spaces.",
"We denote this model structure by $\\Theta _n\\mathcal {S}e\\mathcal {S}p$ and refer to its fibrant objects as $\\Theta _n$ -Segal spaces.",
"However, to get models for $(\\infty ,n)$ -categories, we want to incorporate higher-order completeness conditions as well.",
"Again, we localize with respect to some maps, and to do so, we make use of the underlying simplicial space of a functor $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ .",
"Consider the functor $\\tau _\\Theta \\colon {\\bf \\Delta } \\rightarrow \\Theta _n$ defined by $ \\tau _\\Theta [k] = [k]([0], \\ldots , [0]), $ which by [31] induces a Quillen pair on Reedy model structures $ (\\tau _\\Theta )_\\# \\colon \\mathcal {SS}ets^{\\Delta ^{\\operatorname{op}}} \\rightleftarrows \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}} \\colon \\tau _\\Theta ^*.$ The functor $\\tau _\\Theta ^*$ takes a functor $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ to its underlying simplicial space.",
"Recalling that complete Segal spaces are defined by localizing with respect to $ \\operatorname{Cpt}_{\\bf \\Delta }= \\lbrace E^t \\rightarrow \\Delta [0]^t\\rbrace , $ for $n \\ge 2$ we use the left adjoint functor $(\\tau _\\Theta )_\\#$ to define $ \\operatorname{Cpt}_{\\Theta _n}=\\lbrace (\\tau _\\Theta )_\\# E^t \\rightarrow (\\tau _\\Theta )_\\# \\Delta [0]^t\\rbrace .",
"$ Just as for the Segal conditions, localizing with respect to this map only encodes a completeness condition for objects.",
"Specifically, an object $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ is local with respect to this map when $ X[0] \\simeq X[1]([0])_{\\operatorname{heq}}, $ where the space of homotopy equivalences on the right-hand side can be defined similarly in terms of homotopy equivalences, as for complete Segal spaces, or more formally as $\\operatorname{Map}((\\tau _\\Theta )_\\# E, X)$ .",
"To capture the other necessary completeness conditions, namely that $ X[1]^{(i)} \\simeq X[1]^{(i+1)}_{\\operatorname{heq}} $ for $0<i<n$ , we use the intertwining functor.",
"For example, when $i=2$ , we can define $ X[1]^{(2)}_{\\operatorname{heq}} = \\operatorname{Map}(V[1]((\\tau _\\Theta )_\\# E^t), X).",
"$ Combining with the Segal maps, let $ \\mathcal {T}_1=\\operatorname{Se}_{\\Delta } \\cup \\operatorname{Cpt}_{\\Delta } $ and, for $n \\ge 2$ , $ \\mathcal {T}_n=\\operatorname{Se}_{\\Theta _n} \\cup \\operatorname{Cpt}_{\\Theta _n} \\cup V[1](\\mathcal {T}_{n-1}), $ again recalling that $\\Theta _1=\\Delta $ .",
"Theorem 4.7 [31] Localizing $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ with respect to the set $\\mathcal {T}_n$ results in a cartesian model category that we denote by $\\Theta _n\\mathcal {CSS}$ .",
"We would like to have a good description of the fibrant objects in this model structure.",
"To this end, we first define mapping objects.",
"Definition 4.8 Given a functor $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ and any $(x_0, x_1) \\in X[0]_0 \\times X[0]_0$ , we define the mapping object $M_X^\\Theta (x_0, x_1) \\colon \\Theta _{n-1}^{\\operatorname{op}} \\rightarrow \\mathcal {SS}ets$ , evaluated at any object $c$ of $\\Theta _{n-1}$ , as the pullback of the diagram $ \\lbrace (x_0, x_1)\\rbrace \\rightarrow X[0]\\times X[0] \\leftarrow X[1](c).",
"$ Revisiting the adjunction (REF ), the functor $ \\tau _\\Theta ^* \\colon \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}} \\rightarrow \\mathcal {SS}ets^{\\Delta ^{\\operatorname{op}}} $ is given by $(\\tau _\\Theta ^*X)_m = \\Theta [n]([0], \\ldots , [0])$ , where here $[0]$ is the terminal object of $\\Theta _{n-1}$ .",
"We have the following explicit description of the fibrant objects of the model structure $\\Theta _n\\mathcal {CSS}$ , which we use, for example, in [11].",
"Definition 4.9 A $\\Theta _n$ -space is a functor $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ such that: $X$ is Reedy fibrant; for every $m \\ge 2$ and $c_1, \\ldots , c_m \\in \\operatorname{ob}(\\Theta _{n-1})$ the Segal map $ X[m](c_1, \\ldots , c_m) \\rightarrow X[1](c_1) \\times _{X[0]} \\cdots \\times _{X[0]} X[1](c_m) $ is a weak equivalence of simplicial sets; the underlying simplicial space $\\tau _\\Theta ^* X$ is a complete Segal space; and for every $(x_0, x_1) \\in X[0]_0 \\times X[0]_0$ , the mapping object $M_X^\\Theta (x_0, x_1)$ is a $\\Theta _{n-1}$ -space.",
"Remark 4.10 We have chosen to follow Rezk's original terminology and refer to these objects as “$\\Theta _n$ -spaces\".",
"It is arguably more accurate to call them “complete Segal $\\Theta _n$ -spaces\", a convention we adopt in [7].",
"For the purposes of this paper, however, this specification leads to unwieldy terminology; to avoid having to refer repeatedly to atrocities such as “ complete Segal objects in complete Segal $\\Theta _n$ -spaces\" in what follows, we have chosen to revert back to the more concise name.",
"We retain the specification of “$\\Theta _n$ -Segal space\" when we refer to an object that satisfies the Segal condition but no completeness assumptions.",
"We conclude this section by recalling two different ways to think of a homotopy category of a $\\Theta _n$ -Segal space.",
"Definition 4.11 Let $X$ be a $\\Theta _n$ -Segal space.",
"Then its enriched homotopy category $\\underline{\\operatorname{Ho}}(X)$ has object set $X[0]$ and mapping object $\\underline{\\operatorname{Map}}_{\\underline{\\operatorname{Ho}}(X)}(x_0, x_1) = M_X^\\Theta (x_0, x_1) \\colon \\Theta _{n-1}^{\\operatorname{op}} \\rightarrow \\mathcal {SS}ets.",
"$ Its (ordinary) homotopy category $\\operatorname{Ho}^\\Theta (X)$ has the same objects but has $ \\operatorname{Hom}_{\\operatorname{Ho}^\\Theta (X)}(x_0, x_1) = \\operatorname{Ho}_{\\tau _\\Theta ^*X}(x_0, x_1).",
"$ Alternatively described, $\\operatorname{Ho}^\\Theta (X)$ is the homotopy category of the underlying Segal space of $X$ , in the sense described in Section .",
"We would like to use these definitions to generalize Definition REF to a notion of Dwyer-Kan equivalence for $\\Theta _n$ -spaces, and then prove the analogue of Theorem REF in this context.",
"Because we have need of them in the proof, we first take a detour to review complete Segal objects in $\\Theta _n$ -spaces and the notion of Dwyer-Kan equivalence in that context."
],
[
"Segal and complete Segal objects in $\\Theta _n$ -spaces",
"One feature of $\\Theta _n$ -spaces is that they are suitably equivalent to categories enriched in $\\Theta _{n-1}$ -spaces, following the general principle that $(\\infty ,n)$ -categories should be equivalent to categories enriched in $(\\infty ,n-1)$ -categories.",
"One way to model categories weakly enriched in $\\Theta _{n-1}$ -spaces is via the structure of a complete Segal object in $\\Theta _{n-1}$ -spaces.",
"We give a brief review here, and refer the reader to [11] for more details.",
"The main idea is that, just as a complete Segal space can be thought of as a category weakly enriched in spaces and is given by a functor $W \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ , we can describe a complete Segal object in $\\Theta _n$ -spaces as a functor $W \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\Theta _n\\mathcal {CSS}$ .",
"We emphasize the model structure $\\Theta _n\\mathcal {CSS}$ here because it determines the weak equivalences we use for our Segal conditions, but the objects of the underlying category are functors $W \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ .",
"As with complete Segal spaces and $\\Theta _n$ -spaces, it is helpful to look first at objects that satisfy only the relevant Segal condition.",
"Our first approach uses a straightforward generalization of the definition of Segal space.",
"Definition 5.1 A Segal object in $\\Theta _n$ -spaces is a Reedy fibrant functor $W \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ such that, for every $m \\ge 2$ , the Segal map $ W_m \\rightarrow \\underbrace{W_1 \\times _{W_0} \\cdots \\times _{W_0} W_1}_m $ is a weak equivalence in the model structure $\\Theta _n\\mathcal {CSS}$ .",
"It can be helpful here to think of such functors as $W \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\Theta _n\\mathcal {CSS}$ to emphasize the model structure on the target category.",
"We often refer to Segal objects in $\\Theta _n$ -spaces simply as Segal objects for simplicity.",
"However, there is an equivalent definition that is more widely used in the literature, and which enables a cleaner description of the completeness condition.",
"Here it is helpful to regard functors $W \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ instead as functors $W \\colon \\Delta ^{\\operatorname{op}}\\times \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ .",
"For our alternate definition, which is closely related to Definition REF , we need a notion of mapping object that is analogous to the one given in Definition REF for $\\Theta _n$ -spaces.",
"Definition 5.2 Given a functor $W \\colon \\Delta ^{\\operatorname{op}}\\times \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ and any $x_0, x_1 \\in W([0], [0])_0$ , the mapping object $M^\\Delta _W(x_0, x_1) \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ is defined levelwise by pullbacks $ {M^\\Delta _W(x_0, x_1)(c) [r] [d] & W([1],c) [d] \\\\\\lbrace (x_0, x_1)\\rbrace [r] & W([0], c) \\times W([0],c).",
"}$ The following result is well-known to experts, but we are not aware of a proof in the literature, so we include one here.",
"It is of interest in part due to the subtle role of the two different Reedy structures involved.",
"Proposition 5.4 A Reedy fibrant functor $W \\colon \\Delta ^{\\operatorname{op}}\\times \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ is a Segal object in $\\Theta _n$ -spaces if and only if the following conditions hold: for any $m \\ge 2$ and $c \\in \\operatorname{ob}(\\Theta _n)$ , the Segal map $ W([m], c) \\rightarrow W([1],c) \\times _{W([0], c)} \\cdots \\times _{W([0],c)} W([1],c) $ is a weak equivalence of simplicial sets; and for any $x_0, x_1 \\in W([0],[0])_0$ , the mapping object $M^\\Delta _W(x_0, x_1)$ is a $\\Theta _n$ -space.",
"Suppose that $W$ is a Segal object in $\\Theta _n$ -spaces, so for each $m \\ge 2$ the map $ W_m \\rightarrow W_1 \\times _{W_0} \\cdots \\times _{W_0} W_1 $ is a weak equivalence in $\\Theta _n\\mathcal {CSS}$ .",
"Since $W$ is assumed to be Reedy fibrant, $W_m$ is a $\\Theta _n$ -space for each $m \\ge 0$ [18].",
"Since $\\Theta _n\\mathcal {CSS}$ is obtained as a localized model category, and local weak equivalences between fibrant objects are levelwise weak equivalences, each Segal map above is a levelwise weak equivalence of functors $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ , i.e., the maps as in (REF ) are weak equivalences of simplicial sets.",
"To check (REF ), consider $M_W^\\Delta (x,y)$ for fixed $x,y \\in W([0],[0])_0$ .",
"Since $W$ is assumed to be Reedy fibrant, the right vertical map in (REF ) is a fibration between $\\Theta _n$ -spaces, which are the fibrant objects in $\\Theta _n\\mathcal {CSS}$ .",
"Since the discrete object $\\lbrace (x,y)\\rbrace $ is also a fibrant object in $\\Theta _n\\mathcal {CSS}$ , the pullback must be as well.",
"It follows that $M^\\Delta _W(x,y)$ is fibrant, namely, a $\\Theta _n$ -space.",
"Conversely, suppose conditions (REF ) and (REF ) hold.",
"We first want to show that $W$ is Reedy fibrant as a functor $W \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\Theta _n\\mathcal {CSS}$ .",
"For any $m \\ge 0$ , let $M_mW$ denote the $m$ -th matching object of $W$ ; using the definition of Reedy fibration [18], we need to show that the map $W_m \\rightarrow M_mW$ is a fibration in $\\Theta _n\\mathcal {CSS}$ .",
"Observe that $W_m = \\underline{\\operatorname{Map}}(\\Delta [m], W)$ , the functor $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ defined by $ [p](c_1, \\ldots , c_p) \\mapsto W([m], [p](c_1, \\ldots , c_p)).",
"$ Similarly, $M_mW = \\underline{\\operatorname{Map}}(\\partial \\Delta [m], W)$ .",
"Using the inclusion $\\partial \\Delta [m] \\rightarrow \\Delta [m]$ , one can check that the map $W_m \\rightarrow M_mW$ is indeed a fibration in $\\Theta _n \\mathcal {CSS}$ .",
"Finally, we need to check that for any $m \\ge 2$ the Segal map $ W_m \\rightarrow W_1 \\times _{W_0} \\cdots \\times _{W_0} W_1 $ is a weak equivalence in $\\Theta _n\\mathcal {CSS}$ .",
"We know by assumption that for any $m \\ge 2$ and any object $c$ of $\\Theta _n^{\\operatorname{op}}$ , the map $ W([m], c) \\rightarrow W([1],c) \\times _{W([0],c)} \\cdots \\times _{(W([0],c)} W([1],c) $ is a weak equivalence of simplicial sets.",
"It follows that the Segal map above is a levelwise weak equivalence of simplicial sets, hence also a weak equivalence in $\\Theta _n\\mathcal {CSS}$ .",
"We now incorporate the completeness condition by modifying this second equivalent definition of Segal objects.",
"Analogously to the setting of $\\Theta _n$ -spaces, we make use of an underlying simplicial space functor, which we define as follows.",
"Let $\\tau _\\Delta \\colon \\Delta \\rightarrow \\Delta \\times \\Theta _n$ be given by $[k] \\mapsto ([k], [0])$ .",
"The desired functor is the induced map $ \\tau _\\Delta ^* \\colon \\mathcal {SS}ets^{\\Delta ^{\\operatorname{op}}\\times \\Theta _n^{\\operatorname{op}}} \\rightarrow \\mathcal {SS}ets^{\\Delta ^{\\operatorname{op}}}.",
"$ Definition 5.5 A Reedy fibrant functor $W \\colon \\Delta ^{\\operatorname{op}}\\times \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ is a complete Segal object in $\\Theta _n$ -spaces if: for all $m \\ge 2$ and $c \\in \\operatorname{ob}(\\Theta _n)$ , the map $ W([m],c) \\rightarrow W([1],c)\\times _{W([0],c)} \\cdots \\times _{W([0],c)} W([1],c) $ is a weak equivalence of spaces; for all $x_0,x_1\\in W([0],[0])$ , the functor $M_W^\\Delta (x_0,x_1) \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ is a $\\Theta _n$ -space; the underlying simplicial space $\\tau _\\Delta ^* W$ is a complete Segal space; and for all objects $c \\in \\Theta _n$ , the map $W([0],[0]) \\rightarrow W([0],c)$ is a weak equivalence.",
"Again, we typically refer to these objects simply as complete Segal objects.",
"Theorem 5.6 [11] There is a model structure $\\mathcal {CSS}(\\Theta _nSp)$ on the category of functors $\\Delta ^{\\operatorname{op}}\\times \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ in which the fibrant objects are the complete Segal objects, obtained by a localization of the Reedy model structure.",
"Using the mapping objects $M^\\Delta _W(x,y)$ , we can generalize Definition REF as follows.",
"Here, we define the homotopy category of a Segal object $W$ to be the homotopy category of the underlying Segal space $\\tau _\\Delta ^*W$ , and denote it by $\\operatorname{Ho}^\\Delta (W)$ .",
"Definition 5.7 A map $f \\colon W \\rightarrow Z$ of Segal objects in $\\Theta _nSp$ is a Dwyer-Kan equivalence if: for any objects $x$ and $y$ of $W$ , the induced map $M_W^\\Delta (x,y) \\rightarrow M_Z^\\Delta (fx,fy)$ is a weak equivalence in $\\Theta _nSp$ , and the induced functor $\\operatorname{Ho}^\\Delta (W) \\rightarrow \\operatorname{Ho}^\\Delta (Z)$ is essentially surjective.",
"The following theorem is a generalization of Theorem REF (REF ).",
"Theorem 5.8 [11] A map $f \\colon U \\rightarrow V$ of Segal objects is a Dwyer-Kan equivalence if and only if it is a weak equivalence in the model category $\\mathcal {CSS}(\\Theta _nSp)$ ."
],
[
"Dwyer-Kan equivalences for $\\Theta _n$ -spaces",
"Of key importance in the theory of complete Segal spaces and their relationship with Segal categories are the Dwyer-Kan equivalences, which mirror the natural weak equivalences of simplicial categories that share the same name.",
"The main idea is to generalize the notion of equivalence of categories, namely being fully faithful and essentially surjective, to a more general context.",
"We would like to have a similar notion for $\\Theta _n$ -spaces.",
"While the appropriate definition was given in by Rezk in [31], trying to establish a result analogous to Theorem REF presented some technical difficulties.",
"In this section, we establish these properties, avoiding some of the technical obstacles by using the Quillen equivalence between $\\Theta _n$ -spaces and complete Segal objects in $\\Theta _{n-1}$ -spaces from [11].",
"The following definitions for $\\Theta _n$ -Segal spaces are given in [31].",
"We start with homotopically fully faithful maps, which make use of the mapping objects described in the previous section.",
"Definition 6.1 Let $X$ and $Y$ be $\\Theta _n$ -Segal spaces.",
"A morphism $f \\colon X \\rightarrow Y$ is homotopically fully faithful if for every $x_0, x_1 \\in X[0]$ and every $c \\in \\operatorname{ob}(\\Theta _{n-1})$ the map $ M_X^\\Theta (x_0, x_1)(c) \\rightarrow M_X^\\Theta (fx_0, fx_1)(c) $ is a weak equivalence in $\\mathcal {SS}ets$ .",
"We define essential surjectivity in terms of the homotopy category.",
"Definition 6.2 Let $X$ and $Y$ be $\\Theta _n$ -Segal spaces.",
"A morphism $X \\rightarrow Y$ is essentially surjective if $\\operatorname{Ho}^\\Theta (f) \\colon \\operatorname{Ho}^\\Theta (X) \\rightarrow \\operatorname{Ho}^\\Theta (Y)$ is an essentially surjective functor of categories.",
"We want to consider these notions for more general functors $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ , which we can accomplish via a localization functor.",
"Let us denote by $L_{\\operatorname{Se}}X$ the functorial localization of $X$ in the model structure $\\Theta _n\\mathcal {S}e\\mathcal {S}p$ .",
"Definition 6.3 Suppose $X, Y \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ .",
"A map $X \\rightarrow Y$ is a Dwyer-Kan equivalence if the associated map $L_{\\operatorname{Se}}X \\rightarrow L_{\\operatorname{Se}}Y$ is homotopically fully faithful and essentially surjective.",
"The following result is the analogue of Theorem REF .",
"Theorem 6.4 Let $X, Y \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ be $\\Theta _n$ -Segal spaces.",
"A map $X \\rightarrow Y$ is a Dwyer-Kan equivalence if and only if it is a weak equivalence in $\\Theta _n\\mathcal {CSS}$ .",
"Proving this theorem using the same strategy as the analogous result for complete Segal spaces [32] would be a daunting task, although of interest for the constructions that would need to be made along the way.",
"In that case, Rezk gives an explicit description of a fibrant replacement functor of a Segal space via a Dwyer-Kan equivalence.",
"The higher categorical version of this construction seems quite difficult to produce.",
"However, we can prove the above theorem more efficiently, using the fact that the analogous result is true in the context of complete Segal objects in $\\Theta _{n-1}\\mathcal {CSS}$ .",
"Consider the functor $d \\colon \\Delta \\times \\Theta _{n-1} \\rightarrow \\Theta _n$ , given by $([m],c) \\mapsto [m](c,\\ldots , c)$ .",
"It induces the functor $ \\begin{aligned}d^* \\colon \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}} & \\rightarrow \\mathcal {SS}ets^{\\Delta ^{\\operatorname{op}}\\times \\Theta _{n-1}^{\\operatorname{op}}} \\\\X & \\mapsto \\left(([m], c) \\mapsto X[m](c, \\ldots , c) \\right)\\end{aligned} $ which has a right adjoint $d_*$ given by right Kan extension.",
"Theorem 6.5 [11] The adjoint pair $(d^*, d_*)$ induces a Quillen equivalence $ d^* \\colon \\Theta _nSp\\rightleftarrows \\mathcal {CSS}(\\Theta _{n-1}Sp) \\colon d_*.",
"$ In these two equivalent model structures, we have respective notions of Dwyer-Kan equivalence.",
"The functor $d^*$ is well-behaved with respect to the two, in the following sense.",
"Proposition 6.6 A map $f \\colon X \\rightarrow Y$ of $\\Theta _n$ -Segal spaces is a Dwyer-Kan equivalence if and only if $d^*f \\colon d^*X \\rightarrow d^*Y$ is a Dwyer-Kan equivalence in $\\mathcal {CSS}(\\Theta _{n-1}Sp)$ .",
"We prove in [11] that the functor $d^*$ preserves fibrant objects; looking at the appropriate parts of that proof shows that it takes $\\Theta _n$ -Segal spaces to Segal objects in $\\Theta _{n-1}Sp$ .",
"Thus, since we have assumed that $X$ and $Y$ are fibrant, $d^*f \\colon d^*X \\rightarrow d^*Y$ is a Dwyer-Kan equivalence if and only if it is homotopically fully faithful and essentially surjective, without having to apply the localization functor $L_{\\operatorname{Se}}$ .",
"First, observe that $\\operatorname{Ho}^\\Theta (X) \\cong \\operatorname{Ho}^\\Delta (d^*X)$ , since both are defined in terms of the underlying Segal space of $X$ .",
"Therefore $\\operatorname{Ho}^\\Theta (X) \\rightarrow \\operatorname{Ho}^\\Delta (Y)$ is essentially surjective precisely when $\\operatorname{Ho}^\\Delta (d^*X) \\rightarrow \\operatorname{Ho}^\\Delta (d^*Y)$ is.",
"In [11] we prove that there is a natural isomorphism of mapping objects $ M^\\Theta _X(x_0, x_1)(c) \\cong M^\\Delta _{d^*X}(x_0,x_1)(c) $ for any $x_0, x_1 \\in X[0] = (d^*X)_0$ and $c \\in \\operatorname{ob}(\\Theta _{n-1})$ .",
"It follows that the map $ M^\\Theta _X(x_0, x_1)(c) \\rightarrow M^\\Theta _Y(fx_0,fx_1)(c) $ is a weak equivalence in $\\Theta _{n-1}Sp$ if and only if $ M^\\Delta _{d^*X}(x_0, x_1)(c) \\rightarrow M^\\Delta _{d^*Y}(fx_0,fx_1)(c) $ is.",
"By the previous proposition, we know that $f \\colon X \\rightarrow Y$ is a Dwyer-Kan equivalence in $\\Theta _nSp$ if and only if $d^*(f) \\colon d^*X \\rightarrow d^*Y$ is a Dwyer-Kan equivalence in $\\mathcal {CSS}(\\Theta _{n-1}Sp)$ .",
"Theorem REF says that the latter statement is true if and only if $d^*X \\rightarrow d^*Y$ is a weak equivalence in $\\mathcal {CSS}(\\Theta _{n-1}Sp)$ .",
"Thus, it suffices to prove that $f \\colon X \\rightarrow Y$ is a weak equivalence in $\\Theta _nSp$ if and only if $d^*f \\colon d^*X \\rightarrow d^*Y$ is a weak equivalence in $\\mathcal {CSS}(\\Theta _{n-1}Sp)$ .",
"First, suppose that $f$ is a weak equivalence in $\\Theta _nSp$ .",
"By definition of weak equivalences in a localized model structure, and using the fact that all objects are cofibrant in $\\mathcal {CSS}(\\Theta _{n-1}Sp)$ , it suffices to show that $ \\operatorname{Map}(d^*Y,Z) \\rightarrow \\operatorname{Map}(d^*X, Z) $ is a weak equivalence of simplicial sets for every complete Segal object $Z$ .",
"Using the adjunction $(d^*, d_*)$ , we can consider instead the map of simplicial sets $ \\operatorname{Map}(Y, d_*Z) \\rightarrow \\operatorname{Map}(X, d_*Z).",
"$ We proved in [11] that if $Z$ is a complete Segal object, then $d_*Z$ is a $\\Theta _n$ -space.",
"Since we assumed that $f \\colon X \\rightarrow Y$ is a weak equivalence in $\\Theta _nSp$ , this map is a weak equivalence of simplicial sets, as we needed to show.",
"Conversely, suppose that $d_*f$ is a weak equivalence in $\\mathcal {CSS}(\\Theta _{n-1}Sp)$ , so for any complete Segal object $Z$ , we have a weak equivalence of simplicial sets $ \\operatorname{Map}(d^*Y, Z) \\rightarrow \\operatorname{Map}(d^*X,Z).",
"$ We need to show that $ \\operatorname{Map}(Y,W) \\rightarrow \\operatorname{Map}(X, W) $ is a weak equivalence for any $\\Theta _n$ -space $W$ .",
"Again using the adjunction $(d^*, d_*)$ , it suffices to show that any $\\Theta _n$ -space $W$ can be obtained as $d_*Z$ for some complete Segal object $Z$ .",
"Again using the fact that $d_*$ takes complete Segal objects to $\\Theta _n$ -spaces, define $Z$ by $Z([1],c)= W[1](c)$ for any $c \\in \\operatorname{ob}(\\Theta _{n-1})$ ; the rest of the structure is thus determined by the Segal and completeness conditions.",
"In particular, $Z([0],c)=W[0]$ for any $c \\in \\operatorname{ob}(\\Theta _{n-1})$ .",
"Since the functor $d_*$ is defined via right Kan extension, and we are applying it to a Segal object, we obtain $ \\begin{aligned}(d_*Z)[1](c) & = \\lim _{[1](c) \\rightarrow [p](b, \\ldots b)} Z([p], b) \\\\& \\simeq \\lim _{[1](c) \\rightarrow [p](b, \\ldots b)} \\left(Z([1],b) \\times _{Z([1], b)} \\cdots \\times _{Z([0],b)} Z([1], b) \\right) \\\\& = \\lim _{[1](c) \\rightarrow [p](b, \\ldots b)} \\left(W[1](b) \\times _{W[0]} \\cdots \\times _{W[0]} W[1](b) \\right) \\\\& \\simeq \\lim _{[1](c) \\rightarrow [p](b, \\ldots b)} W[p](b,\\ldots , b) \\\\& = W[1](c).\\end{aligned} $ Thus $d_*Z=W$ , as we wished to show.",
"Now we can prove the following characterization of Dwyer-Kan equivalences between $\\Theta _n$ -spaces, which is a generalization of Theorem REF (REF ).",
"In the proof, we make use of the objects $[1]^{(i)}$ from Example REF .",
"Theorem 6.7 A map $f \\colon X \\rightarrow Y$ of $\\Theta _n$ -spaces is a Dwyer-Kan equivalence if and only if it is a levelwise weak equivalence.",
"First, observe that any levelwise weak equivalence is necessarily a Dwyer-Kan equivalence, so we need only prove the converse statement.",
"Suppose that $f \\colon X \\rightarrow Y$ is a Dwyer-Kan equivalence between $\\Theta _n$ -spaces.",
"Then for any $x,y \\in X[0]_0$ , we have that $ M^\\Theta _X(x,y) \\rightarrow M^\\Theta _Y(fx,fy) $ is a weak equivalence in $\\Theta _{n-1}Sp$ , and that the map $X[0] \\rightarrow Y[0]$ is an isomorphism on components.",
"Recall that $M^\\Theta _X(x,y)$ is a functor $\\Theta _{n-1}^{\\operatorname{op}} \\rightarrow \\mathcal {SS}ets$ , in fact a $\\Theta _{n-1}$ -space when $X$ is a $\\Theta _n$ -space, defined objectwise via pullbacks $ {M^\\Theta _X(x,y)(c) [r] [d] & X[1]([c]) [d]\\\\\\lbrace (x,y)\\rbrace [r] & X[0] \\times X[0]. }",
"$ In particular, we can understand $M^\\Theta _X(x,y)$ by evaluating at the objects $[1]^{(i)}$ for $0 \\le i \\le n$ , using the Segal conditions.",
"Furthermore, the completeness conditions give us weak equivalences $ X[1]^{(n-1)} \\simeq X[1]^{(n)}_{\\operatorname{heq}} $ for $0<i\\le n$ .",
"Thus, it suffices to prove that the maps $X[1]^{(n)} \\rightarrow Y[1]^{(n)}$ and $X[1]^{(n-1)} \\rightarrow Y[1]^{(n-1)}$ are weak equivalences of simplicial sets.",
"Restricting to homotopy equivalences, we take the pullback $ {hM^\\Theta _X(x,y)[1]^{(n-1)} [r] [d] & X[1]^{(n)}_{\\operatorname{heq}} [d] \\\\\\lbrace (x,y)\\rbrace [r] & X[0] \\times X[0],} $ which is a homotopy pullback since the right-hand map is a fibration, following from the Reedy fibrancy of $X$ .",
"Since weak equivalences are preserved by passing to subspaces of homotopy equivalences, by our assumption we know that $ hM^\\Theta _X(x,y)[1]^{(n-1)} \\simeq hM^\\Theta _Y(fx,fy)[1]^{(n-1)}.",
"$ If we precompose with the weak equivalence $X[1]^{(n-1)} \\rightarrow X[1]^{(n)}_{\\operatorname{heq}}$ and the corresponding map for $Y$ , we get a commutative square $ {X[1]^{(n-1)} [r] [d] & X[0] \\times X[0] [d] \\\\Y[1]^{(n-1)} [r] & Y[0] \\times Y[0].}",
"$ Since the fibers of the horizontal maps are weakly equivalent, we can conclude that this diagram is a homotopy pullback square.",
"By our assumption that $X[0] \\rightarrow Y[0]$ induces an isomorphism on components, it follows that the map $X[1]^{(n-1)} \\rightarrow Y[1]^{(n-1)}$ is a weak equivalence.",
"Finally, we consider the diagram $ {X[1]^{(n)} [r] [d] & X[0] \\times X[0] [d] \\\\Y[1]^{(n)} [r] & Y[0] \\times Y[0], } $ which is a pullback with horizontal maps fibrations, since $X$ and $Y$ are assumed to be Reedy fibrant.",
"Therefore, we obtain that the map $X[1]^{(n)} \\rightarrow Y[1]^{(n)}$ is a weak equivalence."
],
[
"Results on diagram categories with discreteness assumptions",
"Since our goal is to develop models for $(\\infty ,n)$ -categories using $\\Theta _n^{\\operatorname{op}}$ -diagrams with discreteness conditions, generalizing the Segal category model, we need to prove analogues of several results that were used to establish a model structure for Segal categories.",
"Especially because we want to develop several different variants (in which we require discreteness at some levels but completeness at others), it is convenient to prove more general results.",
"In this section we consider functors $\\mathcal {C} \\rightarrow \\mathcal {SS}ets$ for some small category $\\mathcal {C}$ , and require the images of some specified objects of $\\mathcal {C}$ to be discrete.",
"For specificity, we give examples throughout this section of how this theory can be applied to the case when $\\mathcal {C} = \\Theta _2^{\\operatorname{op}}$ .",
"Definition 7.1 Let $\\mathcal {C}$ be a small category and $T \\subseteq \\operatorname{ob}(\\mathcal {C})$ .",
"Let $X \\colon \\mathcal {C} \\rightarrow \\mathcal {SS}ets$ .",
"The $T$ -discretization of $X$ , denoted by $(X)_T$ , is the result of replacing the simplicial sets $X(t)$ by their discrete spaces of components for all $t \\in T$ .",
"Example 7.2 In the case of Segal categories, we considered functors $X \\colon \\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ for which $X_0$ is discrete.",
"So for $\\mathcal {C}= \\Delta ^{\\operatorname{op}}$ and $T=\\lbrace [0]\\rbrace $ , the $T$ -discretization is the functor which is denoted by $(-)_r$ in [8].",
"The proof of the following result is straightforward, using arguments like the ones in [8].",
"Proposition 7.3 Let $\\mathcal {SS}ets^{\\mathcal {C}}_{T}$ denote the category of functors $X \\colon \\mathcal {C} \\rightarrow \\mathcal {SS}ets$ with the property that $X(t)$ is discrete for every $t \\in T$ .",
"Then $T$ -discretization defines a right adjoint functor to the inclusion $\\mathcal {SS}ets^{\\mathcal {C}}_{T} \\rightarrow \\mathcal {SS}ets^\\mathcal {C}$ .",
"Given a subset $T$ of the objects of a small category $\\mathcal {C}$ , it is convenient to have notions of skeleton and coskeleton of a functor $X \\colon \\mathcal {C} \\rightarrow \\mathcal {SS}ets$ , given by the left and right adjoint, respectively, of a truncation functor which restricts to diagrams on the full subcategory of $\\mathcal {C}$ with objects in $T$ .",
"To that end, we make the following preliminary definition.",
"Definition 7.4 Let $\\mathcal {C}$ be a small category and $T$ a subset of $\\operatorname{ob}(\\mathcal {C})$ .",
"Let $\\mathcal {C}_T$ denote the full subcategory of $\\mathcal {C}$ whose objects are those in $T$ .",
"The $T$ -truncation $\\operatorname{tr}_T \\colon \\mathcal {SS}ets^\\mathcal {C} \\rightarrow \\mathcal {SS}ets^{\\mathcal {C}_T}$ is the functor induced by precomposition with the inclusion $\\mathcal {C}_T \\rightarrow \\mathcal {C}$ .",
"The category $\\mathcal {SS}ets$ is sufficiently well-behaved that we can invoke the theory of Kan extensions to get the following result.",
"Proposition 7.5 The functor $\\operatorname{tr}_T$ admits a left adjoint $s_T$ and a right adjoint $c_T$ .",
"The following definition allows us to think of these adjoints as functors $\\mathcal {SS}ets^\\mathcal {C} \\rightarrow \\mathcal {SS}ets^\\mathcal {C}$ .",
"Definition 7.6 Let $\\mathcal {C}$ be a small category, $X \\colon \\mathcal {C} \\rightarrow \\mathcal {SS}ets$ a functor, and $T \\subseteq \\operatorname{ob}(\\mathcal {C})$ .",
"The $T$ -skeleton of $X$ is $\\operatorname{sk}_T(X):= s_T \\circ \\operatorname{tr}_T(X)$ and the $T$ -coskeleton of $X$ is $\\operatorname{cosk}_T(X)=c_T \\circ \\operatorname{tr}_T(X)$ .",
"Proposition 7.7 The $T$ -skeleton and $T$ -coskeleton functors define an adjoint pair $ \\operatorname{sk}_T \\colon \\mathcal {SS}ets^\\mathcal {C} \\rightleftarrows \\mathcal {SS}ets^\\mathcal {C} \\colon \\operatorname{cosk}_T.",
"$ Using the above adjunction, for any $X, Y \\colon \\mathcal {C} \\rightarrow \\mathcal {SS}ets$ , we obtain natural isomorphisms $ \\begin{aligned}\\operatorname{Hom}_{\\mathcal {SS}ets^\\mathcal {C}}(\\operatorname{sk}_T(X), Y) & \\cong \\operatorname{Hom}_{\\mathcal {SS}ets^\\mathcal {C}}(s_T \\circ \\operatorname{tr}_T(X), Y) \\\\& \\cong \\operatorname{Hom}_{\\mathcal {SS}ets^{\\mathcal {C}_T}}(\\operatorname{tr}_T(X), \\operatorname{tr}_T(Y)) \\\\& \\cong \\operatorname{Hom}_{\\mathcal {SS}ets^\\mathcal {C}}(X, c_T \\circ \\operatorname{tr}_T(Y)) \\\\& \\cong \\operatorname{Hom}_{\\mathcal {SS}ets^\\mathcal {C}}(X, \\operatorname{cosk}_T(Y).\\end{aligned} $ Example 7.8 When $\\mathcal {C} = \\Delta ^{\\operatorname{op}}$ and $T=\\lbrace [k] \\mid 0 \\le k \\le m\\rbrace $ for some $m \\ge 0$ , then we recover the usual $m$ -skeleton $\\operatorname{sk}_m(X)$ and $m$ -coskeleton $\\operatorname{cosk}_m(X)$ of a simplicial space.",
"When $m=0$ , the 0-skeleton of $X$ is the constant simplicial space on the simplicial set $X_0$ , whereas the 0-coskeleton is given by $\\operatorname{cosk}_0(X)_k = (X_0)^{k+1}$ for each $k \\ge 0$ .",
"Example 7.9 Suppose that $\\mathcal {C} = \\Theta _2^{\\operatorname{op}}$ , and let us consider the coskeleta associated to subsets $T$ of $ S=\\lbrace [0], [1]([0]) \\rbrace \\subseteq \\operatorname{ob}(\\Theta _2^{\\operatorname{op}}).",
"$ We start with the case when $T$ is the subset consisting of the object $[0]$ ; we denote the associated coskeleton functor by $\\operatorname{cosk}_{[0]}$ .",
"Given a functor $X \\colon \\Theta _2^{\\operatorname{op}} \\rightarrow \\mathcal {SS}ets$ , we can use the fact that $\\Theta _2$ is built from $\\Delta $ in particular ways to describe $\\operatorname{cosk}_{[0]}(X)$ .",
"First, when we evaluate at any object of the form $[q]([0], \\ldots , [0])$ , we can use the description of the 0-coskeleton of a simplicial space to see that $ (\\operatorname{cosk}_{[0]}X)[q]([0], \\ldots , [0]) \\cong X[0]^{q+1}.",
"$ In particular, we have $ (\\operatorname{cosk}_{[0]}X)[1]([0]) \\cong X[0]^{2}.",
"$ Now, we can make use of the simplicial structure built into the objects $[1]([c])$ to observe that $ (\\operatorname{cosk}_{[0]}X)[1]([c]) \\cong X[0]^{2c+2}.",
"$ Generalizing to higher $q$ , one can check that $ (\\operatorname{cosk}_{[0]}X)[q]([c_1], \\ldots , [c_q]) \\cong \\left(X[0]^{q+1}\\right)^{c_1 + \\cdots + c_q +q}.",
"$ Now, let us consider instead the case when $T$ is the subset containing only the object $[1]([0])$ .",
"In this situation, the simplicial 0-coskeleton appears in the objects $[1]([c])$ , for any $c \\ge 0$ , in that $ (\\operatorname{cosk}_{[1]([0])}X)[1]([c]) \\cong X[1]([0])^{c+1}.",
"$ At the object $[0]$ , we must have $ (\\operatorname{cosk}_{[1]([0])}X)[0] \\cong \\Delta [0].",
"$ For the objects $[q]([0], \\ldots , [0])$ , we must get $ (\\operatorname{cosk}_{[1]([0])}X)[q]([0], \\ldots , [0]) \\cong X[1]([0])^q.",
"$ The rest of the structure can be deduced combinatorially.",
"Finally, we consider the coskeleton associated to $S$ itself.",
"Here, we get $ (\\operatorname{cosk}_S X)[0] = X[0] $ and $ (\\operatorname{cosk}_S X)[1]([0]) = X[1]([0]).",
"$ It is not hard to check that $ (\\operatorname{cosk}_S X)[q]([0], \\ldots , [0]) \\cong X[1]([0]) \\times _{X[0]} \\cdots \\times _{X[0]} X[1]([0]) $ and that $ (\\operatorname{cosk}_S X)[1]([c]) \\cong X[1]([0])^{c+1}.",
"$ We leave the descriptions upon evaluating at a general $[q]([c_1], \\ldots , [c_q])$ to the reader."
],
[
"Model structures for $\\Theta _n$ -models with discreteness assumptions",
"Now we turn to our question of having models for $(\\infty , n)$ -categories given by functors $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ that satisfy some discreteness conditions.",
"The first such condition we could ask for is on the level of objects, namely that a functor $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ have $X[0]$ a discrete simplicial set.",
"In other words, we want the simplicial set $X[1]([0])= X[1]^{(0)}$ to be discrete, but we also want to ask the same of other $X[1]^{(i)}$ for any $0 \\le i <n$ .",
"Let us apply the results of Section to the category $\\mathcal {C}=\\Theta _n^{\\operatorname{op}}$ and the set $ S= \\lbrace [1]^{(i)} \\mid 0 \\le i <n \\rbrace .",
"$ Definition 8.1 A $\\Theta _n$ -Segal precategory is a functor $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ such that $X[1]^{(i)}$ is discrete for all $[1]^{(i)}$ in $S$ .",
"It is a $\\Theta _n$ -Segal category if it is additionally a $\\Theta _n$ -Segal space.",
"Remark 8.2 Observe that this definition includes an assumption that a $\\Theta _n$ -Segal precategory is Reedy fibrant.",
"As discussed in Remark REF , this choice is less common for Segal categories and their generalizations, but it is convenient for us here, given what we want to prove.",
"As for Segal categories, the model structures that we develop here, with fibrant objects the Reedy fibrant $\\Theta _n$ -Segal precategories, have counterparts whose fibrant objects are projective fibrant instead.",
"We have chosen not to elaborate on this point in this paper, however.",
"However, we can also restrict ourselves only to the objects in some subset $T \\subseteq S$ ; indeed, we give our proofs in this generality.",
"In what follows, we assume that the set $S$ is fixed to be as defined above, but that $T$ is an arbitrary nonempty subset of $S$ ; we impose further conditions on $T$ momentarily.",
"Definition 8.3 Let $T \\subseteq S$ .",
"A $\\Theta _n$ -$T$ -Segal precategory is a functor $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ such that $X[1]^{(i)}$ is discrete for all $[1]^{(i)}$ in $T$ .",
"It is a $\\Theta _n$ -$T$ -Segal category if it is additionally a $\\Theta _n$ -Segal space that is complete for objects $[1]^{(i)}$ in $S \\setminus T$ , in the sense that the map $X[1]^{(i)} \\simeq X[1]^{(i+1)}_{\\operatorname{heq}}$ is a weak equivalence.",
"We denote the category of $\\Theta _n$ -$T$ -Segal precategories by $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_T$ .",
"Remark 8.4 In principle, this definition is sensible for any subset $T \\subseteq S$ .",
"However, we claim that for many choices of $T$ we recover the same objects.",
"Suppose that $[1]^{(i)}$ is an object of $T$ .",
"Then if $X$ is a $\\Theta _n$ -$T$ -Segal category, it follows from the definition that $X[1]^{(i)}$ is a discrete simplicial set, and hence $X[1]^{(i)}_{\\operatorname{heq}}$ must also be discrete.",
"Since we also assume that $X[1]^{(i-1)} \\simeq X[1]^{(i)}_{\\operatorname{heq}}$ , it follows that $X[1]^{(i-1)}$ is homotopy discrete.",
"Indeed, since this weak equivalence is given by a degeneracy map, $X[1]^{(i-1)}$ is a retract of the discrete space $X[1]^{(i)}_{\\operatorname{heq}}$ , hence also discrete.",
"Therefore, it suffices to consider $\\Theta _n$ -$T$ -Segal precategories for which discretization is made “from the bottom up\".",
"Consequently, in what follows, we assume that $T=\\lbrace [1]^{(0)}, \\ldots , [1]^{(j)} \\rbrace $ for some $0 \\le j \\le n-1$ .",
"A key feature that we want for our model structure is that the weak equivalences be the Dwyer-Kan equivalences.",
"However, to make sense of such maps for arbitrary diagrams, we need an appropriate localization functor taking a diagram to one which is a $\\Theta _n$ -$T$ -Segal category.",
"Yet, we need to verify that such a localization results in a $\\Theta _n$ -$T$ -Segal precategory rather than a more general diagram $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ , generalizing the argument for Segal categories in [8].",
"Proposition 8.5 If $X$ is a $\\Theta _n$ -$T$ -Segal precategory, then there exists a functor $L \\colon \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T} \\rightarrow \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T}$ such that $LX$ is a $\\Theta _n$ -$T$ Segal category that is weakly equivalent to $X$ in $\\Theta _n\\mathcal {S}e\\mathcal {S}p$ .",
"Let us first suppose that $T=S$ , so that we discretize at every level.",
"We want to modify the localization functor $L$ in $\\Theta _n\\mathcal {S}e\\mathcal {S}p$ in such a way that if $X[1]^{(i)}$ is discrete for some $i$ , then so is $(LX)[1]^{(i)}$ .",
"We can think of this original localization as occurring in two stages: first, it provides a Reedy fibrant replacement of an object $X$ , and then it gives an additional localization so that the resulting object satisfies the Segal conditions.",
"Let us first look at the Reedy fibrant replacement process.",
"Recall from Section that the generating acyclic cofibrations for the Reedy structure can be taken to be those of the form $ \\partial \\Theta [q](c_1, \\ldots , c_q) \\times \\Delta [m] \\cup \\Theta [q](c_1, \\ldots , c_q) \\times \\Lambda ^k[m] \\rightarrow \\Theta [q](c_1, \\ldots , c_q) \\times \\Delta [m] $ where $m \\ge 1$ , $0 \\le k \\le m$ , and $[q](c_1, \\ldots , c_q) \\in \\operatorname{ob}(\\Theta _n)$ .",
"Thus, a functorial Reedy fibrant replacement in $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_c$ is given by taking iterated pushouts along these maps.",
"However, when $[q](c_1, \\ldots , c_q) = [1]^{(i)}$ for some $0 \\le i < n$ , the resulting pushouts may not satisfy the required discreteness condition on $X[1]^{(i)}$ .",
"For example, if $q=0$ , taking a pushout of $X$ along such a map to get some $X^{\\prime }$ is effectively given by taking a pushout $ {\\Lambda ^k[m] [r] [d] & X[0] [d] \\\\\\Delta [m] [r] & X^{\\prime }[0]. }",
"$ Such a pushout need not be discrete.",
"On the other hand, if $X[0]$ is discrete, then the image of $\\Lambda ^k[m]$ is one of the points of $X[0]$ and therefore this map extends to a map $\\Delta [m] \\rightarrow X[0]$ .",
"In other words, $X$ already satisfies the desired lifting condition with respect to the generating acyclic cofibrations for which $q=0$ , so there is no harm in omitting these maps when taking the localization.",
"A similar argument works for any $[q](c_1, \\ldots , c_q) = [1]^{(i)}$ , so we obtain a Reedy fibrant replacement for $X$ by taking iterated pushouts along the maps $ {\\partial \\Theta [q](c_1, \\ldots , c_q) \\times \\Delta [m] \\cup \\Theta [q](c_1, \\ldots , c_q) \\times V[m,k] [d] \\\\\\Theta [q](c_1, \\ldots , c_q) \\times \\Delta [m]} $ where $m \\ge 1$ , $0 \\le k \\le m$ , and $[q](c_1, \\ldots , c_q) \\in \\operatorname{ob}(\\Theta _n)$ is not an object $[1]^{(i)}$ of $T$ .",
"Now, let us turn to the localization to obtain a Segal $\\Theta _n$ -space.",
"Let us first consider the maps $ G[q](c_1, \\ldots , c_q) \\rightarrow \\Theta [q](c_1, \\ldots , c_q) $ for any $q \\ge 0$ and $c_i \\in \\operatorname{ob}(\\Theta _{n-1})$ .",
"Since these maps are cofibrations between cofibrant objects in the Reedy model structure, and $X$ can now be assumed to be Reedy fibrant, we know that $X$ is local with respect to these maps precisely when a lift exists in any diagram $ {\\partial \\Delta [m] [r] [d] & \\operatorname{Map}(\\Theta [q](c_1, \\ldots , c_q), X) [d] \\\\\\Delta [m] [r] @{-->}[ur] & \\operatorname{Map}(G[q](c_1, \\ldots , c_q), X).}",
"$ The existence of such a lift is equivalent to the existence of a lift in the diagram $ {G[q](c_1, \\ldots , c_q) \\times \\Delta [m] \\cup \\Theta [q](c_1, \\ldots , c_q) \\times \\partial \\Delta [m] [r] [d] & X \\\\\\Theta [q](c_1, \\ldots , c_q) \\times \\Delta [m] @{-->}[ur] & .}",
"$ Thus, it is these maps on the left-hand side which we take pushouts along to obtain a local object.",
"There is potential concern when $q=0$ , but in that case these maps are the identity, since $G[0] =\\Delta [0]$ .",
"Therefore no problems arise when we take pushouts along such maps.",
"When $q=1$ , taking pushouts along these maps again has no effect because $G[1](c) = \\Theta [1](c)$ for any $c \\in \\operatorname{ob}(\\Theta _{n-1})$ .",
"The other maps used in the localization can be shown similarly to present no difficulties, since they are inductively defined, essentially again using the fact that $G[1](c) = \\Theta [1](c)$ for any $c$ .",
"By way of illustration, if $ G[r](d_1, \\ldots , d_r) \\rightarrow \\Theta [r](d_1, \\ldots , d_r) $ are the analogous maps in $\\mathcal {SS}ets^{\\Theta _{n-1}^{\\operatorname{op}}}$ , then we need to localize $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ with respect to the maps $ V[1](G[r](d_1, \\ldots , d_r)) \\rightarrow V[1](\\Theta [r](d_1, \\ldots , d_r)).",
"$ These maps are the identity upon evaluation at $[0]$ , so taking a pushout along them presents no problem at that level.",
"The argument that taking pushouts along these maps when $[r](d_1, \\ldots , d_r)= [1]^{(1)}= [1]([0])$ does not alter discreteness is essentially the same as the argument above for $q=0$ , and with a similar shift for the other $[1]^{(i)}$ .",
"Thus, we have that applying the described modification of Reedy fibrant replacement, followed by the usual $\\Theta _n$ -Segal localization, results in a $\\Theta _n$ -Segal category, as we wished to show.",
"Finally, we consider the case when $T \\ne S$ , so that $T=\\lbrace [1]^{(i)} \\mid 0 \\le i \\le j\\rbrace $ for some fixed $j<n$ .",
"Now, we need to localize further so that at the levels corresponding to elements of $S \\setminus T$ , $LX$ satisfies completeness.",
"Recalling the notation set up in the paragraph before Theorem REF , we define the set $ \\mathcal {T}^{\\prime }_{n,j} = \\operatorname{Cpt}_{\\Theta _n} \\cup V[1](\\operatorname{Cpt}_{\\Theta _{n-1}}) \\cup \\cdots \\cup V[1]^{n-j-1}(\\operatorname{Cpt}_{\\Theta _{j+1}}).",
"$ A similar argument as above shows that localizing with respect to these maps does not affect discreteness at the previously indicated levels, and results in an object with the required completeness conditions, namely, a $\\Theta _n$ -$T$ Segal category.",
"We use this result to make sense of Dwyer-Kan equivalences $X \\rightarrow Y$ between functors $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ that are required to be discrete at those objects $[1]^{(i)}$ in $T$ but that may not be Segal $\\Theta _n$ -spaces.",
"Definition 8.6 A map $X \\rightarrow Y$ in $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T}$ is a Dwyer-Kan equivalence if the map $LX \\rightarrow LY$ is fully faithful and essentially surjective, i.e., a Dwyer-Kan equivalence in the sense of Segal $\\Theta _n$ -spaces.",
"Theorem 8.7 There is a model structure $\\Theta _n\\mathcal {S}e\\mathcal {C}at_T$ on the category of $\\Theta _n$ -$T$ -Segal precategories in which: the weak equivalences are the Dwyer-Kan equivalences; the cofibrations are the monomorphisms; and the fibrant objects are the $\\Theta _n$ -$T$ -Segal categories that are complete at every object $[1]^{(i)}$ in $S \\setminus T$ .",
"The last condition means that a fibrant object $X$ has $X[1]^{(i)}$ discrete when $[1]^{(i)}$ is an object of $T$ , i.e., when $0 \\le i \\le j$ , but for $j>i$ we have that $X[1]^{(i)}$ is weakly equivalent to the space of homotopy equivalences in $X[1]^{(i+1)}$ , just as for $\\Theta _n$ -spaces.",
"Our proof follows the general strategy used to establish the model structure for Segal categories in [8].",
"Some of the proofs there are fairly formal and can be applied nearly identically, so we leave modifying them to our context here as an exercise for the reader.",
"We give proofs, however, for those results whose generalization is less clear, as well as some for which we have found more efficient proofs than the originals.",
"In [10] we proved an analogous result using a different method, and one could take the same approach here.",
"We have chosen former method here because it gives more explicit descriptions of some of the maps used.",
"The first step in proving this theorem is to find candidates for the generating cofibrations and generating acyclic cofibrations.",
"We apply the techniques of Section to modify the Reedy generating cofibrations so that they satisfy the necessary discreteness assumptions.",
"Using the notation there, we let $\\mathcal {C} = \\Theta _n^{\\operatorname{op}}$ and $ T = \\left\\lbrace [1]^{(i)} \\mid 0 \\le i \\le j \\right\\rbrace $ for some fixed $j<n$ .",
"We define a set $I^{n,T}$ of proposed generating cofibrations consisting of the maps $ {(\\partial \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q) \\cup \\Delta [m] \\times \\partial \\Theta [q](c_1, \\ldots , c_q))_T [d] \\\\(\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q))_T} $ for all $m, q \\ge 0$ and $(c_1, \\ldots , c_q) \\in \\operatorname{ob}(\\Theta _{n-1})^{q+1}$ , where $(-)_T$ is the discretization functor from Definition REF .",
"We do not expect such a nice description for the generating acyclic cofibrations, so in line with the approach we used for Segal categories we take $J^{n,T}$ to be the set of isomorphism classes of maps $A \\rightarrow B$ such that the map $A \\rightarrow B$ is a monomorphism and a Dwyer-Kan equivalence, and for all $m, q \\ge 0$ and $(c_1, \\ldots , c_q) \\in \\operatorname{ob}(\\Theta _{n-1})$ , the simplicial sets $A_{[q](c_1, \\ldots , c_q),m}$ and $B_{[q](c_1, \\ldots , c_q),m}$ have only countably many simplices.",
"Observe that $J^{n,T}$ does not at first glance seem to depend on the set $T$ , but the maps in this set are between objects that satisfy the required discreteness conditions at the objects of $T$ .",
"Thus, these sets are in fact different for varying $T$ .",
"The proof of the following result involves some additional techniques, so we defer it to Section .",
"Proposition 8.8 Maps with the right lifting property with respect to $I^{n,T}$ are precisely the maps that are both fibrations and weak equivalences.",
"We turn to some properties of the set $J^{n,T}$ .",
"Proposition 8.9 Any map that is both a cofibration and a weak equivalence can be written as a directed colimit of pushouts along maps in $J^{n,T}$ .",
"Pushouts along maps in $J^{n,T}$ are cofibrations and weak equivalences.",
"Any $J^{n,T}$ -cofibration is an $I^{n,T}$ -cofibration and a weak equivalence.",
"The proof of (REF ) is technical but follows the same line of argument as [8], so we do not repeat it here.",
"To prove (REF ), first notice that any $j \\colon A \\rightarrow B$ in $J^{n,T}$ is an acyclic cofibration in $\\Theta _nSp$ .",
"Since pushouts along maps in $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T}$ preserve the required discreteness conditions, the resulting map is still in $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T}$ .",
"Furthermore, it is an acyclic cofibration in $\\Theta _nSp$ , so a monomorphism which is a Dwyer-Kan equivalence, as we wished to show.",
"Finally, we prove (REF ).",
"By definition and (REF ), a $J^{n,T}$ -cofibration is a map with the left lifting property with respect to the fibrations.",
"Similarly, using Proposition REF , an $I^{n,T}$ -cofibration is a map with the left lifting property with respect to the fibrations which are Dwyer-Kan equivalences.",
"Thus, we need to show that a map with the left lifting property with respect to the fibrations has the left lifting property with respect to the fibrations that are weak equivalences, which is automatic, and is a weak equivalence.",
"Let $f \\colon A \\rightarrow B$ be such a map.",
"By (REF ), a pushout along maps of $J^{n,T}$ is an acyclic cofibration.",
"Therefore, we can use the small object argument to factor $f \\colon A \\rightarrow B$ as $A \\overset{\\simeq }{\\hookrightarrow } A^{\\prime } \\twoheadrightarrow B$ , so there exists a lift in the diagram $ {A [r]^{\\simeq } [d] & A^{\\prime } [d] \\\\B [r]^{\\operatorname{id}} @{-->}[ur] & B.}",
"$ Therefore, $A \\rightarrow B$ is a retract of $A \\rightarrow A^{\\prime }$ and hence a weak equivalence.",
"We now have all the ingredients we need to establish the model structure.",
"We apply the conditions of Theorem REF .",
"The category of $\\Theta _n$ -$T$ -Segal precategories has all small limits and colimits, since they are taken levelwise and therefore do not affect the discreteness assumptions.",
"Similarly, Dwyer-Kan equivalences satisfy the two-out-of-three property and are closed under retracts.",
"The set $I^{n,T}$ permits the small object argument because the generating cofibrations in the Reedy model structure do, and applying the discretization functor does not affect this property.",
"The objects $A$ that appear as the sources of the maps in $J^{n,T}$ are small using the same argument as for Segal categories [8], so the set $J^{n,T}$ permits the small object argument.",
"Thus, condition (REF ) is satisfied.",
"Condition (REF ) is precisely the statement of Proposition REF (REF ).",
"Condition (REF ) and condition (REF )(b) are together precisely the statement of Proposition REF .",
"Finally, we conclude with establishing some additional structure that this model structure possesses.",
"Proposition 8.10 The model structure $\\Theta _n\\mathcal {S}e\\mathcal {C}at_T$ is simplicial and cartesian.",
"We give the proof that the model structure is cartesian; the proof that it is simplicial can be proved similiarly, or using an argument like the one used for Segal categories in [5].",
"We know that every object in $\\Theta _n\\mathcal {S}e\\mathcal {C}at_T$ , in particular the terminal object, is cofibrant, so it remains to check the other two conditions of Definition REF .",
"Consider the category $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ with the model structure whose fibrant objects satisfy the Segal conditions at all levels and completeness conditions at each object $[1]^{(i)}$ in $S \\setminus T$ .",
"This model category, which we denote by $\\mathcal {M}$ here for simplicity, is cartesian, using [31].",
"Note that every weak equivalence in $\\Theta _n\\mathcal {S}e\\mathcal {C}at_T$ is also a weak equivalence in $\\mathcal {M}$ , and similarly for cofibrations, so we can use this structure on $\\mathcal {M}$ to establish the conditions we need.",
"To show that the underlying category $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_T$ is cartesian closed, note that if $X$ and $Y$ are discrete at some level $[1]^{(i)}$ , then so is their product $X \\times Y$ .",
"We claim the same is true of the mapping object $Y^X$ , which is defined by $ (Y^X)_{[m](c_1, \\ldots , c_m)} = \\operatorname{Map}(X \\times \\Theta [m](c_1, \\ldots , c_m), Y).",
"$ A straightforward computation shows that if $X_{[1]^{(i)}}$ and $Y_{[1]^{(i)}}$ are both discrete, then so is $(Y^X)_{[1]^{(i)}}$ .",
"The required compatibility between the cartesian product and mapping object follows because it holds in $\\mathcal {M}$ .",
"Similarly, suppose that $f \\colon X \\rightarrow X^{\\prime }$ and $g \\colon Y \\rightarrow Y^{\\prime }$ are cofibrations in $\\Theta _n\\mathcal {S}e\\mathcal {C}at_T$ , and in particular discrete at every $[1]^{(i)}$ in $T$ .",
"Then the pushout-corner map is again a monomorphism that is discrete at the same levels.",
"Using left properness, which follows since all objects are cofibrant, and the two-out-of-three property, one can check that this map is a weak equivalence if either $f$ or $g$ is.",
"Remark 8.11 When $n=1$ , the previous proposition recovers the result of Simpson that the model structure for Segal categories is cartesian [33]."
],
[
"Comparison of models",
"In this section, we establish Quillen equivalences between the different model structures from the previous section, for varying $T$ , and with $\\Theta _nSp$ .",
"The strategy of proof is very similar to the comparison between the model structures for Segal categories and complete Segal spaces in [8].",
"In this section, let $T_j = \\lbrace [1]^{(i)} \\mid 0 \\le i \\le j \\rbrace $ for a fixed $j <n$ .",
"We want to prove that the inclusion $I$ of the category of $\\Theta _n$ -$T_j$ -Segal precategories into the category of $\\Theta _n$ -$T_{j-1}$ -Segal precategories has a right adjoint, and further that this adjoint pair induces a Quillen equivalence between the model structures for $\\Theta _n$ -$T_j$ -Segal categories and $\\Theta _n$ -$T_{j-1}$ -Segal categories.",
"In other words, for that value of $j$ , we drop the assumption that $X([1]^{(j)})$ be discrete, but ask instead for the corresponding completeness condition.",
"If $j=0$ , then we take $T_{-1} = \\varnothing $ , in which case we get the comparison with $\\Theta _n$ -spaces, for which no spaces in the diagram are required to be discrete.",
"In light of Remark REF , we thus obtain all the comparisons we are interested in.",
"Let $W$ be an object of $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T_{j-1}}$ .",
"Consider the objects $U=\\operatorname{cosk}_{[1]^{(j)}}(W)$ and $V= \\operatorname{cosk}^0_{[1]^{(j)}}(W)$ , where the latter is defined to be the coskeleton of the discrete functor $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ given by $[q](c_1, \\ldots , c_q) \\mapsto W[q](c_1, \\ldots , c_q)_0$ .",
"Define $RW$ to be the pullback in the diagram $ {RW [r] [d] & V [d] \\\\W [r] & U.}",
"$ Note that $RW$ is a $\\Theta _n$ -$T_j$ -Segal precategory, since the effect of this construction is to discretize $W$ at the object $[1]^{(j)}$ .",
"Observe that this construction defines a functor $ R \\colon \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T_{j-1}} \\rightarrow \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T_j}.",
"$ Now, we want to prove the higher analogue of Theorem REF , that this functor $R$ is the right adjoint of a Quillen equivalence.",
"Verifying the following result is a straightforward generalization of the argument used to prove [8].",
"Proposition 9.1 The functor $R \\colon \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T_{j-1}} \\rightarrow \\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}_{T_j}$ is right adjoint to the inclusion $I$ .",
"Now, we need to show that this adjoint pair respects the model structures of interest.",
"Theorem 9.2 The adjoint pair $ @1{I \\colon \\Theta _n\\mathcal {S}e\\mathcal {C}at_{T_j} @<.5ex>[r] & \\Theta _n\\mathcal {S}e\\mathcal {C}at_{T_{j-1}} \\colon R @<.5ex>[l]} $ is a Quillen equivalence.",
"To show that $(I,R)$ is a Quillen pair, we to show that the inclusion map $I$ preserves cofibrations and acyclic cofibrations.",
"It preserves cofibrations because they are precisely the monomorphisms in each model structure; it preserves all weak equivalences, and in particular the acyclic cofibrations, since a map is a weak equivalence in either model structure if and only if it is a Dwyer-Kan equivalence.",
"To show that this Quillen pair is a Quillen equivalence, we need to show that $I$ reflects weak equivalences between cofibrant objects and that for any $\\Theta _n$ -$T_{j-1}$ -Segal category $W$ , the map $I((RW)^c) = IRW \\rightarrow W$ is a weak equivalence in $\\Theta _n\\mathcal {S}e\\mathcal {C}at_{T_j}$ .",
"The fact that $I$ reflects weak equivalences between cofibrant objects follows from the fact that the weak equivalences in each model structure are precisely the Dwyer-Kan equivalences.",
"It remains to show that the map $RW \\rightarrow W$ in the pullback diagram $ {RW [r] [d]_j & V [d] \\\\W [r] & U} $ is a Dwyer-Kan equivalence.",
"By the definition of $RW$ , we have $(RW)[0]_0=W[0]_0$ , so induced map $\\operatorname{Ho}^\\Theta (RW) \\rightarrow \\operatorname{Ho}^\\Theta (W)$ is a bijection on objects, hence essentially surjective.",
"Finally, we need to show that, for any $x,y \\in W[0]_0 = (RW)[0]_0$ , the map $ M^\\Theta _{RW}(x,y) \\rightarrow M^\\Theta _W(x,y) $ is a weak equivalence in $\\Theta _{n-1}\\mathcal {CSS}$ .",
"We claim that it is in fact a levelwise weak equivalence of functors $\\Theta _{n-1}^{\\operatorname{op}} \\rightarrow \\mathcal {SS}ets$ .",
"Since $W$ is assumed to be fibrant, it satisfies the Segal conditions; it follows from the pullback defining it that $RW$ does as well.",
"Thus, it suffices to verify that the map $ M_{RW}^\\Theta (x,y)[1]^{(i)} \\rightarrow M_W^\\Theta (x,y)[1]^{(i)} $ is a weak equivalence of simplicial sets for any $0 \\le i <n-1$ .",
"We hence consider the diagram of homotopy fibers $ {M_{RW}^\\Theta (x,y)[1]^{(i)} [r] [d] & (RW)[1]^{(i)} [r] [d] & (RW)[0] \\times (RW)[0] [d]^= \\\\M_W^\\Theta (x,y)[1]^{(i)} [r] & W[1]^{(i)} [r] & W[0] \\times W[0].",
"}$ First, let us consider $T_0=\\lbrace [0]\\rbrace $ and $T_{-1}=\\varnothing $ .",
"Then $(RW)[1]^{(i)}$ and $W[1]^{(i)}$ differ only in that the former contains degeneracies of the higher-dimensional simplices of $W[0]$ .",
"Since we are taking the homotopy fiber over a 0-simplex of $W[0] \\times W[0]$ , however, these degeneracies do not appear in that homotopy fiber.",
"It follows that the middle vertical map of (REF ) is a weak equivalence, hence also the left-hand vertical map.",
"Now consider $T_j$ for $j \\ge 1$ .",
"Then $W[1]^{(i)}$ is already discrete for each $0 \\le i <j$ , and $(RW)[1]^{(i)} =W[1]^{(i)}$ for these values of $i$ .",
"When $i=j$ , the middle vertical map of (REF ) is given by the inclusion of the discrete subspace $W[1]^{(j)}_0 \\rightarrow W[1]^{(j)}$ , and an argument similar to the one for $T_0$ shows that the induced map on homotopy fibers is a weak equivalence.",
"Finally, when $i>j$ , the middle vertical map of (REF ) is given by the inclusion of a subspace that does not include higher degenerate elements coming from $W[1]^{(j)}$ .",
"It follows that this map is a weak equivalence, hence the left-hand vertical map in (REF ) is also, as we needed to show."
],
[
"Proof of Proposition ",
"In this section, we complete the proof of Proposition REF , which tells us that maps with the right lifting property with respect to $I^{n,T}$ are precisely the maps that are both fibrations and Dwyer-Kan equivalences.",
"Because the two implications are proved quite differently, for clarity we separate them into the following two propositions.",
"Proposition 10.1 If $f \\colon X \\rightarrow Y$ is a map of $\\Theta _n$ -$T$ -Segal precategories with the right lifting property with respect to the maps in $I^{n,T}$ , then it is a fibration and a Dwyer-Kan equivalence.",
"Proposition 10.2 If $f \\colon X \\rightarrow Y$ is a map of $\\Theta _n$ -$T$ -Segal precategories that is both a fibration and a Dwyer-Kan equivalence, then it has the right lifting property with respect to the maps in $I^{n,T}$ .",
"The more difficult of the two is Proposition REF , which requires several preparatory lemmas.",
"For the beginning steps we work incrementally, starting with small values of $n$ , to build intuition for the complicated notation we must inevitably use.",
"Before delving into the details, recall that the definition of Dwyer-Kan equivalence is given in terms of mapping objects in a $\\Theta _n$ -space.",
"For the arguments we make in this section, we want to reformulate this definition somewhat.",
"Given $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ , $c \\in \\operatorname{ob}(\\Theta _{n-1})$ , and $(v_0, v_1) \\in X[0]_0 \\times X[0]_0$ , let $X[1](c)(v_0, v_1)$ be the fiber of the map $X[1](c) \\rightarrow X[0] \\times X[0]$ over $(v_0, v_1)$ .",
"Then it is straightforward to check that $ X[1](c)(v_0, v_1) = M_X^\\Theta (v_0, v_1)(c).",
"$ We use this notation, rather than the mapping object notation, for the remainder of this section.",
"We want to understand the behavior of maps with the right lifting property with respect to maps in $I^{n,T}$ .",
"However, it is easier to get a handle on maps with the right lifting property with respect to the maps in a related set that we denote by $I^{n,T}_f$ , so we first consider these maps, which we develop in some detail.",
"Given any object $[1]^{(i)}$ in $T$ , we can evaluate any representable functor $\\Theta [q](c_1, \\ldots , c_q)$ at $[1]^{(i)}$ to obtain a set that we think of as a doubly constant functor $\\Theta _n^{\\operatorname{op}}\\times \\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {S}ets$ , and we denote it by $\\Theta [q](c_1, \\ldots , c_q)_{[1]^{(i)}}$ .",
"When $i=0$ , we have $\\Theta [q](c_1, \\ldots , c_q)_{[0]} = \\operatorname{Hom}([0], [q](c_1, \\ldots , c_q))$ , which is the set consisting of $q+1$ elements.",
"For any $m \\ge 0$ , object $[q](c_1, \\ldots , c_q)$ of $\\Theta _n$ , and element $[1]^{(i)}$ in $T$ , we have the projection and inclusion $ \\Theta [q](c_1, \\ldots , c_q)_{[1]^{(i)}} \\leftarrow \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q)_{[1]^{(i)}} \\rightarrow \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q).",
"$ Keeping $m$ and $[q](c_1, \\ldots , c_q)$ fixed but varying over all $[1]^{(i)}$ in $T$ , take the diagram given by all such maps and denote its colimit by $Q^{n,T}_{m, \\underline{c}}$ .",
"Denote the colimit of the analogous diagram with $\\Delta [m]$ replaced by $\\partial \\Delta [m]$ by $P^{n,T}_{m, \\underline{c}}$ .",
"There are natural maps $P^{n,T}_{m, \\underline{c}} \\rightarrow Q^{n,T}_{m, \\underline{c}}$ , and it is this collection of maps we want to consider.",
"Specifically, define $ I^{n,T}_{f} = \\lbrace P^{n,T}_{m, \\underline{c}} \\rightarrow Q^{n,T}_{m, \\underline{c}} \\mid m \\ge 0, [q](c_1, \\ldots , c_q) \\in \\operatorname{ob}(\\Theta _n) \\rbrace .",
"$ Remark 10.3 This set of maps can be regarded as a set of generating cofibrations for a model structure for $\\Theta _n$ -$T$ -Segal precategories that is more closely related to the projective model structure.",
"Indeed, these maps are designed to be an appropriate discretization of the generating cofibrations of the projective model structure on $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ .",
"The subscript $f$ is meant to be suggestive of this fact, even though we have removed the corresponding $c$ subscript from the corresponding injective or Reedy version.",
"We refer the reader to [8] for a more detailed motivation for these kinds of maps in the case of Segal categories.",
"Our first step is to obtain a good description of maps $P_{m,\\underline{c}}^{n,T} \\rightarrow X$ and $Q_{m,\\underline{c}}^{n,T} \\rightarrow X$ for general $X$ .",
"Let us first recall the the case where $n=1$ , namely Segal categories, which was treated in [8].",
"With a view toward generalization to higher $n$ , and recalling that $\\Theta _1=\\Delta $ , we denote the representable functor $\\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {S}ets$ on the object $[q]$ by $\\Theta [q]$ , rather than $\\Delta [q]$ .",
"Here, we regard it as a discrete functor $\\Delta ^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ ; we simiarly treat the representable simplicial set $\\Delta [m]$ as a constant functor $\\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ .",
"In this case, there is only one object whose image we discretize, namely $[0]$ , so there is no need to consider different choices of subsets $T$ .",
"We thus simplify the notation for the moment and simply write $P_{m,q}$ and $Q_{m,q}$ .",
"Let us recall some notation.",
"If $X$ is a Segal precategory and $(v_0, \\ldots , v_q) \\in X_0^{q+1}$ , let $X_q(v_0, \\ldots , v_q)$ denote the fiber of the natural map $X_q \\rightarrow X_0^{q+1}$ given by iterated face maps.",
"The following lemma was proved in [8]; we sketch a proof here for the purposes of guiding our generalizations of it.",
"Lemma 10.4 When $n=1$ , for fixed $m, q \\ge 0$ and Segal precategory $X$ , there are isomorphisms $ \\operatorname{Hom}(P_{m,q}, X) \\cong \\coprod _{(v_0, \\ldots , v_q)} \\operatorname{Hom}(\\partial \\Delta [m], X_q(v_0, \\ldots , v_q)) $ and $ \\operatorname{Hom}(Q_{m,q}, X) \\cong \\coprod _{(v_0, \\ldots , v_q)} \\operatorname{Hom}(\\Delta [m], X_q(v_0, \\ldots , v_q)), $ where $(v_0, \\ldots , v_q) \\in X[0]_0^{q+1}$ .",
"We summarize the argument for $Q_{m,q}$ ; the one for $P_{m,q}$ is similar.",
"We have defined $Q_{m,q}$ as the pushout in the diagram $ {\\Delta [m] \\times \\Theta [q]_0 [r] [d] & \\Delta [m] \\times \\Theta [q]_0 [d] \\\\\\Theta [q]_0 [r] & Q_{m,q}. }",
"$ Applying the functor $\\operatorname{Hom}(-,X)$ , we obtain a pullback diagram of sets $ {\\operatorname{Hom}(Q_{m,q}, X) [r] [d] & \\operatorname{Hom}(\\Delta [m] \\times \\Theta [q], X) = X[q]_m [d] \\\\X[0]_0^{q+1} = \\operatorname{Hom}(\\Theta [q]_0, X) [r] & \\operatorname{Hom}(\\Delta [m] \\times \\Theta [q]_0, X) = X[0]_m^{q+1}.}",
"$ But, this pullback can also be described as $ \\coprod _{(v_0, \\ldots , v_q)} \\operatorname{Hom}(\\Delta [m], X_q(v_0, \\ldots , v_q)).",
"$ Now, we want to generalize this argument.",
"Not surprisingly, the combinatorics get quite complicated as we require that more levels of $X$ be discrete.",
"Let us start with $n=2$ and the set $S=\\lbrace [0], [1]([0])\\rbrace $ before attempting a full generalization to higher $n$ and proper subsets of $S$ .",
"For a fixed $m \\ge 0$ and $[q](c_1, \\ldots c_q)$ , we have defined $Q_{m, \\underline{c}}^{2,S}$ to be the colimit of the diagram $ {& \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q) & \\\\\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q)_{[0]} [ur] [d] && \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q)_{[1]([0])} [ul] [d] \\\\\\Theta [q](c_1, \\ldots , c_q)_{[0]} && \\Theta [q](c_1, \\ldots , c_q)_{[1]([0])}.",
"}$ If we focus on the two arrows on the left-hand side of this diagram, and take the pushout thereof, the situation is very similar to the one from the $n=1$ case.",
"Namely, if we apply the functor $\\operatorname{Hom}(-,X)$ to these two arrows, we get a diagram $ {& \\operatorname{Hom}(\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q), X) [d] \\\\\\operatorname{Hom}(\\Theta [q](c_1, \\ldots , c_q)_{[0]}, X) [r] & \\operatorname{Hom}(\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q)_{[0]}, X) } $ that can be rewritten as $ { & X[q](c_1, \\ldots , c_q)_m [d] \\\\X[0]^{q+1}_0 [r] & X[0]^{q+1}_m.}",
"$ The pullback of this diagram is given by $ \\coprod _{(v_0, \\ldots , v_q)} X[q](c_1, \\ldots , c_q)(v_0, \\ldots , v_q)_m, $ where similarly to above, $X[q](c_1, \\ldots , c_q)(v_0, \\ldots , v_q)$ is the fiber of the map $X[q](c_1, \\ldots , c_q) \\rightarrow X_0^{q+1}$ .",
"This pullback is isomorphic to $ \\coprod _{(v_0, \\ldots , v_q)} \\operatorname{Hom}(\\Delta [m], X[q](c_1, \\ldots , c_q)(v_0, \\ldots , v_q)).",
"$ Now, let us treat the pushout of the two right-hand arrows of (REF ) analogously.",
"We first apply the functor $\\operatorname{Hom}(-,X)$ and then describe the pullback of the resulting diagram.",
"So, our first step is to describe the objects of $ {& \\operatorname{Hom}(\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q), X) [d] \\\\\\operatorname{Hom}(\\Theta [q](c_1, \\ldots , c_q)_{[1]([0])}, X) [r] & \\operatorname{Hom}(\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q)_{[1]([0])}, X).}",
"$ Looking at the bottom row, these sets should be described as some coproduct of copies of $X[1]([0])_0$ and $X[1]([0])_m$ , respectively, indexed by maps $[1]([0]) \\rightarrow [q](c_1, \\ldots , c_q)$ in $\\Theta _2$ .",
"We denote by $\\theta (1,\\underline{c})$ the number of such maps.",
"Thus, we can describe this diagram instead as $ { & X[q](c_1, \\ldots , c_q)_m [d] \\\\X[1]([0])_0^{\\theta (1,\\underline{c})} [r] & X[1]([0])_m^{\\theta (1,\\underline{c})}} $ whose pullback is $ \\coprod _{(w_1, \\ldots , w_{\\theta (1,\\underline{c})})} X[q](c_1, \\ldots , c_q)(w_0, \\ldots , w_{\\theta (1,\\underline{c})})_m.",
"$ This pullback can be written alternatively as $ \\coprod _{(w_1, \\ldots , w_{\\theta (1,\\underline{c})})} \\operatorname{Hom}(\\Delta [m],X[q](c_1, \\ldots , c_q)(w_1, \\ldots , w_{\\theta (1,\\underline{c})})).",
"$ Now, if we want to take the colimit of the diagram (REF ), then we can merge these two descriptions to see that $\\operatorname{Hom}(Q_{m,\\underline{c}}^{2,S}, X)$ is isomorphic to $ \\coprod _{(v_0, \\ldots , v_q)} \\coprod _{(w_1, \\ldots , w_{\\theta (q, \\underline{c})})} \\operatorname{Hom}(\\Delta [m], X[q](c_1, \\ldots , c_q)(v_0, \\ldots , v_q)(w_1, \\ldots , w_{\\theta (1,\\underline{c})})).",
"$ We summarize these findings, and the analogous result for $P_{m, \\underline{c}}^{2,S}$ , as well as generalizations for $T \\subseteq S$ , in the following lemma.",
"For notational simplicity, we write $\\underline{v}=(v_0, \\ldots , v_q)$ and $\\underline{w}= (w_1, \\ldots , w_{\\theta (1,\\underline{c})})$ .",
"Lemma 10.6 When $n=2$ , for a fixed $m \\ge 0$ , object $[q](c_1, \\ldots , c_q)$ of $\\Theta _2$ , and functor $X \\colon \\Theta _2^{\\operatorname{op}} \\rightarrow \\mathcal {SS}ets$ , there are natural isomorphisms $ \\operatorname{Hom}(P_{m,\\underline{c}}^{2,S}, X) \\cong \\coprod _{\\underline{v}} \\coprod _{\\underline{w}} \\operatorname{Hom}\\left(\\partial \\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v})(\\underline{w}) \\right) $ and $ \\operatorname{Hom}(Q_{m,\\underline{c}}^{2,S}, X) \\cong \\coprod _{\\underline{v}} \\coprod _{\\underline{w}} \\operatorname{Hom}\\left( \\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v})(\\underline{w}) \\right).",
"$ Discretizing only at $T=\\lbrace [0]\\rbrace $ , we have $ \\operatorname{Hom}(P_{m,\\underline{c}}^{2,T}, X) \\cong \\coprod _{\\underline{v}} \\operatorname{Hom}\\left(\\partial \\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v}) \\right) $ and $ \\operatorname{Hom}(Q_{m,\\underline{c}}^{2,T}, X) \\cong \\coprod _{\\underline{v}} \\operatorname{Hom}\\left( \\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v}) \\right).",
"$ Now let us consider the general case of $Q_{m,\\underline{c}}^{n,T}$ for general $n$ .",
"To this end, let us denote the number of maps $[1]^{(i)} \\rightarrow [q](c_1, \\ldots , c_q)$ by $\\theta (i,\\underline{c})$ .",
"We further denote an element of the set $(X[1]^{(i)}_0)^{\\theta (i,\\underline{c})}$ by $\\underline{v}^{(i)}=(v_1^{(i)}, \\ldots , v_{\\theta (i,\\underline{c})}^{(i)})$ .",
"The argument above generalizes to give a proof of the following lemma.",
"Lemma 10.7 Let $X \\colon \\Theta _n^{\\operatorname{op}}\\rightarrow \\mathcal {SS}ets$ , and let $T=\\lbrace [1]^{(i)} \\mid 0 \\le i \\le j \\rbrace \\subseteq S$ .",
"There are natural isomorphisms $ \\operatorname{Hom}(P_{m,\\underline{c}}^{n,T}, X) \\cong \\coprod _{\\underline{v}^{(0)}} \\cdots \\coprod _{\\underline{v}^{(j)}} \\operatorname{Hom}\\left(\\partial \\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)})\\cdots (\\underline{v}^{(j)})\\right), $ and $ \\operatorname{Hom}(Q_{m,\\underline{c}}^{n,T}, X) \\cong \\coprod _{\\underline{v}^{(0)}} \\cdots \\coprod _{\\underline{v}^{(j)}} \\operatorname{Hom}\\left(\\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)})\\cdots (\\underline{v}^{(j)})\\right).",
"$ Now, for the general case, for any $n$ and $T$ , consider the set $ I_f^{n,T} = \\left\\lbrace P_{m,\\underline{c}}^{n,T} \\rightarrow Q_{m,\\underline{c}}^{n,T} \\mid m \\ge 0, [q](c_1, \\ldots , c_q) \\in \\operatorname{ob}(\\Theta _n) \\right\\rbrace .",
"$ The following result is the main technical point we need to prove Proposition REF .",
"Lemma 10.8 Let $T=\\lbrace [1]^{(i)} \\mid 0 \\le i \\le j \\rbrace \\subseteq S$ , and suppose that $f \\colon X \\rightarrow Y$ is a map of $\\Theta _n$ -$T$ -Segal precategories with the right lifting property with respect to the maps in $I_f^{n,T}$ .",
"Then the map $X[0] \\rightarrow Y[0]$ is surjective and each map $ X[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)}) \\cdots (\\underline{v}^{(j)}) \\rightarrow Y[q](c_1, \\ldots , c_q)(f\\underline{v}^{(0)}) \\cdots (f\\underline{v}^{(j)}) $ is an acyclic fibration of simplicial sets for any object $[q](c_1, \\ldots , c_q)$ of $\\Theta _n$ and every choice of $\\underline{v}^{(i)} \\in (X[1]^{(i)}_0)^{\\theta (i,\\underline{c})}$ for each $0 \\le i \\le j$ .",
"The fact that $X[0] \\rightarrow Y[0]$ is surjective follows from the fact that $f$ has the right lifting property with respect to the map $\\varnothing \\rightarrow \\Theta [0]$ .",
"Thus, we need to show that there is a lift in any diagram $ {\\partial \\Delta [m] [r] [d] & X[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)}) \\cdots (\\underline{v}^{(j)}) [d] \\\\\\Delta [m] [r] @{-->}[ur] & Y[q](c_1, \\ldots , c_q)(f\\underline{v}^{(0)}) \\cdots (f\\underline{v}^{(j)}). }",
"$ By assumption, we know there exist lifts for diagrams $ {P_{m,\\underline{c}}^{n,T} [r] [d] & X [d] \\\\Q_{m,\\underline{c}}^{n,T} [r] @{-->}[ur] & Y.}",
"$ Equivalently, in the diagram $ {\\operatorname{Hom}(Q_{m,\\underline{c}}^{n,T}, X) [r] & P [r] [d] & \\operatorname{Hom}(P_{m,\\underline{c}}^{n,T}, X) [d] \\\\& \\operatorname{Hom}(Q_{m,\\underline{c}}^{n,T} Y) [r] & \\operatorname{Hom}(P_{m,\\underline{c}}^{n,T}, Y),} $ where $P$ denotes the pullback of the right-hand square, the top left-hand map is surjective.",
"Using Lemma REF , we can write $P$ as the pullback of the diagram $ {\\coprod _{\\underline{v}^{(0)}} \\cdots \\coprod _{\\underline{v}^{(j)}} \\operatorname{Hom}\\left(\\Delta [m], Y[q](c_1, \\ldots , c_q)(f\\underline{v}^{(0)}) \\cdots (f\\underline{v}^{(j)})\\right) [d] \\\\\\coprod _{\\underline{v}^{(0)}} \\cdots \\coprod _{\\underline{v}^{(j)}} \\operatorname{Hom}\\left(\\partial \\Delta [m], Y[q](c_1, \\ldots , c_q)(f\\underline{v}^{(0)}) \\cdots (f\\underline{v}^{(j)}) \\right) \\\\\\coprod _{\\underline{v}^{(0)}} \\cdots \\coprod _{\\underline{v}^{(j)}} \\operatorname{Hom}\\left(\\partial \\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)}) \\cdots (\\underline{v}^{(j)}) \\right).",
"[u] } $ On each component, i.e., fixing each $\\underline{v}^{(i)}$ , the surjectivity of the map from $ \\coprod _{\\underline{v}^{(0)}} \\cdots \\coprod _{\\underline{v}^{(j)}} \\operatorname{Hom}\\left(\\Delta [m], X[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)}) \\cdots (\\underline{v}^{(j)})\\right) $ to the appropriate component of $P$ produces exactly our desired lift.",
"Now we use this result to shift our attention back to the maps $I^{n,T}$ .",
"Lemma 10.9 Let $T=\\lbrace [1]^{(i)} \\mid 0 \\le i \\le j \\rbrace \\subseteq S$ , and suppose that $f \\colon X \\rightarrow Y$ is a map of $\\Theta _n$ -$T$ -Segal precategories with the right lifting property with respect to the maps in $I^{n,T}$ .",
"Then $f_0 \\colon X[0] \\rightarrow Y[0]$ is surjective and the maps $ X[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)}) \\cdots (\\underline{v}^{(j)}) \\rightarrow Y[q](c_1, \\ldots , c_q)(f\\underline{v}^{(0)}) \\cdots (f\\underline{v}^{(j)}) $ are acyclic fibrations for any object $[q](c_1, \\ldots , c_q)$ of $\\Theta _n$ and every choice of $\\underline{v}^{(i)} \\in (X[1]^{(i)}_0)^{\\theta (i,\\underline{c})}$ for each $0 \\le i \\le j$ .",
"If $f$ has the right lifting property with respect to the maps in $I^{n,T}$ , then it has the right lifting property with respect to all cofibrations.",
"In particular, it has the right lifting property with respect to the maps in $I_f^{n,T}$ , since it is not hard to check that these maps are all levelwise monomorphisms of simplicial sets.",
"Therefore the result follows from Lemma REF .",
"Finally, we can prove the Proposition REF .",
"Note that the proof strategy is substantially different from the one used for Segal categories in [8].",
"Suppose that $f \\colon X \\rightarrow Y$ is a map with the right lifting property with respect to the maps in $I^{n,T}$ .",
"It follows that $f$ has the right lifting property with respect to all cofibrations, in particular those that are also weak equivalences.",
"Therefore $f$ is a fibration.",
"We need to show that it is a Dwyer-Kan equivalence.",
"First consider the case where $X$ and $Y$ are $\\Theta _n$ -$T$ -Segal categories.",
"Then applying Lemma REF when $q=1$ gives the desired result, after observing that $X[1](c)(v_0, v_1)$ is the union of all the $X[1](c)(v_0, v_1)(\\underline{v}^{(1)}) \\cdots (\\underline{v}^{(j)})$ , and that fibrations are preserved under disjoint union.",
"If $X$ and $Y$ are not $\\Theta _n$ -$T$ -Segal categories, then a Dwyer-Kan equivalence between them is defined in terms of their localizations $LX$ and $LY$ .",
"Therefore, we need to show that the required condition still holds after localizing.",
"Let us recall how the localization is obtained.",
"If $X$ is not local, then we take iterated pushouts $ {\\partial \\Delta [n] \\times \\Theta [q](c_1, \\ldots , c_q) \\cup \\Delta [m] \\times G[q](c_1, \\ldots , c_q) [r] [d] & X [d] \\\\\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q) [r] & X^{\\prime }} $ as well as the analogous pushouts along the localizing maps $V[1](\\mathcal {S}_{n-1})$ and $\\mathcal {T}^{\\prime }_{n,j}$ ; recall that the latter were defined just before Definition REF .",
"We want to show that the induced square on mapping objects is still a pushout square; for simplicity, let us denote the left-hand map above, or any of its analogues for maps in $V[1](\\mathcal {S}_{n-1})$ and $\\mathcal {T}^{\\prime }_{n,j}$ , as $A \\rightarrow B$ , so that we consider the maps on components $ {A[1](c)(v_0, v_1) [r] [d] & X[1](c)(v_0, v_1) [d] \\\\B[1](c)(v_0, v_1) [r] & X^{\\prime }[1](c)(v_0, v_1).}",
"$ We show that this diagram is a homotopy pushout square via an application of Mather's Cube Theorem [25] to the following diagram: $ {A[1](c)(v_0,v_1) [rr] [dr] [dd] && X[1](c)(v_0, v_1) ^{\\prime }[d][dd] [dr] & \\\\& A[1](c) [rr] [dd] && X[1](c) [dd] \\\\B[1](c)(v_0, v_1) ^{\\prime }[r][rr] [dr] && X^{\\prime }[1](c)(v_0, v_1) [dr] & \\\\& B[1](c) [rr] && X^{\\prime }[1](c).}",
"$ We know that the front square is a homotopy pushout, and we want to know that the back square is also; it suffices to show that the top, bottom, and sides of the cube are all homotopy pullback squares.",
"We verify this fact for the top square, namely $ {A[1](c)(v_0, v_1) [r] [d] & X[1](c)(v_0, v_1) [d] \\\\A[1](c) [r] & X[1](c), } $ and leave the argument for the others as an exercise for the reader.",
"First, observe that the right-hand map is an inclusion of components and therefore a fibration, so it suffices to show that the square is an ordinary pullback.",
"Using the descriptions of $A[1](c)(v_0, v_1)$ and $X[1](c)(v_0, v_1)$ as pullbacks, one can check the necessary universal property for $A[1](c)(v_0, v_1)$ .",
"Now, after applying the functorial localization functor to the map $X \\rightarrow Y$ , we obtain that the induced map on mapping objects $ (LX)[1](c)(v_0, v_1) \\rightarrow (LY)[1](c)(v_0, v_1) $ is necessarily a weak equivalence, completing the proof.",
"Now we need to prove the converse, namely Proposition REF .",
"To do so, we can generalize a construction used for the analogous proof when $n=1$ .",
"Suppose $f \\colon X \\rightarrow Y$ is a fibration and a weak equivalence, and that $T=\\lbrace [1]^{(i)} \\mid 0 \\le i \\le j\\rbrace $ , for some $j <n$ .",
"We need to show that $f$ has the right lifting property with respect to the maps in $I^{n,T}$ .",
"First consider the case in which, for every $[1]^{(i)}$ in $T$ , the induced map of discrete simplicial sets $ f \\colon X[1]^{(i)} \\rightarrow Y[1]^{(i)} $ is an isomorphism.",
"First, let us factor $f$ in the Reedy model structure $\\mathcal {SS}ets^{\\Theta _n^{\\operatorname{op}}}$ as $ X \\hookrightarrow Y^{\\prime } \\overset{\\simeq }{\\twoheadrightarrow } Y $ in such a way that $Y^{\\prime }$ still has the required components discrete, for example as in the proof of Proposition REF .",
"Since the map $Y^{\\prime } \\rightarrow Y$ is a Reedy weak equivalence and therefore a Dwyer-Kan equivalence, we can conclude by the two-out-of-three property that $X \\rightarrow Y^{\\prime }$ is also a Dwyer-Kan equivalence.",
"Since $f$ is assumed to be a fibration, a lift exists in the diagram $ {X [r]^= [d]_\\simeq & X [d]^f \\\\Y^{\\prime } [r] @{-->}[ur] & Y.}",
"$ Therefore $f$ is a retract of $Y^{\\prime } \\rightarrow Y$ and therefore a Reedy acyclic fibration.",
"Thus $f$ has the right lifting property with respect to monomorphisms, and in particular to the maps in $I^{n,T}$ .",
"Thus, our result holds when $X$ and $Y$ have isomorphic discrete components.",
"Now, we consider the general case, where for each $[1]^{(i)}$ in $T$ the maps of discrete spaces $X[1]^{(i)} \\rightarrow Y[1]^{(i)}$ are surjective but not necessarily isomorphisms.",
"Define $\\Phi Y$ to be the pullback of the diagram $ {\\Phi Y [r] [d] & Y [d] \\\\\\operatorname{cosk}_T(X) [r] & \\operatorname{cosk}_T(Y). }",
"$ Observe that $ (\\Phi Y)[1]^{(i)} \\cong X[1]^{(i)} $ for every $0 \\le i <n$ , and that for every $[q](c_1, \\ldots , c_q) \\in \\operatorname{ob}(\\Theta _n)$ and every $\\underline{v}^{(i)} \\in (X[1]^{(i)}_0)^{\\theta (i,\\underline{c})}$ for each $0 \\le i \\le j$ , the map $ (\\Phi Y)[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)}) \\cdots (\\underline{v}^{(j)}) \\rightarrow Y[q](c_1, \\ldots , c_q)(\\underline{v}^{(0)}) \\cdots (\\underline{v}^{(j)}) $ is a weak equivalence of simplicial sets.",
"We claim that $X \\rightarrow \\Phi Y$ is both a fibration and a weak equivalence.",
"It is not hard to show that it is a Dwyer-Kan equivalence, so let us show that $X \\rightarrow \\Phi Y$ is a fibration.",
"Let $A \\rightarrow B$ be a generating acyclic cofibration, so we know that a lift exists in any diagram $ {A [d]_\\simeq [r] & X [d]^f \\\\B [r] @{-->}[ur] & Y} $ since $X \\rightarrow Y$ is assumed to be a fibration.",
"But we want to know that this lift is compatible with the factorization of $B \\rightarrow Y$ as the composite $B \\rightarrow \\Phi Y \\rightarrow Y$ , so we want to know that the lift exists in the diagram $ {A [r] [d]_\\simeq & X [d] [dr]^f & \\\\B [r] @{-->}[ur] & \\Phi Y [r] & Y.}",
"$ Since $X \\rightarrow Y$ is assumed to be a fibration, we know that the indicated lift exists, but we want to know that it makes the top left-hand square commute as well, namely that the lift is compatible with the factorization of $B \\rightarrow Y$ as the composite $B \\rightarrow \\Phi Y \\rightarrow Y$ .",
"We know that $\\Phi Y$ agrees with $Y$ except possibly on the spaces corresponding to $[1]^{(i)}$ .",
"Since $A \\rightarrow B$ is a monomorphism and $ X[1]^{(i)} \\cong (\\Phi Y)[1]^{(i)} $ for all $0 \\le i <n$ , a lift exists in any diagram $ {A[1]^{(i)} [r] [d] & X[1]^{(i)} [d] \\\\B[1]^{(i)} [r] @{-->}[ur] & (\\Phi Y)[1]^{(i)}. }",
"$ This lift, together with the lift $B \\rightarrow X$ in the previous diagram, guarantees that the latter lift is compatible with the factorization of $B \\rightarrow Y$ through $\\Phi Y$ .",
"It follows that $X \\rightarrow \\Phi Y$ is a fibration.",
"Now, since $X \\rightarrow \\Phi Y$ is a fibration and a weak equivalence that is the identity on objects, we know from the first part of the proof that it has the right lifting property with respect to the maps in $I^{n,T}$ .",
"Finally, we want to show that the map $\\Phi Y \\rightarrow Y$ has the right lifting property with respect to the maps in $I^{n,T}$ .",
"Thus, we want to show that a lift exists in any diagram of the form $ {\\left( \\partial \\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q) \\cup \\Delta [m] \\times \\partial \\Theta [q](c_1, \\ldots , c_q) \\right)_T [r] [d] & \\Phi Y [d] \\\\\\left(\\Delta [m] \\times \\Theta [q](c_1, \\ldots , c_q) \\right)_T [r] @{-->}[ur] & Y.}",
"$ We work levelwise, evaluating at objects of $\\Theta _n^{\\operatorname{op}}$ .",
"If we evaluate at an object $[1]^{(i)}$ of $T$ , then using the fact that we have discretized at such an object, one can check that the left-hand map is an isomorphism of discrete simplicial sets.",
"Hence, the desired lift exists.",
"If we evaluate at any other object of $\\Theta _n^{\\operatorname{op}}$ , namely one at which we have not discretized, then it follows from the definition of $\\Phi Y$ that the right-hand vertical map is an isomorphism of simplicial sets.",
"Thus, we again get the desired lift.",
"Thus $\\Phi Y \\rightarrow Y$ has the right lifting property with respect to the maps in $I^{n,T}$ ; since we have established the same property for the map $X \\rightarrow \\Phi Y$ , we can conclude that the composite $X \\rightarrow Y$ has the right lifting property with respect to the maps in $I^{n,T}$ , completing the proof."
]
] | 2209.08156 |
[
[
"Field effect two-dimensional electron gases in modulation-doped InSb\n surface quantum wells"
],
[
"Abstract We report on transport characteristics of field effect two-dimensional electron gases in surface indium antimonide quantum wells.",
"A 5 nm thin $n$-InSb capping layer is shown to promote the formation of reliable, low resistance Ohmic contacts to surface InSb quantum wells.",
"High quality single-subband magnetotransport with clear quantized integer quantum Hall plateaus are observed to filling factor $\\nu$=1 in magnetic fields of up to B=18 T. We show that the electron density is gate-tunable, reproducible, and stable from pinch-off to 4$\\times$10$^{11}$ cm$^{-2}$, and peak mobilities exceed 24,000 cm$^2$/Vs.",
"Rashba spin-orbit coupling strengths up to 130 meV$\\cdot$\\r{A} are obtained through weak anti-localization measurements.",
"An effective mass of 0.019$m_e$ is determined from temperature-dependent magnetoresistance measurements, and a g-factor of 41 at a density of 3.6$\\times$10$^{11}$ cm$^{-2}$ is obtained from coincidence measurements in tilted magnetic fields.",
"By comparing two heterostructures with and without a doping layer beneath the quantum well, we find that the carrier density is stable with time when doping in the ternary AlInSb barrier is not present.",
"Finally, the effect of modulation doping on structural asymmetry between the two heterostructures is characterized."
],
[
"Section I: Fabrication methods Section II: Additional magnetotransport characterization Section III: Effective mass and quantum lifetime Section IV: 8-band $\\mathbf {k\\cdot p}$ model of InSb/Al$_{0.1}$ In$_{0.9}$ Sb quantum wells Section V: Coincidence experiment Section VI: Weak anti-localization experiment"
],
[
"Fabrication Methods",
"The fabrication steps towards realization of all Hallbar devices discussed here and in the main text are presented in the following: Samples are cleaned prior to lithography by sonication in acetone and subsequently propanol for 5 minutes each before a final blow dry with nitrogen.",
"Mesa regions are defined with optical lithography using Shipley S1811 photoresist.",
"The resist is spun at 5000 rpm for 60 seconds and baked at 120 $^\\circ $ C for 90 seconds.",
"Following exposure, the photoresist is developed in MF319 developer for one minute.",
"In order to ensure no unintentional thin film of photoresist remains in the exposed regions, samples are ashed in an oxygen plasma at 50 W for twenty seconds prior to wet etching to remove any residual photoresist in the exposed (off-mesa) regions.",
"Wet etching proceeds with a ten second dip in buffered oxide etch (BOE) (1:10) to remove any native oxide on the surface of the sample caused by ashing and exposure to air.",
"The mesa is etched with a solution of H$_2$ O$_2$ :H$_3$ PO$_4$ :C$_6$ H$_8$ O$_7$ :H$_2$ O mixed 3:4:9:44 by volume for approximately 30 seconds or until an etch depth of at least 100 nm has been reached.",
"After etching, the photoresist etch mask is removed by sonication in acetone and isopropanol.",
"Optical lithography for definition of Ohmic contacts uses a bilayer resist recipe of MMA/Shipley.",
"First the MMA (methyl methacrylate) is spun at 5000 rpm for 60 seconds and baked at 150 $^\\circ $ C for 5 minutes.",
"Next the Shipley is spun in the same manner with a bake at 120 $^\\circ $ C for 90 seconds.",
"Optical exposure and development of the sample in MF319 succesfully removes Shipley in regions where Ohmic contacts are to be formed.",
"This exposure and development does not remove the MMA which protects the surface from being etched by the MF319 developer.",
"MMA is subsequently removed by a fifteen minute exposure and development in a solution of isopropanol:H$_2$ O at a 7:3 concentration.",
"The now exposed surfaces are sulphur passivated in a solution of ammonium polysulfide (NH$_4$ )$_2$ S$_x$ for 20 minutes under illumination and at room temperature.",
"Loading the sample into the deposition chamber proceeds immediately after passivation to minimize exposure to air.",
"An angled 45$^\\circ $ deposition of 20/60 nm of Ti/Au is performed in a thermal evaporator.",
"The 60 nm thick HfO$_2$ dielectric layer which isolates the top gate from the quantum well and Ohmic contacts in a gated Hallbar is deposited using atomic layer deposition at 150 $^\\circ $ C; the dielectric breakdown field is $\\sim $ 1.5 MV/cm at $T=1.6$ K. Following deposition, optical lithography with Shipley is used to define vias above the Ohmic contacts.",
"The HfO$_2$ in the exposed vias is etched in BOE at a concentration of 1:10.",
"Following etching, via resist is removed and processing proceeds with optical lithography of the top-gate and bond pads to metallic contacts.",
"A bilayer of MMA/Shipley as discussed for the Ohmic contacts is again used and the Ti/Au (20/60 nm) top-gate and bond pads are similarly deposited in a thermal evaporator at an angle of 45$^\\circ $ .",
"Figure: Electrical circuits for: (a) constant-current four-terminal setup with voltage preamplifiers (◯\\bigcirc ) for measuring V xx V_{xx} and V xy V_{xy}, and (b) constant-voltage two-terminal setup with a current preamplifier (▹\\triangleright ) for measuring differential conductance G=dI/dVG=dI/dV.",
"The ac oscillator (∼\\sim ) outputs a signal ranging from 10 mV to 1 Volt at low frequencies (10--20 Hz)."
],
[
"Additional magnetotransport characterization",
"Figure REF shows the electrical circuits used in experiments.",
"The typical “constant” ac voltage excitation in 2-terminal measurements was 100 µV.",
"The typical “constant” ac current in 4-terminal measurements was 100 nA for $T>1.5$ K and 10 nA for $T<100$ mK.",
"During Hall density and mobility constant-current 4-terminal measurements (Fig.",
"1c and Fig.",
"2c in the main text), the carrier density $n_{2D}$ was kept above 1$\\times 10^{11}$ cm$^{-2}$ at all times.",
"Otherwise, as the sample becomes more resistive, an increasingly significant fraction of the ac signal applied to the 1 MΩ resistor is dropped across the 2DEG.",
"At pinch-off, the ac signal is entirely applied across the 2DEG rather than across the 1 MΩ resistor.",
"Such voltages, which can be larger than the Fermi energy and even the confinement potential of the 2DEG in the InSb quantum well, can cause charging effects that last for the remainder of the cooldown (a thermal cycle to room temperature “resets” the device to its original characteristics).",
"Magnetotransport data of additional Hallbar devices fabricated in G1 and G2 is provided in Figure REF .",
"Magnetotransport characteristics between Hallbars are quite reproducible indicating the quality of growth and fabrication.",
"Furthermore, as discussed in the main text, there are no signs of parasitic parallel conduction or second subband occupation.",
"Figure: Longitudinal resistivity ρ xx \\rho _{xx} and Hall resistance R xy R_{xy} at 1.6 K of additional samples in G1 near 2.7×10 11 2.7\\times 10^{11} cm -2 ^{-2} (a, b, c) and G2 near 2.2×10 11 2.2\\times 10^{11} cm -2 ^{-2} (d, e, f).",
"We observe the oscillation in ρ xx \\rho _{xx} corresponding to ν=2\\nu =2 hit zero resistance, indicating the absence of parasitic conduction.",
"Furthermore, the absence of a second oscillation frequency in all figures is indicative of single-subband occupation."
],
[
"Effective Mass and quantum lifetime",
"In Figure 3a of the main text, the amplitude of oscillations $\\Delta \\rho _{xx}$ is obtained by subtraction of a polynomial background resistance from $\\rho _{xx}$ .",
"Data was taken at a fixed density of $3 \\times 10^{11}$ cm$^{-2}$ .",
"The temperature dependent amplitude of SdH oscillations is described by a thermal damping term $X(T) = \\frac{2\\pi ^2k_BT/\\hbar \\omega _c}{\\sinh (2\\pi ^2k_BT/\\hbar \\omega _c)}$ where $k_B$ is the Boltzmann constant, $T$ is temperature, and $\\omega _c$ is the cyclotron frequency.",
"[1] The effective mass, appearing in the cyclotron frequency, is determined from a least squares fitting of $X(T)$ to the temperature dependent amplitude of an oscillation at a given filling factor.",
"A representative fit is presented in Figure 3b for the oscillation corresponding to $\\nu = 8$ at $B = 1.56$ T. The envelope of SdH oscillations is described by $\\Delta \\rho _{xx} = 4\\rho _{0}X(T)e^{-\\pi /\\omega _c\\tau _q}$ where $\\rho _0$ is the zero field resistivity, $\\omega _c$ is the cyclotron frequency, and $X(T)$ is the thermal dampening term given previously.",
"[2], [3] At low enough temperatures where thermal damping can be neglected, the amplitude of oscillations is described by the Dingle term $e^{-\\pi /\\omega _c\\tau _q}$ .",
"Using a so-called Dingle plot, as shown in Figure 3c in the main text, the quantum lifetime $\\tau _q$ is given by the slope of $\\ln {\\Delta \\rho _{xx}/4\\rho _0X(T)}$ as a function of $1/B$ .",
"Table: Material parameters used in the calculation, based on standard notation."
],
[
"8-BAND k$\\cdot $ p MODEL OF InSb/Al{{formula:ae7cc6f2-bb3a-4b43-969a-851e3053b2b9}} In{{formula:90126690-4cfd-45ef-9367-373ce0089f8a}} Sb quantum well",
"The 8 band $\\mathbf {k\\cdot p}$ model of Livneh et al.",
"is used with the parameters listed in Table REF to estimate the effect of strain and quantum confinement on the in-plane effective mass of InSb.",
"[4], The model has been shown in the past to give very good agreement with the band gaps and absorption spectra of InAs/GaSb, InAs/AlSb and $\\mathrm {InAs/InAs_{1-x}Sb_{x}}$ type II superlattices.",
"[6] Using Eq.",
"C1 of Ref.",
"Livneh2012, $\\gamma _{3}$ of the well material, and the three Luttinger parameters, $\\gamma _{1}$ , $\\gamma _{2}$ and $\\gamma _{3}$ , of the barrier material, are calculated from $\\gamma _{1}$ and $\\gamma _{2}$ of the well, whose values we take from the work of Lawaetz.",
"[7] This reduces systematic errors introduced when Luttinger parameters are taken from more than one source, and is well suited to quantum wells (QWs) with a ternary barrier material since it properly takes band bowing into account.",
"The model also includes interface parameters which are quite significant in the case of the binary/binary T2SLs, but which are negligible in the present case due to the low aluminium concentration in the barriers, whose major constituent is the same as the binary quantum well material.",
"Figure REF compares the in-plane band structures, $E\\left(k_{||}\\right)$ , close to the band gap for relaxed and strained InSb and for a strained $\\mathrm {InSb/In_{0.9}Al_{0.1}Sb}$ superlattice with layer thicknesses of 93 ML / 70 ML (ML = monolayer $\\approx 3\\textrm {Å}$ ), where the strain of -0.53% is provided by pseudomorphic growth on relaxed $\\mathrm {In_{0.9}Al_{0.1}Sb}$ .",
"Because the superlattice layers are quite thick, there is negligible dispersion in the growth direction for the conduction and valence bands shown in Fig.",
"REF , so the superlattice can be viewed as a multiple quantum well (MQW), where the in plane dispersion is the same as for a single QW.",
"When in-plane compressive strain is applied to bulk InSb, as shown in Fig.",
"REF , the hydrostatic component tends to increase the band gap while the uniaxial component tends to reduce it, by splitting the valence band so that the heavy-hole (HH) is uppermost.",
"Hence the band gap exhibits only a small net increase and the HH in-plane dispersion shows a clear anti-crossing with the light-hole (LH).",
"Note that “heavy” and “light” refer to masses in the growth- or z-direction.",
"This behaviour is reflected in the QW, where the valence band edge is HH-like, with a series of closely spaced HH sub-bands whose in-plane dispersions anti-cross with the first LH sub-band.",
"The main difference for the strained QW is that it has a band gap that is 11.3 meV (or 4.6%) larger than that of the strained InSb, due to the additional contribution of quantum confinement.",
"The band gap is 20.9 meV (or 8.8%) larger than that of relaxed InSb.",
"Figure: Comparison of the band structurein the in-plane [100] direction, for relaxed and strained InSband for a strained 93 ML / 70 ML MQW.",
"For bulk InSb, the bands have been shiftedin each case so that the edges of the conduction bands are identicalwith those of a 93 ML / 70 ML MQW with the same in-plane latticeparameter.",
"For the strained cases, the in-plane lattice parameter is that of relaxed In 0.9 Al 0.1 Sb \\mathrm {In_{0.9}Al_{0.1}Sb}.",
"Only the first 3 conduction sub-bands and the first 5 valence sub-bands are shown for the MQW.",
"In the legends, a 0 \\mathrm {a}_{0} is the cubiclattice parameter.In Figure REF (a) the electron effective masses and their ratio are shown for relaxed InSb and the strained QW.",
"They are found by applying the formula, $m^{*}=\\tfrac{\\hbar ^{2}}{|\\partial ^{2}E/\\partial k_{||}^{2}|}$ , to a sixth order polynomial that provides a very good fit to the $\\mathbf {k\\cdot p}$ dispersions in Fig.",
"REF over the range, $|k_{||}|<0.022\\times 2\\pi /\\mathrm {a}_{\\mathrm {InSb}}$ .",
"Based on a simple two band QW Hamiltonian,[8], [9] $H=A\\left(\\sigma _{x}k_{x}-\\sigma _{y}k_{y}\\right)+\\sigma _{z}\\left(\\frac{E_{0}}{2}+Bk_{||}^{2}\\right)+I_{0}Dk_{||}^{2}$ , the in-plane dispersion of the conduction band edge varies as $A^{2}k_{||}^{2}/E_{0}$ with an effective mass, $m^{*}=$$\\hbar ^{2}E_{0}/2A^{2}$ ($\\sigma {}_{i}$ are the Pauli spin matrices, $I_{0}$ is the identity matrix, A is the electron-hole hybridization parameter, $E_{0}$ is the QW band gap, and B, D represent interactions with remote bands which are small and have been ignored).",
"In the limit of infinite well width, $E_{0}\\rightarrow E_{G}$ , where $E_{G}$ is the bulk band gap.",
"Since A scales inversely with the lattice parameter,[10], [9] the electron band edge effective mass in the QW is decreased by 1.06% due to electron-hole hybridization, and increased by 8.8% due to the change in the band gap, giving an overall up shift of 7.7%.",
"This is fairly close to the plotted value of 12.7% in Fig.",
"REF (a), suggesting that the two band model captures the essential physics of the band edge effective mass fairly well, but there may be a small additional contribution due to electron penetration of the barriers.",
"Figure: (a) Calculated band curvature effective massesin terms of the free electron value, m 0 m_{0}, and their ratio forrelaxed InSb and the strained MQW (a InSb =a 0 \\mathrm {a_{InSb}=a_{0}} of InSb)(b) Difference between a parabollic dispersion and the 𝐤·𝐩\\mathbf {k\\cdot p}dispersion of the QW shown in Fig.",
",for different values of the effective mass, m * m^{*}.The rapid increase of the band curvature effective mass with wave vector in Fig.",
"REF (a) shows that strong non-parabolicity exists in the conduction band of both bulk InSb and the QW.",
"For the 2DEG density of $3\\times 10^{11}\\,\\mathrm {cm^{-2}}$ in Fig.",
"3(a) of this letter, the electron Fermi wave vector of $k_{\\mathrm {F}}=0.014\\times 2\\pi /\\mathrm {a}_{\\mathrm {InSb}}$ , corresponds to a band curvature effective mass in the QW of 0.033 $m_{0}$ .",
"This value does not agree with 0.019 $m_{0}$ measured at B = 1.56 T in Fig.",
"3(b), because the magneto-transport assumes a parabollic model, whose mass is used to determine the Landau energies: $E_{N\\uparrow ,\\downarrow }=(N+\\frac{1}{2})\\frac{\\hbar eB}{m^{*}}\\pm \\frac{1}{2}g\\mu _{B}B$ .",
"This parabolic mass value is fitted to the temperature dependent amplitude of the SdH oscillations, where electrons are thermally excited from nearly filled to nearly empty Landau levels.",
"[11] Therefore we need to find a parabolic dispersion that intersects the $\\mathbf {k\\cdot p}$ dispersion close to the Fermi wave vector.",
"At this wave vector, the number of states within the zero field $\\mathbf {k\\cdot p}$ Fermi circle matches the number of filled Landau states.",
"Figure REF (b) shows that the difference between the parabolic and $\\mathbf {k\\cdot p}$ dispersion energies vanishes at $k_F = 0.0137 \\times 2\\pi /a_{||}$Note that $\\mathrm {a}_{||}$ is used here for the QW while $\\mathrm {a}_{\\mathrm {InSb}}$ at $k_{\\mathrm {F}}=0.0137\\times 2\\pi /\\mathrm {a}_{\\mathrm {||}}$ was used earlier, but the difference is small enough to yield the same prefactor close to 0.014. when the parabolic mass is 0.018 $m_{0}$ .",
"If we add the number of states in the next (empty) Landau level at 1.56 T for both spin directions, to take into account their role in the temperature dependence of the SdH oscillations, the wave vector for the circle that includes all of these states increases to $k_{\\mathrm {F}}^{*}=0.0157\\times 2\\pi /\\mathrm {a}_{\\mathrm {||}}$ .",
"Figure REF (b) shows that the effective mass corresponding to this circle increases to 0.0185 $m_{0}$ .",
"Thus an average value close to 0.0183 $m_{0}$ , is expected to correspond to the measured SdH mass.",
"Since the latter was found to be 0.019 $m_{0}$ , the agreement between the $\\mathbf {k\\cdot p}$ model and experiment appears to be quite reasonable."
],
[
"Coincidence Measurement",
"In Figures REF (a) and REF (b) the longitudinal resistivity $\\rho _{xx}$ as a function of the perpendicular magnetic field $B_\\perp $ for different tilt angles $\\theta $ (see inset Fig.S1(b)) is shown for G1 at densities corresponding to (a) 2.8$\\times 10^{11}$ cm$^{-2}$ and (b) 3.6$\\times 10^{11}$ cm$^{-2}$ .",
"At $\\theta = 0^{\\circ }$ , we observe the onset of spin splitting at $\\nu = 5$ around 2 T followed by both even and odd filling factors corresponding to $\\nu = 4, 3, 2$ at higher fields.",
"As the tilt angle is increased, the width of the minima in $\\rho _{xx}$ decreases for even integer filling factors ($\\nu = 2, 4$ ) and increases for odd integer filling factors ($\\nu = 3, 5$ ).",
"Eventually, peaks will coalesce at even integer filling factors as minima in $\\rho _{xx}$ at odd integer filling factors approach their largest widths.",
"The coalescing of peaks in this case corresponds to the crossing of spin split Landau levels of different spin polarizations and is used to determine the effective g-factor.",
"Figure: Coincidence measurement.",
"(a) Longitudinal resistivity versus magnetic field at (a) V g =0.75V_g=0.75 V and (b) V g =0.85V_g=0.85 V is taken at various tilt angles θ\\theta with respect to normal vector of the sample surface.",
"The perpendicular field values B ⊥ B_{\\perp } of peaks in resistivity surround ν=4\\nu =4 in (a) and (b) are plotted versus tilt angle θ\\theta in (c) and (d) respectively.Figures REF (c) and REF (d) show the $B_\\perp $ values of the peaks in the SdH oscillations shown in Figs.",
"REF (a) and REF (b) respectively as a function of tilt angle $\\theta $ .",
"Peaks corresponding to the observed crossing at $\\nu = 4$ in Figs.",
"REF (a) and REF (b) are presented in REF (c) and REF (d) respectively.",
"The evolution of peaks in $\\rho _{xx}$ as a function of $\\theta $ is described by the evolution of the Landau level energy spacing described by $E_N = \\hbar \\omega _c(\\theta )(N + 1/2) \\pm \\frac{1}{2} g^*\\mu _B B_{\\text{tot}}$ where $\\hbar $ is the reduced Plank's constant, $\\omega _c (\\theta )=eB_\\perp (\\theta )/m^*$ is the cyclotron frequency, $N=0,1,2,...$ is an integer, $g^*$ is the effective g-factor, $\\mu _B$ is the Bohr magneton, and $B_{\\text{tot}}$ is the total magnetic field.",
"All scans were taken at two fixed gate voltages ($V_g=0.75$ V and $V_g=0.85$ V), which in this case did not correspond to a fixed density.",
"Operating the piezo-electric rotator stage over the duration of the experiment was observed to change the relation of $n_{2D}(V_g)$ .",
"This particular sample, G1-3, had been otherwise very stable in many cooldowns in two other cryostats.",
"For example, the stable pinch-off curves in Figure 1(c), the stable Landau fan in Figure 2(b), and the temperature dependence of the WAL peak in Section VII of the Supplementary were all performed on sample G1-3, with desnity remaining stable and reproducible for weeks at a time.",
"We thus cannot explain the density instability between scans at different tilt angles (during the scan, the density remains stable throughout), other than perhaps due to the heat pulse generated while the rotator to a different angle $\\theta $ moved between scans.",
"In any case, having measured the carrier density of each $B_\\perp $ scan via the Hall effect, we modeled the density-driven change in the Landau level energy for each scan by using $\\omega _c = eB_{\\perp }/m^*$ and $\\nu = hn_{2D}/eB_\\perp $ for a given filling factor $\\nu $ .",
"With this correction, a best fit of the spin split energy levels (solid lines) to the data (crosses) yielded an effective g-factor of $33\\pm 2$ at $\\nu =4$ in (c) and $41 \\pm 2$ at $\\nu =4$ in (d).",
"Figure: Weak antilocalization measurements of sample G1-3 taken at T=20T=20 mK and T=1.6T=1.6 K. The results of fits to the HLN model (black lines) are reported in figure 4c of the main text."
],
[
"Weak Anti-localization",
"In systems with strong spin-orbit interaction, the longitudinal conductivity in small magnetic fields exhibits a pronounced peak at B = 0 due to the suppression of coherent backscattering.",
"In our measurement, the longitudinal conductivity $\\sigma _{xx}$ (B) is determined from simultaneous measurements of the longitudinal and transverse resistances.",
"As shown in figure 4a of the main text, the conductivity correction $\\Delta \\sigma _{xx} (B) = \\sigma _{xx} (B) - \\sigma _{xx} (0)$ exhibits a peak in both G1 and G2 for various densities.",
"The strength of SOI is quantified from fits of the conductivity correction to the Hikami Larkin Nagaoka model.",
"[13] The conductivity correction of the HLN models reads: $\\Delta \\sigma _{xx} (B) ={}& \\frac{e^2}{2\\pi ^2\\hbar }\\bigg (\\Psi \\big (\\frac{1}{2} + \\frac{H_{\\phi }}{B} + \\frac{H_{SO}}{B}\\big ) + \\frac{1}{2}\\Psi (\\frac{1}{2} + \\frac{H_{\\phi }}{B}+ \\frac{2H_{SO}}{B})\\\\& - \\frac{1}{2}\\Psi (\\frac{1}{2} + \\frac{H_{\\phi }}{B}) - ln(\\frac{H_{\\phi } + H_{SO}}{B}) - \\frac{1}{2}ln(\\frac{H_{\\phi } + 2H_{SO}}{B}) \\nonumber \\\\& + \\frac{1}{2}ln(\\frac{H_{\\phi }}{B})\\bigg ).",
"\\nonumber $ The fit parameters H$_{\\phi }$ and H$_{SO}$ correspond respectively to the phase coherence and spin-orbit effective fields and $\\Psi $ is the Digamma function.",
"The fit parameters can be converted to their corresponding lengths using $l_{\\phi } = \\frac{\\hbar }{4eH_{\\phi }}$ and $l_{SO} = \\sqrt{\\tau _{SO}D}$ where $D$ is the diffusion constant.",
"Figure REF displays fits of the HLN model to the conductivity correction measured in sample G1-3.",
"As reported in the main text and shown here, the phase coherence reached 2.4 µm at 20 mK."
]
] | 2209.08193 |
[
[
"Belief Revision based Caption Re-ranker with Visual Semantic Information"
],
[
"Abstract In this work, we focus on improving the captions generated by image-caption generation systems.",
"We propose a novel re-ranking approach that leverages visual-semantic measures to identify the ideal caption that maximally captures the visual information in the image.",
"Our re-ranker utilizes the Belief Revision framework (Blok et al., 2003) to calibrate the original likelihood of the top-n captions by explicitly exploiting the semantic relatedness between the depicted caption and the visual context.",
"Our experiments demonstrate the utility of our approach, where we observe that our re-ranker can enhance the performance of a typical image-captioning system without the necessity of any additional training or fine-tuning."
],
[
"Introduction",
"Image caption generation is a task that is predominantly at the intersection of the areas of computer vision and natural language processing.",
"The task is primarily aimed at generating a natural language description for a given image.",
"Caption generation systems usually consist of an image encoder that encoders a given image (usually by using a CNN) whose encoding is fed to a decoder (usually by using a generative model such as RNN) to generate a natural language sentence which describes the image succinctly.",
"Some of the most used approaches include a CNN-RNN end-to-end system , , end-to-end systems with attention that attend to specific regions of the image for generation , and systems with reinforcement learning based methods , .",
"Further, recent advances have resulted in end-to-end systems that use Transformer based architecture for language generation and have become the current state-of-the-art , , , .",
"Figure: An overview of our visual semantic re-ranking.",
"We use the visual context from the image to re-rank the most related caption to its visual context.",
"These semantic relatedness measures are learned at a word-to-sentence level.",
"In the example, our visual semantic re-ranker (Visual Beam), as post-processing based approach, are able to re-rank more descriptive caption than Best Beam 5 .While the state-of-the-art models generate captions that are comparable to human level captions, they are known to lack lexical diversity, are often not very distinct, and sound synthetic.",
"We here highlight a few recent approaches that have focused on this problem, these include that uses generative adversarial networks towards generating diverse and human like captions.",
"use beam search with a distractor image to force the model to produce diverse captions by encouraging the models to be discriminative.",
"Other recent works use beam search directly to produce diverse captions by forcing richer lexical word choices , , , .",
"In this work, we follow a similar line of research and focus on the problem of improving diversity and making captions natural and human like and propose a novel re-ranking approach.",
"In this approach, we use $n$ -best reranking with a given beam that explicitly uses semantic correlation between the caption and the visual context through belief revision (an approach inspired from human logic).",
"We refer the reader to Figure REF , where the approach results in the caption that is a) visually relevant and b) most natural and human like.",
"Our main contributions in this paper are: [noitemsep] We demonstrate the utility of Belief Revision framework, which has been shown to work with human judgment, that can be applied to Image Captioning by employing vision-language joint semantic measures using state-of-the-art pre-trained language models.",
"Our approach is a post-processing method and can be used as a drop-in replacement for any caption system.",
"We show that our proposal produces better captions as reported using automated metrics as well as validated by human evaluations."
],
[
"Belief Revision with ",
"In this section, we briefly introduce SimProb which is based on the philosophical intuitions of Belief Revision, an idea that helps to convert similarity measures to probability estimates.",
"introduce a conditional probability model that assumes the preliminary probability result is updated or revised to the degree that the hypothesis proof warrants.",
"The range of revision is based on the informativeness of the argument and its degree of similarity.",
"That is, the similarity to probability conversion can be defined in terms of Belief Revision.",
"Belief Revision is a process of forming a belief by bringing into account a new piece of information.",
"Let us consider the following statements: [noitemsep] 1 Tigers can bite through wire, therefore Jaguars can bite through wire.",
"2 kittens can bite through wire, therefore Jaguars can bite through wire.",
"In the first case, the statement seems logical because it matches our prior belief i.elet@tokeneonedotjaguars are similar to tigers, so we expect them to be able to do similar things.",
"We hence consider that the statement is consistent with our previous belief, and there is no need to revise it.",
"In the second case, the statement is surprising because our prior belief is that kittens are not so similar to jaguars, and thus, not so strong.",
"But if we assume the veracity of the statement, then we need to revise and update our prior belief about the strength of kittens.",
"This work formalizes belief as probabilities and revised belief as conditional probabilities, and provides a framework to compute them based on the similarities of the involved objects.",
"According to the authors, belief revision should be proportional to the similarity of the involved objects (i.elet@tokeneonedotin the example, the statement about kittens and jaguars would cause a stronger belief revision than e.glet@tokeneonedotthe same statement involving a pigeons and jaguars because they are less similar).",
"In our case, we use the same rationale and the same formulas to convert similarity (or relatedness) scores into probabilities suitable for reranking.",
"SimProb Model To obtain the likelihood revisions based on similarity scores, we need three parameters: (1) Hypothesis: prior probabilities, (2) Informativeness: conclusion events and (3) Similarities: measure the relatedness between involved categories.",
"The main goal is to predict a conditional probability of statements, given one or more others.",
"In order to predict the conditional probability of the argument’s conclusion, given its premise or hypothesis, we will need only the prior probabilities of the statements, as well the similarities between the involved categories (e.g.",
"kittens and tigers).",
"We discuss the detail of each component in the SimProb next: Formulation of SimProb The conditional probability $\\operatorname{P}(Q_{c} | Q_{a})$ is expressed in terms of the prior probability of the conclusion statement $\\operatorname{P}(Q_{c})$ , the prior probability of the premise statement $\\operatorname{P}(Q_{a})$ , and the similarity between the conclusion and the premise categories sim(a, c).",
"$\\mathrm {P}\\left(Q_{\\mathrm {c}} \\mid Q_{a}\\right)=\\mathrm {P}\\left(Q_{c}\\right)^{\\alpha }{ }_{\\text{where }} \\alpha =\\left[\\frac{1-\\operatorname{sim}(a, c)}{1+\\operatorname{sim}(a, c)}\\right]^{1-\\mathrm {P}\\left(Q_{a}\\right)}$ Belief Revision Elements As we discussed above, there are two factors to determine the hypothesis probability revision: 1) the sufficient relatedness to the category: as $sim(a,c) \\rightarrow 0$ , $\\alpha \\rightarrow 1$ , and thus $\\operatorname{P}(Q_{c} | Q_{a}) = \\operatorname{P}(Q_{c})$ , i.elet@tokeneonedotno revision takes place, as there are no changes in the original belief.",
"While as $sim(a,c) \\rightarrow 1$ , $\\alpha \\rightarrow 0$ , and the hypothesis probability $\\operatorname{P}(Q_{c})$ is revised and raise closer to 1; 2) the informativeness of the new information $1-\\operatorname{P}(Q_{a})$ : as $\\operatorname{P}(Q_{a}) \\rightarrow 1$ and in consequence is less informative, $\\alpha \\rightarrow 1$ , since there is no new information, and thus no revision is needed either."
],
[
"Problem Formulation",
"Beam search is the dominant method for approximate decoding in structured prediction tasks such as machine translation, speech recognition and image captioning.",
"Larger beam size allows the model to have better exploration in the search space compared to greedy decoding.",
"The main idea of the beam search is to explore the all possible caption in the search space (under a given budged – the beam size) by keeping a set of top candidates.",
"Our goal is to leverage the visual context information of the image to re-rank the candidate sequences obtained through beam search hence, moving the most visually relevant candidate up in the list, as well as moving wrong candidates down.",
"For this purpose, we experiment with different rerankers, based on the relatedness between the candidate caption and the semantic context observed in the image through the idea of Belief Revision."
],
[
"Caption Extraction",
"We employ two recent Transformer based architecture for caption generation to extract the the top candidate captions using different beam sizes ($B=1\\ldots {20}$ ) .",
"The first baseline is based on multi-task model for discriminative Vision and Language BERT that is fine-tuned on 12 downstream tasks.",
"The second baseline is the vanilla Transformer with Meshed-Memory based caption generator with pre-computed top-down visual feature ."
],
[
"Proposed model",
"One approach of using word-level semantic relations, for scene text correction, with the visual context of an image was introduced in , which allows learning semantic correlations between a visual context and a text fragment.",
"The semantic relatedness, in our work, is between a visual context and a given candidate caption (i.elet@tokeneonedotbeam search), using a Belief Revision (BR) via SimProb to re-visit and rank the original beam search based on the similarity to the image objects/labels $c$ (a proxy for image context).",
"The BR in this scenario is a conditional probability, which assumes that the caption preliminary probability (hypothesis) $\\operatorname{P}(w)$ is revised to the degree approved by the semantic similarity with visual context $\\operatorname{sim}(w, c)$ .",
"The final output caption $w$ for a given visual context $c$ is written as: $\\operatorname{P}(w \\mid c)=\\operatorname{P}(w)^{\\alpha }$ where the main components of visual based hypothesis revision: Hypothesis: $\\operatorname{P}(w)$ Informativeness: $1-\\operatorname{P}(c)$ Similarities: $\\alpha =\\left[\\frac{1-\\operatorname{sim}(w, c)}{1+\\operatorname{sim}(w, c)}\\right]^{1-\\operatorname{P}(c)}$ where $\\operatorname{P}(w)$ is the hypothesis probability (beam search candidate caption) and $\\operatorname{P}(c)$ is the probability of the evidence that causes hypothesis probability revision (visual context from the image).",
"We discuss the detail of each component in the SimProb next.",
"Hypothesis: Prior probabilities of original belief.",
"The hypothesis $\\operatorname{P}(w)$ needs to be initialized by a common observation such as language model (LM) trained on general text corpus.",
"Therefore, we employ Generative Pre-trained Transformer (GPT-2) as language model to initialize the hypothesis probability.",
"We compute sentence probability as the mean of LM token probability, since it achieves better results than product probability.",
"Informativeness: Inversely related to the probability of the new information that causes hypothesis revision.",
"We leverage ResNet and Inception-ResNet v2 based Faster R-CNN object detector TensorFlow Object Detection API to extract textual visual context information from the image.",
"We use the classifier probability confidence with a threshold to filter out non exist object in the image.",
"For each image, we extract visual information into two-way (1) top-$k$ concept (2) Multi concept top-3 (label class or object category) visual information (image objects).",
"For the single concept we employ a unigram LM, based on the 3M-token opensubtitles corpus , to initialize the informativeness of the visual information.",
"For multiple concept, we take the mean probability of the three concept.",
"We will discuss in more detail the selection procedure of visual context in the Dataset Section.",
"Similarities: Hypothesis revision is more likely if there is a close relation between the hypothesis and the new information (candidate caption and visual context in our case).",
"We rely on two of the most recent state-of-the-art pre-trained Transformer-based language models to compute the semantic similarity between the caption and its visual context information with contextual embedding: BERT : BERT achieves remarkable results on many sentence level tasks and especially in textual semantic similarity task (STS-B) .",
"Therefore, we fine-tuned BERT$_{\\text{base}}$ on the training dataset, (textual information, 460k captions: 373k for training and 87k for validation) i.elet@tokeneonedotvisual, caption, label [semantically related or not related]), with a binary classification cross-entropy loss function [0,1] where the target is the semantic similarity between the visual and the candidate caption, with batch size 16 for 2 epochs with learning rate $2\\mathrm {e}{-5}$ .",
"RoBERTa : RoBERTa is an improved version of BERT, trained on a large amount of data, using dynamic masking strategies to prevent overfitting.",
"It achieves 2.4% improvement over BERT$_{\\text{Large}}$ in the STS task.",
"Also, show that general-purpose RoBERTa perform well on other specific tasks such as clinical data.",
"Since RoBERTa$_{\\text{Large}}$ is more robust we use an off-the-shelf model tuned on STS-B task.",
"In particular, we follow the traditional approach to compute the semantic similarity with BERT based model with a mean pool, over the last hidden layer, to extract a meaningful vector to compute the cosine distance."
],
[
"Dataset",
"COCO-Caption : It contains around 120k images, each image is annotated with 5 different human-written captions.",
"We use the split provided by , where 5k images are used for testing, 5k for validation, and the rest for model training.",
"Table: Performance of compared baselines on the Karpathy test split with/without Visual semantic Re-ranking.",
"For each base system we report performance using a greedy search and the best beam search.",
"Re-ranking is applied to the top-20 results of each system using BERT or RoBERTa for caption-context similarity.",
"The visual contexts are extracted using ResNet152 and Inception Resnet v2 based Faster R-CNN object detector.",
"We also report results for Bert-based similarity without hypothesis probability (rows marked only sim).Visual Context Dataset: Since there are many publicly dataset for caption, they contain no textual visual information such objects in the image.",
"We enrich COCO-caption with textual visual context information.",
"To automate visual context generation and without the need for a human label, for the training dataset, we use only ResNet152, that have 1000 label classes, to extract top-k 5 label class visual context information for each image in the caption dataset.",
"For testing, we rely only on the top-k visual information as a concept, and we also employ the Inception-ResNet v2 based Faster R-CNN object detector with 80 object classes.",
"In particular, each single annotate caption has five visual context information.",
"We use two approaches to match and filter out not related visual context: 1) Threshold: to filter out the probabilities prediction when the visual classifier not confident.",
"2) Semantic alignment: to match the most related caption to its environmental context.",
"In more details, we use cosine similarity with 840B pre-trained GloVe to match the visual with its context in word level manner.",
"Evaluation Metric: We use the official COCO offline evaluation suite, producing several widely used caption quality metrics: BLEU METEOR , ROUGE and CIDEr .",
"Moreover we also compute the semantic-similarity based metric: BERTscore : BERTscore leverages BERT's pre-trained contextual embedding and matches word candidate/reference sentences using cosine similarity.",
"It uses the token-level similarities between the candidate and reference to compute a sentence similarity score.",
"According to the authors, it surpasses SPICE in human correlation and is more suitable for language generation based visual grounding evaluation ."
],
[
"Results and Discussion",
"We use visual semantic information to re-rank candidate captions produced by out-of-the-box state-of-the-art caption generators.",
"We extract top-20 beam search candidate captions from two state-of-the-art models: VilBERT , fine-tuned on a total of 12 different vision and language datasets such as caption image retrieval and visual question answering, and a specialized caption-based Transformer .",
"Experiments applying different rerankers to the each base system are shown in Table REF .",
"The tested rerankers are: (1) VR$_{\\text{BERT}}$ uses BERT similarity between the candidate caption and the visual context of the image, transforms it to a probability using Equation REF , and combines the result with the original candidate probability to obtain the reranked score.",
"(2) VR$_{\\text{RoBERTa}}$ carries out the same procedure using similarity produced by RoBERTa.",
"A simpler model is also tested –VR$_{\\text{BERT}}$ (only $sim$ ) in Table REF – which replaces Equation REF with $\\operatorname{P}(w \\mid c)=\\operatorname{sim}(w, c)^{\\operatorname{P}(c)}$ , that is, does not rely on the original caption probability.",
"First, we compare our work with the original visual caption re-ranker with multiple word objects as concepts from the image, that extracted via Inception-ResNet v2 based Faster RCNN (i.elet@tokeneonedotperson, van, .etc), $\\text{VR}_{\\text{w-Object}}$ .",
"However, to make a fair comparison, we use the Sentence-RoBERTa$_{\\text{Large}}$ for the sentence semantic similarity model i.elet@tokeneonedot$\\text{cosine}$ (word objects, caption).",
"Secondly, we compare our model against two approaches that uses object information to improve image captioning: First, investigates the benefit of object frequency counts for generating a good captions.",
"We train an LSTM decoder (i.elet@tokeneonedotlanguage generation stage) with an object frequency counts dictionary on the training dataset.",
"The dictionary is a Fully Connected layer, concatenated with the LSTM and a dense layer, that adds more weight to the most frequency counts object are seen by the caption and the visual classifier.",
"Second, that introduce a controllable grounded captions via visual context.",
"We train the last stage (decoder), attention and language model LSTM, on the training dataset to visually ground the generated caption based on the visual context.",
"One observation, in Table REF , is that the benefit of using COCO object as visual context for longer caption, which can increase the chance of re-ranking the most visual related candidate caption, as in the case of VR$_{\\text{W-Object}}$ and VR$_{\\text{RoBERTa-Object}}$ with VilBERT.",
"Figure: Visualization of distribution change in re-rank score on the 40k random sample from the test set.",
"(Left) the score distribution before applying Belief Revision via SimProb with LM-GPT-2 initialization.",
"(Right) the score distribution after applying the revision via similarity RoBERTa Large _{\\text{Large}} with the visual context.Note that we are initializing the single visual context with Unigram Language Model to maximize the visual context score while computing the informativeness, as shown in Figure REF Left a denser SimProb score caption re-ranking.",
"Also, following framework, we investigate the statistical significance of the small changes in BLUE score in the transformer baseline, our VR is significantly better than the best beam result with $\\rho =0.05$ .",
"Figure REF shows the SimProb distribution over 40k samples with pre-trained RoBERTa$_{\\text{Large}}$ similarity score.",
"Applying the Belief Revision via semantic similarity change the score distribution over all the dataset.",
"(Left figure) Before applying the revision, overall re-ranking scores are relatively low, and (Right figure) after the visual context revision, overall scores increased.",
"Figure: Example of captions re-ranked by our visual re-ranker and the original caption by the baseline.",
"Re-ranked captions are more precise, have a higher lexical diversity, or provide more details."
],
[
"Evaluation of Diversity",
"Lexical diversity is a measure of how many different words are used in a text fragment.",
"However, due to, even we get the top-20 candidates from the base systems, many of them are the same or have very small differences (beam search drawback), which will reflect in small performance differences before and after re-ranking.",
"To try to capture the effect of the re-ranking, we also measured lexical diversity of the selected captions using different measures: Type-Token Ratio (TTR): TTR is the number of unique words or types divided by the total number of tokens in a text fragment.",
"Measure of Textual Lexical Diversity (MTLD): MTLD is based on TTR, and measures the average length of subsequences in the text for which a certain TTR is maintained, thus being, unlike TTR, length-invariant.",
"Table REF shows that visual re-ranking selects longer captions and with higher lexical diversity than the base system beam search.",
"Figure REF shows some examples where visual context re-ranking selected captions with more precise lexica (van vs. truck), higher diversity –i.elet@tokeneonedotadding details about objects (white plate vs. plate)– or even selecting a more specific abstraction level (batter, catcher and umpire vs. a group of men).",
"Figure: SimProb score of top-5 Beam search caption re-ranking (Right) with the visual classifier confidence probability without any initialization, (Left) with visual context that initialized by general common observation [email protected]: Measuring the diversity of caption before and after re-ranking.",
"Uniq and WPC columns indicate the average of unique/total words per caption, respectively."
],
[
"Human Evaluation",
"We conducted a human study to investigate human preferences over the visual re-ranked caption.",
"We randomly select 26 test images and gives the 12 reliable human subjects the option to chooses from two captions: (1) Best-beam (BeamS) and (2) Visual R-ranker.",
"We have mixed results as some of the re-ranked captions are grammatically incorrect, like singular instated of plural; sitting on for objects instead of subjects.",
"Overall, we can observe that native speakers 46% agreed with our visual re-ranker.",
"Meanwhile, the result of non-native is 61%.",
"Also, inspired by BERTscore and following that introduce Sentence Semantic Semantic (SSS) for machine translation, we employ sentence-to-sentence semantic similarity score to compare candidate captions with human references.",
"We use pre-trained Sentence-RoBERTa$_{\\text{LARGE}}$ tuned for general STS task.",
"SBERT-sts uses a siamese network to derive meaningful sentence embeddings that can be compared via cosine similarity.",
"Figure REF shows our result with SBERT-sts metric agrees more with human judgment than BERTscore.",
"Figure: Comparison between native human subject, BERTscore and sentence level metric SBERT-sts on test set."
],
[
"Ablation study",
"Belief Revision relies on different block (i.elet@tokeneonedotLM, similarity and visual context) to make the final revision.",
"In this study, we perform an ablation study over a random 100 samples from the test set to investigate the effectiveness of the proposed setup.",
"Table REF shows result with different setting.",
"Language Model Block: One of the principal intentions of initializing the original hypothesis with a LM is the ability to combine different models.",
"We experimented with product probability, however, the mean LM probability achieved better results than the product probability in this task.",
"Similairty Block: The degree of similarity between the caption and its visual context is the most part of the hypothesis revision.",
"Thus, we experimented with light model (Distil SBERT) and unsupervised/supervised Simple Contrastive Sentence Embedding (SimCSE) for learning sentence similarity.",
"The label based training SimCSE uses Natural Language Inference dataset to incorporate supervised sentence pair in contrastive learning.",
"The unsupervised one tries to predict the input sentence itself from the in-batch negatives sample by applying different dropout masks techniques.",
"The result show that unsupervised contrastive learning based similarity preforms well in the case of the longer caption, as shown in VilBERT Table REF .",
"Table: Ablation study with using different information from various baseline in each block (i.elet@tokeneonedotLM, similarity and visual context).Table: Comparison between positive (VR) and Negative Belief Revision (NBR) on the Karpathy split.",
"The NBR uses high similarity VR -high ^{-high} object related to the positive visual but not in the image, low similarity VR -low ^{-low} false positive from the visual classifier, and positive visual via static word level similarity VR -pos ^{-pos}.",
"Boldface fonts reflect improvement over the baseline.Visual Context Block: We experiment with the most recent model Contrastive Language-Image Pre-Training (CLIP) with Zero-Shot Prediction to extract the visual context.",
"Although, CLIP can predict rare object better, there is no improvement over ResNet152 with a huge computational cost."
],
[
"Experiment with Negative Evidence",
"Until now, following we considered only the cases when the visual context increase the belief of the hypothesis (Equation REF ).",
"However, the same authors proposed Equation REF for the case where the absence of evidence leads to decrease the hypothesis probability.",
"$\\operatorname{P}(w \\mid \\lnot c )=1-(1-\\operatorname{P}(w))^{\\alpha }$ In our case, we introduce negative evidence in three ways: False Positive Visual Context (VR$^{-low}$ ): We employ the false-positive produced by the visual classifier as negative information to decrease the hypotheses.",
"In this case, we have low similarities as the relation between the visual context and caption have more distance.",
"Absent Visual Context (VR$^{-high}$ ): The negative information here is a set of visual information extracted from the original visual context (i.elet@tokeneonedotfrom the visual classifier) that does not exist in the image but has some relation.",
"Thus, the visual context produced by the classifier is used as query to a pre-trained 840B GloVe, with cosine similarity, to retrieve the closest visual context in the same semantic space (e.glet@tokeneonedotvisual: river, closest visual: valley).",
"Positive Visual Context (VR$^{-pos}$ ): As the previous two-approaches have unexpected results with low and high similarities as shown in Table REF , we approach this from a positive belief revision perspective but as negative evidence.",
"Until now, all approaches use sentence-level semantic similarity, but in this experiment, we convert the similarity from sentence to word level.",
"For this first, we employ LSTM based CopyRNN keyphrase extractor , that is trained on a combined pre-processed wikidump (i.elet@tokeneonedotkeyword, short sentence) and SemEval 2017 Task 10 (Keyphrases from scientific publications) .",
"Secondly, GloVe is used to compute the cosine similarity with the visual context in a word-level manner.",
"We consider this as negative evidence for the following reasons: (1) the similarity is computed without the context of the sentence and (2) the static embedding is computed without knowing the sense of the word.",
"Finally, we combined VR$^{-joint}$ : the best model VR$^{-pos}$ with the best positive evidence Vil+VR$_{\\text{BERT/RoBERTa}}$ with a simple multiplication.",
"However, there is no improvement over the original positive revision, as shown in Table REF ."
],
[
"Related work",
"Modern sophisticated image captioning systems focus heavily on visual grounding to capture real-world scenarios.",
"Early work builds a visual detector to guide and re-ranked image captioning with global similarity.",
"The work of investigates the informativeness of visual or object information (e.glet@tokeneonedotobject frequency count, size and position) in an end-to-end caption generation.",
"proposes controlled caption language grounding through visual regions from the image.",
"introduces weakly supervised contrastive learning via object context and language modeling (i.elet@tokeneonedotBERT) for caption phrase grounding.",
"Recently, researchers explored visual semantic concept from the image to guide the caption grounding.",
"The work of uses part-of-speech to extract concept visual summary from image to drive and re-rank caption generation.",
"relies on scene concept abstract (object, relationship, and attribute) grounded in the image to learn accurate semantic without labels for image caption.",
"More recently, incorporates different concepts such as scene graph, object, and attribute to learn correct linguistic and visual relevance for better caption language grounding.",
"Inspired by these works, that uses re-ranking via visual information, , , that explored the benefit of object information in image captioning, that benefits of language modeling to extract contextualized word representations and the exploitation of the semantic coherency in caption language grounding , we purpose an object based re-ranker via human inspired logic reasoning with Belief Revision to re-rank the most related caption with contextualized semantic similarity.",
"Unlike these approaches, our methods employ state-of-the-art tools pre-trained models.",
"Therefore, the system will keep improving in the future as better systems become available.",
"In addition, our model can be directly used as a drop-in complement for any caption system that outputs a list of candidate hypothesis."
],
[
"Conclusion",
"In this work, we demonstrate that the Belief Revision approach that works well with human judgment can be applied to Image Captioning by employing human-inspired reasoning via a pre-trained model (i.elet@tokeneonedotGPT, BERT).",
"Belief Revision (BR) is an approach for obtaining the likelihood revisions based on similarity scores via human judgment.",
"We demonstrate the benefits of the approach by showing that two state-of-the-art Transformer-based image captioning results are improved via simple language grounding with visual context information.",
"In particular, we show the accuracy gain in a benchmark dataset using two methods: (1) BR with positive visual evidence (increase the hypothesis) and (2) negative evidence (decrease the hypothesis), with wrong visual i.elet@tokeneonedotfalse positive by the classifier.",
"However, this adaptation could be applied to many re-ranking tasks in NLP (text generation, multimodel MT, lexical selection, etc.)",
"as well as in CV applications such as visual storytelling etc."
]
] | 2209.08163 |
[
[
"A non-Gaussian limit for linear eigenvalue statistics of Hankel matrices"
],
[
"Abstract This article focuses on linear eigenvalue statistics of Hankel matrices with independent entries.",
"Using the convergence of moments we show that the linear eigenvalue statistics of Hankel matrices for odd degree monomials with degree greater than or equal to three does not converge in distribution to a Gaussian random variable.",
"This result is a departure from the known results, Liu, Sun and Wang (2012), Kumar and Maurya (2022), of linear eigenvalue statistics of Hankel matrices for even degree monomial test functions, where the limits were Gaussian random variables."
],
[
"Introduction and main results",
"The study of linear eigenvalue statistics is a popular area of research in random matrix theory.",
"For an $n \\times n$ matrix $A_n$ , the linear eigenvalue statistics of $A_n$ is defined as $ \\mathcal {A}_n(\\phi )= \\sum _{i=1}^{n} \\phi (\\lambda _i),$ where $\\lambda _1 , \\lambda _2 , \\ldots , \\lambda _n$ are the eigenvalues of $A_n$ and $\\phi $ is a `nice' test function.",
"The studies on linear eigenvalue statistics started with the study of central limit theorems for linear eigenvalue statistics of Sample covariance matrix [3], [14].",
"The results have been obtained for other important classes of random matrices and test functions.",
"Notable among them are the results for polynomial test functions on Wigner matrices by Sinai and Soshnikov [21], on tridiagonal matrices by Popescu [18], on Toeplitz matrix by Liu, Sun and Wang [15] and on circulant matrices by Bose et al.",
"[7].",
"For results on fluctuations of linear eigenvalue statistics of Wigner and sample covariance matrices, see [11], [4] and [16].",
"In this paper, we study the linear eigenvalue statistics of Hankel matrices for odd degree monomials and polynomials with odd degree terms.",
"Hankel matrices are given by $H_n=(x_{i+j-1})_{i,j=1}^n$ , where $(x_i)_{i \\ge 1}$ is known as the input sequence.",
"Hankel matrices are an important class of patterned matrices and have wide applications both in pure mathematics and other fields of sciences and engineering.",
"In mathematics, Hankel matrices are best known for their connection to the Hamburger moment problem (see [20]).",
"They also show up in studies of orthogonal polynomials and Pade's approximation [8], and in error-correcting codes [13].",
"Hankel matrices have also found applications in areas as diverse as superconductivity [17], macro-economics [2], image processing [12], spectral learning [5] and spectroscopy [22].",
"Hankel matrices are closely related to another important class of matrices known as Toeplitz matrices and in most cases, their studies go hand in hand.",
"More specifically, for any Toeplitz matrix $T_n$ , $P_nT_n$ is a Hankel matrix and conversely for any Hankel matrix $H_n$ , $P_n H_n$ is a Toeplitz matrix, where $P_n = (\\delta _{i-1, n-j} )_{i,j=1}^n$ is the backward identity permutation.",
"Since $P_n^{-1} = P_n$ , we obtain that any Hankel matrix (and Toeplitz matrix) is of this form.",
"For a sequence of random variables $\\lbrace x_i\\rbrace _{i \\in \\mathbb {Z}}$ , we define the random Toeplitz matrix as $T_n=(x_{i-j})$ and the random Hankel matrix as $H_n=P_nT_n$ .",
"In this article, the Hankel matrices considered are always of the form $H_n= P_nT_n$ .",
"For a random Hankel matrix $H_n$ , we define $ w_p := {\\mbox{Tr}}(A_n^p),$ where $A_n= H_n /\\sqrt{n}$ .",
"To the best of our knowledge, linear eigenvalue statistics of Toeplitz matrices were first studied by Chatterjee in [9] .",
"He showed that for test functions $\\phi (x)=x^{p_n}$ , where $p_n =o(\\log n/ \\log \\log n)$ , the linear eigenvalue statistics of symmetric Toeplitz matrices with Gaussian input entries converge to a Gaussian distribution under the total variation norm.",
"Later in 2012, Liu et al.",
"[15] studied $\\mathcal {A}_n(\\phi )$ for band Toeplitz and band Hankel matrices with independent input sequence $\\lbrace x_i\\rbrace $ obeying the following moment conditions: $\\mbox{E}[x_i]=0, \\ \\mbox{E}[x_i^2]=1 \\ \\forall \\ i \\in \\mathbb {Z}\\text{ and} \\ \\sup _{i \\in \\mathbb {Z}} \\mbox{E}\\big [\\left|x_{i}\\right|^{k}\\big ]=\\alpha _{k}<\\infty \\text{ for } k \\ge 3.$ It was shown in [15] that for $\\phi $ as a monomial and under the normalization $1/\\sqrt{n}$ , the linear eigenvalue statistics of random Toeplitz matrices converge in distribution to a Gaussian random variable.",
"Additionally, for $\\phi (x)= x^p$ , where $p$ is an even natural number, the linear eigenvalue statistics of random Hankel matrices too converge in distribution to a Gaussian random variable, under the normalization $1/\\sqrt{n}$ .",
"In a recent article [1], Kumar and Maurya showed that for odd $p$ , the fluctuations are not sensitive to the normalization $1/\\sqrt{n}$ .",
"The precise statements are as follow.",
"Result 1 Suppose $H_{n}$ is a random Hankel matrix with $w_p={\\mbox{Tr}}(\\frac{H_n}{\\sqrt{n}})^p$ .",
"(Theorem 6.4, [15]) If the entries of $H_n$ satisfies (REF ) and $\\mbox{E}[x_{i}^{4}] = \\kappa \\ \\ \\forall \\ i \\in \\mathbb {Z}$ , then for even $p\\ge 2$ , as $n\\rightarrow \\infty $ , $\\displaystyle \\frac{1}{\\sqrt{n}} \\bigl \\lbrace w_p - \\mbox{E}[w_p]\\bigr \\rbrace \\stackrel{d}{\\rightarrow } N(0,\\sigma _p^2),$ where $ N(0,\\sigma _p^2)$ is a Gaussian random variable with an appropriate covariance structure $\\sigma _p^2$ .",
"(Theorem 3, [1]) If the entries of $H_n$ satisfies (REF ), then for odd $p\\ge 1$ , as $n\\rightarrow \\infty $ , $\\displaystyle \\frac{1}{\\sqrt{n}} \\bigl \\lbrace w_p - \\mbox{E}[w_p]\\bigr \\rbrace \\stackrel{d}{\\rightarrow } 0,$ Our main result provides the limiting behaviour of linear eigenvalue statistics of Hankel matrices for odd degree monomial test functions.",
"Theorem 2 Let $H_n$ be a random Hankel matrix with an input sequence obeying (REF ) and $w_p={\\mbox{Tr}}(\\frac{H_n}{\\sqrt{n}})^p$ .",
"Then for every odd $p \\ge 3$ and $k \\in \\mathbb {N}$ , the limit of $k$ -th moment of $w_p$ are given by $ \\beta _k := \\lim _{n \\rightarrow \\infty } \\mbox{E}[w_p^k]={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\sum _{ \\pi \\in \\mathcal {P}_2(pk)} \\frac{1}{2^{m(\\pi )}}f_k(\\pi ) & \\quad \\mbox{if } k \\text{ is even},\\\\0 & \\quad \\text{if } k \\text{ is odd},\\end{array}\\right.",
"}$ where $f_k(\\pi )$ and $m(\\pi )$ are as given in Definition REF and (REF ), respectively.",
"Furthermore for each odd $p\\ge 3$ , there exist probability measures $\\Gamma _p$ on $\\mathbb {R}$ with moment sequence $\\lbrace \\beta _k \\rbrace $ and any such $\\Gamma _{p}$ has a non-Gaussian distribution with unbounded support.",
"Remark 3 (i) For $p=1$ , $w_1= \\frac{1}{\\sqrt{n}} {\\mbox{Tr}}(H_n) = \\frac{1}{\\sqrt{n}} \\sum _{i=-(n-1) \\atop i \\ odd}^{(n-1)} x_{i}.$ It follows from central limit theorem that $w_1$ converges in distribution to a Gaussian random variable.",
"(ii) In Section REF , we show that $\\lbrace \\beta _{k}\\rbrace $ does not obey Carleman's condition and therefore $\\Gamma _{p}$ might not be unique.",
"Regardless if we assume that $\\Gamma _{p}$ is the unique distribution with moment sequence $\\lbrace \\beta _k\\rbrace $ , then by Theorem REF and moment method, $w_{p} \\stackrel{d}{\\rightarrow } \\Gamma _p$ for any choice of input sequence obeying (REF ).",
"In spite of whether $\\Gamma _{p}$ is unique or not, for each $p \\ge 3$ , $w_p$ does not converge in distribution to a Gaussian random variable.",
"For more details, see Proposition REF and Corollary REF .",
"(iii) If $w_p$ converges in distribution, then the moment sequence of the limiting variable will be $\\lbrace \\beta _k\\rbrace $ and the corresponding density will be symmetric which can also be seen in Figure REF .",
"(iv) We also show that for a real polynomial $Q(x)$ with odd degree terms only, $ \\mbox{E}[{\\mbox{Tr}}\\lbrace Q(A_n)\\rbrace ]^k \\rightarrow \\tilde{\\beta }_k, \\mbox{ as $n\\rightarrow \\infty $,}$ where $\\tilde{\\beta }_k$ is as given in (REF ).",
"If there exists a unique distribution, say $ \\Gamma _Q$ , then ${\\mbox{Tr}}\\lbrace Q(A_n)\\rbrace \\stackrel{d}{\\rightarrow } \\Gamma _Q$ .",
"(v) Our results are in agreement with the simulations in Figure REF .",
"Simulations further suggest that the linear eigenvalue statistics converge to a universal non-Gaussian distribution which is unimodular and absolutely continuous.",
"In [15], the fluctuation of linear eigenvalue statistics of Toeplitz matrix $T_n$ was studied using a trace formula of the following form $ {\\mbox{Tr}}(T_n)^p =\\displaystyle \\sum _{i=1}^{n} \\sum _{j_{1}, \\ldots , j_{p}=-n}^{n} \\prod _{r=1}^{p} a_{j_{r}} \\prod _{\\ell =1}^{p} \\chi _{[1, n]} \\left(i-\\sum _{q=1}^{\\ell }(-1)^{q} j_{q}\\right) \\delta _{0} (\\sum _{q=1}^{p} j_{q}),$ where $\\delta $ is the Dirac function and $\\chi $ is the indicator function.",
"They also derived a closed form of trace formula for Hankel matrices and established the fluctuations of linear eigenvalue statistics of Hankel matrices when the test function is an even degree monomial.",
"In both the situations, the Dirac function appearing in the trace formulas does not depend on “$i$ \".",
"Now in this article, we study the fluctuations of linear eigenvalue statistics of Hankel matrices when the test function is an odd degree monomial.",
"We use a closed form of trace formula (see Result REF ) for Hankel matrix to find the limiting moment sequence of ${\\mbox{Tr}}(\\frac{H_n}{\\sqrt{n}})^p$ .",
"Note from Result REF that for odd $p$ , the Dirac function associated in the trace formula ${\\mbox{Tr}}(\\frac{H_n}{\\sqrt{n}})^p$ depends on “$i$ \".",
"The argument in [15] will not work for the study of linear eigenvalue statistics of Hankel matrices when the test function is an odd degree monomial.",
"For this case, we built a nice connection between the trace formula and a specific type of signed graph to find out the limiting moment sequence.",
"We use some combinatorial arguments to show that the limiting variable is non-Gaussian.",
"Figure: (a) The histogram of w 3 w_3 with i.i.d.",
"centred and normalized U[0,1]U[0,1] as the input sequence.",
"(b) The histogram of w 3 w_3 with i.i.d.",
"standard Gaussian distribution as the input sequence.",
"(c) The histogram of w 7 w_7 with i.i.d.",
"centered and normalized U[0,1]U[0,1] as the input sequence.",
"(d) The histogram of w 7 w_7 with i.i.d.",
"standard Gaussian distribution as the input sequence.",
"In all cases, the size of the Hankel matrix is 200×200200 \\times 200 and a total of 10000 random matrices are taken.Now we briefly outline the rest of the manuscript.",
"In Section we introduce some combinatorial objects and the results associated with them, needed for the proofs of theorems.",
"In Section we prove Theorem REF and discuss the existence and uniqueness of the measure corresponding to the limiting moment sequence.",
"In Section , we study some properties of the limiting distribution of the linear eigenvalue statistics of Hankel matrices."
],
[
"Preliminaries",
"We first introduce certain partitions and some integrals associated with them.",
"Later, we introduce the concept of matching and the trace formula for Hankel matrices.",
"Towards the end of the section, we introduce signed graphs and prove a result on particular types of labelling, named as all-odd labelling and all-even labelling.",
"Definition 4 Consider the set $[n]=\\lbrace 1,2 ,\\ldots , n\\rbrace $ .",
"A partition $\\pi $ of $[n]$ is called a pair-partition if each block of $\\pi $ has exactly two elements.",
"If $i,j$ belong to the same block of $\\pi $ , we write $i \\sim _{\\pi } j$ .",
"The set of all pair-partitions of $[n]$ is denoted by ${\\mathcal {P}}_2(n)$ .",
"Clearly, ${\\mathcal {P}}_2(n)= \\emptyset $ for odd $n$ .",
"For a partition $\\pi $ of $[n]$ , we define a canonical ordering of its blocks by arranging the blocks in the increasing order of their smallest elements.",
"Suppose the partition $\\pi $ contains $k$ blocks $B_1, B_2,\\ldots , B_k$ arranged in the increasing order, then the ordering gives a surjective function, also denoted by $\\pi : [n ]\\rightarrow [k]$ , given by $\\pi (i)=j, \\ \\ \\mbox{if $i \\in B_j$}.$ From a partition $\\pi $ and a family of random variables $\\lbrace y_i \\rbrace _{i \\in [k]}$ , we consider a family of random variables $\\lbrace z_i \\rbrace _{i \\in [n]}$ defined by $z_i=y_{\\pi (i)}$ .",
"Using this, we introduce the following integrals: Definition 5 Let $p,k$ be natural numbers such that $kp$ is even and $\\pi \\in {\\mathcal {P}}_2(kp)$ .",
"Also for $y_1, y_2, \\ldots , y_{\\frac{pk}{2}}$ and $1 \\le r \\le k$ , let $x_r = \\frac{1}{2}\\left(\\sum _{q=1}^{p}(-1)^q y_{\\pi ((r-1) p+q)}+1 \\right)$ .",
"Now for $1 \\le r \\le k$ , we define $U_{r}(p) =\\prod _{l=1}^{p} \\chi _{[0,1]}\\left(x_{r}-\\sum _{q=1}^{l}(-1)^{q} y_{\\pi ((r-1) p+q)}\\right)$ and $ f_k(\\pi ) = \\displaystyle \\int _{[-1,1]^{\\frac{pk}{2}}} \\prod _{r=1}^k U_{r}(p) \\, \\mathrm {d}y_1 \\mathrm {d}y_2\\cdots \\mathrm {d}y_{pk/2}$ , where $\\chi _A$ denotes the indicator function of the set $A$ .",
"Even though, $f_k(\\pi )$ depends on $p$ , we are avoiding it to lighten the notations.",
"Definition 6 Let $p_1,p_2$ be natural numbers such that $p_1+p_2$ is even and $\\pi \\in {\\mathcal {P}}_2(p+q)$ .",
"For $x_1= \\frac{1}{2}\\left(\\sum _{q=1}^{p_1}(-1)^q y_{\\pi (q)}+1 \\right)$ and $x_2= \\frac{1}{2}\\left(\\sum _{q=1}^{p_2}(-1)^q y_{\\pi (p_1+q)}+1 \\right)$ , we define $\\tilde{U}_{p_1}=\\prod _{\\ell =1}^{p_1} \\chi _{[0,1]}\\left(x_{1}-\\sum _{q=1}^{\\ell }(-1)^{q} y_{\\pi (q)}\\right)$ , $\\tilde{U}_{p_2}=\\prod _{\\ell =1}^{p_2} \\chi _{[0,1]}\\left(x_{2}-\\sum _{q=1}^{\\ell }(-1)^{q} y_{\\pi (p_1+q)}\\right)$ and $g_{p_1,p_2}(\\pi )=\\displaystyle \\int _{[-1,1]^{\\frac{p_1+p_2}{2}}} \\tilde{U}_{p_1} \\tilde{U}_{p_2} \\, \\mathrm {d}y_1 \\mathrm {d}y_2 \\cdots \\mathrm {d}y_{\\frac{p_1+p_2}{2}}$ .",
"For a vector $J= (j_1,j_2, \\ldots , j_p) \\in \\mathbb {Z}^p$ , we define the multi-set $S_J$ as $S_J=\\lbrace j_1, j_2, \\ldots , j_p \\rbrace .$ For a sequence of vectors $J_1, J_2, \\ldots $ , we shall use the notation $J_r=(j_1^r, j_2^r,\\ldots , j_p^r)$ to denote the components of $J_r$ and $S_{J_r}$ to denote the multi-set associated with $J_r$ .",
"The concept of matching is an important combinatorial notion connected to random matrices.",
"Here, we define the following two notions of matching connected to Hankel matrices.",
"Definition 7 For $J \\in \\mathbb {Z}^p$ , $j \\in \\mathbb {Z}$ is said to be a self-matched element if $j$ appears at least twice in $S_J$ .",
"For $J_1 \\in \\mathbb {Z}^p$ and $J_2 \\in \\mathbb {Z}^q$ , $j \\in \\mathbb {Z}$ is said to be a cross-matched element if $j \\in S_{J_1} \\cap S_{J_2}$ .",
"Additionally, we say two vectors $J_1$ and $J_2$ are cross-matched if $S_{J_1} \\cap S_{J_2}$ is non-empty.",
"The following definition provides the natural extension of the concept of matching to any partition $\\pi $ of $[p+q]$ .",
"Definition 8 Let $\\pi $ be a partition of $[p+q]$ for fixed $p,q \\in \\mathbb {N}$ .",
"Consider $J_1=(1,2,\\ldots ,p)$ and $J_2=(p+1,p+2,\\ldots ,p+q)$ .",
"Then, we say an element $j \\in J_i$ is self-matched if the intersection of the block of $\\pi $ containing $j$ , and $S_{J_i}$ has cardinality at least two.",
"we say an element $j, 1 \\le j \\le p+q$ is cross-matched if the block of $\\pi $ containing $j$ , has a non-empty intersection with both $S_{J_1}$ and $S_{J_2}$ .",
"We say a partition $\\pi $ is cross-matched if at least one block of $\\pi $ has non-empty intersection with both $S_{J_1}$ and $S_{J_2}$ .",
"Note that if $J_1$ and $J_2$ are as above and $(p+q)$ is odd, then every $\\pi \\in {\\mathcal {P}}_2(p+q)$ is cross-matched.",
"A trace formula for the product of band Hankel matrices was stated in [1].",
"Now in the following result, we recall the trace formula for the product of Hankel matrices by choosing band width equal to the order of matrices.",
"Result 9 (Result 4, [1]) Suppose $H^{(r)}_{n}$ are Hankel matrices with input sequence $\\lbrace x^{(r)}_i\\rbrace _{ i \\in \\mathbb {Z} }$ for $r=1,2, \\ldots $ , respectively.",
"Then $ &{\\mbox{Tr}}(H^{(1)}_{n} H^{(2)}_{n} \\cdots H^{(p)}_{n}) \\nonumber \\\\& \\quad = {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\sum _{i=1}^{n} \\sum _{j_{1}, \\ldots , j_{p}=-n}^{n} \\prod _{r=1}^{p} x^{(r)}_{j_{r}} \\prod _{\\ell =1}^{p} \\chi _{[1, n]} \\left(i-\\sum _{q=1}^{\\ell }(-1)^{q} j_{q}\\right) \\delta _{0} (\\sum _{q=1}^{p}(-1)^{q} j_{q}), &p \\; \\text{even}; \\\\\\displaystyle \\sum _{i=1}^{n} \\sum _{j_{1}, \\ldots , j_{p}=-n}^{n} \\prod _{r=1}^{p} x^{(r)}_{j_{r}} \\prod _{\\ell =1}^{p} \\chi _{[1, n]}\\left(i-\\sum _{q=1}^{\\ell }(-1)^{q} j_{q}\\right) \\delta _{2 i-1-n} (\\sum _{q=1}^{p}(-1)^{q} j_{q}), & p \\; \\text{odd},\\end{array}\\right.",
"}$ where $\\delta _x$ is the Dirac delta function at $x$ and $\\chi $ is the indicator function.",
"Our next definition is connected to the trace formula (REF ).",
"For an odd number $p$ and $i$ such that $1 \\le i \\le n$ , we define $ A_{p,i} &=\\Big \\lbrace (j_{1}, j_2, \\ldots , j_{p}) \\in \\left\\lbrace 0, \\pm 1, \\ldots , \\pm n\\right\\rbrace ^{p}: \\sum _{q=1}^{p} (-1)^{q} j_{q}=2i-1-n\\Big \\rbrace , \\nonumber \\\\A_p &= \\bigcup _{i=1}^{n}A_{p,i}.$ Definition 10 Let $p_1, p_2 \\ldots , p_r$ be finitely many odd natural numbers.",
"We define $B_{p_1, p_2, \\ldots , p_r} \\subseteq A_{p_1} \\times A_{p_2} \\times \\cdots \\times A_{p_r}$ as the set of all $(J_1, J_2, \\ldots , J_r) \\in A_{p_1} \\times A_{p_2} \\times \\cdots \\times A_{p_r} $ , such that each element of the multi-set $\\bigcup _{\\ell =1}^r S_{J_\\ell }$ has cardinality at least two, where $A_{p_i}$ is as defined in (REF ).",
"When $r$ is clear from context and $p_1=p_2=\\cdots = p_r =p$ , we denote $B_{p_1, p_2, \\ldots , p_r}$ simply by $B_{p}$ .",
"For real-valued functions $f$ and $g$ , we say $f=O(g)$ if there exists a constant $C>0$ and $x_0 \\in \\mathbb {R}$ such that $|f(x)| \\le Cg(x)$ for all $x \\ge x_0$ .",
"The next lemma gives the order of cardinality of $B_{p_1 , p_2, \\ldots ,p_r}$ .",
"Lemma 11 For odd natural numbers $p_1 , p_2, \\ldots , p_r$ , the cardinality of $B_{p_1, p_2, \\ldots ,p_r}$ is given by $\\#B_{p_1, p_2, \\ldots ,p_r}= O(n^{\\lfloor \\frac{p_1+p_2+\\cdots + p_r}{2}\\rfloor }),$ where $\\#\\lbrace \\cdot \\rbrace $ denotes the cardinality of the set $\\lbrace \\cdot \\rbrace $ and $\\lfloor x \\rfloor $ denotes the greatest integer less than or equal to $x$ .",
"Consider a vector $(J_1, J_2, \\ldots , J_r) \\in B_{p_1, p_2, \\ldots ,p_r}$ with $J_\\ell = (j_1 ^\\ell , j_2^\\ell ,\\ldots ,j_{p_\\ell }^\\ell )$ for each $1 \\le \\ell \\le r$ .",
"Our objective here is to enumerate the number of possibilities for vectors $(J_1, J_2, \\ldots , J_r) \\in B_{p_1, p_2, \\ldots ,p_r}$ .",
"Define a relation $\\sim $ on $\\lbrace (\\ell ,s) : 1\\le \\ell \\le r, 1\\le s \\le p_\\ell \\rbrace $ by $(\\ell _1,s_1) \\sim (\\ell _2,s_2)$ if $j_{s_1}^{\\ell _1}=j_{s_2}^{\\ell _2}$ .",
"Observe that the number of possible partitions $\\pi $ of $\\lbrace (\\ell ,s) : 1\\le \\ell \\le r, 1\\le s \\le p_\\ell \\rbrace $ is finite.",
"We prove that for each partition $\\pi $ , the number of choices of $(J_1, J_2, \\ldots , J_r)\\in B_{p_1, p_2, \\ldots ,p_r}$ is at most $O(n^{\\lfloor \\frac{p_1+p_2+\\cdots + p_r}{2}\\rfloor })$ .",
"For that, fix a particular partition $\\pi $ and define for each $1 \\le \\ell \\le r$ , $T_\\ell $ as the cardinality of the set $\\lbrace j_s^\\ell \\in S_{J_\\ell }: j_s^\\ell \\notin S_{J_u}, \\, \\forall \\, u < \\ell \\rbrace $ , where $S_{J_\\ell }$ and $S_{J_u}$ are as defined in Definition REF .",
"We enumerate the vectors $(J_1, J_2, \\ldots , J_r)$ by sequentially allotting values for components of each $J_\\ell $ .",
"Notice that the degree of freedom of choosing elements from $A_p$ equals to the degree of freedom of choosing $j_k$ 's freely as once $j_k$ 's are fixed, $i$ becomes fixed accordingly.",
"Therefore, the number of choices for choosing components of $J_1$ is $O(n^{T_1})$ .",
"Continuing forward in the same fashion, it is clear that once $J_1, J_2, \\ldots , J_{\\ell -1}$ are chosen, the number of ways of choosing $J_\\ell \\in A_{p_\\ell }$ is $O(n^{T_\\ell })$ .",
"Thus the total number of possibilities is $O(n^{T_1 + T_2 + \\cdots +T_r})$ .",
"Observe that $T_1 + T_2 + \\cdots +T_r$ is the number of distinct values for components of the vectors $J_\\ell , 1 \\le \\ell \\le r$ , and because each component is repeated at least twice, it follows that $T_1 + T_2 + \\cdots +T_r$ is bounded above by $\\lfloor \\frac{p_1+p_2+\\cdots + p_r}{2} \\rfloor $ .",
"This completes our proof.",
"Recall that in Definition REF , we had imposed the condition that each element of the multi-set $\\bigcup _{\\ell =1}^{r} S_{J_\\ell }$ should be repeated at least twice.",
"Note that, this condition plays a role only in obtaining an upper bound for $T_1 + T_2 + \\cdots +T_r$ .",
"Therefore the cardinality of any subset of $U \\subseteq A_{p_1} \\times A_{p_2} \\times \\cdots \\times A_{p_r}$ is at most $O(n^{T})$ , where $T$ is the maximum of $T_1 + T_2 + \\cdots +T_r$ among elements of $U$ .",
"We summarize this in the following corollary.",
"Corollary 12 Let $p_1 , p_2, \\ldots , p_r$ be finitely many odd natural numbers and $T_1, T_2, \\ldots , T_r$ as defined in the proof of Lemma REF .",
"Let $U_k \\subseteq A_{p_1} \\times A_{p_2} \\times \\cdots \\times A_{p_r}$ be the collection of all elements $(J_1,J_2,\\ldots , J_r)$ such that $T:= T_1 + T_2 +\\cdots + T_r$ is less than or equal to $k$ , where $A_{p_i}$ is as defined in (REF ).",
"Then the cardinality of $U_k$ is given by $\\#U_k= O(n^k).$ Now we introduce certain concepts from graph theory which will be used later in the proofs of theorems.",
"Definition 13 A signed graph is a graph $G=(V,E)$ with a labelling of edges $f:E \\rightarrow \\lbrace +1, -1\\rbrace $ .",
"A signed graph induces a canonical labelling $f_*: V \\rightarrow \\lbrace +1, -1\\rbrace $ on vertices, defined by $f_*(v)=\\prod _{u \\in V: (u,v)\\in E} f(u,v)$ .",
"A labelling $f$ of edges such that the canonical labelling on all vertices are $-1$ is called an all-odd labelling and a labelling such that the canonical labelling on all vertices are +1 is called an all-even labelling.",
"For a given graph $G$ , we denote all-odd labellings and all-even labellings of $G$ by $G^{(ao)}$ and $G^{(ae)}$ , respectively.",
"Lemma 14 Let $G=(V,E)$ be a finite connected graph with no loops and such that $\\#V$ is even.",
"Then $\\#G^{(ao)}= \\# G^{(ae)} = 2^{\\#E-\\#V+1}.$ We prove this lemma only for the all-odd labelling.",
"A similar argument will work for the proof of the all-even labelling.",
"First we prove the lemma for the special case when $G$ is a tree with $\\# V$ as even.",
"We use induction on $k$ , where $2k= \\# V$ .",
"For a tree, we have $\\# E = \\# V -1$ and therefore our aim is to show that for all trees, there exists a unique all-odd labelling.",
"For $k=1$ , the proof is trivial, as there exists only one edge.",
"Suppose the result holds for some $k \\ge 1$ .",
"Let $G$ be a tree such that $\\# V =2k+2$ .",
"A vertex $v \\in V$ , is called a leaf if $\\operatorname{deg}_G(v)=1$ .",
"For a finite tree $G$ , one of the following cases always occur.",
"There exists a vertex $v \\in V$ and $v_1, v_2 \\in V$ such that $(v,v_1), (v,v_2) \\in E$ and $v_1, v_2$ are leaves, there exist $v, v_1 \\in V$ such that $(v,v_1) \\in E, \\operatorname{deg}_G(v)=2$ and $v_1$ is a leaf.",
"Suppose $G$ obeys (i).",
"Consider the graph $G^{\\prime }$ obtained from $G$ by removing $v_1,v_2$ and the edges connected to them.",
"By the induction hypothesis, there exists a unique all-odd labelling $f^{\\prime }$ on $G^{\\prime }$ .",
"We construct a labelling on $G$ by defining $f(u,w)= f^{\\prime }(u,w)$ for all $(u,w) \\in E(G^{\\prime })$ , $f(v,v_1)=-1$ and $f(v,v_2)=-1$ .",
"This ensures that $f$ is an all-odd labelling.",
"Now, suppose $G$ obeys (ii).",
"Consider the graph $G^{\\prime }$ obtained from $G$ by removing $v,v_1$ and the edges connected to them.",
"Again by the induction hypothesis, there exists a unique all-odd labelling $f$ on $G^{\\prime }$ .",
"We define $f(u,w)= f^{\\prime }(u,w)$ for all $(u,w) \\in E(G^{\\prime })$ , $f(v,v_1)=-1$ and $f(v,w)=+1$ , where $(v,w) \\in E(G)$ .",
"To prove that the all-odd labelling on $G$ is unique, notice that given an all-odd labelling $f^\\prime $ on $G^\\prime $ obtained in one of the above ways, there exists a unique labelling $f$ on $G$ such that $f\\big |_{E(G^\\prime )}= f^\\prime $ .",
"Suppose $f_1$ and $f_2$ are distinct all-odd labellings on $G$ .",
"Then $f_1\\big |_{E(G^\\prime )}$ and $f_2\\big |_{E(G^\\prime )}$ are all-odd labellings on $G^\\prime $ .",
"By the induction hypothesis, $f_1\\big |_{E(G^\\prime )}=f_2\\big |_{E(G^\\prime )}$ , which implies that $f_1=f_2$ .",
"Now we prove the general case, using induction on $n= (\\#E - \\#V)$ .",
"The base case here is $n=-1$ , below which the graph $G$ ceases to be connected.",
"For $n=-1$ , $G$ is always a tree and therefore, the result holds.",
"Suppose the result holds for $n=k \\ge -1$ and $\\# V$ as even.",
"Let $G$ be a graph such that $\\#E - \\#V= k+1$ .",
"Since $k+1 \\ge 0$ , there exists at least one cycle in $G$ .",
"Let $(v_1, v_2)$ be an edge on the cycle.",
"Fix a path $P$ from $v_1$ to $v_2$ , other than the edge $(v_1,v_2)$ .",
"Consider the connected graph $G^{\\prime }$ obtained from $G$ by removing the edge $(v_1, v_2)$ .",
"Let $f^{\\prime }$ be an all-odd labelling on $G^{\\prime }$ .",
"For the rest of the proof, we fix this particular choice of path $P$ .",
"We define two labellings $f_1$ and $f_2$ on $G$ as $f_1(u,v)={\\left\\lbrace \\begin{array}{ll}f^{\\prime }(u,v) & \\mbox{if } \\ (u,v) \\in E(G^{\\prime }) \\setminus P \\\\f^{\\prime }(u,v) & \\mbox{if } \\ (u,v) \\in P \\\\+1 & \\mbox{if } \\ u=v_1, v=v_2,\\end{array}\\right.",
"}$ and $f_2(u,v)={\\left\\lbrace \\begin{array}{ll}f^{\\prime }(u,v) & \\mbox{if } \\ (u,v) \\in E(G^{\\prime }) \\setminus P \\\\- f^{\\prime }(u,v) & \\mbox{if } \\ (u,v) \\in P \\\\-1 & \\mbox{if } \\ u=v_1, v=v_2.\\end{array}\\right.",
"}$ Then it follows that $f_1$ and $f_2$ are all-odd labellings on $G^\\prime $ .",
"Furthermore, $f_1$ and $f_2$ are the only possible all-odd labellings on $G$ such that they agree with $f^\\prime $ on $ E(G^{\\prime }) \\setminus P$ .",
"Thus we get $\\# G^{(ao)} \\ge 2 \\times \\# G^{\\prime (ao)}.$ To prove the reverse inequality, we need to show that any all-odd labelling $f$ on $G$ can be obtained from an all-odd labelling $f^\\prime $ on $G^{\\prime }$ such that $f$ and $f^\\prime $ are identical on $E(G^{\\prime }) \\setminus P$ .",
"For $f$ such that $f(v_1,v_2)=+1$ , the appropriate $f^\\prime $ is $f^\\prime (u,v)={\\left\\lbrace \\begin{array}{ll}f(u,v) & \\mbox{if } \\ (u,v) \\in E(G^{\\prime }) \\setminus P \\\\f(u,v) & \\mbox{if } \\ (u,v) \\in P, \\\\\\end{array}\\right.",
"}$ and for $f$ such that $f(v_1,v_2)=-1$ , the appropriate $f^\\prime $ is $f^\\prime (u,v)={\\left\\lbrace \\begin{array}{ll}f(u,v) & \\mbox{if } \\ (u,v) \\in E(G^{\\prime }) \\setminus P \\\\-f(u,v) & \\mbox{if } \\ (u,v) \\in P.",
"\\\\\\end{array}\\right.",
"}$ Thus we get that, $\\# G^{(ao)} &= 2 \\times \\# G^{\\prime (ao)}= 2 \\times 2^{k-1} = 2^k = 2^{\\# E -\\# V +1}.$ This completes the proof of lemma.",
"Figure: An example of constructing all-odd labellings from an all-odd labelling of a subgraph.",
"(a) is a sub-graph of the graphs (b) and (c), with the dotted line representing a path from v 1 v_1 to v 2 v_2.",
"(b) and (c) are all-odd labellings obtained from an all-odd labelling on (a)(a) with PP as the path given by the dotted line.The number $(\\# E -\\# V +1)$ is known as the cyclomatic number or the first Betti number of a connected graph.",
"Now we define a graph associated with pair-partitions.",
"Let $\\pi \\in \\mathcal {P}_2(pk)$ be a pair-partition of $[pk]$ .",
"We construct a graph $G_{\\pi }=(V_{\\pi },E_{\\pi })$ associated with $\\pi $ in the following fashion: We define $V_{\\pi }=\\lbrace 1, 2,\\ldots , k\\rbrace $ and for $r \\ne s$ , $(r,s) \\in E_{\\pi }$ if $\\pi $ contains a block $B$ such that $B=\\lbrace (r-1)p+q_1, (s-1)p +q_2 \\rbrace $ for some $0 \\le q_1,q_2<p$ .",
"A maximal connected subgraph of $G_\\pi $ is called a cluster.",
"For a pair-partition $\\pi \\in \\mathcal {P}_2(pk)$ and integers $j_1, j_2, \\ldots , j_{pk/2}$ , the concept of connectedness and cluster can be extended to vectors $(J_1,J_2,\\ldots , J_k)$ where $J_r = (j_1^r,j_2^r,\\ldots , j_p^r)$ with $j_q^r= j_{\\pi \\left((r-1)p+q\\right)}$ for all $r,q$ and $\\pi $ is the surjective map defined in (REF ).",
"Definition 15 We say two vectors $J_r,J_s$ with $1\\le r,s \\le k$ are connected if the integers $r,s$ are connected in $G_{\\pi }$ via a path.",
"We say vectors $\\lbrace J_{r_1},\\ldots ,J_{r_k}\\rbrace $ form a cluster if $\\lbrace r_1,\\ldots , r_k\\rbrace $ is the vertex set of a cluster in $G_\\pi $ .",
"Note that the definition of connectedness and cluster only depend on the graph $G_{\\pi }$ and is independent of the choice of $j_1,j_2,\\ldots , j_{pk/2}$ .",
"For a given $\\pi \\in \\mathcal {P}_2(pk)$ and an associated graph $G_{\\pi }$ , we define $m(\\pi ) = \\# V_{\\pi }-c= k-c,$ where $c$ is the number of connected components of $G_{\\pi }$ ."
],
[
"Limiting moment sequence",
"We first outline the proof of Theorem REF and discuss its departure from the technique of the proof of Theorem 6.4 of [15], and then we prove Theorem REF .",
"In Section REF , we discuss the existence and uniqueness of measure corresponding to the limiting moment sequence, and in Section REF , we study the fluctuations of linear eigenvalue statistics of Hankel matrices for polynomial test functions.",
"Now we explain how the case of odd degree monomial test functions is significantly difficult than the case of even degree monomials dealt in [15].",
"Recall $w_p$ from (REF ).",
"Using Result REF , we get $\\mbox{E}[w_p^k]= \\frac{1}{n^{\\frac{pk}{2}}}\\displaystyle \\sum _{J_1, \\ldots , J_k} I_{J_1}I_{J_2}\\cdots I_{J_k}\\Delta _{J_1}\\Delta _{J_2}\\cdots \\Delta _{J_k}\\mbox{E}[x_{J_1}x_{J_2}\\cdots x_{J_k}],$ where for each $r=1,2, \\ldots , k$ , $J_r=(j_1^r, j_2^r,\\ldots , j_p^r), \\ I_{J_r} = \\prod _{\\ell =1}^p \\chi _{[1, n]}\\left(i_r-\\sum _{q=1}^{\\ell }(-1)^q j^r_{q}\\right), \\ x_{J_r}= \\prod _{\\ell =1}^{p} x_{j^r_\\ell },$ with $j_q^r \\in \\lbrace -n,-n+1,\\ldots , 0, \\ldots , n\\rbrace $ and $\\Delta _{J_r}= {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\delta _{2 i_r-n-1}\\left( \\sum _{q=1}^{p}(-1)^{q} j_{q}^r\\right) & \\quad \\mbox{if } p \\text{ is odd},\\\\\\displaystyle \\delta _{0}\\left( \\sum _{q=1}^{p}(-1)^{q} j_{q}^r\\right) & \\quad \\text{if } p \\text{ is even}.\\end{array}\\right.",
"}$ For each choice of $J=(J_1,J_2,\\ldots , J_k)$ , we define $j_{(r-1)p+s} :=j^r_s$ for $r=1,2,\\ldots , k$ and $1 \\le s \\le p$ .",
"Now we define a relation $\\sim _J$ on $[pk]$ such that $r_1 \\sim _J r_2$ if $j_{r_1}=j_{r_2}$ .",
"Then (REF ) can be written as $\\mbox{E}[w_p^k]=\\frac{1}{n^{\\frac{pk}{2}}}\\displaystyle \\sum _{\\pi \\in \\mathcal {P}(pk)} \\sum _{J_\\pi }I_{J_1}I_{J_2}\\cdots I_{J_k}\\Delta _{J_1}\\Delta _{J_2}\\cdots \\Delta _{J_k}\\mbox{E}[x_{J_1}x_{J_2}\\cdots x_{J_k}],$ where $\\mathcal {P}(pk)$ is the set of all partitions of $[pk]$ and $J_\\pi $ is the set of all possible $J=(J_1,J_2,\\ldots , J_k)$ such that the block structure of $(J_1,J_2,\\ldots , J_k)$ under the relation $\\sim _J$ is $\\pi $ .",
"In the proof of Theorem REF , we show that only pair-partitions of $[pk]$ contribute to $\\mbox{E}[w_p^k]$ in the limit and thus we argue further with fixed pair-partitions $\\pi $ .",
"For a fixed pair-partition $\\pi $ , $\\lbrace j_q^r\\rbrace $ has $\\frac{pk}{2}$ free choices and we represent each choice as a point in $\\mathbb {R}^{pk/2}$ given by $W=(j_1,j_2,\\ldots , j_{\\frac{pk}{2}})$ .",
"Similarly $\\lbrace i_r\\rbrace $ in (REF ) has $k$ choices which we represent by the point $V=(i_1,i_2,\\ldots , i_k)$ .",
"Consider the $(k+pk/2)$ dimensional space, represented in Figure REF as a plane, with a general element as (V,W).",
"For a pair-partition $\\pi $ , consider the following two systems of equations: $\\sum _{q=1}^{p}(-1)^{q} j_{\\pi ((r-1)p+q)}&=0, \\ \\forall \\ 1 \\le r \\le k, \\\\\\sum _{q=1}^{p}(-1)^{q} j_{\\pi ((r-1)p+q)}&=2 i_r-1-n, \\ \\forall \\ 1 \\le r \\le k ,$ where $\\pi ((r-1)p+q)$ is the image of $(r-1)p+q$ under the surjective map $\\pi $ given in (REF ).",
"Note that for the summand in (REF ) to be non-zero for some point $(V,W) \\in \\mathbb {R}^{k+pk/2}$ , the components of it should obey the systems of equations (REF ) and () for even and odd cases respectively.",
"For odd $p$ and even $k$ , the solution space of the systems of equations () is a $pk/2$ -dimensional spaces represented in Figure 2 as line segment MN.",
"Two differences come up between the $p$ odd and $p$ even cases.",
"The first difference is that for even case, the solution space of (REF ) is independent of $n$ , whereas for the odd case, the solution space MN is changing with respect to $n$ .",
"The second and less obvious difference is the following: Suppose for even values of $p$ and $k$ , $(j_1,j_2,\\ldots , j_{pk/2})$ obeys the system of equations (REF ) for some choice of $V=(i_1,\\ldots ,i_k)$ and $n \\in \\mathbb {N}$ .",
"Then $(j_1,j_2,\\ldots , j_{pk/2})$ is a solution of the system of equations (REF ) for every $m \\ge n$ .",
"This is not true for the odd $p$ case and in fact, a $(j_1, j_2, \\ldots , j_{pk/2})$ cannot be a solution for both odd and even $n$ simultaneously.",
"The challenge of our work is to control, these additional dependencies and show that $\\mbox{E}[w_p^k]$ still converges to a definite integral, just like in the even case.",
"The first issue is taken care of by Proposition REF and the parity dependence is taken care of by Lemma REF .",
"Figure: A pictorial representation of the (k+pk/2)(k+pk/2)-dimensional space corresponding to the solution space of the system of equations ().",
"The line segment IJ corresponds to the solution space of 2i r -1=∑ q=1 p (-1) q y π((r-1)p+q) 2i_r-1=\\sum _{q=1}^{p} (-1)^{q} y_{\\pi ((r-1)p+q)} and the line segment MN corresponds to the solution space of 2i r -1-1 n=∑ q=1 p (-1) q y π((r-1)p+q) 2i_r-1-\\frac{1}{n}=\\sum _{q=1}^{p} (-1)^{q} y_{\\pi ((r-1)p+q)}.Notice in Figure REF that the distance between MN and IJ under the sup norm metric is $1/2n$ and therefore, the distance between them goes to zero as $n \\rightarrow \\infty $ .",
"Recall the definite integral $f_k(\\pi )$ in Definition REF .",
"Considering $f_k(\\pi )$ as a function on $\\mathbb {R}^{pk/2+k}$ , with the first $k$ components representing $\\lbrace x_r\\rbrace $ and the rest $pk/2$ components representing $\\lbrace y_i\\rbrace $ , we obtain that $f_k(\\pi )$ is an integral over IJ.",
"The next proposition shows that $f_k(\\pi )$ can be approximated by appropriate summations on the space MN.",
"Proposition 16 Let $p$ be a fixed odd natural number and $k$ be an even natural number.",
"For $n \\in \\mathbb {N}$ and $\\pi \\in \\mathcal {P}_2(pk)$ , if we define $\\mathcal {R}_n := \\frac{1}{n^{\\frac{pk}{2}}}\\displaystyle {\\sum }_{j_1,j_2,\\ldots , j_{pk/2}=-n}^n \\prod _{r=1}^{k}\\prod _{\\ell =1}^p \\chi _{[1, n]}\\left(i_r-\\sum _{q=1}^{\\ell }(-1)^q j_{\\pi \\left((r-1)p+q\\right)}\\right),$ then as $n \\rightarrow \\infty $ , $\\mathcal {R}_n$ converges to $f_k(\\pi )$ , where $i_r =\\frac{1}{2}\\left( \\sum _{q=1}^{p}(-1)^q j_{\\pi \\left((r-1)p+q\\right)}+n+1 \\right)$ and $f_k(\\pi )$ is as in Definition REF .",
"For $n \\in \\mathbb {N}$ and $\\pi \\in \\mathcal {P}_2(pk)$ , we define $\\mathcal {T}_n= \\frac{1}{n^{\\frac{pk}{2}}}\\displaystyle \\sum _{j_1,j_2,\\ldots , j_{pk/2}=-n}^n \\prod _{r=1}^k\\prod _{\\ell =1}^p \\chi _{[0, 1]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right),$ where $\\tilde{i}_r =\\frac{1}{2}\\left( \\sum _{q=1}^{p}(-1)^q j_{\\pi \\left((r-1)p+q\\right)}+ n \\right)$ .",
"Note that $\\mathcal {T}_n$ is a Riemann sum of the integral $f_k(\\pi )$ .",
"Now $&|\\mathcal {R}_n - \\mathcal {T}_n| \\\\& = \\bigg | \\frac{1}{n^{\\frac{pk}{2}}} \\hspace{-8.0pt}\\displaystyle {\\sum }_{j_1,j_2,\\ldots , \\atop j_{pk/2}=-n}^n \\hspace{-8.0pt}\\left(\\prod _{r=1}^{k}\\prod _{\\ell =1}^p \\chi _{[\\frac{1}{n}, 1]}\\left(\\frac{i_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)-\\prod _{r=1}^{k}\\prod _{\\ell =1}^p \\chi _{[0, 1]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)\\right)\\bigg |\\\\& \\le \\frac{1}{n^{\\frac{pk}{2}}}\\displaystyle \\hspace{-8.0pt}{{\\sum }}_{j_1,j_2, \\ldots , \\atop j_{pk/2}=-n}^n \\hspace{-8.0pt}{{\\sum }}_{r=1}^{k}{{\\sum }}_{\\ell =1}^p\\left|\\left(\\chi _{[\\frac{1}{n}, 1]}\\left(\\frac{i_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)-\\chi _{[0, 1]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)\\right)\\right|.$ Since $\\frac{i_r}{n}=\\frac{\\tilde{i}_r}{n}+\\frac{1}{2n}$ , we have $\\chi _{[\\frac{1}{n}, 1]}\\left(\\frac{i_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right) = \\chi _{\\left[ \\frac{1}{2n}, 1-\\frac{1}{2n}\\right]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right).$ Thus for fixed $r$ and $\\ell $ , $& \\left|\\chi _{[\\frac{1}{n}, 1]} \\left(\\frac{i_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)-\\chi _{[0, 1]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right) \\right|\\nonumber \\\\& = \\left(\\chi _{\\left[0, \\frac{1}{2n} \\right]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)+\\chi _{[1-\\frac{1}{2n}, 1]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)\\right).$ We show that for each choice of $r,\\ell $ , the number of possibilities of $\\lbrace j_1,j_2,\\ldots , j_{pk/2}\\rbrace $ such that the right side of (REF ) is non-zero is of the order $O(n^{pk/2-1})$ .",
"Observe that $\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n} &= \\frac{1}{2}\\left(\\sum _{q=1}^{p} (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}+1\\right)-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\nonumber \\\\&= \\frac{1}{2}\\left(1+\\sum _{q=\\ell +1}^{p} (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right).$ Suppose $j_2,j_3,\\ldots , j_{pk/2}$ are chosen.",
"Since the length of the interval $[0,1/2n]$ is $1/2n$ , we get that there exist at most two values of $j_1=j_{\\pi (1)}$ such that the expression of (REF ) belongs to the interval $[0,1/2n]$ .",
"The same reasoning also implies that there exist at most only two values of $j_1$ such that the expression of (REF ) belongs to the interval $[1-\\frac{1}{2n},1]$ .",
"Therefore there is a reduction in the degree of freedom for choosing $j_k$ 's.",
"As a consequence, $\\frac{1}{n^{pk/2}}\\hspace{-3.0pt}\\sum _{j_1,j_2, \\ldots , \\atop j_{pk/2}=-n}^n\\hspace{-5.0pt}\\left|\\chi _{[\\frac{1}{n}, 1]} \\left(\\frac{i_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)-\\chi _{[0, 1]}\\left(\\frac{\\tilde{i}_r}{n}-\\sum _{q=1}^{\\ell } (-1)^q\\frac{j_{\\pi \\left((r-1)p+q\\right)}}{n}\\right)\\right|= O\\left(\\frac{1}{n}\\right).$ Thus, $|\\mathcal {R}_n - \\mathcal {T}_n|$ converges to zero.",
"Since $\\mathcal {T}_n$ is the Riemann sum of the integral $f_k(\\pi )$ , this proves our result.",
"Now we prove Theorem REF .",
"First note from (REF ) that $(w_p)^k= [{\\mbox{Tr}}(A_n^p)]^k.$ Now using Result REF , we get $\\mbox{E}[w_p^k]= \\mbox{E}\\left[ \\frac{1}{n^{p/2}}\\sum _{J \\in A_p} x_J I_J \\right]^k = \\frac{1}{n^{\\frac{pk}{2}}}\\displaystyle \\sum _{J_1\\in A_p, \\ldots , J_k \\in A_p} I_{J_1}I_{J_2}\\cdots I_{J_k}\\mbox{E}[x_{J_1}x_{J_2}\\cdots x_{J_k}],$ where $A_p$ is as in (REF ) and for each $r=1,2, \\ldots , k$ , $ J_r=(j_1^r, j_2^r,\\ldots , j_p^r), \\ I_{J_r} = \\prod _{\\ell =1}^p \\chi _{[1, n]}\\left(i_r-\\sum _{q=1}^{\\ell }(-1)^q j^r_{q}\\right), \\ x_{J_r}= \\prod _{\\ell =1}^{p} x_{j^r_\\ell }.$ We claim that only pair-partitions of $[pk]$ contribute to (REF ).",
"Consider $S_{J_r}=\\lbrace j^r_1, j^r_2,\\ldots , j^r_p \\rbrace $ for $1 \\le r \\le k$ .",
"For $(J_1,J_2,\\ldots , J_k)$ to have non-zero contribution in (REF ), it is necessary that each element of $S_{J_1}\\cup S_{J_2}\\cup \\cdots \\cup S_{J_k}$ has multiplicity at least two, as $\\mbox{E}[x_j]=0$ for each $j$ .",
"It follows from Corollary REF that if any element has multiplicity greater than or equal to three, then the number of choices for choosing $(J_1,J_2,\\ldots ,J_k)$ would be $O(n^{\\lfloor \\frac{pk-1}{2}\\rfloor })$ .",
"Thus the contribution of all such terms to (REF ) would be $O\\left(\\frac{1}{n^{\\frac{pk}{2}}} \\times n^{\\lfloor \\frac{pk-1}{2}\\rfloor }\\right)= O(n^{-\\frac{1}{2}})$ .",
"This shows that when $k$ is odd, $\\mbox{E}[w_p^k]= o(1)$ .",
"Now suppose $k$ is even.",
"So, from the above discussion it is clear that for a contribution of the order $O(1)$ in $\\mbox{E}[w_p^k]$ , we need to consider only pair-partitions.",
"Since we have $\\mbox{E}[x_i^2]=1$ for all $i$ , (REF ) can be rewritten as $\\mbox{E}[w_p^k] &= \\displaystyle \\frac{1}{n^{pk/2}} \\sum _{J_1,J_2,\\ldots , J_{k}} \\sum _{i_1,\\ldots ,i_k=1}^n I_{J_1}I_{J_2}\\cdots I_{J_k} \\Delta _{J_1} \\Delta _{J_2} \\cdots \\Delta _{J_k}\\nonumber \\\\&=\\displaystyle \\frac{1}{n^{pk/2}}\\sum _{\\pi \\in {\\mathcal {P}}_2(kp)} \\sum _{j_1,j_2,\\ldots , j_{pk/2}=-n \\atop j_1\\ne j_2\\ne \\cdots \\ne j_{pk/2}}^n \\sum _{i_1,\\ldots ,i_k=1}^n I_{J_1}I_{J_2}\\cdots I_{J_k} \\Delta _{J_1} \\Delta _{J_2} \\cdots \\Delta _{J_k},$ where $I_{J_r}$ is as in (REF ) and $\\Delta _{J_r}=\\delta _{2 i_r-n-1}\\left( \\sum _{q=1}^{p}(-1)^{q} j_{q}^r\\right)$ with $j_q^r= j_{\\pi \\left((r-1)p+q\\right)}$ .",
"Henceforth in this proof, we shall use $j_q^r$ to denote $j_{\\pi \\left((r-1)p+q\\right)}$ .",
"For a fixed $\\pi \\in \\mathcal {P}_2(pk)$ , consider the summation in (REF ) corresponding to $\\pi $ , that is, $\\sum _{j_1,j_2,\\ldots , j_{pk/2}=-n \\atop j_1\\ne j_2\\ne \\cdots \\ne j_{pk/2}}^n \\sum _{i_1,\\ldots ,i_k=1}^n I_{J_1}I_{J_2}\\cdots I_{J_k} \\Delta _{J_1} \\Delta _{J_2} \\cdots \\Delta _{J_k}.$ Note that for a fixed $\\pi $ and $j_1,j_2,\\ldots , j_{pk/2}$ , the term $\\Delta _{J_r}$ is non-zero for some choice of $i_r$ , if and only if both the following conditions are satisfied: $-(n-1) \\le \\sum _{q=1}^{p}(-1)^{q} j_q^r \\le n-1$ , $ \\sum _{q=1}^{p}(-1)^{q} j_q^r$ and $n$ have different parity.",
"Therefore (REF ) can be written as $\\mbox{E}[w_p^k]= \\frac{1}{n^{pk/2}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(kp)}\\sum _{j_1,\\ldots , j_{pk/2}=-n \\atop j_1\\ne \\cdots \\ne j_{pk/2}}^n \\prod _{r=1}^k \\left(I_{J_r} \\ \\chi _{[-(n-1),n-1]}\\big \\lbrace \\sum _{q=1}^p (-1)^q j_q^r\\big \\rbrace \\ \\delta _{-1}\\left( (-1)^{\\sum _{q=1}^{p}(-1)^q j_q^r+n}\\right) \\right),$ where $I_{J_r}$ is as in (REF ) with $i_r$ being a function of $j_k$ 's defined by $i_r=\\frac{1}{2}\\left( \\sum _{q=1}^{p}(-1)^q j_q^r+n+1 \\right)$ .",
"Observe that $ \\chi _{\\left[1, n\\right]}\\left(i_r-\\sum _{q=1}^{p}(-1)^q j_q^r\\right)$ is non-zero for some $i_r$ only if (i) is satisfied.",
"Thus, the above equation becomes $\\mbox{E}[w_p^k]&=\\frac{1}{n^{pk/2}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(kp)}\\sum _{j_1,\\ldots , j_{pk/2}=-n \\atop j_1\\ne \\cdots \\ne j_{pk/2}}^n \\prod _{r=1}^k \\left(I_{J_r}\\ \\delta _{-1}\\left( (-1)^{\\sum _{q=1}^{p}(-1)^q j_q^r+n}\\right)\\right)\\nonumber \\\\&=\\frac{1}{n^{pk/2}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(kp)}\\sum _{j_1,\\ldots , j_{pk/2}=-n}^n \\prod _{r=1}^k \\left(I_{J_r}\\ \\delta _{-1}\\left( (-1)^{\\sum _{q=1}^{p}(-1)^q j_q^r+n}\\right)\\right)+o(1),$ where the last equality follows from Corollary REF and our earlier observation that the contribution of partitions other than pair-partitions is $o(1)$ .",
"For the rest of the proof, we fix a partition $\\pi \\in \\mathcal {P}_2(pk)$ .",
"For the chosen $\\pi \\in \\mathcal {P}_2(pk)$ , consider $(J_1, J_2, \\ldots , J_k)$ such that $J_{r}=\\left(j_{1}^{r}, j_{2}^{r}, \\ldots , j_{p}^{r}\\right)$ where $j_q^r=j_{\\pi \\left((r-1)p+q\\right)}$ .",
"For $s \\ne r$ , we define $J_{r,s}$ as the set of all cross-matched elements between $J_r$ and $J_s$ , and $J_{r,r}$ as the set of all self-matched elements in $J_r$ .",
"As $\\sum _{J_{r,r}} (-1)^qj_q^r$ is even, the parity of $\\sum _{q=1}^p (-1)^q j_q^r$ is determined by $\\lbrace \\sum _{s \\ne r} \\sum _{J_{r,s}} (-1)^qj_q^r \\rbrace .$ Condition (ii) imposes that for all $r$ , $\\sum _{q=1}^p j_q^r$ must have same parity.",
"We fix a particular combination of parity for $\\sum _{j_q^r \\in J_{r,s}} (-1)^q j_q^r$ such that $\\sum _{s \\ne r} \\sum _{J_{r,s}}(-1)^q j_q^r$ is odd (or even) for all $r$ .",
"This problem can be translated to assigning a signed labelling for $G_\\pi $ .",
"Notice that choosing the parity of $\\sum _{J_{r,s}} (-1)^q j_q^r$ as odd (or even) is equivalent to labelling the edge $(J_r, J_s) \\in E_{\\pi }$ as $-1$ (or +1) (see Definition REF ).",
"Hence, the condition $\\sum _{s \\ne r} \\sum _{J_{r,s}}(-1)^q j_q^r$ is odd (or even) for all $r$ , is equivalent to finding an all-odd (or all-even) labelling on $G_\\pi $ .",
"Therefore, (REF ) becomes $\\mbox{E}[w_p^k]={\\left\\lbrace \\begin{array}{ll}&\\frac{1}{n^{pk/2}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(kp)} \\sum _{\\gamma \\in G_\\pi ^{(ao)}}\\sum _{j_1,\\ldots , j_{pk/2} }\\prod _{r=1}^k I_{J_r}+o(1)\\quad \\text{if } n \\text{ is even},\\\\&\\frac{1}{n^{pk/2}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(kp)}\\sum _{\\gamma \\in G_\\pi ^{(ae)}}\\sum _{j_1,\\ldots , j_{pk/2}} \\prod _{r=1}^k I_{J_r}+o(1) \\quad \\text{if } n \\text{ is odd},\\end{array}\\right.",
"}$ where $G_\\pi ^{(ao)}$ and $G_\\pi ^{(ae)}$ are the set of all-odd labellings and all-even labellings of graph $G_\\pi $ , respectively.",
"In the above expression, the summation is taken over all $j_1,j_2,\\ldots , j_{pk/2}$ obeying the system of equations $(-1)^{\\sum _{J_{r,s}} (-1)^q j_q^r}= \\gamma \\left((r,s)\\right)\\quad \\text{ for all } (r,s) \\in E_{\\pi }.$ Let $\\gamma $ be either an all-odd labelling or an all-even labelling of $G_\\pi $ .",
"Consider an edge $(r,s) \\in E_\\pi $ .",
"Let $j_u^{r,s} \\in \\lbrace j_1,j_2, \\ldots , j_{pk/2}\\rbrace $ be a cross-matched element in $(J_r,J_s)$ .",
"Observe that the equations in (REF ) are independent in the sense that each $j_q$ appears in at most one equation in (REF ).",
"Therefore, once all other $j_q \\in J_{r,s}$ except $j_u^{r,s}$ are fixed, the parity of $j_u^{r,s}$ is either odd or even, depending on the label $\\gamma (r,s)$ .",
"Furthermore the possible values of $j_u^{r,s}$ such that $\\prod _{r}I_{J_r}$ is non-zero, belongs to an interval.",
"As a consequence for each $(r,s) \\in E_\\pi $ $\\# \\lbrace (J_1, J_2, \\ldots , J_k): \\prod _{r}I_{J_r}=1, j_u^{r,s} \\text{ is odd}\\rbrace =\\# \\lbrace (J_1, J_2, \\ldots , J_k):\\prod _{r}I_{J_r}=1, j_u^{r,s} \\text{ is even}\\rbrace +o(1).$ Thus for a fixed all-odd (all-even) labelling, the reduction on number of possibilities of $(J_1,J_2, \\ldots , J_k)$ is by a factor of $1/2^{\\# E_\\pi }$ , with an error term of the order $o(1)$ .",
"Hence the contribution due to each edge-labelling is $\\sum _{j_1,j_2,\\ldots , j_{pk/2}=-n}^n \\frac{1}{2^{\\# E_{\\pi }}}\\prod _{r=1}^k I_{J_r} + o(1),$ which does not depend on $\\gamma $ .",
"Note from Lemma REF that the number of all-odd (and all-even) labellings of a cluster of $\\pi $ is $2^{\\#E-\\#V+1}$ , where $E$ and $V$ are the edge set and the vertex set of the cluster, respectively.",
"Therefore the number of all-odd (all-even) edge labellings of $G_{\\pi }$ is $2^{\\#E_\\pi -\\#V_\\pi +c}$ , where $c$ is the number of connected components of $G_\\pi $ .",
"Hence we get $\\mbox{E}[w_p^k]&= \\frac{1}{n^{pk/2}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(kp)}2^{\\#E_\\pi -\\#V_\\pi +c}\\sum _{j_1,j_2,\\ldots , j_{pk/2}=-n}^n \\frac{1}{2^{\\#E_\\pi }}\\prod _{r=1}^k I_{J_r} +o(1)\\\\&=\\sum _{\\pi \\in {\\mathcal {P}}_2(kp)}\\frac{1}{2^{m(\\pi )}}\\times \\frac{1}{n^{pk/2}}\\sum _{j_1,j_2,\\ldots , j_{pk/2}=-n}^n \\prod _{r=1}^k I_{J_r}+o(1).$ Finally, (REF ) follows from Proposition REF .",
"This completes the proof.",
"In the following section, we discuss the existence and uniqueness of the measure corresponding to the moment sequence $\\lbrace \\beta _{k}\\rbrace $ , where $\\lbrace \\beta _k\\rbrace $ is as in (REF )."
],
[
"Existence and uniqueness of measure:",
"It is known from Hamburger's theorem (Theorem 3.8, [19]) that a sequence $\\lbrace m_k\\rbrace $ is a moment sequence of a measure on $\\mathbb {R}$ if and only if $\\lbrace m_k\\rbrace $ is a positive semi-definite sequence.",
"We observe that $\\lbrace \\beta _k\\rbrace $ of Theorem REF is a positive semi-definite sequence.",
"Consider a finite sequence $\\lbrace c_1, c_2,\\ldots , c_k \\rbrace $ of complex numbers.",
"Then, $\\sum _{i, j=1}^{k} c_{i} \\overline{c}_{j} \\beta _{i+j}=\\sum _{i, j=1}^{k} c_{i} \\overline{c}_{j}\\lim _{n \\rightarrow \\infty } \\mbox{E}w_p^{i+j}=\\lim _{n \\rightarrow \\infty } \\sum _{i, j=1}^{k} c_{i} \\overline{c}_{j} \\mbox{E}w_p^{i+j}.$ Since $\\lbrace \\mbox{E}w_p^{i}\\rbrace $ is a moment sequence, $\\sum _{i, j=1}^{k} c_{i} \\overline{c}_{j} \\mbox{E}w_p^{i+j} \\ge 0$ for each $n$ and as a result $\\lbrace \\beta _{k}\\rbrace $ is a positive semi-definite sequence.",
"Hence there exist a measure corresponding to $\\lbrace \\beta _{k}\\rbrace $ .",
"In the following lemma, we discuss the uniqueness of the moment sequence $\\lbrace \\beta _{k}\\rbrace $ .",
"Lemma 17 Let $\\lbrace \\beta _k\\rbrace $ be the moment sequence as given in (REF ).",
"Then $\\lbrace \\beta _{k}\\rbrace $ fails to obey Carleman's condition (Lemma 1.2.2, [6]).",
"Consider $f_k(\\pi )$ and $U_r(p)$ as defined in Definition REF .",
"Notice that for an even $k$ , pair-partition $\\pi \\in \\mathcal {P}_2(pk)$ and $y_1, y_2, \\ldots , y_{\\frac{pk}{2}} \\in \\left[-\\frac{1}{8p}, \\frac{1}{8p}\\right]$ , $x_r$ defined as in Definition REF belong to the range $\\left[\\frac{3}{8},\\frac{5}{8}\\right]$ for all $1 \\le r \\le k$ .",
"This implies that for $y_1, y_2, \\ldots , y_{\\frac{pk}{2}} \\in \\left[-\\frac{1}{8p}, \\frac{1}{8p}\\right]$ , $U_r(p)=1$ for each $r$ and consequently, $f_k(\\pi ) \\ge \\left(\\frac{1}{4p}\\right)^{\\frac{pk}{2}}$ and hence $ \\beta _{2k} &= \\sum _{ \\pi \\in \\mathcal {P}_2(2pk)} \\frac{1}{2^{m(\\pi )}}f_{2k}(\\pi ) \\nonumber \\\\& \\ge \\left(\\frac{1}{4p}\\right)^{pk} \\times \\frac{1}{2^{2k}} \\# \\mathcal {P}_2(2pk) = \\left(\\frac{1}{4p}\\right)^{pk} \\times \\frac{1}{2^{2k}} \\frac{(2pk)!}{2^{pk}(pk)!",
"}= \\gamma _{2k}, \\mbox{ say}.$ It follows from Stirling's approximation that the sequence $\\lbrace \\gamma _{2k}\\rbrace $ does not obey Carleman's condition.",
"As a consequence of the inequality (REF ), $\\lbrace \\beta _k\\rbrace $ also fails to obey Carleman's condition.",
"In spite of this, if we additionally assume that $\\Gamma _{p}$ is the unique distribution with moment sequence $\\lbrace \\beta _{k}\\rbrace $ , then by Theorem REF and moment method, $w_{p} \\stackrel{d}{\\rightarrow } \\Gamma _p$ .",
"Clearly, the limit $\\Gamma _p$ is universal.",
"Later, in Section , we show that $\\Gamma _p$ is non-Gaussian and has unbounded support.",
"In the following section, we study the fluctuations of linear eigenvalue statistics of Hankel matrices for polynomial test functions."
],
[
"Polynomial test function:",
"First we recall from (REF ) that for $Q(x)= \\sum _{d=1}^{p} c_d x^d$ , ${\\mbox{Tr}}[Q(A_n)] = \\sum _{d=1}^{p} c_d {\\mbox{Tr}}(A_n^d)= \\sum _{d=1}^{p} c_dw_d= w_Q, \\mbox{ say}.$ The following lemma provides the order of convergence of $\\mbox{E}[w_{p_1} w_{p_2}]$ when one of $p_i$ is even and other is odd.",
"Lemma 18 Let $p_1$ be even and $p_2$ be odd positive integers, then $\\mbox{E}[w_{p_1} w_{p_2}] =O(\\sqrt{n}).$ From Result REF , we get $ \\mbox{E}[w_{p_1} w_{p_2}] = \\frac{1}{n^{\\frac{p_1+p_2}{2}-1}} \\sum _{J_1 \\in A^{\\prime }_{p_1},J_2 \\in A_{p_2}} \\mbox{E}[x_{J_1} x_{J_2}] I_{J_{1}} I_{J_{2}},$ where $I_{J_r}, x_{J_r}$ are as in (REF ), $A_{p_2}$ is as in (REF ) and $A^{\\prime }_{p_1}$ is defined as $ A^{\\prime }_{p_1}=\\Big \\lbrace (j_{1}, j_2, \\ldots , j_{p_1}) \\in \\left\\lbrace 0, \\pm 1, \\ldots , \\pm n\\right\\rbrace ^{p_1}: \\sum _{q=1}^{p_1} (-1)^{q} j_{q}=0\\Big \\rbrace .$ Since the entries $x_i$ satisfy condition (REF ) and $p_2$ is odd, $\\mbox{E}[x_{J_1} x_{J_2}]$ has the maximum contribution if the entries of $J_1$ are pair-matched and one entry of $J_2$ is triple matched with the rest entries as pair-matched.",
"Thus $\\sum _{J_1 \\in A^{\\prime }_{p_1},J_2 \\in A_{p_2}} |\\mbox{E}[x_{J_1} x_{J_2}] I_{J_{1}} I_{J_{2}}| = O(n^{\\frac{p_1}{2} + \\frac{p_2-1}{2} }).$ Now on combining the above expression with (REF ), we get $\\mbox{E}[w_{p_1} w_{p_2}] =O(\\sqrt{n}).$ In general, if at least one of $p_i$ is even for a given set of positive integers $p_1, p_2, \\ldots , p_k$ , then $\\mbox{E}[w_{p_1} w_{p_2} \\cdots w_{p_k}] \\ge O(\\sqrt{n}).$ This shows that if $Q(x)$ is a polynomial with at least one even degree term, then for every $k\\ge 1$ , $\\mbox{E}[{\\mbox{Tr}}[Q(A_n)]]^k$ is divergent.",
"Now suppose $\\displaystyle Q(x)= \\sum _{d=1, \\atop d \\ odd}^{p} c_d x^d$ is a polynomial with odd degree terms only.",
"Then $ \\mbox{E}[w_Q]^k = \\sum _{D_k} c_{p_1} \\cdots c_{p_k} \\mbox{E}[w_{p_1} w_{p_2} \\cdots w_{p_k}],$ where $D_k = \\lbrace (p_1, p_2, \\ldots , p_k) : p_i \\in \\lbrace 1,3, \\ldots , p\\rbrace \\mbox{ for } \\ i=1,2, \\ldots ,k \\rbrace .$ By the similar arguments as used in Theorem REF , we get $ \\Gamma _{p_1,p_2, \\ldots , p_k} := \\lim _{n \\rightarrow \\infty } \\mbox{E}[w_{p_1} w_{p_2} \\cdots w_{p_k}]={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\sum _{ \\pi \\in \\mathcal {P}_2(p_1+ \\cdots +p_k)} \\frac{1}{2^{m(\\pi )}}h_k(\\pi ) & \\quad \\mbox{if } k \\text{ is even},\\\\0 & \\quad \\text{if } k \\text{ is odd},\\end{array}\\right.",
"}$ where $ h_k(\\pi ) $ will be some definite integral similar to $f_k(\\pi )$ , given in Definition REF and $m(\\pi )$ is as given in (REF ).",
"Using the above equation in (REF ), we get $ \\tilde{\\beta }_k := \\lim _{n \\rightarrow \\infty } \\mbox{E}[w_{Q}]^k={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\sum _{D_k} c_{p_1} \\cdots c_{p_k} \\Gamma _{p_1,p_2, \\ldots , p_k} & \\quad \\mbox{if } k \\text{ is even},\\\\0 & \\quad \\text{if } k \\text{ is odd}.\\end{array}\\right.",
"}$ Note from (REF ) that $\\lbrace \\tilde{\\beta }_k\\rbrace $ is a positive semi-definite sequence and therefore there exist measures on $\\mathbb {R}$ with $\\lbrace \\tilde{\\beta }_k\\rbrace $ as its moment sequence.",
"If there exists a unique distribution say, $\\Gamma _Q$ , which corresponds to $\\lbrace \\tilde{\\beta }_k\\rbrace $ , then from the moment method, we have $ w_{Q} \\stackrel{d}{\\rightarrow } \\Gamma _Q \\ \\ \\ \\mbox{ if $Q(x)$ has only odd degree terms},$ where $\\mbox{E}[\\Gamma _Q]^k= \\tilde{\\beta }_k$ ."
],
[
"Properties of $\\Gamma _p$",
"In this section, we study some properties of $\\Gamma _p$ , where $\\Gamma _p$ is any distribution on $\\mathbb {R}$ with moment sequence $\\lbrace \\beta _k\\rbrace $ .",
"The following proposition shows that the moments of $\\Gamma _p$ dominate the moments of the Gaussian distribution.",
"Proposition 19 A distribution $\\Gamma _p$ with moment sequence given by (REF ) is a non-Gaussian distribution.",
"First we recall that the moment sequence of $\\Gamma _p$ is $\\lbrace \\beta _k\\rbrace $ , where $\\lbrace \\beta _k\\rbrace $ is given in (REF ).",
"Note that to show $\\Gamma _p$ is non-Gaussian, it suffices to show that $\\beta _{4} \\ne 3 \\beta _2^2$ .",
"Now we calculate $\\beta _4$ .",
"From (REF ), we have $ \\mbox{E}[w_p^{4}]= \\frac{1}{n^{2p}}\\displaystyle \\sum _{ A_p, A_p, A_p, A_p} \\mbox{E}[x_{J_1}x_{J_2} x_{J_3} x_{J_{4}}] I_{J_1}I_{J_2}I_{J_3} I_{J_{4}},$ where $J_r \\in A_p$ with $A_{p}$ as in (REF ) and $J_r, I_{J_r}$ , $x_{J_r}$ are as given in (REF ) for each $r=1,2, 3,4$ .",
"Recall the notions of connectedness and cluster from Definition REF .",
"Depending on connectedness between $J_{r}$ 's, the following three cases arise in (REF ).",
"Case I.",
"At least one of $J_{r}$ for $r=1,2,3,4$ , is not connected with the remaining ones: Without loss of generality, suppose $J_{1}$ is not connected with other $J_{r}$ 's.",
"Then from the independence of ${x_i}$ , we get $\\frac{1}{n^{2p}}\\displaystyle \\sum _{ A_p, A_p, A_p, A_p} \\mbox{E}[x_{J_1}x_{J_2} x_{J_3} x_{J_{4}}] I_{J_1}I_{J_2}I_{J_3} I_{J_{4}} = \\frac{1}{n^{2p}} \\sum _{ A_p} \\mbox{E}[x_{J_1}] I_{J_1} \\sum _{ A_p, A_p, A_p} \\mbox{E}[x_{J_2} x_{J_3} x_{J_{4}}]I_{J_2}I_{J_3} I_{J_{4}}.$ Note that for each $i$ , $\\mbox{E}[x_i]=0$ and that $J_1$ has odd many components.",
"Corollary REF implies that $\\sum _{ A_p} \\mbox{E}[x_{J_1}] I_{J_1}$ can have maximum contribution of the order $O(n^{\\frac{p-1}{2}})$ .",
"Again, using Corollary REF , we can show that $\\sum _{ A_p, A_p, A_p} \\mbox{E}[x_{J_2} x_{J_3} x_{J_{4}}] I_{J_2}I_{J_3} I_{J_{4}}$ can has contribution of the order at most $O(n^{\\frac{3p-1}{2}})$ .",
"Thus, if $T_1$ is the contribution of this case to $\\mbox{E}[w_p^{4}]$ , then $T_1= o(1).$ Case II.",
"$J_{1}$ is connected with only one of $J_{2}, J_{3}, J_{4}$ and the remaining two of $J_{2}, J_{3}, J_{4}$ are connected only with each other: Without loss of generality, we assume $J_{1}$ is connected only with $J_{2}$ and $J_{3}$ is connected only with $J_{4}$ .",
"So, from the independence of $\\lbrace x_i\\rbrace $ , we get $\\frac{1}{n^{2p}}\\displaystyle \\sum _{ A_p, A_p, A_p, A_p} \\mbox{E}[x_{J_1}x_{J_2} x_{J_3} x_{J_{4}}] I_{J_1}I_{J_2}I_{J_3} I_{J_{4}}&= \\frac{1}{n^{2p}} \\sum _{A_p, A_p} \\mbox{E}[x_{J_1} x_{J_2}] I_{J_1}I_{J_2} \\sum _{A_p, A_p} \\mbox{E}[ x_{J_3} x_{J_{4}}] I_{J_3}I_{J_4} \\nonumber \\\\& = \\mbox{E}[w_p^{2}] \\mbox{E}[w_p^{2}].$ Observe that in this case, there are two more subcases: (i) $J_{1}$ is connected only with $J_{3}$ and $J_{2}$ is connected only with $J_{4}$ , (ii) $J_{1}$ is connected only with $J_{4}$ and $J_{2}$ is connected only with $J_{3}$ .",
"Therefore if we denote the contribution of this case to $\\mbox{E}[w_p^{4}]$ by $T_2$ , then from (REF ), we get $T_2= 3 \\big (\\mbox{E}[w_p^{2}]\\big )^2.$ Case III.",
"$\\lbrace J_{1}, J_{2}, J_{3}, J_{4}\\rbrace $ form a cluster: Suppose $T_3$ is the contribution of this case to $\\mbox{E}[w_p^{4}]$ .",
"Then from the independence of $\\lbrace x_i\\rbrace $ and $\\mbox{E}(x_{i})=0$ , we get $ T_3= \\frac{1}{n^{2p}} \\sum _{( J_{1}, J_{2}, J_{3}, J_{4}) \\in \\tilde{B}_{P_4}} \\mbox{E}[x_{J_1}x_{J_2} x_{J_3} x_{J_{4}}]I_{J_1}I_{J_2}I_{J_3} I_{J_{4}},$ where $J_r=(j^r_1, j^r_2, \\ldots , j^r_p)$ for each $r=1,2,3,4$ and $\\tilde{B}_{P_4}$ is defined as, $\\tilde{ B}_{p_4}=\\lbrace (J_{1}, J_{2}, J_{3}, J_{4}) \\in & \\ A_{p} \\times A_{p} \\times A_{p} \\times A_{p} : \\lbrace J_{1}, J_{2}, J_{3}, J_{4}\\rbrace \\mbox{ form a cluster and entries of } \\\\& \\qquad S_{J_{1}} \\cup S_{J_{2}} \\cup S_{J_{3}} \\cup S_{J_{4}} \\mbox{ have multiplicity greater than or equal to two}\\rbrace .$ Now consider $B^{\\prime }_{P_4}$ , a subset of $\\tilde{B}_{P_4}$ such that $(J_{1}, J_{2}, J_{3}, J_{4}) \\in B^{\\prime }_{P_4}$ if (i) $j_1^1=j^2_1, j^1_2= j_1^3, j^1_3= j_1^4$ , $j_1^1 \\ne j^1_2 \\ne j^1_3$ , (ii) $j^1_{2d}= j^1_{2d+1}$ for all $d=2,3, \\ldots , \\frac{p-1}{2}$ and for $r=2,3,4, \\ j^r_{2d}= j^r_{2d+1}$ for all $d=1,2, \\ldots , \\frac{p-1}{2}$ , (iii) $I_{J_1}I_{J_2}I_{J_3} I_{J_{4}}=1$ .",
"Claim A $\\# B^{\\prime }_{P_4} \\ge \\frac{n^{2p}}{8^3}$ .",
"Proof.",
"Note that if $(J_{1}, J_{2}, J_{3}, J_{4}) \\in B^{\\prime }_{P_4}$ , then from condition (ii), the entries of $J_r$ has the following constraints: $ -j_1^1 +j_2^1 -j_3^1 = 2i_1-1-n, \\ -j_1^1 = 2i_2-1-n, \\ -j_2^1 = 2i_3-1-n, \\ -j_3^1 = 2i_4-1-n,$ where for a fixed $n$ , $j_d^r \\in \\lbrace 0, \\pm 1, \\pm 2, \\ldots , \\pm n \\rbrace $ and $i_r \\in \\lbrace 1, 2, \\ldots , n \\rbrace $ for each $r=1,2,3,4$ .",
"Also observe from the condition (iii) that $I_{J_r} = \\prod _{\\ell =1}^p \\chi _{[1, n]}\\left(i_r-\\sum _{d=1}^{\\ell } (-1)^d j^r_{d}\\right)= 1 \\ \\ \\forall \\ r=1,2,3,4,$ which shows that for all $r=1,2,3,4$ , we have to choose $j_1^r, j^r_2, \\ldots , j^r_p$ from $\\lbrace 0, \\pm 1, \\pm 2, \\ldots , \\pm n \\rbrace $ and $i_r$ from $\\lbrace 1, 2, \\ldots , n \\rbrace $ such that $i_r-\\sum _{d=1}^{\\ell } (-1)^d j^r_{d} \\in \\lbrace 1,2, \\ldots ,n\\rbrace \\ \\ \\forall \\ r=1,2,3,4 \\mbox{ and} \\ \\forall \\ \\ell =1,2, \\ldots , p.$ Now we calculate cardinality of $B^{\\prime }_{P_4} $ .",
"First note from (REF ) that $i_2-i_3+i_4=i_1$ .",
"So, if we choose $i_1,i_2,i_3$ freely, then $i_4$ will be fixed.",
"Let $ \\theta = \\Big \\lbrace (i_1,i_2,i_3,i_4) \\in \\big \\lbrace [7n/16,9n/16] \\cap \\mathbb {Z}\\big \\rbrace ^3 \\times \\lbrace 1,2, \\ldots , n\\rbrace \\; : \\;i_2-i_3+i_4=i_1 \\Big \\rbrace .$ Then $\\# \\theta \\ge (\\frac{n}{8})^3$ (by choosing $i_1,i_2,i_3$ freely), where $[x,y] \\cap \\mathbb {Z}$ denotes the set of integers between $x$ and $y$ .",
"For simplicity of notation, we write $[x,y]$ in place of $[x,y]\\cap \\mathbb {Z}$ .",
"First we calculate the contribution from $J_2$ .",
"Recall from the condition (ii) that $j^2_2=j^2_3, j^2_4=j^2_5, \\ldots , j^2_{p-1}=j^2_p$ .",
"Note that, once $i_2$ is chosen, $j^2_1$ will be fixed by $-j_1^2 = 2i_2-1-n$ with $j^2_1 \\in [-1- n/8, n/8-1] \\subseteq [-n, n]$ .",
"Also $i_2+j^2_1 \\in \\lbrace 1,2, \\ldots , n\\rbrace $ .",
"Now we choose $j^2_2$ freely in $n$ ways from the range $[c^2_1-n, c^2_1-1] \\subseteq [-n, n]$ , where $c^2_1= i_2+j^2_1$ .",
"Then $i_2+j^2_1-j^2_2, \\ i_2+j^2_1-j^2_2+j^2_3 \\in \\lbrace 1,2, \\ldots , n\\rbrace $ ($j^2_2=j^2_3$ ).",
"Once $j^2_2$ is chosen, we choose $j^2_4$ freely in $n$ ways from the range $[c^2_1-n, c^2_1-1]$ .",
"Here note that $i_2-\\sum _{d=1}^{4} (-1)^d j^2_{d}, \\ i_2-\\sum _{d=1}^{5} (-1)^d j^2_{d} \\in \\lbrace 1,2, \\ldots , n\\rbrace $ .",
"By continuing this idea, we can show that $i_2-\\sum _{d=1}^{\\ell } (-1)^d j^2_{d} \\in \\lbrace 1,2, \\ldots , n\\rbrace $ for each $\\ell =6,7, \\ldots , p$ and the total number of degree of freedom is $n^{\\frac{p-1}{2}}$ .",
"Similarly, we can show that for each $J_3$ and $J_4$ , we have $i_3-\\sum _{d=1}^{\\ell } (-1)^d j^3_{d}, \\ i_4-\\sum _{d=1}^{\\ell } (-1)^d j^4_{d} \\in \\lbrace 1,2, \\ldots , n\\rbrace $ for each $\\ell =1, 2, \\ldots , p$ and for each $J_3, J_4$ , the total number of degree of freedom is $n^{\\frac{p-1}{2}}$ .",
"Now we calculate the contribution from $J_1$ .",
"Recall from the condition (i) that $j_1^1=j^2_1, j^1_2= j_1^3, j^1_3= j_1^4$ .",
"So, once $J_2,J_3,J_4$ are chosen, $j^1_1, j^1_2, j^1_3$ will be fixed with $j^1_r \\in [-1- n/8, n/8-1]$ for all $r=1,2,3$ .",
"Note that $i_1,i_2,i_3,i_4$ are also chosen as $(i_1,i_2,i_3,i_4) \\in \\theta $ , where $\\theta $ is given in (REF ).",
"Therefore from the above range of $j_r^1$ and $i_r$ , we can show that, $i_1+j^1_1, i_1+j^1_1-j^1_2, \\ i_1+j^1_1-j^1_2+j^1_3 \\in \\lbrace 1,2, \\ldots , n\\rbrace $ .",
"Now we choose $j^1_4, \\ldots , j^1_p$ .",
"Note from the condition (ii) that $j^1_4=j^1_5, j^1_6=j^1_7, \\ldots , j^1_{p-1}=j^1_p$ .",
"Therefore from the similar idea as used to calculate carnality of $J_2$ , we can show that $i_1-\\sum _{d=1}^{\\ell } (-1)^d j^1_{d} \\in \\lbrace 1,2, \\ldots , n\\rbrace $ for each $\\ell =4,5, \\ldots , p$ and the total number of degree of freedom in $J_1$ is $n^{\\frac{p-3}{2}}$ .",
"Hence $\\# B^{\\prime }_{P_4} = \\theta \\big [ n^{\\frac{p-3}{2}} \\big ] \\big [ n^{\\frac{p-1}{2}} \\big ]^3 \\ge \\frac{n^{2p}}{8^3},$ where the above inequality arises due to $\\theta \\ge (\\frac{n}{8})^3$ .",
"On using the fact that the entries are independent with mean zero and variance one, from (REF ) we get $ \\lim _{n\\rightarrow \\infty } T_3 \\ge \\lim _{n\\rightarrow \\infty } \\frac{1}{n^{2p}} \\# B^{\\prime }_{P_4} \\ge \\frac{1}{8^3} >0,$ where the last inequality arises due to (REF ).",
"Now combining (REF ), (REF ) and (REF ) with (REF ), we get $\\beta _4 > 3\\beta _2^2.$ This completes the proof of Proposition REF .",
"The following corollary is an easy consequence of Proposition REF and Theorem 4.5.2 of [10].",
"Corollary 20 For each odd $p \\ge 3$ , $w_p$ does not converge in distribution to a Gaussian random variable, where $w_p$ is as in (REF ).",
"Our next proposition provides the unbounded support property of $\\Gamma _p$ .",
"Proposition 21 A distribution $\\Gamma _p$ with moment sequence given by (REF ) has unbounded support.",
"First note that to show $\\Gamma _p$ has unbounded support, it suffices to show that $(\\beta _{2k})^{\\frac{1}{k}} \\rightarrow \\infty $ .",
"Now we recall from (REF ) that $ \\mbox{E}[w_p^{2k}]= \\frac{1}{n^{pk}}\\displaystyle \\sum _{J_1\\in A_p, \\ldots , J_{2k} \\in A_p} \\mbox{E}[x_{J_1}x_{J_2}\\cdots x_{J_{2k}}] I_{J_1}I_{J_2}\\cdots I_{J_{2k}}.$ where for $1 \\le r \\le 2k$ , $J_r, I_{J_r}$ and $x_{J_r}$ are as in (REF ) and $A_{p}$ is as in (REF ).",
"Observe from the independence of the entries $\\lbrace x_i\\rbrace $ that if there exist an $\\ell \\in \\lbrace 1,2, \\ldots ,2k\\rbrace $ such that $J_\\ell $ is not connected with any other $J_r$ , then $\\sum _{A_p, \\ldots , A_p} |\\mbox{E}[x_{J_1}x_{J_2}\\cdots x_{J_{2k}}] I_{J_1}I_{J_2}\\cdots I_{J_{2k}}| & \\le \\sum _{ A_p} |\\mbox{E}[x_{J_1}]| \\sum _{ A_p, \\ldots , A_p} |\\mbox{E}[x_{J_2} \\cdots x_{J_{2k}}] | \\\\& \\le O(n^{\\lfloor \\frac{p}{2}\\rfloor }) O(n^{\\lfloor \\frac{ 2pk-p}{2}\\rfloor }),$ which shows that this case has contribution of the order $o(1)$ in $\\mbox{E}[w_p^{2k}]$ .",
"Note that the last inequality in the above expression arises due to the uniform boundedness of moments of $\\lbrace x_i\\rbrace $ and Lemma REF .",
"The above observation shows that the cases which have non-zero contribution in $\\lim _{n\\rightarrow \\infty }\\mbox{E}[w_p^{2k}]$ are the cases when $\\lbrace J_1, J_2, \\ldots , J_{2k}\\rbrace $ decomposes into clusters of size at least two.",
"Hence $\\lim _{n\\rightarrow \\infty }\\mbox{E}[w_p^{2k}] & \\ge \\lim _{n\\rightarrow \\infty }\\frac{1}{n^{pk}} \\sum _{\\pi \\in \\mathcal {P}_2(2k)} \\prod _{i=1}^{k} \\sum _{ J_{y(i)} \\in A_{p},\\ J_{z(i)} \\in A_{p}} \\mbox{E}\\big [x_{J_{y(i)}} x_{J_{z(i)}}\\big ] I_{J_{y(i)}}I_{J_{z(i)}}, \\\\& =\\sum _{\\pi \\in \\mathcal {P}_2(2k)} \\prod _{i=1}^{k} \\lim _{n\\rightarrow \\infty } \\mbox{E}[w^2_p],$ where $\\pi = \\big \\lbrace \\lbrace y(1), z(1) \\rbrace , \\ldots , \\lbrace y(k), z(k) \\rbrace \\big \\rbrace \\in \\mathcal {P}_2(2k)$ .",
"Using Theorem REF , from the above last equation, we get $ \\beta _{2k} \\ge \\sum _{\\pi \\in \\mathcal {P}_2(2k)} \\prod _{i=1}^{k} \\beta _2= \\frac{(2k)!}{k!",
"2^k} (\\beta _2)^k.$ Here note that $\\beta _2>0$ , which can be shown by the similar arguments as used to establish (REF ).",
"Therefore from the above inequality, $(\\beta _{2k})^{\\frac{1}{k}} \\rightarrow \\infty $ .",
"This completes the proof of Proposition REF .",
"Suppose $\\Gamma _p$ is any distribution on $\\mathbb {R}$ with moment sequence $\\lbrace \\beta _k\\rbrace $ , where $\\lbrace \\beta _k\\rbrace $ is as in (REF ).",
"In the following theorem, we derive the covariance structure for $\\lbrace \\Gamma _p : p \\text{ is odd postive interger}\\rbrace $ .",
"Note from Section REF that there could be more than one distribution corresponding to $\\lbrace \\beta _k\\rbrace $ .",
"But the covariance structure is independent of the choice of the distributions.",
"The argument used here is similar to the one used in Theorem REF with some technical changes.",
"Theorem 22 Let $p_1,p_2$ be odd natural numbers.",
"Then for any input sequence $\\lbrace x_i\\rbrace $ obeying (REF ) $\\lim _{n \\rightarrow \\infty } \\mbox{\\rm Cov}(w_{p_1},w_{p_2})= \\frac{1}{2}\\sum _{\\pi \\in {{\\mathcal {P}}}_2(p_1+p_2)} g_{p_1,p_2}(\\pi ),$ where $g_{p_1,p_2}(\\pi )$ is as given in Definition REF .",
"First note from (REF ) and Result REF that $\\lim _{n \\rightarrow \\infty } \\mbox{\\rm Cov}(w_{p_1},w_{p_2})= \\lim _{n \\rightarrow \\infty } \\frac{1}{n^{\\frac{p_1+p_2}{2}}} \\sum _{J_1 \\in A_{p_1},J_2 \\in A_{p_2}} \\left(\\mbox{E}[x_{J_1} x_{J_2}]-\\mbox{E}[x_{J_1}] \\mbox{E}[x_{J_{2}}]\\right) I_{J_{1}} I_{J_{2}},$ where $A_{p_r}$ is as in (REF ) and $J_r, I_{J_r}, x_{J_r}$ are as in (REF ) for $r=1,2$ .",
"Now observe that the summand in (REF ) is non-zero only when each element of $S_{J_1} \\cup S_{J_2}$ is repeated at least twice and there exists at least one cross-matching in $(J_1,J_2)$ .",
"An argument similar to the one employed in the proof of Theorem REF implies that only the pair-partitions of $[p_1+p_2]$ contribute in (REF ).",
"Furthermore, since $p_1,p_2$ are odd, every pair-partition has at least one cross-matching, and hence $\\mbox{E}[x_{J_1}x_{J_2}]=1$ and $\\mbox{E}[x_{J_1}]=\\mbox{E}[x_{J_2}]=0$ in such cases.",
"As a result, (REF ) can be written as $\\lim _{n \\rightarrow \\infty }\\mbox{\\rm Cov}(w_{p_1},w_{p_2})= \\lim _{n \\rightarrow \\infty }\\frac{1}{n^{\\frac{p_1+p_2}{2}}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(p_1+p_2)}\\sum _{j_1,j_2,\\ldots , j_{\\frac{p_1+p_2}{2}}=-n}^n \\sum _{i_1,i_2=1}^n I_{J_1} I_{J_2}\\Delta _{J_1}\\Delta _{J_2},$ where $J_1 = (j_1^1,j_2^1,\\ldots , j_{p_1}^1)$ with $j_s^1= j_{\\pi \\left(s\\right)}$ , $J_2 = (j_1^2,j_2^2,\\ldots , j_{p_2}^2)$ with $j_s^2= j_{\\pi \\left(p_1+s\\right)}$ and for $r=1,2$ , $I_{J_r}= \\prod _{\\ell =1}^{p_r} \\chi _{\\left[\\frac{1}{n}, 1\\right]}\\left(\\frac{i_r}{n}-\\sum _{q=1}^{\\ell }(-1)^q \\frac{j_{\\pi \\left((r-1)p_1+q\\right)}}{n}\\right), \\ \\Delta _{J_r}=\\delta _{\\frac{2i_r}{n}-1-\\frac{1}{n}} \\left( \\sum _{q=1}^{p_r}(-1)^{q} \\frac{j_{\\pi \\left((r-1)p_1+q\\right)}}{n}\\right).$ For a fixed $\\pi \\in {\\mathcal {P}}_2(p_1+p_2)$ and $(J_1,J_2)$ , $\\Delta _{J_i}$ is non-zero only for the solution $i_r \\in [n]$ of the following equation $\\sum _{q=1}^{p_r}(-1)^{q} j_{\\pi \\left((r-1)p_1+q\\right)}= 2 i_r-n-1.$ Consider a combination of $i_r, J_r$ obeying (REF ) such that $I_{J_r}=1$ .",
"Then each term in the product form of $I_{J_r}$ is equal to 1, which in turn implies that $1 \\le i_r - \\sum _{q=1}^{p_r} j_{\\pi \\left((r-1)p_1+q\\right)} \\le n$ .",
"Substituting the expression of $i_r$ from (REF ), it follows that $i_r$ lies between 1 and $n$ .",
"So now our objective is to find integer solutions of (REF ).",
"Again, by (REF ), $i_r$ is an integer if and only if $n$ and $\\sum _{q=1}^{p_r} j_{\\pi \\left((r-1)p_1+q\\right)}$ have different parity.",
"Hence the right side of (REF ) becomes $\\lim _{n \\rightarrow \\infty }\\frac{1}{n^{\\frac{p_1+p_2}{2}}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(p_1+p_2)}\\sum _{j_1,j_2,\\ldots , j_{\\frac{p_1+p_2}{2}}=-n}^n \\prod _{r=1}^2 I_{J_r} \\ \\delta _{-1} \\left((-1)^{\\sum _{q=1}^{p_r}(-1)^q j_{\\pi \\left((r-1)p_1+q\\right)}+n} \\right).$ For $\\pi \\in {\\mathcal {P}}_2(p_1+p_2)$ , consider the sum $\\sum _{q=1}^{p_r}(-1)^{q} j_{\\pi \\left((r-1)p_1+q\\right)}$ .",
"For $r=1,2$ , we shall use $J_{r,r}$ to denote the self-matching elements in $J_r$ and $J_{1,2}$ to denote the cross-matching elements in $(J_1,J_2)$ .",
"Since, the sum $\\sum _{j_q^r \\in J_{r,r}}(-1)^{q} j_q^r$ is always even for $r=1,2$ , the parity of $\\sum _{q=1}^{p_r}(-1)^q j_{\\pi \\left((r-1)p_1+q\\right)}$ is same as the parity of $\\sum _{j_q^r \\in J_{1,2}}(-1)^{q} j_q^r$ for both $r=1,2$ .",
"Now, consider $(i_r,J_r)$ with $I_{J_r}>0$ and $\\sum _{q=1}^{p_r}(-1)^{q} j_{\\pi \\left((r-1)p_1+q\\right)}=2i_r-n-1$ .",
"Let $j_{k_0}$ be a cross-matched element in $(J_1,J_2)$ .",
"Once all other $j_q^r$ are fixed, the parity of $j_{k_0}$ is restricted to either even or odd, depending upon $n$ .",
"Also, once all other $j_q^r$ are fixed, the possible values of $j_{k_0}$ such that $I_{J_1}I_{J_2}=1$ falls in an interval.",
"This reduces the number of possibilities for $j_{k_0}$ by half with an error of at most 1.",
"Thus from (REF ), $ \\lim _{n \\rightarrow \\infty }\\mbox{\\rm Cov}(w_{p_1},w_{p_2})&= \\lim _{n \\rightarrow \\infty } \\Big [\\frac{1}{2} \\times \\frac{1}{n^{\\frac{p_1+p_2}{2}}}\\displaystyle \\sum _{\\pi \\in {\\mathcal {P}}_2(p_1+p_2)}\\sum _{j_1,j_2,\\ldots , j_{\\frac{pk}{2}}=-n}^n \\prod _{r=1}^2 I_{J_r} + \\frac{1}{n^{\\frac{p_1+p_2}{2}}} \\times O(n^{\\frac{p_1+p_2}{2}-1}) \\Big ] \\nonumber \\\\&= \\frac{1}{2}\\sum _{\\pi \\in {\\mathcal {P}}_2(p_1+p_2)} g_{p_1,p_2}(\\pi ),$ where $g_{p_1,p_2}(\\pi )$ is as given in Definition REF .",
"Note that the last expression is obtained by considering $I_{J_1}I_{J_2}$ as a Riemann sum of the integral $g_{p_1,p_2}(\\pi )$ ."
],
[
"Conclusion",
"Our research shows that the behaviour of linear eigenvalue statistics of random Hankel matrices ($w_p$ ) for odd degree monomials with degree greater than or equal to three is significantly different from the behaviour of linear eigenvalue statistics of random Hankel matrices for even degree monomials and that of linear eigenvalue statistics of Toeplitz matrices.",
"First for the monomial test functions, we have shown that the moments of linear eigenvalue statistics of Hankel matrix ($\\mbox{E}[w_p]^k$ ) converge to a limit sequence $\\lbrace \\beta _k \\rbrace $ as $n$ tends to $\\infty $ , where each $\\beta _k$ is finite.",
"We also showed that there exist probability measures on $\\mathbb {R}$ with moment sequence $\\lbrace \\beta _k \\rbrace $ .",
"We proved that any probability measure $\\Gamma _p$ with $\\lbrace \\beta _k \\rbrace $ as moment sequence is non-Gaussian and has unbounded support.",
"The behaviour of linear eigenvalue statistics of Hankel matrices for polynomial test functions were discussed in Section REF .",
"The simulations in Figure REF suggest that the linear eigenvalue statistics converge in distribution to a unique limit which is symmetric, unimodular and absolutely continuous.",
"But establishing it theoretically, is difficult, because the moment sequence $\\lbrace \\beta _k \\rbrace $ might not determine a unique distribution on $\\mathbb {R}$ .",
"Here the moment sequence $\\lbrace \\beta _k \\rbrace $ fails to obey Carleman's condition, see Section REF .",
"In this article, we could not conclude whether there is a unique distribution, which corresponds to $\\lbrace \\beta _k\\rbrace _{k\\ge 1}$ .",
"But we concluded that $w_p$ does not converge in distribution to a Gaussian random variable (Corollary REF ).",
"The question of uniqueness of $\\lbrace \\beta _k \\rbrace $ along with the convergence in distribution of $w_p$ , is still open."
]
] | 2209.08252 |
[
[
"Fluctuations of conserved charges in hydrodynamics and molecular\n dynamics"
],
[
"Abstract I present an overview of recent theoretical results on fluctuations of conserved charges in heavy-ion collisions obtained in relativistic hydrodynamics and molecular dynamics frameworks.",
"In particular, I discuss the constraints on the location of the QCD critical point based on comparisons of experimental data on proton number cumulants with precision calculations of non-critical contributions.",
"Recent developments on critical fluctuations in molecular dynamics simulations are covered as well."
],
[
"Introduction",
"Fluctuations of conserved charges are sensitive probes of QCD matter.",
"In thermodynamic equilibrium within the grand-canonical ensemble, the cumulants of a conserved charge distribution are linked to the susceptibilities, i.e., to the chemical potential derivatives of the partition function, $\\kappa _n \\propto \\frac{\\partial ^n (\\ln Z^{\\rm gce})}{\\partial \\mu ^n}.$ Therefore, fluctuations probe the finer details of the equation of state.",
"In particular, the conserved baryon number plays the role of the order parameter for the hypothetical first-order QCD phase transition at finite baryon densities, and its fluctuations exhibit singular behavior near the critical point (CP) of the transition [1], [2].",
"Critical opalescence is a classical example of such a phenomenon when a usually transparent substance with respect to laser irradiation becomes opaque due to large density fluctuations of macroscopic scales near the CP.",
"The location (and even existence) of the QCD CP and the associated phase structure of QCD matter are one of the most important open questions tackled by relativistic heavy-ion collisions at various energies [3].",
"In contrast to classical fluids, it is impossible to trap a droplet of hot and dense QCD fluid to try to observe critical opalescence.",
"The blob of QCD matter created in heavy-ion collisions quickly expands and hadronises, producing at most a few thousands of (anti)baryons and typically even less.",
"On the other hand, the relative “smallness” of the number of particles created in heavy-ion collisions makes it possible to track each particle in each event (modulo detector efficiency and acceptance limitations), and compute the event-by-event distribution and the associated fluctuation measures directly.",
"In this regard, the cumulants of the proton number – a proxy for the baryon number in the experiment – are sensitive probes of critical behavior [4], in particular the high-order non-Gaussian cumulants [2], [5].",
"The cumulants of proton number are also used to extract other information, and recent works have studied, for instance, the speed of sound [6] or freeze-out temperature [7].",
"Measurements of proton number fluctuations are being performed by several experiments, including ALICE [8], STAR [9], [10], [11], NA61/SHINE [12], and HADES [13].",
"Much attention is devoted to measuring the net-proton kurtosis $\\kappa \\sigma ^2$ at RHIC beam energy scan, which indicated a possible non-monotonic collision energy dependence of this quantity [9].",
"Such a feature has earlier been predicted as a potential signal of the QCD critical point [14].",
"It should be noted, however, that experimental uncertainties are significant, and some of the observed features could be attributed to baryon conservation as opposed to critical point [15].",
"More robust conclusions will become possible once the improved data on $\\kappa \\sigma ^2$ from BES-II becomes available.",
"In the meantime, however, one could explore what can be learned from the more precise available measurements of the second and third-order proton cumulants.",
"These analyses, though, require quantitative comparisons between theory and experiment, which for event-by-event fluctuations involve many caveats (see e.g.",
"[16] for an overview).",
"Addressing these caveats requires extensive dynamical modeling of heavy-ion collisions.",
"Several strategies are being considered to tackle the search for the QCD critical point in heavy-ion collisions within dynamical models: Precision calculation of non-critical proton number fluctuations and its deviations from experimental data.",
"This approach, recently developed in Ref.",
"[17], incorporates essential non-critical contributions to proton number fluctuations (conservation laws, hadronic interactions, momentum cuts) on top of the hydrodynamical background.",
"Recent results are summarized in Sec. .",
"Molecular dynamics with a critical point.",
"Molecular dynamics is a microscopic approach that can address many of the caveats associated with fluctuations.",
"Insights on critical fluctuations using classical molecular dynamics have recently been explored in Ref.",
"[18] and covered in Sec. .",
"Hydrodynamics with critical fluctuations.",
"Ultimately, critical fluctuations should be incorporated into the hydrodynamic framework for heavy-ion collisions [19] as well the particlization procedure [20], allowing one to make testable predictions based on the location of the CP.",
"Such a framework has been under development, for instance, within the BEST Topical Collaboration [21]."
],
[
"Hydrodynamics based analysis of (net-)proton fluctuations",
"One way to search for critical behavior in proton number cumulants is to study deviations of experimental data from baseline predictions that do not incorporate critical fluctuations.",
"The simplest baseline corresponds to uncorrelated proton production, which would yield proton number cumulants consistent with Poisson statistics.",
"However, additional non-critical mechanisms such as baryon number conservation and the repulsive core in baryon-baryon interaction break this assumption and make the non-critical baseline considerably more involved.",
"Relativistic hydrodynamics is considered to be the standard model of heavy-ion collisions and provides a realistic background for calculating the aforementioned non-critical contributions to proton number cumulants.",
"More specifically, the effects of baryon conservation and repulsion are implemented at the Cooper-Frye particlization stage, which is performed either analytically [17], [22] or through Monte Carlo sampling [23].",
"Baryon repulsion is modeled by utilizing the excluded volume model, with baryon excluded volume parameter $b = 1$ fm$^3$ as follows from fits to lattice QCD susceptibilities [24].",
"LHC.",
"Net-proton number cumulants have been measured up to third order by the ALICE Collaboration at the LHC [8], [25].",
"Although fluctuations at the LHC are not expected to be sensitive to the possible presence of the QCD critical point in the baryon-rich regime, the available measurements of a normalized net-proton number variance $\\kappa _2[p-\\bar{p}]/\\langle p+\\bar{p} \\rangle $ appear to establish the relevance of non-critical effects, in particular, that of baryon number conservation [8], [26].",
"Recently, the authors of Ref.",
"[27] pointed out that precision measurements of $\\kappa _2[p-\\bar{p}]/\\langle p+\\bar{p} \\rangle $ at the LHC are also sensitive both to the range of baryon conservation as well as to baryon annihilation in the hadronic phase.",
"Namely, a surplus of annihilation over regeneration leads to an increase of $\\kappa _2[p-\\bar{p}]/\\langle p+\\bar{p} \\rangle $ while reducing the range of correlations associated with baryon conservation brings this quantity down.",
"The data can be described equally well by employing either (1) global baryon conservation and no baryon annihilation or (2) local baryon conservation across $\\Delta y_{\\rm cor} \\sim 3$ units of rapidity and baryon annihilation without regeneration modeled by UrQMD afterburner.",
"The two effects can be constrained experimentally by a combined measurement of $\\kappa _2[p-\\bar{p}]/\\langle p+\\bar{p} \\rangle $ and $\\kappa _2[p+\\bar{p}]/\\langle p+\\bar{p} \\rangle $ , while additional constrains can come from fluctuation measurements involving light nuclei [28].",
"Figure: Dependence of proton cumulant ratios κ 2 [p-p ¯]/〈p+p ¯〉\\kappa _2[p-\\bar{p}]/\\langle p+\\bar{p} \\rangle (left) and κ 2 [p+p ¯]/〈p+p ¯〉\\kappa _2[p+\\bar{p}]/\\langle p+\\bar{p} \\rangle (right) on the pseudorapidity acceptance in 2.76 TeV Pb-Pb collisions calculated in different scenarios regarding the treatment of baryon annihilation and the range of baryon conservation.Taken from .RHIC-BES.",
"Proton number cumulants at RHIC-BES energies ($\\sqrt{s_{\\rm NN}} = 7.7-200$ GeV) have been measured by the STAR Collaboration [9], [10].",
"The non-critical contributions were calculated in Ref.",
"[17] based on hydrodynamic simulations of 0-5% central Au-Au collisions within the MUSIC code [29], where both the ordinary as well as factorial cumulants [30] of (net-)proton number distribution were analyzed.",
"It was shown that multi-proton ($n>3$ ) correlations are small in the non-critical scenario.",
"Thus, the behavior of all the cumulants is driven by two-proton correlations, which are found to be negative.",
"This behavior contrasts critical fluctuations, which would generate sizeable multi-particle correlations [31].",
"Figure: Scaled variance κ 2 /κ 1 \\kappa _2/\\kappa _1 of the proton number distribution in central Au-Au collisions as a function of (left panel) collision energy and (right panel) acceptance in rapidity at s NN =2.4\\sqrt{s_{\\rm NN}} = 2.4 GeV, incorporating various non-critical contributions.Adapted from , .The experimental data of the STAR Collaboration is quantitatively consistent with simultaneous effects of baryon conservation and repulsion at $\\sqrt{s_{\\rm NN}} \\gtrsim 20$ GeV (Fig.",
"REF ), i.e.",
"with non-critical physics.",
"At lower collision energies, however, the data indicate an excess proton correlation over the non-critical baseline.",
"This excess becomes even more prominent when new data from STAR fixed target programme at $\\sqrt{s_{\\rm NN}} = 3$ GeV [11] and HADES experiment at GSI at $\\sqrt{s_{\\rm NN}} = 2.4$ GeV [13] is considered.",
"Additional non-critical contributions such as volume fluctuations and electric charge conservation can influence proton number cumulants.",
"In particular, adding volume fluctuations via an additional parameter can improve the data description at lower collision energies, but it will spoil the agreement at higher collision energies.",
"The effect of exact conservation of electric charge was explored in Ref.",
"[23] and did not yield any improvement in the description of the data at $\\sqrt{s_{\\rm NN}} \\lesssim 20$ GeV.",
"Lower energies.",
"One can conclude that it is challenging to understand the data at $\\sqrt{s_{\\rm NN}} \\lesssim 20$ GeV in terms of non-critical physics.",
"In particular, the HADES data point shows a large proton scaled variance of $\\kappa _2 / \\kappa _1 > 2$ , warranting a closer look at these data.",
"Proton number cumulants in Au-Au collisions at $\\sqrt{s_{\\rm NN}} = 2.4$ GeV have recently been analysed in [32] in the framework of Siemens-Rasmussen-like fireball model, with freeze-out parameters based on Refs.",
"[33], [34].",
"In contrast to the non-critical baseline calculation, this analysis requires no assumptions on baryon number susceptibilities characterizing the emitting source.",
"Instead, one extracts their values from the data by fitting the first four proton cumulants at different rapidity acceptance cuts.",
"In order to minimize the effects of baryon conservation, the cuts were considered up to $y_{\\rm cut} \\le 0.2$ .",
"The experimental data of HADES Collaboration is described by assuming a thermal emission of nucleons from a grand-canonical heat bath (Fig.",
"REF ), provided that the corresponding baryon number susceptibilities of QCD matter are highly non-Gaussian and exhibit the following hierarchy: $\\chi _4^B \\gg -\\chi _3^B \\gg \\chi _2^B \\gg \\chi _1^B$ .",
"This kind of hierarchy of conserved charge susceptibilities can appear in the vicinity of a critical point [2], [35].",
"Therefore, naively, this observation could point to a presence of the QCD critical point close to the HADES chemical freeze-out at $T \\sim 70$ MeV and $\\mu _B \\sim 850-900$ MeV.",
"However, the behavior of proton cumulants at $y_{\\rm cut} > 0.2$ is challenging to describe even qualitatively when one incorporates the effect of exact baryon conservation into the calculations (dashed red line).",
"The above statement applies for $\\kappa _2$ shown in Fig.",
"REF as well as for $\\kappa _3$ and $\\kappa _4$ not shown here, indicating that more theoretical and experimental effort is required to reach a firm conclusion.",
"We emphasize that the challenge in understanding the results in the context of baryon conservation persists already at the second-order, $\\kappa _2/\\kappa _1$ , and should be resolved at this level before turning to third- and fourth-order cumulants."
],
[
"Critical point particle number fluctuations from molecular dynamics",
"Molecular dynamics (MD) is a microscopic approach to studying (non-)equilibrium evolution of dynamical systems and provides an alternative approach to model heavy-ion collisions.",
"Most MD codes such as UrQMD or SMASH are purely hadronic, given that it is challenging to implement both hadron and quark degrees of freedom.",
"Nevertheless, such an approach is better suited to describe non-equilibrium evolution and is arguably preferable for intermediate collision energies.",
"MD simulations already provide useful insights into various non-critical contributions to particle number fluctuations [36], [11], [37].",
"Critical fluctuations can also be studied in MD simulations, as long as one uses an appropriate interaction potential that leads to the presence of a first-order phase transition and a CP.",
"One famous and well-studied example of such a potential is the Lennard-Jones (LJ) potential, which describes fluids exhibiting a liquid-gas type transition.",
"The LJ fluid corresponds to a system of non-relativistic particles with attractive and repulsive interactions.",
"This system is quite different from QCD matter, as QCD contains different degrees of freedom – hadron and partons – on the different sides of the QCD transition.",
"Nevertheless, due to the universality of critical behavior, simulations of the LJ fluid can provide helpful insight into the microscopic development of critical fluctuations.",
"Recent work [18] used MD simulations of the LJ fluid to study the behavior of particle number fluctuations near and away from the CP through a numerical solution of Newton's equations of motion (classical $N$ -body problem).",
"Simulations were performed in a box with periodic boundary conditions and within the microcanonical ensemble.",
"Since the total particle number $N$ is conserved, fluctuations were studied in subsystems by performing cuts either in longitudinal coordinate $z$ or longitudinal velocity $v_z$ .",
"Figure: Dependence of the scaled variance of particle number corrected for global particle number conservation on the subvolume fraction α\\alpha in coordinate (left panel) and momentum (right panel) space calculated using molecular dynamics simulations of the Lennard-Jones fluid near the critical point.Adapted from .The left panel of Fig.",
"REF depicts the results for the scaled variance $\\tilde{\\omega } = \\omega /(1-\\alpha )$ of particle number fluctuations corrected for total particle number conservation by $1-\\alpha $ factor [38], as function of acceptance fraction $\\alpha $ along the $z$ -coordinate.",
"The calculations were performed near the critical point ($T \\simeq 1.06 T_c$ , $n \\simeq n_c$ ), and for different values of the total particle number $N$ .",
"The results are consistent with an approach toward the grand-canonical scaled variance of $\\omega ^{\\rm gce} \\sim 7$ as the particle number (system size) is increased.",
"These observations indicate that the presence of the CP manifests itself in large particle number fluctuations and finite system-size effects, as expected.",
"However, when one studies the fluctuations in the momentum subspace instead, via a cut $|v_z| < v_z^{\\rm cut}$ in longitudinal velocity (right panel of Fig.",
"REF ), the effect of critical fluctuations is completely washed out.",
"This observation reflects that coordinates and momenta in uniform LJ fluid are uncorrelated.",
"Thus, the significant correlations in coordinate space due to CP do not translate into the momentum space.",
"The situation may be different in expanding systems encountered in heavy-ion collisions, where collective velocities generate correlations between coordinates and momenta of emitted particles and may preserve CP signals even in momentum space which is accessible experimentally.",
"Extensions of MD simulations with critical fluctuations to expanding systems will be the subject of future works."
],
[
"Summary",
"The analysis of (net-)proton number cumulants at different collision energies indicates that experimental data are consistent with non-critical physics such as baryon number conservation and short-range repulsion at $\\sqrt{s_{\\rm NN}} \\gtrsim 20$ GeV.",
"In contrast, the data at lower energies indicate significant excess proton correlations over various non-critical baselines, which require further analysis.",
"These observations thus disfavor the existence of QCD CP at small baryon densities, $\\mu _B/T < 2-3$ , consistent with observations from lattice QCD [39], [40], [41].",
"They also indicate that the CP, if it exists, is located in the baryon-rich matter probed by heavy-ion collisions at intermediate energies $\\sqrt{s_{\\rm NN}} \\lesssim 20$ GeV, see Fig.",
"REF for the summary of the available CP constraints.",
"Future efforts in the search for the CP at intermediate collision energies will require improved modeling of CP effects in baryon-rich matter, and new developments in molecular dynamics simulations of critical fluctuations will play a valuable role there."
],
[
"Acknowledgments",
"This work was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-FG02-00ER41132."
]
] | 2209.08233 |
[
[
"On self-similar finite-time blowups of the De Gregorio model on the real\n line"
],
[
"Abstract We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time.",
"These self-similar solutions fall into two classes: the basic class and the general class.",
"The basic class consists of countably infinite solutions that are eigenfunctions of a self-adjoint compact operator.",
"In particular, the leading eigenfunction coincides with the finite-time singularity solution of the De Gregorio model recently obtained by numerical approaches.",
"The general class consists of more complicated solutions that can be obtained by solving nonlinear eigenvalue problems associated with the same compact operator."
],
[
"=1 urlcolor=rust citecolor=dkblue linkcolor=dkblue 1mm 1.04"
]
] | 2209.08232 |
[
[
"ScreenQA: Large-Scale Question-Answer Pairs over Mobile App Screenshots"
],
[
"Abstract We present a new task and dataset, ScreenQA, for screen content understanding via question answering.",
"The existing screen datasets are focused either on structure and component-level understanding, or on a much higher-level composite task such as navigation and task completion.",
"We attempt to bridge the gap between these two by annotating 80,000+ question-answer pairs over the RICO dataset in hope to benchmark the screen reading comprehension capacity."
],
[
"Introduction",
"Mobile app screenshots have been analyzed using machine learning from multiple aspects.",
"These analyses range from pixel level understanding, e.g., layout structural analyses, UI issue detection and correction [18], to UI element semantics, e.g., icon recognition, button action prediction [27], to even higher-level functional analyses such as accessibility support [21], screen description [30], and screen type classification [8].",
"Comparatively, the content understanding aspect is relatively understudied.",
"By content, we mean the information displayed on the screen to convey and satisfy the purpose of using the app.",
"Examples include star ratings from restaurant reviews, messages from chat apps, cuisine ingredients from recipe apps, flight status and in-flight amenities from travel planner apps, etc.",
"Having this capacity of understanding is important for two reasons: First, the sole reason for many apps and app categories to exist is to satisfy users' information need, e.g., weather, map navigation, and news apps.",
"Second, for task completionAlso referred to as automation or app control., which requires the eyes-free agent capacity, the two types of screen understandings — content and action understanding — are inseparable in order to carry out a task successfully.",
"Without knowing a screen state properly, a machine learning agent is unable to self-assess if the action is performed as expected, or unable to provide sufficient feedback to the user to achieve true eyes-free user experience.",
"More intrinsically, from a pure research perspective, we are interested in knowing the limit of machine screen content understandingThis term is analogous to machine reading comprehension from natural language processing.",
"and what constitutes the challenges, given that app screenshots are entirely human artifacts made for convenient comprehension.",
"Accordingly, we annotated the RICO dataset [8] with more than 80,000 question-answer pairs, referred to as Screen Question Answering, or, in short, ScreenQA annotations later in this work, and we will release the dataset in the public domain.URL will be released in a subsequent update of this manuscript.",
"The ScreenQA task requires an agent to answer a user's question by selecting one or multiple UI elements from the given screenshot, as will be formulated in Section .",
"Question answering is employed as a touchstone to sparselyBecause questions are not exhaustively asked against a given screenshot.",
"verify the quality of screen content understanding.",
"To the best of our knowledge, this is the first large-scale questions answering dataset over mobile app screenshots, and the first one to be publicly available.",
"Much inspired by the SQuAD dataset [25], we hope, by releasing this set of annotations, to encourage the community to advance technologies toward better screen content understanding.",
"We anticipate that the advance of such technologies will benefit beyond just the screen UI and the human computer interaction (HCI) domains.",
"As we will discuss in Section , other vision-language related multimodal domains share similar challenges with different emphases on respective modalities and contexts.",
"Comparatively, ScreenQA is language and layout heavy, but it also includes visual ingredients such as icons and symbols as concise representations in place of texts, to declutter the UI.",
"It may also include images or art designs that pose challenges to language centric machine learning agents.",
"The remaining paper is organized in the following way: Section formulates the problem, including the problem description and the evaluation metrics.",
"We discuss relevant prior datasets and annotations in Section to put this work into perspective.",
"Section describes our annotation method.",
"The annotations are then analyzed in Section to provide readers both the qualitative and quantitative views.",
"The paper is concluded in Section with a summary and a remark on future works."
],
[
"Problem Setting",
"We state the problem and define the evaluation metrics in this section."
],
[
"Problem statement",
"The ScreenQA task requires an agent to answer a user's question by selecting relevant UI elements from a given single screenshot.",
"When it comes with multiple relevant UI elements, a list of such UI elements whose contents minimally satisfy the question should be selected and ranked in descending order of relevance to the question, if applicable, or following the common reading order by semantic groups, as will be described in Section REF .",
"This assumes that answers are directly selectable from the screen and logical reasoning and calculation are not needed.",
"If the screenshot does not contain the answers to the question, the agent should respond with “no answer.” This is summarized in Task REF .",
"[htb]ScreenQA Input: a question $Q$ and a screenshot $S$ Output: an answer list $A$ of UI elements selected from $S$ such that their contents minimally satisfy $Q$ .",
"The order of $A$ is further required to be Ranked in descending order of relevance to $Q$ , if applicable.",
"Otherwise, following the common reading order by semantic groups.",
"If no contents in $S$ can satisfies $Q$ , then returns an empty list $A$ ."
],
[
"Properties and terminologies",
"The mobile app UI comes with some nuances.",
"It is worth mentioning a few properties below.",
"View hierarchy, or the structural representation used to render the screen, is not required in Task REF , to be consistent with the human annotation process in Section .",
"View hierarchy usually provides useful UI element candidates, but it may not always be reliable, for example, when using WebView or screen overlays.",
"In such cases, a human annotator can still answer screen questions entirely from pixels without an issue, so we want to benchmark similarly.",
"We leave the choice of dependency on view hierarchies to the modelers and, hence, do not require it.",
"However, this comes with an ambiguity for UI element boundaries.",
"See an example in Figure REF .",
"We devise a more flexible answer matching process to mitigate such an impact, as will be discussed in Section REF .",
"Avoid question answering over long paragraphs.",
"Although it is permissive by Task REF , we discourage annotators from asking such questions during the annotation process.",
"For ScreenQA, we want to focus on learning the relationships between text segments arranged two-dimensionally on the screen, and leave the long paragraph question answering, which investigates the relationships between words, to the traditional NLP domain.",
"Avoid logical reasoning.",
"This task assumes answers can directly be extracted from the screenshot without reasoning, entailment, counting and comparing numbers.",
"This further exclude yes/no and why questions if not explicitly displayed on the screen.",
"The reason is that we want to separate “able to read” and “able to reason” and focus on the former first without generating an over challenging dataset.",
"A few such excluded examples are: counting items, asking about the weather a few days from now, what are the items cheaper than X dollars, etc.",
"Ordered by relevance.",
"The task is designed to enable the eyes-free user experience.",
"That is, a user may not be fully aware of how many relevant answers are displayed on the screen.",
"For example, in Figure REF , when a user asks “What's the temperature on Saturday?”, there are actually two temperatures, high and low, for each day and two Saturdays on the screen.",
"In this case, the two temperatures should just follow the reading order, and the two Saturdays follow the relevance order as a user usually refers to the upcoming Saturday.",
"For a well-designed mobile app, these two usually overlap well and we do not expect a large ambiguity here.",
"Reading order by semantic groups.",
"Sometimes some UI elements are designed as semantic groups and should be referred to together to keep their semantic meaning.",
"For example, in Figure REF , when a user asks “What are the first two movements of deep squat?”, then the answer should be “Deep Squat, 3 sets, 15x”, followed by “Lunge, 3 sets, 10x”.",
"In other words, the common reading order should be based on semantic groups as the unit, rather than simply sorted by the coordinates of UI elements.",
"Note that we set up the problem this way strategically in order to prioritize its solvability considering the progress of current technologies.",
"However, practically, long vs. short texts and retrieval vs. reasoning are naturally mixed together in the daily usage of mobile apps.",
"We will leave this type of composite problems to the future work."
],
[
"Evaluation metrics",
"We consider two types of metrics: 1) Average normalized Discounted Cumulative Gain (Average nDCG) [13], which is commonly used in information retrieval and ranking systems, and 2) Average F1 score, which has been employed in closed-domain question answering problems, such as the SQuAD dataset [25].",
"One major difference between our metrics described below and the commonly used definitions is the unit of predictions.",
"We use the element in the answer list $A$ , described in Task REF , as the unit to determine a hit or a miss for both metrics.",
"Besides, as UI elements can be ambiguous as mentioned in Section REF , we will describe an answer matching algorithm that mitigate such an impact in Section REF ."
],
[
"Average nDCG",
"We use a variant of nDCG that allows varying positions (number of returns) as opposed to a typical fixed position.",
"This is because, unlike the search problem, which is fair to evaluate, say, top-10 retrieved documents across queries, ScreenQA can have different needs of answer lengths across different questions.",
"For example, a question like “what is the address of the store” expects a single returned result.",
"A question like “what are the login options?” expects an enumeration of options on the screen that easily go beyond five.",
"Accordingly, we allow $v$ arying positions as follows: Given a 1-indexed list $A$ , which is the predicted answer for the screen-question pair $(S, Q)$ , and a ground truth answer list $A_g$ for $(S, Q)$ , the Discounted Cumulative Gain at $v$ arying positions (DCG$_v$ ) is computed by: $\\mbox{DCG}_v = \\sum ^{\\Vert A\\Vert }_{i=1} \\frac{r_i}{\\log _2{(i+1)}},$ where $\\Vert \\cdot \\Vert $ is the size of the list argument, $r_i$ is the relevance score for the $i$ -th item of $A$ .",
"We assign the relevance score 1 for a hit and 0 for a miss compared with the ground truth $A^g$ .",
"The corresponding Ideal Discounted Cumulative Gain (IDCG$_v$ ) is computed by: $\\mbox{IDCG}_v = \\sum ^{\\Vert A^g\\Vert }_{i=1} \\frac{1}{\\log _2{(i+1)}}.$ The nDCG$_v$ is then $\\mbox{nDCG}_v = \\frac{\\mbox{DCG}_v}{\\mbox{IDCG}_v}.$ Note that nDCG$_v$ is still between 0 and 1, hence, convenient for comparing scores and computing the average.",
"For a dataset of $N$ examples, each of which is indexed by $i$ and has a predicted answer $A_i$ and $K$ ground truth annotations $A^g_{i, j=1 \\dots K}$ , the average nDCG$_v$ can be computed by $\\mbox{avg}(\\mbox{nDCG}_v) = \\frac{1}{N}\\sum _{i=1}^N \\mbox{max}_j [ \\mbox{nDCG}_v(A_i, A^g_{i,j} ) ].$ We choose a variant nDCG as the metric because 1) we want to measure the quality of the ranking.",
"For example, if one incorrectly predicts the result from the first to the third position, the discount factor brings down the score from 1.0 to only 0.5.",
"2) nDCG has an orthogonal design, which is easier to tweak toward a specific need than the mean average precision (mAP) metric.",
"For example, one can choose to discount faster or slower by changing the base of the denominator $\\log _2(i+1)$ and can choose to penalize irrelevant predictions by assigning negative scores.",
"Mean reciprocal rank (MRR) and mAP are much less controllable in these two aspects.",
"One known drawback of nDCG is that it does not naturally penalize excessive predictions after the last relevant item.",
"We therefore use the average F$_1$ score as a complementary view of the agent performance."
],
[
"Average F$_1$",
"Similar to the definition in SQuAD, the average F$_1$ score is computed as below, following the same notation as in (REF ): $\\mbox{avg}(\\mbox{F}_1) = \\frac{1}{N}\\sum _{i=1}^N \\mbox{max}_j [ \\mbox{F}_1(A_i, A^g_{i,j} ) ].$ Note that F$_1$ does not concern ranking.",
"For some cases, such as enumeration questions, this is desirable, as the ranking order is merely the reading order, even if the item order is permuted, the answer quality is in general not compromised, hence, reasonable to be assigned the same evaluation score.",
"On the contrary, if relevance ranking is important, such as in Figure REF , then nDCG provides a better view.",
"Since both types of questions exist in the ScreenQA annotations, it is more complete to evaluate against both metrics.",
"Also note that the unit of precision and recall computation is based on items in $A$ , unlike SQuAD, which uses words as the unit instead.",
"We describe how to compare items in an answer $A$ with the ground truth $A^g$ in the next section."
],
[
"Answer matching",
"As mentioned in Section REF , the segmentation of UI elements provided in the predicted answer list $A$ may not coincide with the UI elements in the ground truth list $A^g$ .",
"Yet, if the overall answers are the same, the segmentation difference should not affect the evaluation score.",
"Therefore, we use the following empirical procedure to mitigate such an impact, using an illustrated example (each capitalized character is a word token): $A &= [\"AB\", \"B\", \"BB\", \"CBA\"] \\\\A^g &= [\"AB\", \"BC\", \"AB\"],$ Concatenate items in $A$ into a single item list $A^c = [``ABBBBCBA\"]$ .",
"Iterate through each $g \\in A^g$ and check if $g$ is contained in any item in $A^c$ .",
"If so, mark the $g$ as HIT ($\\text{\\ding {51}}$ ) and mark the corresponding matched word token in the original $A$ and remove the matched part and split the remaining parts in $A^c$ .",
"Otherwise, mark the $g$ as MISS ($\\text{\\ding {55}}$ ).",
"In this example, when $g = \"AB\"$ , it is a HIT: $A &= [``A_\\text{\\ding {51}}B_\\text{\\ding {51}}\", ``B\", ``BB\", ``CBA\"] \\\\A^c &= [``BBBCBA\"] \\\\A^g &= [``AB\"_\\text{\\ding {51}}, ``BC\", ``AB\"].$ Then when $g = ``BC\"$ , it is a HIT.",
"Note that the item in $A^c$ is split into two because of matching in the middle: $A &= [``A_\\text{\\ding {51}}B_\\text{\\ding {51}}\", ``B\", ``BB_\\text{\\ding {51}}\", ``C_\\text{\\ding {51}}BA\"] \\\\A^c &= [``BB\", ``BA\"] \\\\A^g &= [``AB\"_\\text{\\ding {51}}, ``BC\"_\\text{\\ding {51}}, ``AB\"].$ Last, when $g = ``AB\"$ again, it is a MISS, $A$ and $A^c$ unchanged, hence, omitted: $A^g &= [``AB\"_\\text{\\ding {51}}, ``BC\"_\\text{\\ding {51}}, ``AB\"_\\text{\\ding {55}}].$ Finally, iterate through each $a \\in A$ .",
"If any $a$ has at least one word token marked as HIT, then the whole $a$ is a HIT, otherwise, a MISS.",
"$A &= [``AB\"_\\text{\\ding {51}}, ``B\"_\\text{\\ding {55}}, ``BB\"_\\text{\\ding {51}}, ``CBA\"_\\text{\\ding {51}}].$ This procedure converts $A$ and $A^g$ into lists of HITs and MISSes.",
"Then the evaluation metrics in (REF ) and (REF ) can be applied.",
"Note that this procedure is not order invariant.",
"This in turn makes the F$_1$ score not entirely order independent if any UI element ambiguity happens.",
"This choice is to avoid the permutation complexity in evaluation.",
"In practice, this is rarely an issue because when the ambiguity happens, the UI elements involved are almost always tightly close to each other, making their order practically fixed.",
"See Case 3 in Figure REF as an example."
],
[
"Related Datasets and Annotations",
"ScreenQA has two aspects: multimodality and question answering.",
"We discuss related problems and datasets from these two aspects and focus our survey on datasets that are 1) human annotated and 2) released to the public domain."
],
[
"Multimodality",
"Mobile app screenshots contain nearly all possible representation of information through pixels.",
"Most commonly, the information is majorly by text, blended with icons, symbols, and images.Also videos, if we consider consecutive screenshots.",
"We leave out the video modality here in the context of annotating the underlying RICO screenshots.",
"We discuss three related multimodal domains."
],
[
"Screen UI for mobile apps",
"For data released in the public domain, the RICO dataset [8] is, to the best of our knowledge, still the largest collection of mobile app screenshots [7].",
"It contains 66k unique screenshots and their corresponding view hierarchies from 9.7k Android apps spanning 27 app categories.",
"Its overall approach extended ERICA [9], which is an interactive trace recording tool and also released 3k traces for 18k unique screenshots from 1k Android apps for the search intent.",
"LabelDroid [5] and [4] by the same authors released a dataset of 55k UI screenshots from 25 categories of 7.7k top-downloaded Android apps.",
"Annotations and the corresponding problems can be roughly categorized by the scope of the contexts.",
"At the UI element level, [27] annotated 77 icon types by shape, 15 out of which are additionally annotated with 38 semantic types, reaching about total 500k unique annotations.",
"This work is further concerned with how UI elements are associated with companion labels such that the screen understanding between UI elements can be established.",
"CLAY [18] attempted to resolve the layout and view hierarchy denoising problem, annotating 60k RICO screenshots, a total of 1.4M UI elements with bounding boxes and types.",
"[21] annotated 163k free-from descriptions for 61k UI elements from 22k RICO screenshots.",
"At the single-screen level, [30] collected text summarizations for screens, consisting of 112k screen descriptions across 22k RICO screenshots.",
"At the multi-screen level, one challenging direction is screen navigation, which requires the understanding of screen states, feasible action spaces of the current screen, and overall task goals.",
"Since multiple types of understandings are involved, this problem is not strictly focused on screen content understanding.",
"PixelHelp [19] contains 187 multi-step instructions over 780 screenshots for four task categories.",
"MoTIF [2] contains 6k fine-grained instructions mixed with infeasible ones, over for 125 apps spanning 15 app categories.",
"From the data perspective, annotating this type of problem is labor intensive and usually does not scale well.",
"In comparison, the ScreenQA dataset is single-screen, focused on screen contents, and based on the RICO screenshots."
],
[
"Document image understanding",
"Document image understandingAlso referred to as document analysis and recognition (DAR) or simply document understanding.",
"concerns understanding documents represented in pixels or scanned, photographed formats.",
"This domain is similar to mobile app screens for its text-heavy and non-sequential nature.",
"The most noticeable dataset is RVL-CDIP [11], a 400k-image subset from IIT-CDIP [17], a collection of low-resolution noisy documents, with balanced 16 document-level classes.",
"FUNSD [14] extracted a 199 scanned form images from RVL-CDIP and annotated them with bounding boxes and 4 text-segment-level classes.",
"SROIE [12] has 1k scanned receipt images for text localization, OCR, and key information extraction of 4 entity types.",
"CORD [24] contains 11k scanned receipt images, annotated with 9 classes and 54 subclasses for text segments in OCR boxes.",
"These earlier works are more about classification for text segments or for the whole document image.",
"A more recent work, DocVQA [22], uses a question answering format for span/segment extraction, with an annotation of 50k questions over 12k rectified, higher resolution document images.",
"DocVQA is highly related to ScreenQA for its 2D arrangement of texts and for its extractive question answering format.",
"We believe that the techniques developed for screens and document images are cross applicable."
],
[
"Visual question answering",
"Visual question answering (VQA) [1] and screen UI are oftentimes mentioned together, especially in the latter community, because of their vision-language multimodal nature.",
"However, VQA is distinctively different from screen understanding for two reasons: 1) The visual context for VQA is usually light in, or even free from, any text, while screen UI is the opposite, and 2) The images for VQA are typically photos of natural or daily scenes with objects, while screen UIs are information oriented and arranged in a certain visual structure.",
"There are some VQA variants comparatively closer to screen UI, to mention a few: VQA for texts on objects in photos, e.g., VizWiz [10] and TextVQA [26], and VQA for figures and charts, e.g., DVQA [15], FigureQA [16], and LEAF-QA [3].",
"These VQA tasks may appear as part of screens but in general are different problems."
],
[
"Question answering",
"Question answering tasks can be categorized by 1) open- or closed-domain, 2) answer formats and 3) main capacities to evaluate.Here we only include one or two examples per format and per capacity.",
"This is by no means to be comprehensive.",
"The common answer formats include span [25], entity [28], multiple choice [23], and generation [31].",
"The capacities to evaluate range from reading comprehension [32], multi-hop reasoning [33], [6], logic reasoning [34], and commonsense reasoning [29].",
"From this question answering perspective, ScreenQA is a closed-domain question answering task that expects answers by span (or UI element phrase) selection for screen reading comprehension.",
"As described in Section , we instructed the data annotators to avoid multi-hop, mathematical counting, and logic reasoning, in order to focus on the fundamental screen comprehension capacity."
],
[
"Annotation Method",
"We perform several steps to collect the ScreenQA annotations, as depicted in Figure REF .",
"Each step is described below.",
"Figure: View hierarchies (VHs) are overlaid on the screenshots with class names and the first few characters printed to assist annotators to determine whether the VHs for the main contents are in sync.a) The VH of some UI elements may be occluded.b) The VH of the hamburger menu is ghosting.Since the VH for the main content stays intact, data annotators are instructed to mark this as in-sync.c) Although the app bar and the orange plus button are in sync, the bounding boxes in the main content area are all out of sync.This is instructed to mark as out-of-sync and are excluded from further annotations for questions and answers.Figure: Data annotation interfaces for question and answer collection.a) Question annotation was performed in a sequential manner, the later and non-overlapping annotators can see all previous questions to diversify question framing and avoid duplication.We also used the sequential process to provide more feedback and training to the annotators for quality improvement.b) The answer annotators were tasked to determine if the question is valid and if the question is answerable from the screen context.If both are positive, the annotators need to answer the questions by 1) selecting or drawing the bounding boxes of UI elements, 2) Fill the text for each selected/drawn bounding box on right right, and 3) ranking them appropriately.The annotators were also tasked to review and make necessary corrections if the question has grammatical errors or typos."
],
[
"Pre-filtering",
"The pre-filtering stage filters out 1) screenshots from non-English appsThis is different from “non-English screenshots”, as translation and dictionary apps could pose confusion., and 2) screenshots whose view hierarchies (VHs) are out of sync with the main contents.",
"It is a known issue that in the RICO dataset, some screenshots and their corresponding view hierarchies are not perfectly in sync: there exists certain time difference between view hierarchy extraction and screenshot capturing.",
"We want to remove those screenshots to ensure that all ScreenQA annotations are not subject to such data noises.",
"Classifying the sync quality is tricky, even for human readers.",
"One may not be able to differentiate between occlusion, ghosting, and the actual out-of-sync.",
"See Figure REF for examples.",
"Accordingly, we instructed the annotators to focus on the main content area of the screen and make sure the bounding boxes in that area are not corrupted, as this is where most contents of interest and questions come from.",
"We use 27 annotators to perform this step.",
"Among RICO's 66k unique screenshots, about 11k screenshots are from non-English apps, and about 13k screenshots have out-of-sync view hierarchies.This out-of-sync number is different from [20] because we focus on the main content area.",
"With the union of these two filtered out, there remains about 51k screenshots from English apps with in-sync VHs."
],
[
"Question annotations",
"For question annotation, we asked the annotators to frame questions given a screenshot as the context.",
"The annotators were expected to compose 1) natural, daily-life questions as if using the app.",
"2) The composed questions should inquire information that can directly read off from the screen and 3) should not require logical reasoning, counting and calculation, mathematical comparison, etc.",
"We further required the annotators 4) not to ask questions about any advertisement on the screen.",
"The annotation UI is depicted in Figure REF .",
"We asked the annotators to compose up to five questions given a screenshot in the first pass.",
"In the second pass, we asked for up to three questions given a screenshot and the questions previously composed.",
"Each pass involved one annotator for each screenshot and whoever annotated the screenshot before is excluded from being assigned to the same screenshot.",
"This ensures that every screenshot is assigned precisely two annotators to compose questions.",
"We chose this sequential process 1) to avoid tricky deduplication of similar questions, and 2) to encourage annotators to diversify their questions.",
"Note that the same set of annotators were involved in the both passes such that each annotator had an opportunity to develop its own question style in the first pass before seeing others' in the second pass.",
"This makes sure that we still have certain numbers of question styles in the dataset before they converge to each other in repeated passes.",
"We again involved the 27 annotators.",
"The first pass of question annotation generated 46k questions.",
"The second pass added additional 36k questions.",
"These amount to a total of 82k questions, leaving about 15k screenshots with no questions annotated, due to lack of interesting contents."
],
[
"Answer annotations",
"We used the total 82k questions of 35k screenshots from the previous two-pass question annotation step to further annotate the corresponding answers.",
"The annotator who composed the question is excluded from annotating its own answer to avoid potential biases.",
"The answer annotation UI is shown in Figure REF .",
"Given an example, which contains a screenshot and a question, the annotators are tasked to Fix any grammatical errors or typos of the given question without altering its intention.",
"Answer the question, based on the context of the given screenshot, by 1) selecting bounding boxes from the underlying view hierarchy leaf nodes that contain the relevant answers, or drawing bounding boxes if no suitable leaf nodes can be used, and 2) ranking the answers in descending order of relevance if applicable, or by the common reading order.",
"Consider two exceptions: 1) The question may be incomprehensible or 2) the screenshot does not contain the answer to the question, due to the questioner's lack of understanding of the app.",
"Then the example should be marked as “invalid questions” and “not answerable from the screenshot”, respectively.",
"As this step of annotation is more subjective and requires better understanding of the presented apps, we used a selected pool of annotators to perform this step of annotations.",
"We then filtered out the questions marked as “invalid question” to produce the overall ScreenQA annotations."
],
[
"Annotation Analysis",
"We analyze the annotations of questions and answers in this section.",
"Table: Question category distribution and examples.Figure: Histograms for number of composed questions and number of bounding boxes in answers.a) The two question annotation passes were capped at five and three questions, respectively, resulting in the maximum eight questions in total.b) The cases when a single bounding box forms a sufficient answer amount to 92% of the questions, hence removed from the chart for the clarity of the long tail.Anything beyond 11 bounding boxes is less than 0.05%, accumulatively less than 0.1%."
],
[
"Question analysis",
"We collected overall 82k questions over 36k unique screenshots from RICO.",
"Among the 82k questions, there are 46k unique questions.Note that it is natural and valid to ask the same common questions over various screenshots, for example, “Which option is selected on the screen?” and “What is the email address?” Some screenshots receive more questions because they usually contain more information to be asked about.",
"Yet, the histogram still exhibits a reasonable exponential decay with a mild slope, as depicted in Figure REF .",
"To further understand what questions have been asked, we categorize the questions using regular expressions based on a list of empirically determined question categories.",
"The categories are meant to provide a rough overview of the question annotations and by no means to provide a precise categorization.",
"The distribution and examples by these categories are tabulated in Table REF .",
"Note that the questions were not composed at the annotators' full discretion: They are conditioned on the given screenshots.",
"That is to say, the distribution is implicitly influenced by the RICO crawling process.",
"For example, as RICO crawled screen traces from freshly installed apps and did not login an account, a noticeable number of the screen traces end at a login page.",
"This in turn translates to a higher percentage of questions asked about app names, email addresses, permissions to login, etc."
],
[
"Answer analysis",
"We analyze the answer annotations in two aspects: 1) How often do we need more than one bounding box and its text to answer the question, and 2) How often do human annotators find the view hierarchy useful to provide a minimal answer to the question.",
"Figure REF illustrates the histogram of number of bounding boxes used in each answer.",
"About 92% of questions can be answered fully using a single bounding box.",
"Among these single-bounding-box answers, 52% uses a VH leaf node directly, while 48% uses a manually drawn bounding box.",
"If we consider all questions together, still 52% uses VH leaf nodes entirely, while 47% uses manually drawn bounding boxes.",
"That is, for about half of the total number of screenshots, human annotators preferred to manually draw the bounding boxes in order to provide answers that minimally satisfy the question.",
"This observation reflects the necessity not to require the view hierarchy input for ScreenQA as described in Task REF .",
"Interestingly, there exist some cases, about 0.5% of the questions, that the human annotators used a mixture of VH leaf nodes and manually drawn bounding boxes as their full answer.",
"By inspecting those cases, we found that these usually happen 1) when the answer is an enumeration of “inhomogeneous” options that are organized differently on the screen, such as using email vs. other APIs to login, and 2) when an answer needs multiple parts to be complete, such as a date consisting of year, month, and day scattered on the calendar UI, and a temperature or a measurement requiring a number followed by the corresponding unit.",
"These parts may not be displayed in the same way, resulting in lack of useful VH leaf nodes for some of the parts."
],
[
"Conclusion",
"In this work, we proposed the ScreenQA task.",
"We annotated a large-scale ScreenQA dataset, which contains more than 80,000 question-answer pairs.",
"Compared to other vision-language multimodal problems, such as document image understanding and visual question answering, ScreenQA poses its unique challenges: rich in text, diverse in apps, and blended with icons and symbols.",
"We hope to use the ScreenQA task and the dataset to encourage the community to look into this screen content understanding problem, as it enables new technologies and new user experiences."
],
[
"Acknowledgements",
"The authors would like to thank Srinivas Sunkara for his valuable discussions and comments on this manuscript."
]
] | 2209.08199 |
[
[
"Intrinsically Motivated Reinforcement Learning based Recommendation with\n Counterfactual Data Augmentation"
],
[
"Abstract Deep reinforcement learning (DRL) has been proven its efficiency in capturing users' dynamic interests in recent literature.",
"However, training a DRL agent is challenging, because of the sparse environment in recommender systems (RS), DRL agents could spend times either exploring informative user-item interaction trajectories or using existing trajectories for policy learning.",
"It is also known as the exploration and exploitation trade-off which affects the recommendation performance significantly when the environment is sparse.",
"It is more challenging to balance the exploration and exploitation in DRL RS where RS agent need to deeply explore the informative trajectories and exploit them efficiently in the context of recommender systems.",
"As a step to address this issue, We design a novel intrinsically ,otivated reinforcement learning method to increase the capability of exploring informative interaction trajectories in the sparse environment, which are further enriched via a counterfactual augmentation strategy for more efficient exploitation.",
"The extensive experiments on six offline datasets and three online simulation platforms demonstrate the superiority of our model to a set of existing state-of-the-art methods."
],
[
"Introduction",
"Recently, deep reinforcement learning (DRL) receives increasing interests in RS because of its capability in capturing users' dynamic interests [1].",
"Current DRL-based RS can be generally categorized into three streams: value-based methods, policy-based methods and hybrid methods.",
"One of the representatives of value-based methods would be the Deep Q-learning (DQN), in which [2] bring it into news recommendation.",
"However, deep Q-learning based methods require the “maximize” operation over the action space (i.e., all the candidate items) which is not traceable and may induce agent stuck problem [3].",
"Policy-gradient methods can mitigate such problem but suffers the high variance problem as the optimization is based on last step's trajectory which could be distinct from previous trajectories [4].",
"Hybrid method is the combination of policy-gradient and value-based methods.",
"It aims to reduce the variance for policy-gradient by introducing the value-based method [5] and gains more attention [6], [7], [8], [9].",
"However, the user-item interactions are commonly sparse.",
"It hinders policy optimisations to find the rewards via exploration as well as maximizing the performance via exploitation.",
"Specifically, as DRL relies on carefully engineering environment rewards that are extrinsic to the agents, the sparsity barely provides dense reward signals (i.e., most of the reward signals might be missing because of the highly incomplete interactions and user feedback).",
"Hence, new exploration strategies would be a choice to encourage agent to discover a wider range of states and formulate richer interaction trajectories [10].",
"Recent literature [11], [12] show that exploration is effective to reduce the model uncertainty in regions of sparse rewards or user interactions.",
"Moreover, most of existing works in DRL RS apply $\\epsilon $ -greedy as the exploration strategy that agent has $\\epsilon $ possibility to conduct exploration randomly [1].",
"However, random exploration increases the training time and the uncertainty and may not be able to explore enough informative interaction trajectories.",
"Moreover, it will cost considerable amount of trials and it is not feasible to be applied for highly sparse user feedback in the recommender systems as it also requires significant amount of trials for exploitation which is known as exploration and exploitation trade-off.",
"Differently, several attempts have been made from the different perspective of data augmentation to relief the sparsity.",
"Experience replay is widely used in DRL methods which empowers agent to learn by reusing past interaction trajectories.",
"However, the experience replay can only promote certain trajectory to be replayed [13].",
"The policy learning process may be harmed if the generated trajectories are not informative.",
"Recent studies also investigate to equip data augmentation with causality by governing to generate informative trajectories.",
"For instance, [14] designs a simple counterfactual method by measuring the embedding changing to generate new user sequence.",
"Moreover, [15] considers the embedding contains two parts which are dispensable or indispensable items related to the final recommended items by leveraging the causality.",
"By replacing dispensable items, it can generate more user sequence but with the same performance.",
"The main limitation with these approaches, however, is that they assume an embedding of state space, while state is dynamic and updated after each interaction.",
"Moreover, the agent never knows the ground-truth (i.e., user's final choice) during the online interactions.",
"Hence, it is impractical to leverage the ground-truth to determine the embedding difference or indispensable items as existing works did.",
"In order to address the above issues, we propose a new end to end model namely Intrinsically Motivated Reinforcement Learning with Counterfactual Augmentation (IMRL) from two aspects: augmenting informative trajectories and a new exploration strategy.",
"We design an novel empowerment-based exploration strategy to encourage agent to explore the potentially informative interaction trajectories in the sparse environment.",
"Moreover, we elaborate a new counterfactual data augmentation method for DRL RS to augment those newly explored informative trajectories so that they can have a higher exposure probability thus boost the final performance.",
"In summary, we make the following contributions in this paper: We propose a novel DRL method IMRL which can augment trajectories which is causally valid but never seen by the agent to relief the data sparsity problem.",
"Moreover, we also introduce an adaptive threshold to dynamically control the boundary of the informative as the learning process in DRL is evolutionary.",
"We design an empowerment-driven exploration strategy for IMRL to help explore those un-explored but potentially informative interaction trajectories.",
"Our experiments show that the designed exploration strategy can boost the final performance in the online simulation platforms.",
"We have conduct extensive experiments in both offline and online settings and show the superiority of IMRL.",
"We have conduct offline experiments with six well-known datasets and online experiments in three public simulation platforms.",
"Reinforcement learning based recommender systems learn from interactions through a Markov Decision Process (MDP).",
"Given a recommendation problem consisting of a set of users $\\mathcal {U} = \\lbrace u_0,u_1,\\cdots $ $u_n\\rbrace $ , a set of items $\\mathcal {I} = \\lbrace i_0,i_1,\\cdots i_m\\rbrace $ and user's demographic information $\\mathcal {D}=\\lbrace d_0,d_1,\\cdots ,d_n\\rbrace $ , MDP can be represented as a tuple $(\\mathcal {S},\\mathcal {A},\\mathcal {P},\\mathcal {R},\\gamma )$ , where each represents the following, $\\mathcal {S}$ denotes the state space, which is the combination of the subsets of $\\mathcal {I}$ and $\\mathcal {D}$ , it represents user's previous interactions and demographic information.",
"Based on that, it can be written as a composition form: $\\mathcal {S} = \\mathcal {S}^1 \\oplus \\mathcal {S}^2 \\oplus \\cdots \\mathcal {S}^n$ for a fixed $n$ which represents the dynamic count of components [16]; $\\mathcal {A}$ is the action space, which represents agent's selection during recommendation based on the state space $\\mathcal {S}$ .",
"Similarly, it can also be written as a composition form: $\\mathcal {A} = \\mathcal {A}^1 \\oplus \\mathcal {A}^2 \\oplus \\cdots \\mathcal {A}^n$ ; $\\mathcal {P}$ is the set of transition probabilities for state transfer based on the action received which also refers to users' behavior probabilities, it is worth to mention that $\\mathcal {P}$ will not be estimated in this study as we are using model-free reinforcement learning approach; $\\mathcal {R}$ is a set of rewards received from users, which are used to evaluate the action taken by the recommendation system, with each reward being a binary value to indicate user's click; $\\gamma $ is a discount factor; $\\gamma \\in [0,1]$ for balancing the future reward and current reward.",
"Given a user $u_0$ and the state $s_0$ observed by the agent (or the recommendation system), which includes a subset of item set $\\mathcal {I}$ and user's demographic information $d_0$ , a typical recommendation iteration for user $u_0$ goes as follows.",
"First, the agent makes an action $a_0$ based on the recommend policy $\\pi _0$ under the observed initial state $s_0$ and receives the corresponding reward $r_0$ .",
"Then, the agent generates a new policy $\\pi _1$ based on the received reward $r_0$ and determines the new state $s_1$ based on the probability distribution $p(s_{new}|s_0,a_0)\\in \\mathcal {P}$ .",
"The cumulative reward after $k$ iterations is as follows: $r_c = \\sum _{k=0} \\gamma ^{k}r_k$"
],
[
"Local Causal Models",
"Structural Causal Models (SCMs) [17] can be represented as a tuple: $\\mathcal {M}_t(V_t,U_t,F)$ with the following components based on the state and action composition form at timestamp $t$ .",
"It normally represents as a directed acyclic graph (DAG) $\\mathcal {G}$ with the following components, $V_t = \\lbrace s_t^1, s_t^2, \\cdots , s_t^n, a_t^1,\\cdots ,a_t^m, s_{t+1}^1,\\cdots ,s_{t+1}^n\\rbrace $ represents the nodes in DAG $\\mathcal {G}$ .",
"$U_t = \\lbrace u^1, \\cdots , u^{2n+m}\\rbrace $ is a set of noise variables, one for each in $V_t$ .",
"It is determined by the initial state, past actions and environment.",
"We assume that the noise variable is time independent.",
"Which implies that $U_t \\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}U_{t+1}$ .",
"$F$ is a set of functions that: $U_t \\times \\text{Parentage}(V_t) \\rightarrow V_t$ where $\\text{Parentage}(\\cdot )$ means the parent node of $\\cdot $ .",
"where we assume that the dynamic count of state is $n$ and $m$ for action, state observed at timestamp $t$ is written as $s_t$ which is the composition of $\\lbrace s_t^1, s_t^2, \\cdots , s_t^n\\rbrace $ .",
"Local causal models is an extension to the SCM which only considers about the local causal effect [18].",
"Local causal model can be represented as $\\mathcal {M}_t^\\mathcal {L}(V_t^\\mathcal {L},U_t^\\mathcal {L},F^\\mathcal {L})$ with DAG $\\mathcal {G}^\\mathcal {L}$ from the global causal model $\\mathcal {M}_t$ in the subspace $\\mathcal {L}$ with the same components with extra constraint, $& \\text{Parentage}(V_t^\\mathcal {L}) = \\text{Parentage}(V_t|(s_t,a_t)\\in \\mathcal {L}), \\\\& \\text{Parentage}(U_t^\\mathcal {L}) = \\text{Parentage}(U_t|(s_t,a_t)\\in \\mathcal {L}).$ Moreover, the local causal model requires the set of edges in $\\mathcal {G}$ to be structurally minimal [19]."
],
[
"Methodology",
"In this section, we will briefly explain the proposed approach Intrinsically Motivated Reinforcement Learning with Counterfactual Augmentation (IMRL) for reinforcement learning based recommendation which can address the sparse interactions problem in DRL RS.",
"We are addressing such problem from two aspects according to the aforementioned challenges: i) use a novel data augmentation method to generate more potentially informative interaction trajectories by using using counterfactual reasoning; ii) design a new exploration strategy by introducing an intrinsic reward signal to encourage agent to conduct the exploration.",
"Hence, the proposed IMRL consists two main components: Counterfactual reasoning for data augmentation and Intrinsically Motivated Exploration."
],
[
"Counterfactual Data Augmentation",
"Formally, given an arbitrary trajectory $\\tau :(s,a,r,s^{\\prime })$ that sampled from the replay buffer, $r$ is the reward signals received by agent when action $a$ is executed at state $s$ .",
"Given the large candidate item set situation, most of the trajectories are not informative (i.e., $r$ is zero).",
"With the zero reward situation, the informative trajectories are hard to be sampled because the number of those non-informative trajectories is significantly larger.",
"Hence, augment those informative trajectories to increase the possibility to be sampled is a straightforward solution.",
"We assume that state $s_{t+1}$ satisfy the SCM: $s_{t+1} = f(s_t,a_t,U_{t+1}),$ where $f(\\cdot )$ represents the causal mechanism, $a_t$ is the action at timestamp $t$ , and $U_{t+1}$ is the noise term which is independent of $(s_t,a_t)$ .",
"Our main idea is to estimate the causal mechanism $f(\\cdot )$ and generate more data that unseen but causally valid.",
"To achieve the goal, the simple solution is to work out the causal mechanism $f(\\cdot )$ firstly.",
"However, estimate the global $f(\\cdot )$ is challenging [20].",
"Instead, inspired by recent advance in local causal models [21], [18], we decide to estimate the local causal mechanism $f_l(\\cdot )$ .",
"Local causal model assumes that there exists a local DAG $\\mathcal {G}^{\\mathcal {L}(i\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}j)}$ in a subspace $\\mathcal {L}^{i\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}j}$ such that $\\mathcal {L}^{i\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}j} \\subset \\mathcal {L}$ where $\\mathcal {L}: \\mathcal {S}\\times \\mathcal {A}$ .",
"It satisfies the following condition: $s_{t+1}^j \\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}V_t^i | \\text{Parentage}(s_{t+1}^j)\\setminus V_t^i, (s_t,a_t) \\in \\mathcal {L}^{j\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}i},$ Specifically, in recommender systems, there is a large subspace of states in which the users' previous interests will not affect the final recommendation as user's interest are dynamics.",
"Hence, if we focus on the subspace $(s_t,a_t)\\in \\mathcal {L}^{(j\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}i)}$ , we can formulate a local causal model $\\mathcal {M}_t^{\\mathcal {L}^{(j\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}i)}}$ that local DAG $\\mathcal {G}^{\\mathcal {L}^{(j\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}i)}}$ contains no edge from $V_t^i$ to $s_{t+1}^j$ .",
"It implies that the local DAG $\\mathcal {G}^{\\mathcal {L}^{(j\\protect \\mathchoice{\\protect \\mathrel {\\unknown.",
"{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.",
"{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}i)}}$ is strictly sparser than global DAG $\\mathcal {G}$ .",
"With such property, we now can uses the local causal model to conduct the data augmentation to relief the sparsity problem in DRL RS.",
"Consider a counterfactual question “What if user $u$ interested in item $j$ instead of item $i$ at timestamp $t$ ?”.",
"It can be described in causal form - “What if component $s_t^i$ had value $x$ instead of $y$ at timestamp $t$ ?”.",
"It can be solved by using Pearl’s do-calculus to the causal model $\\mathcal {M}$ to obtain a sub-model, $\\mathcal {M}_{\\text{do}(s_t^i=x)}^\\mathcal {L} = (V,U,F_x) \\text{ where } F_x = F \\setminus f^i \\cup \\lbrace s_t^i = x\\rbrace .$ Moreover, the incoming edge to $s_t^i$ will be removed from $\\mathcal {G}_{\\text{do}(s_t^i=x)}$ .",
"Now we will utilize the local causal model to generate data that unseen by agent but causally valid.",
"In order to achieve that, we will augment the data based on the counterfactual modification with subset of the causal factor at timestamp $t$ , and keep the reaming factors unchanged.",
"Such augmentation process can use the counterfactual model $\\mathcal {M}_{\\text{do}(s_t^i=x)}^\\mathcal {L}$ to modify the causal factors $s_t^{i\\cdots j}$ and re-generate the corresponding children in the DAG which would increase the However, such process is computational expensive in our recommendation scenario as it requires re-sampling for the children in DAG.",
"Inspired by the idea for collaborative filtering and the state composition form which was mentioned previously.",
"We can simplify the process by omitting the sampling process.",
"Specifically, the core of the augmentation is to estimate the $\\mathcal {M}_{\\text{do}(s_t^i=x)}^\\mathcal {L}$ .",
"It can be easily to get by assuming that similar users will have similar interests which is the idea of collaborative filtering.",
"Under such assumption, we can get $\\mathcal {M}_{\\text{do}(s_t^i=x)}^\\mathcal {L}$ by replacing the causal independent component of $s_t$ using the local causal model.",
"For example, those interaction histories that not affect current recommendation which could be recognized as causally independent with the current action $a_t$ .",
"The overall algorithmFor set-based representations can be found in alg:augmentation.",
"[ht] Inputinput Fnfunction isend Augmentation ($\\tau _1,\\tau _2$ ) $s_1,a_1,s_1^{\\prime } \\leftarrow \\tau _1$ , $s_2,a_2,s_2^{\\prime }\\leftarrow \\tau _2$ $m_1, m_2 \\leftarrow $ ind($s_1, a_1$ ), ind($s_2, a_2$ ) $d \\leftarrow $ random sample from ($m_1 \\cap m_2$ ) $s_g,a_g,s_g^{\\prime } = s_1,a_1,s_1^{\\prime }.$ copy() $s_g[d],a_g[d],s_g^{\\prime }[d] = s_2[d],a_2[d],s_g^{\\prime }[d]$ $\\tau _g \\leftarrow s_g,a_g,s_g^{\\prime },R(s_g,a_g)$ $D.append(\\tau _g)$ if $s_g[d], a_g[d], s^{\\prime }_g[d]$ exists else $D$ $D$ ; ind(s,a) Generate the adjacency matrix $M$ by using $s$ and $a$ Find the connected (dependent) components set $C$ index of independent components in $\\mathcal {G}\\setminus C$ Counterfactual Data Augmentation"
],
[
"Intrinsically Motivated Exploration",
"The second aspect we use to address the sparsity is the intrinsically motivated exploration strategy.",
"We propose to use the empowerment to represent the intrinsical motivation.",
"It can boost agent's exploration capability so that more states can be reachable which can be used to produce corresponding potential informative interaction trajectories.",
"Empowerment is an information-theoretic method where an agent executes a sequence of $k$ actions $\\textbf {a}^k \\in \\mathcal {A}$ when in state $s \\in \\mathcal {S}$ according to a explore policy $\\pi _{\\textit {empower}}(s,\\textbf {a}^k)$ (we use $\\pi _e(s,\\textbf {a}^k)$ to shorten the notation) which is a conditional probability distribution: $\\pi _e: \\mathcal {S}\\times \\mathcal {A}\\rightarrow [0,1]$ .",
"The agent’s aim is to identify an optimal policy $\\pi _e$ that maximizes the mutual information $I[\\textbf {a}^k, s^{\\prime }|s]$ between the action sequence $\\textbf {a}^k$ and the state $s^{\\prime }$ to which the environment transitions after executing the sequence $\\textbf {a}$ in current state $s$ , formulated as: $\\overline{\\mathbb {E}}(s) & = \\max _{\\pi _e} I[\\textbf {a}^k, s^{\\prime }|s] \\\\ &= \\max _{\\pi _e}\\mathbb {E}_{\\pi _e(s,\\textbf {a}^k)\\mathcal {P}(s,\\textbf {a}^k,s^{\\prime })}\\log \\bigg [\\frac{p(\\textbf {a}^k,s^{\\prime }|s)}{\\pi _e(\\textbf {a}^k,s)}\\bigg ].$ Here, $\\overline{\\mathbb {E}}(s)$ refers to the optimal empowerment value and $\\mathcal {P}(s,\\textbf {a}^k,s^{\\prime })$ to the probability of transitioning to $s^{\\prime }$ after executing the action sequence $\\textbf {a}^k$ in state $s$ , where $\\mathcal {P}:\\mathcal {S}\\times \\mathcal {A}\\times \\mathcal {S} \\rightarrow [0,1]$ .",
"Importantly, $p(\\textbf {a}^k,s^{\\prime }|s) = \\frac{\\mathcal {P}(s,\\textbf {a}^k,s^{\\prime })\\pi _e(\\textbf {a}^k,s)}{\\sum _{\\textbf {a}^k} \\mathcal {P}(s,\\textbf {a}^k,s^{\\prime })\\pi _e(\\textbf {a}^k,s)}$ is the inverse dynamics model of $\\pi _e$ .",
"The optimal empowerment values are obtained by the policy $\\pi ^*$ that maximizes $\\mathbb {E}^{\\pi ^*}(s)$ .",
"However, the above definition of empowerment is more general than RL setting as it considers $k$ -step policy but RL normally considers single step policy.",
"Moreover, estimate the $k$ -step empowerment is challenging.",
"Hence, we use $k=1$ in this study to narrow down the empowerment into RL setting which only consider one step further.",
"Blahut-Arimoto algorithm [22], [23] shows that empowerment can be solved in low-dimensional discrete settings.",
"Moreover, [24] uses the parametric function approximators to estimate the empowerment in high-dimensional and continuous state-action spaces.",
"It provides the theoretical guarantee for using empowerment in recommender systerms as the state-action spaces are high-dimensional [1].",
"There are two possibilities for utilizing empowerment in RL: Find high mutual information between actions and the subsequent state that achieved by that action.",
"Train a behavioral policy to take an action in each state such that the expected empowerment value of the next state is highest.",
"Both approaches are feasible for normal reinforcement learning setting which can encourage agent to take an action that can result in the maximum number of future states.",
"But there is some conceptual difference between them.",
"Second approach seeks states with a large number of reachable next states [25], [26].",
"The first approach aims to find high mutual information between actions and subsequent state which is not necessarily the same as seeking high empowered state [24].",
"The first approach can be achieved by transferring state and its subsequent states' representations into KL divergence and minimizing it [27].",
"However, the transformation introduces extra complexity and information lost which may affect the performance.",
"The second approach uses the behavior policy to explore the highly empowerment states would be more suitable and simple for our setting.",
"The majority reason is that we are using model-free approach to solve the problem.",
"The model-free RL methods maintain two policy which are target policy $\\pi $ and behavior policy $\\pi _e$ .",
"Second approach is more suitable for the model-free approaches as it does not require extra computational cost to traverse all of the subsequent states and calculate the KL divergence.",
"It would be easily adopts into the existing RL frameworks.",
"Hence, the goal of the MDP process with the empowerment can be rewritten as, $\\max _{\\pi _b} \\mathbb {E}_{\\pi _b,\\mathcal {P}} \\bigg [\\sum _{t=0}^\\infty \\gamma ^t (\\alpha \\cdot R(s_t,a_t) + \\beta \\cdot \\frac{p(a_t|s_{t+1},s_t)}{\\pi _b(a_t,s_t)}\\bigg ]$ where $\\pi _b$ is the behavior policy, $\\alpha , \\beta $ are constants used to balance instant reward and empowerment.",
"We use the empowerment as the extra term to the reward signal $R(s_t,a_t)$ to encourage agent for exploration."
],
[
"Training Procedure",
"From information-theoretic prospective, optimizing for empowerment is equivalent to optimize the inverse dynamics [26], [28] based on the distribution $\\pi _e(s,a)$ .",
"Hence, we introduce the inverse dynamics into the objective function to calculate the empowerment.",
"Our method is build up on Soft Actor-Critic (SAC) [29] with temperature tuning and deterministic policy.",
"The overall training algorithm can be found in alg:train.",
"We follow the same training strategy as standard SAC algorithm.",
"However, as the empowerment is introduced, we modify the objective function to ensure the empowerment term can be optimized.",
"We use several function approximators to learn different components in the proposed method.",
"Value function $V$ is parametrized by $\\psi $ , Q-function is parametrized by $\\theta $ , target policy is parametrized by $\\phi $ and inverse dynamics is parameterized by $\\xi $ .",
"As we are using an off-policy algorithm where the transition probability is not be learned, we use the $\\mathcal {P}$ to represent the state transition probability in the environment.",
"The soft Q-function can be trained by minimizing the following objective function: $J_Q(\\theta ) = \\mathbb {E}_{(s_t,a_t)\\sim \\mathcal {D}}\\big [Q_\\theta (s_t,a_t) - (r(s_t,a_t) + \\gamma V_{\\psi }(s_{t+1}))^2\\big ].$ The target function $V_\\psi $ can be optimized by minimizing: $J_V(\\psi ) = \\mathbb {E}_{s_t\\sim \\mathcal {D}} \\Big [V_\\psi (s_t) -\\mathbb {E}_{a_t\\sim \\pi _\\phi }\\big (Q_\\theta (s_t,a_t)+\\underbrace{\\beta g(s_t,a_t)}_{\\text{policy}}\\big )^2\\Big ],$ where $\\beta $ is a constant used to balance the empowerment.",
"Different from the origin SAC algorithm, we replace the policy term from $-\\log \\pi _\\phi (s_t,a_t)$ to $g(s_t,a_t)$ to consider the empowerment where $g(s_t,a_t)$ is defined as: $g(s_t,a_t) = \\mathbb {E}_{\\mathcal {P}(s^{\\prime }|s_t,a_t)}\\big [\\log p_\\xi (a_t|s^{\\prime },s_t) -\\log \\pi _\\phi (s_t,a_t)\\big ].$ Note that, different from $s_t$ , the $s^{\\prime }$ represent all the possible subsequent states where $a_t$ is executed in state $s_t$ at timestamp $t$ .",
"[ht] mycommfont Inputinput each episode till converge $s_0 \\leftarrow $ initiate state each environment step $a_t \\sim \\pi _{\\phi }(a_t|s_t)$ $r_t \\leftarrow \\mathcal {R}(s_t,a_t)$ $s_{t+1} \\leftarrow \\mathcal {P}(s_{t+1}|s_t,a_t)$ *Sample from environment $\\mathcal {D} \\leftarrow \\mathcal {D} \\cup \\lbrace s_t,a_t,r_t,s_{t+1}\\rbrace $ $r_t \\ge T$ Augmentation($\\mathcal {D}$ ) Update $T$ based on eq:threhold each gradient step $\\theta \\leftarrow \\theta - \\lambda _Q \\nabla _{\\theta } J_Q(\\theta )$ $\\psi \\leftarrow \\psi - \\lambda _\\psi \\nabla _\\psi J_V(\\psi )$ $\\phi \\leftarrow \\phi - \\lambda _\\phi \\nabla _\\phi J_\\pi (\\phi )$ $\\xi \\leftarrow \\xi - \\lambda _\\xi \\nabla _\\xi J_p(\\xi )$ $\\alpha \\leftarrow \\alpha - \\lambda \\nabla _\\alpha J(\\alpha )$ Overall training algorithm Similarly, the optimization of the policy $\\pi (\\phi )$ can be written as: $J_\\pi (\\phi ) = -\\mathbb {E}_{s_t\\sim \\mathcal {D}}\\Big [\\mathbb {E}_{a_t\\sim \\pi _\\phi }\\big [\\beta g(s_t,a_t)+Q_\\theta (s_t,a_t)\\big ]\\Big ],$ where apply the same substitution.",
"The inverse dynamic $p(\\xi )$ will be updated based on: $J_p(\\xi ) = -\\mathbb {E}_{\\pi _\\phi }\\big [\\log {p_\\xi }(a_t|s^{\\prime },s_t)\\big ].$ Lastly, the temperature parameter will be adjusted automatically by using the following entropy method [30]: $J(\\alpha ) = -\\mathbb {E}_{a_t\\sim \\pi _\\phi }\\big [\\alpha \\log \\pi _\\phi (s_t,a_t) + \\alpha \\mathcal {H}\\big ].$ Note that, SAC uses exponentially averaged value $\\psi ^{\\prime }$ to stabilize the training process [31].",
"The update rule can be written as: $\\psi ^{\\prime } \\leftarrow \\lambda _{\\psi ^{\\prime }} \\psi + (1-\\lambda _{\\psi ^{\\prime }})\\psi ^{\\prime }$ .",
"Moreover, we conduct the data augmentation to the replay buffer after each interaction to generate causally valid unseen trajectories.",
"Such augmentation can provide more trajectories at the early stage to increase the samples.",
"Specifically, most of model parameters are learned by sampling from the replay buffer $\\mathcal {D}$ .",
"The training process can be described as searching states or state-action pairs in the $\\mathcal {D}$ to update the target policy such that the received reward is maximized.",
"As the augmentation introduces more samples into the replay buffer, the gradient update process have a higher chance to achieve a better policy.",
"It is worth to mention that, we are only augmenting those informative trajectories.",
"However, the definition of the informative trajectories are highly depends on the learning progress.",
"We believe that every trajectories with non-zero reward are informative at the early stage but are harmful when the final target policy close optimal.",
"Hence, we selectively conduct the augmentation with replay buffer to ensure that those zero-reward trajectories will not be augmented to increase sparse.",
"However, as the interaction goes forward, the way we determine informative trajectories is changing.",
"Some trajectories is informative at the early stage as agent need to explore all the possibilities.",
"At the later stage, agent will pursue higher rewarding trajectories which makes those low-rewarding trajectories less useful.",
"In such situation, we design a adaptive threshold to evaluate the trajectory is worth for augmentation or not.",
"The designed adaptive threshold is intuitive where moving average is used.",
"It can be represented as, $T = \\sigma /\\lambda _d \\text{ if } T \\le T_{max} \\text{ else } T_{max}$ where $\\sigma $ is a customized constant to determinate the initial value of the threshold and the decay rate $\\lambda _d \\in (0,1]$ .",
"The decay rate will decrease when number of episode increase.",
"With a set of value $(\\sigma , \\lambda _d)$ , we can achieve a monotone increasing threshold.",
"Ideally, we start with the $\\sigma = 1$ and $\\lambda _d = 1$ as initial values.",
"$T_{max}$ is an environment constant which represents the maximum reward that agent can achieve each step."
],
[
"Experiments",
"In this section, we conduct the experiments to answer three main research questions: RQ1: Does IMRL outperform than existing DRL approaches in both offline and online settings?",
"RQ2: Can IMRL help to relief the sparsity interaction problem in DRL RS in online simulation environments?",
"RQ3: How each components contribute to the final performance in online simulation environments?"
],
[
"Experiment Setup",
"In order to demonstrate the superiority of IMRL, we've conducted the experiments on both offline setting and online simulation setting.",
"We use six public available datasets: MoveLens-20Mhttps://grouplens.org/datasets/movielens/20m/ is a dataset about the user behavior of watching movies.",
"Librarythinghttp://cseweb.ucsd.edu/~jmcauley/datasets.html#social_data is a dataset about book review information.",
"Book-crossinghttp://www2.informatik.uni-freiburg.de/ cziegler/BX/is a dataset related to book preference.",
"Netflix Prizehttps://www.kaggle.com/netflix-inc/netflix-prize-data is a dataset from Netflix yearly competition for recommendation.",
"Amazon-CDhttps://jmcauley.ucsd.edu/data/amazon/ is e-commerce datasets which contains user's purchase behavior.",
"GoodReadshttps://www.goodreads.com/ is a book dataset.",
"The statistics of those datasets are summarized in tab:stat.",
"Table: Statistics of the datasets used in our offline experiments.Moreover, due to the special interaction logic in reinforcement learning based methods.",
"Extra data preparation process is required to ensure that the agent can interact with such offline datasets.",
"We follow the same strategy in previous work [32] that transfer those datasets into reinforcement learning environments so that IMRL can interact with."
],
[
"Baselines and offline evaluation metrics",
"The following baselines are selected which contain both non-reinforcement learning based methods and reinforcement learning based methods: SASRec [33] is a well-known baseline for sequential recommendation method that utilize the self-attention mechanism.",
"CASR [14] is a counterfactual data augmentation method for sequential recommendation.",
"As CASR only conduct the augmentation, we select the STAMP [34] to make recommedation which is described in CASR.",
"CauseRec [15] is a counterfactual sequence generation method for sequential recommendation.",
"CoCoRec [35] is a category-aware collaborative method for sequential recommendation.",
"DEERS [34] is a reinforcement learning based recommendation method that considers both positive and negative feedback.",
"KGRL [6] is a reinforcement learning based method that utilize the capability of GCN to process the knowledge graph information.",
"TPGR [7] is a model that uses reinforcement learning and binary tree for large-scale interactive recommendation.",
"PGPR [36]is a knowledge-aware model that employs reinforcement learning for explainable recommendation.",
"It is worth to mention that, because of the different training paradigm of those two kinds of methods (i.e., supervised learning and reinforcement learning), we are not able to guarantee that the comparison with those existing non-reinforcement learning based state-of-the-art methods are strictly fair.",
"We've conducted those supervised learning based methods in the same setting as well as those reinforcement learning based methods.",
"Recall, precision, nDCG are selected as the evaluation metrics.",
"Table: The overall results of our model comparison with several state-of-the-art models in different datasets.",
"The result was reported based on top-20 recommendation and highest results are in bold and second highest is marked by *"
],
[
"Online Simulation",
"Different from offline datasets, online simulation platforms are all based on gymhttps://gym.openai.com which is a standard toolkit for reinforcement learning research.",
"We conduct the online experiments on three widely used public simulation platforms: VirtualTB [37], RecSim [38] and RecoGym [39], which mimic online recommendations in real-world applications.",
"VirtualTB is a real-time simulation platform for recommendation, where the agent recommend items based on users' dynamic interests.",
"VirtualTB uses a pre-trained generative adversarial imitation learning (GAIL) to generate different users who have both static interest and dynamic interest.",
"Moreover, the interactions between users and items are generated by GAIL as well.",
"Benefit from that, VirualTB can provide a large number of users and the corresponding interactions to simulate the real-world scenario.",
"VirtualTB would generate different users each time after the initialization and the dynamic interest will change after each single interaction.",
"RecSim is a configurable platform for authoring simulation environments that naturally supports sequential interaction with users in recommender systems.",
"RecSim differs from VirtualTB in containing different, simpler tasks but fewer users and items.",
"The task we decide to use in RecSim, namely interest evolution.",
"The interest evolution encourages the agent to explore and fulfill the user's interest without further exploitation.",
"Figure: Overall results for online simulation environments.",
"Episode represents the number of test episode.RecoGym is a small platform, where users have no long-term goals.",
"Different from RecSim and VirtualTB, RecoGym is designed for computational advertising.",
"Similar with RecSim, RecoGym uses the click or not to represent the reward signal.",
"Moreover, similar with RecSim, users in those two environments do not contain any dynamic interests."
],
[
"Baselines for online simulation",
"In our online simulation experiments, all the baselines are reinforcement learning based.",
"Hence, those non-reinforcement learning based methods are ignored as they are not able to interact with the gym-based environment.",
"It is worth to mention that, some methods require extra side information from the environment which is not exist in those three platforms.",
"Hence, we have to remove those components to ensure the comparison is fair (i.e., every method receive the same kind of state representation).",
"The major evaluation metric used for online simulation is determined by the platform which is the Click-Through-Rate (CTR).",
"IMRL is implemented by using Pytorch [40] and all experiments are conducted on a server with two Intel Xeon E5-2697 v2 CPUs with 4 NVIDIA TITAN X Pascal GPUs, 2 NVIDIA TITAN RTX, 2 NVIDIA RTX A5000 and 768 GB memory.",
"We provide details about model parameters for reproducibility.",
"We set the hidden unit to 256 for the actor network and the critic network, respectively.",
"Learning rate, $\\gamma $ , and size of replay buffer are set to $0.0003$ , $0.99$ and $1e^6$ , respectively, during experiments.",
"The training episode is set to 1e6 and test is conducted every 10 episodes in VirtualTB.",
"And the training episode is set to $10,000$ for RecoGym and RecSim and tests are condcuted every 10 episodes."
],
[
"Offline Experiments",
"The fully results could be found in tab:result.",
"We find that our method IMRL are generally outperforms than all those existing state-of-the-art methods both in non-reinforcement learning based methods and reinforcement learning methods.",
"Notice that IMRL does not beat CauseRec in two datasets but still outperform than all the others.",
"Although IMRL does not has a better precision than CauseRec in Book-Crossing, we can find that recall and nDCG is better than CauseRec.",
"The same situation also happens in Netflix where the nDCG of IMRL is lower than CauseRec while precision and recall is better than CauseRec."
],
[
"Online Simulations (RQ2)",
"We also report the performance of those selected reinforcement learning based baselines in three online simulation environments.",
"The results can be found in fig:resultonline.",
"As we can find that, IMRL is outperform than all the others in all of selected three simulation platforms.",
"The performance in RecoGym and RecSim are quite close as those two environment are very small where does not require a complex exploration policy.",
"Hence, we will focus on the later discussion in VirtualTB as it has a more complex environment which is more similar with the real-world situation.",
"The simplest way to evaluate the sparsity is be relived or not is to evaluate the speed of the model that tend to converge.",
"In reinforcement learning, the way we use to measure the sparsity is the number of useful samples are fed into the agent via replay buffer or sampled from the environment.",
"Hence, dense environment can boost the model to converge at the early stage.",
"In fig:virtualtb, we can find that IMRL has a outstanding speed of convergence than other methods in VirtualTB which shows that it can overcome the sparse environment.",
"In RecoGym and RecSim, IMRL also demonstrates a considerable improvement when comparing with those baselines.",
"The majority reason is that, RecoGym and RecSim are small environment which contains only a few items and users.",
"The sparsity is not serious and can be handled by random exploration.",
"Figure: Ablation study on VirtualTB."
],
[
"Ablation Study (RQ3)",
"In order to answer RQ3, we conduct the experiments with the two major components in IMRL which are empowerment and augmentation.",
"The result of such study can be found in fig:ablation where IMRL-E denotes IMRL without empowerment and IMRL-A denotes IMRL without augmentation.",
"Moreover, we also investigate the effect of different strategy of empowerment in IMRL.",
"Specifically, we also investigate the KL-divergence approach mentioned in Section sec:kl.",
"We use IMRL-KL to represent such method.",
"We can find that both components play an important role in IMRL and contribute to the final performance jointly.",
"Furthermore, we can see that IMRL-KL does not perform as well as others.",
"One of the possible reason is that the information lost during the transformation during the calculation of KL-divergence.",
"Hence, we can infer that our approach of using the empowerment is better than KL-Divergence."
],
[
"Related Work",
"In this section, we will briefly review two topics which related to our work: reinforcement learning based recommendation and causality in recommender systems.",
"Reinforcement learning based recommendation.",
"[2] introduce the DRL into RS by using DQN to conduct the news recommendation.",
"It uses double DQN to build the user's profile and design a activeness score to evaluate user is active or not.",
"[34] extend this method by introducing the negative feedback.",
"[32] uses cascading DQN and design generative user model method to handle the unknown reward situation.",
"[41] design a scaleable policy-gradient based methods for recommendation by introducing a policy correction gradient estimator to reduce the variance.",
"[4] design a Pairwise Policy Gradient method for recommendation to reduce the variance as well.",
"[7] propose a tree-based method for large-scale interactive recommendation by utilizing the actor-critic algorithm.",
"[6] integrates the knowledge graph into the actor-critic structure and uses graph convolutional network to capture the information.",
"[36] design a knowledge graph based environment for explainable recommendation.",
"[9] uses the inverse reinforcement learning to avoid the elaborate reward function in online recommendation.",
"[42] uses SAC to conduct the multi-module recommendation via multi-agent approach.",
"Causality in recommender systems.",
"Causality receives significant research interest in recent literature about recommendation.",
"It has been widely used for debias or data augmentation for RS.",
"[43] use model-agnostic counterfactual reasoning to address the popularity bias in RS.",
"Similarly, [44] propose causal intervention to relief the popularity bias as well.",
"[14] design a counterfactual data augmentation methods by measuring embedding difference to generate new user sequence.",
"Differently, [15] separate the users' historical actions into dispensable and indispensable items where dispensable items can be omitted without affecting the final recommendation results.",
"New user sequences are generated by replacing the dispensable items.",
"In recent years, causality shows a strong connection with the reinforcement learning as both of them can affect the status of the input [45].",
"[46] use the reinforcement learning to conduct the causal discovery where employs actor-critic algorithm to discover different DAG structure.",
"[47] propose a meta reinforcement learning framework to conduct the causal reasoning by exploring different causal structures.",
"[48] uses causal inference to determine the unobserved confounders to improve the performance of the imitation learning.",
"[49] utilizes casual inference to build an explainable reinforcement learning model."
],
[
"Conclusion",
"In this paper, we propose IMRL to address the sparse interaction problem in DRL RS from two folds which are quantity and quality.",
"We proposes a counterfactual based method to augment informative interaction trajectories and empowerment based exploration to boost the possibility of finding high quality trajectories.",
"We have conducted the experiments on both offline datasets and online simulation platforms to demonstrate the superiority of the proposed method.",
"In the future, we are planning to further investigate power of empowerment and new solutions for reliving the sparse interaction for DRL RS.",
"Moreover, how to stabilize the training process of DRL in RS is another challenges that we are targeting.",
"[Figure: NO_CAPTION Siyu Wang is a PhD student with School of Computer Science and Engineering, The University of New South Wales (UNSW), Australia.",
"Her research interests are casual inference and recommender systems.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION"
]
] | 2209.08228 |
[
[
"DiPietro-Hazari Kappa: A Novel Metric for Assessing Labeling Quality via\n Annotation"
],
[
"Abstract Data is a key component of modern machine learning, but statistics for assessing data label quality remain sparse in literature.",
"Here, we introduce DiPietro-Hazari Kappa, a novel statistical metric for assessing the quality of suggested dataset labels in the context of human annotation.",
"Rooted in the classical Fleiss's Kappa measure of inter-annotator agreement, the DiPietro-Hazari Kappa quantifies the the empirical annotator agreement differential that was attained above random chance.",
"We offer a thorough theoretical examination of Fleiss's Kappa before turning to our derivation of DiPietro-Hazari Kappa.",
"Finally, we conclude with a matrix formulation and set of procedural instructions for easy computational implementation."
],
[
"Introduction",
"While the datasets powering state-of-the-art machine learning models have grown exponentially in size, the metrics used to assess the quality of these datasets have remained stagnant and unnuanced.",
"Here, we present a novel statistical metric capable of assessing dataset quality for supervised learning tasks.",
"Datasets used in supervised learning tasks consist of a target label and a vector of features.",
"While the target label is generally treated as a ground truth, it is often created by unwritten heuristics used when assembling the dataset, such as the source of the data or keywords used to obtain it.",
"These heuristics are imperfect, and it is important to quantitatively assess their quality.",
"One method for doing this is to randomly select a subset of the data and have multiple human annotators label it.",
"Generally, these annotations are then assessed via measures of inter-annotator agreement.",
"If annotators are generally in strong agreement with each other over the aggregate of the dataset, then the inter-annotator agreement is high.",
"If they are not, then the inter-annotator agreement is low.",
"Datasets with high inter-annotator agreement are assumed to be of high quality, while those without are assumed to be of low quality.",
"Unfortunately, current methods of inter-annotator agreement do not taken into account the suggested label for each piece of data.",
"In other words, annotators can entirely disagree with heuristic-suggested labels, but, so long as they agree with each other, the inter-annotator agreement is high.",
"However, clearly the suggested label would not be of high quality if all of the annotators (uniformly) disagreed with it.",
"So, traditional measures of inter-annotator agreement have little value in this context.",
"It's worth noting that this suggested label need not come from some simple dataset generation heuristic.",
"Indeed, a statistical metric for assessing labels in the context of human annotation could also be used to assess the performance of any proposed label, including those produced from machine learning classification models on unlabeled, novel data.",
"Note that this assumes that human annotators perform well on the task being assessed, as it is pointless to assess label quality using human annotators if the annotators themselves are incapable.",
"The requirements for an inter-annotator agreement metric assessing pre-labeled data are simple: if annotators agree on the suggested label (and, by extension, with each other), then the suggested label is good.",
"if annotators disagree with the suggested label but do not agree with each other on their label of choice, the suggested label is poor.",
"if annotators disagree with the suggested label and agree with each other on their label of choice, the suggested label is very poor.",
"This paper proposes the DiPietro-Hazari Kappa ($\\kappa _{DH}$ ), a novel statistical measure that assesses the quality of suggested dataset labels in the context of inter-annotator agreement, quantitatively implementing the three requirements outlined above.",
"We begin with an intuitive presentation of Fleiss's Kappa, a common measure of inter-annotator agreement that serves as the foundation for our derivation.",
"We then turn to a theoretical presentation of our novel metric."
],
[
"Fleiss's Kappa: Building an Intuition",
"The DiPietro-Hazari Kappa is heavily rooted in the intuition of the Fleiss's Kappa ($\\kappa $ ) metric [1].",
"Thus, we offer an extensive presentation of Fleiss's Kappa below.",
"Suppose we have $n$ pieces of data, each denoted $d_1, \\dots , d_n$ .",
"Each piece of data is assessed by $N$ annotators.",
"There are $m$ possible categories, each denoted $c_1, \\dots , c_m$ .",
"Consider the function $P(d_i, c_j)$ .",
"Given a data point $d_i$ and a category $c_j$ , this function yields the number of annotators that placed $d_i$ in category $c_j$ ."
],
[
"Expected Pair Agreement",
"The first step of computing Fleiss's Kappa is to find the proportion of total annotations that each category accounts for.",
"In other words, what is the likelihood that a human annotator will place a piece of data in each category if they label in accordance with the distribution of annotations over the entire dataset?",
"We denote this value for each category as $C_1, \\dots , C_m$ .",
"To compute these values, we use $\\textbf {C_j = \\frac{1}{Nn} \\sum _{i=0}^n P(d_i, c_j)}$ Now, based on this distribution, what is the chance that two annotators agree on any category by sheer randomness?",
"Well, there is a $C_j$ chance of a single annotator predicting category $j$ .",
"So, there is a $C_j \\cdot C_j=C_j^2$ chance of two annotators both predicting category $j$ , assuming they label independently and in accordance with the distribution of labels over the dataset.",
"By extension, there is a $C_E=\\sum _{j=0}^m C_j^2$ chance that two annotators agree on any category based on our distribution.",
"So, $C_E$ serves as a useful baseline for measuring inter-annotator agreement: agreement is only meaningful if it is happening at a rate above what we would expect from random chance."
],
[
"Observed Pair Agreement",
"The second step of computing Fleiss's Kappa is to find the proportion of total possible annotator pairs that actually agreed for each data point.",
"We denote these values as $R_1, \\dots , R_n.$ These values are computed by dividing the number of annotator pairs that empirically agreed by the total number of possible pairs.",
"For each data point, there are ${N \\atopwithdelims ()2}$ possible annotator pairs.",
"For a data point $d_i$ , there are $\\sum _{j=0}^m {P(d_i, c_j) \\atopwithdelims ()2}$ pairs that actually agreed.",
"Then, we define this proportion as follows $R_i = \\frac{1}{{N \\atopwithdelims ()2}} \\sum _{j=0}^m {P(d_i, c_j) \\atopwithdelims ()2}.$ Now, we compute $\\overline{R}$ , which is the the average of $R_1, \\dots , R_n$ .",
"In other words, $\\overline{R}$ is the observed rate of annotator agreement across the entire dataset."
],
[
"Agreement Observed above Random Chance",
"Now, we have two key values.",
"We have $C_E$ , which indicates the rate of annotator agreement that is expected by sheer chance given the annotation distribution over the entire dataset.",
"We also have $\\overline{R}$ , which represents the observed rate of annotator agreement across our dataset.",
"Consider the value $1-C_E$ .",
"1 indicates perfect annotator agreement–$100\\%$ of annotators agreed with each other.",
"Thus, $1-C_E$ indicates the maximum attainable agreement above random chance.",
"Now, consider the value $\\overline{R}-C_E$ .",
"This demonstrates the agreement that was obtained in practice above sheer randomness.",
"We may now compute Fleiss's Kappa as $\\kappa =\\frac{\\overline{R}-C_E}{1-C_E}$ Again, the denominator indicates the agreement that is achievable above random chance.",
"The numerator indicates the agreement that was achieved in practice above random chance.",
"As a result, Fleiss's Kappa is the proportion of possible performance above random chance that was achieved.",
"Thus, a Fleiss's Kappa of 1 indicates that we achieved perfect agreement, whereas a negative Fleiss's Kappa indicates that our inter-annotator agreement underperformed what would be expected by chance."
],
[
"DiPietro-Hazari Kappa: Derivation and Intuition",
"The DiPietro-Hazari Kappa calculation proceeds similarly as above, except with the addition of suggested labels that we would like to assess.",
"Suppose we have $n$ pieces of data, each denoted $d_1, \\dots , d_n$ .",
"Each piece of data is assessed by $N$ annotators.",
"There are $m$ possible categories, each denoted $c_1, \\dots , c_m$ .",
"We also have $n$ “proposed labels”, each denoted $\\ell _1, \\dots , \\ell _n$ where each $\\ell _i \\in \\lbrace c_j\\rbrace _{j=1}^m$ denotes the proposed category label of $d_i$ .",
"Define the function $P(d_i, c_j)$ as above.",
"Define the function $Q(\\ell _i, c_j) = 1$ if $\\ell _i = c_j$ and 0 otherwise."
],
[
"Expected Correct Pair Agreement",
"First, compute the proportion of total annotations that each category accounts for, as is done for Fleiss's Kappa.",
"We denote this value for each category as $C_1, \\dots , C_m$ .",
"To compute these values, we use $\\textbf {C_j = \\frac{1}{Nn} \\sum _{i=0}^n P(d_i, c_j)}$ Next, compute the proportion of proposed labels that each category accounts for, denoted $L_1, \\dots , L_m$ , as follows $L_j = \\frac{1}{n} \\sum _{i=0}^n Q(\\ell _i, c_j)$ Now, $L_j$ gives us the chance that a randomly selected point was given a proposed label of category $c_j$ if labeled at random given the distribution of suggested labels over the dataset.",
"$C_j$ gives us the chance that an annotator randomly guesses category $c_j$ .",
"Then, the chance that an annotator randomly guesses a category $c_j$ and is correct is $C_j \\cdot L_j$ .",
"The chance that two annotators guess category $c_j$ and are correct (and also agree with each other) is $C_j^2 \\cdot L_j$ .",
"Then, the chance that two annotators agree on the correct label for any category by sheer chance is $C_E = \\sum _{j=1}^m C_j^2 \\cdot L_j.$"
],
[
"Expected Incorrect Pair Agreement",
"We would like to find the chance that two annotators disagree with the proposed label but agree with each other by chance.",
"First, we have $\\sum _{j=1, j\\ne {j^{\\prime }}}^{m} C_j^2$ which yields the chance that two annotators agree at random on a category that is not $c_{j^{\\prime }}$ .",
"Recall that $L_{j^{\\prime }}$ gives the chance that a randomly selected point was given a proposed label category of $c_{j^{\\prime }}$ .",
"So, the probability that a randomly selected point is given a proposed label of $c_{j^{\\prime }}$ and two annotators agree on any category that isn't $c_{j^{\\prime }}$ by sheer chance is $L_{j^{\\prime }}\\sum _{j=1, j\\ne {j^{\\prime }}}^{m} C_j^2.$ Then, the chance that two annotators agree on a label other than the proposed label for any category by sheer chance is $C_F$ , defined as follows $C_F = \\sum _{i=1}^m\\left( L_i\\sum _{j=1, j\\ne i}^{m} C_j^2 \\right)$"
],
[
"Observed Correct Pair Agreement",
"Now, we would like to find the proportion of annotator pairs that agreed with each other and the proposed label for each data point.",
"We denote this value $R_1, \\dots , R_n$ .",
"This value is computed by dividing the number of pairs that agreed on the correct label by the number of total possible pairs.",
"For each data point, there are ${N \\atopwithdelims ()2}$ possible annotator pairs.",
"For a data point $d_i$ , there are ${P(d_i, \\ell _i) \\atopwithdelims ()2}$ pairs that actually agreed on the correct label.",
"Then, we have $R_i = \\frac{1}{{N \\atopwithdelims ()2}} {P(d_i, \\ell _i) \\atopwithdelims ()2}$ Now, we compute $\\overline{R}$ , which indicates the average of $R_1, \\dots , R_n$ .",
"In other words, $\\overline{R_E}$ is the rate of annotator agreement on the proposed label across the entire dataset."
],
[
"Observed Incorrect Pair Agreement",
"Next, we would like to find the proportion of annotator pairs that agreed with each other but disagreed with the proposed label.",
"We denote this value $S_1, \\dots , S_n$ .",
"For each data point, there are ${N \\atopwithdelims ()2}$ possible annotator pairs.",
"For a data point $d_i$ , there are $\\sum _{j=1, c_j \\ne \\ell _i}^n {P(d_i,c_j) \\atopwithdelims ()2}$ pairs that agreed on a specific label other than the proposed label.",
"Then, we have $S_i = \\frac{1}{{N \\atopwithdelims ()2}} \\left( \\sum _{j=1, c_j \\ne \\ell _i}^n {P(d_i,c_j) \\atopwithdelims ()2} \\right)$ Now, we compute $\\overline{S}$ , which indicates the average of $S_1, \\dots , S_n$ .",
"In other words, $\\overline{S}$ is the rate of annotator agreement on a label other than the proposed label, measured across the entire dataset."
],
[
"Agreement Differential Obtained above Chance",
"Recall that $C_E$ indicates the proportion of annotators that we expect to agree with each other and the proposed label by sheer chance.",
"Similarly, recall that $C_F$ indicates the proportion of annotators that we expect to agree with each other but disagree with the proposed label by sheer chance.",
"Then, we refer to $C_E - C_F$ as the expected annotator agreement differential.",
"Consider the value $1-(C_E-C_F)$ .",
"1 indicates perfect annotator agreement on the correct label and no annotator agreement on the incorrect label.",
"Thus, $1-(C_E-C_F)$ indicates the maximum attainable agreement differential above random chance.",
"Note that $\\overline{R}-\\overline{S}$ indicates the annotator agreement differential observed in practice.",
"Then, $(\\overline{R}-\\overline{S}) - (C_E-C_F)$ indicates the agreement differential that was obtained in practice above sheer randomness.",
"Hence, we construct the DiPietro-Hazari Kappa so that it indicates the proportion of possible agreement differential above chance that was achieved in practice.",
"$\\kappa _{DH} = \\frac{(\\overline{R}-\\overline{S})-(C_E-C_F)}{1-(C_E-C_F)}$"
],
[
"DiPietro-Hazari Kappa: Matrix Formulation for Easy Computation",
"In this section, we offer a concise matrix formulation of the DiPietro-Hazari Kappa to enable convenient computational implementation.",
"Let there be $n$ pieces of data $d_1, \\dots , d_n$ with $m$ possible categories, $c_1, \\dots , c_m$ .",
"Each piece of data is assessed by $N$ annotators; there are $n$ proposed labels as well, denoted $\\ell _1, \\dots , \\ell _n$ , where each $\\ell _i \\in \\lbrace c_j\\rbrace _{j=1}^m$ denotes the proposed category label of $d_i$ .",
"Let $A$ define the $n \\times m$ matrix where each $a_{ij}$ indicates the number of annotators that placed $d_i$ in $c_j$ .",
"Let $B$ denote the $n \\times m$ matrix where $b_{ij}=1$ if $\\ell _i = c_j$ and 0 otherwise.",
"Define $\\oplus $ as the Hadamard product.",
"Let $f(P\\mapsto Q)$ define the element-wise matrix map $p_{ij} \\mapsto {p_{ij} \\atopwithdelims ()2}$ where $p_{ij} \\in P$ , ${p_{ij} \\atopwithdelims ()2} = q_{ij} \\in Q$ .",
"Let $\\text{row\\_sum}(P)$ and $\\text{col\\_sum}(P)$ denote row-wise and column-wise matrix summation functions respectively.",
"Now, we may compute $\\kappa _{DH}$ via the procedural instructions in (14).",
"$\\begin{split}C & \\leftarrow (\\text{col\\_sum}(A)) \\cdot \\frac{1}{Nn} \\\\L & \\leftarrow (\\text{col\\_sum}(B)) \\cdot \\frac{1}{n} \\\\C_E & \\leftarrow \\text{row\\_sum}(C \\oplus C \\oplus L) \\\\T & \\leftarrow \\begin{bmatrix}C \\\\\\vdots \\\\C\\\\\\end{bmatrix}_{m \\times m} \\oplus \\left(\\mathbf {J}_{m \\times m} - \\mathbf {I}_{m \\times m}\\right)\\\\C_F & \\leftarrow \\text{col\\_sum} ( \\text{row\\_sum}(T \\oplus T) \\oplus L^T )\\\\\\overline{R} & \\leftarrow \\text{col\\_sum} \\left( \\text{row\\_sum} ( f(A) \\oplus B ) \\cdot \\frac{1}{{N \\atopwithdelims ()2}} \\right) \\cdot \\frac{1}{n}\\\\\\overline{S} & \\leftarrow \\text{col\\_sum} \\left( \\text{row\\_sum} ( f(A) \\oplus (\\mathbf {J}_{n \\times m} - B) ) \\cdot \\frac{1}{{N \\atopwithdelims ()2}} \\right) \\cdot \\frac{1}{n}\\\\\\kappa _{DH} & \\leftarrow \\frac{(\\overline{R}-\\overline{S})-(C_E-C_F)}{1-(C_E-C_F)}\\end{split}$ These instructions are implemented as a high-efficiency Python function, available at https://github.com/dandip/DH_Kappa."
],
[
"Conclusion",
"Here, we presented thorough theoretical outlines of Fleiss's Kappa, as well as our novel metric DiPietro-Hazari Kappa.",
"To our knowledge, this is the first statistical measure that uses inter-annotator agreement to assess the quality of a dataset's labels (which may be generated by a heuristic or inferenced by a model).",
"As the importance of dataset quality becomes increasingly apparent in machine learning research, this metric has the potential to serve as a commonplace benchmark in supervised learning literature."
],
[
"Acknowledgments",
"We thank Ziray Hao, John McCambridge, and Alexander “Sasha” Kokoshinskiy for valuable discussions concerning the creation of this metric."
]
] | 2209.08243 |
[
[
"Summary Report of the Topical Group on Physics Education, Community\n Engagement Frontier (CEF4/CommF4) Snowmass 2021"
],
[
"Abstract An essential companion to the development and advancement of the field of Particle Physics is a strong program in physics education at all levels, that can attract entry level students across the full demographic spectrum and provide them with the education, training and skills needed to advance to successful careers in Science, Technology, Engineering and Mathematics (STEM) and other fields.",
"This report summarizes the work of several investigative teams that have reviewed and assessed current opportunities in physics education across K-12, undergraduate, graduate and postdoctoral domains, including national and international linkages.",
"From these assessments, recommendations have been put forward aimed to innovate educationally in strategic ways to strengthen ties between the research community and teachers, between the academic community and the private sector, and through both domestic and international connections."
],
[
"Introduction",
"To have in place the needed educated and trained workforce for currently planned and future experiments in high energy particle physics and ancillary fields, the attraction to and education of students for a broad range of careers in STEM is both necessary and essential.",
"The Challenge: Traditional STEM educational efforts have provided a high-quality workforce, but with rather selective demographics.",
"And if left unchanged, the character of the workforce will likely remain stagnant and generally discouraging to the participation of women and those from underrepresented groups.",
"The Opportunity: The Snowmass 2021 Process, with its 10-year planning and 20-year vision, offers a creative opportunity to assess the challenges, build on what is currently working very well, and frame a structure to broader opportunity for young researchers to join the exciting particle physics field through an expanded range of educational opportunities.",
"The Approach and Organization of this Topical Group: To identify what is working and the shortfalls, and to recommend actions to be taken, the Community Engagement Frontier Topical Group on Physics Education has viewed the physics education process in a systemic way as indicated schematically in Figure REF .",
"The schema highlights the educational process from the level of incoming K-12 students and then upward through undergraduate and graduate education, postdoctoral and faculty education.",
"Figure: A Schematic Representation of Particle Physics EducationA very large and demographically diverse population of students who might consider and potentially enter STEM fields enters at the base of the pyramid.",
"For a variety of reasons, educational, cultural and societal, the fraction of students that choose STEM careers and are able proceed upward to research careers becomes reduced at each step.",
"To assess these issues, the Physics Education Topical Group formed four Working groups (A through D) organized by community interest, all of which have submitted contributed papers to the arXiv and include: Group A.",
"Opportunities for Particle Physics Engagement in K-12 Schools and Undergraduate Education [1]; Group B.",
"Transforming U.S.",
"Particle Physics Education: A Snowmass 2021 Study [2]; Group C. Broadening the Scope of Education, Career and Open Science in HEP [3]; and Group D. The Necessity of International Particle Physics Opportunities for American Education [4].",
"The Organization of this Report: In this Topical Group Summary Report, we follow the schematic (and systemic) flow up the Pyramid of Figure REF and highlight briefly important issues identified by each group including selected recommendations.",
"The education process arises naturally at the K-12 and undergraduate levels where the challenge is to attract and engage young students to interest in STEM and physics, and then to hold and enrich their interest and provide needed scaffolding to help sustain their upward path to research careers.",
"Greater depth and insight will be found in each of the associated and referenced Contributed Papers indicated above [1], [2], [3], [4].",
"At the end of this report, we provide potential linkages of this Physics Education Topical Group with the other Topical Groups of the Community Engagement Frontier of Snowmass 2021, and conclude with a final comment."
],
[
"Opportunities for Particle Physics Engagement in K-12 Schools and Undergraduate Education",
"While many (particularly young) students might show an early interest and aptitude for science and mathematics at the elementary level, the structures are not necessarily in place to capture, nurture and develop such nascent interest.",
"In the contributed paper [1] it was found that joint activities of academia and K-12 educators and pupils should focus on how scientists develop knowledge and the essence of the knowledge acquired so far.",
"It should connect to sister STEM fields, to their successful outreach programs and with an emphasis of a holistic view on STEM.",
"It is important that K-12 activities address all students and teachers, adapting to their appropriate level of scientific literacy, and be mindful of biases in program designs that might filter student participation by race, gender, or socioeconomic background.",
"The paper [1] makes a number of recommendations that can be condensed into: Formation at the local level of collaborative communities (\"fora\") of STEM experts of all backgrounds (physicists, engineers, technicians, students and K-12 teachers and students - to form direct linkages among researchers and educators for dialog, collaboration, mutual enrichment and mentoring among members.",
"These fora should be responsive to local community need.",
"These fora should not be isolated or self-isolating.",
"To aid in interaction and information transfer, they should be supported by a nationally, or even internationally, organised online repository for sharing resources; The fora can only be created and sustained if a minimal amount of support for coordination and logistics is made available.",
"For the sustainability, it is important to have a steady source for this support, which can come from colleges, universities or institutes, but might also come from or be supplemented by outreach support in research grants.",
"An important ingredient for sustainable fora is that the efforts of the participants are appropriately and regularly recognised."
],
[
"Educational Opportunities at the Undergraduate, Graduate and Postdoctoral Level",
"This critical time domain for the development and pursuit of careers in particle physics and ancillary fields has traditionally been the purview of university and laboratory groups in the US and abroad.",
"In the words of the engaged group of young researchers working on this topic: \"Graduate school (and undergraduate school to a lesser extent) is where researchers acquire most of the technical skills required for research, develop scientific problem solving abilities, learn how to establish themselves in their field, and begin developing their career.",
"It is unfortunate, then, that the skills gained by physicists during their formal education are often mismatched with the skills actually required for a successful career in physics.\"",
"These words suggest that there is clearly work to do to strengthen this key educational domain."
],
[
"Online Survey of Educational Experiences in the HEP Community",
"The group performed an online survey of the U.S. particle physics community to gain background on how practitioners in the field at all levels assessed their own physics education and training, with the aim to identify areas of challenge and inform potential improvements for the future.",
"The group's report [2] is detailed and replete with findings and recommendations, from which we have identified the following succinct elements: Graduate programs in particle physics should normalize training for a broad range of STEM careers with appropriate formal courses, rather than forcing students to resort prematurely to self-teaching or peer learning.",
"The courses should provide strong grounding in particle physics and mathematics, but also computation, statistics and instrumentation, with the aim to benefit careers in physics, industry or education.",
"Correspondingly, Universities should provide undergraduate students with a more complete picture of what particle physics trained researchers do.",
"A realistic view of common career paths post baccalaureate and post graduate school should be presented, including for theoretical and experimental positions as well as non-academic careers.",
"Undergraduate participation in the survey was rather minimal, driven in part by lack of information.",
"With support from Professional Societies and Physics Departments, connections and networking opportunities should be developed to strengthen connections for undergraduate students with HEPA community activities.",
"And the HEPA Community should actively plan to perform a future survey with emphasis on undergraduate participation, to assess and strengthen links to students at that academic level.",
"."
],
[
"A New Perspective on the Masters Degree",
"As noted in several of the recommendations derived from the comprehensive survey cited above and detailed in the Contributed Paper [2], a potentially important professional level post baccalaureate, the Masters Degree, is often overlooked or ignored by the physics community which tends to be \"Ph D\" driven.",
"The Masters level has several important attributes worthy of community consideration.",
"(1) It is a level where more extensive cross-disciplinary (elective) course work is possible, providing potential branches to applied math, statistics, computer science, engineering and nuclear medicine.",
"(2) It is a level of potential participation by and engagement with students from the private sector who enroll for technical advancement with support from their companies.",
"This nexus can provide a bridge between students following an academic path with those already in commercial applications sectors, opening up dialog and potential career opportunities that might otherwise be overlooked.",
"(3) The Masters Level also affords an intermediate (and potentially achievable) academic target for students from groups traditionally underrepresented in physics and for whom a PhD in physics might seem an unlikely goal.",
"And (4) enriched programs at the Masters Level can lead to collaborative opportunities across academia, which is the central topic of the next section.",
"Universities, especially also non-research universities, should consider setting up Masters Degree programs in particle physics and related areas, such as hardware and software technology for Big Science experiments."
],
[
"Collaborative Opportunities Across Academia",
"The emphases of this group were to review the challenges and develop strategies to help transform the particle physics field into a stronger and more diverse ecosystem of talent and expertise, with the expectation of long-lasting scientific and societal benefits.",
"Regarded as central were the building of collaborative bridges between faculty and experimental programs at R1 institutions (major research universities and national laboratories) with those at R2 institutions (that provide training up to masters degrees), Predominantly Undergraduate Institutions (PUIs) and Community Colleges (CC)).",
"The intended aims over the next decade are to: Expand the benefits of faculty collaboration and research opportunities across the broad spectrum of academia and give equivalence: opportunities for all in technical and scientific leadership on projects and with appropriate recognition for contributions.",
"In corollary, conduct a study of new models of collaboration or cooperation that would allow R2/PUI/CC faculty and their students to collaborate in demonstrably effective ways in experiments.",
"This includes addressing the challenges of teaching loads, student training and funding availability that directly impact consequential participation.",
"Broadly accessible data and analysis platforms will benefit student access and participation.",
"In the true spirit of Open Science, the HEP community should define, with cogent arguments, what should be the scope of making our data and resources publicly available, and the hardware, software and person-power costs associated with such implementation.",
"Machine Learning (ML) is becoming an integral part of physics research.",
"Many critical HEP algorithms for triggering, reconstruction, and analysis rely on ML and there are entire conferences and summer schools dedicated to this crosscutting field.",
"Despite the relevance and importance of this research, pursuing a career at the intersection of these fields remains a tenuous and undefined endeavor.",
"The current mindset in the field is that highly specialized skills such as software and firmware development and instrumentation development are not broadly recognized as “physics” work.",
"This perception of what it means to ‘be a physicist’ must be challenged lest it continue to be an impediment.",
"Confronting and addressing these issues would encourage an influx of new workforce into the field, help retain those who are in the field, and equip those who might seek careers outside of HEP.",
"Qualification for HEP faculty jobs should not be based solely on physics analysis but must be expanded to to include computing, software and/or hardware contributions."
],
[
"International Opportunities for Particle Physics Education",
"Particle physics is a global endeavor.",
"No one institution or nation can assemble the resources or expertise needed to explore the frontiers of the field.",
"The diversity of national, social and cultural backgrounds present in the experiments and labs enriches the pool of intellectual thought and solidifies the validity of their scientific findings.",
"And no one country or region has a monopoly on good ideas or approaches to strengthen the educational experience for studwnts.",
"But there are noteworthy examples of excellent ongoing efforts, which this working group reviewed in detail.",
"Their line of thought has been worked out in a contributed paper [4] and hasled to several noteworthy recommendations.",
"U.S. based pre-university particle physics collaborations, such as QuarkNet and other outreach programmes, should expand collaboration with international partners, such as the International Particle Physics Outreach Group (IPPOG), the CERN Beamline for Schools (BL4S) and Teacher summer school programmes in Europe, and should collaborate with partners in the developing world, such as the African School of Fundamental Physics and Applications.",
"The participation in the Global Cosmics portal should be enhanced by developing low-cost cosmic ray detectors for educational use.",
"Student exchange programs should be fostered and supported, such as the NSF Research Experience for Undergraduates (REU) program, which funds participation of U.S. students in the CERN Summer Student program, and the DoE-INFN summer student exchange program between the U.S. and Italy.",
"Where possible these should be extended, in particular with student exchange programs and summer schools in developing countries, such as the African School of Fundamental Physics and Applications."
],
[
"Interconnections and Synergies with Community Engagement Topical Groups",
"Physics Education (CEF4) has important interconnections to other Snowmass 2021 Topical Groups within the Community Engagement Frontier.",
"With CEF1, the Topical Group on Applications and Industry:.",
"Opportunties are potentially vast, linking education with technological innovation.",
"National Laboratories are a natural training ground for student research opportunities.",
"And at the Masters Degree level, universities and colleges are a fertile training ground to mix students moving up to STEM careers with students from industry who seek to further enrich their technological training.",
"This is a melting pot worth pursiuing energetically and behooves universities and colleges to consider the Masters Degree in Physics in a new light.",
"With CEF2, the Topical Group on Career Pipeline and Development: Physics Education (CEF4) is central to creating a skilled workforce pipeline to all HEP Frontiers and beyond for STEM areas in industry.",
"All recommendations in CEF2 are strongly endorsed by this CEF4 Working Group.",
"Beyond regular course curriculum, Software Training programs and Open Science activities [3] can go a long way to attracting talent in HEP as well as preparing HEP talent for STEM industry jobs.",
"Strong participation by faculties and students of PUIs and CCs in HEP programs [5] will enable a diverse and inclusive STEM workforce for HEP and industry.",
"With CEF3, the Topical Group on Diversity and Inclusion: With the inclusion of a well-educated and demographically and geographically diverse group of students, the person power needed for current and future experimental and scientific challenges will be in place to meet the needs of particle physics community over the next two decades.",
"With CEF5, the Topical Group on Public Education and Outreach: K/12 education is fundamentally public education at its most fundamental level.",
"With students excited and engaged, and their bringing that interest and excitement home to their families, the thrill of particle physics and the possibilities of future discoveries will help build the strong supportive community needed to sustain the science.",
"With CEF6, the Topical Group on Public Policy and Government Engagement: Particle Physics is an exciting, yet esoteric field.",
"It is therefore challenging to translate that excitement to those who make public policy and provide funding support, when they are under myriad pressures from across the political spectrum.",
"However, hand-in-hand with the expansive educational opportunities that discovery science affords, and through mention of the excitement and participation of students across the nation, the conversational optics can be refocused in constructive ways, to the benefit of the field and nation.",
"With CEF7 the Topical Group on Environmental and Societal Impacts: At the time of Fermilab's founding near the village of Weston, Illinois, the land was prairie.",
"Since that time, the laboratory has had programs on environment and ecology that are particularly accessible to younger students and to engage them in the natural world.",
"These programs, coordinated through the Lederman Science Center at Fermilab, are a wonderful model of the interconnection of particle physicists, education, and local schools and communities [6], [7].",
"While there are outreach aspects to such programs, the latter reference is intrinsically educational by design."
],
[
"Conclusions",
"To support a compelling program of scientific discovery in the long run, a robust education program in physics, mathematics and the sciences is an essential companion.",
"Such a program should provide students across the demographic spectrum ample basis of opportunity to enter particle physics and ancillary fields to engage in and benefit from the science.",
"The ten-year program and twenty-year vision of Snowmass 2021 affords a strategic opportunity and time window within which both the science and the education process can evolve holistically and constructively.",
"This requires Community Engagement at all educational levels, which can only be sustained when the people involved are appropriately recognised, credited and supported for their contributions."
]
] | 2209.08225 |
[
[
"Pole position of $\\Lambda(1405)$ measured in $d(K^-,n)\\pi\\Sigma$\n reactions"
],
[
"Abstract We measured a set of $\\pi^\\pm\\Sigma^\\mp$, $\\pi^0\\Sigma^0$, and $\\pi^-\\Sigma^0$ invariant mass spectra below and above the $\\bar{K}N$ mass threshold in $K^-$-induced reactions on deuteron.",
"We deduced the $S$-wave $\\bar{K}N\\rightarrow\\pi\\Sigma$ and $\\bar{K}N\\rightarrow\\bar{K}N$ scattering amplitudes in the isospin 0 channel in the framework of a $\\bar{K}N$ and $\\pi\\Sigma$ coupled channel.",
"We find that a resonance pole corresponding to $\\Lambda(1405)$ is located at 1417.7$^{+6.0}_{-7.4}$(fitting errors)$^{+1.1}_{-1.0}$(systematic errors) + $[-26.1^{+6.0}_{-7.9}$(fitting errors)$^{+1.7}_{-2.0}$(systematic errors)]$i$ MeV/$c^2$, closer to the $\\bar{K}N$ mass threshold than the value determined by the Particle Data Group."
],
[
"Introduction",
"$\\Lambda (1405)$ is a well-known hyperon resonance with strangeness $-1$ , spin-parity 1/2$^-$ , and isospin 0 ($I$ = 0).",
"It is classified as the first orbital excited state in the constituent quark model.",
"However, the properties of $\\Lambda (1405)$ are not easily explained, such as the fact that it has the lightest mass among the negative parity baryons even though it contains a heavier strange quark, and the large mass difference it exhibits compared to that for the so-called spin–orbit partner state of $\\Lambda (1520)$ .",
"It has been argued that $\\Lambda (1405)$ is a bound state of an anti-kaon ($\\bar{K}$ ) and a nucleon ($N$ ) since it is located just below the $\\bar{K}N$ mass threshold, Dalitz and Tuan first predicted a possible quasi-bound state of $\\bar{K}N$ with $I$ = 0 in 1959, based on low-energy $K^-$ -proton scattering experiments [1], [2].",
"The first observation of a hyperon resonance sitting just below the $\\bar{K}N$ mass threshold in $\\pi ^-\\Sigma ^+/\\pi ^+\\Sigma ^-$ invariant mass spectra was reported in 1961 [3].",
"Since then, several sets of experimental data on $\\Lambda (1405)$ have been reported [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].",
"Dalitz and Deloff deduced a resonance energy and width of 1406.5 $\\pm $ 4.0 MeV and 50 $\\pm $ 2 MeV by analyzing the measured $\\pi ^-\\Sigma ^+$ mass spectrum [7] based on $\\bar{K}N$ scattering theory [17].",
"The latest edition of the Review of Particle Physics [18] gives average values of 1405.1$^{+1.3}_{-0.9}$ MeV and 50.5 $\\pm $ 2.0 MeV, including two later works [19], [20] which demonstrate that a so-called phenomenological approach giving the $\\Lambda (1405)$ mass at $\\sim $ 1405 MeV [21], [22] is favorable for fitting the $(\\pi \\Sigma )^0$ invariant mass spectra from $K^-$ stopped on $^4$ He [23] and proton–proton collisions (HADES) [12].",
"A recent review on $\\Lambda (1405)$ is available in Ref. [24].",
"Over the last two decades, there have been intensive discussions about the so-called chiral unitary approach, which is a coupled-channel meson–baryon scattering theory employing chiral Lagrangians.",
"Several calculations indicate that there are two resonance poles between the $\\pi \\Sigma $ and $\\bar{K}N$ mass thresholds [25], [26], [27], [28], [29], where the higher pole, coupled to $\\bar{K}N$ , is located at around 1420 MeV or greater.",
"The chiral unitary approach is in contradiction with the phenomenological approach.",
"There is a discussion of differences in different theoretical treatments of chiral unitary approaches and the phenomenological approach [30].",
"The experimental situation is also controversial.",
"Recent measurements of $(\\pi \\Sigma )^0$ mass spectra have been reported in photo-induced reactions on protons [10], [13], [14], [15], [16] and proton–proton collisions [11], [12].",
"The CLAS collaboration reported precise $\\pi ^-\\Sigma ^+$ , $\\pi ^+\\Sigma ^-$ , and $\\pi ^0\\Sigma ^0$ spectra for a wide range of incident photon energies [13], [14].",
"Theoretical analyses have been made on these data and reproduced the spectral shapes fairly well, even though they involved many parameters [31] and/or reaction diagrams [32].",
"The HADES collaboration reported invariant mass spectra of $\\pi ^-\\Sigma ^+$ , $\\pi ^+\\Sigma ^-$ , and their sum [12].",
"Their spectral shapes were different from those for photo-production.",
"In particular, they observed peaks even below 1400 MeV.",
"Theoretical analyses of these spectra have also been made [33], [20].",
"However, the locations of $\\Lambda (1405)$ determined by a chiral unitary model [33] and a phenomenological model [20] are not compatible with each other.",
"Therefore, experimental data to directly determine the $\\bar{K}N$ scattering amplitude coupled to $\\Lambda (1405)$ are required."
],
[
"Experiment",
"We carried out an experimental study of kaon-induced $\\pi \\Sigma $ production via $d(K^-,n)\\pi \\Sigma $ reactions [34].",
"Our expectation was to measure a reaction sequence consisting of a 1-GeV/$c$ incident negative kaon knocking out a neutron at a very forward angle (less than 6 degrees in the laboratory frame) from a deuteron, with the $\\bar{K}$ recoiled backward reacting with the residual nucleon ($N_2$ ) to produce $\\pi $ and $\\Sigma $ , as shown in the reaction diagram in the inset of Fig.",
"REF .",
"In the second step of the reaction sequence, $\\bar{K}N_2\\rightarrow \\pi \\Sigma $ scattering takes place even below the $\\bar{K}N$ mass threshold.",
"Since the typical momentum of a recoiled $\\bar{K}$ is as low as $\\sim $ 250 MeV/$c$ for a $\\pi \\Sigma $ invariant mass of around 1405 MeV/$c^2$ , $S$ -wave scattering is expected to be dominant.",
"We measured the $\\pi \\Sigma $ invariant mass spectra, from which we deduced the $\\bar{K}N$ scattering amplitude in the $I$ = 0 channel.",
"Figure: Schematic illustration of the experimental setup.",
"CDS: cylindrical detector system, CDH: cylindrical scintillator hodoscope, CDC: cylindrical drift chamber, D 2 _2 TGT: liquid deuterium target, T0: time zero counter, BPD: backward proton detector, BPC: backward proton drift chamber, DEF: beam defining counter, FDC: forward drift chamber, NC: neutron counter array, CVC: charged particle veto counter, and PC: proton counter array.",
"The reaction diagram expected for d(K - ,N)πΣd(K^-,N)\\pi \\Sigma is shown in the inset.The experiment was performed at the K1.8BR beam line [35] of the Japan Proton Accelerator Research Complex (J-PARC).",
"Negatively charged kaons delivered from K1.8BR were incident on a liquid deuterium (D$_2$ ) target of 125 mm thickness.",
"The momentum of the incident kaons was analyzed by the K1.8BR-D5 magnetic spectrometer.",
"A schematic layout of the experimental setup [36], [37] is illustrated in Fig.",
"REF .",
"A time-zero counter (T0), placed 1100 mm upstream from the D$_2$ target, defined a time origin triggered by the incident kaon for time-of-flight measurements of scattered particles.",
"A drift chamber (BPC) and scintillator hodoscopes (BPD) were placed 143.2 mm and 482.5 mm upstream from the D$_2$ target center, respectively.",
"Kaon beam tracks were measured by the BPC, which were used to determine the reaction vertex.",
"The BPD and the BPC were used to detect backward emitted protons from $\\pi ^0\\Sigma ^0$ productions, as mentioned later.",
"The kaon beam was finally defined by a beam defining counter (DEF) placed just in front of the D$_2$ target.",
"The integrated luminosity of the kaon beam used in the present analysis was ($5927\\pm 158$ ) [($8078\\pm 248$ )] $\\mu $ b$^{-1}$ , which was the product of the beam intensity 5.56$\\times $ 10$^{10}$ , the number of target deuterons 4.82$\\times $ 10$^{23}$ [6.03$\\times $ 10$^{23}$ ], efficiencies of the beam line detectors (28.1$\\pm $ 1.7)% [(31.5$\\pm $ 1.9)%)], trigger system (95.2$\\pm $ 2.6)%, and the data acquisition system (76.5$\\pm $ 5.9)%.",
"Here, the numbers in square brackets are for the $\\pi ^0\\Sigma ^0$ mode.",
"Charged particles from the D$_2$ target were measured by a cylindrical detector system (CDS), consisting of a cylindrical drift chamber (CDC) and scintillator hodoscopes (CDHs) surrounding the D$_2$ target.",
"An efficiency of the CDC for one-charged particle tracking was estimated to be (97.7$\\pm $ 0.4)%.",
"The CDS was operated in a solenoid magnet with a magnetic field of 0.714 Tesla.",
"Scattered neutrons were detected by neutron counters (NCs), consisting of an array of 112 plastic scintillator slabs (200 mm width, 1500 mm height, and 50 mm thickness each), placed approximately 15 m from the D$_2$ target.",
"Since the solid angle of the NCs seen from the target is (21.5$\\pm $ 0.2)% msr, the angular coverage for the emitted neutrons is less than 6 degrees.",
"The detection efficiency of the NC was estimated to be (31.7$\\pm $ 1.6)%.",
"It was measured by finding a neutron at the NC to the predicted neutron emission direction in the $p(K^-,\\bar{K}^0)n$ reaction, where a backward-recoiled $\\bar{K}^0$ was reconstructed by the CDS.",
"In reality, a factor of (91.9$\\pm $ 0.7)% due to a charged-particle veto counter (CVC) placed in front of the NC to veto charged particles, was multiplied as an effective efficiency of the NC.",
"Charged particles emitted in a forward angle, including the incident beams, were swept out by a dipole magnet placed behind the solenoid magnet.",
"Protons knocked out from deuterons by the incident kaon beam were bent by the dipole magnet in the opposite direction of the beam.",
"The time-of-flight of the knocked-out proton was measured by proton counters (PC), consisting of hodoscopes of 27 scintillator slabs (100 mm width, 1500 mm height, and 50 mm thickness each), which were placed beside the CVC.",
"The solid angle of the PC is slightly momenum dependent and is typically 22.6 msr.",
"The trajectory of each scattered proton was determined using the position information from the reaction vertex at the target, a drift chamber (FDC) placed at the entrance of the dipole magnet, and the hit slab of the PC.",
"A tracking efficiency for the proton is estimated to be (81.9$\\pm $ 4.2)%.",
"We measured the $d(K^-,p)\\pi ^-\\Sigma ^0$ reaction, where the $p$ was detected by the PC and the two $\\pi ^-$ s were detected by the CDS.",
"We measured the $\\pi ^\\pm \\Sigma ^\\mp $ production associated with a knocked-out neutron detected by the NC, where $\\pi ^+$ and $\\pi ^-$ were detected by the CDS and the missing neutron was identified separately in a $d(K^-,n\\pi ^+\\pi ^-)$ missing mass spectrum, as shown in Fig.",
"REF .",
"In these modes, three background processes are relevant as they all give the same final state of $n\\pi ^+\\pi ^-n_{\\rm miss}$ , where $n_{\\rm miss}$ represents the neutron identified in the $d(K^-,n\\pi ^+\\pi ^-)$ missing mass spectrum: (1) $K^-d\\rightarrow n\\bar{K}^0n_{\\rm miss}$ , (2) $K^-d\\rightarrow \\pi ^-\\Sigma ^+n_{\\rm miss}$ , and (3) $K^-d\\rightarrow \\pi ^+\\Sigma ^-n_{\\rm miss}$ .",
"In (2) and (3), $\\pi ^\\mp \\Sigma ^\\pm $ are produced with an incident $K^-$ interacting with a bound proton in a deuteron.",
"A neutron from the $\\Sigma $ decay is emitted at a forward angle and detected by the NC.",
"The $n_{\\rm miss}$ is a spectator neutron in the processes.",
"They are the so-called one-step $\\pi ^\\mp \\Sigma ^\\pm $ production processes, which occurs in a quite differrent kinematical region compared with that for the two-step process that we concern in the present article.",
"Above three processes can be excluded since we can identify the $\\bar{K}^0$ , $\\Sigma ^+$ , and $\\Sigma ^-$ peaks in the invariant mass spectra of $\\pi ^+\\pi ^-$ , $n\\pi ^+$ , and $n\\pi ^-$ , as shown in Fig.",
"REF (b), (c), and (d), respectively.",
"We obtained the $\\pi ^\\pm \\Sigma ^\\mp $ missing mass spectra in the $d(K^-,n){\\pi ^\\pm \\Sigma ^\\mp }$ reactions separately, as we will show later.",
"The production ratio of $\\pi ^-\\Sigma ^+$ to $\\pi ^+\\Sigma ^-$ was obtained to reproduce the $\\Sigma ^\\pm $ peak and its kinematic reflection (continuum-like distribution).",
"The decomposed $d(K^-,n\\pi ^\\pm )$ missing mass spectra are shown in Fig.",
"REF (a) and (b), respectively.",
"Figure: (a) Missing mass spectrum of d(K - ,nπ + π - )d(K^-,n\\pi ^+\\pi ^-) togather with an illustration of a typical signal event topology.",
"The missing neutron (n miss n_{\\rm miss}) is identified separately.",
"The dashed lines indicate a selected neutron mass region.",
"(b), (c), and (d) Invariant mass spectra of π + π - \\pi ^+\\pi ^-, nπ + n\\pi ^+, and nπ - n\\pi ^-, respectively, in the d(K - ,nπ + π - )n miss d(K^-,n\\pi ^+\\pi ^-)n_{\\rm miss} reactions.",
"The peak regions shown with the dashed lines were excluded as K 0 K^0 and one-step π ∓ Σ ± \\pi ^\\mp \\Sigma ^\\pm production processes as typical event topologies are illustrated, respectively.Figure: (a) and (b) Decomposed Σ ± \\Sigma ^\\pm peaks in the d(K - ,nπ ∓ )d(K^-,n\\pi ^\\mp ) missing mass spectra, respectively.",
"(c) Invariant mass spectrum of π - \\pi ^- and pp measured by CDS and BPD/BPC, respectively.",
"A Λ\\Lambda peak was selected for the π 0 Σ 0 \\pi ^0\\Sigma ^0 mode (vertical lines) as the Σ 0 \\Sigma ^0 immediately decays into Λ+γ\\Lambda +\\gamma .",
"A typical event topology for the π 0 Σ 0 \\pi ^0\\Sigma ^0 mode is illustrated in the figure.",
"(d) Missing mass spectrum of d(K - ,nΛ)d(K^-,n\\Lambda ).The expected π 0 \\pi ^0, π 0 γ\\pi ^0\\gamma , and background components (BG) are overlaid as histograms.",
"See text for BG.",
"A π 0 γ\\pi ^0\\gamma region (0.18–0.3 GeV/c 2 c^2) was gated for the π 0 Σ 0 \\pi ^0\\Sigma ^0 mode.",
"(e) Scatter plot of two possible d(K - ,pπ - )d(K^-,p\\pi ^-) missing masses for the π - Σ 0 \\pi ^-\\Sigma ^0 mode.",
"A typical event topology for the π - Σ 0 \\pi ^-\\Sigma ^0 mode is illustrated in the figure.",
"A Σ 0 \\Sigma ^0 mass region was selected, as indicated by the blue lines.",
"(f) Missing mass spectrum of d(K - ,pπ - π - )d(K^-,p\\pi ^-\\pi ^-) (crosses) and selected Σ 0 \\Sigma ^0 region in (e) (histogram).",
"A pγp\\gamma peak was selected to identify the π - Σ 0 \\pi ^-\\Sigma ^0 mode (vertical lines).In the $\\pi ^0\\Sigma ^0$ production, $\\Sigma ^0$ immediately decays to $\\Lambda \\gamma $ .",
"The $\\Lambda $ hyperon decays to $\\pi ^-$ and a proton.",
"The $\\pi ^-$ is emitted in a wide angular region and could be detected by the CDS.",
"While, the proton is generally emitted backward because most of the momentum of the $\\pi ^0\\Sigma ^0$ that recoiled backward in the $d(K^-,n)$ reaction is carried by the heavier particle.",
"We measured the time-of-flight of the backward proton, detected by the BPC and the BPD.",
"We identified the decaying $\\Lambda $ in the invariant mass spectrum reconstructed from the measured momenta of the $\\pi ^-$ and the proton [Fig.",
"REF (c)].",
"Then, the missing mass spectrum of $d(K^-,n\\Lambda )$ was obtained as shown in Fig.",
"REF (d).",
"The missing $\\pi ^0$ , $\\pi ^0\\gamma $ , and background components (BG) contributions were decomposed based on a Monte Carlo simulation, as indicated in the figure, which were 12%, 70%, and 18%, respectively.",
"Here, hyperon ($Y$ )-production processes that associate with a backward proton ($K^-d\\rightarrow p(Y\\pi )^-$ ) and those induced by quasi-free backward kaons that react with an another deuteron ($d^\\prime $ ) in the deuterium target ($K^-d\\rightarrow \\bar{K}X, \\bar{K}d^\\prime \\rightarrow YX^\\prime $ ) are taken into accout as the BG components.",
"By gating the mass window for 0.18 to 0.3 GeV/$c^2$ in the spectrum, we obtained the $\\pi ^0\\Sigma ^0$ mode with only a small amount of contamination from the $\\pi ^0\\Lambda $ mode and background components, which were reduced to be 1.0% and 3.9%, respectively.",
"The contribution of the contamination is subtracted in the present $\\pi ^0\\Sigma ^0$ missing mass spectrum.",
"The $\\pi ^-\\Sigma ^0$ mode was identified by selecting the $\\Sigma ^0$ and $p\\gamma $ mass regions in a scatter plot of the two possible $d(K^-,p\\pi ^-)$ missing masses and the $d(K^-,p\\pi ^-\\pi ^-)$ missing mass spectrum, as shown in Fig.",
"REF (e) and (f), respectively.",
"The missing $p\\gamma $ mass distribution is isolated since $\\Sigma ^0$ is moving slowly."
],
[
"$\\pi \\Sigma $ mass spectra",
"The mass spectra of $\\pi ^\\pm \\Sigma ^\\mp $ , $\\pi ^0\\Sigma ^0$ , and $\\pi ^-\\Sigma ^0$ were obtained, as shown in Fig.",
"REF .",
"Errors in the vertical axes include statistical errors, scaling factor errors, and systematic uncertainties.",
"The scaling factor errors arise from uncertainties in corrections of the target thickness and beam intensity, the efficiencies of the data acquisition system, detectors, and event selections in the analysis codes, and the geometrical acceptances and efficienceies of the relevant particle detectors.",
"Geometrical acceptances of the detector setup for $\\pi ^\\pm $ , $\\pi ^-p$ , and $2\\pi ^-$ from $\\pi ^\\pm \\Sigma ^\\mp $ , $\\pi ^0\\Sigma ^0$ , and $\\pi ^-\\Sigma ^0$ , respectively, are shown in Fig.",
"REF (d), which were evaluated by Monte Carlo simulations in conditions that knocked-out neutron and proton are detected at the NC and the PC, respectively.",
"The scaling factor errors relative to the obtained cross sections for $\\pi ^{\\pm }\\Sigma ^{\\mp }$ , $\\pi ^0\\Sigma ^0$ , and $\\pi ^-\\Sigma ^0$ are estimated to be 5.8%, 6.2%, and 3.5%, respectively.",
"For the $\\pi ^\\pm \\Sigma ^\\mp $ spectra, the fitting errors to separate the two modes as described in Fig.",
"REF (a) and (b) are also taken into account, which are dominant sources of systematic uncertainties in estimations of the cross sections.",
"The fitting errors relative to the cross sections are typically 7% and 6% at the $K^-p$ /$K^0n$ mass thresholds for the $\\pi ^\\pm \\Sigma ^\\mp $ modes, respectively.",
"In the case of the $\\pi ^0\\Sigma ^0$ mode, the fitting error to decompose the $\\pi ^0\\Lambda $ mode and the other background mentioned in the previous section is dependent on the missing mass.",
"It is typically 1.5% at the $\\bar{K}N$ mass threshold.",
"On the other hand it is 13% at around 1475 MeV/$c^2$ , where contamination of the BG conponents is maximum.",
"The statistical and total errors are shown separately as inner and outer bars in Fig.",
"REF (a) and (b), while only the total errors are shown in Fig.",
"REF (c).",
"Figure: Measured spectra of (a) π ± Σ ∓ \\pi ^\\pm \\Sigma ^\\mp , (b) π 0 Σ 0 \\pi ^0\\Sigma ^0 and π - Σ 0 \\pi ^-\\Sigma ^0, and (c) π 0 Σ 0 \\pi ^0\\Sigma ^0 and (π + Σ - +π - Σ + -π - Σ 0 )/2(\\pi ^+\\Sigma ^-+\\pi ^-\\Sigma ^+-\\pi ^-\\Sigma ^0)/2.",
"(d) Acceptances of the detectors for π ± \\pi ^\\pm , π - p\\pi ^-p, and 2π - 2\\pi ^- from π ± Σ ∓ \\pi ^\\pm \\Sigma ^\\mp , π 0 Σ 0 \\pi ^0\\Sigma ^0, and π - Σ 0 \\pi ^-\\Sigma ^0, respectively.Statistical and total errors are shown separately as inner and outer bars in (a) and (b), while only total errors are shown in (c).The vertical thin lines shows the K - pK^-p and K 0 nK^0n mass thresholds.We observed different line shapes in the $\\pi ^\\pm \\Sigma ^\\mp $ modes [Fig.",
"REF (a)].",
"Since both the $I$ = 0 and 1 amplitudes contribute to the modes, the difference is due to interference between the two amplitudes.",
"In the $\\pi ^-\\Sigma ^+$ mode, we find a bump around 1450 MeV/$c^2$ with a small shoulder below the $K^-p$ mass threshold.",
"On the other hand, the $\\pi ^+\\Sigma ^-$ spectrum shows a broad distribution with a maximum strength just below the $K^-p$ mass threshold.",
"The $\\pi ^0\\Sigma ^0$ and $\\pi ^-\\Sigma ^0$ modes [Fig.",
"REF (b)] contain only the $I$ = 0 and 1 amplitudes, respectively.",
"The strength of the $\\pi ^-\\Sigma ^0$ spectrum is smaller than that of the $\\pi ^0\\Sigma ^0$ spectrum.",
"We find that the $I$ = 0 amplitude is dominant, particularly below the $\\bar{K}N$ mass threshold.",
"We find no structure at around 1385 MeV/$c^2$ in the $\\pi ^-\\Sigma ^0$ , where we might expect a structure of the $\\Sigma ^*(1385)$ resonance.",
"This fact suggests dominance of $S$ -wave $\\pi \\Sigma $ production in the present reactions, since $\\Sigma ^*(1385)$ decays into a $P$ -wave $\\pi \\Sigma $ state.",
"The $\\pi \\Sigma $ production cross sections can be described with $T_1^I$ and $T_2^{I^\\prime }$ as follows; $\\frac{d\\sigma }{d\\Omega }(\\pi ^\\pm \\Sigma ^\\mp )\\propto \\left|C_1^0T_2^{I^\\prime =0}\\mp {C_1^1T_2^{I^\\prime =1}}\\right|^2,\\\\\\frac{d\\sigma }{d\\Omega }(\\pi ^-\\Sigma ^0)\\propto \\left|C_1^1T_2^{I^\\prime =1}\\right|^2,\\\\\\frac{d\\sigma }{d\\Omega }(\\pi ^0\\Sigma ^0)\\propto \\left|C_1^0T_2^{I^\\prime =0}\\right|^2,\\\\C_1^0=\\frac{3T_1^{I=0}-T_1^{I=1}}{4\\sqrt{3}},\\ \\ C_1^1=\\frac{T_1^{I=1}+T_1^{I=1}}{4}.$ Here, $T_1^I$ and $T_2^{I^\\prime }$ represent the scattering amplitude of the first-step and second-step two-body $K^-N_1\\rightarrow \\bar{K}N$ and $\\bar{K}N_2\\rightarrow \\pi \\Sigma $ reactions with isospin $I$ and $I^\\prime $ , respectively.",
"The coefficients are determined by the sums of the products of the Clebsch–Gordan coefficients in terms of the isospin in the possible processes in the two-step reaction, as described as follows: $\\sum _{m_X,I,m,I^\\prime ,m^\\prime }{\\langle \\frac{1}{2}m_{N_1}\\frac{1}{2}m_{N_2}|00\\rangle \\langle \\frac{1}{2}m_{\\bar{K}}\\frac{1}{2}m_{N}|Im\\rangle }\\nonumber \\\\\\times \\langle \\frac{1}{2}m_{K^-}\\frac{1}{2}m_{N_1}|Im\\rangle {T_1^I} \\nonumber \\\\\\times \\langle \\frac{1}{2}m_{\\pi }\\frac{1}{2}m_{\\Sigma }|I^\\prime {m^\\prime }\\rangle \\langle \\frac{1}{2}m_{\\bar{K}}\\frac{1}{2}m_{N_2}|I^\\prime {m^\\prime }\\rangle {T_2^{I^\\prime }},$ where $m_{X=N_1, N_2, \\bar{K}, N, K^-, \\pi , \\Sigma }$ is a $z$ -component of the isospin of a relevant particle $X$ .",
"Then, one finds a relation among the four reaction cross sections as $\\frac{1}{2}\\frac{d\\sigma }{d\\Omega }(\\pi ^+\\Sigma ^-+\\pi ^-\\Sigma ^+-\\pi ^-\\Sigma ^0)=\\frac{d\\sigma }{d\\Omega }(\\pi ^0\\Sigma ^0).$ We confirmed the relationship, as demonstrated in Fig.",
"REF (c)."
],
[
"Discussion",
"Several authors have discussed $\\pi \\Sigma $ production associated with nucleon emission in kaon induced reactions on deuterons [38], [39], [40], [41], [42], and hence we describe the $\\pi \\Sigma $ spectral shape assuming that the two-step reaction is dominant when the knocked-out nucleon is emitted at a very forward angle.",
"We neglect the direct production of $\\pi \\Sigma $ by collisions of incident $K^-$ with nucleons in deuteron as its contribution is negligibly small at the very forward angle of knocked-out neutron.",
"Then, the $\\pi \\Sigma $ production cross section can be described as $\\frac{d^2\\sigma }{dM_{\\pi \\Sigma }d\\Omega _n}\\sim \\left|\\langle {n\\pi \\Sigma }|T_2G_0(\\bar{K},N_2)T_1|K^-\\Phi _d\\rangle \\right|^2,\\\\T_2=T_2^{I^\\prime }(\\bar{K}N_2,\\pi \\Sigma ),\\\\T_1=T_1^I(K^-N_1,\\bar{K}N),$ where $|K^-\\Phi _d\\rangle $ and $|n\\pi \\Sigma \\rangle $ denote the initial $K^-$ and deuteron and final $n\\pi \\Sigma $ wave functions, respectively.",
"$G_0(\\bar{K},N_2)$ is the Green's function which describes the intermediate $\\bar{K}$ propagation between the two vertices.",
"More detailed expressions can be found in Refs.",
"[40], [38], [42].",
"The cross section can be simplified by a factorization approximation, as follows: $\\frac{d^2\\sigma }{dM_{\\pi \\Sigma }d\\Omega _n}\\approx \\left|T_2^{I^\\prime }\\right|^2F_{\\rm res}(M_{\\pi \\Sigma }),\\\\F_{\\rm res}(M_{\\pi \\Sigma })=\\left|\\int {G_0T_1^I\\Phi _d(q_{N_2})d^3q_{N_2}}\\right|^2.$ Here, $q_{N_2}$ is the momentum of the residual nucleon.",
"In this way, the $\\pi \\Sigma $ spectrum can be decomposed into $T_2^{I^\\prime }$ and the response function $F_{\\rm res}$ .",
"Using the $K^-N\\rightarrow \\bar{K}N$ scattering amplitudes based on a partial wave analysis [43] and the deuteron wave function $\\Phi _d$ [44], we evaluate $F_{\\rm res}$ as a function of the $\\pi \\Sigma $ mass $M_{\\pi \\Sigma }$ , as shown by the dashed line in Fig.",
"REF (b).",
"Here, we took 3 degrees as a typical scattering angle of the knocked-out nucleon in the laboratory frame.",
"The line shapes of the $\\pi \\Sigma $ mass spectra above the $\\bar{K}N$ mass threshold are characterized by $F_{\\rm res}$ , the distribution of which reflects the Fermi motion of a nucleon in the dueteron.",
"For $S$ -wave $T_2^{I^\\prime }$ , we consider the $\\bar{K}N$ -$\\pi \\Sigma $ coupled channel $T$ matrix.",
"The diagonal and off-diagonal matrix elements can be parametrized similarly to the case in Ref.",
"[48] as $T_2^{I^\\prime }(\\bar{K}N,\\bar{K}N)=\\frac{A^{I^\\prime }}{1-iA^{I^\\prime }k_2+\\frac{1}{2}A^{I^\\prime }R^{I^\\prime }k_2^2},\\\\T_2^{I^\\prime }(\\bar{K}N,\\pi \\Sigma )=\\frac{e^{i\\delta ^{I^\\prime }}}{\\sqrt{k_1}}\\frac{\\sqrt{{\\rm Im}A^{I^\\prime }-\\frac{1}{2}|A^{I^\\prime }|^2{\\rm Im}R^{I^\\prime }k_2^2}}{1-iA^{I^\\prime }k_2+\\frac{1}{2}A^{I^\\prime }R^{I^\\prime }k_2^2},$ where $A^{I^\\prime }$ , $R^{I^\\prime }$ , and $\\delta ^{I^\\prime }$ are the complex scattering length, complex effective range, and real phase, respectively.",
"$k_1$ and $k_2$ are respectively the momenta of $\\pi $ and $\\bar{K}$ in the center of mass frame.",
"Here, $k_2$ becomes a pure imaginary number below the $\\bar{K}N$ mass threshold, to satisfy analytic continuity.",
"Figure: (a) Experimental resolution as a function of the πΣ\\pi \\Sigma mass.",
"(b) Calculated πΣ\\pi \\Sigma spectrum to fit the measured spectra in the II = 0 channel.",
"The solid thick and thin lines are the spectrum with and without the resolution function convoluted, respectively.",
"The response function F res F_{\\rm res} is shown as a dashed line in arbitrary units.",
"(c) Deduced scattering amplitude of K ¯N→K ¯N\\bar{K}N\\rightarrow \\bar{K}N in the II = 0 channel.",
"The real and imaginary parts are shown as solid and dashed lines, respectively.",
"The vertical thin lines show the K - pK^-p and K 0 nK^0n mass thresholds.We demonstrate the fitting result for the $\\pi \\Sigma $ ($I$ = 0) channel, as shown in Fig.",
"REF (b).",
"$A^0$ and $R^0$ are determined to fit the measured $\\pi ^0\\Sigma ^0$ and $(\\pi ^+\\Sigma ^-+\\pi ^-\\Sigma ^+-\\pi ^-\\Sigma ^0)/2$ spectra, simultaneously.",
"We took the $\\bar{K}N$ mass threshold at the average of $K^-p$ and $K^0n$ since the differential cross sections of $K^-n\\rightarrow K^-n$ [45] and $K^-p\\rightarrow K^0n$ [46] are almost equal at a neutron forward angle at an incident kaon momentum of $\\sim $ 1 GeV/$c$ .",
"However, we took into account the differences from the fitting results for the cases of the $K^-p$ and $K^0n$ mass thresholds as systematic errors.",
"In the present fitting, $\\delta ^{I^\\prime }$ could not be determined since it deos not appear explicitly in the fitting function that depends on $|T_2^{I^\\prime }(\\bar{K}N,\\pi \\Sigma )|^2$ .",
"In the fitting, the experimental resolution function [Fig.",
"REF (a)] was convoluted with the calculated spectrum and the vertical scale is arbitrarily adjusted.",
"We obtained $A^0 = [-1.12\\pm 0.11(\\mbox{fit})^{+0.10}_{-0.07}(\\mbox{syst.",
"})]$ + $[0.84\\pm 0.12(\\mbox{fit})^{+0.08}_{-0.07}(\\mbox{syst.",
"})]i$ fm, $R^0 = [-0.18\\pm 0.31\\mbox{(fit)}^{+0.08}_{-0.06}(\\mbox{syst.",
"})]$ + $[-0.40\\pm 0.13\\mbox{(fit)}\\pm 0.09(\\mbox{syst.",
"})]i$ fm, where the fitting errors are indicated as “(fit)”.",
"As mentioned above, the differences of the different $\\bar{K}N$ mass threshold were taken into account as systematic errors indicated as “(syst.)”.",
"The reduced chi-square was 1.76 with 24 degrees of freedom.",
"The present scattering length is smaller than a recent theoretical calculation, $-1.77+1.08i$ , which is based on the lattice QCD [47].",
"The thick and thin solid lines in Fig.",
"REF (b) show the resolution-convoluted and no-resolution-convoluted spectra, respectively, calculated with the best fit values.",
"The energy dependence of the deduced $T_2^0(\\bar{K}N,\\bar{K}N)$ is shown in Fig.",
"REF (c).",
"We find a zero-crossing in the real part and a bump in the imaginary part at the same place.",
"This is a typical structure of a resonance.",
"We find a resonance pole at $1417.7^{+6.0}_{-7.4}(\\mbox{fit})^{+1.1}_{-1.0}(\\mbox{syst.",
"})$ + $[-26.1^{+6.0}_{-7.9}(\\mbox{fit})^{+1.7}_{-2.0}(\\mbox{syst.",
"})]i$ MeV/$c^2$ in the $I$ = 0 channel of the $\\bar{K}N\\rightarrow \\bar{K}N$ scattering.",
"The errors are estimated by fluctuations of the pole position due to the errors for the best fit values of $A^0$ and $R^0$ .",
"The real part of the deduced pole is closer to the $K^-p$ mass threshold than the so-called PDG value of 1405.1 MeV/$c^2$ .",
"It is worthy of evaluating the following quantity, $|T_2^0(\\bar{K}N,\\bar{K}N)|^2/|T_2^0(\\bar{K}N,\\pi \\Sigma )|^2\\sim 2.2^{+1.0}_{-0.6}$ (fit)$\\pm 0.3$ (syst.)",
"at the pole energy, which corresponds to the ratio of the two partial widths in the Flatté formula [49], [50].",
"This suggests that the coupling of $\\Lambda (1405)$ to $\\bar{K}N$ is predominant, which does not contradict a picture of $\\Lambda (1405)$ as a $\\bar{K}N$ -bound state.",
"Meißner and Hyodo have reviewed and discussed the pole structure of the $\\Lambda (1405)$ region based on chiral unitary approaches with a constraint on the scattering length obtained from kaonic hydrogen atom $X$ -ray data by the SIDDHARTA collaboration [51], [52][53].",
"They collected four sets of two poles deduced by several authors in the relevant region.",
"Poles 1 and 2 are the so-called higher and lower poles, respectively, which are thought to be coupled to $\\bar{K}N$ and $\\pi \\Sigma $ , respectively.",
"The suggested higher poles are located at the region of 1421–1434 MeV on the real axis and 10–26 MeV on the imaginary axis in the complex energy plane.",
"The pole position determined by the present experiment is consistent to the higher poles though it is located at slightly smaller and larger values for the real and imaginary parts, respectively.",
"A lattice QCD calculation has reported two poles and the so-called higher pole is located at $1430-22i$ MeV/$c^2$ [54].",
"Our result is smaller and similar in real and imaginary part, respectively.",
"Recently, Anisovich $et\\ al.$ reported one single pole of $\\Lambda (1405)$ contribution to fit the data of $\\gamma $ and $K^-$ induced reactions on proton and the kaonic hydrogen atom, as $1422\\pm 3-(21\\pm 3)i$ MeV/$c^2$ [55].",
"The present result is consistent with the reported pole position."
],
[
"Conclusion",
"We measured $\\pi ^\\pm \\Sigma ^\\mp $ , $\\pi ^0\\Sigma ^0$ , and $\\pi ^-\\Sigma ^0$ mass spectra below and above the $\\bar{K}N$ mass threshold in $d(K^-,N)\\pi \\Sigma $ reactions at a forward angle, of $N$ knocked out by an incident kaon momentum of 1 GeV/$c$ .",
"We obtained decomposed $\\pi \\Sigma $ spectra in terms of $I$ = 0 and 1, and confirmed a relation between the four reactions with respect to the isospin states.",
"We find that the $I$ = 0 amplitude is dominant.",
"We demonstrated that the $\\pi \\Sigma $ spectral shape in the $I$ = 0 channel is well reproduced by the two-step reaction of a neutron knocked out at a forward angle by an incident negative kaon and a recoiled $\\bar{K}$ reacting with a residual nucleon in deuteron to produce $\\pi \\Sigma $ in the $I$ = 0 state.",
"We deduced the two-body $\\bar{K}N$ scattering amplitude in the $I$ = 0 channel around the $\\bar{K}N$ mass threshold, from which we find a resonance pole at $1417.7^{+6.0}_{-7.4}(\\mbox{fit})^{+1.1}_{-1.0}(\\mbox{syst.",
"})$ + $[-26.1^{+6.0}_{-7.9}(\\mbox{fit})^{+1.7}_{-2.0}(\\mbox{syst.",
"})]i$ MeV/$c^2$ .",
"The present data provide fundamental information on the $\\bar{K}N$ interaction and kaonic nuclei [56], [57]."
],
[
"Acknowledgements",
"The authors would like to express their thanks to the J-PARC PAC members and the crews of the J-PARC accelerator and hadron facility group for their encouragement, support, and stable delivery of beams for the E31 experiment.",
"We are grateful to Professor D. Jido, Dr. T. Sekihara, and Professor J. Yamagata-Sekihara for their support since the planning stage of the E31 experiment.",
"We are grateful to Professor K. Miyagawa and Dr. H. Kamano for their contributions to the calculations of the $\\pi \\Sigma $ spectral shapes.",
"The present work was supported by MEXT Grants-in-Aid of Innovative Area No.",
"21105003, No.",
"18H05402, and a Grant-in-Aid of Scientific Research A No.",
"16H02188 and S No.",
"22H04940."
]
] | 2209.08254 |
[
[
"Bayesian Image-on-Scalar Regression with a Spatial Global-Local\n Spike-and-Slab Prior"
],
[
"Abstract In this article, we propose a novel spatial global-local spike-and-slab selection prior for image-on-scalar regression.",
"We consider a Bayesian hierarchical Gaussian process model for image smoothing, that uses a flexible Inverse-Wishart process prior to handle within-image dependency, and propose a general global-local spatial selection prior that extends a rich class of well-studied selection priors.",
"Unlike existing constructions, we achieve simultaneous global (i.e, at covariate-level) and local (i.e., at pixel/voxel-level) selection by introducing `participation rate' parameters that measure the probability for the individual covariates to affect the observed images.",
"This along with a hard-thresholding strategy leads to dependency between selections at the two levels, introduces extra sparsity at the local level, and allows the global selection to be informed by the local selection, all in a model-based manner.",
"We design an efficient Gibbs sampler that allows inference for large image data.",
"We show on simulated data that parameters are interpretable and lead to efficient selection.",
"Finally, we demonstrate performance of the proposed model by using data from the Autism Brain Imaging Data Exchange (ABIDE) study.",
"To the best of our knowledge, the proposed model construction is the first in the Bayesian literature to simultaneously achieve image smoothing, parameter estimation and a two-level variable selection for image-on-scalar regression."
],
[
"Introduction",
"With the explosive growth in the amount of image data collected for various medical research there comes an increasing interest in discovering the relation between the image data and potential covariates measured on the same set of subjects.",
"Image-on-scalar regression models have drawn increasing attention for this purpose, see [40], [46], [23], [45], [44], among others.",
"These models present several challenges: spatial dependency in the image data can be highly complex and hard to model; image data can be composed by a large number of pixels/voxels and lead to extremely large covariance matrices with heavy computational burden; covariates can have partial influence on the image responses, i.e., they can affect only a few pixels/voxels in the image, making it hard to distinguish such covariates from noisy ones.",
"Conventional approaches for image-on-scalar regression are based on mass univariate analysis (MUA), for example by running pixel/voxel-wise independent general linear models to generate maps for statistics of interest, and then applying methods for post-inference [40], [15].",
"These methods are computationally efficient and have well-studied theoretical properties.",
"However, they completely ignore the spatial dependency within the images and are generally not optimal in regards to statistical power [9].",
"To address these shortcomings, recent approaches consider the image data as realizations of functions on a given domain and apply functional data analysis (FDA) methods that use basis expansions and spatially-varying coefficients to account for dependency within and across images [46], [23], [44].",
"Joint uncertainty quantification for all model parameters, however, is difficult to achieve for these methods in the frequentist literature.",
"Here, we consider a Bayesian hierarchical Gaussian process (GP) model for image smoothing that avoids assumptions on functional forms and that uses a flexible Inverse-Wishart process to handle within-image dependency.",
"This modeling structure extends an approach proposed by [43] for longitudinal data to the case of image data.",
"An important aspect in image-on-scalar regression is the selection and interpretation of influential covariates.",
"Ideally, one may want a covariate to be influential for the whole image.",
"In practice, however, the covariate can only partially affect the image, i.e., being influential only for a few pixels/voxels.",
"We refer to the aspect of selecting whether a covariate is influential for the images as “global\", and to the aspect of selecting which pixels/voxels are affected by the covariate as “local\".",
"In the Bayesian framework, a global selection prior was proposed by [29] and allows coefficients to be non-zero constant, spatially-varying function, or zero constant.",
"For local selection, a common way of selecting pixels/voxels uses a two-component mixture prior, which models the spatially-varying coefficient via a latent continuous process and a binary selection indicator process, see [34], [31], [13], [22], [8].",
"Recently [20] proposed a soft-threshold Gaussian process prior that does not make use of the indicator process but rather achieves local sparsity by thresholding.",
"This idea can be traced back to the earlier research of [27], who used a hard-threshold prior for longitudinal data to introduce sparsity at each time point.",
"Overall, none of these prior constructions achieve simultaneous global- and local-level selection of the coefficients.",
"Finally, in the more general framework of function-on-scalar regression, in which we consider images as 2/3-dimensional functions, Bayesian approaches employ basis functions and functional principal component analysis to model the within-function dependency, see for example [7], [21].",
"Built on the basis functions domain, this framework can be computationally more efficient.",
"However, the covariates' effects are assumed on the basis functions, instead of directly on the observed functions.",
"As a consequence, selection relies on the choice of the basis functions, especially for high-dimensional functional responses, and a two-level selection becomes less intuitive, as the local level selection would require all basis functions to be set to 0 at some specific pixels/voxels.",
"We propose a spatial global-local spike-and-slab process prior for image-on-scalar regression that broadly relates to a rich class of well-studied local selection priors.",
"We achieve simultaneous global and local selection by introducing participation rate parameters, that measure the probability for the individual covariates to affect the observed images, and employing hard thresholding.",
"The proposed prior performs bi-level selection, allowing the global selection to be informed by the local selection.",
"We design an efficient Gibbs sampler that allows inference for large image data and use simulated data to show that prior parameters are interpretable and lead to efficient selection.",
"We also demonstrate the performance of the proposed model with respect to MUA methods.",
"Finally, we apply our method to data from the Autism Brain Imaging Data Exchange (ABIDE) study [11].",
"Results show that modeling dependency in the data leads to more localized selection.",
"The rest of the paper is organized as follows.",
"In section , we introduce the proposed method, the prior construction and the sampler procedure.",
"In Section , we conduct simulations and compare the proposed approach with widely used MUA methods.",
"In Section , we apply the method to image data from the ABIDE study."
],
[
"Bayesian Image-on-Scalar Regression",
"Suppose $n$ images $Y_i(\\cdot )$ are observed on a $K$ -dimensional common domain $\\mathbf {S} \\subseteq \\mathbb {R}^{K}$ , each associated with a $q$ -dimensional covariate vector $\\mathbf {x}_i = (x_{i1}, \\ldots , x_{iq})^T$ , for $i = 1, \\ldots , n$ .",
"We begin with a Bayesian hierarchical model for image responses and scalar covariates $& Y_i( \\mathbf {s} ) = Z_i(\\mathbf {s}) + \\epsilon _{i,\\mathbf {s}}, \\quad \\epsilon _{i,\\mathbf {s}} \\stackrel{i.i.d}{\\sim } N\\left( 0, \\sigma ^2_\\epsilon \\right), \\quad \\mathbf {s} \\in \\mathbf {S} \\\\& Z_i(\\cdot ) \\sim \\mathcal {GP}\\left( \\mu _i(\\cdot ), \\Sigma \\left(\\cdot , \\cdot \\right)\\right), \\quad \\mu _i(\\cdot ) = \\beta _0(\\cdot ) + \\sum ^q_{j=1} x_{ij} \\beta _j(\\cdot ), \\quad i = 1, \\ldots n,$ where the noise-free mean surface of $Y_i(\\cdot )$ , $Z_i(\\cdot )$ , is modeled by a Gaussian process (GP) with covariate-dependent, subject-specific mean $\\mu _i(\\cdot )$ and a common covariance surface $\\Sigma (\\cdot , \\cdot )$ for all images, $\\beta _0(\\cdot )$ is the intercept coefficient image and $\\lbrace \\beta _j(\\cdot )\\rbrace ^q_{j=1}$ are the coefficient images linking covariates $\\mathbf {x}_i$ with $\\mu _i(\\cdot )$ .",
"Here we assume Gaussian errors $\\epsilon _{i,\\mathbf {s}} \\sim N\\left( 0, \\sigma ^2_\\epsilon \\right)$ independently across both location $\\mathbf {s}$ and subject $i$ , with $\\sigma ^2_\\epsilon \\sim \\text{Inverse-Gamma}(a_\\epsilon , b_\\epsilon )$ .",
"Throughout this article, we assume $\\mathbf {S}$ to be a compact set.",
"Equations (REF )-() define a Bayesian hierarchical model for image data.",
"This model has considerable virtues: it enables simultaneous smoothing of individual observations and borrowing of information across observations, while being flexible through Bayes nonparametrics and interpretable.",
"It has been widely used to analyze functional data.",
"For example, in the case of longitudinal data (i.e., $K = 1$ ), [43] focus on a mean-covariance structure in the absence of covariates, with a common mean function, [25] study conditional quantiles of $Y_i(\\cdot )$ by altering the Gaussian error to asymmetric Laplace, and [32] consider a sparse Bayesian infinite factor model for $Z_i(\\cdot )$ ."
],
[
"Spatial Global-Local Spike-and-Slab Prior",
"We are interested in performing selection at both image and location (pixel/voxel) levels, while estimating the model coefficients and the covariance structure.",
"We achieve this via a novel spatial global-local spike-and-slab (SGLSS) prior for the coefficient images $\\left\\lbrace \\beta _j(\\cdot )\\right\\rbrace ^{q}_{j=1}$ .",
"Here, selection at the global level represents the covariate selection, as eliminating a covariate would zero out the entire coefficient image, while the local level refers to individual locations (pixels/voxels) in the coefficient image.",
"The proposed SGLSS prior is a three-component process consisting of a continuous process, a local-level discrete selection process, and a global-level indicator: $\\begin{aligned}\\beta _j(\\cdot ) & = \\tilde{\\beta }_j(\\cdot ) \\times \\tau _j(\\cdot ) \\times I \\left( \\pi _j \\ge d\\right), \\\\& = \\left[\\tau _j(\\cdot )I \\left( \\pi _j \\ge d\\right)\\right] \\tilde{\\beta }_j(\\cdot ) + \\left[ 1 - \\tau _j(\\cdot )I \\left( \\pi _j \\ge d\\right)\\right] \\delta _0,\\end{aligned} $ with $\\delta _0$ a point mass distribution at 0, $\\tilde{\\beta }_j(\\cdot )$ the continuous process, $\\tau _j(\\cdot )$ the discrete local selection process and $I( \\pi _j \\ge d)$ the global indicator.",
"Construction (REF ) is completed by choosing priors for $\\tilde{\\beta }_j(\\cdot )$ , $\\tau _j(\\cdot )$ , and $\\pi _j$ .",
"The high dimensionality of image data poses substantial challenges to computational efficiency, particularly for Markov Chain Monte Carlo methods.",
"To achieve computational scalability, we adopt the following prior setting that exploits conjugacy: $\\begin{aligned}\\tilde{\\beta }_j(\\mathbf {s}) | \\tau _j(\\mathbf {s}) & \\sim \\tau _j(\\mathbf {s}) N\\left( \\mu _{0j}(\\mathbf {s}), \\sigma ^2_{0j}(\\mathbf {s})\\right) + (1 - \\tau _j(\\mathbf {s}))\\delta _0, \\quad \\mathbf {s}\\in \\mathbf {S}\\\\\\tau _j( \\mathbf {s}) | \\pi _j & \\sim \\text{Bernoulli}\\left(\\pi _j \\right), \\\\\\pi _j & \\sim \\text{Beta}(a_\\pi , b_\\pi ).\\end{aligned}$ The SGLSS prior for $\\beta _j(\\cdot )$ with the specification of Equation (REF ) can be re-written into a classic spike-and-slab prior as $\\begin{aligned}\\beta _j(\\mathbf {s}) | \\tau _j(\\mathbf {s}), \\pi _j & \\sim \\left[\\tau _j(\\mathbf {s}) I\\left( \\pi _j \\ge d \\right)\\right] N\\left( \\mu _{0j}(\\mathbf {s}), \\sigma ^2_{0j}(\\mathbf {s})\\right) + \\left[1 - \\tau _j(\\mathbf {s}) I\\left( \\pi _j \\ge d \\right)\\right]\\delta _0, \\\\\\tau _j( \\mathbf {s}) | \\pi _j & \\sim \\text{Bernoulli}\\left(\\pi _j \\right), \\\\\\pi _j & \\sim \\text{Beta}(a_\\pi , b_\\pi ).\\end{aligned}$ As for the intercept image $\\beta _0(\\mathbf {s})$ , we fix $\\tau _0(\\mathbf {s})I(\\pi _0\\ge d) = 1$ for all $\\mathbf {s} \\in \\mathbf {S}$ since this term is typically always included in the model.",
"The proposed SGLSS prior construction has rich connections with a wide range of existing priors, as we point out in the section below.",
"The parameter $\\pi _j$ , which we call `participation rate', has the interpretation that $\\pi _j$ percent of the $j$ th coefficient image is expected to be non-zero, and can also be interpreted as the probability that $x_j$ has an influence on the observed images.",
"The parameter $d$ defines the threshold at which we include covariate $x_j$ , i.e., if $d\\times 100$ percent of its corresponding coefficient images are expected to be non-zero.",
"Therefore, the threshold parameter $d$ controls selection at the “global\" level, leading to the exclusion of those covariates with low participation rates.",
"Here, without a priori information, we use a common $d$ for all covariates.",
"When a priori information is available, covariate-dependent parameters $d_j$ 's can be specified.",
"The parameter $\\pi _j$ , along with the hard-thresholding structure, introduces extra sparsity at local level, as it can be seen by calculating the expectation of the global-local selection indicator $\\nonumber E\\left[ I(\\pi _j \\ge d) \\tau _j(\\mathbf {s}) \\right] & = E_{\\pi _j} \\left[ E\\left[ I(\\pi _j \\ge d) \\tau _j(\\mathbf {s}) \\right] | \\pi _j \\right] = E_{\\pi _j}\\left[ I(\\pi _j \\ge d) \\pi _j \\right]\\\\\\nonumber & = \\int ^1_d \\pi _j \\frac{1}{B(a_{\\pi }, b_{\\pi })} \\pi _j^{a_{\\pi } -1 } (1-\\pi _j)^{b_{\\pi } -1 } d \\pi _j \\\\& = \\frac{a_\\pi }{a_\\pi +b_\\pi } \\left[ 1 - F_{\\text{Beta}}(d)\\right] = E\\left[\\tau _j(\\mathbf {s})\\right] \\left[ 1 - F_{\\text{Beta}}(d)\\right] , $ where $F_{\\text{Beta}}(\\cdot )$ is the cumulative distribution function of $\\text{Beta}(a_{\\pi }+1, b_{\\pi })$ .",
"Hence, the extra factor $\\left[ 1 - F_{\\text{Beta}}(d)\\right] \\le 1$ in Equation (REF ), which is strictly decreasing in $d$ , introduces more sparsity in the coefficient images on average from a prior perspective.",
"The participation rate parameter $\\pi _j$ and the threshold parameter $d \\in [0,1]$ establish a bridge between global and local level selection, favorably endowing existing local level selection prior with simultaneous two-level selection, allowing global selection to be informed by the selection at the local level.",
"In practice, image data are typically observed at a grid of discretized locations.",
"Given a vector of $p$ locations of interest $\\vec{\\mathbf {s}} = (\\mathbf {s}_1, ..., \\mathbf {s}_p)$ , we use $Z_i(\\vec{\\mathbf {s}}) = (Z_i(\\mathbf {s}_1), ..., Z_i(\\mathbf {s}_p))$ to denote a $p$ -by-1 vector of the process values at location $\\mathbf {s} \\in \\vec{\\mathbf {s}}$ and $\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) = \\lbrace \\Sigma (\\mathbf {s},\\mathbf {s}^{\\prime }); \\mathbf {s}, \\mathbf {s}^{\\prime } \\in \\vec{\\mathbf {s}} \\rbrace $ a $p$ -by-$p$ matrix of the within-image covariance.",
"We remark that the proposed method is applicable to discretized locations in a general $K$ -dimensional domain, i.e., the vector $\\vec{\\mathbf {s}}$ is not limited to integers nor needs to be equidistant."
],
[
"Relationship to existing literature",
"The proposed SGLSS process prior broadly relates to a wide range of existing priors that can be obtained for different choices of the parameter $d$ and prior choices for $\\tilde{\\beta }_j(\\cdot )$ and $\\tau _j(\\cdot )$ .",
"In particular, for $d=1$ the use of continuous priors on $\\pi _j$ leads to $I\\left( \\pi _j \\ge 1\\right) = 0$ with probability 1, thus no covariate will be included in the model almost surely.",
"This degenerate SGLSS process prior results in a high-dimensional extension of the mean-covariance smoothing model of [43] for the following choice of prior $\\beta _0(\\cdot ) = \\tilde{\\beta }_0(\\cdot ) \\sim \\mathcal {GP}\\left( \\mu _0(\\cdot ), \\frac{1}{c}\\Sigma ( \\cdot , \\cdot ) \\right),$ extending the mean-covariance smoothing structure from one-dimensional time-series to high-dimensional images.",
"When $d = 0$ is specified, the three-component mixture prior in Equation (REF ) degenerates to a two-component mixture of the type $\\beta _j(\\cdot ) = \\tilde{\\beta }_j(\\cdot ) \\times \\tau _j(\\cdot ) \\times 1.$ This degenerate construction naturally relates to the prior constructions used in scalar-on-image regression, and easily accommodates spatially dependent priors on $\\tilde{\\beta }_j(\\cdot )$ and $\\tau _j(\\cdot )$ [34], [31], [13], [22], [8].",
"We note, however, that incorporating spatially-correlated priors for the regression coefficients within our general global-local construction poses substantial computational challenges and a more careful interpretation of the prior parameters.",
"See also the Conclusion section.",
"Thresholding priors can also be accommodating.",
"For example, by using an autoregressive process for $\\tilde{\\beta }_j(\\cdot )$ and setting the hard threshold, $\\tau _j(\\cdot ) = I( \\tilde{\\beta }_j(\\cdot )>d_j)$ , the prior in [27] can be obtained, and by setting $\\tilde{\\beta }_j(\\cdot ) = \\text{sgn}\\left( z_j(\\cdot )\\right)\\left(\\left|z_j(\\cdot )\\right| - \\lambda _j \\right)$ , with threshold $\\tau _j(\\cdot ) = I\\left( \\left|z_j(\\cdot ) \\right| > \\lambda _j \\right)$ and $z_j(\\cdot ) \\sim \\mathcal {GP}$ , the prior in [20] can be obtained.",
"Another partially reproducible prior is the global level selection prior of [29], $\\beta _j(\\cdot ) = \\gamma _{1j} \\left(\\beta _{0} + \\gamma _{2j} z_j(\\cdot )\\right),$ that sets $\\beta _j(\\cdot )$ to a constant non-zero coefficient, a spatially varying process $( \\beta _0 + z_j(\\cdot ))$ or a zero image.",
"Since the SGLSS prior does not distinguish various types of included coefficient images, such as a non-zero constant versus a spatially varying process, it cannot fully recover the prior of [29].",
"However, it can distinguish zero images, when $\\pi _j = 0$ with all indicators $\\tau _j(\\vec{s}) = 0$ , and full images, when $\\pi _j = 1$ with all indicators $\\tau _j(\\vec{s}) = 1$ .",
"We also mention the closely related bi-level selection priors for covariates with group structure, see [35], [42], [6], [24].",
"These priors also deal with a two-level selection, but the global level is a group of covariates and the local level is the single covariate in the group.",
"For example, [35] proposed a linear regression model for identifying pathways, i.e, groups of genes, related to a particular phenotype, and a two-layer selection of pathways and genes.",
"The two selection priors, however, are not linked and this leads to the necessity of constraining the prior set of possible configurations, to avoid selection of an empty group.",
"[6] addressed this issue by adding an indicator correction procedure to do post-inference for group selection.",
"[42] and [24] followed the idea of Bayesian group lasso and conducted inference based on posterior median estimators of coefficients.",
"Unlike these constructions, the proposed SGLSS prior construction naturally leads to dependency between selections at the global and local levels, via the participation rate parameter, $\\pi $ , therefore preventing the selection of empty “groups\", i.e., covariates with no effects on the images.",
"Furthermore, with the bi-level selection priors, a group is selected in the regression if at least one of its members has non-zero effect, while in the proposed SGLSS prior construction the selection at global level is based on the probability of a covariate to affect the image, as measured by the participation rate."
],
[
"Inverse-Wishart Process prior",
"For the covariance surface $\\Sigma (\\cdot , \\cdot )$ , pre-specified parametric kernels such as the Matérn or squared exponential kernel lack flexibility, and the possible misspecification may introduce considerable bias that hampers inference.",
"We employ a flexible process, called the Inverse-Wishart process (IWP), to mitigate this concern.",
"As a nonparametric generalization of the finite-dimensional Inverse-Wishart (IW) distribution, the IWP has been used in time series to capture time-varying volatility and co-volatility [28], [14], [39], [18], and as a flexible prior for covariance kernels in functional data analysis [43].",
"Existing literature, such as [47], [43], often uses a one-dimensional support; we instead define an IWP for a general $K$ -dimensional index set $\\mathbf {S}$ in the following sense.",
"Definition 1 An Inverse-Wishart process is a stochastic process $\\Sigma = (\\Sigma (\\mathbf {s}, \\mathbf {s}^{\\prime }): (\\mathbf {s}, \\mathbf {s}^{\\prime }) \\in \\mathbf {S} \\times \\mathbf {S})$ indexed by $\\mathbf {S} \\times \\mathbf {S}$ such that the random matrix $\\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) = ((\\!\\:\\Sigma ({\\mathbf {s}}_i, {\\mathbf {s}}_j)\\!\\:))_{i,j}$ possesses an Inverse-Wishart distribution for any $\\vec{\\mathbf {s}} = (\\mathbf {s}_1, \\ldots , \\mathbf {s}_p)$ and $p \\in \\mathbb {N}$ with $\\mathbf {s}_i \\in \\mathbf {S}$ for $i,j=1,\\ldots ,p$ .",
"The matrix-valued $\\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})$ in the definition are finite-dimensional marginals evaluated on $\\vec{\\mathbf {s}}$ .",
"An IW distribution is determined by two parameters: the degrees of freedom and a scale matrix that is symmetric and positive semi-definite.",
"However, we shall follow the parameterization in [10], denoted by $\\text{IW}(\\delta , \\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}))$ with $\\delta $ a positive integer and scale matrix $\\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})$ .",
"This parameterization guarantees a crucial consistency property of IW after marginalization.",
"Let $\\Psi : \\mathbf {S} \\times \\mathbf {S} \\rightarrow \\mathbb {R}$ be a symmetric and positive semi-definite mapping, i.e., the matrix $\\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) = ((\\!\\:\\Psi (\\mathbf {s}_i, \\mathbf {s}_j)\\!\\:))_{i, j=1, \\ldots , p}$ is symmetric and positive semi-definite for any $\\vec{\\mathbf {s}} = (\\mathbf {s}_1, \\ldots , \\mathbf {s}_p)$ .",
"By the Kolmogorov extension theorem, there exists an IWP for integer $\\delta > 0$ and $\\Psi $ , which we denote by $\\text{IWP}\\left(\\delta , \\Psi (\\cdot , \\cdot ) \\right)$ ; see Lemma 2 in the Appendix of [47] for an elaborate proof when the index set is $\\mathbb {N}$ and Proposition 1 in [43] for a related discussion.",
"We typically choose $\\delta > 4$ to ensure marginals of an IWP have finite second moments.",
"We put this IWP prior on $\\Sigma (\\cdot ,\\cdot )$ , $\\Sigma (\\cdot , \\cdot ) \\sim \\text{IWP}\\left(\\delta , \\Psi (\\cdot , \\cdot )\\right) ,$ and choose the Matérn covariance function for $\\Psi (\\cdot ,\\cdot )$ , $\\begin{aligned}\\Psi (\\mathbf {s}, \\mathbf {s}^{\\prime }) & = \\text{Matérn}\\left( || \\mathbf {s} - \\mathbf {s}^{\\prime } ||_{l_2}; \\sigma ^2_s, \\rho , \\nu \\right), \\quad \\mathbf {s},\\mathbf {s}^{\\prime } \\in {\\mathbf {S}} \\\\& = \\frac{\\sigma ^2_s}{\\Gamma (\\nu ) 2^{\\nu -1}} \\left( \\sqrt{2\\nu }\\frac{ || \\mathbf {s} - \\mathbf {s}^{\\prime } ||_{l_2} }{\\rho }\\right)^{\\nu } K_{\\nu } \\left(\\sqrt{2 \\nu } \\frac{|| \\mathbf {s} - \\mathbf {s}^{\\prime } ||_{l_2}}{\\rho }\\right) ,\\end{aligned}$ where $||\\cdot ||_{l_2}$ is the $l_2$ norm.",
"For a given vector of locations $\\vec{\\mathbf {s}}$ , the prior leads to $\\Sigma \\left(\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}\\right) \\sim \\text{IW}(\\delta , \\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})).$ In the applications reported below, we fix $\\nu = 5/2$ , following [43], to have the analytical forms for both the Matérn kernel and its gradients, facilitating computation for large covariance matrices in image data, and choose the other two hyperparameters $(\\sigma ^2_s, \\rho )$ by minimizing the mean square error between an empirical covariance estimate, obtained as the MUA estimate, and the Matérn$(\\sigma ^2_s, \\rho , 5/2)$ kernel."
],
[
"Posterior Inference",
"We derive an efficient Gibbs sampler for the proposed hierarchical model with SGLSS prior.",
"Posterior sampling proceeds in three main steps as follows, with detailed derivations provided in the Supplement.",
"Update the BHM parameters $\\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^n_{i=1}$ and $\\sigma ^2_\\epsilon $ conditional on $\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=1} $ and $\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})$: Evaluated on locations $\\vec{\\mathbf {s}}$ , Equations (REF ) and () yield $\\begin{aligned}& Y_i(\\vec{\\mathbf {s}}) | Z_i(\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon \\sim \\text{MVN}( Z_i(\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon I_p), \\quad \\sigma ^2_\\epsilon \\sim \\text{Inverse-Gamma}(a_\\epsilon , b_\\epsilon ),\\\\& Z_i(\\vec{\\mathbf {s}}) | \\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace _{j=1}^q, \\Sigma ( \\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) \\sim \\text{MVN}( \\mu _i(\\vec{\\mathbf {s}}), \\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})), \\quad \\mu _i(\\vec{\\mathbf {s}}) = \\beta _0(\\vec{\\mathbf {s}}) + \\sum ^q_{j=1}x_{ij} \\beta _j(\\vec{\\mathbf {s}}).",
"\\\\\\end{aligned}$ In view of conjugacy, we sample $ Z_i(\\vec{\\mathbf {s}}) | Y_i(\\vec{\\mathbf {s}}), \\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace _{j=1}^q, \\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon \\sim MVN\\left( \\mu _{Z_i}, V_{Z_i}\\right),$ with $V_{Z_i} = \\left( \\sigma ^{-2}_\\epsilon I_p + \\Sigma (\\vec{\\mathbf {s}},\\vec{ \\mathbf {s}})^{-1}\\right)^{-1}$ and $\\mu _{Z_i} = V_{Z_i} \\left( \\sigma ^{-2}_\\epsilon Y_i(\\vec{\\mathbf {s}}) + \\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})^{-1} \\mu _i(\\vec{\\mathbf {s}})\\right)$ independently for $i = 1, \\ldots , n$ , and $\\begin{aligned}& \\sigma ^2_{\\epsilon } | \\lbrace Y_i(\\mathbf {\\vec{s}})\\rbrace _{i=1}^n, \\lbrace Z_i(\\mathbf {\\vec{s}})\\rbrace _{i=1}^n \\sim \\\\& {\\hspace{40.0pt}}\\text{Inverse-Gamma}\\left( a_\\epsilon + \\frac{np}{2}, b_{\\epsilon }+ \\frac{1}{2}\\sum ^n_{i=1} \\left( Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right)^T\\left(Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right) \\right).\\end{aligned}$ Update the SGLSS prior parameters $\\left\\lbrace \\beta _j(\\vec{\\mathbf {s}}), \\tau _j(\\vec{\\mathbf {s}}),\\pi _j\\right\\rbrace ^q_{j=0}$ conditional on $\\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^n_{i=1}$ and $\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})$: In this step, we first sample the indicators $\\left\\lbrace \\tau _j(\\vec{\\mathbf {s}})\\right\\rbrace ^q_{j=1}$ and update $\\left\\lbrace \\pi _j\\right\\rbrace ^q_{j=1}$ to obtain selection indicators at both global and local levels.",
"This is achieved via a blocked Gibbs strategy as in [29] and location-wise Gibbs updates similar to [4] and [34], adapted to the SGLSS prior.",
"We use a blocked Gibbs sampler with respect to each feature $j \\ge 1$ .",
"Denote $\\tilde{Z}_{ij}(\\vec{\\mathbf {s}}) = Z_i(\\vec{\\mathbf {s}}) - \\sum _{j^{\\prime } \\ne j} x_{ij^{\\prime }} \\beta _{j^{\\prime }}(\\vec{\\mathbf {s}})$ .",
"Equations () and (REF ) lead to a location-wise model, $ \\begin{aligned}\\tilde{Z}_{ij}(\\mathbf {s}) | & \\tilde{\\beta }_j(\\mathbf {s}), \\tau _j(\\mathbf {s}) = 1, \\Sigma (\\mathbf {s},\\mathbf {s}) \\sim N( x_{ij} \\tilde{\\beta }_j(\\mathbf {s}), \\Sigma (\\mathbf {s},\\mathbf {s})), \\\\& \\tilde{\\beta }_j(\\mathbf {s}) | \\tau _j(\\mathbf {s}) = 1 \\sim N( \\mu _{0j}(\\mathbf {s}), \\sigma ^2_{0j}(\\mathbf {s})), \\\\\\tilde{Z}_{ij}(\\mathbf {s}) | & \\tau _j(\\mathbf {s}) = 0, \\Sigma (\\mathbf {s},\\mathbf {s}) \\sim N( 0, \\Sigma (\\mathbf {s},\\mathbf {s})).\\end{aligned}$ The location-wise Bayes factor can be obtained by integrating out $\\tilde{\\beta }_j(\\mathbf {s})$ , $ \\begin{aligned}& \\theta _j(\\mathbf {s}) = \\\\& \\frac{ \\prod ^n_{i=1} p\\left( \\tilde{Z}_{ij}(\\mathbf {s}) | \\tau _j(\\mathbf {s}) = 0, \\Sigma (\\mathbf {s},\\mathbf {s}),\\pi _j\\right)p(\\tau _j(\\mathbf {s}) = 0 | \\pi _j)}{\\left\\lbrace \\int \\prod ^n_{i=1} p\\left( \\tilde{Z}_{ij}(\\mathbf {s}) | \\tilde{\\beta }_j(\\mathbf {s}) , \\tau _j(\\mathbf {s}) = 1, \\Sigma (\\mathbf {s},\\mathbf {s}),\\pi _j\\right) p(\\tilde{\\beta }_j(\\mathbf {s})) d\\tilde{\\beta }_j(\\mathbf {s}) \\right\\rbrace p(\\tau _j(\\mathbf {s}) = 1 | \\pi _j)} \\\\ \\\\= & \\\\& \\frac{1-\\pi _j}{ \\pi _j \\times \\left( \\sigma ^2_{0j}(\\mathbf {s})\\right)^{-\\frac{1}{2}}\\exp \\left\\lbrace - \\frac{1}{2} \\left(\\mu _{0j}^2(\\mathbf {s}) /\\sigma ^{2}_{0j}(\\mathbf {s})\\right)\\right\\rbrace \\times \\left( \\tilde{\\nu }_{j}(\\mathbf {s})\\right)^{\\frac{1}{2}} \\exp \\left\\lbrace \\frac{1}{2} \\tilde{m}^2_{j}(\\mathbf {s})\\tilde{\\nu }_{j}(\\mathbf {s}) \\right\\rbrace },\\end{aligned}$ with $\\begin{aligned}\\tilde{\\nu }_{j}(\\mathbf {s}) & = \\left[ \\sum ^n_{i=1}x^2_{ij}/\\Sigma (\\mathbf {s},\\mathbf {s}) + 1/\\sigma ^{2}_{0j}(\\mathbf {s}) \\right]^{-1}, \\\\ \\tilde{m}_{j}(\\mathbf {s}) & = \\sum ^n_{i=1} x_{ij}\\tilde{Z}_{ij}(\\mathbf {s}) / \\Sigma (\\mathbf {s},\\mathbf {s}) + \\mu _{0j}(\\mathbf {s})/\\sigma ^{2}_{0j}(\\mathbf {s}).\\end{aligned}$ This Bayes factor allows us to sample local selection indicators $\\tau _j(\\vec{\\mathbf {s}})$ and participation rates $\\pi _j$ from the conditional posterior distributions $\\begin{aligned}& \\tau _{j}(\\mathbf {s}) | \\lbrace \\mathbf {\\beta }_{j^{\\prime }}(\\vec{\\mathbf {s}})\\rbrace _{j^{\\prime }\\ne j}, \\lbrace Z_i(\\vec{\\mathbf {s}})\\rbrace ^n_{i=1},\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}), \\pi _j \\sim \\text{Bernoulli}\\left( \\frac{1}{1 + \\theta _{j}(\\mathbf {s})}\\right),\\\\& \\pi _j | \\tau _{j}(\\vec{\\mathbf {s}}) \\sim \\text{Beta}\\left( a_{\\pi _j} + \\sum _{\\mathbf {s} \\in \\vec{\\mathbf {s}}} \\tau _{j}(\\mathbf {s}), b_{\\pi _j} + p - \\sum _{\\mathbf {s} \\in \\vec{\\mathbf {s}}} \\tau _{j}(\\mathbf {s})\\right),\\end{aligned}$ which also gives samples of global selection indicators $I\\left( \\pi _j \\ge d \\right)$ .",
"When $j = 0$ , both $\\tau _0(\\vec{\\mathbf {s}})$ and $\\pi _j$ are fixed at 1.",
"Conditional on selection indicators at the two levels, we sample the coefficient image $\\beta _j(\\vec{\\mathbf {s}})$ for $j \\ge 0$ as follows.",
"If $\\tau _j(\\mathbf {s}) \\times I\\left(\\pi _j \\ge d \\right) = 0$ , we set $\\beta _j(\\mathbf {s}) = 0$ ; otherwise, using Equation (REF ) we sample $\\tilde{\\beta }_j(\\mathbf {s})$ from $\\tilde{\\beta }_j(\\mathbf {s}) | \\lbrace \\mathbf {\\beta }_{j^{\\prime }}(\\vec{\\mathbf {s}})\\rbrace _{j^{\\prime }\\ne j}, \\lbrace Z_i(\\vec{\\mathbf {s}})\\rbrace ^n_{i=1},\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) \\sim N(\\tilde{\\nu }_{j}(\\mathbf {s}) \\tilde{m}_{j}(\\mathbf {s}),\\tilde{\\nu }_{j}(\\mathbf {s})),$ and set $\\beta _j(\\mathbf {s}) = \\tilde{\\beta }_j(\\mathbf {s}).$ Note that the covariate $x_{i0} = 1$ when sampling the intercept $\\beta _0(\\vec{\\mathbf {s}})$ .",
"The joint update of coefficient images and selection indicators avoids reversible jump [30].",
"Update the IWP prior parameter $\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})$ conditional on $\\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^n_{i=1}$ and $\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0}$: Equations () and (REF ) lead to the conditional posterior distribution, $\\begin{aligned}&\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) | \\lbrace Z_i(\\vec{\\mathbf {s}})\\rbrace ^n_{i=1},\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0} \\sim \\\\& {\\hspace{50.0pt}} \\text{IW}\\left( n + \\delta , \\sum ^n_{i=1}\\left(Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}})\\right)\\left( Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}}) \\right)^T + \\Psi (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) \\right),\\end{aligned}$ where $\\mu _i(\\vec{\\mathbf {s}}) = \\beta _0(\\vec{\\mathbf {s}}) + \\sum ^q_{j=1}x_{ij} \\beta _j(\\vec{\\mathbf {s}}).$ At each iteration of the MCMC algorithm, for each covariate, the local level informs the selection at the global level, as the local selection indicator is sampled first and used to calculate the participation rate.",
"The covariate is then selected if the participation rate is greater than $d$ .",
"Meanwhile, at the next iteration, the participation rate serves as the prior in the Binomial-Beta conjugate update of the local level indicators, providing feedback from the local level at the previous iteration.",
"At convergence, global-level selection is done by calculating the marginal posterior probabilities of inclusion (MPPIs) of $I(\\pi _j \\ge d)$ .",
"Following [1], we use the median probability model and include the covariate if $\\mathrm {MPPI}>0.5$ , i.e., if more than half of the posterior samples give $I(\\pi _j \\ge d) = 1$ .",
"Similarly, local-level selection for covariate $j$ is determined by thresholding the MPPIs of $I(\\pi _j \\ge d)\\tau _j(\\mathbf {s})$ at 0.5.",
"Commonly used values can be specified for the sparsity parameter $d$ , such as $d =0.05$ or $d = 0.1$ , to induce a desired level of sparsity.",
"See results from the applications below and the supplementary material.",
"Given the selected covariates and locations, we estimate the corresponding $\\beta _j({\\mathbf {s}})$ via posterior means obtained from the MCMC samples.",
"We also estimate $ \\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^n_{i=1}$ , $\\sigma ^2_{\\epsilon },$ and $\\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})$ via the posterior means.",
"We recommend setting the initial value of $\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0}$ at $\\hat{\\mathbf {\\beta }}_{\\text{MUA}}(\\vec{\\mathbf {s}})$ , and the initial value of $\\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})$ at $\\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})$ .",
"Aside from the initial values, the sampler needs very little tuning.",
"We provide a python implementation where we have optimized the sampling of $Z_i(\\cdot )$ and $\\Sigma (\\cdot , \\cdot )$ by utilizing various matrix decompositions to avoid redundant matrix inversions through equivalent formulations, and further building upon pytorch, which allows automated efficient large matrix operations.",
"We remark here that the independent prior specified in Equation (REF ) allows a parallel update for the $\\left(\\beta _j(\\mathbf {s}), \\tau _j(\\mathbf {s}) \\right)$ parameters at each local pixel/voxel $\\mathbf {s} \\in \\mathbf {S}$ in each iteration of the MCMC, therefore reducing the computational cost from a multivariate Gaussian, roughly $O(|\\mathbf {S}|^3)$ for the inverse of covariance matrix, to $|\\mathbf {S}|$ univariate Gaussian."
],
[
"Simulation Study",
"In this section we conduct simulations to assess the performances of the proposed method, which we call BHM, and perform comparisons with alternative approaches.",
"We set the number of covariates to $q=15$ and the sample size to $n =100$ and generate image data (i.e., $K=2$ ) from model (REF )-() using a 30-by-30 grid, i.e., $p = 900$ .",
"We sample the coefficient images $\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^{15}_{j=1}$ and the intercept $\\beta _0(\\vec{\\mathbf {s}})$ similarly to [23].",
"At covariate level, we induce sparsity by sampling $\\tilde{\\beta }_j(\\vec{\\mathbf {s}})$ for $j \\in \\lbrace 0,1,2,3,4,5,6,7,8\\rbrace $ from a $\\mathcal {GP}(\\mathbf {0}, \\Sigma _{\\beta })$ , with $\\Sigma _{\\beta }$ specified by the Matérn kernel, and setting the remaining $\\beta _j(\\mathbf {s}) = \\tilde{\\beta }_j(\\mathbf {s}) = 0, \\forall \\mathbf {s} \\in \\vec{\\mathbf {s}}, j \\in \\lbrace 9,10,11,12,13,14,15\\rbrace $ .",
"At location level, we first rescale the images as $\\beta _j(\\vec{\\mathbf {s}}) = \\frac{ \\tilde{\\beta }_j(\\vec{\\mathbf {s}}) + \\text{sign}( \\tilde{\\beta }_j(\\mathbf {s}^{\\prime })) \\left| \\tilde{\\beta }_j(\\mathbf {s}^{\\prime }) \\right|}{2\\left| \\tilde{\\beta }_j(\\mathbf {s}^{\\prime }) \\right| }, ~~\\mathbf {s}^{\\prime } = \\operatornamewithlimits{argmax}_{s \\in \\vec{\\mathbf {s}}} \\left| \\tilde{\\beta }_j(\\mathbf {s}) \\right|,$ which excludes zeros introduced by randomness and then consider two different scenarios to introduce sparsity.",
"In the first scenario we set $10\\%$ randomly chosen elements of $\\beta _{2}(\\vec{\\mathbf {s}})$ and $\\beta _{7}(\\vec{\\mathbf {s}})$ to zero, $20\\%$ randomly chosen elements of $\\beta _{3}(\\vec{\\mathbf {s}})$ and $\\beta _{8}(\\vec{\\mathbf {s}})$ to zero, $30\\%$ randomly chosen elements of $\\beta _{4}(\\vec{\\mathbf {s}})$ to zero and $40\\%$ randomly chosen elements of $\\beta _{5}(\\vec{\\mathbf {s}})$ .",
"The second scenario addresses a more realistic and challenging case, in which signals are clustered and influential covariates only affect a small portion of the images, hardly distinguishable from the noise covariates.",
"In this case, we randomly select a square with $\\pi $ percent pixels/voxels being non-zero.",
"We consider two settings, $\\pi _j \\approx 10\\%$ and $\\pi _j \\approx 20\\%$ $(j = \\lbrace 1,2,3,4,5,6,7,8\\rbrace )$ .",
"Figures REF ,REF and REF show some example images of both noise-free images and coefficient images generated from the three scenarios.",
"Next, we generate the covariates $x_{i,j}$ 's, including both continuous and discrete variables.",
"We generate $x_{i,j}$ with $j = 1,2,3,4,5$ from $N(0,1)$ , to obtain continuous features, and $x_{i,j}$ , with $j = 6,7,8$ , from Bernoulli$(0.5)$ , for the discrete features.",
"We add noisy features generated from $N(0,1)$ , for $j=9,\\ldots ,15$ .",
"Finally, we sample the noise-free mean surface $Z_i(\\vec{\\mathbf {s}})$ from Equation () using a Matérn kernel for the covariance matrix $\\Sigma $ , and the image data $Y_i(\\vec{\\mathbf {s}})$ from Equation (REF ) with $\\epsilon _{i,\\mathbf {s}}$ sampled from $N(0,1)$ for $i=1,\\ldots ,n$ .",
"Below we report results using Matérn kernels of the type $\\Sigma = \\Sigma _\\beta = $ Matérn$(1, 1/4, 5/2)$ .",
"Figure: First simulated scenario: Example images of Z(𝐬 →)Z(\\vec{\\mathbf {s}}) and β(𝐬 →)\\beta (\\vec{\\mathbf {s}}), where 1st row is generated data; 2nd row is the estimates of BHM; and 3rd row is the estimates of MUA.Figure: Second simulated scenario (π=9%)(\\pi = 9\\%): Example images of Z(𝐬 →)Z(\\vec{\\mathbf {s}}) and β(𝐬 →)\\beta (\\vec{\\mathbf {s}}), where 1st row is generated data; 2nd row is the estimates of BHM; and 3rd row is the estimates of MUA.Figure: Second simulated scenario (π≈18.8%)(\\pi \\approx 18.8\\%): Example images of Z(𝐬 →)Z(\\vec{\\mathbf {s}}) and β(𝐬 →)\\beta (\\vec{\\mathbf {s}}), where 1st row is generated data; 2nd row is the estimates of BHM; and 3rd row is the estimates of MUA."
],
[
"Prior Specification",
"For the prior specification, we use a weakly-informative prior on the participation rate parameters $\\lbrace \\pi _j\\rbrace _{j=1}^{q}$ in Equation (REF ), by setting $a_\\pi = b_\\pi =1$ .",
"For the threshold $d$ , we report the results for a conventional sparsity level, $d=0.05$ , and then discuss sensitivity in the supplementary material.",
"We center the slab distribution in Equation (REF ) at $\\lbrace \\mu _{0j}(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0} = 0$ , as commonly done with spike-and-slab priors, and set $\\lbrace \\sigma ^2_{0j}(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0} = 1$ (see supplementary material for a sensitivity analysis).",
"As previously discussed, we derive empirical estimates of the Matérn parameters $(\\sigma ^2_s,\\rho )$ in Equation (REF ) by minimizing the mean square error between the sample covariance estimate and the Matérn kernel.",
"This provides a prior with the closest kernel to the empirical covariance matrix by Frobenius norm and prevents singularity issues caused by $n < p$ .",
"We set $\\delta =5$ for the IWP prior in Equation (REF ), following [43].",
"Finally, we set a weakly-informative Inverse-Gamma prior on the noise variance $\\sigma ^2_{\\epsilon , \\mathbf {s}}$ by setting $a_{\\epsilon } = b_{\\epsilon } = 1$ ."
],
[
"Results",
"All results we report were obtained by running MCMC chains with 2000 iterations and 500 burn-in.",
"A single chain took around 8 minutes to run on a 6-core 2.6GHz Intel(R) core i7 CPU.",
"For each chain, convergence was assessed by inspecting the MCMC traces, and more formally using the Geweke test [12] to check for signs of non-convergence of the individual parameters.",
"As an example, the z-scores from the Geweke test were 0.9603 for $\\left\\lbrace \\tau _j(\\vec{\\mathbf {s}})\\right\\rbrace ^{15}_{j=1}$ and 1.0581 for $\\left\\lbrace \\pi _j \\right\\rbrace ^{15}_{j=1}$ , clearly indicating that the MCMC chains were run for a sufficient number of iterations.",
"We evaluated performance for variable selection and parameter estimation.",
"For variable selection, we calculated $F_1 = 2\\cdot \\frac{\\text{Precision}\\cdot \\text{Recall}}{\\text{Precision} + \\text{Recall}}, \\quad \\text{Precision} = \\frac{TP}{TP + FP}, \\quad \\text{Recall} = \\frac{TP}{TP + FN}.$ For parameter estimation, we evaluated performances by calculating mean squared errors (MSEs) as $\\text{MSE} =\\frac{1}{|\\mathbf {A}| } \\sum _{\\mathbf {a} \\in \\mathbf {A}} (F(\\mathbf {a}) - \\hat{F}(\\mathbf {a}))^2,$ where $F(\\cdot )$ and $\\hat{F}(\\cdot )$ represent the true and estimated parameters, respectively, and $\\mathbf {a}$ represents the vector of the related indices and/or locations, e.g.",
"$i = 1,...,n; j = 1,...,q; \\mathbf {s}, \\mathbf {s}^{\\prime } \\in \\vec{\\mathbf {s}}$ .",
"We report the accumulated MSE of all coefficient images $\\left\\lbrace \\beta _j(\\vec{\\mathbf {s}}) \\right\\rbrace ^{15}_{j=0}$ as a summary measure of performance.",
"We first showcase inference from our BHM model on one simulated data set and then perform comparisons on 50 replicated data sets.",
"Figures REF ,REF and REF show estimates from BHM of the example images for one data set from each of the three simulated scenarios.",
"The MUA estimates are also shown, for comparison.",
"Results show that the coefficient images estimates can capture the pixel-level information relatively well, even with the location-wise independent priors (REF )-(REF ) for spatially-dependent coefficient images.",
"Tables REF reports precision, recall and $F_1$ scores for both global and local selections and Table REF reports the MSEs for the parameters of interests.",
"The proposed method performs well at the global level selection, leading to precision, recall and $F_1$ scores all relatively high.",
"At the local level selection, some differences are noted among the different simulated scenarios, in particular in the second scenario with the sparser case $\\pi = 9\\%$ , as in this scenario influential covariates are closer to noisy ones.",
"Also, results vary with the covariates' types, with coefficient images for discrete covariates being challenging for local selection, as shown by the lower recalls and $F_1$ scores.",
"The MSEs of all parameters of interests are relatively small, demonstrating that BHM can estimate those parameters relatively well.",
"Results for BHM with $d = 0.01$ and $ d=0.1$ are reported in the supplementary material.",
"As expected, as $d$ increases, precision tends to increase and recall tends to decrease.",
"However, when there exists a ‘good separation’ between the influential and noisy covariates, like in the first scenario and the second scenario with $\\pi =$ 18.8%, good performances overall can be observed for different choices of $d$ .",
"Table: BHM (d=0.05d=0.05): Global-local selection for a representative datasetTable: BHM (d=0.05d=0.05): MSEs for a representative dataset"
],
[
"Performance comparisons",
"Next, we simulate 50 replicated data sets, according to the same settings described above, and compare the performance of BHM with MUA methods.",
"MUA approaches fit independent linear regressions at each location $\\mathbf {s}$ , to estimate coefficient images, and rely on post-inference to do variable selection and smoothing.",
"For global selection, we first use Simes test [33] to convert multiple p-values at each location to one single p-value for the whole coefficient image, and then control the False Discovery Rate (FDR) at $0.05$ for the 15 coefficient images.",
"We implement three different FDR control procedures, the Benjamini–Hochberg (BH) procedure [2], the Benjamini–Yekutieli (BY) procedure [3], both implemented in R via the function `p.adjust', and another Benjamini-Hochberg procedure described in [36], implemented in the R package `fdrtool', which estimates the proportion of null features from data.",
"We denote the third procedure by MUA (SBH).",
"Local level selection is achieved by applying these three procedures to control the FDR at $0.05$ for each coefficient image.",
"Table REF reports precision, recall and $F_1$ scores for both global-level selection and local-level selection, averaged over 50 replicates from the first scenario, for MUA (BH), MUA (BY), MUA (SBH) and BHM with SGLSS prior and $d=0.05$ .",
"For the global level selection, all the methods achieve similarly high values for all three metrics, indicating that the influential covariates can potentially be well distinguished from the noisy ones.",
"For the local level selection, we observe BHM ($d=0.05$ ) and MUA (SBH) have similar performance with respect to the continuous covariates, $\\tau _{1,2,3,4,5}(\\vec{\\mathbf {s}})$ .",
"Meanwhile, when it comes to the discrete covariates $\\tau _{6,7,8}(\\vec{\\mathbf {s}})$ , BHM ($d = 0.05$ ) achieves higher averaged $F_1$ scores than the other methods, due to a relatively better balance between precision and recall.",
"As for the other MUA approaches, MUA (BH) and MUA (BY) have higher precision but much lower recall, leading to lower $F_1$ scores, especially for the discrete covariates.",
"Table REF reports the three metrics for the second scenario with two settings, $\\pi = 9\\%$ and $\\pi \\approx 18.8\\%$ .",
"As noted above, this scenario is more challenging since influential covariates are closer to the noisy ones, especially for the sparser case $\\pi = 9\\%$ .",
"However, results are relatively consistent with the previous setting.",
"For the global level selection, all methods achieve comparably high metrics.",
"We notice some precision-recall trade-offs, while high $F_1$ scores result from a relative balance between precision and recall.",
"For the local level selection, BHM ($d=0.05$ ) obtains similarly high $F_1$ scores as MUA (SBH) and MUA (BH) on the continuous covariates, and higher $F_1$ scores on the discrete covariates.",
"Although the precision of BHM($d=0.05$ ) is not as high as the other methods, its recall is relatively higher, leading to comparable $F_1$ scores.",
"At the same time, although MUA (BY) has the lowest $F_1$ scores, it has the highest precision in both settings.",
"Table REF report the MSEs for the parameters of interest and their standard errors (SE).",
"The MUA estimators are best linear unbiased estimators (BLUE) at each location, and indeed lead to relatively accurate estimates for the coefficient images $\\left\\lbrace \\beta _j(\\vec{\\mathbf {s}}) \\right\\rbrace ^{15}_{j=0}$ .",
"Meanwhile, the proposed BHM with SGLSS prior can return comparably good estimates since the global level indicators can exclude noisy covariates, leading to zero errors when global-level selection is done correctly.",
"In addition, given its hierarchical structure, BHM also produces estimates for noise-free mean surface $\\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^{100}_{i=1}$ and covariance surface $\\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})$ , which are shown to be relatively accurate.",
"Table: First simulated scenario: Global and local selection for 50 replicatesTable: Second simulated scenario: Global and local selection for 50 replicatesTable: MSEs for 50 replicates"
],
[
"Real Data Application",
"We demonstrate the proposed model using the Autism Brain Imaging Data Exchange (ABIDE) study of [11].",
"The study collected resting-state fMRI data from 17 experiment sites including 1112 subjects, with the aim of improving the understanding of neurophysiological mechanisms.",
"For each subject, rs-fMRI data were recorded over time, along with the subject's information such as age, gender, intelligence quotient, etc.",
"To aid computations, we reduced the image size by summarizing the fMRI data to voxel-level imaging statistics and then considered individual brain networks instead of the whole brain image.",
"Following [16] and [45], we used the pipeline of [5] to preprocess the data and then considerded a parcellation of the brain as defined by the Automated Anatomical Labeling [37] to select networks.",
"The selected networks are described in Table REF , and are known to be associated with cognitive ability based on previous research [38], [41], [17], [19], [45].",
"As for the covariates, those collected in the 17 experiment sites include diagnostic, age, gender and full-scale intelligence quotient scores (FIQ).",
"The FIQ scores were assessed differently across sites, including DAS-II, WASI, WISC, WAIS, RAVENS and STANFORD scales.",
"After removing missing values, we ended up with 1001 subjects.",
"We standardized the continuous variables, age and FIQ scores, to put them on the same scale with the discrete indicators, diagnostic and gender, which we left unchanged.",
"We included a vector of ones as the intercept to account for those potentially influential covariates which are not available in the study.",
"We then applied BHM with the SGLSS prior, separately, to the four selected networks.",
"Table: Networks of interestsWe specified the threshold $d$ at the conventional sparsity level $d = 0.05$ .",
"As for the slab prior specification, we set $\\lbrace \\mu _{0j}(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0} = 0$ and specified $\\lbrace \\sigma ^2_{0j}(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0} = 1$ .",
"We ran MCMC chains with 2000 iterations and 500 burnin.",
"On average, the z-scores from the Geweke test were 0.9616 for $\\left\\lbrace \\tau _j(\\vec{\\mathbf {s}})\\right\\rbrace ^{4}_{j=1}$ and 1.1858 for $\\left\\lbrace \\pi _j \\right\\rbrace ^{4}_{j=1}$ , indicating that the MCMC chains were run for a sufficient number of iterations.",
"The MCMC chain took around 35 seconds per iteration on two 20-core 2.4 GHz Intel(R) Xeon CPUs for networks with nearly $10,000$ voxels.",
"Table REF shows selection results for BHM and the MUA methods.",
"For local level selection, $\\pi $ denotes the ratio of selected voxels for MUA-based methods and the posterior mean of the participation rate $\\pi _j$ for BHM.",
"The check marks denote whether the covariate is selected at global level.",
"For global level selection, all methods agree on selecting the covariate age as influential for human cognitive ability, which makes sense because as people age the brain naturally changes, along with its cognitive functions.",
"The difference is in the selection of the FIQ scores.",
"The MUA-based methods tend to include the FIQ in the model, while the proposed BHM $(d=0.05)$ considers FIQ to be related only to Visual network and Dorsal network.",
"At the local level selection, the MUA with Benjamini-Hochberg based procedures tend to select the covariates at almost all voxels.",
"Meanwhile, MUA with Benjamini-Yekutieli procedure tends to select the covariates at fewer voxels than the MUA (BH) especially when it comes to the FIQ scores.",
"The BHM-based methods tend to have similar results as the MUA (BY) on the selection of age.",
"As for the selection of FIQ, however, BHM with $d=0.05$ tends to select even fewer voxels.",
"Furthermore, when fitting the BHM model we noticed that a more stringent threshold of $d = 0.1$ would exclude FIQ for all networks entirely, while a less stringent threshold of $d = 0.01$ would include FIQ for all networks but only for very few selected voxels.",
"These results suggest that, although FIQ score may somehow be related to brain signals, this relationship can be hard to recover in this application, possibly because different experimental sites use different standards to measure this covariate.",
"Figure REF shows the selected voxels for the covariate age by MUA (SBH) (Left), MUA (BY) (Middle) and BHM ($d=0.05$ ) (Right).",
"We observe a decreasing number of voxels selected by the methods, and similar local selective results between MUA (BY) and BHM ($0.05$ ), both of which tend to have a more sparse selection with selected voxels mainly in the central portions of the regions in the functional networks.",
"Table REF reports the ratios of region included in the local selection, showing consistent selection results for BHM.",
"We note that the final selection is determined by the posterior summary, i.e.",
"the median rule, based on the posterior samples, while the threshold $I(\\pi \\ge d)$ takes effect at each iteration.",
"Hence, although the global indicator can guarantee $\\pi \\ge d$ at each iteration, the final ratios of selected voxels/pixels are not necessarily greater than $d$ , as it is evident from the results.",
"We report results for BHM with $d = 0.01$ and $d=0.1$ in the supplementary material and note here that BHM maintains highly consistent local selection results with different specification of $d$ , i.e.",
"when Ceneus R is considered to be affected by FIQ, BHM ($d =0.01$ ) selects $1.6\\%$ of the region and BHM ($d=0.05)$ selects $1.77\\%$ ; when Temporal Mid R is considered to be affected by FIQ, BHM ($d =0.01$ ) selects $1.7\\%$ of the region and BHM ($d=0.05)$ selects $1.5\\%$ .",
"These results also confirm the previous observation that FIQ scores, converted from different standards, may not show strong relationship to the brain regions.",
"Table: Selection results for the four networksTable: Ratios of Region included within each networksFigure: Selected voxels for covariate `age', by MUA with SBH (Left), MUA with BY (Middle) and BHM with SGLSS (d=0.05d = 0.05) (Right).",
"Figures are plotted using the R package threeBrain by"
],
[
"Concluding remarks",
"In this article, we have extended to image data a Bayesian hierarchical Gaussian process (GP) model that uses a flexible Inverse-Wishart process prior to handle within-image dependency, and have proposed a novel spatial global-local spike-and-slab prior that broadly relates to a rich class of well-studied selection priors.",
"The proposed prior construction achieves simultaneous global (i.e, at covariate-level) and local (i.e., at pixel/voxel-level) selection via participation rate parameters that measure the probability for the individual covariates to affect the observed images.",
"We have used hard-thresholding to decide whether a covariate should be included in the model and have shown on simulated data that parameters are interpretable and lead to efficient selection.",
"The introduced participation rate and threshold parameters establish a bridge between global and local level selection, allowing global selection to be informed by the selection at the local level.",
"This framework can be applied to more general functional data applications.",
"There are several interesting future directions to extend our model.",
"Our rationale for choosing an independent prior on spatial coefficients $\\beta _j(s)$ has been largely computational.",
"Our efficient Gibbs sampler takes advantage of this independent prior and only requires to invert the $|\\mathbf {S}|-by-|\\mathbf {S}|$ covariance matrix once for the noise-free mean surface $Z(\\cdot )$ , roughly $O(|\\mathbf {S}|^3$ ) at each Gibbs iteration.",
"In the application to real data, our model is able to handle relatively large datasets, with $S$ about 10,000 voxels, and $n$ about 1,000 subject images.",
"With a dependent prior, we would not be able to parallelize computations, which would result into having to calculate the inversion of at most $|\\mathbf {S}|-by-|\\mathbf {S}|$ covariance matrices for $q$ covariates at each iteration, with roughly a $O( |\\mathbf {S}|^3q )$ complexity at each iteration.",
"This would have been infeasible for our application.",
"In addition, a construction with a dependent prior would require a more careful interpretation of the participation rate parameters $\\pi _j$ 's, which measure the probability for the individual covariates to affect the observed image under the assumption of independence.",
"Given these challenges, we have decided to leave the investigation of dependent priors to future work.",
"We note, however, that, even though we do not explicitly account for dependency among the coefficients, our model borrows information across voxels via the use of the spatial Gaussian process prior $\\mathcal {GP}\\left(\\mu (\\cdot ), \\Sigma (\\cdot , \\cdot )\\right)$ on $Z_i$ .",
"In the applications of this paper, when investigating the role of the parameter $d$ and the sensitivity of the results to the specification of this parameter, we found the case $d=0$ interesting.",
"In this degenerate case the model includes all the covariates at each iteration, to explain the observed images, and the traces of the parameters $\\pi _j$ inform us on the relative importance of the individual covariates.",
"These trace plots provide an empirical tool that might be helpful in the choice of $d$ , particularly in cases where a separation among the traces is observed.",
"In the Supplementary Material we show these plots for one of the simulated scenarios used in this paper, along with comments on how the plots can guide the user in the choice of $d$ .",
"We remark, however, that this procedure is ad-hoc and cannot be used as a general method, in particular as the behavior of the trace plots is application-dependent and a clear separation of the traces might not always be observed.",
"We leave further investigation of the role and properties of the parameter $d$ to future work.",
"In the absence of prior information, we recommend to view $d$ as conventional sparsity parameter and use standard values, i.e.",
"$d = 0.05$ or $d = 0.1$ .",
"Our sensitivity analyses in the simulations and real data application have shown good performances overall for different choices of $d$ , with highly consistent local selection results.",
"Finally, our proposed global-local selection prior construction can be potentially useful for other modeling settings, such as function-on-scalar and network-on-scalar regressions."
],
[
"S.1. Markov Chain Monte Carlo Sampling (MCMC)",
"In this section, we provide the detailed derivations for the Gibbs sampler in Section REF .",
"Update the BHM parameters $\\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^n_{i=1}$ and $\\sigma ^2_\\epsilon $ conditional on $\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0} $ and $\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})$: Evaluated on locations $\\vec{\\mathbf {s}}$ , Equations (REF ) and () yield $\\begin{aligned}& Y_i(\\vec{\\mathbf {s}}) | Z_i(\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon \\sim \\text{MVN}( Z_i(\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon I_p), \\quad \\sigma ^2_\\epsilon \\sim \\text{Inverse-Gamma}(a_\\epsilon , b_\\epsilon ),\\\\& Z_i(\\vec{\\mathbf {s}}) | \\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace _{j=1}^q, \\Sigma ( \\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) \\sim \\text{MVN}( \\mu _i(\\vec{\\mathbf {s}}), \\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})), \\quad \\mu _i(\\vec{\\mathbf {s}}) = \\beta _0(\\vec{\\mathbf {s}}) + \\sum ^q_{j=1}x_{ij} \\beta _j(\\vec{\\mathbf {s}}),\\end{aligned}$ leading to a Normal Inverse-Gamma conjugacy.",
"With $\\mu _i(\\vec{\\mathbf {s}}) = \\beta _0(\\vec{\\mathbf {s}}) + \\sum ^q_{j=1} x_{ij}\\beta _j(\\vec{\\mathbf {s}})$ , we have $\\begin{aligned}\\raggedleft & p(Z_i(\\vec{\\mathbf {s}}) | Y_i(\\vec{\\mathbf {s}}),\\left\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\right\\rbrace ^q_{j=0}, \\Sigma \\left(\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}\\right), \\sigma ^2_\\epsilon )\\\\\\propto & p(Y_i(\\vec{\\mathbf {s}}) | Z_i(\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon ) p\\left( Z_i(\\vec{\\mathbf {s}}) | \\left\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\right\\rbrace ^q_{j=0}, \\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) \\right) \\\\\\propto & \\exp \\left\\lbrace - \\frac{1}{2} \\left( Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right)^T \\sigma ^{-2}_\\epsilon I_p \\left( Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}}) \\right)\\right\\rbrace \\exp \\left\\lbrace - \\frac{1}{2}\\left( Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}})\\right)^T \\right.",
"\\\\& \\quad \\times \\left.\\Sigma ^{-1}(\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})\\left(Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}}) \\right)\\right\\rbrace \\\\\\propto & \\exp \\left\\lbrace - \\frac{1}{2} \\left[ Z^T_{i}(\\vec{\\mathbf {s}})\\left( \\sigma ^{-2}_\\epsilon I_p + \\Sigma ^{-1}(\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})\\right)Z_i(\\vec{\\mathbf {s}}) - 2Z^T_i(\\vec{\\mathbf {s}})\\left( \\sigma ^{-2}_\\epsilon Y_i(\\vec{\\mathbf {s}}) \\right.",
"\\right.",
"\\right.",
"\\\\& \\quad \\left.",
"\\left.",
"\\left.",
"+ \\Sigma ^{-1}(\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) \\mu _i(\\vec{\\mathbf {s}})\\right)\\right]\\right\\rbrace \\end{aligned}$ which gives to the posterior distribution, $\\begin{aligned}& Z_i(\\vec{\\mathbf {s}}) | Y_i(\\vec{\\mathbf {s}}), \\mu _i(\\vec{\\mathbf {s}}), \\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon \\sim MVN\\left( \\mu _{Z_i}(\\vec{\\mathbf {s}}), V_{Z_i}(\\vec{\\mathbf {s}})\\right), \\\\& V_{Z_i}(\\vec{\\mathbf {s}}) = \\left( \\sigma ^{-2}_\\epsilon I_p + \\Sigma ^{-1}(\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})\\right)^{-1}, \\\\& \\mu _{Z_i}(\\vec{\\mathbf {s}}) = V_{Z_i}(\\vec{\\mathbf {s}}) \\left( \\sigma ^{-2}_\\epsilon Y_i(\\vec{\\mathbf {s}}) + \\Sigma ^{-1}(\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})\\mu _i(\\vec{\\mathbf {s}})\\right).\\end{aligned}$ The posterior distribution of the corresponding variance component is given by $\\begin{aligned}&p(\\sigma ^2_\\epsilon | \\lbrace Y_i(\\vec{\\mathbf {s}})\\rbrace _{i=1}^n, \\lbrace Z_i(\\vec{\\mathbf {s}})\\rbrace _{i=1}^n ) \\propto \\prod ^n_{i=1}p( Y_i(\\vec{\\mathbf {s}}) | Z_i(\\vec{\\mathbf {s}}), \\sigma ^2_\\epsilon ) p(\\sigma ^2_{\\epsilon }) \\\\\\propto & {\\hspace{10.0pt}} |\\sigma ^2_{\\epsilon } I_p |^{-\\frac{n}{2}} \\exp \\left\\lbrace -\\frac{1}{2} \\sum ^n_{i=1} \\left( Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right)^T \\sigma ^{-2}_\\epsilon I_p \\left(Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right) \\right\\rbrace \\\\& {\\hspace{30.0pt}} \\times \\left( \\sigma ^{-2}_{\\epsilon }\\right)^{a_\\epsilon + 1} \\exp \\left\\lbrace - \\sigma ^{-2}_\\epsilon b_{\\epsilon }\\right\\rbrace \\\\\\propto & {\\hspace{10.0pt}} \\left( \\sigma ^{-2}_\\epsilon \\right)^{\\frac{np}{2} + a_\\epsilon + 1} \\exp \\left\\lbrace - \\sigma ^{-2}_\\epsilon \\left[ b_{\\epsilon }+ \\frac{1}{2}\\sum ^n_{i=1} \\left( Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right)^T\\left(Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right)\\right]\\right\\rbrace ,\\end{aligned}$ which gives $\\begin{aligned}&\\sigma ^2_{\\epsilon } | \\lbrace Y_i(\\vec{\\mathbf {s}})\\rbrace _{i=1}^n, \\lbrace Z_i(\\vec{\\mathbf {s}})\\rbrace _{i=1}^n \\sim \\\\& {\\hspace{40.0pt}} \\text{InverseGamma}\\left( a_\\epsilon + \\frac{np}{2}, b_{\\epsilon }+ \\frac{1}{2}\\sum ^n_{i=1} \\left( Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right)^T\\left(Y_i(\\vec{\\mathbf {s}}) - Z_i(\\vec{\\mathbf {s}})\\right) \\right).\\end{aligned}$ Update the SGLSS prior parameters $\\left\\lbrace \\beta _j(\\vec{\\mathbf {s}}), \\tau _j(\\vec{\\mathbf {s}}),\\pi _j\\right\\rbrace ^q_{j=0}$ conditional on $\\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^n_{i=1}$ and $\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})$: In this step, we first sample the indicators $\\left\\lbrace \\tau _j(\\vec{\\mathbf {s}})\\right\\rbrace ^q_{j=1}$ and update $\\left\\lbrace \\pi _j\\right\\rbrace ^q_{j=1}$ to obtain selection indicators at both global and local levels.",
"This is achieved via a blocked Gibbs strategy and has the Beta Binomial conjugacy when integrating out $\\tilde{\\beta }_j(\\vec{\\mathbf {s}})$ .",
"Using the blocked Gibbs sampler with respect to each feature $j$ , we have $\\tilde{Z}_{ij}(\\vec{\\mathbf {s}}) = Z_i(\\vec{\\mathbf {s}}) - \\sum _{j^{\\prime } \\ne j} x_{ij^{\\prime }}\\mathbf {\\beta }_{j^{\\prime }}(\\vec{\\mathbf {s}})$ .",
"Furthermore, we denote $\\tilde{z}_{ij\\mathbf {s}} = \\tilde{Z}_{ij}(\\mathbf {s}), \\tilde{\\beta }_{j\\mathbf {s}} = \\tilde{\\beta }_j(\\mathbf {s})$ , $\\mu _{0j\\mathbf {s}} = \\mu _{0j}(\\mathbf {s}), \\sigma ^2_{0j\\mathbf {s}} = \\sigma ^2_{0j}(\\mathbf {s}), \\sigma ^2_{\\mathbf {s}} = \\Sigma (\\mathbf {s}, \\mathbf {s}), \\mathbf {s} \\in \\vec{\\mathbf {s}}$ and $\\tilde{z}_{\\cdot j\\mathbf {s}} = \\lbrace \\tilde{z}_{ij\\mathbf {s}}\\rbrace ^n_{i=1}$ for short, where $\\mathbf {s}$ is one location in the vector of locations $\\vec{\\mathbf {s}}$ .",
"Based on Equation (REF ), we calculate the marginal posterior probability of $\\tau _j(\\mathbf {s}) = 1$ by integrating out $\\tilde{\\beta }_{j\\mathbf {s}}$ , $&& p\\left( \\tau _j(\\mathbf {s}) = 1 | \\tilde{z}_{\\cdot j\\mathbf {s}} \\right) = \\dfrac{ \\int p\\left(\\tau _j(\\mathbf {s}) = 1, \\tilde{\\beta }_{j\\mathbf {s}} | \\tilde{z}_{\\cdot j\\mathbf {s}}\\right) d\\tilde{\\beta }_{j\\mathbf {s}} }{ p\\left(\\tau _j(\\mathbf {s}) = 0, \\tilde{\\beta }_{j\\mathbf {s}} = 0 | \\tilde{z}_{\\cdot j\\mathbf {s}}\\right) + \\int p\\left(\\tau _j(\\mathbf {s}) = 1, \\tilde{\\beta }_{j\\mathbf {s}} | \\tilde{z}_{\\cdot j\\mathbf {s}}\\right) d\\tilde{\\beta }_{j\\mathbf {s}} }.$ Substituting $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{405.6487pt}\\begin{aligned}\\int p\\left(\\tau _j(\\mathbf {s}) = 1, \\tilde{\\beta }_{j\\mathbf {s}} | \\tilde{z}_{\\cdot j\\mathbf {s}}\\right) d\\tilde{\\beta }_{j\\mathbf {s}}& = \\frac{1}{p\\left( \\tilde{z}_{\\cdot j\\mathbf {s}} \\right)} \\int p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 1, \\tilde{\\beta }_{j\\mathbf {s}} \\right) p\\left( \\tilde{\\beta }_{j\\mathbf {s}}\\right) p(\\tau _j(\\mathbf {s}) = 1 | \\pi _j)d\\tilde{\\beta }_{j\\mathbf {s}}\\end{aligned}}$ and $p\\left(\\tau _j(\\mathbf {s}) = 0, \\tilde{\\beta }_{j\\mathbf {s}} = 0 | \\tilde{z}_{\\cdot j\\mathbf {s}}\\right)= \\frac{1}{p\\left( \\tilde{z}_{\\cdot j\\mathbf {s}} \\right)} p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 0, \\tilde{\\beta }_{j\\mathbf {s}} = 0 \\right) p(\\tau _j(\\mathbf {s}) = 0 | \\pi _j)$ yields $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{405.6487pt}p\\left( \\tau _j(\\mathbf {s}) = 1 | \\tilde{z}_{\\cdot j\\mathbf {s}} \\right) = \\dfrac{ \\int p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 1, \\tilde{\\beta }_{j\\mathbf {s}} \\right) p\\left( \\tilde{\\beta }_{j\\mathbf {s}}\\right)d\\tilde{\\beta }_{j\\mathbf {s}} \\times \\pi _j }{ p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 0, \\tilde{\\beta }_{j\\mathbf {s}} = 0 \\right)\\times (1 - \\pi _j) + \\int p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 1, \\tilde{\\beta }_{j\\mathbf {s}} \\right) p\\left( \\tilde{\\beta }_{j\\mathbf {s}}\\right)d\\tilde{\\beta }_{j\\mathbf {s}} \\times \\pi _j }.",
"}$ For $p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 0, \\tilde{\\beta }_{j\\mathbf {s}} = 0 \\right)$ , we have $p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 0, \\tilde{\\beta }_{j\\mathbf {s}} = 0 \\right) =\\underbrace{\\left(2\\pi \\sigma ^2_{\\mathbf {s}}\\right)^{-\\frac{n}{2}} \\exp \\left\\lbrace - \\frac{1}{2} \\left( \\sum ^n_{i=1}\\tilde{z}^2_{ij\\mathbf {s}}/\\sigma ^2_{\\mathbf {s}} \\right)\\right\\rbrace }_{\\text{common factor (CF)}}.$ For $ \\int p\\left(\\tilde{z}_{\\cdot j\\mathbf {s}} | \\tau _j(\\mathbf {s}) = 1, \\tilde{\\beta }_{j\\mathbf {s}} \\right) p\\left( \\tilde{\\beta }_{j\\mathbf {s}}\\right)d\\tilde{\\beta }_{j\\mathbf {s}}$ , we have $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{384.2974pt}\\begin{aligned}& \\int \\left( 2\\pi \\sigma _{\\mathbf {s}}^2 \\right)^{-\\frac{n}{2}} \\exp \\left\\lbrace -\\frac{1}{2} \\sum ^n_{i=1} \\left( \\tilde{z}_{ij \\mathbf {s}} - x_{ij}\\tilde{\\beta }_{j\\mathbf {s}}\\right)^2/\\sigma ^{2}_{\\mathbf {s}} \\right\\rbrace \\left(2\\pi \\sigma ^2_{0j\\mathbf {s}}\\right)^{-\\frac{1}{2}} \\exp \\left\\lbrace - \\frac{1}{2} \\left(\\tilde{\\beta }_{j\\mathbf {s}} -\\mu _{0j\\mathbf {s}}\\right)^2/\\sigma ^{2}_{0j\\mathbf {s}} \\right\\rbrace d \\tilde{\\beta }_{j\\mathbf {s}} \\\\& = \\left( 2\\pi \\sigma ^2_{\\mathbf {s}}\\right)^{-\\frac{n}{2}} \\left(2\\pi \\sigma ^2_{0j\\mathbf {s}}\\right)^{-\\frac{1}{2}}\\int \\exp \\left\\lbrace - \\frac{1}{2} \\left[ \\tilde{\\beta }^2_{j\\mathbf {s}}\\left( \\sum ^n_{i=1} x^2_{ij} / \\sigma ^2_{\\mathbf {s}}\\right) - 2\\tilde{\\beta }_{j\\mathbf {s}}\\left( \\sum ^n_{i=1} x_{ij}\\tilde{z}_{ij\\mathbf {s}} / \\sigma ^{2}_{\\mathbf {s}} \\right) \\right.",
"\\right.",
"\\\\& \\left.",
"\\left.",
"+ \\sum ^n_{i=1} \\tilde{z}^2_{ij\\mathbf {s}}/ \\sigma ^{2}_{\\mathbf {s}} \\right] -\\frac{1}{2} \\left[ \\tilde{\\beta }^2_{j\\mathbf {s}}/\\sigma ^2_{0j\\mathbf {s}} - 2\\tilde{\\beta }_{j\\mathbf {s}}\\left( \\mu _{0j\\mathbf {s}} /\\sigma ^{2}_{0j\\mathbf {s}} \\right) + \\mu ^2_{0j\\mathbf {s}}/\\sigma ^{2}_{0j\\mathbf {s}}\\right]\\right\\rbrace d\\tilde{\\beta }_{j\\mathbf {s}} \\\\& = \\left(2\\pi \\sigma ^2_{\\mathbf {s}}\\right)^{-\\frac{n}{2}} \\exp \\left\\lbrace - \\frac{1}{2} \\left( \\sum ^n_{i=1}\\tilde{z}^2_{ij\\mathbf {s}}/\\sigma ^2_{\\mathbf {s}} \\right)\\right\\rbrace \\left(2\\pi \\sigma ^2_{0j\\mathbf {s}}\\right)^{-\\frac{1}{2}}\\exp \\left\\lbrace - \\frac{1}{2} \\left(\\mu _{0j\\mathbf {s}}^2 /\\sigma ^{2}_{0j\\mathbf {s}}\\right)\\right\\rbrace \\\\& \\times \\int \\exp \\left\\lbrace - \\frac{1}{2} \\left[ \\tilde{\\beta }^2_{j\\mathbf {s}} \\underbrace{\\left( \\sum ^n_{i=1}x^2_{ij}/\\sigma ^{2}_{\\mathbf {s}} + 1/\\sigma ^{2}_{0j\\mathbf {s}}\\right)}_{\\tilde{v}^{-1}_{j\\mathbf {s}}} - 2\\tilde{\\beta }_{j\\mathbf {s}} \\underbrace{\\left( \\sum ^n_{i=1} x_{ij}\\tilde{z}_{ij\\mathbf {s}} /\\sigma ^{2}_{\\mathbf {s}} + \\mu _{0j\\mathbf {s}}/\\sigma ^{2}_{0j\\mathbf {s}}\\right)}_{\\tilde{m}_{j\\mathbf {s}}}\\right]\\right\\rbrace d\\tilde{\\beta }_{j\\mathbf {s}} \\\\& = \\left(2\\pi \\sigma ^2_{\\mathbf {s}}\\right)^{-\\frac{n}{2}} \\exp \\left\\lbrace - \\frac{1}{2} \\left( \\sum ^n_{i=1}\\tilde{z}^2_{ij\\mathbf {s}}/\\sigma ^2_{\\mathbf {s}} \\right)\\right\\rbrace \\left(2\\pi \\sigma ^2_{0j\\mathbf {s}}\\right)^{-\\frac{1}{2}}\\exp \\left\\lbrace - \\frac{1}{2} \\left(\\mu _{0j\\mathbf {s}}^2 /\\sigma ^{2}_{0j\\mathbf {s}}\\right)\\right\\rbrace \\\\& \\times \\int \\left( 2\\pi \\tilde{\\nu }_{j\\mathbf {s}}\\right)^{-\\frac{1}{2}} \\exp \\left\\lbrace - \\frac{1}{2}\\left( \\tilde{\\beta }^2_{j\\mathbf {s}} - 2\\tilde{\\beta }_{j\\mathbf {s}} \\tilde{v}_{j\\mathbf {s}} \\tilde{m}_{j\\mathbf {s}} + \\tilde{\\nu }^2_{j\\mathbf {s}}\\tilde{m}^2_{j\\mathbf {s}}\\right)/\\tilde{\\nu }_{j\\mathbf {s}} \\right\\rbrace d\\tilde{\\beta }_{j\\mathbf {s}} \\\\& \\times \\left( 2\\pi \\tilde{\\nu }_{j\\mathbf {s}}\\right)^{\\frac{1}{2}} \\exp \\left\\lbrace \\frac{1}{2} \\tilde{m}^2_{j\\mathbf {s}}\\tilde{\\nu }_{j\\mathbf {s}} \\right\\rbrace \\\\& = \\underbrace{\\left(2\\pi \\sigma ^2_{\\mathbf {s}}\\right)^{-\\frac{n}{2}} \\exp \\left\\lbrace - \\frac{1}{2} \\left( \\sum ^n_{i=1}\\tilde{z}^2_{ij\\mathbf {s}}/\\sigma ^2_{\\mathbf {s}} \\right)\\right\\rbrace }_{\\text{common factor (CF)}} \\times \\underbrace{\\left( \\sigma ^2_{0j\\mathbf {s}}\\right)^{-\\frac{1}{2}}\\exp \\left\\lbrace - \\frac{1}{2} \\left(\\mu _{0j\\mathbf {s}}^2 /\\sigma ^{2}_{0j\\mathbf {s}}\\right)\\right\\rbrace }_{\\text{prior factor (PF)}} \\\\& \\times \\left( \\tilde{\\nu }_{j\\mathbf {s}}\\right)^{\\frac{1}{2}} \\exp \\left\\lbrace \\frac{1}{2} \\tilde{m}^2_{j\\mathbf {s}}\\tilde{\\nu }_{j\\mathbf {s}} \\right\\rbrace .\\end{aligned}}$ Combining both, we obtain that $&& p\\left( \\tau _j(\\mathbf {s}) = 1 | \\tilde{z}_{\\cdot j\\mathbf {s}} \\right) \\\\& = & \\frac{ \\text{CF} \\times \\text{PF} \\times \\left( \\tilde{\\nu }_{j\\mathbf {s}}\\right)^{\\frac{1}{2}} \\exp \\left\\lbrace \\frac{1}{2} \\tilde{m}^2_{j\\mathbf {s}}\\tilde{\\nu }_{j\\mathbf {s}} \\right\\rbrace \\pi _j }{\\text{CF} \\times (1-\\pi _j) + \\text{CF} \\times \\text{PF} \\times \\left( \\tilde{\\nu }_{j\\mathbf {s}}\\right)^{\\frac{1}{2}} \\exp \\left\\lbrace \\frac{1}{2} \\tilde{m}^2_{j\\mathbf {s}}\\tilde{\\nu }_{j\\mathbf {s}} \\right\\rbrace \\pi _j} \\\\& = & \\frac{ 1 }{ 1 + \\theta _{j\\mathbf {s}}},$ with $\\theta _{j\\mathbf {s}} = \\frac{1-\\pi _j}{ \\pi _j \\times \\left( \\sigma ^2_{0j\\mathbf {s}}\\right)^{-\\frac{1}{2}}\\exp \\left\\lbrace - \\frac{1}{2} \\left(\\mu _{0j\\mathbf {s}}^2 /\\sigma ^{2}_{0j\\mathbf {s}}\\right)\\right\\rbrace \\times \\left( \\tilde{\\nu }_{j\\mathbf {s}}\\right)^{\\frac{1}{2}} \\exp \\left\\lbrace \\frac{1}{2} \\tilde{m}^2_{j\\mathbf {s}}\\tilde{\\nu }_{j\\mathbf {s}} \\right\\rbrace },$ and $\\tilde{\\nu }_{j\\mathbf {s}} = \\left[ \\sum ^n_{i=1}x^2_{ij}/\\sigma ^{2}_{\\mathbf {s}} + 1/\\sigma ^{2}_{0j\\mathbf {s}} \\right]^{-1}, \\quad \\tilde{m}_{j\\mathbf {s}} = \\sum ^n_{i=1} x_{ij}\\tilde{z}_{ij\\mathbf {s}} /\\sigma ^{2}_{\\mathbf {s}} + \\mu _{0j\\mathbf {s}}/\\sigma ^{2}_{0j\\mathbf {s}}.$ This gives the posterior distribution for $\\tau _{j}(\\mathbf {s})$ as a Bernoulli distribution, and leads to the Beta distribution for $\\pi _j$ by counting the $\\tau _{j}(\\vec{\\mathbf {s}})$ samples $\\pi _j | \\tau _j(\\vec{\\mathbf {s}}) \\sim \\text{Beta}\\left( a_{\\pi _j} + \\sum _{\\mathbf {s} \\in \\vec{\\mathbf {s}}} \\tau _{j}(\\mathbf {s}), b_{\\pi _j} + p - \\sum _{\\mathbf {s} \\in \\vec{\\mathbf {s}}} \\tau _{j}(\\mathbf {s})\\right),$ as written in Equation (REF ).",
"Conditional on selection indicators at the two levels, we sample the coefficient image $\\beta _j(\\vec{\\mathbf {s}})$ as summarized in Section REF .",
"Update the IWP prior parameter $\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})$ conditional on $\\left\\lbrace Z_i(\\vec{\\mathbf {s}})\\right\\rbrace ^n_{i=1}$ and $\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0}$: This is a Gibbs step from Multivariate Normal Inverse Wishart conjugacy.",
"We have $\\begin{aligned}& p\\left( \\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) | \\lbrace Z_i(\\vec{\\mathbf {s}}\\rbrace ^n_{i=1}, \\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0}, \\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) \\right) \\\\\\propto & {\\hspace{10.0pt}} \\prod ^n_{i=1} p\\left( Z_i(\\vec{\\mathbf {s}}) | \\left\\lbrace \\beta _j(\\vec{\\mathbf {s}})\\right\\rbrace ^q_{j=0}, \\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) \\right) p \\left( \\Sigma ( \\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) \\right) \\\\\\propto & {\\hspace{10.0pt}} \\prod ^n_{i=1} \\left\\lbrace |\\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})|^{-\\frac{1}{2}} \\exp \\left\\lbrace -\\frac{1}{2} \\left( Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}})\\right)^T\\Sigma ^{-1}(\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})\\left(Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}}) \\right)\\right\\rbrace \\right\\rbrace \\\\& {\\hspace{100.0pt}} \\times |\\Sigma (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}})|^{-\\frac{(\\delta +p+1)}{2} }\\exp \\left\\lbrace -\\frac{1}{2} \\text{tr}\\left( \\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) \\Sigma ^{-1}(\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})\\right)\\right\\rbrace \\end{aligned}$ $\\begin{aligned}& \\propto |\\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}})|^{-\\frac{(n+\\delta +p + 1)}{2}} \\exp \\left\\lbrace - \\frac{1}{2}\\text{tr}\\left[\\left( \\sum ^n_{i=1}\\left(Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}}) \\right)\\left( Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}}) \\right)^T \\right.",
"\\right.",
"\\right.",
"\\\\& {\\hspace{220.0pt}} \\left.",
"\\left.",
"\\left.",
"+ \\Psi (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) \\right)\\Sigma ^{-1}(\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) \\right]\\right\\rbrace ,\\end{aligned}$ which gives the Inverse Wishart distribution $\\begin{aligned}& \\Sigma (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) | \\lbrace Z_i(\\vec{\\mathbf {s}})\\rbrace ^n_{i=1}, \\lbrace \\beta _j(\\vec{\\mathbf {s}})\\rbrace ^q_{j=0},\\Psi (\\vec{\\mathbf {s}}, \\vec{\\mathbf {s}}) \\sim \\\\& {\\hspace{80.0pt}} \\text{IW}\\left( n + \\delta , \\sum ^n_{i=1}\\left(Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}}) \\right)\\left( Z_i(\\vec{\\mathbf {s}}) - \\mu _i(\\vec{\\mathbf {s}}) \\right)^T + \\Psi (\\vec{\\mathbf {s}},\\vec{\\mathbf {s}}) \\right).\\end{aligned}$ with, again, $\\mu _i(\\mathbf {s}) = \\beta _0(\\vec{\\mathbf {s}}) + \\sum ^{q}_{j=1} x_{ij}\\beta _j(\\mathbf {s})$ .",
"Tables REF and REF report precision, recall and $F_1$ scores for both global and local selections for our method with $d=0.01, 0.05$ and $d=0.1$ .",
"Clearly, a lower $d$ tends to include more covariates, potentially selecting noisy covariates.",
"On the contrary, a higher $d$ tends to exclude more covariates, potentially selecting out influential covariates.",
"Correspondingly, we can observe that as $d$ increases, precision tends to increase and recall tends to decrease especially in the more challenging second simulated scenario.",
"In addition, when there exists a `good separation' between the influential and noisy covariates, like in the first scenario and the second scenario with $\\pi \\approx 18.8\\%$ , good performances overall can be observed for different choices of $d$ .",
"This is also reflected in the MSE estimates of the model parameters shown in Tables REF and REF .",
"Table: First simulated scenario: Global-local selection for a representative datasetTable: First simulated scenario: MSEs for a representative datasetTable: Second simulated scenario: Global-local selection for a representative datasetTable: Second simulated scenario: MSEs for a representative datasetAs pointed out in the paper, the sparsity parameter $d$ has the interpretation that covariates affecting more than $d$ percent of the images are included in the model.",
"The challenge in specifying $d$ is to distinguish the low values of the participation parameters $\\pi _j$ 's corresponding to the noisy covariates from those of the partially influential covariates, i.e., those covariates that affect a small number of voxels/pixels.",
"To better understand the role of the sparsity parameter $d$ , we found helpful to look the MCMC trace plots of $\\left\\lbrace \\pi _j\\right\\rbrace ^{15}_{j=1}$ for the specification $d=0$ .",
"In this degenerate case we have $I(\\pi _j \\ge 0) = 1$ for all $j$ and the model includes all the covariates at each iteration to explain the observed images.",
"Figure REF (a) shows these traces for a representative simulated case from setting 1.",
"In this figure, traces corresponding to fully non-zero image coefficients (influential covariates) are in red, those for partially influential covariates are in blue and those for the noisy covariates are in black.",
"We observe a clear separation between the three types of traces.",
"In particular, we have $\\pi \\in [0,0.2]$ for the noisy covariates, $\\pi \\in [0.6, 0.9]$ for the partially influential covariates and $\\pi \\approx 1$ for the covariates affecting the whole images.",
"This plot suggests that values of $d$ lower than .6 can be good choices, as they separate noisy covariates from the fully and partially influential ones.",
"Clearly, smaller values of $d$ , as those we use in the paper and the sensitivity analysis above, are preferred, as they induce sparsity, excluding covariates effecting very low portions of the images and including those effecting larger portions.",
"As further evidence, in Figure REF we show MCMC traces obtained by fitting our model for a grid of values $d \\in [0.1,0.9]$ .",
"These figures confirm that the model is relatively insensitive to a range of choices $d\\in [0.1, 0.6]$ and support the selection of a reasonably small $d$ , which separates the traces.",
"On the contrary, when the choice of $d$ is too large, i.e., $d \\ge 0.7$ , the covariates get in and out from the model, introducing large fluctuations in several of the traces.",
"As a word of caution, we remark that the trace plots we show here are meant to provide an empirical tool that might be helpful in the choice of $d$ , particularly in cases where a separation among the traces is observed.",
"However, this procedure is ad-hoc and cannot be used as a general method, in particular as the behavior of the trace plots is application-dependent and a clear separation of the traces might not always be observed.",
"In such cases, and in the absence of prior information, we recommend to view $d$ as conventional sparsity parameter and use standard values, i.e.",
"$d = 0.05$ or $d = 0.1$ .",
"Figure: Trace plots of π j j=1 15 \\left\\lbrace \\pi _j \\right\\rbrace ^{15}_{j=1} for d=0d = 0.",
"Traces corresponding to fully non-zero image coefficients (influential covariates) are in red, those for partially influential covariates are in blue and those for the noisy covariates are in black.Figure: Trace plots of π j j=1 15 \\left\\lbrace \\pi _j \\right\\rbrace ^{15}_{j=1} for a grid of d∈[0,1]d \\in [0,1].",
"Traces corresponding to fully non-zero image coefficients (influential covariates) are in red, those for partially influential covariates are in blue and those for the noisy covariates are in black.In addition to the hard-threshold $d$ , the influential model parameters are the means $\\left\\lbrace \\mu _{0j}( \\vec{\\mathbf {s}})\\right\\rbrace ^{15}_{j=0}$ and variance parameters $\\left\\lbrace \\sigma ^2_{0j} (\\vec{\\mathbf {s}}) \\right\\rbrace ^{15}_{j=0}$ of the slab prior distributions.",
"As seen in Equation (REF ), the slab prior is involved in the calculation of the Bayes factors, therefore informing the local selection.",
"Conventional specifications use slab normal distributions with means 0 and variance parameters in the range $\\sigma ^2_0 \\in [1,100]$ .",
"We compare the performances of different $\\sigma ^2_0$ using the first simulated scenario used in the paper, and fixed $d = 0.05$ .",
"Tables REF and REF report the averaged precisions, recalls and $F_1$ scores for both global and local selection and MSEs for parameters of interest calculated over the 50 replicates.",
"We observe that the proposed method shows consistent good performances at the global level selection, achieving similarly high values in precision, recall and $F_1$ scores, with all prior specifications.",
"At the local level, results show a trade-off between precision and recall, in that larger prior variances tend to achieve higher precisions but lower recalls.",
"As for parameter estimation, the MSEs of the parameters of interests are relatively similar for different slab prior specifications, showing the estimations are not very sensitive to the slab prior specification.",
"Table: Global and local selection for the sensitivity analysis - 50 replicatesTable: MSEs for sensitivity analysis - 50 replicatesTable REF and REF report selection results for $d = 0.01, 0.05$ and $0.1$ , along with results from MUA (BY) as a comparison.",
"We observe that, as expected, a more stringent threshold of $d = 0.1$ would exclude FIQ for all networks entirely and a less stringent threshold of $d = 0.01$ would include FIQ for all networks but only very few voxels selected.",
"Without over-interpreting the results, we remark that at the local selection level the ratios of region included are relatively consistent when the covariates are included, i.e.",
"for Cuneus R, for both $d= 0.01$ and $d=0.05$ , the ratio is around $1.7\\%$ .",
"Table: Selection results for the four networksTable: Ratios of Region included within each networks"
]
] | 2209.08234 |
[
[
"Site-Net: Using global self-attention and real-space supercells to\n capture long-range interactions in crystal structures"
],
[
"Abstract Site-Net is a transformer architecture that models the periodic crystal structures of inorganic materials as a labelled point set of atoms and relies entirely on global self-attention and geometric information to guide learning.",
"Site-Net processes standard crystallographic information files to generate a large real-space supercell, and the importance of interactions between all atomic sites is flexibly learned by the model for the prediction task presented.",
"The attention mechanism is probed to reveal Site-Net can learn long-range interactions in crystal structures, and that specific attention heads become specialized to deal with primarily short- or long-range interactions.",
"We perform a preliminary hyperparameter search and train Site-Net using a single graphics processing unit (GPU), and show Site-Net achieves state-of-the-art performance on a standard band gap regression task."
],
[
"Introduction",
"The application of machine learning to materials science has enabled a new paradigm of high throughput property prediction for the screening and identification of new materials.",
"Prediction pipelines based on machine learning models are significantly less computationally intensive than DFT and other physical simulations in much the same way that computational methods can be a faster and cheaper alternative to synthetic investigations [1], [2].",
"Consequently, the material discovery process can be significantly accelerated by initial screening and recommendation by machine learning models, which may lead to subsequent validation of promising candidates through physical modelling, and the final demonstration of discovery through preparation in the laboratory [3].",
"Machine learning models that rely only on the elemental composition have been widely successful and have been applied to a range of property prediction tasks [4], [5].",
"The elemental composition of materials is often the most well-characterised feature, and the fixed and limited number of elements mean that the compositions can be readily embedded as a fixed length vector that is amenable to most machine learning methods.",
"However, many properties are strongly dependent on the crystal structure, and composition-based methods do not distinguish between materials with similar or identical compositions yet different crystal structures, such as polymorphs.",
"A classic example is graphite and diamond, which both have a trivial elemental composition of pure carbon but have wildly different physical properties (e.g., band gap, electrical resistivity, thermal conductivity) [6].",
"The challenge of creating a suitable representation of the crystal structure prevents directly embedding crystal structures for use in property prediction tasks.",
"Specifically, periodic crystal structures have an unbounded number of atoms, and the conventional methods used to describe periodic systems are challenging to represent appropriately for a machine learning algorithm.",
"There is an infinite number of possible unit cells that can be chosen for a given crystal structure, and the varying number of atomic sites between the unit cells of different crystal structures makes it difficult to construct a representation using a fixed length vector.",
"Further, any representation of the unit cell that uses a coordinate system must be invariant to rigid transformations, otherwise simple rotation and translation can lead to different predictions for different descriptions of the same crystal structure [7].",
"Treating the crystal structure as a graph and using convolution neural networks has shown promising results for predicting properties [8], [9], [10], [11], and such models now outperform composition-only models where sufficient structural data is available.",
"However, these models rely on an explicitly defined cutoff distance or number of neighbours to define a meaningful interaction between atomic sites in the crystal structure.",
"These graph learning methods were initially applied to molecules, where short-range chemical bonds are often much more significant than long-range interactions [12].",
"However, extended inorganic solids have many competing interactions at a range of length scales, and many functional properties arise from long-range features of the crystal structure.",
"In this report, we present Site-Net, a point-set model based on global self-attention augmented with pairwise interactions, where all atomic sites are free to interact with each other.",
"Site-Net uses a physically motivated representation of the crystal structure as a point set of atomic sites, which is separated into “site features” containing chemical information (about elements), and “interaction features” containing geometric information (about positions).",
"The set of atomic sites is directly ingested without any pre-defined connections, and the importance of interactions between all atomic sites is flexibly learned by the model through global self-attention.",
"The attention mechanism is probed to reveal Site-Net learns long-range interactions, and that specific attention heads become specialized to deal with primarily short- or long-range interactions.",
"This learning leads to state-of-the-art performance, which we assess using the band gap regression task from Matbench [13], where Site-Net achieves a mean absolute error of 0.251 eV on an 80:20 train:test split of the dataset."
],
[
"Representation construction and featurization",
"Periodic crystal structures have an infinite number of atoms and are thus commonly described in a more compact form by choosing an appropriate description (e.g., a unit cell) that can be infinitely tiled in 3 dimensions.",
"There are infinite possible choices of valid unit cells, and although there are several conventions for arriving at a useful unit cell to humans, defining a canonical unit cell for a given crystal structure that is robust to noise is a challenging problem [7].",
"Unfortunately, the lack of a unique unit cell causes issues for training machine learning models, whereas model predictions for a given crystal structure should not be influenced by the arbitrary choice of unit cell.",
"Notably, if the goal is to predict the properties of a crystal structure, two different choices of unit cell for the same crystal structure should lead to the same prediction.",
"With Site-Net, we sidestep this problem by working with a large set of atoms without symmetry, and assume the set is large enough to capture most relevant features without suffering from finite size effects.",
"Site-Net is able to ingest crystallographic information files (CIF) that are commonly used to represent crystal structures and generate an appropriate representation for training machine learning models (fig:cif2features).",
"Any conventional unit cell from a crystallographic database is transformed into a primitive unit cell in $P1$ (i.e., all symmetry constraints are removed, and the atoms are all listed explicitly).",
"This minimal $P1$ unit cell is then iteratively tiled to generate a large set of atoms (fig:cif2featuresc).",
"While Site-Net avoids the need for a canonical choice of unit cell, there is nevertheless a soft requirement to provide each atomic site in the crystal the largest local environment possible.",
"Accordingly, the aforementioned supercell is created to explicitly include longer range interactions with higher order images of the minimal $P1$ unit cell.",
"In this work, we show Site-Net performs well with a set of 100 atoms to work within the memory constraints of a single consumer graphics processing unit (GPU), though this is only a technical constraint, and the model performance should improve with increasing number of atoms and the consequently more rich structural context from considering longer range interactions.",
"The set of atoms is generated by determining the optimal transformation of the minimal $P1$ unit cell to the largest possible supercell that is approximately cubic and contains less than 100 atoms.",
"The creation of appropriate supercells that are roughly cubic remains an open challenge [14], and most methods seek to optimize for a given volume, rather than number of atoms.",
"As this work seeks to optimize for a given number of atoms, we perform supercell construction using an algorithm developed here for this task (sec:supercellalgo).",
"If exactly 100 atoms cannot be achieved, zero padding is used to bring all atom sets to the same size.",
"Crystal structures with minimal $P1$ unit cells containing more than 100 atoms were not considered during training due to memory constraints on a single GPU, though these 3245 instances in the Matbench band gap dataset account for only 3% of the total instances (fig:sizelimitmatbench).",
"These same large crystal structures were still included during testing and performance evaluation, and thus will penalise the model if it cannot learn about larger structures.",
"The resulting set of $\\sim $ 100 atoms, roughly cubic in shape, is featurized into two distinct tensors that separately encode elemental information and spatial information.",
"The elemental information is encoded as a vector of atomic site features (fig:cif2featuresd), consisting of the identities of elements in the crystal structure, along with related properties of these elements.",
"These elemental properties (e.g., atomic radius) can be manually defined, though we also include a learned embedding unique to each element, similar in concept to word embeddings with word2vec [15], where the tokens are chemical elements.",
"For every chemical element, Site-Net stores a unique vector that is updated during model training; the length of the elemental vectors is a hyperparameter of the model (Tab:hparam.",
"In the present implementation of Site-Net, the raw site features are represented by a tensor of dimension [100,101], comprising the number of sites (100), and the elemental features associated with each of the sites (101 in this report).",
"The spatial information is encoded in the interaction features using a full pairwise interaction matrix between these sites (fig:cif2featurese).",
"The core of the interaction features is the full real-space Euclidean distance matrix of all atoms (respecting periodic boundary conditions), which ensures the spatial relationships of all atoms in the crystal structure are encoded [16], [17].",
"Figure: Site-Net transforms a crystal structure into a fixed length global vector that can be used for downstream tasks (such as property prediction).",
"This is simplified here for illustrative purposes.",
"(a) The minimal P1P1 unit cell is used to generate site features, shown here for Li 2 _2O, which has 2 chemically equivalent Li sites, and one O site.",
"(b) These vectors of site features are then passed through a set of neural networks (including self-attention blocks) to create new context-enriched site features that are imbued with knowledge of their chemical and structural environment.",
"(c) These context-enriched site features are then compressed by a permutation-invariant function (such as taking the mean) to generate a fixed length global vector that describes the entire crystal structure.",
"Without the multiple layers of self-attention to enrich the context of the features, mean pooling of the raw features into a global vector would otherwise cause too much information loss for useful property predictions."
],
[
"Self-attention as a mechanism to create context-enriched site features",
"Site-Net is a set transformer [18] architecture that takes a crystal structure, constructs a point set from the atomic sites in the unit cell, and processes the point set into a fixed length global feature vector representing the entire crystal structure, which is suitable for downstream property prediction (fig:modelconcept).",
"Rather than encoding spatial information through a coordinate system (e.g., PointNet [19]), a matrix of pairwise interactions is incorporated into the custom attention mechanism to iteratively encode the spatial information into the point-set.",
"Once a fixed length global vector has been attained, property prediction is performed through the application of standard dense neural network layers.",
"However, the compression into a fixed length global feature vector necessary for property prediction is performed using permutation-invariant aggregation (here we use mean pooling), which leads to significant information loss.",
"For example, taking the mean over the initial site features would destroy most relevant crystallographic information.",
"Accordingly, Site-Net passes the initial site features through multiple attention layers to enrich the atomic site features with the context of their local chemical and structural environment (fig:modeloutline).",
"These context-enriched site features retain enough structural information following compression to allow useful property predictions.",
"Figure: A simplified architecture of Site-Net shows the dataflow from atoms in a crystal structure, to a downstream property prediction task.",
"(a) The site features and interaction features are generated from the list of atoms in a large supercell (for example, Li 2 _2O).",
"These are passed to attention blocks, which are used to progressively enrich atomic site features with contextual information from neighbouring atomic sites.",
"(b) The first attention block generates new site and interaction features, which are sequentially fed into subsequent attention blocks.",
"After passing through the final attention block, the site feature outputs from all attention blocks are concatenated together.",
"(d) The concatenated site features are then passed to a pooling layer for permutation-invariant aggregation, where the mean is taken to produce a fixed length global feature vector that describes the crystal in its totality.",
"(e) This global feature vector is then processed downstream through prediction neural network layers to generate predictions for a property of interest.Before being passed to an attention block, the raw site features ($S^{\\circ }_{i,f}$ ) and raw interaction features ($I^{\\circ }_{i,j,f}$ ) are reprocessed to an auxiliary embedding.",
"Here, $i$ and $j$ are atomic site identities, and $f$ is the featurization dimension.",
"The auxiliary embedding is likely to depend on the prediction task, so the lengths of the featurization dimensions are tunable hyperparameters to allow the Site-Net model flexibility to find an optimal representation or dimensionality for a given task.",
"This is accomplished by a single neural network layer preceding the first attention block, which ingests the raw site features ($S^{\\circ }_{i,f}$ ) and interaction features ($I^{\\circ }_{i,j,f}$ ) and generates processed analogues of the correct dimensionality.",
"Specifically, the raw site feature tensor $S^{\\circ }_{i,f}$ [100, 101] is transformed to $S_{i,f}$ [100, $\\lambda $ ], and the raw interaction features tensor $I^{\\circ }_{i,j,f}$ [100, 100, 2] is transformed to $I_{i,j,f}$ [100, 100, $\\mu $ ].",
"These dimensions are consistent across the attention blocks for both input and output.",
"In the final model presented here, the hyperparameters found after a preliminary search are $\\lambda $ = 90 and $\\mu $ = 48 (Tab:hparam).",
"Starting from the raw site features in a crystal structure, site features enriched with the context of their local environment are constructed using a sequence of self-attention blocks, where the site features are iteratively replaced with a weighted aggregation of the pairwise interactions with all other atomic sites in the crystal structure representation (fig:attentionmechanism).",
"This process replaces the purely elemental features of atomic sites with the aggregation of their local environment, and thus encodes information about the crystal structure into the context-enriched site features.",
"This aggregation function does not depend on the ordering of atomic sites, and is thus permutation invariant in the same way the global feature vector is produced from the site features.",
"At a conceptual level, self-attention is a learned permutation- invariant function that prioritizes the most important interactions when constructing the new enriched site features.",
"$S_{i,f} &\\in \\mathbb {R}^{N \\times \\lambda } \\\\I_{i,j,f} &\\in \\mathbb {R}^{N \\times N \\times \\mu } \\\\B_{i,j,*} = S_{i,*} \\mathbin \\Vert I_{i,j,*} \\mathbin \\Vert S_{j,*}&, \\text{ where } B_{i,j,f} \\in \\mathbb {R}^{N \\times N \\times (2\\lambda + \\mu )}$ To begin, the site features $S_{i,f}$ (eq:S) and interaction features $I_{i,j,f}$ (eq:I) for each pair of atoms are concatenated to create bond features $B_{i,j,f}$ (eq:B).",
"The bond feature vector $B_{i,j,*}$ captures interactions between atomic sites $i$ and $j$ , and is an ordered combination of a site vector $S_{i,*}$ , followed by the interaction vector $I_{i,j,*}$ , and then $S_{j,*}$ .",
"Here, an asterisk ($\\ast $ ) denotes the span of an index.",
"Importantly, because the order of the atom pairs is preserved, these bond features are directional ($B_{i,j,*} \\ne B_{j,i,*}$ ).",
"Assembling all the bond feature vectors into the complete bond features tensor $B_{i,j,f}$ leads to a unified representation of the crystal structure (fig:attentionmechanismc).",
"This is carried forward and subsequently used to derive new context-enriched site features $S^\\prime _{i,f}$ and new context-enriched interaction features $I^\\prime _{i,j,f}$ (fig:attentionmechanism).",
"Figure: The mechanism of the Site-Net attention block, whose purpose is to enrich the atomic site features with context of their local environments, is illustrated using the simplified example of the minimal P1P1 unit cell of Li 2 _2O.Each attention block ingests (a) the site features S i,j S_{i,j} and (b) the interaction features I i,j,f I_{i,j,f}, which are concatenated to generate the bond features.",
"(c) The bond features B i,j,f B_{i,j,f} are a unified representation of every ordered pair of atoms and the interaction between them.These set of bond vectors B i,j,* B_{i,j,*} then go through a series of transformations to eventually generate the new site features S i,j ' S^\\prime _{i,j} and new interaction features I i,j,f ' I^\\prime _{i,j,f}.",
"(d) The attention features A i,j,f F A^F_{i,j,f} are derived from the learned function g F g^{F} and serve as precursors to the new site features.Self-attention mechanism is used to combine these attention features as a weighted sum to form S i,j ' S^\\prime _{i,j}.",
"(e) The relative contribution of each attention feature is dictated by the scalar attention weights A i,j,f W A^W_{i,j,f}.",
"These weights are derived from the learned function g W g^W, and represent the strength of the influence of a particular attention feature on the local environment.",
"(f) New site features S i,j ' S^\\prime _{i,j} produced through such a self-attention mechanism represent local environments of the atomic sites instead of solely elemental properties.",
"(g) The new interaction features I i,j,f ' I^\\prime _{i,j,f} are derived from the learned function g I g^I to compress the bond features B i,j,f B_{i,j,f} back down to the proper dimensionality, and are enriched by the context of atoms connected by the interactions.$A^F_{i,j,*} = g^F(B_{i,j,*})&, \\text{ where } A^F_{i,j,f} \\in \\mathbb {R}^{N \\times N \\times \\lambda } \\\\a^W_{i,j} = \\frac{e^{g^W(B_{i,j,*})+\\delta _{ij}}}{\\sum _j e^{g^W(B_{i,j,*})+\\delta _{ij}}}&, \\text{ where } A^W_{i,j} \\in \\mathbb {R}^{N \\times N} \\text{ and } \\delta _{ij} \\text{ is the Kronecker delta} \\\\S^\\prime _{i,*} = \\sum _j a^W_{i,j}A^F_{i,j,*}&, \\text{ where } S^\\prime _{i,f} \\in \\mathbb {R}^{N \\times \\lambda }$ Global self-attention is used to generate new context-enriched site features $S^\\prime _{i,j}$ .",
"In this implementation, we introduce intermediate attention features $A^F_{i,j,f}$ (fig:attentionmechanismd, eq:Af), and attention weights $a^W_{i,j}$ (fig:attentionmechanisme, eq:Aw).",
"The vectors $A^F_{i,j,*}$ have the same dimension as site feature vectors and are obtained from bond vectors $B_{i,j,*}$ by means of a fully connected neural network $g^F: \\mathbb {R}^{2\\lambda +\\mu } \\xrightarrow{} \\mathbb {R}^{\\lambda }$ .",
"The relative importance of site $j$ to $i$ based on their interaction is captured by the scalar attention weights $a^W_{i,j}$ (fig:attentionmechanisme, eq:Aw), which are computed using another fully connected neural network $g^W: \\mathbb {R}^{2\\lambda +\\mu } \\xrightarrow{} \\mathbb {R}$ .",
"The number of layers and the number of neurons per layer for $g^W$ are hyperparameters of the model.",
"The resulting scalar values $g^W (B_{i,j,*})$ are normalized using the softmax function (eq:Aw).",
"For every atomic site $i$ , this softmax normalization ensures that the weights $a^W_{i,j}$ over all atomic sites $j$ sum to 1.",
"As a consequence of the softmax normalization to generate attention weights $a^W_{i,j}$ , the resulting distribution of weights is conceptually similar to a probability distribution over all neighbours, where the attention weights $a^W_{i,j}$ represent the significance of neighbour $j$ to $i$ .",
"Critically, the exponential nature of softmax is likely important to discard many of the negligible contributions that will be present when considering all pairwise interactions in Site-Net.",
"To improve training, the attention weights for $i=j$ are increased by 1 to bias the model towards preserving the identity.",
"Finally, the new context-enriched site feature vector $S^\\prime _{i,*}$ is a sum of vectors $A^F_{i,j,*}$ weighted by scalars $a^W_{i,j}$ (eq:Sprime).",
"In simple terms, each atomic site has a vector representing its chemical and geometric configuration, which is subsequently replaced by the mean of all vectors for every neighbour and itself.",
"This mean is modified by the relative importance of every site.",
"As a consequence, the new site features are no longer a descriptor of a single site.",
"Rather, they are representations of every local environment in the crystal structure.",
"With repeated attention blocks, the representation of each individual site feature becomes more abstract.",
"$I^\\prime _{i,j,*} = g^I(B_{i,j,*}), \\text{ where } I^\\prime _{i,j,f} \\in \\mathbb {R}^{N \\times N \\times \\mu }$ The bond features are also used to produce new interaction features $I^\\prime _{i,j,f}$ .",
"In comparison to the bond features, the obtaining of new interaction features is straightforward.",
"The new interaction features and the bond features are of the same dimension, so new interaction features are obtained (eq:IPrime) by passing the bond features through a single feed forward layer ($g^I$ ) so they are of the expected dimensionality.",
"These new interaction features contain the information of the two sites connected by that interaction and serve a similar role to residual connections, as they preserve this information for subsequent attention blocks.",
"With respect to the overall architecture, we have described the process of performing single-headed attention.",
"This process can be generalized to multi-headed attention, where multiple sets of attention feature and attention weight tensors are independently computed inside the same attention block and then concatenated.",
"The use of more attention heads allow more attention operations to be performed in parallel, where each head can focus on a specific group of interactions.",
"Similarly, the number of attention blocks can be increased to achieve more abstract features, as attention is performed on the outputs of the last attention block.",
"The preliminary hyperparameter search performed on the band gap prediction task revealed that 2 attention blocks and 3 attention heads is a reasonable balance able to achieve state-of-the-art performance (Tab:hparam)."
],
[
"Post-attention processing and pooling",
"The new site features and interaction features generated by the attention block are fed into the next attention block to repeat this process of contextual enrichment through the construction of higher-level features.",
"After passing through all attention blocks, the separate site feature outputs from each and every attention block are concatenated together in preparation for pooling to a fixed length global feature vector by taking the mean of all sites.",
"Further, a final pre-pooling step is performed to minimize the information loss of the subsequent pooling.",
"Here, a simple neural network whose size is a hyperparameter of the model is used to process the concatenated site features from the attention blocks to an auxiliary embedding for pooling, much in the same way that a single neural network layer was used to reprocess the raw site features and interaction features to an auxiliary embedding for performing attention.",
"After taking the mean of all sites to produce the fixed length global feature vector, obtaining a property prediction is straightforward a process with a sequence of feed forward neural network layers.",
"An explicit process flow diagram for the entire architecture can be found in the Supporting Information (fig:Sflowchart)."
],
[
"Implementation",
"To generate the representations of the crystals, CIF files are first converted into Pymatgen structure objects [20].",
"From these Pymatgen structure objects, the crystal structure can be featurized using featurization libraries such as matminer [21] and dscribe [22].",
"The models were developed using pytorch [23] combined with the use of the pytorch lightning framework [24] to provide automatic GPU training and data management.",
"To handle the varying sizes of crystal structures in the dataset, the tensor representation of atom sets with less than 100 atoms were zero-padded to ensure all tensors were of the same size with respect to the forward pass.",
"All operations performed on padding are excluded from training.",
"Hyperparameter tuning was handled via hyperopt [25] using the Ray Tune [26] distributed hyperparameter tuning framework as a front end.",
"The hyperparameters that performed best on the validation set when trained on the training dataset are benchmarked on the holdout dataset.",
"The hyperparameter search was performed on the Barkla compute cluster using a single Tesla P100 GPU with 16 GB of VRAM; the batch size was limited by the available VRAM.",
"A preliminary hyperparameter search was performed by sequentially training 30 models for 24 hours each, using previous models to inform future hyperparameter choices.",
"The best set of hyperparameters was then carried forward for longer training of the final models presented here (Tab:hparam).",
"The model is sensitive to the choice of hyperparameters, and based on the limited search performed here, these hyperparameters are likely far from optimal and will allow considerable model improvement in the future.",
"Table: Site-Net hyperparameter search space and final values for the reported band gap prediction task.",
"The model was generally sensitive to hyperparameters, which were fixed after achieving best-in-class performance in a preliminary search.",
"All hyperparameters were optimized using Ray Tune , except where noted as fixed.",
"Given the sensitivity of the model to hyperparameter choice and the large search space available, further hyperparameter tuning will undoubtedly improve model performance."
],
[
"Results and Discussion",
"The performance of Site-Net was assessed using the band gap regression task from Matbench, a materials benchmarking test suite [13].",
"The first fold of the preset cross validation pipeline was used for this benchmarking, and consists of 106113 crystal structures and associated band gap energies.",
"The training set was 80% of the available data and the test set was 20%.",
"Within the training data, 80% was used for training, while 20% of the training data was used for a validation score for hyperparameter optimisation.",
"As training was done using a single GPU, it was computationally unfeasible to run a separate hyperparameter search over all five Matbench data folds.",
"Further, using a single hyperparameter set on all five folds would be unsuitable owing to data leakage, as training data from the first fold—on which hyperparameters are determined—is cycled into test data of the other 4 folds.",
"The Matbench band gap dataset poses unique challenges as it contains a smooth continuum of positive band gap energies together with a large number of zeros.",
"We employ a custom activation function to address this unique property of the dataset, wherein negative predictions of band gap were clamped to zero while preserving the gradient to allow the model to recover from false zero predictions.",
"Given negative band gaps are non-physical, we thus treat negative predictions as a level of confidence in the classification of zero rather than an “overshoot” that needs to be corrected.",
"Figure: Training and prediction performance of Site-Net on the Matbench band gap prediction task.",
"(a) The learning curve exhibits smooth monotonic loss per epoch, with no overtraining.",
"The mean absolute error (MAE) reaches a plateau at 0.251 eV after ∼\\sim 500 epochs, which is ∼\\sim 7 days of training.",
"(b) The parity plot reveals the model is consistent across band gap values, and has an associated MAE of 0.251 eV.",
"Colour is used to represent the number of materials at a particular coordinate; the peak at the origin is outside the bounds of the scaling used due to the high number of materials in the dataset with a band gap of exactly zero.Training Site-Net on the band gap regression task leads to a smooth, monotonic learning curve that steadily converges to a plateau; models did not exhibit divergent overtraining behaviour (fig:performancea).",
"Despite its complexity, the model trains to a stable state and does not suffer from problems typically encountered with continued training where the validation score begins to diverge.",
"Site-Net achieves a mean absolute error (MAE) of 0.251 eV on the band gap regression task, and performance of the model is consistent across band gap values (fig:performanceb).",
"Even with only a preliminary hyperparameter search, Site-Net currently demonstrates competitive performance with the highest performing algorithms on the leaderboard.",
"For example, CGCNN [30] has a reported MAE of 0.297 eV as of this report, and ALIGNN [10], which uses second and third order interactions such as angles and solid angles between atoms, is the highest performing algorithm with a reported MAE of 0.186 eV.",
"Examining the attention weights of the trained band gap model for all pairs of atomic sites in the test dataset allows interrogation of the model to investigate the importance of pairwise atomic interactions at different distances (fig:attentionheads).",
"The attention heads of the first attention block focus on atomic sites that are close together ($<$ 5 Å), which is consistent with local interactions being important to material properties.",
"The second attention head within the first block notably contains more long-range interactions, suggesting that model training specialised the attention head for this purpose while other heads were more focused on the local environment.",
"Enforcing a cutoff limit of 5 Å on the range of the attention and retraining the model decreases performance (MAE 0.291 eV), confirming that interactions beyond this distance meaningfully contribute to model predictions.",
"Meanwhile, the attention weights of the second attention block are much higher at longer distances.",
"This is consistent with focusing on higher-order correlations, as features entering the second attention head are more context-enriched after passing through the first attention block.",
"Notably, the model learns that the majority of significant interactions are at short range but is able to nevertheless capture significant interactions at longer distances, without having to define beforehand what constitutes a meaningful interaction.",
"This is consistent with the decrease in performance seen when a cutoff distance is enforced.",
"Enforcing a 5 Å distance cutoff limit to the attention in Site-Net decreases model performance to levels to graph-based models with the same cutoff (MAE 0.291 eV).",
"Figure: The attention weights of the trained band gap model for all pairs of atomic sites in the test dataset as a function of inter-atomic distance.To visualize the ∼\\sim 10 7 ^7 attention weights, 2-dimensional histograms are constructed by ordinally binning by the interatomic distance, and then normalizing such that the sum of all attention weights at any one distance bin sum to 1.The colour corresponds to the proportion of attention weights that lie within a given bin.The number of attention heads (3 in this report) and attention blocks (2 in this report) are hyperparameters of the model and were chosen by a preliminary hyperparameter search.",
"The attention heads of the first attention block focus on atomic sites that are close together (<<5 Å), which is consistent with local interactions being important to material properties.",
"The second attention head notably contains more long-range interactions, suggesting that model training specialised the attention head for this purpose while other heads were more focused on the local environment.",
"The attention weights of the second attention block are much higher at longer distances.",
"This is consistent with focusing on high-order correlations, as features entering the second attention head are more context-enriched after passing through the first attention block.While Site-Net demonstrates excellent performance on the band gap prediction task, not all tasks are expected to benefit from identical model features.",
"Accordingly, most model features of Site-Net used in this report were deliberately chosen to be tuneable hyperparameters that can be learned (e.g., the learned elemental embedding), but some initial site features and interaction features were defined manually and may not be optimal.",
"For example, construction of models without the Coulomb matrix in the interaction features resulted in marginal decrease in model performance on the band gap regression task, while models trained without the real-distance matrix led to reasonable training but poor test set performance.",
"Further, the hyperparameter search was only preliminary owing to the large search space, and thus model hyperparameters used here could be far from optimal.",
"For example, the attention weights demonstrate similar distance-dependent behaviour in attention head 1 and attention head 3 (fig:attentionheads), so the removal of an attention head may only marginally decrease performance while significantly reducing model training time.",
"We have shown here that Site-Net is effective at operating on ordered crystal structures, but owing to the construction of a large supercell and removal of symmetry, the same process can also be used to examine disordered crystal structures.",
"Disordered materials could either be treated directly (e.g., using the raw atom positions from a molecular dynamics simulation), or treated by constructing multiple ordered supercells (e.g., using Pymatgen [20]) and generating predictions for all supercell approximants.",
"We note the predictions on the set of ordered supercells could be aggregated and subsequently interrogated using simple statistics to infer the reliability of the predictions."
],
[
"Computational Considerations",
"The model is computationally intensive and has a quadratic dependence on the number of atoms in the crystal in terms of VRAM and computational load.",
"The current model ingests unit cells of a maximum size of 100 atoms, and was trained using 16 GB of VRAM, which can run on a single desktop GPU.",
"While this was done as a proof of concept, and the model size in this report was limited by the amount of GPU VRAM, there is no fundamental limitation to the number of atoms that can be ingested beyond the amount of computational resources available to operate the larger attention blocks.",
"100 atoms was chosen for models trained and reported here because $>$ 97% of structures in the band gap dataset have unit cells with less than 100 atoms (fig:sizelimitmatbench).",
"However, the cutoff limit of 100 atoms should also be appropriate on more general tasks using other datasets.",
"Similar examination of $\\sim $ 200 000 crystal structures in the Inorganic Crystal Structure Database (ICSD) demonstrates a cutoff limit of 100 atoms would allow training on 92% of the crystal structures in the ICSD (fig:sizelimitICSD).",
"A cutoff limit of 100 atoms provides a balance by having sufficiently large local environment for any atomic site as well as including nearly all of the data in the training set, while avoiding prohibitive batch sizes and training times.",
"It is expected that increasing the number of atoms will increase performance up to a certain point, as the larger local environments in larger supercell models will allow the explicit encoding of increasingly long-range interactions.",
"There are several straightforward ways to achieve larger models.",
"In the most simple case, larger supercells of 300 atoms could easily be treated by running on a larger GPU, for example using a high-performance computing cluster with a 128 GB GPU.",
"Being able to run models with a size limit of 300 atoms allows using 99.96% of all the data for the dataset used here (fig:sizelimitmatbench), generates supercells that have a large number of atoms to encode long-range interactions (fig:atomsincell, and generates well-behaved pseudo-cubic supercells (fig:cubicdeviation).",
"Straightforward changes to the architecture can also be made to achieve larger models.",
"The first is to split the parameters of the model across multiple GPUs to increase the available VRAM and speed up training.",
"Alternatively, parameters of the model could be offloaded from VRAM to system RAM or even high speeds solid state drives [31].",
"These methodologies combined would allow a site-net implementation to scale to a local environment of arbitrary size.",
"More significant improvement in performance will come from better handling of the tensors in the model.",
"Zero padding used in the current model means that the largest crystal in the dataset that determines the VRAM and computational requirements, and all crystal structures must be scaled up to the largest member.",
"An implementation with full support for variable sized tensors will considerably reduce VRAM requirements, as it removes the need for zero-padding [32].",
"Specifically, for the Matbench band gap prediction task, with 100 atom supercells as used here, the VRAM requirement would be reduced by a factor of 5.",
"This would allow either training the same model 5 times faster, or training a model that is 5 times larger (e.g., 500 atoms).",
"These improvements will become increasingly large more significant with larger supercells.",
"This has further benefits, because atomic sites generated in supercells could then also be treated more efficiently.",
"Rather than calculating the attention weights explicitly on all atomic sites in the supercell, calculation of the attention weights and training could be performed for only the unique atomic sites in the initial minimal $P1$ unit cell.",
"Specifically, in the interaction features tensor $I_{i,j,f}$ , the length of $i$ would be equal to the number of atomic sites in the minimal $P1$ unit cell, and the length of $j$ would be equal to the total number of atoms in the supercell.",
"Finally, after Site-Net becomes more mature and the reasonable ranges of hyperparameters are outlined, the hyperparameter search space can likely be reduced, which will lead to much faster model training."
],
[
"Invariance under unit cell transformations",
"Every physical crystal structure can be represented using many possible unit cells or supercells.",
"For example, the choice of unit cell setting in triclinic crystal systems is not straightforward [33], and non-standard representations can be preferred in some circumstances (e.g., the use of hexagonal unit cells as opposed to primitive rhombohedral unit cells [34]).",
"Transforming between unit cells changes various parameters in the CIF, such as the atomic coordinates and unit cell parameters.",
"If these parameters form part of the input of a machine learning model, then the choice of unit cell can lead to different predictions.",
"This issue has gained recent attention in the literature and has prompted assessment of existing models [7], [35], [36].",
"The design of the Site-Net model ensures that model predictions do not change under translations and unimodular transformations, described below.",
"These transformations do not change the volume of the unit cell or supercell, but they nevertheless lead to unit cells that are very distinct (fig:invariance).",
"Importantly, the types and quantities of crystallographic sites remain unchanged by these transformations, so while the order of crystallographic sites may change, the same set of inputs $S_{i,f}$ and $I_{i,j,f}$ will be processed by a model that is invariant to permutations.",
"In the case of translation, which is more straightforward, the number of crystallographic sites $S^t_{i,f}$ in the translated unit cell and their identities remain the same, but their ordering might change.",
"Formally, it means that there is a permutation $\\pi $ over the sites such that $S_{i,f}$ and $S^t_{\\pi (i),f}$ are equal.",
"Furthermore, since the distances are computed under periodic boundary conditions (i.e., the distance between any two sites is always the distance between an atomic site and the closest site from any self or image unit cell), the resulting interaction features $I^t_{i,j,f}$ will be identical to $I_{i,j,f}$ after the rearrangement $I^t_{\\pi (i),\\pi (j),f}$ .",
"Thus, the tensors $B_{i,j,f}$ and $B^{t}_{i,j,f}$ , which constitute the only input to Site-Net, are identical up to a permutation.",
"Since all operations performed in Site-Net are permutation-invariant, we arrive at the same predictions.",
"The same reasoning applies in the case of unimodular transformations, where we show that the number of crystallographic sites and their identities are preserved.",
"A crystallographic lattice defined by the lattice vectors $V = [\\vec{a}, \\vec{b}, \\vec{c}]$ can be generated using different sets of vectors.",
"A classical result from lattice theory states that multiplication of $V$ by a unimodular matrix $U$ (i.e., a matrix with integer coefficients and the determinant $\\pm 1$ ) leads to vectors $V^\\prime = VU$ that also generate the initial lattice [37].",
"As the point lattice before and after transformation are identical, both unit cells (as fundamental domains) will have the same volume and contain a unique representative of every crystallographic site.",
"Therefore, the new sites $S^u_{i,f}$ of the unit cell after a unimodular transformation are identical to the original sites $S_{\\pi (i),f}$ after application of a suitable permutation $\\pi $ of indices.",
"Similarly to the case of translations, we can conclude that the the tensors $B_{i,j,f}$ and $B^{u}_{\\pi (i),\\pi (j),f}$ are the same, which leads to identical predictions produced by the Site-Net model.",
"Finally, it is important to note that Site-Net is, by design, not invariant to scale.",
"Site-Net is designed to update its predictions by incorporating increasingly long-range interactions.",
"Accordingly, as the supercell size is increased, attention heads will be able to examine more interactions at longer radial distances, and we expect convergence at some sufficiently long distance when all meaningful interactions are considered.",
"Figure: Translations, reflections, and unimodular transformations of the unit cell parameters do not change the computed interaction features.",
"We show several transformations of a hypothetical two-dimensional unit cell containing 3 crystallographic sites.",
"For sites on corners and edges, a position is chosen arbitrarily; equivalent choices are shown with transparency.Despite apparently distinct arrangements of the sites, the pairwise distances remain the same."
],
[
"Conclusions",
"We present Site-Net, a transformer model for learning structure–property relationships in extended inorganic solids.",
"Site-Net processes standard crystallographic information files , and uses a physically motivated representation of the crystal structure as a point set of atomic sites.",
"As many physical phenomena in extended inorganic solids arise from long-range interactions and features of the crystal structure, we build a large supercell to encode this information explicitly.",
"Critically, the set of atomic sites is directly ingested without any pre-defined connections, and the importance of interactions between all atomic sites is flexibly learned by the model for the prediction task presented.",
"The relevant structural information will differ between property prediction tasks, and the use of a custom global self-attention mechanism on all pairwise interactions of atomic sites allows Site-Net to identify important interactions and effectively deal with the all-to-all connectivity that would otherwise be overwhelming.",
"The attention mechanism in Site-Net works by iteratively replacing the atomic sites with context-enriched versions of themselves, which are created by aggregating the most important structural information from all other atomic sites in the crystal structure present in the supercell.",
"The use of attention in Site-Net allows interrogation of the learning by examining the weights assigned to interactions at different interatomic distances.",
"We show that for the band gap prediction task performed here, Site-Net learns from interactions that are beyond the nearest neighbour atomic sites, and that attention heads performing the attention calculations become specialized to deal with primarily short- or long-range interactions.",
"Further, training Site-Net where the attention has an artificial distance cutoff limit of 5 Å decreases model performance, confirming that including longer range interactions within a crystal structure meaningfully contributes to property predictions of extended inorganic materials.",
"We demonstrate the effectiveness of Site-Net through a band gap prediction task, as this task is heavily studied and commonly used as a benchmark for model performance.",
"As a proof of concept, we build small supercells of 100 atoms and train Site-Net using a single consumer graphics processing unit (GPU).",
"Site-Net achieves a mean absolute error (MAE) of 0.251 eV using the Matbench band gap regression dataset, and performance of the model is consistent across band gap values.",
"Even after only a preliminary hyperparameter search and using small supercells of 100 atoms, Site-Net demonstrates competitive performance with the highest performing algorithms on the Matbench leaderboard.",
"The performance of Site-Net is likely to improve following a more extensive hyperparameter search, and through the use of larger supercells.",
"Both paths to improvement can be easily accommodated through changes to the way calculations are handled internally as well as through the use of larger or parallel GPUs.",
"Importantly, we show that explicit incorporation of long-range interactions through the use of supercells can improve the performance of machine learning models that use crystal structure to predict properties of extended inorganic solids.",
"Given that many physical properties result from long-range features and/or the extended nature of a crystal structure, the performance of other models on many prediction tasks may be likewise improved through similar methods, particularly where the models rely solely on the primitive unit cell."
],
[
"Acknowledgements",
"Work was performed using Barkla, part of the High Performance Computing facilities at the University of Liverpool, UK.",
"The authors thank the Leverhulme Trust for funding via the Leverhulme Research Centre for Functional Materials Design.",
"MWG thanks the Ramsay Memorial Fellowships Trust for funding through a Ramsay Trust Memorial Fellowship.",
"Supporting information: Site-Net: Using global self-attention and real-space supercells to capture long-range interactions in crystal structures"
],
[
"Construction of roughly cubic supercells with a limited number of atoms",
"Site-net operates by recursively aggregating sites with their local environment, so it is critical that the local environment of an atomic site is well-behaved at radial distances examined by the attention heads.",
"Accordingly, there is a soft requirement to provide each atomic site in the crystal the largest local environment possible, so a supercell is created to explicitly include longer range interactions with images of the minimal $P1$ unit cell.",
"Ideally, there should be no edge effects or finite size effects owing to the construction of the supercell; the local environment of any atomic site within the minimal $P1$ unit cell should be equivalent to looking out into the infinite crystal structure.",
"In practice, the size of the model is limited by computational resources, and the distance at which these edge effects begin to contribute is defined here as the self-intersection limit, which is half the shortest distance from any site to its own image in a neighbouring unit cell (or supercell).",
"As an example, an attention head operating on an atomic site at the centre of an orthorhombic unit cell would examine all interactions out to the edge of the unit cell, after which there are no interactions to consider for attention along that direction, resulting in a self-intersection limit equal to half the shortest unit cell parameter.",
"It is possible that overtraining could result from the model learning the edge effects if making use of interactions beyond this range.",
"To maximize the self-intersection limit, all crystal structures are transformed to the largest possible supercell that is approximately cubic and contains an appropriate number of atoms – fewer than a specified limit.",
"We present below a simple algorithm for this task.",
"To explore how the supercells behave at different size limits, we build supercells of 100 atoms and 300 atoms for crystal structures within the Matbench band gap prediction dataset, and examine key features relevant to Site-Net (fig:atomsincell,fig:self-intersection,fig:cubicdeviation,fig:sizelimitmatbench).",
"In the current implementation of Site-Net, 300 atoms represents a rough maximum supercell size that could be handled by a single GPU.",
"For every crystal structure, we change the maximum size limit of the supercell and then examine (a) the resulting number of atoms (fig:atomsincell), (b) the self-intersection limit at which edge-effects will begin to contribute (fig:self-intersection), and (c) the deviation of the supercell from an ideal cube (fig:cubicdeviation).",
"Importantly, all features become more well-behaved with increasing supercell size, and a greater portion of the dataset becomes available for training (fig:sizelimitmatbench).",
"We now formally describe the transformation creating an approximately cubic supercell.",
"To begin, let $V^P$ be the matrix where each row is a basis vector of the minimal $P1$ unit cell.",
"We perform Gram–Schmidt orthogonalization procedure on $V^P$ to obtain the decomposition $V^P = RQ$ , where $R$ is the upper triangular matrix and $Q$ is an orthogonal matrix, i.e., $QQ^T = I$ .",
"Note that normalization is not performed, thus $R$ has values of 1 on the diagonal entries.",
"Given that $R^{-1}V^P = Q$ , the transformation $R^{-1}$ creates an orthogonal unit cell.",
"This is the orthogonalization component of our transformation.",
"Next, $Q$ is used to compute the scaling component of the supercell transformation, denoted as $S$ .",
"The shortest unit cell basis vector from $Q$ is iteratively incremented until any further incrementation would bring the number of atoms in the unit cell above the specified limit (100 in this work); $S$ is the diagonal matrix that performs the scaling of $Q$ , and the $i$ -th diagonal entry $s_{i,i}$ encodes the number of times the $i$ -th lattice vector should be repeated to form the supercell.",
"Finally, by combining $R^{-1}$ and $S$ into $SR^{-1}$ , we compute a transformation matrix that converts the minimal $P1$ unit cell $V^P$ into an approximately cubic supercell denoted $V^S$ .",
"Note that $R$ is invertible, as it has only values of 1 on the diagonal, so its determinant is 1, and its inverse is also an upper triangular matrix.",
"The combined transformation $SR^{-1}$ will have non-zero values on the diagonal; however, the matrix entries are likely to be non-integer, which is problematic because only non-singular integer matrices are guaranteed to create a valid supercell.",
"Accordingly, the entries of $SR^{-1}$ are rounded to the nearest non-zero integer to obtain the cubic supercell lattice parameters $C^S$ , where $V^S = \\lfloor SR^{-1}\\rceil V^P$ .",
"This whole procedure ensures the supercell $V^S$ is valid, approximately cubic, and contains less than a specified number of atoms.",
"Importantly, performing the orthogonalization and scaling together is much more versatile than performing these transformations serially, as rounding matrix elements does not need to be performed until the end of the process.",
"Effectively, performing both operations together means the minimal $P1$ unit cell $V^P$ can be tiled along directions not parallel to the unit cell basis vectors.",
"Accordingly, the final supercell is expected to be a better approximation of a cube, which will have the largest possible self-intersection limit.",
"top=2cm,bottom=2cm"
]
] | 2209.08190 |
[
[
"Generalized derivations of Hom-Jordan algebras"
],
[
"Abstract In this paper, we give some properties of generalized derivation algebras of Hom-Jordan algebras.",
"In particular, we show that $GDer(V) = QDer(V) + QC(V)$, the sum of the quasiderivation algebra and the quasicentroid.",
"We also prove that $QDer(V)$ can be embedded as derivations into a larger Hom-Jordan algebra.",
"General results on centroids of Hom-Jordan algebras are also developed in this paper."
],
[
"Introduction",
"In 1932, the physicist P. Jordan proposed a program to discover a new algebraic setting for quantum mechanics, so Jordan algebras were created in this proceeding.",
"Moreover, Jordan algebras were truned out to have illuminating connections with many areas of mathematics.",
"A.",
"A. Albert renamed them Jordan algebras and developed a successful structure theory for Jordan algebras over any field of characteristic zero in [1].",
"Hom-type algebras have been studied by many authors in [2], [3], [8], [5].",
"In general, Hom-type algebras are a kind of algebras which are obtained by twisting the identity defining the structure with one or several linear maps(called the twisted maps) on the basis of the original algebras.",
"The concept of Hom-Jordan algebras was first introduced by A. Makhlouf in his paper [9].",
"He introduced a Hom-Jordan algebra and showed that it fits with the Hom-associative structure, that is a Hom-associative algebra leads to a Hom-Jordan algebra by considering a plus algebra.",
"As is well known, derivations and generalized derivations play an important role in the research of structure and property in algebraic system.",
"The research on generalized derivations was started by Leger and Luks in [7].",
"They gave many properties about generalized derivations of Lie algebras.",
"From then on, many authors generalized the results in [7] to other algebras.",
"R. X. Zhang and Y.",
"Z. Zhang generalized it to Lie superalgebras in [13].",
"And L. Y. Chen, Y. Ma and L. Ni generalized the results to Lie color algebras successfully in [4].",
"J. Zhou, Y. J. Niu and L. Y. chen generalized the above results to Hom-Lie algebras in [14].",
"As for Jordan algebras, A. I. Shestakov generalized the results into Jordan superalgebras in [12].",
"He showed that for semi-simple Jordan algebras over any field of characteristic 0 or simple Jordan algebras over any field of characteristic other than 2, the generalized derivations are the sum of the derivations and the centroids with some exceptions.",
"The purpose of this paper is to generalized some beautiful results in [14] and [11] to Hom-Jordan algebras.",
"We proceed as follow.",
"In Section , we recall some basic definitions and propositions which will be used in what follows.",
"In Section , we'll give some properties about generalized derivations of Hom-Jordan algebras and their Hom-subalgebras.",
"In Section , we show that quasiderivations of a Hom-Jordan algebra can be embedded as derivations into a larger Hom-Jordan algebra and obtain a direct sum decomposition of $Der(\\breve{V})$ when the centralizer of $V$ equals to zero.",
"In Section , we give some propositions with respect to the centroids of Hom-Jordan algebras."
],
[
"Preliminaries",
"Definition 2.1 [10] An algebra $J$ over a field $\\rm {F}$ is a Jordan algebra satisfying for any $x, y \\in J$ , $x \\circ y = y \\circ x$ ; $(x^{2} \\circ y) \\circ x = x^{2} \\circ (y \\circ x)$ .",
"Definition 2.2 [9] A Hom-Jordan algebra over a field $\\rm {F}$ is a triple $(V, \\mu , \\alpha )$ consisting of a linear space $V$ , a bilinear map $\\mu : V \\times V \\rightarrow V$ which is commutative and a linear map $\\alpha : V \\rightarrow V$ satisfying for any $x, y \\in V$ , $\\mu (\\alpha ^{2}(x), \\mu (y, \\mu (x, x))) = \\mu (\\mu (\\alpha (x), y), \\alpha (\\mu (x, x))),$ where $\\alpha ^{2} = \\alpha \\circ \\alpha $ .",
"Definition 2.3 A Hom-Jordan algebra $(V, \\mu , \\alpha )$ is called multiplicative if for any $x, y \\in V$ , $\\alpha (\\mu (x, y)) = \\mu (\\alpha (x), \\alpha (y))$ .",
"Definition 2.4 [6] A subspace $W \\subseteq V$ is a Hom-subalgebra of $(V, \\mu , \\alpha )$ if $\\alpha (W) \\subseteq W$ and $\\mu (x, y) \\in W,\\quad \\forall x, y \\in W.$ Definition 2.5 [6] A subspace $W \\subseteq V$ is a Hom-ideal of $(V, \\mu , \\alpha )$ if $\\alpha (W) \\subseteq W$ and $\\mu (x, y) \\in W,\\quad \\forall x \\in W, y \\in V.$ Definition 2.6 [6] Let $(V, \\mu , \\alpha )$ and $(V^{^{\\prime }}, \\mu ^{^{\\prime }}, \\beta )$ be two Hom-Jordan algebras.",
"A linear map $\\phi : V \\rightarrow V^{^{\\prime }}$ is said to be a homomorphism of Hom-Jordan algebras if $\\phi (\\mu (x, y)) = \\mu ^{^{\\prime }}(\\phi (x), \\phi (y))$ ; $\\phi \\circ \\alpha = \\beta \\circ \\phi $ .",
"Lemma 2.7 [6] Let $(V, \\mu , \\alpha )$ be a Hom-Jordan algebra over a field $\\rm {F}$ , we define a subspace $\\mathcal {W}$ of $End(V)$ where $\\mathcal {W} = \\lbrace w \\in End(V) | w \\circ \\alpha = \\alpha \\circ w\\rbrace $ , $\\sigma : \\mathcal {W} \\rightarrow \\mathcal {W}$ is a linear map satisfying $\\sigma (w) = \\alpha \\circ w$ .",
"A triple $(\\mathcal {W}, \\nu , \\sigma )$ , where the multiplication $\\nu : \\mathcal {W} \\times \\mathcal {W} \\rightarrow \\mathcal {W}$ is defined for $w_{1}, w_{2} \\in \\mathcal {W}$ by $\\nu (w_{1}, w_{2}) = w_{1} \\circ w_{2} + w_{2} \\circ w_{1},$ is a Hom-Jordan algebra over $\\rm {F}$ .",
"A triple $(\\mathcal {W}, \\nu ^{^{\\prime }}, \\sigma )$ , where the multiplication $\\nu ^{^{\\prime }} : \\mathcal {W} \\times \\mathcal {W} \\rightarrow \\mathcal {W}$ is defined for $w_{1}, w_{2} \\in \\mathcal {W}$ by $\\nu ^{^{\\prime }}(w_{1}, w_{2}) = w_{1} \\circ w_{2} - w_{2} \\circ w_{1},$ is a Hom-Lie algebra over $\\rm {F}$ .",
"Definition 2.8 [6] For any nonnegative integer $k$ , a linear map $D : V \\rightarrow V$ is called an $\\alpha ^{k}$ -derivation of the Hom-Jordan algebra $(V, \\mu , \\alpha )$ , if $D \\circ \\alpha = \\alpha \\circ D$ ; $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y)),\\quad \\forall x, y \\in V$ .",
"Lemma 2.9 [6] For any $D \\in Der_{\\alpha ^{k}}(V)$ and $D^{^{\\prime }} \\in Der_{\\alpha ^{s}}(V)$ , define their commutator $\\nu ^{^{\\prime }}(D, D^{^{\\prime }})$ as usual: $\\nu ^{^{\\prime }}(D, D^{^{\\prime }}) = D \\circ D^{^{\\prime }} - D^{^{\\prime }} \\circ D.$ Then $\\nu ^{^{\\prime }}(D, D^{^{\\prime }}) \\in Der_{\\alpha ^{k + s}}(V)$ .",
"Let $(V, \\mu , \\alpha )$ be a multiplicative Hom-Jordan algebra.",
"We denote the set of all $\\alpha ^{k}$ -derivations by $Der_{\\alpha ^{k}}(V)$ , then $Der(V) = \\dotplus _{k \\ge 0}Der_{\\alpha ^{k}}(V)$ provided with the commutator and the following map $\\sigma : Der(V) \\rightarrow Der(V);\\quad \\sigma (D) = \\alpha \\circ D$ is a Hom-subalgebra of $(\\mathcal {W}, \\nu ^{^{\\prime }}, \\sigma )$ according to Lemma REF and is called the derivation algebra of $(V, \\mu , \\alpha )$ .",
"Definition 2.10 For any nonnegative integer $k$ , a linear map $D \\in End(V)$ is said to be a generalized $\\alpha ^{k}$ -derivation of $(V, \\mu , \\alpha )$ , if there exist two linear maps $D^{^{\\prime }}, D^{^{\\prime \\prime }} \\in End(V)$ such that $D \\circ \\alpha = \\alpha \\circ D,\\; D^{^{\\prime }} \\circ \\alpha = \\alpha \\circ D^{^{\\prime }},\\; D^{^{\\prime \\prime }} \\circ \\alpha = \\alpha \\circ D^{^{\\prime \\prime }}$ ; $\\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D^{^{\\prime }}(y)) = D^{^{\\prime \\prime }}(\\mu (x, y)),\\quad \\forall x, y \\in V$ .",
"Definition 2.11 For any nonnegative integer $k$ , a linear map $D \\in End(V)$ is said to be an $\\alpha ^{k}$ -quasiderivation of $(V, \\mu , \\alpha )$ , if there exist a linear map $D^{^{\\prime }}\\in End(V)$ such that $D \\circ \\alpha = \\alpha \\circ D,\\; D^{^{\\prime }} \\circ \\alpha = \\alpha \\circ D^{^{\\prime }}$ ; $\\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y)) = D^{^{\\prime }}(\\mu (x, y)),\\quad \\forall x, y \\in V$ .",
"Let $GDer_{\\alpha ^{k}}(V)$ and $QDer_{\\alpha ^{k}}(V)$ be the sets of generalized $\\alpha ^{k}$ -derivations and of $\\alpha ^{k}$ -quasiderivations, respectively.",
"That is $GDer(V) = \\dotplus _{k \\ge 0}GDer_{\\alpha ^{k}}(V),\\quad QDer(V) = \\dotplus _{k \\ge 0}QDer_{\\alpha ^{k}}(V).$ It's easy to verify that both $GDer(V)$ and $QDer(V)$ are Hom-subalgebras of $(\\mathcal {W}, \\nu ^{^{\\prime }}, \\sigma )$ (See Proposition REF ).",
"Definition 2.12 For any nonnegative integer $k$ , a linear map $D \\in End(V)$ is said to be an $\\alpha ^{k}$ -centroid of $(V, \\mu , \\alpha )$ , if $D \\circ \\alpha = \\alpha \\circ D$ ; $\\mu (D(x), \\alpha ^{k}(y)) = \\mu (\\alpha ^{k}(x), D(y)) = D(\\mu (x, y)),\\quad \\forall x, y \\in V$ .",
"Definition 2.13 For any nonnegative integer $k$ , a linear map $D \\in End(V)$ is said to be an $\\alpha ^{k}$ -quasicentroid of $(V, \\mu , \\alpha )$ , if $D \\circ \\alpha = \\alpha \\circ D$ ; $\\mu (D(x), \\alpha ^{k}(y)) = \\mu (\\alpha ^{k}(x), D(y)),\\quad \\forall x, y \\in V$ .",
"Let $C_{\\alpha ^{k}}(V)$ and $QC_{\\alpha ^{k}}(V)$ be the sets of $\\alpha ^{k}$ -centroids and of $\\alpha ^{k}$ -quasicentroids, respectively.",
"That is $C(V) = \\dotplus _{k \\ge 0}C_{\\alpha ^{k}}(V),\\quad QC(V) = \\dotplus _{k \\ge 0}QC_{\\alpha ^{k}}(V).$ Definition 2.14 For any nonnegative integer $k$ , a linear map $D \\in End(V)$ is said to be an $\\alpha ^{k}$ -central derivation of $(V, \\mu , \\alpha )$ , if $D \\circ \\alpha = \\alpha \\circ D$ ; $\\mu (D(x), \\alpha ^{k}(y)) = D(\\mu (x, y)) = 0,\\quad \\forall x, y \\in V$ .",
"We denote the set of all $\\alpha ^{k}$ -central derivations by $ZDer_{\\alpha ^{k}}(V)$ , then $ZDer(V) = \\dotplus _{k \\ge 0}ZDer_{\\alpha ^{k}}(V)$ .",
"According to the definitions, it's easy to show that $ZDer(V) \\subseteq Der(V) \\subseteq QDer(V) \\subseteq GDer(V) \\subseteq End(V)$ .",
"Definition 2.15 Let $(V, \\mu , \\alpha )$ be a Hom-Jordan algebra.",
"If $Z(V) = \\lbrace x \\in V | \\mu (x, y) = 0,\\quad \\forall y \\in V\\rbrace $ , then $Z(V)$ is called the centralizer of $(V, \\mu , \\alpha )$ ."
],
[
"Generalized derivation algebras and their subalgebras",
"At first, we give some basic properties of center derivation algebras, quasiderivation algebras and generalized derivation algebras of a Hom-Jordan algebra.",
"Proposition 3.1 Suppose that $(V, \\mu , \\alpha )$ is a multiplicative Hom-Jordan algebra.",
"Then the following statements hold: $GDer(V)$ , $QDer(V)$ and $C(V)$ are Hom-subalgebras of $(\\mathcal {W}, \\nu ^{^{\\prime }}, \\sigma )$ ; $ZDer(V)$ is a Hom-ideal of $Der(V)$ .",
"(1).",
"Suppose that $D_{1} \\in GDer_{\\alpha ^{k}}(V)$ , $D_{2} \\in GDer_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , $&\\mu (\\sigma (D_{1})(x), \\alpha ^{k + 1}(y)) = \\mu (\\alpha \\circ D_{1}(x), \\alpha ^{k + 1}(y)) = \\alpha (\\mu (D_{1}(x), \\alpha ^{k}(y)))\\\\&= \\alpha (D_{1}^{^{\\prime \\prime }}(\\mu (x, y)) - \\mu (\\alpha ^{k}(x), D_{1}^{^{\\prime }}(y))) = \\sigma (D_{1}^{^{\\prime \\prime }})(\\mu (x, y)) - \\mu (\\alpha ^{k + 1}(x), \\sigma (D_{1}^{^{\\prime }})(y)).$ Since $\\sigma (D_{1}^{^{\\prime \\prime }}), \\sigma (D_{1}^{^{\\prime }}) \\in End(V)$ , we have $\\sigma (D_{1}) \\in GDer_{\\alpha ^{k + 1}}(V)$ .",
"$&\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y))\\\\&= \\mu (D_{1} \\circ D_{2}(x), \\alpha ^{k + s}(y)) - \\mu (D_{2} \\circ D_{1}(x), \\alpha ^{k + s}(y))\\\\&= D_{1}^{^{\\prime \\prime }}(\\mu (D_{2}(x), \\alpha ^{s}(y))) - \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}^{^{\\prime }}(\\alpha ^{s}(y))) - D_{2}^{^{\\prime \\prime }}(\\mu (D_{1}(x), \\alpha ^{k}(y))) +\\\\ &\\mu (\\alpha ^{s}(D_{1}(x)), D_{2}^{^{\\prime }}(\\alpha ^{k}(y)))\\\\&= D_{1}^{^{\\prime \\prime }}(D_{2}^{^{\\prime \\prime }}(\\mu (x, y)) - \\mu (\\alpha ^{s}(x), D_{2}^{^{\\prime }}(y))) - \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}^{^{\\prime }}(\\alpha ^{s}(y)))\\\\&- D_{2}^{^{\\prime \\prime }}(D_{1}^{^{\\prime \\prime }}(\\mu (x, y)) - \\mu (\\alpha ^{k}(x), D_{1}^{^{\\prime }}(y))) + \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}^{^{\\prime }}(\\alpha ^{k}(y)))\\\\&= D_{1}^{^{\\prime \\prime }} \\circ D_{2}^{^{\\prime \\prime }}(\\mu (x, y)) - D_{1}^{^{\\prime \\prime }}(\\mu (\\alpha ^{s}(x), D_{2}^{^{\\prime }}(y))) - \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}^{^{\\prime }}(\\alpha ^{s}(y)))\\\\&- D_{2}^{^{\\prime \\prime }} \\circ D_{1}^{^{\\prime \\prime }}(\\mu (x, y)) + D_{2}^{^{\\prime \\prime }}(\\mu (\\alpha ^{k}(x), D_{1}^{^{\\prime }}(y))) + \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}^{^{\\prime }}(\\alpha ^{k}(y)))\\\\&= D_{1}^{^{\\prime \\prime }} \\circ D_{2}^{^{\\prime \\prime }}(\\mu (x, y)) - \\mu (D_{1}(\\alpha ^{s}(x)), \\alpha ^{k}(D_{2}^{^{\\prime }}(y))) - \\mu (\\alpha ^{k + s}(x), D_{1}^{^{\\prime }}(D_{2}^{^{\\prime }}(y)))\\\\&- \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}^{^{\\prime }}(\\alpha ^{s}(y))) - D_{2}^{^{\\prime \\prime }} \\circ D_{1}^{^{\\prime \\prime }}(\\mu (x, y)) + \\mu (D_{2}(\\alpha ^{k}(x)), \\alpha ^{s}(D_{1}^{^{\\prime }}(y)))\\\\&+ \\mu (\\alpha ^{k + s}(x), D_{2}^{^{\\prime }}(D_{1}^{^{\\prime }}(y))) + \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}^{^{\\prime }}(\\alpha ^{k}(y)))\\\\&= D_{1}^{^{\\prime \\prime }} \\circ D_{2}^{^{\\prime \\prime }}(\\mu (x, y)) - D_{2}^{^{\\prime \\prime }} \\circ D_{1}^{^{\\prime \\prime }}(\\mu (x, y)) - \\mu (\\alpha ^{k + s}(x), D_{1}^{^{\\prime }}(D_{2}^{^{\\prime }}(y)))\\\\&+ \\mu (\\alpha ^{k + s}(x), D_{2}^{^{\\prime }}(D_{1}^{^{\\prime }}(y)))\\\\&= \\nu ^{^{\\prime }}(D_{1}^{^{\\prime \\prime }}, D_{2}^{^{\\prime \\prime }})(\\mu (x, y)) - \\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}^{^{\\prime }}, D_{2}^{^{\\prime }})(y)).$ Since $\\nu ^{^{\\prime }}(D_{1}^{^{\\prime \\prime }}, D_{2}^{^{\\prime \\prime }}), \\nu ^{^{\\prime }}(D_{1}^{^{\\prime }}, D_{2}^{^{\\prime }}) \\in End(V)$ , we have $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in GDer_{\\alpha ^{k + s}}(V)$ .",
"Therefore, $GDer(V)$ is a Hom-subalgebra of $(\\mathcal {W}, \\nu ^{^{\\prime }}, \\sigma )$ .",
"Similarly, we have $QDer(V)$ is a Hom-subalgebra of $(\\mathcal {W}, \\nu ^{^{\\prime }}, \\sigma )$ .",
"Suppose that $D_{1} \\in C_{\\alpha ^{k}}(V)$ , $D_{2} \\in C_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , $\\sigma (D_{1})(\\mu (x, y)) = \\alpha \\circ D_{1}(\\mu (x, y)) = \\alpha (\\mu (D_{1}(x), \\alpha ^{k}(y))) = \\mu (\\sigma (D_{1})(x), \\alpha ^{k + 1}(y)).$ Similarly, we have $\\sigma (D_{1})(\\mu (x, y)) = \\mu (\\alpha ^{k + 1}(x), \\sigma (D_{1})(y))$ .",
"Hence, we have $\\sigma (D_{1}) \\in C_{\\alpha ^{k + 1}}(V)$ .",
"$&\\nu ^{^{\\prime }}(D_{1}, D_{2})(\\mu (x, y))\\\\&= D_{1} \\circ D_{2}(\\mu (x, y)) - D_{2} \\circ D_{1}(\\mu (x, y))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y))) - D_{2}(\\mu (D_{1}(x), \\alpha ^{k}(y)))\\\\&= \\mu (D_{1}(D_{2}(x)), \\alpha ^{k + s}(y)) - \\mu (D_{2}(D_{1}(x)), \\alpha ^{k + s}(y))\\\\&= \\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y)).$ Similarly, we have $\\nu ^{^{\\prime }}(D_{1}, D_{2})(\\mu (x, y)) = \\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y))$ .",
"Hence, we have $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in C_{\\alpha ^{k + s}}(V)$ .",
"Therefore, we have $C(V)$ is a Hom-subalgebras of $(\\mathcal {W}, \\nu ^{^{\\prime }}, \\sigma )$ .",
"(2).",
"Suppose that $D_{1} \\in ZDer_{\\alpha ^{k}}(V)$ , $D_{2} \\in Der_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , $\\sigma (D_{1})(\\mu (x, y)) = \\alpha \\circ D_{1}(\\mu (x, y)) = 0.$ $\\sigma (D_{1})(\\mu (x, y)) = \\alpha \\circ D_{1}(\\mu (x, y)) = \\alpha (\\mu (D_{1}(x), \\alpha ^{k}(y))) = \\mu (\\sigma (D_{1})(x), \\alpha ^{k + 1}(y)).$ Hence, $\\sigma (D_{1}) \\in ZDer_{\\alpha ^{k + 1}}(V)$ .",
"$&\\nu ^{^{\\prime }}(D_{1}, D_{2})(\\mu (x, y))\\\\&= D_{1} \\circ D_{2}(\\mu (x, y)) - D_{2} \\circ D_{1}(\\mu (x, y))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y)) + \\mu (\\alpha ^{s}(x), D_{2}(y))) = 0.$ $&\\nu ^{^{\\prime }}(D_{1}, D_{2})(\\mu (x, y))\\\\&= D_{1} \\circ D_{2}(\\mu (x, y)) - D_{2} \\circ D_{1}(\\mu (x, y))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y)) + \\mu (\\alpha ^{s}(x), D_{2}(y))) - D_{2}(\\mu (D_{1}(x), \\alpha ^{k}(y)))\\\\&= \\mu (D_{1}(D_{2}(x)), \\alpha ^{k + s}(y)) + \\mu (D_{1}(\\alpha ^{s}(x)), \\alpha ^{k}(D_{2}(y))) - \\mu (D_{2}(D_{1}(x)), \\alpha ^{k + s}(y))\\\\&- \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y)))\\\\&= \\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y)).$ Hence, we have $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in ZDer_{\\alpha ^{k + s}}(V)$ .",
"Therefore, $ZDer(V)$ is a Hom-ideal of $Der(V)$ .",
"Proposition 3.2 Let $(V, \\mu , \\alpha )$ be a multiplicative Hom-Jordan algebra, then $\\nu ^{^{\\prime }}(Der(V), C(V)) \\subseteq C(V)$ ; $\\nu ^{^{\\prime }}(QDer(V), QC(V)) \\subseteq QC(V)$ ; $\\nu ^{^{\\prime }}(QC(V), QC(V)) \\subseteq QDer(V)$ ; $C(V) \\subseteq QDer(V)$ ; $QDer(V) + QC(V) \\subseteq GDer(V)$ ; $C(V) \\circ Der(V) \\subseteq Der(V)$ .",
"(1).",
"Suppose that $D_{1} \\in Der_{\\alpha ^{k}}(V)$ , $D_{2} \\in C_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , we have $&\\nu ^{^{\\prime }}(D_{1}, D_{2})(\\mu (x, y))\\\\&= D_{1} \\circ D_{2}(\\mu (x, y)) - D_{2} \\circ D_{1}(\\mu (x, y))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y))) - D_{2}(\\mu (D_{1}(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D_{1}(y)))\\\\&= \\mu (D_{1}(D_{2}(x)), \\alpha ^{k + s}(y)) + \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y))) - \\mu (D_{2}(D_{1}(x)), \\alpha ^{k + s}(y))\\\\&- \\mu (D_{2}(\\alpha ^{k}(x)), \\alpha ^{s}(D_{1}(y)))\\\\&= \\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y)).$ Similarly, we have $\\nu ^{^{\\prime }}(D_{1}, D_{2})(\\mu (x, y)) = \\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y))$ .",
"Hence, $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in C_{\\alpha ^{k + s}}(V)$ .",
"Therefore $\\nu ^{^{\\prime }}(Der(V), C(V)) \\subseteq C(V)$ .",
"(2).",
"Suppose that $D_{1} \\in QDer_{\\alpha ^{k}}(V)$ , $D_{2} \\in QC_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , we have $&\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y))\\\\&= \\mu (D_{1} \\circ D_{2}(x), \\alpha ^{k + s}(y)) - \\mu (D_{2} \\circ D_{1}(x), \\alpha ^{k + s}(y))\\\\&= D_{1}^{^{\\prime }}(\\mu (D_{2}(x), \\alpha ^{s}(y))) - \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y))) - \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y)))\\\\&= D_{1}^{^{\\prime }}(\\mu (\\alpha ^{s}(x), D_{2}(y))) - \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y))) - \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y)))\\\\&= \\mu (D_{1}(\\alpha ^{s}(x)), \\alpha ^{k}(D_{2}(y))) + \\mu (\\alpha ^{k + s}(x), D_{1}(D_{2}(y))) - \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y)))\\\\&- \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y)))\\\\&= \\mu (\\alpha ^{k + s}(x), D_{1}(D_{2}(y))) - \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y)))\\\\&= \\mu (\\alpha ^{k + s}(x), D_{1}(D_{2}(y))) - \\mu (D_{2}(\\alpha ^{k}(x)), \\alpha ^{s}(D_{1}(y)))\\\\&= \\mu (\\alpha ^{k + s}(x), D_{1}(D_{2}(y))) - \\mu (\\alpha ^{k + s}(x), D_{2}(D_{1}(y)))\\\\&= \\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y)).$ Hence, we have $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in QC_{\\alpha ^{k + s}}(V)$ .",
"So $\\nu ^{^{\\prime }}(QDer(V), QC(V)) \\subseteq QC(V)$ .",
"(3).",
"Suppose that $D_{1} \\in QC_{\\alpha ^{k}}(V)$ , $D_{2} \\in QC_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , we have $&\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y))\\\\&= \\mu (D_{1} \\circ D_{2}(x), \\alpha ^{k + s}(y)) - \\mu (D_{2} \\circ D_{1}(x), \\alpha ^{k + s}(y))\\\\&= \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y))) - \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y)))\\\\&= \\mu (D_{2}(\\alpha ^{k}(x)), \\alpha ^{s}(D_{1}(y))) - \\mu (D_{1}(\\alpha ^{s}(x)), \\alpha ^{k}(D_{2}(y)))\\\\&= \\mu (\\alpha ^{k + s}(x), D_{2}(D_{1}(y))) - \\mu (\\alpha ^{k + s}(x), D_{1}(D_{2}(y)))\\\\&= -\\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y)),$ i.e., $\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y)) + \\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y)) = 0$ .",
"Hence, we have $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in QDer_{\\alpha ^{k + s}}(V)$ , which implies that $\\nu ^{^{\\prime }}(QC(V), QC(V)) \\subseteq QDer(V)$ .",
"(4).",
"Suppose that $D \\in C_{\\alpha ^{k}}(V)$ .",
"Then for any $x, y \\in V$ , we have $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) = \\mu (\\alpha ^{k}(x), D(y)).$ Hence, we have $\\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y)) = 2D(\\mu (x, y)),$ which implies that $D \\in QDer_{\\alpha ^{k}}(V)$ .",
"So $C(V) \\subseteq QDer(V)$ .",
"(5).",
"Suppose that $D_{1} \\in QDer_{\\alpha ^{k}}(V)$ , $D_{2} \\in QC_{\\alpha ^{k}}(V)$ .",
"Then for any $x, y \\in V$ , we have $&\\mu ((D_{1} + D_{2})(x), \\alpha ^{k}(y))\\\\&= \\mu (D_{1}(x), \\alpha ^{k}(y)) + \\mu (D_{2}(x), \\alpha ^{k}(y))\\\\&= D_{1}^{^{\\prime }}(\\mu (x, y)) - \\mu (\\alpha ^{k}(x), D_{1}(y)) + \\mu (\\alpha ^{k}(x), D_{2}(y))\\\\&= D_{1}^{^{\\prime }}(\\mu (x, y)) - \\mu (\\alpha ^{k}(x), (D_{1} - D_{2})(y)).$ Since $D_{1}^{^{\\prime }}$ , $D_{1} - D_{2} \\in End(V)$ , we have $D_{1} + D_{2} \\in GDer_{\\alpha ^{k}}(V)$ .",
"So $QDer(V) + QC(V) \\subseteq GDer(V)$ .",
"(6).",
"Suppose that $D_{1} \\in C_{\\alpha ^{k}}(V)$ , $D_{2} \\in Der_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , we have $&D_{1} \\circ D_{2}(\\mu (x, y))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y)) + \\mu (\\alpha ^{s}(x), D_{2}(y)))\\\\&= \\mu (D_{1}(D_{2}(x)), \\alpha ^{k + s}(y)) + \\mu (\\alpha ^{k + s}(x), D_{1}(D_{2}(y))),$ which implies that $D_{1} \\circ D_{2} \\in Der_{\\alpha ^{k + s}}(V)$ .",
"So $C(V) \\circ Der(V) \\subseteq Der(V)$ .",
"Theorem 3.3 Suppose that $(V, \\mu , \\alpha )$ is a multiplicative Hom-Jordan algebra.",
"Then $GDer(V) = QDer(V) + QC(V)$ .",
"According to Proposition REF (5), we need only to show that $GDer(V) \\subseteq QDer(V) + QC(V)$ .",
"Suppose that $D \\in GDer_{\\alpha ^{k}}(V)$ .",
"Then there exist $D^{^{\\prime }},\\; D^{^{\\prime \\prime }} \\in End(V)$ such that $D^{^{\\prime \\prime }}(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D^{^{\\prime }}(y)),\\quad \\forall x, y \\in V.$ Since $\\mu $ is commutative, we have $D^{^{\\prime \\prime }}(\\mu (y, x)) = \\mu (D^{^{\\prime }}(y), \\alpha ^{k}(x)) + \\mu (\\alpha ^{k}(y), D(x)),\\quad \\forall x, y \\in V,$ which implies that $D^{^{\\prime }} \\in GDer_{\\alpha ^{k}}(V)$ .",
"Moreover, we have $&\\mu \\left(\\frac{D + D^{^{\\prime }}}{2}(x), \\alpha ^{k}(y)\\right) + \\mu \\left(\\alpha ^{k}(x), \\frac{D + D^{^{\\prime }}}{2}(y)\\right)\\\\&= \\frac{1}{2}(\\mu (D(x), \\alpha ^{k}(y)) + \\mu (D^{^{\\prime }}(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y)) + \\mu (\\alpha ^{k}(x), D^{^{\\prime }}(y)))\\\\&= D^{^{\\prime \\prime }}(\\mu (x, y)),$ which implies $\\frac{D + D^{^{\\prime }}}{2} \\in QDer_{\\alpha ^{k}}(V)$ .",
"$&\\mu \\left(\\frac{D - D^{^{\\prime }}}{2}(x), \\alpha ^{k}(y)\\right)\\\\&= \\frac{1}{2}(\\mu (D(x), \\alpha ^{k}(y)) - \\mu (D^{^{\\prime }}(x), \\alpha ^{k}(y)))\\\\&= \\frac{1}{2}(D^{^{\\prime \\prime }}(\\mu (x, y)) - \\mu (\\alpha ^{k}(x), D^{^{\\prime }}(y)) - D^{^{\\prime \\prime }}(\\mu (x, y)) + \\mu (\\alpha ^{k}(x), D(y)))\\\\&= \\mu \\left(\\alpha ^{k}(x), \\frac{D - D^{^{\\prime }}}{2}(y)\\right),$ which implies that $\\frac{D - D^{^{\\prime }}}{2} \\in QC_{\\alpha ^{k}}(V)$ .",
"Hence, $D = \\frac{D + D^{^{\\prime }}}{2} + \\frac{D - D^{^{\\prime }}}{2} \\in QDer(V) + QC(V)$ , i.e., $GDer(V) \\subseteq QDer(V) + QC(V)$ .",
"Therefore, $GDer(V) = QDer(V) + QC(V)$ .",
"Proposition 3.4 Suppose that $(V, \\mu , \\alpha )$ is a multiplicative Hom-Jordan algebra where $V$ can be decomposed into the direct sum of two Hom-ideals, i.e., $V = V_{1} \\oplus V_{2}$ and $\\alpha $ is surjective.",
"Then we have $Z(V) = Z(V_{1}) \\oplus Z(V_{2})$ ; If $Z(V) = \\lbrace 0\\rbrace $ , then we have $Der(V) = Der(V_{1}) \\oplus Der(V_{2})$ ; $GDer(V) = GDer(V_{1}) \\oplus GDer(V_{2})$ ; $QDer(V) = QDer(V_{1}) \\oplus QDer(V_{2})$ ; $C(V) = C(V_{1}) \\oplus C(V_{2})$ .",
"(1).",
"Obviously, $Z(V_{1}) \\cap Z(V_{2}) = \\lbrace 0\\rbrace $ .",
"And it's easy to show that $Z(V_{i})(i = 1, 2)$ are Hom-ideals of $Z(V)$ .",
"For all $z_{i} \\in Z(V_{i})(i = 1, 2)$ , take $x = x_{1} + x_{2}$ where $x_{1} \\in J_{1}$ , $x_{2} \\in J_{2}$ .",
"We have $\\mu (z_{1} + z_{2}, x) = \\mu (z_{1} + z_{2}, x_{1} + x_{2}) = \\mu (z_{1}, x_{1}) + \\mu (z_{2}, x_{2}),$ since $z_{i} \\in Z(V_{i})(i = 1, 2)$ , we have $\\mu (z_{i}, x_{i}) = 0,\\; i = 1, 2.$ Hence, $\\mu (z_{1} + z_{2}, x) = 0,$ which implies that $z_{1} + z_{2} \\in Z(V)$ , i.e., $Z(V_{1}) \\oplus Z(V_{2}) \\subseteq Z(V)$ .",
"On the other hand, for all $a \\in Z(V)$ , suppose that $a = a_{1} + a_{2}$ where $a_{i} \\in V_{i}(i = 1, 2)$ .",
"Then for all $x_{1} \\in V_{1}$ , $\\mu (a_{1}, x_{1}) = \\mu (a - a_{2}, x_{1}) = \\mu (a, x_{1})$ since $a \\in Z(V)$ , we have $\\mu (a, x_{1}) = 0.$ Hence, $\\mu (a_{1}, x_{1}) = 0,$ which implies that $a_{1} \\in Z(V_{1})$ .",
"Similarly, we have $a_{2} \\in Z(V_{2})$ .",
"Hence, $Z(V) \\subseteq Z(V_{1}) \\oplus Z(V_{2})$ .",
"Therefore, we have $Z(V) = Z(V_{1}) \\oplus Z(V_{2})$ .",
"(2).",
"$\\bf {Step 1}$ .",
"We'll show that $\\forall i = 1, 2$ , $D(V_{i}) \\subseteq V_{i},\\quad \\forall D \\in Der_{\\alpha ^{k}}(V)(k \\ge 0)$ .",
"Suppose that $x_{i} \\in V_{i}$ , then $\\mu (D(x_{1}), \\alpha ^{k}(x_{2})) = D(\\mu (x_{1}, x_{2})) - \\mu (\\alpha ^{k}(x_{1}), D(x_{2})) \\in V_{1} \\cap V_{2} = 0,$ since $V_{1}$ , $V_{2}$ are Hom-ideals of $(V, \\mu , \\alpha )$ .",
"Suppose that $D(x_{1}) = u_{1} + u_{2}$ where $u_{1} \\in V_{1}$ , $u_{2} \\in V_{2}$ .",
"Then $\\mu (u_{2}, \\alpha ^{k}(x_{2})) = \\mu (u_{1} + u_{2}, \\alpha ^{k}(x_{2})) = \\mu (D(x_{1}), \\alpha ^{k}(x_{2})) = 0,$ which implies that $u_{2} \\in Z(V_{2})$ since $\\alpha $ is surjective.",
"Note that $Z(V) = \\lbrace 0\\rbrace $ , we have $Z(V_{i}) = \\lbrace 0\\rbrace (i = 1, 2)$ .",
"Hence, $u_{2} = 0$ .",
"That is to say $D(x_{1}) \\in V_{1}$ .",
"Similarly, we have $D(x_{2}) \\in V_{2}$ .",
"$\\bf {Step 2}$ .",
"We'll show that $Der_{\\alpha ^{k}}(V_{1}) \\dotplus Der_{\\alpha ^{k}}(V_{2}) \\subseteq Der_{\\alpha ^{k}}(V)(k \\ge 0)$ .",
"For any $D \\in Der_{\\alpha ^{k}}(V_{1})$ , we extend it to a linear map on $V$ as follow $D(x_{1} + x_{2}) = D(x_{1}),\\quad \\forall x_{1} \\in V_{1}, x_{2} \\in V_{2}.$ Then for any $x, y \\in V$ , suppose that $x = x_{1} + x_{2}$ , $y = y_{1} + y_{2} \\in V$ , where $x_{1}, y_{1} \\in V_{1}$ , $x_{2}, y_{2} \\in V_{2}$ , we have $D(\\mu (x, y)) = D(\\mu (x_{1} + x_{2}, y_{1} + y_{2})) = D(\\mu (x_{1}, y_{1}) + \\mu (x_{2}, y_{2})) = D(\\mu (x_{1}, y_{1})),$ $&\\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y))\\\\&= \\mu (D(x_{1} + x_{2}), \\alpha ^{k}(y_{1} + y_{2})) + \\mu (\\alpha ^{k}(x_{1} + x_{2}), D(y_{1} + y_{2}))\\\\&= \\mu (D(x_{1}), \\alpha ^{k}(y_{1})) + \\mu (\\alpha ^{k}(x_{1}), D(y_{1})),$ since $D \\in Der_{\\alpha ^{k}}(V_{1})$ , $D(\\mu (x_{1}, y_{1})) = \\mu (D(x_{1}), \\alpha ^{k}(y_{1})) + \\mu (\\alpha ^{k}(x_{1}), D(y_{1})),$ we have $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y)),$ which implies that $D \\in Der_{\\alpha ^{k}}(V)$ , i.e., $Der_{\\alpha ^{k}}(V_{1}) \\subseteq Der_{\\alpha ^{k}}(V)$ .",
"Moreover, $D \\in Der_{\\alpha ^{k}}(V_{1})$ if and only if $D(x_{2}) = 0,\\; \\forall x_{2} \\in V_{2}$ .",
"Similarly, we have $Der_{\\alpha ^{k}}(V_{2}) \\subseteq Der_{\\alpha ^{k}}(V)$ and $D \\in Der_{\\alpha ^{k}}(V_{2})$ if and only if $D(x_{1}) = 0,\\; \\forall x_{1} \\in V_{1}$ .",
"Then we have $Der_{\\alpha ^{k}}(V_{1}) + Der_{\\alpha ^{k}}(V_{2}) \\subseteq Der_{\\alpha ^{k}}(V)$ and $Der_{\\alpha ^{k}}(V_{1}) \\cap Der_{\\alpha ^{k}}(V_{2}) = \\lbrace 0\\rbrace $ .",
"Hence, $Der_{\\alpha ^{k}}(V_{1}) \\dotplus Der_{\\alpha ^{k}}(V_{2}) \\subseteq Der_{\\alpha ^{k}}(V)$ .",
"$\\bf {Step 3}$ .",
"We'll prove that $Der_{\\alpha ^{k}}(V_{1}) \\dotplus Der_{\\alpha ^{k}}(V_{2}) = Der_{\\alpha ^{k}}(V)$ .",
"Suppose that $D \\in Der_{\\alpha ^{k}}(V)$ .",
"Set $x = x_{1} + x_{2}, x_{i} \\in V_{i}$ .",
"Define $D_{1}, D_{2}$ as follows $\\left\\lbrace \\begin{aligned}D_{1}(x_{1} + x_{2}) = D(x_{1}),\\\\D_{2}(x_{1} + x_{2}) = D(x_{2}).\\end{aligned}\\right.$ Obviously, $D = D_{1} + D_{2}$ .",
"For any $u_{1}, v_{1} \\in V_{1}$ , $&D_{1}(\\mu (u_{1}, v_{1})) = D(\\mu (u_{1}, v_{1})) = \\mu (D(u_{1}), \\alpha ^{k}(v_{1})) + \\mu (\\alpha ^{k}(u_{1}), D(v_{1}))\\\\&= \\mu (D_{1}(u_{1}), \\alpha ^{k}(v_{1})) + \\mu (\\alpha ^{k}(u_{1}), D_{1}(v_{1})).$ Hence, $D_{1} \\in Der_{\\alpha ^{k}}(V_{1})$ .",
"Similarly, $D_{2} \\in Der_{\\alpha ^{k}}(V_{2})$ .",
"Therefore, $Der_{\\alpha ^{k}}(V_{1}) \\dotplus Der_{\\alpha ^{k}}(V_{2}) = Der_{\\alpha ^{k}}(V)$ as a vector space.",
"Hence, we have $&Der(V) = \\dotplus _{k \\ge 0}Der_{\\alpha ^{k}}(V) = \\dotplus _{k \\ge 0}(Der_{\\alpha ^{k}}(V_{1}) \\dotplus Der_{\\alpha ^{k}}(V_{2}))\\\\&= (\\dotplus _{k \\ge 0}Der_{\\alpha ^{k}}(V_{1})) \\dotplus (\\dotplus _{k \\ge 0}Der_{\\alpha ^{k}}(V_{2})) = Der(V_{1}) \\dotplus Der(V_{2}).$ $\\bf {Step 4}$ .",
"We'll show that $Der(V_{i})(i = 1, 2)$ are Hom-ideals of $Der(V)$ .",
"Suppose that $D_{1} \\in Der_{\\alpha ^{k}}(V_{1})$ , $D_{2} \\in Der_{\\alpha ^{s}}(V)$ .",
"Then for any $x_{1}, y_{1} \\in V_{1}$ , we have $&\\sigma (D_{1})(\\mu (x_{1}, y_{1})) = \\alpha \\circ D_{1}(\\mu (x_{1}, y_{1})) = \\alpha (\\mu (D_{1}(x_{1}), \\alpha ^{k}(y_{1})) + \\mu (\\alpha ^{k}(x_{1}), D_{1}(y_{1})))\\\\&= \\mu (\\sigma (D_{1})(x_{1}), \\alpha ^{k + 1}(y_{1})) + \\mu (\\alpha ^{k + 1}(x_{1}), \\sigma (D_{1})(y_{1})),$ which implies that $\\sigma (D_{1}) \\in Der_{\\alpha ^{k + 1}}(V_{1})$ .",
"$&\\nu ^{^{\\prime }}(D_{1}, D_{2})(\\mu (x_{1}, y_{1}))\\\\&= D_{1} \\circ D_{2}(\\mu (x_{1}, y_{1})) - D_{2} \\circ D_{1}(\\mu (x_{1}, y_{1}))\\\\&= D_{1}(\\mu (D_{2}(x_{1}), \\alpha ^{s}(y_{1})) + \\mu (\\alpha ^{s}(x_{1}), D_{2}(y_{1}))) - D_{2}(\\mu (D_{1}(x_{1}), \\alpha ^{k}(y_{1})) + \\mu (\\alpha ^{k}(x_{1}), D_{1}(y_{1})))\\\\&= \\mu (D_{1}(D_{2}(x_{1})), \\alpha ^{k + s}(y_{1})) + \\mu (\\alpha ^{k}(D_{2}(x_{1})), D_{1}(\\alpha ^{s}(y_{1}))) + \\mu (D_{1}(\\alpha ^{s}(x_{1})), \\alpha ^{k}(D_{2}(y_{1})))\\\\&+ \\mu (\\alpha ^{k + s}(x_{1}), D_{1}(D_{2}(y_{1}))) - \\mu (D_{2}(D_{1}(x_{1})), \\alpha ^{k + s}(y_{1})) - \\mu (\\alpha ^{s}(D_{1}(x_{1})), D_{2}(\\alpha ^{k}(y_{1})))\\\\&- \\mu (D_{2}(\\alpha ^{k}(x_{1})), \\alpha ^{s}(D_{1}(y_{1}))) - \\mu (\\alpha ^{k + s}(x_{1}), D_{2}(D_{1}(y_{1})))\\\\&= \\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x_{1}), \\alpha ^{k + s}(y_{1})) + \\mu (\\alpha ^{k + s}(x_{1}), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y_{1})),$ which implies that $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in Der_{\\alpha ^{k + s}}(V_{1})$ .",
"So $Der(V_{1})$ is a Hom-ideal of $Der(V)$ .",
"Similarly, we have $Der(V_{2})$ is a Hom-ideal of $Der(V)$ .",
"Therefore, $Der(V) = Der(V_{1}) \\oplus Der(V_{2})$ .",
"(3), (4), (5) similar to the proof of (2).",
"Theorem 3.5 Let $(V, \\mu , \\alpha )$ be a multiplicative Hom-Jordan algebra, $\\alpha $ a surjection and $Z(V)$ the centralizer of $(V, \\mu , \\alpha )$ .",
"Then $\\nu ^{^{\\prime }}(C(V), QC(V)) \\subseteq End(V, Z(V))$ .",
"Moreover, if $Z(V) = \\lbrace 0\\rbrace $ , then $\\nu ^{^{\\prime }}(C(V), QC(V)) = \\lbrace 0\\rbrace $ .",
"Suppose that $D_{1} \\in C_{\\alpha ^{k}}(V)$ , $D_{2} \\in QC_{\\alpha ^{s}}(V)$ and $x \\in V$ .",
"Since $\\alpha $ is surjective, there exists $y^{^{\\prime }} \\in V$ such that $y = \\alpha ^{k + s}(y^{^{\\prime }})$ for any $y \\in V$ .",
"$&\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), y)\\\\&= \\mu (D_{1} \\circ D_{2}(x), \\alpha ^{k + s}(y^{^{\\prime }})) - \\mu (D_{2} \\circ D_{1}(x), \\alpha ^{k + s}(y^{^{\\prime }}))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y^{^{\\prime }}))) - \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y^{^{\\prime }})))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y^{^{\\prime }}))) - \\mu (D_{1}(\\alpha ^{s}(x)), \\alpha ^{k}(D_{2}(y^{^{\\prime }})))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y^{^{\\prime }}))) - D_{1}(\\mu (\\alpha ^{s}(x), D_{2}(y^{^{\\prime }})))\\\\&= D_{1}(\\mu (D_{2}(x), \\alpha ^{s}(y^{^{\\prime }})) - \\mu (\\alpha ^{s}(x), D_{2}(y^{^{\\prime }})))\\\\&= 0,$ which implies that $\\nu ^{^{\\prime }}(D_{1}, D_{2})(x) \\in Z(V)$ , i.e., $\\nu ^{^{\\prime }}(C(V), QC(V)) \\subseteq End(V, Z(V))$ .",
"Furthermore, if $Z(V) = \\lbrace 0\\rbrace $ , it's clearly that $\\nu ^{^{\\prime }}(C(V), QC(V)) = \\lbrace 0\\rbrace $ .",
"Theorem 3.6 Let $(V, \\mu , \\alpha )$ be a multiplicative Hom-Jordan algebra.",
"Then $ZDer(V) = C(V) \\cap Der(V)$ .",
"Assume that $D \\in C(V)_{\\alpha ^{k}} \\cap Der_{\\alpha ^{k}}(V)$ .",
"Then for any $x, y \\in V$ , we have $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y))$ since $D \\in Der_{\\alpha ^{k}}(V)$ .",
"$D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) = \\mu (\\alpha ^{k}(x), D(y))$ since $D \\in C_{\\alpha ^{k}}(V)$ .",
"Hence we have $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) = 0$ , which implies that $D \\in ZDer_{\\alpha ^{k}}(V)$ .",
"Therefore, $C(V) \\cap Der(V) \\subseteq ZDer(V)$ .",
"On the other hand, assume that $D \\in ZDer_{\\alpha ^{k}}(V)$ .",
"Then for any $x, y \\in V$ , we have $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) = 0,$ hence we have $\\mu (\\alpha ^{k}(x), D(y)) = \\mu (D(y), \\alpha ^{k}(x)) = 0.$ Therefore $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y))$ which implies that $D \\in Der_{\\alpha ^{k}}(V)$ .",
"And $D(\\mu (x, y)) = \\mu (D(x), \\alpha ^{k}(y)) = \\mu (\\alpha ^{k}(x), D(y))$ which implies that $D \\in C_{\\alpha ^{k}}(V)$ .",
"Therefore $D \\in C(V)_{\\alpha ^{k}} \\cap Der_{\\alpha ^{k}}(V)$ , i.e., $ZDer(V) \\subseteq C(V) \\cap Der(V)$ .",
"Hence, we have $ZDer(V) = C(V) \\cap Der(V)$ .",
"Theorem 3.7 Let $(V, \\mu , \\alpha )$ be a multiplicative Hom-Jordan algebra.",
"Then $(QC(V), \\nu , \\sigma )$ is a Hom-Jordan algebra.",
"According to Lemma REF (1), we need only to show that $QC(V)$ is a Hom-subalgebra of $(\\mathcal {W}, \\nu , \\sigma )$ .",
"Suppose that $D_{1} \\in QC_{\\alpha ^{k}}(V)$ , $D_{2} \\in QC_{\\alpha ^{s}}(V)$ .",
"Then for any $x, y \\in V$ , we have $&\\mu (\\sigma (D_{1})(x), \\alpha ^{k + 1}(y)) = \\mu (\\alpha \\circ D_{1}(x), \\alpha ^{k + 1}(y)) = \\alpha (\\mu (D_{1}(x), \\alpha ^{k}(y)))\\\\&= \\alpha (\\mu (\\alpha ^{k}(x), D_{1}(y))) = \\mu (\\alpha ^{k + 1}(x), \\sigma (D_{1})(y)),$ which implies that $\\sigma (D_{1}) \\in QC_{\\alpha ^{k + 1}}(V)$ .",
"$&\\mu (\\nu (D_{1}, D_{2})(x), \\alpha ^{k + s}(y))\\\\&= \\mu (D_{1} \\circ D_{2}(x), \\alpha ^{k + s}(y)) + \\mu (D_{2} \\circ D_{1}(x), \\alpha ^{k + s}(y))\\\\&= \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y))) + \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y)))\\\\&= \\mu (D_{2}(\\alpha ^{k}(x)), \\alpha ^{s}(D_{1}(y))) + \\mu (D_{1}(\\alpha ^{s}(x)), \\alpha ^{k}(D_{2}(y)))\\\\&= \\mu (\\alpha ^{k + s}(x), D_{2}(D_{1}(y))) + \\mu (\\alpha ^{k + s}(x), D_{1}(D_{2}(y)))\\\\&= \\mu (\\alpha ^{k + s}(x), \\nu (D_{1}, D_{2})(y)),$ which implies that $\\nu (D_{1}, D_{2}) \\in QC_{\\alpha ^{k + s}}(V)$ .",
"Therefore, $QC(V)$ is a Hom-subalgebra of $(\\mathcal {W}, \\nu , \\sigma )$ , i.e., $(QC(V), \\nu , \\sigma )$ is a Hom-Jordan algebra.",
"Theorem 3.8 Suppose that $(V, \\mu , \\alpha )$ be a multiplicative Hom-Jordan algebra over a field $\\rm {F}$ of characteristic other than 2.",
"$(QC(V), \\nu ^{^{\\prime }}, \\sigma )$ is a Hom-Lie algebra if and only if $(QC(V), \\iota , \\sigma )$ is a Hom-associative algebra where $\\iota (D_{1}, D_{2}) = D_{1} \\circ D_{2},\\quad \\forall D_{1}, D_{2} \\in QC(V)$ .",
"If $\\alpha $ is a surjection and $Z(V) = \\lbrace 0\\rbrace $ , then $(QC(V), \\nu ^{^{\\prime }}, \\sigma )$ is a Hom-Lie algebra if and only if $\\nu ^{^{\\prime }}(QC(V), QC(V)) = \\lbrace 0\\rbrace $ .",
"(1).",
"$(\\Leftarrow )$ .",
"For all $D_{1}, D_{2} \\in QC(V)$ , we have $D_{1} \\circ D_{2} \\in QC(V)$ , $D_{2} \\circ D_{1} \\in QC(V)$ .",
"So $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in QC(V)$ .",
"Moreover, $\\sigma (D_{1}) \\in QC(V)$ .",
"Therefore, $(QC(V), \\nu ^{^{\\prime }}, \\sigma )$ is a Hom-Lie algebra.",
"$(\\Rightarrow )$ .",
"For all $D_{1}, D_{2} \\in QC(V)$ , we have $\\iota (D_{1}, D_{2}) = \\frac{1}{2}(\\nu ^{^{\\prime }}(D_{1}, D_{2}) + \\nu (D_{1}, D_{2}))$ .",
"According to Theorem REF , we have $\\nu (D_{1}, D_{2}) \\in QC(V)$ .",
"Note that $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in QC(V)$ , we have $\\iota (D_{1}, D_{2}) \\in QC(V)$ .",
"Hence, $(QC(V), \\iota , \\sigma )$ is a Hom-associative algebra.",
"(2).",
"$(\\Rightarrow )$ .",
"Suppose that $D_{1} \\in QC_{\\alpha ^{k}}(V)$ , $D_{2} \\in QC_{\\alpha ^{s}}(V)$ and $x \\in V$ .",
"Since $\\alpha $ is a surjection, there exists $y^{^{\\prime }} \\in V$ such that $y = \\alpha ^{k + s}(y^{^{\\prime }})$ for any $y \\in V$ .",
"Note that $(QC(V), \\nu ^{^{\\prime }}, \\sigma )$ is a Hom-Lie algebra, then $\\nu ^{^{\\prime }}(D_{1}, D_{2}) \\in QC_{\\alpha ^{k + s}}(V)$ .",
"Then $\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), y) = \\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y^{^{\\prime }})) = \\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y^{^{\\prime }})).$ Note that $&\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y^{^{\\prime }}))\\\\&= \\mu (D_{1} \\circ D_{2}(x), \\alpha ^{k + s}(y^{^{\\prime }})) - \\mu (D_{2} \\circ D_{1}(x), \\alpha ^{k + s}(y^{^{\\prime }}))\\\\&= \\mu (\\alpha ^{k}(D_{2}(x)), D_{1}(\\alpha ^{s}(y^{^{\\prime }}))) - \\mu (\\alpha ^{s}(D_{1}(x)), D_{2}(\\alpha ^{k}(y^{^{\\prime }})))\\\\&= \\mu (D_{2}(\\alpha ^{k}(x)), \\alpha ^{s}(D_{1}(y^{^{\\prime }}))) - \\mu (D_{1}(\\alpha ^{s}(x)), \\alpha ^{k}(D_{2}(y^{^{\\prime }})))\\\\&= \\mu (\\alpha ^{k + s}(x), D_{2} \\circ D_{1}(y^{^{\\prime }})) - \\mu (\\alpha ^{k + s}(x), D_{1} \\circ D_{2}(y^{^{\\prime }}))\\\\&= -\\mu (\\alpha ^{k + s}(x), \\nu ^{^{\\prime }}(D_{1}, D_{2})(y^{^{\\prime }})).$ Hence, we have $\\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), y) = \\mu (\\nu ^{^{\\prime }}(D_{1}, D_{2})(x), \\alpha ^{k + s}(y^{^{\\prime }})) = 0$ , which implies that $\\nu ^{^{\\prime }}(D_{1}, D_{2})(x) \\in Z(V)$ .",
"Note that $Z(V) = \\lbrace 0\\rbrace $ , we have $\\nu ^{^{\\prime }}(D_{1}, D_{2})(x) = 0$ , i.e., $\\nu ^{^{\\prime }}(D_{1}, D_{2}) = 0$ .",
"Therefore, $\\nu ^{^{\\prime }}(QC(V), QC(V)) = \\lbrace 0\\rbrace $ .",
"$(\\Leftarrow )$ .",
"Obviously."
],
[
"The quasiderivations of Hom-Jordan algebras",
"In this section, we will prove that the quasiderivations of $(V, \\mu , \\alpha )$ can be embedded as derivations in a larger Hom-Jordan agebra and obtain a direct sum decomposition of $Der(\\breve{V})$ when the centralizer of $(V, \\mu , \\alpha )$ is equal to zero.",
"Proposition 4.1 Suppose that $(V, \\mu , \\alpha )$ is a Hom-Jordan algebra over $\\rm {F}$ and $t$ is an indeterminate.",
"We define $\\breve{V} := \\lbrace \\sum (x \\otimes t + y \\otimes t^{2}) | x, y \\in V\\rbrace $ , $\\breve{\\alpha }(\\breve{V}) := \\lbrace \\sum (\\alpha (x) \\otimes t + \\alpha (y) \\otimes t^{2}) | x, y \\in V\\rbrace $ and $\\breve{\\mu }(x \\otimes t^{i}, y \\otimes t^{j}):= \\mu (x,y) \\otimes t^{i + j}$ where $x, y \\in V, i, j \\in \\lbrace 1, 2\\rbrace $ .",
"Then $(\\breve{V}, \\breve{\\mu }, \\breve{\\alpha })$ is a Hom-Jordan algebra.",
"It's obvious that $\\breve{\\mu }$ is a bilinear map and commutative since $\\mu $ is a bilinear map and commutative.",
"For any $x \\otimes t^{i}, y \\otimes t^{j} \\in \\breve{V}$ , we have $&\\breve{\\mu }(\\breve{\\alpha }^{2}(x \\otimes t^{i}), \\breve{\\mu }(y \\otimes t^{j}, \\breve{\\mu }(x \\otimes t^{i}, x \\otimes t^{i})))\\\\&= \\mu (\\alpha ^{2}(x), \\mu (y, \\mu (x, x))) \\otimes t^{3i + j}\\\\&= \\mu (\\mu (\\alpha (x), y), \\alpha (\\mu (x, x))) \\otimes t^{3i + j}\\\\&= \\breve{\\mu }(\\breve{\\mu }(\\breve{\\alpha }(x \\otimes t^{i}), y \\otimes t^{j}), \\breve{\\alpha }(\\breve{\\mu }(x \\otimes t^{i}, x \\otimes t^{i}))).$ Therefore, $(\\breve{V}, \\breve{\\mu }, \\breve{\\alpha })$ is a Hom-Jordan algebra.",
"For convenience, we write $xt(xt^{2})$ in place of $x \\otimes t(x \\otimes t^{2})$ .",
"If $U$ is a subspace of $V$ such that $V = U \\dotplus \\mu (V, V)$ , then $\\breve{V} = Vt + Vt^{2} = Vt + Ut^{2} + \\mu (V, V)t^{2}.$ Now we define a map $\\varphi : QDer(V) \\rightarrow End(\\breve{V})$ satisfying $\\varphi (D)(at + ut^{2} + bt^{2}) = D(a)t + D^{^{\\prime }}(b)t^{2},$ where $D \\in QDer_{\\alpha ^{k}}(V)$ , and $D^{^{\\prime }}$ is in Definition REF (2), $a \\in V, u \\in U, b \\in \\mu (V, V)$ .",
"Proposition 4.2 $V, \\breve{V}, \\varphi $ are defined as above.",
"Then $\\varphi $ is injective and $\\varphi (D)$ does not depend on the choice of $D^{^{\\prime }}$ .",
"$\\varphi (QDer(V)) \\subseteq Der(\\breve{V})$ .",
"(1).",
"If $\\varphi (D_{1}) = \\varphi (D_{2})$ , then for all $a \\in V, u \\in U, b \\in \\mu (V, V)$ , we have $\\varphi (D_{1})(at + ut^{2} + bt^{2}) = \\varphi (D_{2})(at + ut^{2} + bt^{2}),$ which implies that $D_{1}(a)t + D_{1}^{^{\\prime }}(b)t^{2} = D_{2}(a)t + D_{2}^{^{\\prime }}(b)t^{2}.$ Hence we have $D_{1}(a) = D_{2}(a),$ which implies that $D_{1} = D_{2}$ .",
"Therefore, $\\varphi $ is injective.",
"Suppose that there exists $D^{^{\\prime \\prime }}$ such that $\\varphi (D)(at + ut^{2} + bt^{2}) = D(a)t + D^{^{\\prime \\prime }}(b)t^{2}$ and $\\mu (D(x), \\alpha ^{k}(y)) + \\mu (\\alpha ^{k}(x), D(y)) = D^{^{\\prime \\prime }}(\\mu (x, y))$ , then we have $D^{^{\\prime \\prime }}(\\mu (x, y)) = D^{^{\\prime }}(\\mu (x, y)),$ which implies that $D^{^{\\prime \\prime }}(b) = D^{^{\\prime }}(b)$ .",
"Hence, we have $\\varphi (D)(at + ut^{2} + bt^{2}) = D(a)t + D^{^{\\prime \\prime }}(b)t^{2} = D(a)t + D^{^{\\prime }}(b)t^{2}.$ That is to say $\\varphi (D)$ does not depend on the choice of $D^{^{\\prime }}$ .",
"(2).",
"$\\forall i + j \\ge 3$ , we have $\\breve{\\mu }(x \\otimes t^{i}, y \\otimes t^{j}) = \\mu (x, y) \\otimes t^{i + j} = 0$ .",
"Hence, we need only to show that $\\varphi (D)(\\breve{\\mu }(xt, yt)) = \\breve{\\mu }(\\varphi (D)(xt), \\breve{\\alpha }^{k}(yt)) + \\breve{\\mu }(\\breve{\\alpha }^{k}(xt), \\varphi (D)(yt)).$ For all $x, y \\in V$ , we have $&\\varphi (D)(\\breve{\\mu }(xt, yt))\\\\&= \\varphi (D)(\\mu (x, y)t^{2}) = D^{^{\\prime }}(\\mu (x, y))t^{2}\\\\&= \\mu (D(x), \\alpha ^{k}(y))t^{2} + \\mu (\\alpha ^{k}(x), D(y))t^{2}\\\\&= \\breve{\\mu }(D(x)t, \\alpha ^{k}(y)t) + \\breve{\\mu }(\\alpha ^{k}(x)t, D(y)t)\\\\&= \\breve{\\mu }(\\varphi (D)(xt), \\breve{\\alpha }^{k}(yt)) + \\breve{\\mu }(\\breve{\\alpha }^{k}(xt), \\varphi (D)(yt)).$ Hence, $\\varphi (D) \\in Der_{\\breve{\\alpha }^{k}}(\\breve{V})$ .",
"Therefore, $\\varphi (QDer(V)) \\subseteq Der(\\breve{V})$ .",
"Proposition 4.3 Let $(V, \\mu , \\alpha )$ be a Hom-Jordan algebra.",
"$Z(V) = \\lbrace 0\\rbrace $ and $\\breve{V}, \\varphi $ are defined as above.",
"Then $Der(\\breve{V}) = \\varphi (QDer(V)) \\dotplus ZDer(\\breve{V})$ .",
"Assume that $xt + yt^{2} \\in Z(\\breve{V})$ , then for any $x^{^{\\prime }}t + y^{^{\\prime }}t^{2} \\in \\breve{V}$ , we have $0 = \\breve{\\mu }(xt + yt^{2}, x^{^{\\prime }}t + y^{^{\\prime }}t^{2}) = \\mu (x, x^{^{\\prime }})t^{2},$ which implies that $\\mu (x, x^{^{\\prime }}) = 0$ .",
"Note that $Z(V) = \\lbrace 0\\rbrace $ , we have $x = 0$ .",
"Hence, $Z(\\breve{V}) \\subseteq Vt^{2}$ .",
"Obviously, $Vt^{2} \\subseteq Z(\\breve{V})$ .",
"Therefore, $Z(\\breve{V}) = Vt^{2}$ .",
"Suppose that $g \\in Der_{\\breve{\\alpha }^{k}}(\\breve{V})$ , $at^{2} \\in Z(\\breve{V})$ .",
"Since $\\alpha $ is surjective, $\\breve{\\alpha }$ is also surjective.",
"For any $xt + yt^{2} \\in \\breve{V}$ , there exists $x^{^{\\prime }}t + y^{^{\\prime }}t^{2} \\in \\breve{V}$ such that $xt + yt^{2} = \\breve{\\alpha }^{k}(x^{^{\\prime }}t + y^{^{\\prime }}t^{2})$ .",
"$&\\breve{\\mu }(g(at^{2}), xt + yt^{2}) = \\breve{\\mu }(g(at^{2}), \\breve{\\alpha }^{k}(x^{^{\\prime }}t + y^{^{\\prime }}t^{2}))\\\\&= g(\\breve{\\mu }(at^{2}, x^{^{\\prime }}t + y^{^{\\prime }}t^{2})) - \\breve{\\mu }(\\breve{\\alpha }^{k}(at^{2}), g(x^{^{\\prime }}t + y^{^{\\prime }}t^{2}))\\\\&= 0,$ which implies that $g(at^{2}) \\in Z(\\breve{V})$ .",
"Therefore, $g(Z(\\breve{V})) \\subseteq Z(\\breve{V})$ .",
"Hence, we have $g(Ut^{2}) \\subseteq g(Z(\\breve{V})) \\subseteq Z(\\breve{V}) = Vt^{2}$ .",
"Now we define a map $f : Vt + Ut^{2} + \\mu (V, V)t^{2} \\rightarrow Vt^{2}$ by $f(x) =\\left\\lbrace \\begin{aligned}g(x) \\cap Vt^{2},\\quad x \\in Vt\\\\g(x),\\quad x \\in Ut^{2}\\\\0,\\quad x \\in \\mu (V, V)t^{2}\\end{aligned}\\right.$ It's clear that $f$ is linear.",
"Note that $f(\\breve{\\mu }(\\breve{V}, \\breve{V})) = f(\\mu (V, V)t^{2}) = 0,$ $\\breve{\\mu }(f(\\breve{V}), \\breve{\\alpha }^{k}(\\breve{V})) \\subseteq \\breve{\\mu }(Vt^{2}, \\alpha ^{k}(V)t + \\alpha ^{k}(V)t^{2}) = 0,$ we have $f \\in ZDer_{\\breve{\\alpha }^{k}}(\\breve{V})$ .",
"Since $(g - f)(Vt) = g(Vt) - f(Vt) = g(Vt) - g(Vt) \\cap Vt^{2} = g(Vt) - Vt^{2} \\subseteq Vt,$ $(g - f)(Ut^{2}) = 0,$ $(g - f)(\\mu (V, V)t^{2}) = g(\\breve{\\mu }(\\breve{V}, \\breve{V})) \\subseteq \\breve{\\mu }(\\breve{V}, \\breve{V}) = \\mu (V, V)t^{2},$ hence there exist $D, D^{^{\\prime }} \\in End (V)$ such that $\\forall a \\in V, b \\in \\mu (V, V)$ , $(g - f)(at) = D(a)t, (g - f)(bt^{2}) = D^{^{\\prime }}(b)t^{2}.$ Since $g - f \\in Der_{\\breve{\\alpha }^{k}}(\\breve{V})$ , we have $(g - f)(\\breve{\\mu }(a_{1}t, a_{2}t)) = \\breve{\\mu }((g - f)(a_{1}t), \\breve{\\alpha }^{k}(a_{2}t)) + \\breve{\\mu }(\\breve{\\alpha }^{k}(a_{1}t), (g - f)(a_{2}t)),$ which implies that $D^{^{\\prime }}(\\mu (a_{1}, a_{2}))t^{2} = \\mu (D(a_{1}), \\alpha ^{k}(a_{2}))t^{2} + \\mu (\\alpha ^{k}(a_{1}), D(a_{2}))t^{2}.$ Hence, we have $D \\in QDer_{\\alpha ^{k}}(V) \\subseteq QDer(V)$ .",
"Therefore, $g - f = \\varphi (D) \\in \\varphi (QDer(V))$ , which implies that $Der(\\breve{V}) \\subseteq \\varphi (QDer(V)) + ZDer(\\breve{V})$ .",
"According to Proposition REF (2), we have $Der(\\breve{V}) = \\varphi (QDer(V)) + ZDer(\\breve{V})$ .",
"$\\forall f \\in \\varphi (QDer(V)) \\cap ZDer(\\breve{V})$ , there exists $D \\in QDer(V)$ such that $f = \\varphi (D)$ .",
"For all $a \\in V, b \\in \\mu (V, V), u \\in U$ , $f(at + bt^{2} + ut^{2}) = \\varphi (D)(at + bt^{2} + ut^{2}) = D(a)t + D^{^{\\prime }}(b)t^{2}.$ On the other hand, for any $xt + yt^{2} \\in \\breve{V}$ , there exists $x^{^{\\prime }}t + y^{^{\\prime }}t^{2} \\in \\breve{V}$ such that $xt + yt^{2} = \\breve{\\alpha }^{k}(x^{^{\\prime }}t + y^{^{\\prime }}t^{2})$ since $\\breve{\\alpha }$ is surjective.",
"Then we have $&\\breve{\\mu }(f(at + bt^{2} + ut^{2}), xt + yt^{2}) = \\breve{\\mu }(f(at + bt^{2} + ut^{2}), \\breve{\\alpha }^{k}(x^{^{\\prime }}t + y^{^{\\prime }}t^{2}))\\\\&= f(\\breve{\\mu }(at + bt^{2} + ut^{2}, x^{^{\\prime }}t + y^{^{\\prime }}t^{2}))\\\\&= 0$ since $f \\in ZDer(\\breve{V})$ .",
"Hence, $f(at + bt^{2} + ut^{2}) \\in Z(\\breve{V}) = Vt^{2}$ .",
"Therefore, $D(a) = 0$ , i.e., $D = 0$ .",
"Hence, $f = 0$ .",
"Therefore, $Der(\\breve{V}) = \\varphi (QDer(V)) \\dotplus ZDer(\\breve{V})$ ."
],
[
"The centroids of Hom-Jordan algebras",
"Proposition 5.1 Suppose that $(V, \\mu , \\alpha )$ is a simple multiplicative Hom-Jordan algebra over an algebraically closed field $\\rm {\\mathbb {F}}$ of characteristic 0.",
"If $C(V) = \\rm {\\mathbb {F}}id$ , then $\\alpha = id$ .",
"For all $k \\in \\mathbb {N}^{+}$ , $\\forall \\; 0 \\ne \\psi \\in C_{\\alpha ^{k}}(V)$ , we have $\\psi = p\\;id,\\quad p \\in \\rm {\\mathbb {F}},\\;p \\ne 0$ .",
"So $\\forall x, y \\in V$ , $p\\mu (x, y) = \\psi (\\mu (x, y)) = \\mu (\\psi (x), \\alpha ^{k}(y)) = \\mu (px, \\alpha ^{k}(y)) = p\\mu (x, \\alpha ^{k}(y)),$ which implies that $\\mu (x, y) = \\mu (x, \\alpha ^{k}(y))$ for all $k \\in \\mathbb {N}^{+}$ .",
"Since $\\rm {\\mathbb {F}}$ is algebraically closed, $\\alpha $ has an eigenvalue $\\lambda $ .",
"We denote the corresponding eigenspace by $E_{\\lambda }(\\alpha )$ .",
"So $E_{\\lambda }(\\alpha ) \\ne 0$ .",
"Let $k = 1$ .",
"For any $x \\in E_{\\lambda }(\\alpha )$ , $y \\in V$ , we have $\\alpha (\\mu (x, y)) = \\mu (\\alpha (x), \\alpha (y)) = \\mu (\\lambda x, \\alpha (y)) = \\lambda \\mu (x, \\alpha (y)) = \\lambda \\mu (x, y),$ which implies that $\\mu (x, y) \\in E_{\\lambda }(\\alpha )$ .",
"Moreover, for any $x \\in E_{\\lambda }(\\alpha )$ , we have $\\alpha (\\alpha (x)) = \\lambda ^{2}x = \\lambda \\alpha (x),$ which implies that $\\alpha (x) \\in E_{\\lambda }(\\alpha )$ , i.e., $\\alpha (E_{\\lambda }(\\alpha )) \\subseteq E_{\\lambda }(\\alpha )$ .",
"So $E_{\\lambda }(\\alpha )$ is a Hom-ideal of $(V, \\mu , \\alpha )$ .",
"Since $(V, \\mu , \\alpha )$ is simple, we have $E_{\\lambda }(\\alpha ) = V$ , i.e., $\\alpha = \\lambda id$ .",
"Then for any $x, y \\in V, k = 1$ , we have $\\mu (x, y) = \\mu (x, \\alpha (y)) = \\mu (x, \\lambda y) = \\lambda \\mu (x, y),$ which implies that $\\lambda = 1$ , i.e., $\\alpha = id$ .",
"Proposition 5.2 Let $(V, \\mu , \\alpha )$ be a multiplicative Hom-Jordan algebra.",
"If $\\alpha $ is a surjection, then $V$ is indecomposable if and only if $C(V)$ does not contain idempotents except 0 and $id$ .",
"If $(V, \\mu , \\alpha )$ is perfect(i.e., $V = \\mu (V, V)$ ), then every $\\psi \\in C_{\\alpha ^{k}}(V)$ is $\\alpha ^{k}$ -symmetric with respect to any invariant form on $V$ .",
"(1).",
"$(\\Rightarrow )$ .",
"If there exists $\\psi \\in C_{\\alpha ^{k}}(V)$ is an idempotent and satisfies that $\\psi \\ne 0,\\;id$ , then $\\psi ^{2}(x) = \\psi (x),\\quad x \\in V$ .",
"For any $x \\in ker(\\psi ), y \\in V$ , we have $\\psi (\\mu (x, y)) = \\mu (\\psi (x), \\alpha ^{k}(y)) = \\mu (0, \\alpha ^{k}(y)) = 0,$ which implies that $\\mu (x, y) \\in ker(\\psi )$ .",
"Moreover, $\\psi (\\alpha (x)) = \\alpha (\\psi (x)) = \\alpha (0) = 0$ , so $\\alpha (x) \\in ker(\\psi )$ , i.e., $\\alpha (ker(\\psi )) \\subseteq ker(\\psi )$ .",
"Hence, $ker(\\psi )$ is a Hom-ideal of $(V, \\mu , \\alpha )$ .",
"For any $y \\in Im(\\psi )$ , there exists $y^{^{\\prime }} \\in V$ such that $y = \\psi (y^{^{\\prime }})$ .",
"And for any $z \\in V$ , there exists $z^{^{\\prime }} \\in V$ such that $z = \\alpha ^{k}(z^{^{\\prime }})$ since $\\alpha $ is surjective.",
"Then $\\mu (y, z) = \\mu (\\psi (y^{^{\\prime }}), \\alpha ^{k}(z^{^{\\prime }})) = \\psi (\\mu (y^{^{\\prime }}, z^{^{\\prime }})),$ which implies that $\\mu (y, z) \\in Im(\\psi )$ .",
"Moreover, $\\alpha (y) = \\alpha (\\psi (y^{^{\\prime }})) = \\psi (\\alpha (y^{^{\\prime }}))$ , so $\\alpha (y) \\in Im(\\psi )$ , i.e., $\\alpha (Im(\\psi )) \\subseteq Im(\\psi )$ .",
"Hence, $Im(\\psi )$ is a Hom-ideal of $(V, \\mu , \\alpha )$ .",
"$\\forall x \\in ker(\\psi ) \\cap Im(\\psi )$ , $\\exists x^{^{\\prime }} \\in V$ such that $x = \\psi (x^{^{\\prime }})$ .",
"So $0 = \\psi (x) = \\psi ^{2}(y) = \\psi (y)$ , which implies that $x = 0$ .",
"Hence, $ker(\\psi ) \\cap Im(\\psi ) = \\lbrace 0\\rbrace $ .",
"We have a decomposition $x = (x - \\psi (x)) + \\psi (x)$ , $\\forall x \\in V$ .",
"So we have $V = ker(\\psi ) \\oplus Im(\\psi )$ .",
"Contradiction.",
"$(\\Leftarrow )$ .",
"Suppose that $V = V_{1} \\oplus V_{2}$ where $V_{i}$ are Hom-ideals of $(V, \\mu , \\alpha )$ .",
"Then for any $x \\in V$ , $\\exists x_{i} \\in V_{i}$ such that $x = x_{1} + x_{2}$ .",
"Define $\\psi : V \\rightarrow V$ by $\\psi (x) = \\psi (x_{1} + x_{2}) = x_{1} - x_{2}$ .",
"It's obvious that $\\psi ^{2}(x) = \\psi (x)$ .",
"For any $x, y \\in V$ , suppose that $x = x_{1} + x_{2},\\;y = y_{1} + y_{2}$ , $\\psi (\\mu (x, y)) = \\psi (\\mu (x_{1} + x_{2}, y_{1} + y_{2})) = \\psi (\\mu (x_{1}, y_{1}) + \\mu (x_{2}, y_{2})) = \\mu (x_{1}, y_{1}) - \\mu (x_{2}, y_{2}),$ $\\mu (\\psi (x), y) = \\mu (\\psi (x_{1} + x_{2}), y_{1} + y_{2}) = \\mu (x_{1} - x_{2}, y_{1} + y_{2}) = \\mu (x_{1}, y_{1}) - \\mu (x_{2}, y_{2}),$ we have $\\psi (\\mu (x, y)) = \\mu (\\psi (x), y).$ Similarly, we have $\\psi (\\mu (x, y)) = \\mu (x, \\psi (y))$ .",
"So $\\psi \\in C_{\\alpha ^{0}}(V) \\subseteq C(V)$ .",
"Contradiction.",
"(2).",
"Let $f$ be an invariant $\\rm {F}$ -bilinear form on $V$ .",
"Then we have $f(\\mu (x, y), z) = f(x, \\mu (y, z))$ for any $x, y, z \\in V$ .",
"Since $(V, \\mu , \\alpha )$ is perfect,let $\\psi \\in C_{\\alpha ^{k}}(V)$ , then for any $a, b, c \\in V$ we have $&f(\\psi (\\mu (a, b)), \\alpha ^{k}(c)) = f(\\mu (\\alpha ^{k}(a), \\psi (b)), \\alpha ^{k}(c)) = f(\\alpha ^{k}(a), \\mu (\\psi (b), \\alpha ^{k}(c)))\\\\&= f(\\alpha ^{k}(a), \\mu (\\alpha ^{k}(b), \\psi (c))) = f(\\mu (\\alpha ^{k}(a), \\alpha ^{k}(b)), \\psi (c)) = f(\\alpha ^{k}(\\mu (a, b)), \\psi (c)).$ Proposition 5.3 Let $(V, \\mu , \\alpha )$ be a Hom-Jordan algebra and $I$ an $\\alpha $ -invariant subspace of $V$ where $\\alpha |_{I}$ is surjective.",
"Then $Z_{V}(I) = \\lbrace x \\in V | \\mu (x, y) = 0,\\;\\forall y \\in I\\rbrace $ is invariant under $C(V)$ .",
"So is any perfect Hom-ideal of $(V, \\mu , \\alpha )$ .",
"For any $\\psi \\in C_{\\alpha ^{k}}(V)$ and $x \\in Z_{V}(I)$ , $\\forall y \\in I$ , $\\exists y^{^{\\prime }} \\in I$ such that $y = \\alpha ^{k}(y^{^{\\prime }})$ since $\\alpha |_{I}$ is surjective, we have $\\mu (\\psi (x), y) = \\mu (\\psi (x), \\alpha ^{k}(y^{^{\\prime }})) = \\psi (\\mu (x, y^{^{\\prime }})) = \\psi (0) = 0,$ which implies that $\\psi (x) \\in Z_{V}(I)$ .",
"So $Z_{V}(I)$ is invariant under $C(V)$ .",
"Suppose that $J$ is a perfect Hom-ideal of $(V, \\mu , \\alpha )$ .",
"Then $J = \\mu (J, J)$ .",
"For any $y \\in J$ , there exist $a, b \\in J$ such that $y = \\mu (a, b)$ , then we have $\\psi (y) = \\psi (\\mu (a, b)) = \\mu (\\psi (a), \\alpha ^{k}(b)) \\in \\mu (J, J) = J.$ So $J$ is invariant under $C(V)$ .",
"Theorem 5.4 Suppose that $(V_{1}, \\mu _{1}, \\alpha _{1})$ and $(V_{2}, \\mu _{2}, \\alpha _{2})$ are two Hom-Jordan algebras over field $\\rm {F}$ with $\\alpha _{1}$ is surjective.",
"Let $\\pi : V_{1} \\rightarrow V_{2}$ be an epimorphism of Hom-Jordan algebras.",
"Then for any $f \\in End_{\\rm {F}}(V_{1}, ker(\\pi )) := \\lbrace g \\in \\mathcal {W}_{1} | g(ker(\\pi )) \\subseteq ker(\\pi )\\rbrace $ , there exists a unique $\\bar{f} \\in \\mathcal {W}_{2}$ satisfying $\\pi \\circ f = \\bar{f} \\circ \\pi $ where $\\mathcal {W}_{i}(i = 1, 2)$ are defined as Lemma REF .",
"Moreover, the following results hold: The map $\\pi _{End} : End_{\\rm {F}}(V_{1}, ker(\\pi )) \\rightarrow \\mathcal {W}_{2}$ , $f \\mapsto \\bar{f}$ is a Hom-algebra homomorphism with the following proporties, where $(End_{\\rm {F}}(V_{1}, ker(\\pi )), \\circ , \\sigma _{1})$ and $(\\mathcal {W}_{2}, \\circ , \\sigma _{2})$ are two Hom-associative algebras and $\\sigma _{1} : End_{\\rm {F}}(V_{1}, ker(\\pi )) \\rightarrow End_{\\rm {F}}(V_{1}, ker(\\pi )),\\;f \\mapsto \\alpha _{1} \\circ f$ and $\\sigma _{2} : \\mathcal {W}_{2} \\rightarrow \\mathcal {W}_{2},\\;g \\mapsto \\alpha _{2} \\circ g$ .",
"$\\pi _{End}(Mult(V_{1})) = Mult(V_{2})$ , $\\pi _{End}(C(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi ))) \\subseteq C(V_{2})$ .",
"By restriction, there is a Hom-algebra homomorphism $\\pi _{C} : C(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi )) \\rightarrow C(V_{2}),\\quad f \\mapsto \\bar{f}.$ If $ker(\\pi ) = Z(V_{1})$ , then every $\\varphi \\in C(V_{1})$ leaves $ker(\\pi )$ invariant.",
"Suppose that $V_{1}$ is perfect and $ker(\\pi ) \\subseteq Z(V_{1})$ .",
"Then $\\pi _{C} : C(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi )) \\rightarrow C(V_{2}),\\quad f \\mapsto \\bar{f}$ is injective.",
"If $V_{1}$ is perfect, $Z(V_{2}) = \\lbrace 0\\rbrace $ and $ker(\\pi ) \\subseteq Z(V_{1})$ , then $\\pi _{C} : C(V_{1}) \\rightarrow C(V_{2})$ is a Hom-algebra homomorphism.",
"For any $y \\in V_{2}$ , there exists $x \\in V_{1}$ such that $y = \\pi (x)$ since $\\pi $ is an epimorphism.",
"Then for any $f \\in End_{\\rm {F}}(V_{1}, ker(\\pi ))$ , define $\\bar{f} : V_{2} \\rightarrow V_{2}$ by $\\bar{f}(y) = \\pi (f(x))$ where $x$ satisfies that $y = \\pi (x)$ .",
"It's obvious that $\\pi \\circ f = \\bar{f} \\circ \\pi $ .",
"$&(\\alpha _{2} \\circ \\bar{f}) \\circ \\pi = \\alpha _{2} \\circ (\\bar{f} \\circ \\pi ) = \\alpha _{2} \\circ (\\pi \\circ f) = (\\alpha _{2} \\circ \\pi ) \\circ f = (\\pi \\circ \\alpha _{1}) \\circ f = \\pi \\circ (\\alpha _{1} \\circ f)\\\\&= \\pi \\circ (f \\circ \\alpha _{1}) = (\\pi \\circ f) \\circ \\alpha _{1} = (\\bar{f} \\circ \\pi ) \\circ \\alpha _{1} = \\bar{f} \\circ (\\pi \\circ \\alpha _{1}) = \\bar{f} \\circ (\\alpha _{2} \\circ \\pi ) = (\\bar{f} \\circ \\alpha _{2}) \\circ \\pi .$ Since $\\pi $ is surjective, we have $\\alpha _{2} \\circ \\bar{f} = \\bar{f} \\circ \\alpha _{2}$ , i.e., $\\bar{f} \\in \\mathcal {W}_{2}$ .",
"If there exist $\\bar{f}$ and $\\bar{f}^{^{\\prime }}$ such that $\\pi \\circ f = \\bar{f} \\circ \\pi $ and $\\pi \\circ f = \\bar{f}^{^{\\prime }} \\circ \\pi $ , then we have $\\bar{f} \\circ \\pi = \\bar{f}^{^{\\prime }} \\circ \\pi $ .",
"For any $y \\in V_{2}$ , there exists $x \\in V_{1}$ such that $y = \\pi (x)$ .",
"Hence, $\\bar{f}(y) = \\bar{f}(\\pi (x)) = \\bar{f}^{^{\\prime }}(\\pi (x)) = \\bar{f}^{^{\\prime }}(y)$ .",
"So $\\bar{f} = \\bar{f}^{^{\\prime }}$ .",
"(1).",
"For all $f, g \\in End_{\\rm {F}}(V_{1}, ker(\\pi ))$ , we have $\\pi \\circ (f \\circ g) = (\\pi \\circ f) \\circ g = (\\bar{f} \\circ \\pi ) \\circ g = \\bar{f} \\circ (\\pi \\circ g) = \\bar{f} \\circ (\\bar{g} \\circ \\pi ) = (\\bar{f} \\circ \\bar{g}) \\circ \\pi ,$ which implies that $\\pi _{End}(f \\circ g) = \\bar{f} \\circ \\bar{g} = \\pi _{End}(f) \\circ \\pi _{End}(g)$ .",
"$\\pi \\circ (\\alpha _{1} \\circ f) = (\\pi \\circ \\alpha _{1}) \\circ f = (\\alpha _{2} \\circ \\pi ) \\circ f = \\alpha _{2} \\circ (\\pi \\circ f) = \\alpha _{2} \\circ (\\bar{f} \\circ \\pi ) = (\\alpha _{2} \\circ \\bar{f}) \\circ \\pi ,$ which implies that $\\pi _{End}(\\sigma _{1}(f)) = \\alpha _{2} \\circ \\bar{f} = \\sigma _{2}(\\pi _{End}(f))$ , i.e., $\\pi _{End} \\circ \\sigma _{1} = \\sigma _{2} \\circ \\pi _{End}$ .",
"Therefore, $\\pi _{End}$ is a Hom-algebra homomorphism.",
"(a).",
"For all $x \\in V_{1}$ , $z \\in ker(\\pi )$ , we have $\\pi (L_{x}(z)) = \\pi (\\mu _{1}(x, z)) = \\mu _{2}(\\pi (x), \\pi (z)) = \\mu _{2}(\\pi (x), 0) = 0,$ which implies that $L_{x}(z) \\in ker(\\pi )$ , i.e., $L_{x} \\in End_{\\rm {F}}(V_{1}, ker(\\pi ))$ .",
"Moreover, we have $\\pi \\circ L_{x} = L_{\\pi (x)} \\circ \\pi $ .",
"So $\\pi _{End}(L_{x}) = L_{\\pi (x)} \\in Mult(V_{2})$ .",
"Hence, $\\pi _{End}(Mult(V_{1})) \\subseteq Mult(V_{2})$ .",
"On the other hand, for any $L_{y} \\in Mult(V_{2})$ , $\\exists x \\in V_{1}$ such that $y = \\pi (x)$ .",
"So $L_{y} = L_{\\pi (x)} = \\pi _{End}(L_{x}) \\in \\pi _{End}(Mult(V_{1})),$ which implies that $Mult(V_{2}) \\subseteq \\pi _{End}(Mult(V_{1}))$ .",
"Therefore, $\\pi _{End}(Mult(V_{1})) = Mult(V_{2})$ .",
"For any $\\varphi \\in C_{\\alpha _{1}^{k}}(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi ))$ , $\\forall x^{^{\\prime }}, y^{^{\\prime }}\\in V_{2}$ , $\\exists x, y \\in V_{1}$ such that $x^{^{\\prime }} = \\pi (x),\\; y^{^{\\prime }} = \\pi (y)$ , then we have $&\\bar{\\varphi }(\\mu _{2}(x^{^{\\prime }}, y^{^{\\prime }})) = \\bar{\\varphi }(\\mu _{2}(\\pi (x), \\pi (y))) = \\bar{\\varphi }(\\pi (\\mu _{1}(x, y))) = \\pi (\\varphi (\\mu _{1}(x, y))) = \\pi (\\mu _{1}(\\varphi (x), \\alpha _{1}^{k}(y)))\\\\&= \\mu _{2}(\\pi (\\varphi (x)), \\pi (\\alpha _{1}^{k}(y))) = \\mu _{2}(\\bar{\\varphi }(\\pi (x)), \\alpha _{2}^{k}(\\pi (y))) = \\mu _{2}(\\bar{\\varphi }(x^{^{\\prime }}), \\alpha _{2}^{k}(y^{^{\\prime }})).$ Similarly, we have $\\bar{\\varphi }(\\mu _{2}(x^{^{\\prime }}, y^{^{\\prime }})) = \\mu _{2}(\\alpha _{2}^{k}(x^{^{\\prime }}), \\bar{\\varphi }(y^{^{\\prime }}))$ .",
"Hence, $\\bar{\\varphi } \\in C_{\\alpha _{2}^{k}}(V_{2})$ .",
"Therefore, $\\pi _{End}(C(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi ))) \\subseteq C(V_{2})$ .",
"(b).",
"$\\forall f, g \\in C(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi ))$ , we have $\\pi \\circ (f \\circ g) = (\\pi \\circ f) \\circ g = (\\bar{f} \\circ \\pi ) \\circ g = \\bar{f} \\circ (\\pi \\circ g) = \\bar{f} \\circ (\\bar{g} \\circ \\pi ) = (\\bar{f} \\circ \\bar{g}) \\circ \\pi ,$ which implies that $\\pi _{C}(f \\circ g) = \\bar{f} \\circ \\bar{g} = \\pi _{C}(f) \\circ \\pi _{C}(g)$ .",
"$\\pi \\circ (\\alpha _{1} \\circ f) = (\\pi \\circ \\alpha _{1}) \\circ f = (\\alpha _{2} \\circ \\pi ) \\circ f = \\alpha _{2} \\circ (\\pi \\circ f) = \\alpha _{2} \\circ (\\bar{f} \\circ \\pi ) = (\\alpha _{2} \\circ \\bar{f}) \\circ \\pi ,$ which implies that $\\pi _{C}(\\sigma _{1}(f)) = \\alpha _{2} \\circ \\bar{f} = \\sigma _{2}(\\pi _{C}(f))$ , i.e., $\\pi _{C} \\circ \\sigma _{1} = \\sigma _{2} \\circ \\pi _{C}$ .",
"Therefore, $\\pi _{C}$ is a Hom-algebra homomorphism.",
"(c).",
"If $ker(\\pi ) = Z(V_{1})$ , $\\forall x \\in ker(\\pi ), \\varphi \\in C_{\\alpha _{1}^{k}}(V_{1})$ , for any $y \\in V_{1}$ there exists $y^{^{\\prime }} \\in V_{1}$ such that $y = \\alpha _{1}^{k}(y^{^{\\prime }})$ since $\\alpha _{1}$ is surjective.",
"We have $\\mu _{1}(\\varphi (x), y) = \\mu _{1}(\\varphi (x), \\alpha _{1}^{k}(y^{^{\\prime }})) = \\varphi (\\mu _{1}(x, y^{^{\\prime }})) = \\varphi (0) = 0,$ which implies that $\\varphi (x) \\in Z(V_{1}) = ker(\\pi )$ .",
"Hence, every $\\varphi \\in C(V_{1})$ leaves $ker(\\pi )$ invariant.",
"(2).",
"If $\\bar{\\varphi } = 0$ for $\\varphi \\in C_{\\alpha _{1}^{k}}(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi ))$ , then $\\pi (\\varphi (V_{1})) = \\bar{\\varphi }(\\pi (V_{1})) = 0$ .",
"Hence, $\\varphi (V_{1}) \\subseteq ker(\\pi ) \\subseteq Z(V_{1})$ .",
"Hence, $\\varphi (\\mu _{1}(x, y)) = \\mu _{1}(\\varphi (x), \\alpha _{1}^{k}(y)) = 0$ .",
"Note that $V_{1}$ is perfect, we have $\\varphi = 0$ .",
"Therefore, $\\pi _{C} : C(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi )) \\rightarrow C(V_{2}),\\quad f \\mapsto \\bar{f}$ is injective.",
"(3).",
"$\\forall y \\in V_{2}, \\exists y^{^{\\prime }} \\in V_{1}$ such that $y = \\pi (y^{^{\\prime }})$ .",
"For all $x \\in Z(V_{1})$ , $\\mu _{2}(\\pi (x), y) = \\mu _{2}(\\pi (x), \\pi (y^{^{\\prime }})) = \\pi (\\mu _{1}(x, y^{^{\\prime }})) = \\pi (0) = 0,$ which implies that $\\pi (x) \\in Z(V_{2})$ .",
"So $\\pi (Z(V_{1})) \\subseteq Z(V_{2}) = \\lbrace 0\\rbrace $ .",
"Therefore, $Z(V_{1}) \\subseteq ker(\\pi )$ .",
"Note that $ker(\\pi ) \\subseteq Z(V_{1})$ , we have $Z(V_{1}) = ker(\\pi )$ .",
"For all $\\varphi \\in C_{\\alpha _{1}^{k}}(V_{1}), x \\in Z(V_{1}), y \\in V_{1}$ , $\\exists y^{^{\\prime }} \\in V_{1}$ such that $y = \\alpha _{1}^{k}(y^{^{\\prime }})$ , $\\mu _{1}(\\varphi (x), y) = \\mu _{1}(\\varphi (x), \\alpha _{1}^{k}(y^{^{\\prime }})) = \\varphi (\\mu _{1}(x, y^{^{\\prime }})) = \\varphi (0) = 0,$ which implies that $\\varphi (x) \\in Z(V_{1}) = ker(\\pi )$ .",
"So $\\varphi (ker(\\pi )) \\subseteq ker(\\pi )$ , i.e., $\\varphi \\in End_{\\rm {F}}(V_{1}, ker(\\pi ))$ .",
"So $C(V_{1}) \\subseteq End_{\\rm {F}}(V_{1}, ker(\\pi ))$ .",
"Therefore, $C(V_{1}) \\cap End_{\\rm {F}}(V_{1}, ker(\\pi )) = C(V_{1})$ .",
"According to (1) (b), we have $\\pi _{C} : C(V_{1}) \\rightarrow C(V_{2})$ is a Hom-algebra homomorphism."
]
] | 1906.04551 |
[
[
"Using generative modelling to produce varied intonation for speech\n synthesis"
],
[
"Abstract Unlike human speakers, typical text-to-speech (TTS) systems are unable to produce multiple distinct renditions of a given sentence.",
"This has previously been addressed by adding explicit external control.",
"In contrast, generative models are able to capture a distribution over multiple renditions and thus produce varied renditions using sampling.",
"Typical neural TTS models learn the average of the data because they minimise mean squared error.",
"In the context of prosody, taking the average produces flatter, more boring speech: an \"average prosody\".",
"A generative model that can synthesise multiple prosodies will, by design, not model average prosody.",
"We use variational autoencoders (VAEs) which explicitly place the most \"average\" data close to the mean of the Gaussian prior.",
"We propose that by moving towards the tails of the prior distribution, the model will transition towards generating more idiosyncratic, varied renditions.",
"Focusing here on intonation, we investigate the trade-off between naturalness and intonation variation and find that typical acoustic models can either be natural, or varied, but not both.",
"However, sampling from the tails of the VAE prior produces much more varied intonation than the traditional approaches, whilst maintaining the same level of naturalness."
],
[
"Introduction",
"Prosody in natural human speech varies predictably based on contextual factors.",
"However, it also varies arbitrarily, or due to unknown factors [1].",
"Text-to-speech (TTS) voices are typically designed to synthesise a single most likely rendition of a given sentence.",
"While many methods have been proposed to add control to TTS voices, often they do not take this arbitrary variation into account.",
"In contrast, we focus on designing TTS voices that are able to produce any viable prosodic realisation of a given sentence in isolation.",
"Such a system could be driven by contextual information (e.g.",
"provided by a dialogue system) to produce more appropriate prosodic renditions.",
"However, we here focus on the task of producing random (but acceptable) prosodic renditions given an isolated sentence.",
"Since neural statistical parametric speech synthesis (SPSS) became the leading paradigm in speech synthesis research [2] most TTS voices have used static plus dynamic features optimised with mean squared error, followed by maximum likelihood parameter generation (MLPG) and post-filtering [3].",
"These methods are a legacy of hidden Markov model (HMM) SPSS [4], where the problem of oversmoothing was observed and methods were developed to mitigate it.",
"Oversmoothing of acoustic features is still an issue in neural SPSS, due to a combination of assumptions made in designing models [5].",
"Here we focus on prosody (and specifically on modelling intonation, which is the F0 component of prosody) where the symptom of oversmoothing is flatter, more average prosody.",
"We argue that a model designed to synthesise distinct renditions will, by design, not model average prosody.",
"Variational autoencoders (VAEs) are a class of generative models that can learn a smooth latent space approximating the true latent factors of the data.",
"Therefore, we use a VAE [6] to tackle the problem of average prosody, using the latent space to capture otherwise unaccounted-for variation.",
"We propose that by sampling from the low-probability regions of the VAE's prior we can generate idiosyncratic prosodic renditions."
],
[
"Related work",
"Methods for control of SPSS voices roughly fall into two categories: explicitly labelled control and latent control.",
"The former is typically expensive because labelling is labour-intensive, although this can be automated at the expense of accuracy [7], [8].",
"Labelling requires a concrete and consistent schema that can be followed by human annotators.",
"For many aspects of variation in speech this is challenging, a clear example being emotion labelling [9].",
"For example, categorical emotions (e.g.",
"happy or sad) may be too coarse, and appraisal-based measures (e.g.",
"arousal or valence) may be too complex or ambiguous for labellers [10].",
"Additionally, there is the question of elicitation: should natural speech be annotated, or should the variation of interest be elicited (e.g., acted) and assumed to be correct?",
"It has been shown that unsupervised methods can achieve similar results to supervised control [11], which may be related to the challenge of accurately labelling variation in real data, as discussed above.",
"Discriminant condition codes, first proposed for speech recognition [12] have proved useful for multi-speaker TTS [13].",
"The same method has been applied in an unsupervised fashion [14], allowing for control of arbitrary variation.",
"While these methods have been shown to have a probabilistic interpretation [11], they do not model uncertainty or guarantee smoothness in the latent space.",
"As we discuss in Section , this smoothness is important for determining what corresponds to an idiosyncratic (and thus more varied) rendition of a sentence.",
"Tacotron [15] is a sequence-to-sequence model, for which style control using “global style tokens” (GST) has been proposed [16].",
"GSTs produce high quality speech, and can be predicted from text [17].",
"However, individual GSTs cannot be effectively used to produce distinct styles as they are trained as weighted combinations; using individual GSTs leads to significantly degraded audio quality.",
"We expect a random weighting of the tokens will also produce degraded naturalness, since there is no smoothness constraint.",
"VAEs have been demonstrated for speech synthesis [18], [19], voice conversion [20], and intonation modelling [21].",
"Discrete representations have also been incorporated into the VAE framework [22], [23].",
"An experiment with VQ-VAE [23] demonstrated that phones can be learnt with unsupervised training, a result promising for potentially learning discrete prosodic styles.",
"However, in this work we use a continuous latent space.",
"The recently-introduced clockwork hierarchical VAE (CHiVE) [24] is similar to the model we propose here, however our VAE does not make use of the clockwork hierarchical structure and we only predict intonation, while CHiVE predicts F0, duration, and C0.",
"Since we consider isolated sentences, we are not concerned with a single “best” output of our system.",
"Prior work using VAEs has focused on modelling segmental features [23], [18], with some applications to intonation modelling, e.g.",
"for style transfer [24] and predicting latents from text [21].",
"However, our method moves towards TTS systems that can synthesise multiple distinct prosodic renditions (in an unsupervised framework and without the need for control)."
],
[
"Average prosody",
"While many methods have been proposed to add control, there is a more fundamental issue, known as oversmoothing, which leads to flatter, more boring prosody.",
"Typical SPSS uses either feedforward neural networks, or recurrent neural networks (RNNs) to map from a linguistic specification to acoustic features.",
"This mapping is learnt by minimising the mean squared error (MSE) against the ground truth acoustics.",
"MSE is equivalent to minimising the negative log-likelihood (NLL) of a unit-variance Gaussian.",
"This has two effects on such SPSS models: they learn the mean of the data, and are sensitive to outliers.",
"By modelling the mean, SPSS models over-smooth the acoustics – in the context of prosody this is known as average prosody.",
"Methods such as the $\\epsilon $ -contaminated Gaussian [25] exist to handle outliers.",
"However, to fix both issues, it is common to collect speech that is as controlled and consistent as possible in terms of style.",
"Training data with a single style results in models which produce more natural speech [26], but it also limits the voice's stylistic range.",
"If we are interested in producing more varied style/prosody/intonation we need more varied data, but this must then be handled appropriately by our model.",
"Generative models, such as Mixture density networks (MDN) [27], have the ability to handle multiple modes.",
"MDNs parameterise a Gaussian mixture model (GMM) for each acoustic frame which can help with oversmoothing of spectral features [28].",
"However, for prosodic features, we are interested in fixing oversmoothing over a longer timescale, for which frame-level GMMs are less suitable.",
"Instead, we use variational autoencoders which model a distribution in an abstract (latent) space at whichever timescale is preferred."
],
[
"Variational autoencoders",
"Variational autoencoders (VAEs) [6] are a class of latent variable models, i.e.",
"they learn some unsupervised latent representation of the data.",
"They consist of an encoder and a decoder: the encoder parameterises the approximate posterior $q_{{\\phi }}(\\mathbf {z}\\mid \\mathbf {x})$ , which is an approximation of $p_{{\\theta }}(\\mathbf {z}\\mid \\mathbf {x})$ – the underlying factors that describe the data.",
"The decoder is trained to reconstruct the input signal $\\mathbf {x}$ from this latent space, i.e.",
"given a sample from the posterior $\\tilde{\\mathbf {z}} \\sim q_{{\\phi }}(\\mathbf {z}\\mid \\mathbf {x})$ , we reconstruct $\\bar{\\mathbf {x}} \\sim p_{{\\theta }}(\\mathbf {x}\\mid \\tilde{\\mathbf {z}})$ .",
"The encoder and decoder are trained jointly by maximising the evidence lower bound (ELBO), L(, ; x) = -KL(q(zx) p(z)) + Eq(zx) [ p(xz) ] The first term in the ELBO enforces a prior on the approximate posterior, while the second term measures reconstruction error.",
"The Kullback-Leibler (KL) divergence term – used to enforce the prior – puts a cost on using the latent space.",
"This cost on transmitting information through the latent space can encourage the approximate posterior to collapse to the prior, thus encoding no information: posterior collapse.",
"KL-cost annealing is a common way to mitigate posterior collapse [29], where the KL term is down-weighted at the start of training, reducing the cost of encoding information in the latent space.",
"Here we consider conditional VAEs [30], which model F0 conditioned on linguistic features.",
"We use a sentence-level approximate posterior, although a sequence of phrase- or syllable-level latents would be a reasonable alternative.",
"We use an isotropic Gaussian prior $p(\\mathbf {z}) = \\mathcal {N}(\\mathbf {z}; \\mathbf {0}, \\mathbf {1})$ , which gives an analytical form of the KL term.",
"Enforcing a Gaussian prior gives another useful quality: the single mode and smooth pdf means the distance of $q_{{\\phi }}(\\mathbf {z}\\mid \\mathbf {x})$ from the prior mean $\\mathbf {0}$ will be inversely proportional to the similarity of $\\mathbf {x}$ and the largest mode in the data (e.g., the most common prosodic style).",
"That is, the most idiosyncratic $\\mathbf {x}$ will be far from the peak at $\\mathbf {0}$ .",
"This is helpful for our interest in varied prosodic renditions; we can generate varied prosodic renditions using the decoder by sampling low-density regions in the prior.",
"Thus we define two models that use only the VAE decoder, zpeak = 0 xpeak p(xzpeak) ztail vMF(=0) xtail(r) p(xr ztail) where $\\bar{\\mathbf {x}}_{\\textsc {peak}}$ should correspond to the most common mode, i.e.",
"style.",
"Due to the uni-modal prior $p(\\mathbf {x})$ , $\\bar{\\mathbf {x}}_{\\textsc {peak}}$ may instead correspond to an average of multiple styles, i.e.",
"average prosody.",
"Our proposed model uses $\\mathbf {z}_{\\textsc {tail}}$ (uniform samples on a hypersphere's surfaceSampled from a von Mises-Fisher distribution ($vMF$ ) with uniform concentration – wikipedia.org/wiki/Von_Mises–Fisher_distribution) to produce idiosyncratic renditions $\\bar{\\mathbf {x}}_{\\textsc {tail}(r)}$ , where the larger the radius $r$ is the more unlikely the rendition."
],
[
"Systems",
"We focus on modelling intonation, though in the future we plan to extend this to complete prosodic modelling (F0, duration and energy).",
"Modelling only F0 limits the range of variation we can achieve, but reduces the risk of producing unnatural speech: spectral features and durations are taken from natural speech in our experiments, with full TTS left for future work.",
"We use the WORLD vocoder [31], for analysis and synthesis.",
"Our modelsCode is available at github.com/ZackHodari/average_prosody are implemented in PyTorch [32].",
"Figure: Illustration of our models, where only the first three are trained models.",
"vae–peak and vae–tail are different configurations of the VAE model.",
"Blue: learned modules.",
"Green: frame-level inputs.",
"Orange: frame-level predictions.",
"Yellow: sentence-level latent space.We use the same basic recurrent architecture for all trainable modules in Figure REF : a feedforward layer with 256 units, followed by three uni-directional recurrent layers using gated recurrent cells (GRUs) [33] with 64 units, finally any outputs used are projected to the required output dimension.",
"We use 600-dimensional linguistic labels from the standard Unilex question-set and 9 frame-level positional features with min-max normalisation as in the standard Merlin recipe [34].",
"The model predicts logF0, delta (velocity), and delta-delta (acceleration) features with mean-variance normalisation.",
"We use Adam [35] with an initial learning of 0.005, which is increased linearly for the first 1000 batches, and then decayed proportional to the inverse square of the number of batches [36], where our batch size is 32.",
"Early stopping is used based on validation performance.",
"MLPG [37] is used to generate the F0 contour from the dynamic features; predicted standard deviations are used by the MDN, and all other models use the global standard deviation of the training data.",
"The MDN uses four mixture components, whose variances are floored at $10^{-4}$ .",
"To synthesise from the MDN, we use the most likely component sequence (i.e.",
"argmax) to select means and variances used in MLPG.",
"Systems vae–peak and vae–tail in the list below are identical apart from the use of different sampling schemes (see Section ).",
"Their shared model uses a 16-dimensional isotropic Gaussian as the approximate posterior.",
"The latent sample $\\tilde{\\mathbf {z}}$ is broadcast to frame-level and input to the decoder, along with the linguistic features.",
"The decoder predicts static and dynamic logF0 features; as such the reconstruction loss is MSE.",
"The KL-divergence term is weighted by zero during the first epoch and increased linearly to 0.01 over 40 epochs.",
"Using this annealing schedule, the model converged to a KL-divergence of 3.13.",
"Standard RNN-based SPSS model, using MSE.",
"MDN with 4 mixture components, using NLL.",
"VAE decoder using $\\mathbf {z}_{\\textsc {peak}}$ , i.e.",
"the zero vector.",
"VAE decoder using $\\mathbf {z}_{\\textsc {tail}}$ with $r=3$ , i.e.",
"points on the surface of a hypersphere with radius 3.",
"Natural F0.",
"A quadratic polynomial fitted to natural F0.",
"F0 from rnn, scaled vertically by a factor of 3."
],
[
"Purpose of baselines",
"baseline sets a lower bound on naturalness (and variedness): no matter how much variation a system produces, its naturalness should never fall below that of baseline.",
"An upper-bound is copy–synth: no system should be more natural than this, but might sound more varied even though it is unclear whether this would be favoured by listeners.",
"Because we expect that adding more variation will degrade naturalness, we wish to quantify this.",
"rnn–scaled is intended as a lower-bound on naturalness using a naïve method for increasing variation, similar to variance scaling [38].",
"rnn–scaled is intended to demonstrate that vae–tail can produce the same amount of perceived variation but without sacrificing as much naturalness.",
"In this study, setting the amount of perceived variation in rnn–scaled and vae–tail was calibrated in a pilot listening test by the authors, where we attempted to match the level of variation to copy–synth."
],
[
"Hypotheses",
" vae–tail will be much more varied than the typical SPSS systems (rnn, vae–peak, mdn).",
"rnn–scaled, vae–tail, and copy–synth will have the same level of variedness.",
"rnn, vae–peak, and mdn will have a similar level of variedness, where mdn is more varied than the other two.",
"vae–tail will have slightly lower naturalness than the typical SPSS systems (rnn, vae–peak, mdn).",
"vae–tail will be much more natural than the varied baseline rnn–scaled."
],
[
"Data",
"Our choice of training data is motivated by the need for prosodic variation: if the data is very stylistically consistent, there will be too little variation for the VAE to capture in its latent space.",
"We therefore use the Blizzard Challenge 2018 dataset [39] provided by Usborne Publishing.",
"The data consists of stories read in an expressive style for a 4–6 year old audience, with some character voices.",
"Many of the stories include substantial amounts of direct speech.",
"In total it contains 6.5 hours (7,250 sentences) of professionally-recorded speech from a female speaker of standard southern British English.",
"The training-validation-test split described in Watts et al.",
"[14] is used."
],
[
"Evaluation",
"We want to evaluate the amount of variation produced by the systems described.",
"However, variation alone is not a guarantee of “better” speech synthesis [40].",
"For this reason we evaluate quality along with variation.",
"To determine quality we measure naturalness using a standard mean opinion score test, where users were asked to “rate the naturalness” on a 5-point Likert scale.",
"Evaluating variation is less straightforward.",
"We employed a preference test where two systems were compared side by side for the same sentence.",
"Users were asked to choose “which sentence has more varied intonation”, where one sentence must be be marked as “more flat”, and the other as “more varied”.",
"Due to the large number of pairs for 7 systems, we excluded baseline in the pairwise test, as it is clear from the speech samplesSpeech samples available at github.com/ZackHodari/average_prosody that it would be the least varied.",
"However, without baseline in the variation test we lose our lower-bound on variation.",
"We randomly selected 32 test sentences of between 7 and 11 words (1.4 to 4.8 seconds).",
"The naturalness test was performed before the preference test.",
"As there were 22 screens to be completed for each sentence it was necessary to split the test into two halves using a simple 2x2 Latin square between-subjects design.",
"In total we used 30 participants, 15 per listener group, the test took 45 minutes and participants were paid £8."
],
[
"Naturalness test",
"A summary of the naturalness ratings is provided in Figure REF .",
"We perform a Wilcoxon rank-sums significance test between all pairs of systems in the naturalness test, followed by Holm-Bonferroni correction.",
"This statistical analysis is the same as for the Blizzard challenge [41].",
"vae–tail, rnn, mdn, and vae–peak form a group for which we did not find any significant difference.",
"All other system pairs are significantly different, with a corrected p-value of less than 0.00001.",
"Figure: Naturalness results.",
"Solid red lines are medians, dashed green lines are means (cannot be used for statistical comparison), blue boxes show the 25th and 75th percentiles, and whiskers show the range of the ratings, excluding outliers which are plotted with ++.",
"Ordered according to the variation test."
],
[
"Variation test",
"While it is not guaranteed that human preferences are self-consistent, or globally consistentAs described by Arrow's impossibility theorem [42]., we see that the results in Figure REF do form a consistent ordering from most flat to most varied: rnn $\\rightarrow $ vae–peak $\\rightarrow $ mdn $\\rightarrow $ vae–tail $\\rightarrow $ copy–synth $\\rightarrow $ rnn–scaled.",
"However, relative variedness is sometimes inconsistent, e.g.",
"while rnn–scaled is more varied than copy–synth (5th row), we see that the difference between copy–synth and rnn (13th row) is greater than the difference between rnn–scaled and rnn (15th row).",
"We perform a binomial significance test for the 15 pairs in the listening test, followed by Holm-Bonferroni correction.",
"With the correction we find that (rnn, vae–peak), (vae–peak, mdn), and (copy–synth, rnn–scaled) did not show a significant difference: this is indicated by the colouring of those pairs in Figure REF .",
"All other pairs are significantly different, with a corrected p-value of less than 0.0002.",
"Figure: Pairwise variedness results.",
"Pairs are ordered such that the more varied system is on the left.",
"The top 5 rows give the pairs that are consecutive in the ordering, with following rows showing systems that are increasingly further apart in the ordering.",
"We did not find a significant difference for the pairs marked in a lighter colour."
],
[
"Naturalness–Variedness trade-off",
"While this ordering supports our expectations, we cannot clearly comment on their support of our hypotheses in Section as the relative variedness between systems is not clear.",
"Additionally, we would like to clearly compare the trade-off between increasing intonation variation and naturalness.",
"This requires us to represent the pairwise preferences in Figure REF along a single axis.",
"We could approach this using multi-dimensional scaling (MDS) [43]; however, the pairwise preferences correspond to directed edges, not distances.",
"Instead, we formulate the problem as a system of linear equationsWe thank Erfan Loweimi and Gustav Henter for insightful discussions that led to this formulation of the problem..",
"Here, the variables are the positions of each system in the dimension of relative variedness, and each equation describes the “excess preference” of a system pair (the difference between the two system's average preference).",
"This system can be solved using ordinary least squares: $Ax = b \\qquad \\qquad x = (A^TA)^{-1}A^Tb$ where $A \\in \\lbrace -1, 0, 1\\rbrace ^{15 \\times 6}$ and $b \\in \\mathbb {R}^{15 \\times 1}$ encode the pairwise results in Figure REF .",
"Given the solution ($x \\in \\mathbb {R}^{6 \\times 1}$ ) we plot naturalness against relative variedness in Figure REF .",
"Systems to the left have flatter intonation, and systems to the right have more varied intonation.",
"This axis represents human preference and is not intended to be a perceptual scale.",
"In Figure REF , we see that vae–tail is much more varied than the typical SPSS systems (H1).",
"It is also clear that our calibration favoured less variation in vae–tail than copy–synth (rejecting H2), thus we cannot make broad statements about the naturalness-variedness trade-off.",
"However, based on the significant drop in naturalness from rnn to rnn–scaled, and the clustering over relative variedness, we believe that vae–tail would still be significantly more natural than rnn–scaled even if it matched copy–synth's level of variation.",
"rnn, vae–peak, and mdn are clustered along the axis of relative variation, with mdn being significantly more varied, but only by a small amount (H3).",
"Demonstrating that all systems suffer from oversmoothing of F0 to a similar extent.",
"While the mean naturalness of vae–tail is lower than rnn, vae–peak, and mdn, the means cannot be directly compared, and no significant difference was found in Section REF .",
"Rejecting H4 suggests we can produce more varied intonation without sacrificing naturalness.",
"However, we expect that with the ideal calibration we may see some slight degradation in naturalness of vae–tail.",
"We do observe that vae–tail is much more natural than rnn–scaled (H5).",
"Figure: Histogram of logF0 values for each system over all the listening test material.",
"Ordered according to the variation test.Figure: Naturalness-variedness trade-off.",
"Ideally as we increase the amount of prosodic variation our system will not decrease in naturalness.",
"Note that naturalness comparisons can only be made using the significance results in Section ."
],
[
"Calibration",
"The horizontal axis in Figure REF shows vae–tail having much greater perceived intonation variation than mdn, while the logF0 histograms in Figure REF shows them as having the same amount of objective variation – variance of logF0 predictions for the listening test stimuli.",
"This demonstrates that objective measures do not necessarily correspond to perceived variation, which is exactly what makes calibration of vae–tail and rnn–scaled difficult.",
"Figure REF shows that vae–tail has a narrower histogram than copy–synth, however as objective measures do not necessarily correspond to perceived variation we chose not to rely on objective measures for calibration.",
"We have demonstrated the ability to produce varied intonation while maintaining the same level of naturalness, thus mitigating average prosody.",
"However, we have not demonstrated vae–tail's ability to produce multiple distinct prosodic renditions.",
"In Figure REF we present a density of 10,000 F0 contours $\\bar{\\mathbf {x}}_{\\textsc {tail}(3)}$ produced using samples $\\mathbf {z}_{\\textsc {tail}} \\sim vMF(\\kappa =0)$ .",
"As expected, the F0 contours produced vary smoothly, but more importantly we see that they vary between multiple distinct contours.",
"For this sentence we see that there may be three distinct contours.",
"We are interested in evaluating the distinctiveness of multiple different samples from vae–tail; however, this is out of the current scope.",
"While mdn is also a generative model, sampling from the frame-level GMMs is not straightforward.",
"MLPG can be used to select the single best trajectory [37].",
"But to produce multiple renditions from mdn we must choose a sequence of Gaussian components.",
"However, randomly choosing components produces noisy F0 contours, and using the same component for the entire sequence does not produce distinct performances.",
"This is likely because the components don't represent modes of the data, but behave in a similar way to the $\\epsilon $ -contaminated Gaussian distribution [25].",
"Figure: Density of F0 predictions made by vae–tail for the sentence \"Goldilocks skipped around a corner and saw...\""
],
[
"Conclusion",
"We have demonstrated that output from typical RNN and MDN models exhibits flat intonation.",
"Additionally, we have provided evidence that sampling from the tails of a VAE prior produces speech that is much more varied than typical SPSS while maintaining the same level of naturalness.",
"In future we plan to undertake a full evaluation of this trade-off, to determine if and when this method begins to improve or degrade in quality.",
"In future work, we plan to: use MUSHRA in place of a preference test; use a neural vocoder; make use of seq2seq models with attention instead of upsampling the linguistic features; predict other prosodic features; and make use of either a discrete latent space [22] or a mixture model VAE prior [44].",
"Acknowledgements: Zack Hodari was supported by the EPSRC Centre for Doctoral Training in Data Science, funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016427/1) and the University of Edinburgh.",
"Oliver Watts was supported by EPSRC Standard Research Grant EP/P011586/1."
]
] | 1906.04233 |
[
[
"Tuning morphology and thermal transport of asymmetric smart polymer\n blends by macromolecular engineering"
],
[
"Abstract A grand challenge in designing polymeric materials is to tune their properties by macromolecular engineering.",
"In this context, one of the drawbacks that often limits broader applications under high temperature conditions is their poor thermal conductivity $\\kappa$.",
"Using molecular dynamics simulations, we establish a structure-property relationship in hydrogen bonded polymer blends for possible improvement of $\\kappa$.",
"For this purpose, we investigate two experimentally relevant hydrogen bonded systems -- one system consists of short poly({N}-acryloyl piperidine) (PAP) blended with longer chains of poly(acrylic acid) (PAA) and the second system is a mixture of PAA and short poly(acrylamide) (PAM) chains.",
"Simulation results show that PAA-PAP blends are at the onset of phase separation over the full range of PAP monomer mole fraction $\\phi_{PAP}$, which intensifies even more for $\\phi_{PAP} > 0.3$.",
"While PAA and PAP interact with preferential hydrogen bonding, phase separation is triggered by the dominant van der Waals attraction between the hydrophobic side groups of PAP.",
"However, if PAP is replaced with PAM, which has a similar chemical structure as PAP without the hydrophobic side group, PAA-PAM blends show much improved solubility.",
"Better solubility is due to the preferential hydrogen bonding between PAA and PAM.",
"As a result, PAM oligomers act as cross-linking bridges between PAA chains resulting in a three dimensional highly cross-linked network.",
"While $\\kappa$ for PAA-PAP blends remain almost invariant with $\\phi_{PAP}$, PAA-PAM systems show improved $\\kappa$ with increasing PAM concentration and also with respect to PAA-PAP blends.",
"Consistent with the theoretical prediction for the thermal transport of amorphous polymers, we show that $\\kappa$ is proportional to the materials stiffness, i.e., the bulk modulus K and sound velocity v of PAA-PAM blends."
],
[
"Introduction",
"Polymers are ubiquitous in our everyday life, finding a variety of applications ranging from physics to materials science and chemistry to biology [1], [2], [3], [4], [5], [6].",
"The properties of polymers are intimately linked to large conformational and compositional fluctuations.",
"Because of the molecular flexibility, polymer conformations can be tuned almost at will for desired applications and thus provide a robust platform for advanced functional materials design.",
"However, one of the drawbacks of standard commodity polymeric materials is the poor thermal conductivity $\\kappa $ in their amorphous states [7], [8], [9].",
"This is partially because of rather weak van der Waals interactions dictating polymer properties.",
"Therefore, it is desirable to tune $\\kappa $ of polymeric materials, especially when they are used in high temperature environments.",
"One of the standard protocols to improve $\\kappa $ of polymeric materials is blending them with materials having $\\kappa $ values exceeding the thermal conductivity of metals, such as carbon based materials [10], [11], [12], [13].",
"In this context, following the arguments of continuum theory, one should expect to increase $\\kappa $ of polymer composites with increasing concentration of the high $\\kappa $ guest.",
"Moreover, establishing a tunable structure-function relationship in these composite materials is often difficult because they exhibit large spatial and temporal heterogeneities.",
"Furthermore, a significant improvement in $\\kappa $ also requires concentrations of external guest material exceeding their percolation threshold, thus also losing the inherent property and flexibility of the host polymeric systems.",
"Therefore, a more attractive protocol to tune $\\kappa $ is to strengthen microscopic interactions within the polymer system itself.",
"Here, smart polymers may serve as ideal candidates.",
"A polymer is referred to as “smart\", when they exhibit fast responsiveness to a change in their environment in solutions and are typically dictated by hydrogen bonding whose strength typically falls within the range of 4-8 $k_{\\rm _B}T$ , thus exceeding significantly the van der Waals pair interactions that are only of the order of less than $k_{\\rm _B}T$ [1], [2], [4], [5].",
"Therefore, the solvent-free dry states of these hydrogen bonded polymers are dictated by strong inter-polymer interactions.",
"In this context, there is considerable interest to study thermal transport in tunable polymer materials [14], [15], [16], smart hydrogels [17], [18], concentrated polymer solutions [19], and solid polyelectrolytes [20].",
"In particular, a recent work uses the idea of hydrogen bonding as a tunable interaction to propose a wide range of polymer blends, where a significant increase in $\\kappa $ was observed [14].",
"One of the interesting systems this experiment proposes is an asymmetric blend of poly(N-acryloyl piperidine) (PAP) and poly(acrylic acid) (PAA) with M$_{\\rm w}^{\\rm PAA} \\gg $ M$_{\\rm w}^{\\rm PAP}$ , where M$_{\\rm w}$ is the molecular weight.",
"Schematic representations of chemical structures of PAA and PAP are shown in Figures REF (a-b).",
"Figure: Schematic representations of-- (a) poly(acrylic acid) (PAA),(b) poly(N-acryloyl piperidine) (PAP) and (c) poly(acrylamide) (PAM) systems.The length of PAA was chosen as N ℓ PAA =30N_{\\ell }^{\\rm PAA} = 30, the lengths of PAP and PAM is taken as 3.Highlighted blue regions are possible groups forming hydrogen bonding, while the black and red regions arehydrophobic in nature.It was reported that, for a PAP monomer mole fraction $\\phi _{\\rm PAP} \\sim 0.3$ , $\\kappa $ increases by a factor of about 6$-$ 7 times with respect to $\\kappa \\sim 0.25$ Wm$^{-1}$ K$^{-1}$ of pure PAA (or pure PAP) [14].",
"This increase was attributed to the strong PAA-PAP hydrogen bonding, which also demands as prerequisite that the binary solution of PAA and PAP are fairly miscible.",
"In a separate experimental study, however, it was found that PAA and PAP phase separate, while no variation in $\\kappa $ was observed over the full range of $\\phi _{\\rm PAP}$ [15].",
"If a system is phase separated, it is expected to show reduced $\\kappa $ because of the weakened interfacial interaction between two phase separated regions.",
"However, because of the delicate interplay between the van der Waals and hydrogen bonding interactions in these systems, it is not always straightforward to predict molecular level morphology and its connection to thermal transport, which is the motivation behind this study.",
"To best of our knowledge, there is no theoretical/computational work addressing polymer blends in their dry states for tunable $\\kappa $ .",
"In this work, we present molecular dynamics simulation results establishing a structure-property relationship in polymer blends.",
"For this purpose, we investigate two experimentally relevant systems$-$ one system is a simulation replica of the PAA-PAP blend reported earlier [14] and the second system consists of a PAA and poly(acrylamide) (PAM) blend, where PAM has a very similar chemical structure as PAP without the hydrophobic carbon ring, see Figure REF (c).",
"The remainder of the paper is organized as follows: In section we sketch our methodology.",
"Results and discussions are presented in sections and finally the conclusions are drawn in section ."
],
[
"Method and model",
"In this work all-atom molecular dynamics simulations are performed at two stages: the GROMACS 4.6 package [21] was used for the equilibration and structural analysis of the polymer blends, while the LAMMPS package [22] was used for the thermal transport calculations.",
"GROMACS simulations are performed in the NpT ensemble, where N is the number of particles in a system, p is the isotropic pressure, and T is the temperature.",
"T= 600 K is set for the initial simulations using a Berendsen thermostat with a coupling constant 2 ps.",
"This ensures that the polymer blends are in their melt states.",
"p is kept at 1 bar using a Berendsen barostat with a coupling time of 0.5 ps [23].",
"Electrostatics are treated using the particle mesh Ewald method [24].",
"The interaction cutoff for non-bonded interactions is chosen as 1.0 nm.",
"The simulation time step is chosen as $\\Delta t = 1$ fs and the equations of motion are integrated using the leap-frog algorithm [25].",
"All bond vibrations are constrained using a LINCS algorithm [26].",
"We investigate two different polymer systems, namely PAA-PAP and PAA-PAM blends.",
"Configurations consist of a total of 200 chains, where the length of PAA is chosen as $N_{\\ell }^{\\rm PAA} = 30$ , while the lengths of both PAP and PAM oligomers are taken as $N_{\\ell }^{\\rm PAP}$ (or $N_{\\ell }^{\\rm PAM}$ ) = 3.",
"In a series of simulations, the PAP monomer mole fractions $\\phi _{\\rm PAP}$ and PAM monomer mole fractions $\\phi _{\\rm PAM}$ are varied between $0.0-1.0$ , where $\\phi _i = 0.0$ corresponds to pure PAA and $\\phi _i = 1.0$ represents pure PAP or PAM systems.",
"The standard OPLS force field was used for PAA and PAP systems [27].",
"For PAM we have used modified parameters that were developed by two of us earlier [28].",
"While both these force fields were previously used for the study of polymers in solution, in the supplementary materials we provide evidence that these force fields are also suitable to study dry polymer films.",
"After an initial equilibration of 20 ns, production runs are performed for 60 ns and the configurations were saved each 50 fs for the calculation of the structural properties.",
"These simulation runs are at least one order of magnitude larger than the longest relaxation time $\\tau \\sim 5$ ns of a PAA chain with $N_{\\ell }^{\\rm PAA} = 30$ and T = 600 K, which is estimated using the end-to-end distance auto-correlation function $\\left<{\\bf R}(t)\\cdot {\\bf R}(0) \\right> \\sim e^{-t/\\tau }$ .",
"During the production runs, observables such as the radial distribution function ${\\rm g}_{ij} (r)$ and the number of hydrogen bonds (h$-$ bond) for different solution components are calculated.",
"H$-$ bond are calculated using the standard geometrical criterion implemented in GROMACS, i.e., a hydrogen bond exists when the donor-acceptor distance is $\\le 0.35$ nm and the acceptor-donor-hydrogen angle is $\\le 30^{\\circ }$ .",
"The final configurations from the GROMACS simulations were first quenched down to $T = 300$ K and then imported in LAMMPS for $\\kappa $ calculations.",
"Note that both pure PAA and PAP have their glass transition temperatures $T_{\\rm g} \\sim 373$ K [14], while $T_{\\rm g}$ of bulk PAM exceeds 430 K [29].",
"Therefore, without attempting to identify a precise $T_{\\rm g}$ value for different blends and following a simple combination rule, one expects $T = 300$ K to be well below their corresponding $T_{\\rm g}$ .",
"This, however, also assumes a priory that the bulk solution is miscible.",
"Figure: Part (a) shows the time equilibration of temperature difference between the hot T hot T_{\\rm hot} and the cold T cold T_{\\rm cold}regions of a PAA-PAM blend for φ PAM =0.286\\phi _{\\rm PAM} = 0.286.",
"Part (b) shows the steady-state temperature profile withbin index along the z-z-axis of the simulation domain.",
"Note that while we sub-divide simulation domain intoeight chunks, nine chunks are shown because of the periodic boundary condition.For the calculation of $\\kappa $ , we employ a non-equilibrium method [30].",
"In this method, a heat flux $J$ through the system is generated over a simulation time $\\mathcal {T}$ in the microcanonical ensemble by swapping atomic velocities between a hot and a cold region of the simulation box, $J=\\frac{1}{2\\,A\\mathcal {T}} {\\sum _{\\text{swaps}}\\Delta E_{\\text{kin}}},$ where $\\Delta E_{\\text{kin }}$ denotes the kinetic energy exchanged per swap, $A$ is the cross sectional area perpendicular to the direction of heat flow, and the factor 2 accounts for the two directions of heat flow present in systems with periodic boundaries.",
"As a result of velocity swapping, once the system reaches its steady-state, a temperature gradient $\\Delta T/\\Delta z$ along the transport direction $z$ can be extracted and the thermal conductivity $\\kappa $ is calculated by applying Fourier's law of heat conduction, $\\kappa =\\frac{J}{\\left|\\Delta T/\\Delta z\\right|}.$ Here we chose to divide the simulation box into eight slabs of equal width along the $z-$ direction.",
"This will lead to at least $\\sim 1500$ atoms per slab.",
"Note that when dealing with PAA-PAP blends special care needs to be taken in choosing the slab width because of the phase separation.",
"Velocity swapping was performed between the slowest atom in the center slab and the fastest atom in the slab at the cell boundary.",
"This swapping was performed every 20 fs with $\\Delta t = 0.2$ fs.",
"After an initial steady-state equilibration for $10^{6}$ time-steps (see Figure REF (a)), the heat flux was computed over a simulation time of $5\\cdot 10^{5}$ time-steps.",
"Finally, a linear fit of the temperature profile as a function of slab index was used to calculate $\\kappa $ by means of Eq.",
"(REF ), as shown in Figure REF (b).",
"We have also attempted to calculate $\\kappa $ using the equilibrium Kubo-Green method in LAMMPS [31], which, however, overestimates $\\kappa $ by about an order of magnitude.",
"This can be attributed to the heat-flux autocorrelation function routine of the LAMMPS code that only considers pair-wise interactions.",
"Systems with many-body interactions may, therefore, lead to problems in $\\kappa $ calculations.",
"More specifically, Kubo-Green should give the same $\\kappa $ values as in a non-equilibrium method.",
"This, however, also require to properly accounting angular and dihedral interactions in the all-atom force fields for the heat-flux calculations [32].",
"This was also identified earlier for the simulations of carbon based materials [33]."
],
[
"Morphology of polymer blends in the melt state",
"We start our discussion by investigating the molecular level morphologies of PAA-PAP blends at T = 600 K. For this purpose, we calculate pair correlation functions ${\\rm g}_{ij} (r)$ between different solution components.",
"Because the properties of these systems are dictated by hydrogen bonding, ${\\rm g}_{ij}(r)$ are calculated only between oxygen and hydrogen of PAA and oxygen and nitrogen of PAP, see highlighted blue components in Figure REF .",
"In Figures REF (a-c), we show ${\\rm g}_{ij} (r)$ between different monomeric species for two different $\\phi _{\\rm PAP}$ .",
"A closer look at the data for $\\phi _{\\rm PAP} = 0.024$ (black curves in Figures REF (a-c)) reveals two important length scales: (a) The most prominent fluid structure is observed for $r \\le 1$ nm represented by the correlation peaks.",
"(b) ${\\rm g}_{ij} (r) \\rightarrow 1$ for $r \\ge 1.5$ nm, which is the typical correlation length in the hydrogen bonded systems [35].",
"Furthermore, the long tail decay, as observed for $\\phi _{\\rm PAP} = 0.320$ (red curves in Figures REF (a-c)), indicates that the system is at the onset of phase separation.",
"Figure: Radial distribution functions g ij (r){\\rm g}_{\\rm ij}(r) between three different monomeric pairsin polymer blends consisting of poly(N-acrylyol piperidine) (PAP) and poly(acrylic acid) (PAA): (a)PAA-PAA, (b) PAA-PAP, and (c) PAP-PAP.",
"Results are shown for two different PAP monomer molar fractionsφ PAP \\phi _{\\rm PAP} under their melt states at a temperature of T=600=600 K and ambient pressure.Number of repeat units of PAP and PAA was chosen to be 3 and 30, respectively.For the calculation of g ij (r){\\rm g}_{\\rm ij}(r), we only consider oxygens and hydrogen of PAA and oxygen andnitrogen of PAP, as highlighted by blue species in Figure .",
"Inset in part (c) is the enlarged viewof g PAP - PAP (r){\\rm g}_{\\rm PAP-PAP}(r) for φ PAP =0.320\\phi _{\\rm PAP} = 0.320 highlighting first peak height.",
"Parts (d-f) shows thecumulative integral of the second virial coefficient 𝒱 ij (r)\\mathcal {V}_{ij} (r) between solution components.Inset in part (d) highlights the diverging 𝒱 PAA - PAA (r)\\mathcal {V}_{\\rm PAA-PAA} (r) for φ PAP =0.320\\phi _{\\rm PAP} = 0.320.The pair correlation function ${\\rm g}_{\\rm ij}(r)$ not only gives information about the pairwise solution structure, it also provides information about solution thermodynamics via the second virial coefficient, $\\mathcal {V}_{ij}(r) &=& 2\\pi \\int _0^{\\infty } \\left[1 - e^{-{\\rm V}_{ij}(r)/k_{\\rm B}T} \\right] r^2 dr \\nonumber \\\\&=& 2\\pi \\int _0^{\\infty } \\left[1 - {\\rm g}_{ij}(r) \\right] r^2 dr.$ This assumes $V_{ij}(r) = - k_{\\rm B}T \\ln \\left[{\\rm g}_{ij}(r)\\right]$ [34].",
"In polymer science, $\\mathcal {V}_{ij}$ is also known as the excluded volume and is defined by the plateau value of the cumulative integral ${\\mathcal {V}}_{ij}(r)$ for $r$ value above the correlation length.",
"For example, the interaction between $i$ and $j$ is repulsive when $\\mathcal {V}_{ij} > 0$ and attractive when $\\mathcal {V}_{ij} < 0$ .",
"When $\\mathcal {V}_{ij} = 0$ , the long range energetic attraction gets exactly canceled by the short range entropic repulsion, which is also known as the “so called\" $\\Theta -$ point (or a critical point) that is dictated by large diverging fluctuations.",
"Note that the convergence of ${\\mathcal {V}}_{ij}(r)$ for large $r$ values suffers from severe system size effects, especially for multi-component solutions [35].",
"Moreover, in this study we have chosen system sizes to be large enough to avoid system size effects.",
"Figures REF (d-f) show ${\\mathcal {V}}_{ij}(r)$ for three different pairs of PAA-PAP blends.",
"For $\\phi _{\\rm PAP} = 0.024$ it can be appreciated that both PAA-PAA and PAA-PAP interactions are weak, while the PAP-PAP interaction is highly attractive as indicated by large negative value of ${\\mathcal {V}}_{\\rm PAP-PAP}(r)$ .",
"Almost equal preference for the interactions between PAA-PAA and PAA-PAP are not surprising, given that these species are hydrogen bonded.",
"Moreover, the dominant contribution of the PAP-PAP interaction comes from the van der Waals interaction between the hydrophobic side ring of PAP (highlighted by red in Figure REF ).",
"We also want to emphasize that even when van der Waals interaction between two individual particles is rather weak (i.e., less than $k_{\\rm B}T$ ), collectively they may result in several $k_{\\rm B}T$ of interaction strength, as seen here for the PAP-PAP coordination.",
"Furthermore, a diverging ${\\mathcal {V}}_{\\rm PAA-PAP}(r)$ for $\\phi _{\\rm PAP} = 0.320$ further indicates phase separation in PAA-PAP blend.",
"Figure: Radial distribution function g ij (r){\\rm g}_{\\rm ij}(r) between three different monomeric pairsin polymer blends for two different systems.",
"One system is a blend oftrimer of poly(N-acrylyol piperidine) (PAP) is blended in with poly(acrylic acid) (PAA) withlength N ℓ PAA =30N_{\\ell }^{\\rm PAA} = 30 (red curves) and the second systems is a mixture of PAA of same length withtrimer of poly(acrylamide) (PAM) (black curves).",
"The data is shown for 0.320 monomer mole fraction ofshorter species, T=600=600 K and ambient pressure.",
"Parts (a-c) show-- PAA-PAA, PAA-ii and i-ii-i structures,with the unit ii can either be PAP or PAM as specified in the legend.Solution processing of polymers with distinct nanoscopic interfaces, as in the case of phase separation, and their use for tunable thermal, mechanical, optical and/or rheological properties is always a paramount challenge.",
"Therefore, it is desirable to have better miscible systems for advanced applications.",
"In this context, since the phase separation in a PAA-PAP blend is dictated by interactions between the hydrophobic side groups of PAP, one possibility to improve solubility of a blend might be to remove the side carbon ring in a PAP monomer structure.",
"Here PAM may serve as an ideal candidate (see Figure REF (c)).",
"PAM is an easy replacement because it has a similar monomer structure as PAP without the carbon ring.",
"Additionally, PAM is a water soluble polymer [28], unlike PAP [15] that is hydrophobic.",
"The added advantage of using PAM arises from more possibility of hydrogen bonds in comparison to PAP, thus forming stronger contacts between two particles.",
"In Figure REF we show a component-wise ${\\rm g}_{ij}(r)$ for two different blends.",
"It can be appreciated that the system shows better tail convergence for PAA-PAM systems in comparison to PAA-PAP blends, i.e., ${\\rm g}_{ij}(r)=1$ around the correlation length of 1.5 nm.",
"This indicates a much better solubility in the system, as expected by the structural tuning of the monomer units discussed in the preceding paragraph.",
"Figure: Space filling representation of the simulation snapshots for PAA-PAP blend part(a) and PAA-PAM blend part(b).The data is shown for 0.320 monomer mole fraction of shorter species, T=600=600 K and ambient pressure.",
"Yellow spheresrepresent PAA atoms and blue spheres show PAP atoms (panel (a)) and PAM atoms (panel (b)).An illustration of molecular level morphologies in two blends are shown in Figure REF .",
"It is evident that PAA-PAM is more homogeneous, while PAA-PAP shows distinct islands.",
"Having discussed morphologies of smart polymer blends, we now move to understand the correlation between morphologies and $\\kappa $ in the dry states of these systems."
],
[
"Hydrogen bonding and thermal transport in polymer blends below the glass transition temperature",
"Hydrogen bonding (H$-$ bond) is an important molecular level interaction in these smart materials [1], [2], [4], [5], [36].",
"Therefore, we now investigate possible H$-$ bonds between different solution species.",
"In Figure REF , we show the fraction of H$-$ bonds $\\phi ^{\\rm H-bond}_{{\\rm PAA}-i}$ between PAA and the other species $i$ , which can be either PAP or PAM.",
"In a nutshell, if $\\phi ^{\\rm H-bond}_{{\\rm PAA}-i}$ is above the blue line (i.e., linear extrapolation between two concentrations with unity slope and zero intercept), there is an excess of H$-$ bonds for a given monomer mole fraction $\\phi _i$ .",
"Figure: Fraction of hydrogen bond φ PAA -i H- bond \\phi ^{\\rm H-bond}_{{\\rm PAA}-i} between cross components of two polymerblends (PAA-PAP and PAA-PAM blends) as a function of monomer mole fraction φ i \\phi _i, with ii can either bePAP or PAM.",
"Blue dashed line is a linear interpolation between two values of φ i \\phi _i with unit slope andzero intercept.",
"The data is shown for the dry state of polymer blends for T=300=300 K and ambient pressure.For the PAA-PAP systems (red stars in Figure REF ) it can be seen that even when PAA and PAP phase separate, there is an excess concentration of H$-$ bonds between PAA and PAP.",
"This is because both PAA and PAP form isolated islands having their side groups dangling within the interface between two phase separated regions.",
"Therefore, they can still facilitate interfacial H$-$ bonds acting as adhesive contacts between two islands.",
"Here it is also worth mentioning that the phase separation, as observed in PAA-PAP case, may not be a standard spinodal decomposition [37].",
"More specifically, the PAP oligomers are glued together by their hydrophobic contacts leading to the phase separation.",
"To better understand the thermodynamic origin of the phase separation in PAA-PAP blends a more detailed investigation is needed, which is beyond the scope of the present study.",
"Furthermore, we expect these small islands to coarsen over longer simulation times even if they are driven by weak surface tension.",
"Moreover, our simulations already show a clear signature that the PAA-PAP systems are at the onset of phase separation (see Figure REF ).",
"Figure: Thermal transport coefficient κ\\kappa forPAA-PAP blend as a function of PAP monomer mole fraction φ PAP \\phi _{\\rm PAP}.The data is shown for the dry state (below the glass transition temperature) of polymer blends fortemperature T=300=300 K and ambient pressure.",
"A typical error of 10% is estimated from fivedifferent κ\\kappa calculations using different random seeds during microcanonical simulations.For comparison, we have also included experimentally reported κ\\kappa values obtained forPAA-PAP blends and the homopolymer data for pure PAA (φ PAP =0.0\\phi _{\\rm PAP}=0.0)and for pure PAP (φ PAP =1.0\\phi _{\\rm PAP}=1.0) .Having two glued regions, as in the case of PAA-PAP blends, does not necessarily mean that one can also expect to have a variation in $\\kappa $ within the intermediate mixing ratios of $\\phi _{\\rm PAP}$ .",
"Instead the overall behavior is expected to be dominated by $\\kappa $ values of two individual components, with rather weak interfacial interactions.",
"Therefore, following the simple mixing rule, one should only expect a smooth interpolation of $\\kappa $ between two pure phases of PAA and PAP.",
"Indeed, as shown by the simulation data in Figure REF (red stars), $\\kappa $ varies rather monotonically with $\\phi _{\\rm PAP}$ .",
"It should also be noted that$-$ for the pure phases of PAA and PAP, our calculated values $\\kappa \\sim 0.32$ Wm$^{-1}$ K$^{-1}$ (for $\\phi _{\\rm PAP} = 0.0$ ) and $\\kappa \\sim 0.27$ Wm$^{-1}$ K$^{-1}$ (for $\\phi _{\\rm PAP} = 1.0$ ) are in good agreement with the experimental data, see Figure REF [14], [15].",
"For example, one experiment reported $\\kappa \\sim 0.22$ Wm$^{-1}$ K$^{-1}$ (for $\\phi _{\\rm PAP} = 0.0$ ) and $\\kappa \\sim 0.20$ Wm$^{-1}$ K$^{-1}$ (for $\\phi _{\\rm PAP} = 1.0$ ) [14], while another set of experimental data reported $\\kappa \\sim 0.37$ Wm$^{-1}$ K$^{-1}$ (for $\\phi _{\\rm PAP} = 0.0$ ) and $\\kappa \\sim 0.16$ Wm$^{-1}$ K$^{-1}$ (for $\\phi _{\\rm PAP} = 1.0$ ) [15].",
"Furthermore, our simulation data for intermediate mixing ratios of $\\phi _{\\rm PAP}$ is in clear contradiction with one set of earlier published experimental data [14] (see the data set corresponding to the black circles in Figure REF ), while it is in agreement with another set of experimental observations [15].",
"For the PAA-PAM system, we observe a higher $\\phi ^{\\rm H-bond}_{{\\rm PAA}-{\\rm PAM}}$ (black filled circles in Figure REF ).",
"This excess is also coupled with an improvement in $\\kappa $ for PAA-PAM blends in comparison to PAA-PAP systems, see corresponding data with black filled circles in Figure REF (a).",
"Figure: Part (a) shows comparative data of thermal transport coefficient κ\\kappa obtained from simulations forPAA-PAM and PAA-PAP blends as a function of monomer mole fraction φ i \\phi _i, with ii can either bePAP or PAP.",
"The data is shown for the dry state (below glass transition temperature) of polymer blends fortemperature T=300=300 K and ambient pressure.",
"A typical error of 10% is estimated from fivedifferent κ\\kappa calculations using different random seeds during microcanonical simulations.For comparison, we have also included experimentally reported κ\\kappa values obtained forpure PAA (φ PAP =0.0\\phi _{\\rm PAP}=0.0) , and pure PAM (φ PAP =1.0\\phi _{\\rm PAP}=1.0) .Lines are drawn to guide the eye.Parts (b-c) show κ\\kappa as a function of bulk modulus KK and sound velocity v=K/ρv = \\sqrt{K/\\rho } calculated for both blends,where ρ\\rho is the mass density.",
"The black solid lines are linear fits to the PAA-PAM data.The improvement of $\\kappa $ for PAA-PAM as compared to PAA-PAP is not surprising, given that PAA and PAM are fairly miscible because of preferential H$-$ bonding between PAA and PAM (see Figure REF and Figure REF ).",
"Moreover, to further investigate the tunability of $\\kappa $ , we have also looked into the minimal thermal conductivity model [15], [38], [39].",
"Within this theory for amorphous polymer (as in our cases), $\\kappa $ relates directly to the materials stiffness, thus is also related to the glass transition temperature $T_{\\rm g}$ of amorphous systems [14] and the sound wave velocity $v$ .",
"The higher the stiffness (or $v$ ), the larger the corresponding $\\kappa $ [40].",
"In this context and for the pure phases of PAA, PAP and PAM, we find that our calculated $\\kappa $ values follow the trend $\\kappa ^{\\rm PAM} > \\kappa ^{\\rm PAA} > \\kappa ^{\\rm PAP}$ and are consistent with $T_{\\rm g}^{\\rm PAM} > T_{\\rm g}^{\\rm PAA} > T_{\\rm g}^{\\rm PAP}$ , see the supplementary Fig.",
"1 and Table I.",
"Furthermore, the sound velocity can be estimated using the Newton-Laplace equation $v = \\sqrt{K/\\rho }$ , where $\\rho $ is the mass density and $K$ is the bulk modulus.",
"Here, volume fluctuations are used to calculate $K$ from NpT simulations using the expression $K = k_{\\rm B}T \\frac{\\left<V\\right>}{\\left<V^2\\right> - \\left<V\\right>^2}.$ As expected from the theory [15], [38], [39] and observed earlier in experiments [15], our simulation data for PAA-PAM blends show that $\\kappa $ is proportional to $K$ and $v$ , see black symbols in Figures REF (b-c).",
"Additionally, the lack of correlation between $\\kappa $ and $K$ (or $v$ ) for PAA-PAP is due to the phase separation in the systems, see red symbols in Figures REF (b-c).",
"From the $K$ and $v$ values, we have also estimated the typical range of Debye temperatures $\\Theta _{\\rm D}$ using the expression in Ref.",
"[41].",
"$\\Theta _{\\rm D}$ ranges between 200$-$ 250 K for different blends, while simulations are conducted at $T= 300$ K. Therefore, quantum effects can be neglected [42], thus classical molecular dynamics is an appropriate technique for these systems.",
"Figure REF (a) also shows that $\\kappa $ for PAA-PAM systems first weakly increases and then decreases again, attaining a maximum around $\\phi _{\\rm PAM} \\sim 0.30$ (see black filled circles).",
"On the other hand, $\\kappa $ for PAA-PAP systems decreases with $\\phi _{\\rm PAP}$ (see red stars).",
"Here the PAA-PAM H$-$ bonds are preferred (over PAA-PAA and PAM-PAM H$-$ bonds) because the maximum possible H$-$ bonds between a PAA and a PAM monomer is about four, which collectively can lead to more than $10 k_{\\rm B}T$ energy per contact.",
"On the other hand, two PAA or two PAM monomers can maximally have one or two possible H$-$ bonds between them, respectively [28], thus leading to lesser contact energy between same monomeric species.",
"The strongly H$-$ bonded contact between PAA and PAM can also explain the non-monotonous variation of $\\kappa $ with $\\phi _{\\rm PAM}$ (see black solid circles in Figure REF (a)).",
"In this context, we find that the short PAM oligomers act as cross-linking bridges between two (or more) PAA monomers belonging to different polymers.",
"A simplified schematic of this bridging scenario is presented in Figure REF .",
"Figure: A schematic representation of short PAM oligomers (blue lines) forming bridges betweenPAA monomers (yellow lines).Here the degree of cross linking is dictated by $\\phi _{\\rm PAM}$ .",
"The higher the $\\phi _{\\rm PAM}$ upto a threshold concentration, the larger the bridging connectivity and thus increased stiffness (as estimated from $K$ ) of materials.",
"This increased $K$ then leads to elevated $\\kappa $ .",
"Furthermore, the maximum value of $\\kappa $ is observed around $\\phi _{\\rm PAM} \\sim 0.3-0.4$ .",
"This is expected because when a small amount of PAM are blended in the PAA material, each PAM oligomer will bind to more than one PAA monomer to reduce the binding free energy [2], [43].",
"It should also be noted that the size of a PAM trimer is of the order of 0.75 nm, which is also typically equivalent to 2$-$ 3 times the PAA monomer size (see Figure REF ).",
"This length scale consistency also leads to almost perfect packing for PAA-PAM systems and thus forming three dimensional cross-linking networks.",
"Moreover, when $\\phi _{\\rm PAM}$ is increased above a threshold value (for example $\\phi _{\\rm PAM} > 0.4-0.5$ ) the effect is expected to be diluted because almost all PAA monomer will have at least one PAM to bind.",
"Therefore, $\\kappa $ values will then be dominated by the pure phases of the individual polymers, see Figure REF (a).",
"Lastly we would also like to emphasize that even when PAA-PAP blends phase separate, leading to no noteceable change in $\\kappa $ with varying $\\phi _{\\rm PAP}$ (see Figure REF ), di-block copolymer architectures (consisting of PAA and PAP blocks) may lead to interesting lamellar mesophases [44], [45] and, therefore, providing another route towards the tunability of $\\kappa $ in amorphous polymers.",
"In this context, layered superlattices (or thermal band gap materials) have been shown to exhibit interesting thermal behavior because of their interfacial properties [46].",
"Here, however, it should also be noted that in lamellar phases of di-block copolymers, unlike superlattices, phonon interference may not be expected to give a dominant contribution because the phonon mean free path is vanishingly small in amorphous solids and polymers [15], [39]."
],
[
"Conclusions and outlook",
"In this work, we have used molecular dynamics simulations to study thermal transport of asymmetric smart polymer blends and its connection to underlying macromolecular morphologies.",
"For this purpose, we investigate two experimentally relevant polymer blends.",
"Our structural analysis suggests that$-$ while a system of PAA-PAP blends are at the onset of phase separation, a system of PAA and PAM is fairly well miscible with significant excess hydrogen bonded interaction between cross species.",
"The short PAM chains act as cross-linking bridges between monomers of different PAA chains forming a three dimensional (hydrogen bonded) highly cross-linked smart polymer network, thus increasing materials stiffness and improved thermal transport coefficient $\\kappa $ .",
"We want to emphasize that the absolute values of $\\kappa $ calculated in our simulations are within the experimental uncertainty and also consistent with the stiffness measurements.",
"A rather generic picture emerging of these results is that $\\kappa $ may be tuned when a system satisfies a few key conditions: miscibility, preferential hydrogen bonding, and the formation of cross-linking networks.",
"Although we have presented data for PAA-PAM systems with improved $\\kappa $ , our simulation results indicate a rather generic design principle for plastic materials with the improvement and tunability of $\\kappa $ .",
"To validate the protocol presented in this work, more detailed experimental synthesis, characterization, and their thermal transport measurements are needed on a broader spectrum of polymeric materials.",
"Moreover, results presented here may serve as a guiding principle for the operational understanding and functional design of advanced materials with tunable properties."
],
[
"Acknowledgement",
"D.M.",
"thank Kurt Kremer and Carlos M. Marques for fruitful continual collaborations that lead to the understanding of smart polymers presented here.",
"We thank Derek Fujimoto for useful discussions.",
"We further acknowledge support from Compute Canada where simulations were performed."
]
] | 1906.04224 |
[
[
"Linear and Nonlinear Fractional Elliptic Problems"
],
[
"Abstract This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs.",
"The emphasis is on the interplay between regularity, including boundary behavior, and approximability by piecewise linear finite element methods.",
"We discuss several error estimates on graded meshes, and computational challenges associated to implementing and solving efficiently the ensuing integral equations, along with numerical experiments."
],
[
"Introduction",
"Diffusion, which is one of the most common physical processes, is the net movement of particles from a region of higher concentration to a region of lower concentration.",
"The assumption that particles respond to Brownian motion leads to classical models of diffusion [36], that have been well studied for a long time.",
"Fick's first law states that the magnitude of the diffusive flux is proportional to the concentration gradient; by now, it is clear that such a constitutive relation is a questionable model for numerous phenomena [55].",
"When the associated underlying stochastic process is not given by Brownian motion, the diffusion is regarded as anomalous.",
"In particular, anomalous superdiffusion refers to situations that can be modeled using fractional spatial derivatives or fractional spatial differential operators.",
"Integer-order differentiation operators are local because the derivative of a function at a given point depends only on the values of the function in an infinitesimal neighborhood of it.",
"In contrast, fractional-order derivatives are nonlocal, integro-differential operators.",
"A striking example of such an operator is the fractional Laplacian of order $s \\in (0,1)$ , which we will denote by $(-\\Delta )^s$ , and is given by $ (-\\Delta )^s u (x) := C_{d,s} \\, \\text{P.V.",
"}\\int _{\\mathbb {R}^d}\\frac{u(x)-u(y)}{|x-y|^{d+2s}} \\; dy , \\quad C_{d,s} := \\frac{2^{2s} s \\Gamma (s+\\frac{d}{2})}{\\pi ^{d/2} \\Gamma (1-s)} .$ We refer to [63] for an illustration of how the heat equation involving the fractional Laplacian arises from a simple random walk with jumps.",
"The nonlocal structure of the operator (REF ) is apparent: to evaluate $(-\\Delta )^s u$ at a spatial point, information involving all spatial points is needed.",
"This work deals with fractional diffusion.",
"Our main goal is to review finite element methods (FEMs) to approximate solution of elliptic problems involving $(-\\Delta )^s$ or related operators on bounded domains.",
"We shall not discuss methods for the spectral fractional Laplacian; the surveys [10], [52] offer comparison between such an operator and the fractional Laplacian (REF ) and review other numerical methods.",
"We point out that the fractional Laplacian (REF ) of order $s \\in (0,1)$ is the infinitesimal generator of a $2s$ -stable Lévy process.",
"In this regard, problems on a bounded domain with homogeneous Dirichlet boundary conditions arise when the process is killed upon exiting the domain.",
"Throughout this work, we assume that $\\Omega $ is a bounded Lipschitz domain.",
"Whenever additional assumptions on $\\partial \\Omega $ are required, we shall state them explicitly.",
"Even though there is a wide variety of numerical methods for fractional-order problems available in the literature [52], in this work we shall focus on piecewise linear finite element methods as in [10].",
"We emphasize the interplay between regularity, including boundary behavior, and approximability.",
"In fact, the convergence rates achievable for the fractional elliptic PDEs discussed below, both linear and nonlinear, are limited by the presence of an algebraic boundary layer regardless of the regularity of $\\partial \\Omega $ and the polynomial degree for shape regular elements.",
"The paper is organized as follows.",
"sec:linear deals with the homogeneous Dirichlet problem for the fractional Laplacian in $\\Omega $ ; we discuss regularity of solutions and discuss both theoretical and computational aspects of conforming finite element discretizations.",
"We also comment on some recent applications of this approach and on an alternative nonconforming FEM based on a Dunford-Taylor representation of the weak form of $(-\\Delta )^s$ .",
"Afterwards, in sec:obstacle we address the obstacle problem for the fractional Laplacian.",
"To derive optimal convergence estimates, we focus on weighted Sobolev regularity, where the weight is a power of the distance to the boundary of $\\Omega $ .",
"These estimates follow from a precise quantification of boundary regularity of solutions and how solutions detach from the obstacle.",
"Finally, sec:NMS deals with fractional minimal graphs, which in fact are subgraphs that minimize a suitable nonlocal perimeter.",
"We formulate a variational form for this problem, which is nonlinear and degenerate.",
"We report on approximation properties of a conforming finite element scheme, and show convergence rates with respect to a novel geometric quantity.",
"The paper concludes with a couple of computational explorations of the behavior of fractional minimal graphs for $d=2$ ."
],
[
"Linear Problems",
"In this section we consider the homogeneous Dirichlet problem for the fractional Laplacian (REF ).",
"Given $f : \\Omega \\rightarrow \\mathbb {R}$ , one seeks a function $u$ such that $\\left\\lbrace \\begin{array}{rl}(-\\Delta )^s u = f &\\mbox{ in }\\Omega , \\\\u = 0 &\\mbox{ in }\\Omega ^c :={\\mathbb {R}^d}\\setminus \\Omega .\\end{array} \\right.$"
],
[
"Variational Formulation",
"The natural variational framework for (REF ) is within the fractional Sobolev space ${\\widetilde{H}}^s(\\Omega )$ , that is defined by ${\\widetilde{H}}^s(\\Omega ):= \\lbrace v \\in H^s({\\mathbb {R}^d}) \\colon \\operatorname{supp}v \\subset \\overline{\\Omega } \\rbrace .$ We refer to [13] for definitions and elementary properties of fractional Sobolev spaces.",
"Here we just state that on the space ${\\widetilde{H}}^s(\\Omega )$ , because of the fractional Poincaré inequality, the natural inner product is equivalent to $ (v,w)_s := \\frac{C_{d,s}}{2} \\iint _{Q_\\Omega } \\frac{(v(x)-v(y))(w(x) - w(y))}{|x-y|^{d+2s}} \\; dx \\; dy,$ where $Q_\\Omega = ({\\mathbb {R}^d}\\times {\\mathbb {R}^d}) \\setminus (\\Omega ^c\\times \\Omega ^c)$ .",
"The corresponding norm is $\\Vert v\\Vert _{{\\widetilde{H}}^s(\\Omega )} := (v,v)_s^{1/2}$ .",
"The duality pairing between ${\\widetilde{H}}^s(\\Omega )$ and its dual $H^{-s}(\\Omega )$ shall be denoted by $\\langle \\cdot , \\cdot \\rangle $ .",
"In view of (REF ) we see that, whenever $v \\in {\\widetilde{H}}^s(\\Omega )$ then $(-\\Delta )^s v \\in H^{-s}(\\Omega )$ and that $(v,w)_s = \\langle (-\\Delta )^s v, w \\rangle , \\quad \\forall w \\in {\\widetilde{H}}^s(\\Omega ).$ Therefore, given $f \\in H^{-s}(\\Omega )$ , the weak formulation of (REF ) reads: find $u \\in {\\widetilde{H}}^s(\\Omega )$ such that $ (u, v)_s = \\langle f, v \\rangle \\quad \\forall v \\in {\\widetilde{H}}^s(\\Omega ).$ Existence and uniqueness of weak solutions, and stability of the solution map $f \\mapsto u$ , follow straightforwardly from the Lax-Milgram Theorem."
],
[
"Regularity",
"A priori, it is not clear how smooth weak solutions are.",
"If $f$ is more regular than $H^{-s}(\\Omega )$ , then $u$ could be expected to be more regular than ${\\widetilde{H}}^s(\\Omega )$ .",
"We now review some results regarding regularity of solutions to problem (REF ).",
"Since our main interest is to derive convergence rates for finite element discretizations, we shall focus on Sobolev regularity estimates.",
"Recently, using Fourier analytical tools, Grubb [47] obtained estimates of solutions in terms of the so-called Hörmander $\\mu $ -spaces [48], but such estimates can be reinterpreted in terms of standard Sobolev spaces.",
"A drawback of the following result from [47] is that it assumes the domain $\\Omega $ to have smooth boundary, which is a too restrictive condition for finite element applications.",
"Theorem 2.1 (regularity on smooth domains) Let $\\Omega $ be a domain with $\\partial \\Omega \\in C^\\infty $ , $s \\in (0,1)$ , $f \\in H^r(\\Omega )$ for some $r\\ge -s$ , $u$ be the solution of (REF ) and $\\gamma = \\min \\lbrace s + r, 1/2 -\\varepsilon \\rbrace $ , with $\\varepsilon > 0$ arbitrarily small.",
"Then, $u \\in \\widetilde{H}^{s + \\gamma }(\\Omega )$ and the following regularity estimate holds: $\\Vert u \\Vert _{\\widetilde{H}^{s + \\gamma }(\\Omega )} \\le C(\\Omega ,d,s,\\gamma ) \\Vert f \\Vert _{H^r(\\Omega )}.$ A rather surprising feature of the previous result is that no matter how smooth the right hand side $f$ is, we cannot guarantee that solutions are any smoother than $\\widetilde{H}^{s + 1/2 - \\varepsilon }(\\Omega )$ .",
"Indeed, because the fractional Laplacian is an operator of order $2s$ , it could be expected to have a lift of order $2s$ .",
"As the following example [42] shows, such a reasoning is incorrect, and T:reggrubb is sharp.",
"Example 2.2 (limited regularity) Consider $\\Omega = B(0,1) \\subset {\\mathbb {R}^d}$ and $f \\equiv 1$ .",
"Then, the solution to (REF ) is given by $ u(x) = \\frac{\\Gamma (\\frac{d}{2})}{2^{2s} \\Gamma (\\frac{d+2s}{2})\\Gamma (1+s)} ( 1- |x|^2)^s_+,$ where $t_+ =\\max \\lbrace t,0\\rbrace $ .",
"As ex:nonsmooth illustrates, rough boundary behavior causes the reduced Sobolev regularity of solutions.",
"Ros-Oton and Serra [59] studied problem (REF ) using potential theory tools, and were able to obtain a fine characterization of boundary behavior of solutions, that led them to deduce Hölder regularity estimates.",
"In particular, it turns out that the asymptotic expansion $ u (x) \\approx d(x, \\partial \\Omega )^s \\varphi (x),$ where $\\varphi $ is a smooth function, is generic.",
"The Hölder estimates from [59] give rise to Sobolev estimates for solutions in terms of Hölder norms of the data.",
"To capture the boundary behavior, reference [2] introduced fractional weighted norms, where the weight is a power of the distance to the boundary, and developeds estimates in such norms.",
"We denote $\\delta (x,y) := \\min \\big \\lbrace \\textrm {dist}(x, \\partial \\Omega ), \\textrm {dist}(y, \\partial \\Omega ) \\big \\rbrace $ and, for $\\ell = k + s$ , with $k \\in \\mathbb {N}$ and $s \\in (0,1)$ , and $\\kappa \\ge 0$ , we define the norm $\\Vert v \\Vert _{H^{\\ell }_\\kappa (\\Omega )}^2 := \\Vert v \\Vert _{H^k (\\Omega )}^2 + \\sum _{|\\beta | = k }\\iint _{\\Omega \\times \\Omega } \\frac{|D^\\beta v(x)-D^\\beta v(y)|^2}{|x-y|^{d+2s}} \\, \\delta (x,y)^{2\\kappa } \\, dy \\, dx$ and the associated space $ H^\\ell _\\kappa (\\Omega ) := \\left\\lbrace v \\in H^\\ell (\\Omega ) \\colon \\Vert v \\Vert _{H^\\ell _\\kappa (\\Omega )} < \\infty \\right\\rbrace .$ The regularity estimate in the weighted Sobolev scale (REF ) reads as follows.",
"Theorem 2.3 (weighted Sobolev estimate) Let $\\Omega $ be a bounded, Lipschitz domain satisfying the exterior ball condition, $s\\in (0,1)$ , $f \\in C^{1-s}(\\overline{\\Omega })$ and $u$ be the solution of (REF ).",
"Then, for every $\\varepsilon >0$ we have $u \\in \\widetilde{H}^{s+1-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )$ and $\\Vert u\\Vert _{\\widetilde{H}^{s+1-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )} \\le \\frac{C(\\Omega ,d,s)}{\\varepsilon } \\Vert f \\Vert _{C^{1-s}(\\overline{\\Omega })}.$ For simplicity, the theorem above was stated using the weight $\\kappa = 1/2 -\\varepsilon $ ; a more general form of the result can be found also in [10].",
"We point out that, for its application in finite element analysis, such a choice is optimal.",
"In principle, increasing the exponent $\\kappa $ of the weight allows for a higher differentiability in the solution, with no restriction on $\\kappa $ above (as long as $f$ is sufficiently smooth).",
"However, when exploiting this weighted regularity by introducing approximations on a family of shape-regular and graded meshes, the order of convergence (with respect to the number of degrees of freedom) is only incremented as long as $\\kappa < 1/2$ .",
"It is worth pointing out that T:weightedregularity is valid for Lipschitz domains satisfying an exterior ball condition.",
"Although such a condition on the domain is much less restrictive than the $C^\\infty $ requirement in T:reggrubb, for polytopal domains it implies convexity.",
"For that reason, we present here a result of an ongoing work [15], that characterizes regularity of solutions in terms of Besov norms.",
"Theorem 2.4 (regularity on Lipschitz domains) Let $\\Omega $ be a bounded Lipschitz domain, $s \\in (0,1)$ and $f \\in H^r(\\Omega )$ for some $r \\in (-s,0]$ .",
"Then, the solution $u$ to (REF ) belongs to the Besov space $B^{s+t}_{2,\\infty }(\\Omega )$ , where $t= \\min \\lbrace s + r -\\varepsilon , 1/2 \\rbrace $ , with $\\varepsilon > 0$ arbitrarily small, with $ \\Vert u \\Vert _{B^{s+t}_{2,\\infty }(\\Omega )} \\le C(\\Omega ,d,s,t) \\Vert f\\Vert _{H^r(\\Omega )}.$ Consequently, by an elementary embedding, we deduce $ \\Vert u \\Vert _{H^{s+\\gamma }(\\Omega )} \\le \\frac{C(\\Omega ,d,s,\\gamma )}{\\varepsilon } \\Vert f\\Vert _{H^r(\\Omega )},$ where $\\gamma = \\min \\lbrace s+r, 1/2\\rbrace - \\varepsilon $ is `almost' as in T:reggrubb.",
"We briefly outline the main ideas in the proof of T:Besovregularity, which follows a technique proposed by Savaré [61] for local problems.",
"The point is to use the classical Nirenberg difference quotient method, and thus bound a certain Besov seminorm of the solution $u$ .",
"Let $t \\in (0,1)$ and $D$ be a set generating ${\\mathbb {R}^d}$ and star-shaped with respect to the origin (for example, a cone).",
"Then the functional $ [v]_{s+t,2,\\Omega } := \\sup _{h \\in D \\setminus \\lbrace 0 \\rbrace }\\frac{1}{|h|^t} |v-v(\\cdot +h)|_{H^s(\\Omega )}$ induces the standard seminorm in the Besov space $B^{s+t}_{2,\\infty }(\\Omega )$ .",
"Because $\\Omega $ is a Lipschitz domain, it satisfies a uniform cone property; upon a partition of unity argument, this gives (finitely many) suitable sets $D$ where translations can be taken.",
"For a localized translation operator $T_h$ , it is possible to prove a bound of the form $ |T_h u - u|_s^2 \\le C \\, |h|^s \\, |u|_s^2,$ which, in view of (REF ), yields $u \\in B^{3s/2}_{2,\\infty }(\\Omega )$ .",
"Once this estimate has been obtained, a bootstrap argument leads to (REF ).",
"Moreover, a refined estimate in $B^{3s/2}_{2,\\infty }(\\Omega )$ reads $|u|_{B^{3s/2}_{2,\\infty }(\\Omega )} \\lesssim \\Vert f\\Vert _{B^{-s/2}_{2,1}(\\Omega )},$ and interpolation with $|u|_{\\widetilde{H}^s(\\Omega )} \\lesssim \\Vert f\\Vert _{H^{-s}(\\Omega )}$ yields the following result [15].",
"Theorem 2.5 (lift theorem on Lipschitz domains) Let $\\Omega $ be a bounded Lipschitz domain, $s \\in (0,1)$ and $f \\in H^r(\\Omega )$ for some $r \\in (-s,-s/2]$ .",
"Then, the solution $u$ to (REF ) belongs to the Sobolev space $\\widetilde{H}^{r+2s}(\\Omega )$ , with $\\Vert u\\Vert _{\\widetilde{H}^{r+2s}(\\Omega )} \\lesssim \\Vert f\\Vert _{H^{r}(\\Omega )}.$"
],
[
"Finite element discretization",
"In this section, we discuss a direct finite element method to approximate (REF ) using piecewise linear continuous functions.",
"We consider a family $\\lbrace \\mathcal {T}_h\\rbrace _{h>0}$ of conforming and simplicial meshes of $\\Omega $ , that we assume to be shape-regular, namely, $\\sigma := \\sup _{h>0} \\max _{T \\in \\mathcal {T}_h} \\frac{h_T}{\\rho _T} <\\infty ,$ where $h_T = \\mbox{diam}(T)$ and $\\rho _T $ is the diameter of the largest ball contained in $T$ .",
"As usual, the subindex $h$ denotes the mesh size, $h = \\max _{T \\in \\mathcal {T}_h} h_T$ and we take elements to be closed sets.",
"We denote by $\\mathcal {N}_h$ the set of interior vertices of $\\mathcal {T}_h$ , by $N$ the cardinality of $\\mathcal {N}_h$ , and by $\\lbrace \\varphi _i \\rbrace _{i=1}^N$ the standard piecewise linear Lagrangian basis, with $\\varphi _i$ associated to the node $\\texttt {x}_i \\in \\mathcal {N}_h$ .",
"Thus, the set of discrete functions is $ \\mathbb {V}_h := \\left\\lbrace v \\in C_0(\\Omega ) \\colon v = \\sum _{i=1}^N v_i \\varphi _i \\right\\rbrace ,$ and is clearly conforming: $\\mathbb {V}_h \\subset {\\widetilde{H}}^s(\\Omega )$ for all $s \\in (0,1)$ .",
"With the notation described above, the discrete counterpart to (REF ) reads: find $u_h \\in \\mathbb {V}_h$ such that $ (u_h, v_h)_s = \\langle f, v_h \\rangle \\quad \\forall v_h \\in \\mathbb {V}_h.$ Because $u_h$ is the projection of $u$ onto $\\mathbb {V}_h$ with respect to the ${\\widetilde{H}}^s(\\Omega )$ -norm, we have the best approximation property $ \\Vert u - u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )} = \\min _{v_h \\in \\mathbb {V}_h} \\Vert u - v_h \\Vert _{{\\widetilde{H}}^s(\\Omega )}.$ Therefore, in order to obtain a priori rates of convergence in the energy norm, it suffices to bound the distance between the discrete spaces and the solution.",
"Although the bilinear form $(\\cdot , \\cdot )_s$ involves integration on $\\Omega \\times {\\mathbb {R}^d}$ , one can apply an argument based on a fractional Hardy inequality, to prove that the energy norm may be bounded in terms of fractional–order norms on $\\Omega $ (see [2]).",
"It follows that bounding errors within $\\Omega $ leads to error estimates in the energy norm.",
"A technical aspect of fractional-order seminorms is that they are not additive with respect to domain decompositions.",
"With the goal of deriving interpolation estimates, we define the star or first ring of an element $T \\in \\mathcal {T}_h$ by $S^1_T := \\bigcup \\left\\lbrace T^{\\prime } \\in \\mathcal {T}_h\\colon T\\cap T^{\\prime }\\ne \\emptyset \\right\\rbrace .$ We also introduce the star of $S^1_T$ (or second ring of $T$ ), $S^2_T := \\bigcup \\left\\lbrace T^{\\prime } \\in \\mathcal {T}_h\\colon S^1_T \\cap T^{\\prime }\\ne \\emptyset \\right\\rbrace ,$ and the star of the node $\\texttt {x}_i \\in \\mathcal {N}_h$ , $S_i := \\mbox{supp}(\\varphi _i)$ .",
"Faermann [37] proved the localization estimate $|v|_{H^s(\\Omega )}^2 \\le \\sum _{T \\in \\mathcal {T}_h} \\left[ \\int _T \\int _{S^1_T} \\frac{|v (x) - v (y)|^2}{|x-y |^{d+2s}} \\; dy \\; dx + \\frac{C(d,\\sigma )}{s h_T^{2s}} \\Vert v \\Vert ^2_{L^2(T)} \\right] \\quad \\forall v \\in H^s(\\Omega ).$ This inequality shows that to estimate fractional seminorms over $\\Omega $ , it suffices to compute integrals over the set of patches $\\lbrace T \\times S^1_T \\rbrace _{T \\in \\mathcal {T}_h}$ plus local zero-order contributions.",
"Bearing this in mind, one can prove the following type of estimates for suitable quasi-interpolation operators (see, for example, [16], [27]).",
"Proposition 2.6 (interpolation estimates on quasi-uniform meshes) Let $T \\in , $ s (0,1)$, $ (s, 2]$, and $ h$ be a suitable quasi-interpolation operator.If $ v H(S2T)$, then\\begin{equation} \\int _T \\int _{S^1_T} \\frac{|(v-\\Pi _hv) (x) - (v-\\Pi _hv) (y)|^2}{|x-y|^{d+2s}} \\, d y \\, d x \\le C \\, h_T^{2(\\ell -s)} |v|_{H^\\ell (S^2_T)}^2.\\end{equation}where $ C = C(,d,s,, )$.Therefore, for all $ v H()$, it holds\\begin{equation} | v - \\Pi _hv|_{H^s(\\Omega )} \\le C(\\Omega ,d,s,\\sigma , \\ell ) \\, h^{\\ell -s} |v|_{H^\\ell (\\Omega )}.\\end{equation}$ The statement () in prop:appSZ could have also been obtained by interpolation of standard integer-order interpolation estimates.",
"However, the technique of summing localized fractional-order estimates also works for graded meshes (cf.",
"(REF ) and (REF ) below).",
"Combining estimate () with the best approximation property (REF ) and the regularity estimates described in sec:regularitylinear, we can derive convergence rates.",
"Concretely, the estimates from T:reggrubb and T:Besovregularity translate into a priori rates for quasi-uniform meshes.",
"However, optimal application of T:weightedregularity requires a certain type of mesh grading.",
"In two-dimensional problems ($d=2$ ), this can be attained by constructing graded meshes in the spirit of [46].",
"In addition to shape regularity, we assume that the family $\\lbrace \\mathcal {T}_h\\rbrace $ satisfies the following property: there is a number $\\mu \\ge 1$ such that given a parameter $h$ representing the meshsize at distance 1 to the boundary $\\partial \\Omega $ and $T\\in \\mathcal {T}_h$ , we have $ h_T \\le C(\\sigma )\\begin{dcases}h^\\mu , & T \\cap \\partial \\Omega \\ne \\emptyset , \\\\h \\textrm {dist}(T,\\partial \\Omega )^{(\\mu -1)/\\mu }, & T \\cap \\partial \\Omega = \\emptyset .\\end{dcases}$ The number of degrees of freedom is related to $h$ by means of the parameter $\\mu $ because (recall that $d=2$ ) $N = \\dim \\mathbb {V}_h \\approx \\begin{dcases}h^{-2}, & \\mu < 2, \\\\h^{-2} | \\log h |, & \\mu = 2, \\\\h^{-\\mu }, & \\mu > 2.\\end{dcases}$ Also, $\\mu $ needs to be related to the exponent $\\kappa $ used in the Sobolev regularity estimate (cf.",
"T:weightedregularity).",
"It can be shown that the choice $\\mu = 2$ , that corresponds to $\\kappa = 1/2$ , yields optimal convergence rates in terms of the dimension of $\\mathbb {V}_h$ .",
"We also remark that, as discussed in [13], for three-dimensional problems ($d=3$ ), the grading strategy (REF ) becomes less flexible, and yields lower convergence rates.",
"For optimal mesh grading beyond $\\mu =2$ for both $d=2,3$ we need to break the shape regularity assumption and resort to anisotropic finite elements.",
"They in turn are less flexible in dealing with the isotropic fractional norm of $H^s(\\Omega )$ and its localization [37].",
"This important topic remains open.",
"Quasi-interpolation estimates in weighted Sobolev spaces (REF ) can be derived in the same way as in prop:appSZ.",
"More precisely, the weighted counterparts to () and () read $ \\int _T \\int _{S^1_T} \\frac{|(v-\\Pi _hv) (x) - (v-\\Pi _hv) (y)|^2}{|x-y|^{d+2s}} \\, d y \\, d x \\le Ch_T^{2(\\ell -s-\\kappa )} |v|_{H^\\ell _\\kappa (S^2_T)}^2,$ for all $v \\in H^\\ell _\\kappa (S^2_T)$ and $ | v - \\Pi _hv|_{H^s(\\Omega )} \\le C h^{\\ell -s-\\alpha } |v|_{H^\\ell _\\kappa (\\Omega )} \\quad \\forall v \\in H^\\ell _\\kappa (\\Omega ),$ respectively.",
"The constants above are $C = C(\\Omega ,d,s,\\sigma ,\\ell ,\\kappa )$ .",
"We collect all the convergence estimates in the energy norm –involving quasi-uniform and graded meshes– in a single statement [2].",
"Theorem 2.7 (energy error estimates for linear problem) Let $u$ denote the solution to (REF ) and denote by $u_h \\in \\mathbb {V}_h$ the solution of the discrete problem (REF ), computed over a mesh $ consisting of elements with maximum diameter $ h$.",
"If $ f L2()$, we have$$\\Vert u - u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )} \\le C(\\Omega ,d,s,\\sigma ) \\, h^\\alpha |\\log h| \\, \\Vert f\\Vert _{L^2(\\Omega )},$$where $ = {s, 1/2 }$.Additionally, if $ R2$, $ f C1-s()$ and the family $ {Th}$ satisfies (\\ref {eq:H}) with $ = 2$, we have$$\\Vert u - u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )} \\le C(\\Omega ,s,\\sigma ) \\, h |\\log h| \\Vert f\\Vert _{C^{1-s}(\\overline{\\Omega })}.$$$ To illustrate that T:convlinear is sharp, we solve the problem from ex:nonsmooth on the discrete spaces (REF ) using a family of quasi-uniform meshes and a family of meshes graded according to (REF ).",
"In tab:ejemplo, we report computational convergence rates in the energy norm for several values of $s$ .",
"We observe good agreement with the rates predicted by T:convlinear.",
"Table: Computational rates of convergence (with respect to hh) for the problem from ex:nonsmooth in d=2d=2 dimensions.",
"Rates using quasi-uniform meshes are listed in the second row, while rates using graded meshes, with μ=2\\mu = 2 in (), are reported in the third row."
],
[
"Computational challenges",
"Having at hand theoretical estimates for finite element discretizations of (REF ), we still need to address how to compute discrete solutions and, in particular, how to accelerate the assembly and solution of the discrete system that arises.",
"Matrix assembly.",
"We first comment on key aspects of the finite element implementation for problems in dimension $d = 2$ .",
"If $\\mathbf {U}= (u_i)_{i=1}^N$ and $u_h = \\sum _{i=1}^N u_i \\varphi _i$ , it follows from (REF ) and (REF ) that the linear finite element system reads $\\mathbf {A}\\mathbf {U}= \\mathbf {F}$ with stiffness matrix $\\mathbf {A}$ and right-hand side vector $\\mathbf {F}$ given by $\\mathbf {A}_{ij} = (\\varphi _i, \\varphi _j)_s, \\quad \\mathbf {F}_i = \\langle f, \\varphi _i \\rangle .$ Computation of the stiffness matrix is not an easy task.",
"There are two numerical difficulties in taking a direct approach.",
"In first place, the bilinear form $(\\cdot , \\cdot )_s$ requires integration on unbounded domains; we point out that –at least for homogeneous problems as the ones considered here– integration over $\\Omega \\times \\Omega ^c$ can be reduced to a suitable integration over $\\Omega \\times \\partial \\Omega $ by using the Divergence Theorem [4].",
"Secondly, suitable quadrature rules to compute the stiffness matrix entries are required.",
"To handle the singular (non-integrable) kernel $|x|^{-d-2s}$ , one could use techniques from the boundary element method [26], [60]; we refer to [1] for details.",
"Compression.",
"Note that, independently of $s$ , finite element spaces (REF ) give rise to full stiffness matrices.",
"Indeed, for any pair of nodes $\\texttt {x}_i, \\texttt {x}_j$ such that $S_i \\cap S_j = \\emptyset $ , $\\mathbf {A}_{ij} = - C_{d,s} \\iint _{S_i \\times S_j} \\frac{\\varphi _i(x) \\; \\varphi _j(y)}{|x-y|^{d+2s}} \\; dy \\; dx < 0.$ Thus, computation of the stiffness matrix $\\mathbf {A}$ involves a large number of far-field interactions, that is, elements $\\mathbf {A}_{ij}$ for $\\texttt {x}_i$ and $\\texttt {x}_j$ sufficiently far.",
"However, these elements should be significantly smaller than the ones that involve nodes close to one another.",
"In [4], [64] the cluster method from the boundary element literature was applied and, instead of computing and storing all individual elements from $\\mathbf {A}$ , far field contributions are replaced by suitable low-rank blocks.",
"The resulting data-sparse representation has $\\mathcal {O}(N \\log ^\\alpha N)$ complexity for some $\\alpha \\ge 0$ .",
"Reference [50] shows that the inverse of $\\mathbf {A}$ can be represented using the same block structure as employed to compress the stiffness matrix.",
"Preconditioning.",
"There are also issues to be addressed regarding the solution of the dense matrix equation $\\mathbf {A}\\mathbf {U}= \\mathbf {F}$ .",
"The use of matrix factorization to solve such a system has complexity $\\mathcal {O}(N^3)$ .",
"As an alternative, one can use a conjugate gradient method and thereby the number of iterations needed for a fixed tolerance scales like $\\sqrt{\\kappa (\\mathbf {A})}$ , where $\\kappa (\\mathbf {A})$ is the condition number of $\\mathbf {A}$ and satisfies [5] $\\kappa (\\mathbf {A}) = \\mathcal {O}\\left( N^{2s/d} \\left( \\frac{h_{max}}{h_{min}} \\right)^{d-2s} \\right).$ Therefore, for two-dimensional problems, we deduce $\\kappa (\\mathbf {A}) = \\mathcal {O}(h^{-2s})$ for quasi-uniform meshes, while $\\kappa (\\mathbf {A}) = \\mathcal {O}(h^{-2} |\\log h|^s)$ for meshes graded according to (REF ) with $\\mu = 2$ .",
"In the latter case, diagonal preconditioning allows us to recover the same condition number as for uniform meshes [5].",
"Recently, there have been some advances in the development of preconditioners for fractional diffusion.",
"For instance, multigrid preconditioners were mentioned in [4], while operator preconditioners were studied in [45].",
"We now briefly comment on some features of an additive Schwarz preconditioner of BPX-type [17] (see also [38]).",
"Assume we have a hierarchy of discrete spaces $\\mathbb {V}_0 \\subset \\ldots \\mathbb {V}_J = \\mathbb {V}$ , with mesh size $h_j = \\gamma ^{2j}$ , and let $\\iota _j : \\mathbb {V}_j \\rightarrow \\mathbb {V}$ be the inclusion operator.",
"The basic ingredients needed to apply the general theory for additive Schwarz preconditioners are: [leftmargin=*] Stable decomposition: for every $v\\in \\mathbb {V}$ , there exists a decomposition $v = \\sum _{j=0}^J v_j$ with $v_j \\in \\mathbb {V}_j$ such that $ \\sum _{j=0}^J h_j^{-2s} \\Vert v_j\\Vert _{L^2(\\Omega )}^2 \\le c_0 \\Vert v\\Vert _{{\\widetilde{H}}^s(\\Omega )}^2.$ A fundamental ingredient to prove this estimate for polyhedral domains is the optimal regularity pickup estimate for Lipschitz domains of T:lift, that allows us to perform an Aubin-Nitsche duality argument.",
"Boundedness: for every $v = \\sum _{j=0}^J v_j$ with $v_j \\in \\mathbb {V}_j$ , $ \\Vert \\sum _{j=0}^J v_j \\Vert _{{\\widetilde{H}}^s(\\Omega )}^2 \\le c_1 \\sum _{j=0}^J h_j^{-2s} \\Vert v_j\\Vert _{L^2(\\Omega )}^2.$ As usual, boundedness of multilevel decompositions can be proved by estimating how much scales interact (i.e., using a strengthened Cauchy-Schwarz inequality).",
"Nonlocality adds some difficulties to the derivation of such an estimate, because one cannot integrate by parts elementwise.",
"The argument in [17] is based on the Fourier representation of the fractional Laplacian.",
"The conditions (REF ) and (REF ) imply that the preconditioner $\\mathbf {B}:= \\sum _{j=0}^J h_j^{2s-d} \\iota _j \\iota ^{\\prime }_j$ satisfies $\\kappa (\\mathbf {B}\\mathbf {A}) \\le \\frac{c_0}{c_1}$ for graded bisection grids [17].",
"We illustrate this statement in Table REF for $\\Omega =(-1,1)^2$ , $f=1$ and $s=0.9, 0.5, 0.1$ .",
"We observe a mild increase of iteration counts but rather robust performance with respect to $s$ .",
"Table: Number of iterations for Gauss-Seidel (GS), conjugate gradient (CG)and preconditioned CG with BPX preconditioner (PCG).",
"Stopping criteria is∥𝐀𝐔-𝐅∥ 2 ∥𝐅∥ 2 <10 -6 \\frac{\\Vert \\mathbf {A}\\mathbf {U}- \\mathbf {F}\\Vert _2}{\\Vert \\mathbf {F}\\Vert _2} < 10^{-6}."
],
[
"Applications and related problems",
"Numerical methods for fractional diffusion models have been extensively studied recently.",
"Let us mention some applications on linear elliptic problems of the approach treated in this section: [leftmargin=*] A posteriori error analyisis and adaptivity: The reduced regularity of solutions of (REF ) and the high computational cost of assembling the stiffness matrix $\\mathbf {A}$ motivate the pursue of suitable adaptive finite element methods.",
"A posteriori error estimates of residual type have been proposed and analyzed in [4], [39], [44], [58], and gradient-recovery based estimates in [64].",
"Adaptivity is however a topic of current research [38], [39], [44].",
"Eigenvalue problems: The fractional eigenvalue problem arises, for example, in quantum mechanics problems in which the Brownian-like quantum paths are replaced by Lévy-like ones in the Feynman path integral [51].",
"Reference [18] studies conforming finite element approximations and applies the Babuška-Osborn theory [7], thereby obtaining convergence rates for eigenfunctions (in the energy and in the $L^2$ -norms) and eigenvalues.",
"Other methods, implemented on one-dimensional problems, include finite differences [35] and matrix methods [43], [65].",
"Control problems: Finite element methods for linear-quadratic optimal control problems involving the fractional Laplacian (REF ) have been studied recently.",
"In these problems, the control may be located inside [31] or outside the domain [6].",
"Non-homogeneous Dirichlet conditions: A mixed method for the non-homogeneous Dirichlet problem for the integral fractional Laplacian was proposed in [3].",
"Such a method is based on weak enforcement of the Dirichlet condition and using a suitable non-local derivative [32] as a Lagrange multiplier.",
"To circumvent approximating the nonlocal derivative, [6] proposed approximations of the non-homogeneous Dirichlet problem by a suitable Robin exterior value problem."
],
[
"Nonconforming FEM: Dunford-Taylor approach",
"We finally report on a finite element approach for (REF ) proposed in [11] and based on the Fourier representation of the ${\\widetilde{H}}^s(\\Omega )$ -inner product: $(v,w)_s = \\int _{{\\mathbb {R}^d}} |\\xi |^s {F}(v) |\\xi |^s \\overline{{F}(w)} d\\xi = \\int _{{\\mathbb {R}^d}} {F}((-\\Delta )^s v)(\\xi ) \\overline{{F}(w(\\xi ))} d\\texttt {x}.$ This expression can be equivalently written as $ (v,w)_s = \\frac{2\\sin (s\\pi )}{\\pi }\\int _0^\\infty t^{1-2s} \\int _{{\\mathbb {R}}^d} \\big ( -\\Delta (I-t^2\\Delta )^{-1} v \\big ) w \\, dx dt.$ To see this, use Parseval's formula to obtain $\\int _{{\\mathbb {R}}^d} \\big ( -\\Delta (I-t^2\\Delta )^{-1} v \\big ) w \\, dx = \\int _{{\\mathbb {R}}^d}\\frac{|\\xi |^2}{1+t^2|\\xi |^2} {F}(v)(\\xi ) \\overline{{F}(w)(\\xi )} d\\xi ,$ followed by the change of variables $z=t|\\xi |$ , which converts the repeated integrals in the expression for $(v,w)_s$ into separate integrals, one of them being $\\int _0^\\infty \\frac{z^{1-2s}}{1+z^2} dz = \\frac{\\pi }{2\\sin (s \\pi )}.$ Although identity (REF ) is not an integral representation of the operator $(-\\Delta )^s$ , but rather of the bilinear form $(\\cdot , \\cdot )_s$ , we regard it as a Dunford-Taylor representation.",
"To set up this formal calculation in the correct functional framework, given $u\\in {\\widetilde{H}}^s(\\Omega )\\subset L^2({\\mathbb {R}}^d)$ and $t>0$ , let $v(u,t) \\in H^{2+s}({\\mathbb {R}}^d)$ be the solution to $v - t^2 \\Delta v = -u$ in ${\\mathbb {R}^d}$ , or equivalently $v = -(I-t^2\\Delta )^{-1}u$ .",
"Therefore, $\\Delta v = t^{-2} (v+u)$ and $(u,w)_s = \\frac{2\\sin (s \\pi )}{\\pi } \\int _0^\\infty t^{-1-2s} \\langle u + v(u,t) , w \\rangle \\; dt \\quad \\forall u,w \\in {\\widetilde{H}}^s(\\Omega ).$ This representation is the starting point of a three-step numerical method [11].",
"[leftmargin=*] Sinc quadrature: the change of variables $t = e^{-y/2}$ yields $ (u,w)_s = \\frac{\\sin (s\\pi )}{\\pi } \\int _{-\\infty }^\\infty e^{sy} \\langle u + v(u,t(y)) , w \\rangle \\; dy$ Thus, given an integer $N>0$ and a set of points $\\lbrace y_j\\rbrace _{j=-N}^N$ with uniform spacing $\\approx N^{-1}$ , the sinc quadrature $Q_s(u,w)$ approximation of $(\\cdot ,\\cdot )_s$ is given by $Q_s(u,w) = \\frac{\\sin (s\\pi )}{N \\pi } \\sum _{j = -N}^{N} e^{sy_j}\\langle u + v(u,t(y_j)), w \\rangle .$ Domain truncation: We stress that, in spite of $u$ being supported in $\\Omega $ , $v(u,t)$ is supported in all of ${\\mathbb {R}^d}$ for all $t$ ; hence, some truncation is required.",
"The method from [11] considers, for given $M>0$ , a family of balls $B^M(t)$ that contain $\\Omega $ and whose radius depends on $M$ and $t$ and can be computed a priori.",
"Finite element approximation: Finally, a standard finite element discretization on $B^M(t)$ is performed.",
"This requires meshes that fit $\\Omega $ and $B^M(t) \\setminus \\Omega $ exactly – a non-trivial task; let us denote the discrete spaces on $\\Omega $ and $B^M(t)$ by $\\mathbb {V}_h$ and $\\mathbb {V}_h^M$ , respectively.",
"Given $\\psi \\in L^2({\\mathbb {R}^d})$ , $t>0$ and $M>0$ , we define $v_h^M = v_h^M(\\psi , t)\\in \\mathbb {V}_h^M$ to be the unique solution of $\\int _{B^M(t)} v_h^M w_h + t^2 \\nabla v_h^M \\cdot \\nabla w_h \\; dx = - \\int _{B^M(t)} \\psi w_h \\; dx \\quad \\forall \\, w_h \\in \\mathbb {V}_h^M.$ The fully discrete bilinear form reads: $a_{N,M}(u_h, w_h) := \\frac{\\sin (s\\pi )}{N \\pi } \\sum _{j =-N}^{N} e^{sy_j} \\langle u_h + v_h^M(u_h,t(y_j)), w_h\\rangle \\quad \\forall \\, u_h, w_h \\in \\mathbb {V}_h.$ Using a Strang's type argument to quantify the consistency errors generated by the three steps above, one obtains the a priori estimate [11] $\\Vert u - u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )} \\le C \\left( e^{-c\\sqrt{N}} + e^{-cM} + h^{\\beta -s} |\\log h| \\right) \\Vert u \\Vert _{{\\widetilde{H}}^\\beta (\\Omega )},$ where $\\beta \\in (s,3/2)$ .",
"Choosing $\\beta = s + 1/2 -\\varepsilon $ , which is consistent with T:reggrubb and T:Besovregularity, $M = \\mathcal {O}(|\\log h|)$ and $N = \\mathcal {O}(|\\log h|^2)$ gives the convergence rate $\\Vert u - u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )} \\le C h^{\\min \\lbrace s,\\frac{1}{2}\\rbrace } |\\log h| \\, \\Vert f\\Vert _{L_2(\\Omega )}.$ This is similar to the rate obtained in T:convlinear for quasi-uniform meshes.",
"To the best of the authors' knowledge, implementation of this approach over graded meshes, while feasible in theory, has not yet been pursued in practice."
],
[
"Fractional Obstacle Problem",
"In this section we review finite element methods for the solution of the obstacle problem for the integral fractional Laplacian which, from now on, we shall simply refer to as the fractional obstacle problem.",
"The fractional obstacle problem appears, for example, in optimal stopping times for jump processes.",
"In particular, it is used in the modeling of the rational price of perpetual American options [28].",
"More precisely, if $u$ represents the rational price of a perpetual American option where the assets prices are modeled by a Lévy process $X_t$ , if $\\chi $ denotes the payoff function, then $u$ solves a fractional obstacle problem with obstacle $\\chi $ .",
"An a posteriori error analysis of approximations of variational inequalities involving integral operators on arbitrary bounded domains was performed in [58].",
"We also comment on two recent works related to the approach we review here.",
"Reference [20] deals with finite element discretizations to obstacle problems involving finite and infinite-horizon nonlocal operators.",
"The experiments shown therein were performed on one-dimensional problems with uniform meshes, and indicate convergence with order $h^{1/2}$ in the energy norm.",
"A theoretical proof of that convergence order was obtained in [12], where approximations using the approach discussed in sec:dunford-taylor were considered.",
"We also refer to [44] for computational comparisons between adaptive strategies and uniform and graded discretizations in two-dimensional problems."
],
[
"Variational formulation",
"As before, we assume that $\\Omega \\subset {\\mathbb {R}^d}$ is an open and bounded domain and, for the sake of applying weighted regularity estimates, we assume that $\\Omega $ has a Lipschitz boundary and satisfies the exterior ball condition.",
"Given $s \\in (0,1)$ and functions $f: \\Omega \\rightarrow \\mathbb {R}$ and $\\chi : \\overline{\\Omega }\\rightarrow \\mathbb {R}$ , with $\\chi < 0$ on $\\partial \\Omega $ , the obstacle problem is a constrained minimization problem on ${\\widetilde{H}}^s(\\Omega )$ associated with a quadratic functional.",
"Defining the admissible convex set ${\\mathcal {K}}:= \\left\\lbrace v \\in {\\widetilde{H}}^s(\\Omega ): v \\ge \\chi \\mbox{ a.e in } \\Omega \\right\\rbrace ,$ the solution to the fractional obstacle problem is $u=\\textrm {argmin}_{v\\in {\\mathcal {K}}} \\mathcal {J}(v)$ where $\\mathcal {J}(v) := \\frac{1}{2} \\Vert v \\Vert _{{\\widetilde{H}}^s(\\Omega )}^2 - \\langle f, v \\rangle .$ Existence and uniqueness of solutions is standard.",
"Taking first variation of $\\mathcal {J}$ , we deduce that such a minimizer $u \\in {\\mathcal {K}}$ solves the variational inequality $(u,u-v)_s \\le \\langle f, u-v \\rangle \\quad \\forall v \\in {\\mathcal {K}}.$ It can be shown [57] that, if $f \\in L^p(\\Omega )$ for $p > d/2s$ , then the solution to the obstacle problem is indeed a continuous function, and that it satisfies the complementarity condition $\\min \\left\\lbrace \\lambda , u-\\chi \\right\\rbrace = 0 \\quad \\mbox{ a.e in } \\Omega , \\mbox{ where } \\quad \\lambda := (-\\Delta )^su - f.$ For our discussion, we assume that $f$ is such that the solution is defined pointwise, and consequently, we define the coincidence (or contact) and non-coincidence sets, $\\Lambda := \\lbrace x \\in \\Omega : u(x) = \\chi (x) \\rbrace , \\quad N := \\Omega \\setminus \\Lambda .$ The complementarity condition (REF ) can be succinctly expressed as $\\lambda \\ge 0$ in $\\Lambda $ and $\\lambda = 0$ in $N$ .",
"The set $\\partial \\Lambda $ , where the solution detaches from the obstacle, is the free boundary."
],
[
"Regularity",
"The following regularity results for solutions to the fractional obstacle problem are instrumental for error analysis.",
"We recall our assumption that the obstacle $\\chi $ is a continuous function and strictly negative on $\\partial \\Omega $ : $ \\varrho := \\textrm {dist}\\left( \\lbrace \\chi >0\\rbrace , \\partial \\Omega \\right) > 0.$ Furthermore, we shall assume that $f \\ge 0$ .",
"Heuristically, these assumptions should guarantee that the behavior of solutions near $\\partial \\Omega $ is dictated by a linear problem and that the nonlinearity is confined to the interior of the domain.",
"Finally, to derive regularity estimates, we assume that the data satisfy $\\chi \\in C^{2,1}(\\Omega ), \\quad f \\in \\mathcal {F}_s(\\overline{\\Omega }) = \\begin{dcases}C^{2,1-2s+\\epsilon }(\\overline{\\Omega }), & s\\in \\left(0,\\frac{1}{2}\\right], \\\\C^{1,2-2s+\\epsilon }(\\overline{\\Omega }), & s \\in \\left(\\frac{1}{2},1\\right),\\end{dcases}$ where $\\epsilon >0$ is sufficiently small, so that $1-2s+\\epsilon $ is not an integer.",
"Under these conditions, Caffarelli, Salsa and Silvestre [23] proved that the solution to the problem posed in the whole space (with suitable decay conditions at infinity) is of class $C^{1,s}({\\mathbb {R}}^d)$ .",
"It is worth examining the limiting cases $s=1$ and $s=0$ .",
"The former corresponds to the classical obstacle problem whose solutions are of class $C^{1,1}({\\mathbb {R}}^d)$ .",
"The latter reduces to $\\min \\lbrace u-\\chi ,u-f\\rbrace =0$ whose solutions are just of class $C^{0,1}({\\mathbb {R}}^d)$ .",
"The regularity of [23] is thus a natural intermediate result.",
"We emphasize that deriving interior regularity estimates for (REF ) from this result, which is valid for problems posed in ${\\mathbb {R}^d}$ , is not as straightforward as for classical problems.",
"Indeed, the nonlocal structure of $(-\\Delta )^s$ implies that, if $0\\le \\eta \\le 1$ is a smooth cut-off function such that $\\eta =1$ in $\\lbrace \\chi > 0 \\rbrace $ , then $(-\\Delta )^s(\\eta u) \\ne \\eta (-\\Delta )^su \\quad \\mbox{in } \\lbrace \\eta = 1 \\rbrace .$ To overcome this difficulty, reference [16] proceeds as follows.",
"Given a set $D$ such that $\\lbrace \\chi >0\\rbrace \\subset D \\subset \\Omega $ , one can define a cutoff $\\eta $ such that $D \\subset \\lbrace \\eta = 1 \\rbrace $ and split the space roughly into a region where $\\eta = 1$ , a region where $\\eta = 0$ and a transition region.",
"In the first two regions, $(-\\Delta )^s(\\eta u)$ essentially coincides with a convolution operator with kernel $|z|^{-d-2s}$ but regularized at the origin, while the latter region is contained in the non-contact set $N$ and allows one to invoke interior regularity estimates for linear problems involving $(-\\Delta )^s$ .",
"An important outcome is that solutions to fractional obstacle problems are more regular near the free boundary ($C^{1,s}$ ) than near the domain boundary ($C^{0,s}$ ).",
"This is critical for approximation.",
"Alternatively, one may invoke the Caffarelli-Silvestre extension [24] to obtain local regularity estimates [23].",
"Since the extension problem involves a degenerate elliptic equation with a Muckenhoupt weight of class $A_2$ that depends only on the extended variable, one needs to combine fine estimates for degenerate equations with the translation invariance in the $x$ -variable of the Caffarelli-Silvestre weight.",
"Once the interior regularity of solutions is established, one can invoke the Hölder boundary estimates for linear problems [59] and perform an argument similar to the one in [2] to deduce weighted Sobolev regularity estimates [16].",
"Theorem 3.1 (weighted Sobolev regularity for the obstacle problem) Let $\\Omega $ be a bounded Lipschitz domain satisfying the exterior ball condition, $s \\in (0,1)$ , and $\\chi \\in C^{2,1}(\\Omega )$ satisfying (REF ).",
"Moreover, let $0 \\le f \\in \\mathcal {F}_s(\\overline{\\Omega })$ and $u \\in {\\widetilde{H}}^s(\\Omega )$ be the solution to (REF ).",
"For every $\\varepsilon >0$ we have that $u \\in {\\widetilde{H}}^{s+1-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )$ with the estimate $ \\Vert u\\Vert _{{\\widetilde{H}}^{s+1-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )} \\le \\frac{C(\\chi , s, d, \\Omega , \\varrho , \\Vert f \\Vert _{\\mathcal {F}_s(\\overline{\\Omega })})}{\\varepsilon }.$ We have stated the estimate in T:regularityobstacle in weighted spaces because we are interested in the application of that result for finite element schemes over graded meshes.",
"With the same arguments as in [16], it can be shown that the solution to the fractional obstacle problem (REF ) satisfies $u \\in {\\widetilde{H}}^{s+1/2-\\varepsilon }(\\Omega )$ and $\\Vert u\\Vert _{{\\widetilde{H}}^{s+1/2-\\varepsilon }(\\Omega )} \\le \\frac{C(\\chi , s, d, \\Omega , \\varrho , \\Vert f \\Vert _{\\mathcal {F}_s(\\overline{\\Omega })})}{\\varepsilon }.$ A similar result, for the obstacle problem for a class of integro-differential operators, was obtained in [12].",
"In the case of purely fractional diffusion (i.e., problems without a second-order differential operator), the estimate builds on [47] (cf.",
"T:reggrubb).",
"Therefore, we point out that using T:Besovregularity, the requirement that $\\Omega $ be a $C^\\infty $ domain in [12] can be relaxed to $\\Omega $ being Lipschitz."
],
[
"Finite element discretization",
"We consider the same finite element setting as in sec:FElinear: let $\\mathbb {V}_h$ be linear Lagrangian finite element spaces as in (REF ) over a family of conforming and simplicial meshes $.",
"An instrumental tool in the analysis we review here is the interpolation operator $ h: L1() Vh$ introduced in \\cite {ChenNochetto} that, besides satisfying (\\ref {eq:interpolation}), is {\\it positivity preserving}: it satisfies $ hv 0$ for all $ v 0$.",
"Such a property yields that, for every $ v K$,\\begin{equation} \\Pi _hv \\ge \\Pi _h\\chi \\ \\mbox{ in } \\Omega .\\end{equation}$ We therefore define the discrete admissible convex set ${\\mathcal {K}}_h := \\left\\lbrace v_h \\in \\mathbb {V}_h: v_h \\ge \\Pi _h\\chi \\mbox{ in } \\Omega \\right\\rbrace ,$ and consider the discrete fractional obstacle problem: find $u_h \\in {\\mathcal {K}}_h$ such that $(u_h,u_h-v_h)_s \\le \\langle f, u_h-v_h \\rangle \\quad \\forall v_h \\in {\\mathcal {K}}_h.$ We illustrate the delicate interplay between regularity and approximability next.",
"We exploit that $u$ is both globally of class $C^{0,s}(\\overline{\\Omega })$ , via graded meshes as in the linear problem, and locally of class $C^{1,s}(\\Omega )$ .",
"First, we split the error as $\\Vert u-u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )}^2 = (u - u_h, u - I_h u)_s + (u - u_h, I_h u - u_h)_s$ , use Cauchy-Schwarz inequality and the interpolation estimate (REF ) to deduce $\\frac{1}{2} \\Vert u-u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )}^2 \\le C h^{2(1-2\\varepsilon )} \\Vert u \\Vert ^2_{\\widetilde{H}^{1+s-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )} + (u - u_h, I_h u - u_h)_s.$ This is a consequence of Theorem REF and the use of graded meshes with parameter $\\mu =2$ as in the linear theory of Section REF .",
"For the remaining term we integrate by parts and utilize the discrete variational inequality (REF ) to arrive at $\\begin{aligned}( u - u_h &, I_h u - u_h )_s \\le \\int _\\Omega (I_h u - u_h) \\big ((-\\Delta )^su -f \\big ) \\\\& = \\int _\\Omega \\Big [ (u -\\chi ) + \\underbrace{(I_h \\chi - u_h)}_{\\le 0} + \\big (I_h (u - \\chi ) - (u -\\chi ) \\big ) \\Big ] \\Big (\\underbrace{(-\\Delta )^su -f}_{\\ge 0} \\Big ).\\end{aligned}$ Invoking the complementarity condition (REF ), we obtain $(u - u_h, I_h u - u_h)_s \\le \\sum _{T \\in \\int _T \\big ( I_h (u - \\chi ) - (u - \\chi ) \\big ) \\big ((-\\Delta )^su -f \\big ).We next observe that the integrand does not vanish only for elements T in the vicinity of the free boundary, namely T^{\\prime }s for which u\\ne \\chi and (-\\Delta )^su \\ne f. Exploiting that u\\in C^{1,s}(\\Omega ), we infer that (-\\Delta )^su -f \\in C^{0,1-s}(\\Omega ), whence\\big | \\big ( (-\\Delta )^su -f \\big ) \\big ( I_h (u - \\chi ) -(u - \\chi ) \\big ) \\big | \\le C h^{2} .This yields the following optimal energy error estimate.",
"We refer to \\cite {BoNoSa18} for details.",
"}$ Theorem 3.2 (error estimate for obstacle problem) Let $u$ be the solution to (REF ) and $u_h$ be the solution to (REF ), respectively.",
"Assume that $\\chi \\in C^{2,1}(\\Omega )$ satisfies (REF ) and that $f \\in \\mathcal {F}_s(\\overline{\\Omega })$ .",
"If $d=2$ , $\\Omega $ is a convex polygon, and the meshes satisfy the grading hypothesis (REF ) with $\\mu = 2$ , then we have that $ \\begin{aligned}& \\Vert u-u_h\\Vert _{{\\widetilde{H}}^s(\\Omega )} \\le C h |\\log h| & (s \\ne 1/2), \\\\& \\Vert u-u_h\\Vert _{\\widetilde{H}^{1/2}(\\Omega )} \\le C h |\\log h|^2 & (s = 1/2),\\end{aligned} $ where $C>0$ depends on $\\chi $ , $s$ , $d$ , $\\Omega $ , $\\varrho $ and $\\Vert f \\Vert _{\\mathcal {F}_s(\\overline{\\Omega })}$ .",
"We conclude this section with a computational example illustrating the qualitative behavior of solutions.",
"Further experiments can be found in [16].",
"Figure: Discrete solutions to the fractional obstacle problem for s=0.1s=0.1 (left), s=0.5s=0.5 (center) and s=0.9s=0.9 (right) computed with graded meshes with h=2 -5 h = 2^{-5}.",
"Top: lateral view.",
"Bottom: top view, with the discrete contact set highlighted.Example 3.3 (qualitative behavior) Consider problem (REF ), posed in the unit ball $B_1 \\subset \\mathbb {R}^2$ , with $f = 0$ and the obstacle $\\chi (x_1, x_2) = \\frac{1}{2} - \\sqrt{ \\left(x_1 - \\frac{1}{4}\\right)^2 + \\frac{1}{2} x_2^2}.$ fig:qualitative shows solutions for $s \\in \\lbrace 0.1, 0.5, 0.9 \\rbrace $ on meshes graded according to (REF ) with $\\mu =2$ .",
"The coincidence set $\\Lambda $ , which contains a neighborhood of the singular point $(1/4,0)$ is displayed in color in the bottom view.",
"It can be observed that, while for $s=0.9$ the discrete solution resembles what is expected for the classical obstacle problem, the solution for $s=0.1$ is much flatter in the non-coincidence set $N$ .",
"Because $f \\ge 0$ , the solution $u$ satisfies $u \\ge 0$ .",
"Therefore, the solution $u$ approaches $\\chi _+$ in the limit $s \\rightarrow 0$ , while $u$ is expected to touch the obstacle only at the singular point and detach immediately in the diffusion limit $s=1$ .",
"The discrete nonlinear system has been solved using a semi-smooth Newton method."
],
[
"Fractional minimal graphs",
"In this section we discuss the fractional minimal graph problem.",
"The line of study of this nonlinear fractional problem, that can be regarded as a nonlocal version of the classical Plateau problem, and other related problems, began with the seminal works by Imbert [49] and Caffarelli, Roquejoffre and Savin [22].",
"As a motivation for the notion of fractional minimal sets, we show how the fractional perimeter arises in the study of a nonlocal version of the Ginzburg-Landau energy, extending a well-known result for classical minimal sets [56].",
"Let $\\Omega \\subset {\\mathbb {R}^d}$ be a bounded set with Lipschitz boundary, $\\varepsilon > 0$ and define the energy $\\mathcal {J}_{\\varepsilon }[u;\\Omega ] := \\frac{\\varepsilon }{2} \\int _\\Omega |\\nabla u(x)|^2 \\; dx + \\frac{1}{\\varepsilon }\\int _{\\Omega } W(u(x)) \\; dx,$ where $W(t) = \\frac{1}{4}(1-t^2)^2$ is a double-well potential.",
"Then, for every sequence $\\lbrace u_\\varepsilon \\rbrace $ of minimizers of $\\mathcal {J}_{\\varepsilon }[u;\\Omega ]$ with uniformly bounded energies there exists a subsequence $\\lbrace u_{\\varepsilon _k} \\rbrace $ such that $ u_{\\varepsilon _k} \\rightarrow \\chi _E - \\chi _{E^c} \\quad \\mbox{in } L^1(\\Omega ),$ where $E$ is a set with minimal perimeter in $\\Omega $ .",
"We now consider a different regularization term: given $s \\in (0,1/2)$ , we set $\\mathcal {J}^s_{\\varepsilon }[u;\\Omega ] := \\frac{1}{2}\\iint _{Q_{\\Omega }} \\frac{|u(x) - u(y)|^2}{|x-y|^{n+2s}} \\; dx dy + \\frac{1}{\\varepsilon ^{2s}} \\int _{\\Omega } W(u(x)) \\; dx,$ where ${Q_{\\Omega } = \\left( {\\mathbb {R}^d}\\times {\\mathbb {R}^d}\\right) \\setminus \\left( {\\Omega }^c \\times {\\Omega }^c \\right)}$ as in (REF ).",
"The first term in the definition of $\\mathcal {J}^s_\\varepsilon $ involves the $H^s({\\mathbb {R}^d})$ -norm of $u$ , except that the interactions over $\\Omega ^c \\times \\Omega ^c$ are removed; for a minimization problem in $\\Omega $ , these are indeed fixed.",
"As proved in [62], for every sequence $\\lbrace u_\\varepsilon \\rbrace $ of minimizers of $\\mathcal {J}^s_{\\varepsilon }$ with uniformly bounded energies there exists a subsequence $\\lbrace u_{\\varepsilon _k} \\rbrace $ such that $ u_{\\varepsilon _k} \\rightarrow \\chi _E - \\chi _{E^c} \\quad \\mbox{in } L^1(\\Omega ) \\quad \\mbox{as } \\varepsilon _k \\rightarrow 0^+.$ However, instead of minimizing the perimeter in $\\Omega $ , here the set $E$ is a $s$ -minimal set in $\\Omega $ , because it minimizes the so-called fractional perimeter $P_s(E,\\Omega )$ among all measurable sets $F \\subset {\\mathbb {R}^d}$ such that $F \\setminus \\Omega = E \\setminus \\Omega $ .",
"In [22] this notion of fractional perimeter (also known as nonlocal perimeter) was proposed, and nonlocal minimal set problems were studied.",
"We refer to [19] and [29] for nice introductory expositions to the topic and applications."
],
[
"Formulation of the problem and regularity",
"Our goal is to compute fractional minimal graphs, that is, to study the nonlocal minimal surface problem under the restriction of the domain being a cylinder.",
"Concretely, from now on we consider $\\Omega ^{\\prime } = \\Omega \\times \\mathbb {R}$ with $\\Omega \\subset {\\mathbb {R}^d}$ being a bounded Lipschitz domain.",
"We assume that the exterior datum is the subgraph of some uniformly bounded function $g: {\\mathbb {R}^d}\\setminus \\Omega \\rightarrow \\mathbb {R}$ , $ E_0 := \\left\\lbrace (x^{\\prime }, x_{d+1}) \\colon x_{d+1} < g(x^{\\prime }), \\; x^{\\prime } \\in {\\mathbb {R}^d}\\setminus \\Omega \\right\\rbrace .$ The fractional minimal graph problem consists in finding a locally $s$ -minimal set $E$ in $\\Omega ^{\\prime }$ such that $E \\setminus \\Omega ^{\\prime } = E_0$ .",
"We refer to [53] for details on why the notion of locally $s$ -minimality is the `correct' one.",
"Under the conditions described above, it can be shown that minimal sets need to be subgraphs, that is, $E \\cap \\Omega ^{\\prime } = \\left\\lbrace (x^{\\prime }, x_{d+1}) \\colon x_{d+1} < u(x^{\\prime }), \\; x^{\\prime } \\in \\Omega \\right\\rbrace $ for some function $u$ (cf.",
"[54]).",
"We shall refer to such a set $E$ as a nonlocal minimal graph in $\\Omega $ .",
"In order to find nonlocal minimal graphs, we introduce the space $\\mathbb {V}^g := \\lbrace v \\colon {\\mathbb {R}^d}\\rightarrow \\mathbb {R}\\; \\colon \\; v\\big |_\\Omega \\in W^{2s}_1(\\Omega ), \\ v = g \\text{ in } {\\Omega }^c\\rbrace $ (we write $\\mathbb {V}^0$ whenever $g \\equiv 0$ ) and, considering the weight function $F_s \\colon \\mathbb {R}\\rightarrow \\mathbb {R}$ , $ F_s(\\rho ) := \\int _0^\\rho \\frac{\\rho -r}{\\left( 1+r^2\\right)^{(d+1+2s)/2}} dr.$ we define the energy functional $I_s[u] := \\iint _{Q_{\\Omega }} F_s\\left(\\frac{u(x)-u(y)}{|x-y|}\\right) \\frac{1}{|x-y|^{d+2s-1}} \\;dxdy.$ In [54] it is shown that finding nonlocal minimal graphs is equivalent to minimizing the energy $I_s$ over $\\mathbb {V}^g$ .",
"Existence of solution $u$ follows from the existence of locally $s$ -minimal sets [53], while uniqueness is a consequence of $I_s$ being strictly convex.",
"We also point out that for any function $v \\colon {\\mathbb {R}^d}\\rightarrow \\mathbb {R}$ , its energy $I_s[v]$ is closely related to certain ${W^{2s}_1}$ -seminorms [14]: $ \\begin{aligned}& |v|_{W^{2s}_1(\\Omega )} \\le C_1 + C_2 I_s[v],& I_s[v] \\le C_3 \\iint _{Q_{\\Omega }} \\frac{|v(x)-v(y)|}{|x-y|^{d+2s}} dxdy .\\end{aligned} $ To give a clearer picture of the nonlocal minimal graph problem, we compare it to its classical counterpart.",
"Given a bounded domain $\\Omega \\subset {\\mathbb {R}^d}$ with sufficiently smooth boundary, and $g \\colon \\partial \\Omega \\rightarrow {\\mathbb {R}^d}$ , the classical Plateau problem consists in finding $u \\colon \\Omega \\rightarrow {\\mathbb {R}^d}$ that minimizes the graph surface area functional $ I [u] := \\int _\\Omega \\sqrt{1 + |\\nabla u (x)|^2 } \\, dx$ among those functions $u \\in H^1(\\Omega )$ satisfying $u = g$ on $\\partial \\Omega $ .",
"By taking first variation of $I$ , it follows that the minimizer $u$ satisfies $ \\int _\\Omega \\frac{\\nabla u(x) \\cdot \\nabla v (x)}{\\sqrt{1 + |\\nabla u (x)|^2 }} \\, dx = 0 \\quad \\forall \\, v \\in H^1_0(\\Omega ).$ The left hand side in (REF ) consists of an $H^1$ -inner product between $u$ and $v$ , with a possibly degenerate weight that depends on $u$ .",
"For the nonlocal problem, after taking first variation of $I_s$ in (REF ), we obtain that $u$ is a minimizer if and only if $a_u(u,v) = 0 \\quad \\mbox{ for all } v \\in \\mathbb {V}^0,$ where the bilinear form $a_u \\colon \\mathbb {V}^g \\times \\mathbb {V}^0 \\rightarrow \\mathbb {R}$ is given by $ a_u(w,v) := \\iint _{Q_{\\Omega }} \\widetilde{G}_s\\left(\\frac{u(x)-u(y)}{|x-y|}\\right) \\frac{(w(x)-w(y))(v(x)-v(y))}{|x-y|^{d+1+2s}}dx dy,$ and $\\widetilde{G}_s(\\rho ) = \\int _0^1 (1+ \\rho ^2 r^2)^{-(d+1+2s)/2} dr$ and hence it satisfies $\\rho \\widetilde{G}_s(\\rho ) = G_s(\\rho ) = F^{\\prime }_s(\\rho )$ .",
"Similar to (REF ), the left hand side $a_u(u,v)$ in (REF ) is a weighted $H^{s+\\frac{1}{2}}$ -inner product with a possibly degenerate weight depending on $u$ .",
"As for the regularity of nonlocal minimal graphs, the following result is stated in [21] and builds on the arguments in [9], [41].",
"Theorem 4.1 (interior smoothness of nonlocal minimal graphs) Assume $E \\subset \\mathbb {R}^{d+1}$ is a locally $s$ -minimal set in $\\Omega ^{\\prime } = \\Omega \\times \\mathbb {R}$ , given by the subgraph of a measurable function $u$ that is bounded in an open set $\\Lambda \\supset \\Omega $ .",
"Then, $u \\in C^\\infty (\\Omega )$ .",
"Remark 4.2 (stickiness) thm:smoothness does not address boundary regularity.",
"By using (REF ), it can be easily proved that $u \\in W^{2s}_1(\\Omega )$ but, because $2s < 1$ , this does not even guarantee that $u$ has a trace on $\\partial \\Omega $ .",
"In fact, nonlocal minimal graphs can develop discontinuities across $\\partial \\Omega $ .",
"Furthermore, nonlocal minimal graphs generically exhibit this sticky behavior [33], [34]."
],
[
"Finite element discretization",
"In this section, we review the finite element discretization of the nonlocal minimal graph problem proposed in [14] and discuss its convergence and error estimates.",
"For simplicity, we assume that $\\mbox{supp}(g) \\subset \\Lambda $ for some bounded set $\\Lambda $ .",
"As before, we take a family $\\lbrace \\mathcal {T}_h\\rbrace _{h>0}$ of conforming, simplicial and shape-regular meshes on $\\Lambda $ , which we impose to mesh $\\Omega $ exactly.",
"To account for non-zero boundary data, we make a slight modification on (REF ) to define the discrete spaces $\\mathbb {V}_h := \\lbrace v \\in C(\\Lambda ) \\colon v|_T \\in \\mathcal {P}_1 \\; \\forall T \\in \\mathcal {T}_h\\rbrace ,$ and we define $\\mathbb {V}_h^g := \\lbrace v_h \\in \\mathbb {V}_h \\colon \\ v_h|_{\\Lambda \\setminus \\Omega } = \\Pi _h^c g\\rbrace , \\quad \\mathbb {V}_h^0 := \\lbrace v_h \\in \\mathbb {V}_h \\colon \\ v_h|_{\\Lambda \\setminus \\Omega } = 0\\rbrace ,$ where $\\Pi _h^c$ denotes the Clément interpolation operator in $\\Omega ^c$ .",
"With the notation introduced above, the discrete problem seeks $u_h \\in \\mathbb {V}^g_h$ such that $a_{u_h}(u_h, v_h) = 0 \\quad \\mbox{for all } v_h \\in \\mathbb {V}^0_h.$ Existence and uniqueness of solutions to this discrete problem follow directly from (REF ) and the strict convexity of $I_s$ .",
"To prove the convergence of the finite element scheme, the approach in [14] consists in proving that the discrete energy is consistent and afterwards using a compactness argument.",
"Theorem 4.3 (convergence for the nonlocal minimal graph problem) Let $s \\in (0,1/2)$ , $\\Omega $ be a bounded Lipschitz domain and $g$ be uniformly bounded and satisfying $\\mbox{supp}(g) \\subset \\Lambda $ for some bounded set $\\Lambda $ .",
"Let $u$ and $u_h$ be, respectively, the solutions to (REF ) and (REF ).",
"Then, it holds that $\\lim _{h \\rightarrow 0} I_s[u_h] = I_s[u] \\quad \\mbox{ and } \\quad \\lim _{h \\rightarrow 0} \\Vert u - u_h \\Vert _{W^{2r}_1(\\Omega )} = 0 \\quad \\forall r \\in [0,s).$ The theorem above has the important feature of guaranteeing convergence without any regularity assumption on the solution.",
"However, it does not offer convergence rates.",
"We now show estimates for a geometric notion of error that mimics the one analyzed in [40] for the classical Plateau problem (see also [8], [30]).",
"Such a notion of error is given by $\\begin{aligned}e^2(u,u_h) & :=\\int _{\\Omega } \\ \\Big | \\widehat{\\nu }(\\nabla u) - \\widehat{\\nu }(\\nabla u_h) \\Big |^2 \\;\\frac{Q(\\nabla u) + Q(\\nabla u_h)}{2} \\ dx , \\\\& = \\int _{\\Omega } \\ \\Big ( \\widehat{\\nu }(\\nabla u) - \\widehat{\\nu }(\\nabla u_h) \\Big ) \\cdot \\ \\Big ( \\nabla (u-u_h), 0 \\Big ) dx ,\\end{aligned}$ where $Q({a}) = \\sqrt{1+|{a}|^2}$ , $\\widehat{\\nu }({a}) = \\frac{({a},-1)}{Q({a})}$ .",
"Because $\\widehat{\\nu }(\\nabla u)$ is the normal unit vector on the graph of $u$ , the quantity $e(u,u_h)$ is a weighted $L^2$ -discrepancy between the normal vectors.",
"For the nonlocal minimal graph problem, [14] introduced ${\\begin{aligned}e_s(u,u_h) := \\left( \\widetilde{C}_{d,s} \\iint _{Q_{\\Omega }} \\Big ( G_s\\left(d_u(x,y)\\right) - G_s\\left(d_{u_h}(x,y)\\right) \\Big ) \\frac{d_{u-u_h}(x,y)}{|x-y|^{d-1+2s}} dxdy \\right)^{1/2} ,\\end{aligned}}$ where $G_s(\\rho ) = F_s^{\\prime }(\\rho )$ , the constant $\\widetilde{C}_{d,s} = \\frac{1 - 2s}{\\alpha _{d}}$ , $\\alpha _{d}$ is the volume of the $d$ -dimensional unit ball and $d_v$ is the difference quotient of the function $v$ , $d_v(x,y) := \\frac{v(x)-v(y)}{|x-y|}.$ In [14], this novel quantity $e_s(u,u_h)$ is shown to be connected with a notion of nonlocal normal vector, and its asymptotic behavior as $s\\rightarrow 1/2^-$ is established.",
"Theorem 4.4 (asymptotics of $e_s$ ) For all $u,v \\in H^1_0(\\Lambda )$ , we have $\\lim _{s \\rightarrow {\\frac{1}{2}}^-} e_s(u,v) = e(u,v).$ A simple `Galerkin orthogonality' argument allows to derive an error estimate for $e_s(u,u_h)$ (cf.",
"[14]).",
"Theorem 4.5 (geometric error) Under the same hypothesis as in thm:consistency, it holds that $ \\begin{aligned}e_s(u,u_h) &\\le C (d,s) \\, \\inf _{v_h \\in \\mathbb {V}_h^g} \\left( \\iint _{Q_{\\Omega }} \\frac{|(u-v_h)(x)-(u-v_h)(y)|}{|x-y|^{d+2s}} dxdy \\right) ^{1/2}.\\end{aligned} $ Therefore, to obtain convergence rates with respect to $e_s(u,u_h)$ , it suffices to prove interpolation estimates for the nonlocal minimizer.",
"Although minimal graphs are expected to be discontinuous across the boundary, we still expect that $u \\in BV(\\Lambda )$ in general.",
"Under this circumstance, the error estimate (REF ) leads to $e_s(u,u_h) \\le C(d,s) \\, h^{1/2-s} |u|^{1/2}_{BV(\\Lambda )}.$"
],
[
"Numerical experiments",
"We conclude by presenting several numerical experiments and discussing the behavior of nonlocal minimal graphs.",
"The first example we compute is on a one-dimensional domain, and is proposed and theoretically studied in [33] as an illustration of stickiness phenomena.",
"Example 4.6 (stickiness in 1D) Let $\\Omega = (-1,1)$ and $g(x) = \\textrm {sign}(x)$ for $x \\in \\Omega ^c$ .",
"Discrete nonlocal minimal graphs for $s\\in \\lbrace 0.1, 0.25,0.4\\rbrace $ are shown in F:Ex-stickiness (left).",
"Figure: Left: plot of u h u_h for s=0.1,0.25,0.4s = 0.1, 0.25, 0.4 (from left to right) in Ex:1dstickiness.",
"Right: plot of u h u_h for uniform meshes with h=2 -5 h = 2^{-5} in Ex:2dannulus.Although our method requires discrete functions to be continuous across the boundary of $\\Omega $ , the presented 1D picture clearly suggests a stickiness phenomenon.",
"In addition, the plot also indicates that stickiness becomes more observable when $s$ gets closer to 0.",
"Example 4.7 (stickiness in an annulus) Let $\\Omega = B_1 \\setminus \\overline{B}_{1/2} \\subset \\mathbb {R}^2$ , where $B_r$ denotes an open ball with radius $r$ centered at the origin, and let $g = \\chi _{B_{1/2}}$ .",
"The discrete nonlocal minimal graph for $s = 0.25$ is plotted in F:Ex-stickiness (right).",
"In this example, stickiness is clearly observed on $\\partial B_{1/2}$ , while the stickiness on $\\partial B_1$ is less noticeable.",
"Example 4.8 (effect of $s$ ) Let $\\Omega = B_1 \\subset \\mathbb {R}^2$ , $g = \\chi _{B_{3/2}\\setminus \\Omega }$ .",
"F:NMS-Exmultis shows minimizers for several values of $s$ .",
"As $s \\rightarrow 1/2$ , the nonlocal minimal graphs get closer to the classical minimal graph, which is trivially constant in $\\Omega $ .",
"On the other hand, as $s$ decreases we observe a stronger jump across $\\partial \\Omega $ .",
"Figure: Plot of u h u_h for s=0.01,0.1,0.25,0.4,0.49s = 0.01, 0.1, 0.25, 0.4, 0.49 (from left to right) with uniform h=2 -5 h = 2^{-5} in Ex:2dcircle.In the end, we make brief comments on the computational side for the examples presented above.",
"In all of the experiments for fractional minimal graphs, we use Newton's method to solve the nonlinear equation (REF ).",
"Although the Jacobian matrix $\\mathbf {A}_{u_k}$ in the iterative process corresponds to an $H^{s+1/2}-$ inner product $a_{u_k} (w,v)$ with degenerate weights, our experiments indicate its condition number behaves like $\\kappa (\\mathbf {A}_{u_k}) \\approx \\mathcal {O}\\left( N^{2(s+\\frac{1}{2})/d} \\right)$ for $u_k$ whose gradient blows up near the boundary $\\partial \\Omega $ , quasi-uniform meshes and dimensions $d=1,2$ .",
"This behavior is the same as the one in linear fractional diffusion with order $s+\\frac{1}{2}$ .",
"However, the degenerate weight does bring more difficulties in preconditioning.",
"Another thing worth to be pointed out is the use of Dirichlet condition for our discrete space $\\mathbb {V}_h^g$ requires the discrete function $u_h$ to be continuous across $\\partial \\Omega $ .",
"Due to the stickiness phenomenon in rem:stickiness, this may not be true for the solution $u$ of the minimal graph problem.",
"Fortunately, this does not preclude the convergence in `trace blind' fractional Sobolev spaces $W^{2s}_1(\\Omega )$ , and we are still able to capture discontinuities across the boundary in practice.",
"While permitting discontinuities would be desirable, it has conflicts with using Newton's method in solving (REF ) because the bilinear form $a_u(w,v)$ in (REF ) may not be well-defined.",
"The question of how to solve the nonlinear equation (REF ) faster when allowing discontinuous across $\\partial \\Omega $ is still under investigation.",
"In this section we review finite element methods for the solution of the obstacle problem for the integral fractional Laplacian which, from now on, we shall simply refer to as the fractional obstacle problem.",
"The fractional obstacle problem appears, for example, in optimal stopping times for jump processes.",
"In particular, it is used in the modeling of the rational price of perpetual American options [28].",
"More precisely, if $u$ represents the rational price of a perpetual American option where the assets prices are modeled by a Lévy process $X_t$ , if $\\chi $ denotes the payoff function, then $u$ solves a fractional obstacle problem with obstacle $\\chi $ .",
"An a posteriori error analysis of approximations of variational inequalities involving integral operators on arbitrary bounded domains was performed in [58].",
"We also comment on two recent works related to the approach we review here.",
"Reference [20] deals with finite element discretizations to obstacle problems involving finite and infinite-horizon nonlocal operators.",
"The experiments shown therein were performed on one-dimensional problems with uniform meshes, and indicate convergence with order $h^{1/2}$ in the energy norm.",
"A theoretical proof of that convergence order was obtained in [12], where approximations using the approach discussed in sec:dunford-taylor were considered.",
"We also refer to [44] for computational comparisons between adaptive strategies and uniform and graded discretizations in two-dimensional problems."
],
[
"Variational formulation",
"As before, we assume that $\\Omega \\subset {\\mathbb {R}^d}$ is an open and bounded domain and, for the sake of applying weighted regularity estimates, we assume that $\\Omega $ has a Lipschitz boundary and satisfies the exterior ball condition.",
"Given $s \\in (0,1)$ and functions $f: \\Omega \\rightarrow \\mathbb {R}$ and $\\chi : \\overline{\\Omega }\\rightarrow \\mathbb {R}$ , with $\\chi < 0$ on $\\partial \\Omega $ , the obstacle problem is a constrained minimization problem on ${\\widetilde{H}}^s(\\Omega )$ associated with a quadratic functional.",
"Defining the admissible convex set ${\\mathcal {K}}:= \\left\\lbrace v \\in {\\widetilde{H}}^s(\\Omega ): v \\ge \\chi \\mbox{ a.e in } \\Omega \\right\\rbrace ,$ the solution to the fractional obstacle problem is $u=\\textrm {argmin}_{v\\in {\\mathcal {K}}} \\mathcal {J}(v)$ where $\\mathcal {J}(v) := \\frac{1}{2} \\Vert v \\Vert _{{\\widetilde{H}}^s(\\Omega )}^2 - \\langle f, v \\rangle .$ Existence and uniqueness of solutions is standard.",
"Taking first variation of $\\mathcal {J}$ , we deduce that such a minimizer $u \\in {\\mathcal {K}}$ solves the variational inequality $(u,u-v)_s \\le \\langle f, u-v \\rangle \\quad \\forall v \\in {\\mathcal {K}}.$ It can be shown [57] that, if $f \\in L^p(\\Omega )$ for $p > d/2s$ , then the solution to the obstacle problem is indeed a continuous function, and that it satisfies the complementarity condition $\\min \\left\\lbrace \\lambda , u-\\chi \\right\\rbrace = 0 \\quad \\mbox{ a.e in } \\Omega , \\mbox{ where } \\quad \\lambda := (-\\Delta )^su - f.$ For our discussion, we assume that $f$ is such that the solution is defined pointwise, and consequently, we define the coincidence (or contact) and non-coincidence sets, $\\Lambda := \\lbrace x \\in \\Omega : u(x) = \\chi (x) \\rbrace , \\quad N := \\Omega \\setminus \\Lambda .$ The complementarity condition (REF ) can be succinctly expressed as $\\lambda \\ge 0$ in $\\Lambda $ and $\\lambda = 0$ in $N$ .",
"The set $\\partial \\Lambda $ , where the solution detaches from the obstacle, is the free boundary."
],
[
"Regularity",
"The following regularity results for solutions to the fractional obstacle problem are instrumental for error analysis.",
"We recall our assumption that the obstacle $\\chi $ is a continuous function and strictly negative on $\\partial \\Omega $ : $ \\varrho := \\textrm {dist}\\left( \\lbrace \\chi >0\\rbrace , \\partial \\Omega \\right) > 0.$ Furthermore, we shall assume that $f \\ge 0$ .",
"Heuristically, these assumptions should guarantee that the behavior of solutions near $\\partial \\Omega $ is dictated by a linear problem and that the nonlinearity is confined to the interior of the domain.",
"Finally, to derive regularity estimates, we assume that the data satisfy $\\chi \\in C^{2,1}(\\Omega ), \\quad f \\in \\mathcal {F}_s(\\overline{\\Omega }) = \\begin{dcases}C^{2,1-2s+\\epsilon }(\\overline{\\Omega }), & s\\in \\left(0,\\frac{1}{2}\\right], \\\\C^{1,2-2s+\\epsilon }(\\overline{\\Omega }), & s \\in \\left(\\frac{1}{2},1\\right),\\end{dcases}$ where $\\epsilon >0$ is sufficiently small, so that $1-2s+\\epsilon $ is not an integer.",
"Under these conditions, Caffarelli, Salsa and Silvestre [23] proved that the solution to the problem posed in the whole space (with suitable decay conditions at infinity) is of class $C^{1,s}({\\mathbb {R}}^d)$ .",
"It is worth examining the limiting cases $s=1$ and $s=0$ .",
"The former corresponds to the classical obstacle problem whose solutions are of class $C^{1,1}({\\mathbb {R}}^d)$ .",
"The latter reduces to $\\min \\lbrace u-\\chi ,u-f\\rbrace =0$ whose solutions are just of class $C^{0,1}({\\mathbb {R}}^d)$ .",
"The regularity of [23] is thus a natural intermediate result.",
"We emphasize that deriving interior regularity estimates for (REF ) from this result, which is valid for problems posed in ${\\mathbb {R}^d}$ , is not as straightforward as for classical problems.",
"Indeed, the nonlocal structure of $(-\\Delta )^s$ implies that, if $0\\le \\eta \\le 1$ is a smooth cut-off function such that $\\eta =1$ in $\\lbrace \\chi > 0 \\rbrace $ , then $(-\\Delta )^s(\\eta u) \\ne \\eta (-\\Delta )^su \\quad \\mbox{in } \\lbrace \\eta = 1 \\rbrace .$ To overcome this difficulty, reference [16] proceeds as follows.",
"Given a set $D$ such that $\\lbrace \\chi >0\\rbrace \\subset D \\subset \\Omega $ , one can define a cutoff $\\eta $ such that $D \\subset \\lbrace \\eta = 1 \\rbrace $ and split the space roughly into a region where $\\eta = 1$ , a region where $\\eta = 0$ and a transition region.",
"In the first two regions, $(-\\Delta )^s(\\eta u)$ essentially coincides with a convolution operator with kernel $|z|^{-d-2s}$ but regularized at the origin, while the latter region is contained in the non-contact set $N$ and allows one to invoke interior regularity estimates for linear problems involving $(-\\Delta )^s$ .",
"An important outcome is that solutions to fractional obstacle problems are more regular near the free boundary ($C^{1,s}$ ) than near the domain boundary ($C^{0,s}$ ).",
"This is critical for approximation.",
"Alternatively, one may invoke the Caffarelli-Silvestre extension [24] to obtain local regularity estimates [23].",
"Since the extension problem involves a degenerate elliptic equation with a Muckenhoupt weight of class $A_2$ that depends only on the extended variable, one needs to combine fine estimates for degenerate equations with the translation invariance in the $x$ -variable of the Caffarelli-Silvestre weight.",
"Once the interior regularity of solutions is established, one can invoke the Hölder boundary estimates for linear problems [59] and perform an argument similar to the one in [2] to deduce weighted Sobolev regularity estimates [16].",
"Theorem 3.1 (weighted Sobolev regularity for the obstacle problem) Let $\\Omega $ be a bounded Lipschitz domain satisfying the exterior ball condition, $s \\in (0,1)$ , and $\\chi \\in C^{2,1}(\\Omega )$ satisfying (REF ).",
"Moreover, let $0 \\le f \\in \\mathcal {F}_s(\\overline{\\Omega })$ and $u \\in {\\widetilde{H}}^s(\\Omega )$ be the solution to (REF ).",
"For every $\\varepsilon >0$ we have that $u \\in {\\widetilde{H}}^{s+1-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )$ with the estimate $ \\Vert u\\Vert _{{\\widetilde{H}}^{s+1-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )} \\le \\frac{C(\\chi , s, d, \\Omega , \\varrho , \\Vert f \\Vert _{\\mathcal {F}_s(\\overline{\\Omega })})}{\\varepsilon }.$ We have stated the estimate in T:regularityobstacle in weighted spaces because we are interested in the application of that result for finite element schemes over graded meshes.",
"With the same arguments as in [16], it can be shown that the solution to the fractional obstacle problem (REF ) satisfies $u \\in {\\widetilde{H}}^{s+1/2-\\varepsilon }(\\Omega )$ and $\\Vert u\\Vert _{{\\widetilde{H}}^{s+1/2-\\varepsilon }(\\Omega )} \\le \\frac{C(\\chi , s, d, \\Omega , \\varrho , \\Vert f \\Vert _{\\mathcal {F}_s(\\overline{\\Omega })})}{\\varepsilon }.$ A similar result, for the obstacle problem for a class of integro-differential operators, was obtained in [12].",
"In the case of purely fractional diffusion (i.e., problems without a second-order differential operator), the estimate builds on [47] (cf.",
"T:reggrubb).",
"Therefore, we point out that using T:Besovregularity, the requirement that $\\Omega $ be a $C^\\infty $ domain in [12] can be relaxed to $\\Omega $ being Lipschitz."
],
[
"Finite element discretization",
"We consider the same finite element setting as in sec:FElinear: let $\\mathbb {V}_h$ be linear Lagrangian finite element spaces as in (REF ) over a family of conforming and simplicial meshes $.",
"An instrumental tool in the analysis we review here is the interpolation operator $ h: L1() Vh$ introduced in \\cite {ChenNochetto} that, besides satisfying (\\ref {eq:interpolation}), is {\\it positivity preserving}: it satisfies $ hv 0$ for all $ v 0$.",
"Such a property yields that, for every $ v K$,\\begin{equation} \\Pi _hv \\ge \\Pi _h\\chi \\ \\mbox{ in } \\Omega .\\end{equation}$ We therefore define the discrete admissible convex set ${\\mathcal {K}}_h := \\left\\lbrace v_h \\in \\mathbb {V}_h: v_h \\ge \\Pi _h\\chi \\mbox{ in } \\Omega \\right\\rbrace ,$ and consider the discrete fractional obstacle problem: find $u_h \\in {\\mathcal {K}}_h$ such that $(u_h,u_h-v_h)_s \\le \\langle f, u_h-v_h \\rangle \\quad \\forall v_h \\in {\\mathcal {K}}_h.$ We illustrate the delicate interplay between regularity and approximability next.",
"We exploit that $u$ is both globally of class $C^{0,s}(\\overline{\\Omega })$ , via graded meshes as in the linear problem, and locally of class $C^{1,s}(\\Omega )$ .",
"First, we split the error as $\\Vert u-u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )}^2 = (u - u_h, u - I_h u)_s + (u - u_h, I_h u - u_h)_s$ , use Cauchy-Schwarz inequality and the interpolation estimate (REF ) to deduce $\\frac{1}{2} \\Vert u-u_h \\Vert _{{\\widetilde{H}}^s(\\Omega )}^2 \\le C h^{2(1-2\\varepsilon )} \\Vert u \\Vert ^2_{\\widetilde{H}^{1+s-2\\varepsilon }_{1/2-\\varepsilon }(\\Omega )} + (u - u_h, I_h u - u_h)_s.$ This is a consequence of Theorem REF and the use of graded meshes with parameter $\\mu =2$ as in the linear theory of Section REF .",
"For the remaining term we integrate by parts and utilize the discrete variational inequality (REF ) to arrive at $\\begin{aligned}( u - u_h &, I_h u - u_h )_s \\le \\int _\\Omega (I_h u - u_h) \\big ((-\\Delta )^su -f \\big ) \\\\& = \\int _\\Omega \\Big [ (u -\\chi ) + \\underbrace{(I_h \\chi - u_h)}_{\\le 0} + \\big (I_h (u - \\chi ) - (u -\\chi ) \\big ) \\Big ] \\Big (\\underbrace{(-\\Delta )^su -f}_{\\ge 0} \\Big ).\\end{aligned}$ Invoking the complementarity condition (REF ), we obtain $(u - u_h, I_h u - u_h)_s \\le \\sum _{T \\in \\int _T \\big ( I_h (u - \\chi ) - (u - \\chi ) \\big ) \\big ((-\\Delta )^su -f \\big ).We next observe that the integrand does not vanish only for elements T in the vicinity of the free boundary, namely T^{\\prime }s for which u\\ne \\chi and (-\\Delta )^su \\ne f. Exploiting that u\\in C^{1,s}(\\Omega ), we infer that (-\\Delta )^su -f \\in C^{0,1-s}(\\Omega ), whence\\big | \\big ( (-\\Delta )^su -f \\big ) \\big ( I_h (u - \\chi ) -(u - \\chi ) \\big ) \\big | \\le C h^{2} .This yields the following optimal energy error estimate.",
"We refer to \\cite {BoNoSa18} for details.",
"}$ Theorem 3.2 (error estimate for obstacle problem) Let $u$ be the solution to (REF ) and $u_h$ be the solution to (REF ), respectively.",
"Assume that $\\chi \\in C^{2,1}(\\Omega )$ satisfies (REF ) and that $f \\in \\mathcal {F}_s(\\overline{\\Omega })$ .",
"If $d=2$ , $\\Omega $ is a convex polygon, and the meshes satisfy the grading hypothesis (REF ) with $\\mu = 2$ , then we have that $ \\begin{aligned}& \\Vert u-u_h\\Vert _{{\\widetilde{H}}^s(\\Omega )} \\le C h |\\log h| & (s \\ne 1/2), \\\\& \\Vert u-u_h\\Vert _{\\widetilde{H}^{1/2}(\\Omega )} \\le C h |\\log h|^2 & (s = 1/2),\\end{aligned} $ where $C>0$ depends on $\\chi $ , $s$ , $d$ , $\\Omega $ , $\\varrho $ and $\\Vert f \\Vert _{\\mathcal {F}_s(\\overline{\\Omega })}$ .",
"We conclude this section with a computational example illustrating the qualitative behavior of solutions.",
"Further experiments can be found in [16].",
"Figure: Discrete solutions to the fractional obstacle problem for s=0.1s=0.1 (left), s=0.5s=0.5 (center) and s=0.9s=0.9 (right) computed with graded meshes with h=2 -5 h = 2^{-5}.",
"Top: lateral view.",
"Bottom: top view, with the discrete contact set highlighted.Example 3.3 (qualitative behavior) Consider problem (REF ), posed in the unit ball $B_1 \\subset \\mathbb {R}^2$ , with $f = 0$ and the obstacle $\\chi (x_1, x_2) = \\frac{1}{2} - \\sqrt{ \\left(x_1 - \\frac{1}{4}\\right)^2 + \\frac{1}{2} x_2^2}.$ fig:qualitative shows solutions for $s \\in \\lbrace 0.1, 0.5, 0.9 \\rbrace $ on meshes graded according to (REF ) with $\\mu =2$ .",
"The coincidence set $\\Lambda $ , which contains a neighborhood of the singular point $(1/4,0)$ is displayed in color in the bottom view.",
"It can be observed that, while for $s=0.9$ the discrete solution resembles what is expected for the classical obstacle problem, the solution for $s=0.1$ is much flatter in the non-coincidence set $N$ .",
"Because $f \\ge 0$ , the solution $u$ satisfies $u \\ge 0$ .",
"Therefore, the solution $u$ approaches $\\chi _+$ in the limit $s \\rightarrow 0$ , while $u$ is expected to touch the obstacle only at the singular point and detach immediately in the diffusion limit $s=1$ .",
"The discrete nonlinear system has been solved using a semi-smooth Newton method."
],
[
"Fractional minimal graphs",
"In this section we discuss the fractional minimal graph problem.",
"The line of study of this nonlinear fractional problem, that can be regarded as a nonlocal version of the classical Plateau problem, and other related problems, began with the seminal works by Imbert [49] and Caffarelli, Roquejoffre and Savin [22].",
"As a motivation for the notion of fractional minimal sets, we show how the fractional perimeter arises in the study of a nonlocal version of the Ginzburg-Landau energy, extending a well-known result for classical minimal sets [56].",
"Let $\\Omega \\subset {\\mathbb {R}^d}$ be a bounded set with Lipschitz boundary, $\\varepsilon > 0$ and define the energy $\\mathcal {J}_{\\varepsilon }[u;\\Omega ] := \\frac{\\varepsilon }{2} \\int _\\Omega |\\nabla u(x)|^2 \\; dx + \\frac{1}{\\varepsilon }\\int _{\\Omega } W(u(x)) \\; dx,$ where $W(t) = \\frac{1}{4}(1-t^2)^2$ is a double-well potential.",
"Then, for every sequence $\\lbrace u_\\varepsilon \\rbrace $ of minimizers of $\\mathcal {J}_{\\varepsilon }[u;\\Omega ]$ with uniformly bounded energies there exists a subsequence $\\lbrace u_{\\varepsilon _k} \\rbrace $ such that $ u_{\\varepsilon _k} \\rightarrow \\chi _E - \\chi _{E^c} \\quad \\mbox{in } L^1(\\Omega ),$ where $E$ is a set with minimal perimeter in $\\Omega $ .",
"We now consider a different regularization term: given $s \\in (0,1/2)$ , we set $\\mathcal {J}^s_{\\varepsilon }[u;\\Omega ] := \\frac{1}{2}\\iint _{Q_{\\Omega }} \\frac{|u(x) - u(y)|^2}{|x-y|^{n+2s}} \\; dx dy + \\frac{1}{\\varepsilon ^{2s}} \\int _{\\Omega } W(u(x)) \\; dx,$ where ${Q_{\\Omega } = \\left( {\\mathbb {R}^d}\\times {\\mathbb {R}^d}\\right) \\setminus \\left( {\\Omega }^c \\times {\\Omega }^c \\right)}$ as in (REF ).",
"The first term in the definition of $\\mathcal {J}^s_\\varepsilon $ involves the $H^s({\\mathbb {R}^d})$ -norm of $u$ , except that the interactions over $\\Omega ^c \\times \\Omega ^c$ are removed; for a minimization problem in $\\Omega $ , these are indeed fixed.",
"As proved in [62], for every sequence $\\lbrace u_\\varepsilon \\rbrace $ of minimizers of $\\mathcal {J}^s_{\\varepsilon }$ with uniformly bounded energies there exists a subsequence $\\lbrace u_{\\varepsilon _k} \\rbrace $ such that $ u_{\\varepsilon _k} \\rightarrow \\chi _E - \\chi _{E^c} \\quad \\mbox{in } L^1(\\Omega ) \\quad \\mbox{as } \\varepsilon _k \\rightarrow 0^+.$ However, instead of minimizing the perimeter in $\\Omega $ , here the set $E$ is a $s$ -minimal set in $\\Omega $ , because it minimizes the so-called fractional perimeter $P_s(E,\\Omega )$ among all measurable sets $F \\subset {\\mathbb {R}^d}$ such that $F \\setminus \\Omega = E \\setminus \\Omega $ .",
"In [22] this notion of fractional perimeter (also known as nonlocal perimeter) was proposed, and nonlocal minimal set problems were studied.",
"We refer to [19] and [29] for nice introductory expositions to the topic and applications."
],
[
"Formulation of the problem and regularity",
"Our goal is to compute fractional minimal graphs, that is, to study the nonlocal minimal surface problem under the restriction of the domain being a cylinder.",
"Concretely, from now on we consider $\\Omega ^{\\prime } = \\Omega \\times \\mathbb {R}$ with $\\Omega \\subset {\\mathbb {R}^d}$ being a bounded Lipschitz domain.",
"We assume that the exterior datum is the subgraph of some uniformly bounded function $g: {\\mathbb {R}^d}\\setminus \\Omega \\rightarrow \\mathbb {R}$ , $ E_0 := \\left\\lbrace (x^{\\prime }, x_{d+1}) \\colon x_{d+1} < g(x^{\\prime }), \\; x^{\\prime } \\in {\\mathbb {R}^d}\\setminus \\Omega \\right\\rbrace .$ The fractional minimal graph problem consists in finding a locally $s$ -minimal set $E$ in $\\Omega ^{\\prime }$ such that $E \\setminus \\Omega ^{\\prime } = E_0$ .",
"We refer to [53] for details on why the notion of locally $s$ -minimality is the `correct' one.",
"Under the conditions described above, it can be shown that minimal sets need to be subgraphs, that is, $E \\cap \\Omega ^{\\prime } = \\left\\lbrace (x^{\\prime }, x_{d+1}) \\colon x_{d+1} < u(x^{\\prime }), \\; x^{\\prime } \\in \\Omega \\right\\rbrace $ for some function $u$ (cf.",
"[54]).",
"We shall refer to such a set $E$ as a nonlocal minimal graph in $\\Omega $ .",
"In order to find nonlocal minimal graphs, we introduce the space $\\mathbb {V}^g := \\lbrace v \\colon {\\mathbb {R}^d}\\rightarrow \\mathbb {R}\\; \\colon \\; v\\big |_\\Omega \\in W^{2s}_1(\\Omega ), \\ v = g \\text{ in } {\\Omega }^c\\rbrace $ (we write $\\mathbb {V}^0$ whenever $g \\equiv 0$ ) and, considering the weight function $F_s \\colon \\mathbb {R}\\rightarrow \\mathbb {R}$ , $ F_s(\\rho ) := \\int _0^\\rho \\frac{\\rho -r}{\\left( 1+r^2\\right)^{(d+1+2s)/2}} dr.$ we define the energy functional $I_s[u] := \\iint _{Q_{\\Omega }} F_s\\left(\\frac{u(x)-u(y)}{|x-y|}\\right) \\frac{1}{|x-y|^{d+2s-1}} \\;dxdy.$ In [54] it is shown that finding nonlocal minimal graphs is equivalent to minimizing the energy $I_s$ over $\\mathbb {V}^g$ .",
"Existence of solution $u$ follows from the existence of locally $s$ -minimal sets [53], while uniqueness is a consequence of $I_s$ being strictly convex.",
"We also point out that for any function $v \\colon {\\mathbb {R}^d}\\rightarrow \\mathbb {R}$ , its energy $I_s[v]$ is closely related to certain ${W^{2s}_1}$ -seminorms [14]: $ \\begin{aligned}& |v|_{W^{2s}_1(\\Omega )} \\le C_1 + C_2 I_s[v],& I_s[v] \\le C_3 \\iint _{Q_{\\Omega }} \\frac{|v(x)-v(y)|}{|x-y|^{d+2s}} dxdy .\\end{aligned} $ To give a clearer picture of the nonlocal minimal graph problem, we compare it to its classical counterpart.",
"Given a bounded domain $\\Omega \\subset {\\mathbb {R}^d}$ with sufficiently smooth boundary, and $g \\colon \\partial \\Omega \\rightarrow {\\mathbb {R}^d}$ , the classical Plateau problem consists in finding $u \\colon \\Omega \\rightarrow {\\mathbb {R}^d}$ that minimizes the graph surface area functional $ I [u] := \\int _\\Omega \\sqrt{1 + |\\nabla u (x)|^2 } \\, dx$ among those functions $u \\in H^1(\\Omega )$ satisfying $u = g$ on $\\partial \\Omega $ .",
"By taking first variation of $I$ , it follows that the minimizer $u$ satisfies $ \\int _\\Omega \\frac{\\nabla u(x) \\cdot \\nabla v (x)}{\\sqrt{1 + |\\nabla u (x)|^2 }} \\, dx = 0 \\quad \\forall \\, v \\in H^1_0(\\Omega ).$ The left hand side in (REF ) consists of an $H^1$ -inner product between $u$ and $v$ , with a possibly degenerate weight that depends on $u$ .",
"For the nonlocal problem, after taking first variation of $I_s$ in (REF ), we obtain that $u$ is a minimizer if and only if $a_u(u,v) = 0 \\quad \\mbox{ for all } v \\in \\mathbb {V}^0,$ where the bilinear form $a_u \\colon \\mathbb {V}^g \\times \\mathbb {V}^0 \\rightarrow \\mathbb {R}$ is given by $ a_u(w,v) := \\iint _{Q_{\\Omega }} \\widetilde{G}_s\\left(\\frac{u(x)-u(y)}{|x-y|}\\right) \\frac{(w(x)-w(y))(v(x)-v(y))}{|x-y|^{d+1+2s}}dx dy,$ and $\\widetilde{G}_s(\\rho ) = \\int _0^1 (1+ \\rho ^2 r^2)^{-(d+1+2s)/2} dr$ and hence it satisfies $\\rho \\widetilde{G}_s(\\rho ) = G_s(\\rho ) = F^{\\prime }_s(\\rho )$ .",
"Similar to (REF ), the left hand side $a_u(u,v)$ in (REF ) is a weighted $H^{s+\\frac{1}{2}}$ -inner product with a possibly degenerate weight depending on $u$ .",
"As for the regularity of nonlocal minimal graphs, the following result is stated in [21] and builds on the arguments in [9], [41].",
"Theorem 4.1 (interior smoothness of nonlocal minimal graphs) Assume $E \\subset \\mathbb {R}^{d+1}$ is a locally $s$ -minimal set in $\\Omega ^{\\prime } = \\Omega \\times \\mathbb {R}$ , given by the subgraph of a measurable function $u$ that is bounded in an open set $\\Lambda \\supset \\Omega $ .",
"Then, $u \\in C^\\infty (\\Omega )$ .",
"Remark 4.2 (stickiness) thm:smoothness does not address boundary regularity.",
"By using (REF ), it can be easily proved that $u \\in W^{2s}_1(\\Omega )$ but, because $2s < 1$ , this does not even guarantee that $u$ has a trace on $\\partial \\Omega $ .",
"In fact, nonlocal minimal graphs can develop discontinuities across $\\partial \\Omega $ .",
"Furthermore, nonlocal minimal graphs generically exhibit this sticky behavior [33], [34]."
],
[
"Finite element discretization",
"In this section, we review the finite element discretization of the nonlocal minimal graph problem proposed in [14] and discuss its convergence and error estimates.",
"For simplicity, we assume that $\\mbox{supp}(g) \\subset \\Lambda $ for some bounded set $\\Lambda $ .",
"As before, we take a family $\\lbrace \\mathcal {T}_h\\rbrace _{h>0}$ of conforming, simplicial and shape-regular meshes on $\\Lambda $ , which we impose to mesh $\\Omega $ exactly.",
"To account for non-zero boundary data, we make a slight modification on (REF ) to define the discrete spaces $\\mathbb {V}_h := \\lbrace v \\in C(\\Lambda ) \\colon v|_T \\in \\mathcal {P}_1 \\; \\forall T \\in \\mathcal {T}_h\\rbrace ,$ and we define $\\mathbb {V}_h^g := \\lbrace v_h \\in \\mathbb {V}_h \\colon \\ v_h|_{\\Lambda \\setminus \\Omega } = \\Pi _h^c g\\rbrace , \\quad \\mathbb {V}_h^0 := \\lbrace v_h \\in \\mathbb {V}_h \\colon \\ v_h|_{\\Lambda \\setminus \\Omega } = 0\\rbrace ,$ where $\\Pi _h^c$ denotes the Clément interpolation operator in $\\Omega ^c$ .",
"With the notation introduced above, the discrete problem seeks $u_h \\in \\mathbb {V}^g_h$ such that $a_{u_h}(u_h, v_h) = 0 \\quad \\mbox{for all } v_h \\in \\mathbb {V}^0_h.$ Existence and uniqueness of solutions to this discrete problem follow directly from (REF ) and the strict convexity of $I_s$ .",
"To prove the convergence of the finite element scheme, the approach in [14] consists in proving that the discrete energy is consistent and afterwards using a compactness argument.",
"Theorem 4.3 (convergence for the nonlocal minimal graph problem) Let $s \\in (0,1/2)$ , $\\Omega $ be a bounded Lipschitz domain and $g$ be uniformly bounded and satisfying $\\mbox{supp}(g) \\subset \\Lambda $ for some bounded set $\\Lambda $ .",
"Let $u$ and $u_h$ be, respectively, the solutions to (REF ) and (REF ).",
"Then, it holds that $\\lim _{h \\rightarrow 0} I_s[u_h] = I_s[u] \\quad \\mbox{ and } \\quad \\lim _{h \\rightarrow 0} \\Vert u - u_h \\Vert _{W^{2r}_1(\\Omega )} = 0 \\quad \\forall r \\in [0,s).$ The theorem above has the important feature of guaranteeing convergence without any regularity assumption on the solution.",
"However, it does not offer convergence rates.",
"We now show estimates for a geometric notion of error that mimics the one analyzed in [40] for the classical Plateau problem (see also [8], [30]).",
"Such a notion of error is given by $\\begin{aligned}e^2(u,u_h) & :=\\int _{\\Omega } \\ \\Big | \\widehat{\\nu }(\\nabla u) - \\widehat{\\nu }(\\nabla u_h) \\Big |^2 \\;\\frac{Q(\\nabla u) + Q(\\nabla u_h)}{2} \\ dx , \\\\& = \\int _{\\Omega } \\ \\Big ( \\widehat{\\nu }(\\nabla u) - \\widehat{\\nu }(\\nabla u_h) \\Big ) \\cdot \\ \\Big ( \\nabla (u-u_h), 0 \\Big ) dx ,\\end{aligned}$ where $Q({a}) = \\sqrt{1+|{a}|^2}$ , $\\widehat{\\nu }({a}) = \\frac{({a},-1)}{Q({a})}$ .",
"Because $\\widehat{\\nu }(\\nabla u)$ is the normal unit vector on the graph of $u$ , the quantity $e(u,u_h)$ is a weighted $L^2$ -discrepancy between the normal vectors.",
"For the nonlocal minimal graph problem, [14] introduced ${\\begin{aligned}e_s(u,u_h) := \\left( \\widetilde{C}_{d,s} \\iint _{Q_{\\Omega }} \\Big ( G_s\\left(d_u(x,y)\\right) - G_s\\left(d_{u_h}(x,y)\\right) \\Big ) \\frac{d_{u-u_h}(x,y)}{|x-y|^{d-1+2s}} dxdy \\right)^{1/2} ,\\end{aligned}}$ where $G_s(\\rho ) = F_s^{\\prime }(\\rho )$ , the constant $\\widetilde{C}_{d,s} = \\frac{1 - 2s}{\\alpha _{d}}$ , $\\alpha _{d}$ is the volume of the $d$ -dimensional unit ball and $d_v$ is the difference quotient of the function $v$ , $d_v(x,y) := \\frac{v(x)-v(y)}{|x-y|}.$ In [14], this novel quantity $e_s(u,u_h)$ is shown to be connected with a notion of nonlocal normal vector, and its asymptotic behavior as $s\\rightarrow 1/2^-$ is established.",
"Theorem 4.4 (asymptotics of $e_s$ ) For all $u,v \\in H^1_0(\\Lambda )$ , we have $\\lim _{s \\rightarrow {\\frac{1}{2}}^-} e_s(u,v) = e(u,v).$ A simple `Galerkin orthogonality' argument allows to derive an error estimate for $e_s(u,u_h)$ (cf.",
"[14]).",
"Theorem 4.5 (geometric error) Under the same hypothesis as in thm:consistency, it holds that $ \\begin{aligned}e_s(u,u_h) &\\le C (d,s) \\, \\inf _{v_h \\in \\mathbb {V}_h^g} \\left( \\iint _{Q_{\\Omega }} \\frac{|(u-v_h)(x)-(u-v_h)(y)|}{|x-y|^{d+2s}} dxdy \\right) ^{1/2}.\\end{aligned} $ Therefore, to obtain convergence rates with respect to $e_s(u,u_h)$ , it suffices to prove interpolation estimates for the nonlocal minimizer.",
"Although minimal graphs are expected to be discontinuous across the boundary, we still expect that $u \\in BV(\\Lambda )$ in general.",
"Under this circumstance, the error estimate (REF ) leads to $e_s(u,u_h) \\le C(d,s) \\, h^{1/2-s} |u|^{1/2}_{BV(\\Lambda )}.$"
],
[
"Numerical experiments",
"We conclude by presenting several numerical experiments and discussing the behavior of nonlocal minimal graphs.",
"The first example we compute is on a one-dimensional domain, and is proposed and theoretically studied in [33] as an illustration of stickiness phenomena.",
"Example 4.6 (stickiness in 1D) Let $\\Omega = (-1,1)$ and $g(x) = \\textrm {sign}(x)$ for $x \\in \\Omega ^c$ .",
"Discrete nonlocal minimal graphs for $s\\in \\lbrace 0.1, 0.25,0.4\\rbrace $ are shown in F:Ex-stickiness (left).",
"Figure: Left: plot of u h u_h for s=0.1,0.25,0.4s = 0.1, 0.25, 0.4 (from left to right) in Ex:1dstickiness.",
"Right: plot of u h u_h for uniform meshes with h=2 -5 h = 2^{-5} in Ex:2dannulus.Although our method requires discrete functions to be continuous across the boundary of $\\Omega $ , the presented 1D picture clearly suggests a stickiness phenomenon.",
"In addition, the plot also indicates that stickiness becomes more observable when $s$ gets closer to 0.",
"Example 4.7 (stickiness in an annulus) Let $\\Omega = B_1 \\setminus \\overline{B}_{1/2} \\subset \\mathbb {R}^2$ , where $B_r$ denotes an open ball with radius $r$ centered at the origin, and let $g = \\chi _{B_{1/2}}$ .",
"The discrete nonlocal minimal graph for $s = 0.25$ is plotted in F:Ex-stickiness (right).",
"In this example, stickiness is clearly observed on $\\partial B_{1/2}$ , while the stickiness on $\\partial B_1$ is less noticeable.",
"Example 4.8 (effect of $s$ ) Let $\\Omega = B_1 \\subset \\mathbb {R}^2$ , $g = \\chi _{B_{3/2}\\setminus \\Omega }$ .",
"F:NMS-Exmultis shows minimizers for several values of $s$ .",
"As $s \\rightarrow 1/2$ , the nonlocal minimal graphs get closer to the classical minimal graph, which is trivially constant in $\\Omega $ .",
"On the other hand, as $s$ decreases we observe a stronger jump across $\\partial \\Omega $ .",
"Figure: Plot of u h u_h for s=0.01,0.1,0.25,0.4,0.49s = 0.01, 0.1, 0.25, 0.4, 0.49 (from left to right) with uniform h=2 -5 h = 2^{-5} in Ex:2dcircle.In the end, we make brief comments on the computational side for the examples presented above.",
"In all of the experiments for fractional minimal graphs, we use Newton's method to solve the nonlinear equation (REF ).",
"Although the Jacobian matrix $\\mathbf {A}_{u_k}$ in the iterative process corresponds to an $H^{s+1/2}-$ inner product $a_{u_k} (w,v)$ with degenerate weights, our experiments indicate its condition number behaves like $\\kappa (\\mathbf {A}_{u_k}) \\approx \\mathcal {O}\\left( N^{2(s+\\frac{1}{2})/d} \\right)$ for $u_k$ whose gradient blows up near the boundary $\\partial \\Omega $ , quasi-uniform meshes and dimensions $d=1,2$ .",
"This behavior is the same as the one in linear fractional diffusion with order $s+\\frac{1}{2}$ .",
"However, the degenerate weight does bring more difficulties in preconditioning.",
"Another thing worth to be pointed out is the use of Dirichlet condition for our discrete space $\\mathbb {V}_h^g$ requires the discrete function $u_h$ to be continuous across $\\partial \\Omega $ .",
"Due to the stickiness phenomenon in rem:stickiness, this may not be true for the solution $u$ of the minimal graph problem.",
"Fortunately, this does not preclude the convergence in `trace blind' fractional Sobolev spaces $W^{2s}_1(\\Omega )$ , and we are still able to capture discontinuities across the boundary in practice.",
"While permitting discontinuities would be desirable, it has conflicts with using Newton's method in solving (REF ) because the bilinear form $a_u(w,v)$ in (REF ) may not be well-defined.",
"The question of how to solve the nonlinear equation (REF ) faster when allowing discontinuous across $\\partial \\Omega $ is still under investigation."
]
] | 1906.04230 |
[
[
"On the Vector Space in Photoplethysmography Imaging"
],
[
"Abstract We study the vector space of visible wavelength intensities from face videos widely used as input features in Photoplethysmography Imaging (PPGI).",
"Based upon theoretical principles of Group invariance in the Euclidean space we derive a change of the topology where the corresponding distance between successive measurements is defined as geodesic on a Riemannian manifold.",
"This lower dimensional embedding of the sensor signal unifies the invariance properties with respect to translation of the features as discussed by several former approaches.",
"The resulting operator acts implicit on the feature space without requiring any kind of prior knowledge and does not need parameter tuning.",
"The resulting feature's time varying quasi-periodic shaping naturally occurs in form of the canonical state space representation according to the known Diffusion process of blood volume changes.",
"The computational complexity is low and the implementation becomes fairly simple.",
"During experiments the operator achieved robust and competitive estimation performance of heart rate from face videos on two public databases."
],
[
"Introduction",
"Nearly 70 years ago, short after the end of world war II in 1948, Norbert Wiener published, his sociopolitical often controversial discussed work, Cybernetics or Control and Communication in the Animal and the Machine [1].",
"During this period and short after, mankind already discovered most of the important principles found today in everyday technology.",
"Although we have not seen yet a fully functional realization of Wieners visions of self-regulating mechanisms, but we are able to trace ongoing and fast emerging progress in the computational interpretation of sensors, signals and systems which yields at least to a direction of a soupçon of artifical intelligenz.",
"As part of natural social interaction the human face with it's contingent of verbal activity, non-verbal behaviour as well as it's appearance is reflecting the majority of interpretable signal sources by computer systems.",
"In contrast to the non-verbal behavioral signals the facial appearance is able to provide changes in peripheral nervous system as well as central nervous system surrogate states by the analysis of skin blood perfusion.",
"Though the tiny intensity changes can not be perceived by human eyes.",
"However with the help of opto-electronic circuitry this information has become accessible.",
"Scientifically a plain disposability is nothing special.",
"Under the context of non-obtrusive remote measurability this new technology enables acquisitions of biological and cognitive human data under exceptional situations potentially leading to new findings and application fields.",
"Basically, nearly every tasks targeting on the specific dependent variables, formally captured by cable mounted sensors sticked onto human skin, can be sensed by the camera based counterpart as well.",
"The possible range of new applications and analysis legitimates to name its potential as ground breaking.",
"The topic is traditional anchored in the medical sciences.",
"However, currently the focus elementary changed into direction of computer vision.",
"Here the role of physiological states has a large impact.",
"It is primarily used during human state computing tasks, where the radiation unnoticeable transports information from face without contact holding states of affective nature.",
"During the last years measuring blood volume changes and heart rate measurements from facial images gained attention at top computer vision conferences [2], [3], [4], [5], [6], [7] frequently.",
"Most of these contributions focus on how to cope with motion like head pose variations and facial expressions since any kind of motion on a specific skin region of interest will destroy the raw signal in a way that no reliable information can be extracted anymore.",
"Beside from being able to estimate vitality parameters like heart rate and respiration, the functional survey of wounds as well as quantification of allergic skin reaction [8] are further topics of discovered employments of skin blood perfusion analysis.",
"Recently, prediction of emotional states, stress [9], [10], [11], fatigue [12] and sickness [13] became interesting new achievements in this area, pushing the focus of this technology further towards human-machine interaction.",
"In contrast to the genuine medical use-case of the technology, in computer vision and human-machine interaction we can't expect any cooperative behavior of the user without introducing lack of convenience and a reduction of the general user acceptance.",
"Further, beyond any well tempered clinical and laboratory like scenarios, the majority application will face strong challenging environmental changes and differences much more quite common.",
"Thus, there's an emerging demand to produce better features and models significant more robust to nuisance factors, still preserving the desired target information.",
"To reach such a formulation a fundamental profound understanding of the underlying optical and mathematical properties is one of the current foci of this research discipline.",
"The main contribution of this work is a mathematically analysis of the PPGI's feature space determined over facial skin pixels.",
"The general aim is to study the properties and behaviour of the features with respect to the influence of the actions of the Group acting on this space when induced by natural head and face motion.",
"As result a new feature operator is developed and evaluated against common operators on various data sets.",
"These sets are collected to explicitly study the influence of typical nuisance factors.",
"To support an efficient dissemination and to speed up the research progress in this field we have encapsulated all feature operators and the reference implementation of the stochastic model of blood volume changes into a new object-orientated MATLAB toolbox.",
"The code is public available under http://bit.ly/PPGI-Toolbox.",
"The toolbox provides the necessary code to reproduce all results presented in this work.",
"The outline of this work is as follows.",
"From the historical genuine up to the development of the state of the art in computer vision and bio-medical engineering, the methodology of heart rate estimation from face videos will be reviewed.",
"Followed by theoretical aspects, the feature space will be analyzed and the proposed methodology mathematically described.",
"Based upon an extensive evaluation on different databases the results will be presented and finally discussed."
],
[
"Related Work",
"The historical genuine of the term Photoplethysmography, short PPG, dates back to the late first half of the 20th century, when the two scientists Molitor and Kniazak [14] recorded peripheral circulatory changes in animals.",
"A year later, Hertzman [15] introduced the term Photoelectric Plethysmograph as \"the amplitude of volume pulse as a measure of the blood supply of the skin\".",
"Hertzman's instrumentation comprised mainly of a tungsten arc lamp and a photomultiplier tube.",
"During the same time the methodology was described by a German scientist in a medical journal [16].",
"However, today it is not reproducible anymore who really invented this technology.",
"In literature Hertzman is accredited with this reputation usually.",
"Around 50 years later the advancement to the classical PPG, the camera based PPGI (with I for Imaging) method, was introduced by the pioneering work of Blazek [17].",
"The basic principle behind the measurement of blood volume changes in the skin by means of PPGI (as well as PGG too) is the fact that the oxygen binding ferrous protein complex hemoglobin in the blood absorbs specific frequency bands of light many times more strongly than the remaining skin tissues.",
"Accordingly, tiny intensity changes can be observed over specific frequency bands (e.q.",
"the density of spectral lines of the emission spectra of iron) as oscillation caused by the quasi periodic rhythm of the human heart.",
"In PGG a part of the skin surface is illuminated by dedicated light sources like illumination panels consisting of LED.",
"In PPGI a common CCD camera is used as detector and the illumination can be as well as a common ambient light for which the intensity of backscattered optical radiation, eq.",
"reflected light, is calculated [18], [19], [20].",
"In general the computational pipeline to determine vitality parameters and its derivatives from blood volume changes can be regarded as classical signal processing chain.",
"Typically, from a skin region of interest (ROI) features are calculated, filtered and analyzed by spectral methods [19], [21], [20].",
"The first published visualization of pulsatile skin perfusion patterns in the time and frequency domain is given by Blazek [17].",
"However motion of the skin ROI [19] and micro motion of the head due to cardiac activity [22], [23] inherently induces artifacts into the extracted signal, especially when lighting is neither uniform nor orthogonal [24].",
"Canceling motion artifacts during signal processing became one of the most important aspect for reliable skin blood perfusion measurements.",
"An early idea of skin ROI motion compensating is to track every skin pixel position by optical flow methods directly in the image plane [19].",
"However this doesn't account for any change of illumination.",
"Poh et al.",
"[21] proposed to extract motion components in the signal by blind source separation using Independent Component Analysis (ICA) over the different color channels.",
"Wedekind et al.",
"[25] compared ICA in multiple setting and Principal Component Analysis and showed limitations of either transform.",
"Further, the ICs cannot be obtained in a deterministic order [26].",
"A solution to this problem is discussed by Macwan et al.",
"[27].",
"Tarassenko et al.",
"[28] tried to cope with light flicker by using an auto-regressive modeling and pole cancellation.",
"De Haan and Jeanne [29] and De Haan and Van Leest [30] proposed to map the PPGI-signals by linear combination of RGB data to a direction that is orthogonal to motion induced artifacts.",
"An alternative approach, which does not require skin-tone or pulse-related priors in contrast to the channel mapping algorithms, determines the spatial subspace of skin-pixels and measure its temporal rotation for signal extraction [31].",
"Tulyakov et al.",
"[4] proposed matrix completion to jointly estimate reliable regions and heart rate estimates whereby Li et al.",
"[2] applied an adaptive least square approach to extract robust pulse frequencies.",
"Both reported performance gains similar to De Haan and Jeanne [29].",
"Interestingly they used the often criticized compressed videos of the MAHNOB-HCI database [32] during their experiments.",
"This leaves reasonable doubts on the validity of results since it is well known that any kind of image compression will destroy the underlying tiny perfusion signal [33].",
"Wang et al.",
"[34] reported an orthogonal behavior of skin color and motion artifacts derived by optical properties but introduced a static projection operator for feature transformation and represented their results on private data.",
"An entirely different model was introduced by Pilz et al.",
"[5].",
"Here, the quasi-periodic nature of the blood volume changes is modeled as stochastic resonator based upon a diffusion process.",
"A group theoretic deviated feature transform for motion compensation is introduced by Pilz et al.",
"[6].",
"Following the popularity of Deep Learning Chen and McDuff [7] claim to outperform recent algorithms using a convolutional network architecture (CNN) for modelling motion representation.",
"However, they also reported some findings on the compressed videos of the MAHNOB-HCI database and they didn't provide their CNN implementation or at least the trained model yet."
],
[
"Methodology",
"Meanwhile, it is well understood that subject motion and fast strong changes of illumination alters the distribution of pixel intensity negatively making it quite difficult to extract skin perfusion signals from video images.",
"It is assumed that the perfusion signals exits in either case and its further assumed that it is combined together with the distribution of intensity belonging to motion forces by some unknown operator.",
"Following the basic principles of the Hilbert projection theorem.",
"If $\\mathcal {V}$ is a closed subspace of the Hilbert Space $\\mathcal {H}$ and $x \\in \\mathcal {H}$ , then there is a unique element $\\hat{x} \\in \\mathcal {V}$ such that $\\Vert x-\\hat{x} \\Vert = \\underset{y\\in \\mathcal {V}}{\\textrm {inf }} \\Vert x-y \\Vert $ and only if $\\hat{x} \\in \\mathcal {V} $ and $(x-\\hat{x}) \\in \\mathcal {V}^{\\perp }$ where $\\mathcal {V}^{\\perp }$ is the orthogonal complement to $\\mathcal {V}$ in$\\mathcal {H}$ .",
"Then, the current paradigm in understanding PPGI signal components in the feature space assumes that blood volume changes exist in a lower dimensional space where this space is orthogonal to any kind of motion induces signal components.",
"Derived from optical properties by De Haan et al.",
"and Wang et al.",
"[29], [34] and from Group theoretic principals by Pilz et al.",
"[6] this can be expressed as $\\vec{c}=\\vec{p}+\\vec{m}$ .",
"$\\vec{p}$ and $\\vec{m}$ are two orthogonal vectors with $\\vec{p} \\cdot \\vec{m}^T=0$ , thus statistically linear independent.",
"This principal is illustrate in Figure REF .",
"In the following we explain the Group theoretic principals behind motion robust sensing of blood volume changes as introduced by Pilz et al.",
"[6].",
"Based upon the analysis of the properties of the resulting linear operator we demonstrate that there exists an equivalent implicit operator.",
"This operator maps the observation onto an embedded Riemannian submanifold of the Euclidean space.",
"And we show that the corresponding directional statistics evolve in form of the previously published Diffusion process model as function of time [5].",
"Figure: The actual state-of-the-art paradigm in understanding major PPGI signal components in the feature space.",
"With c →=p →+m →\\vec{c}=\\vec{p}+\\vec{m}, p →\\vec{p} and m →\\vec{m} are two orthogonal vectors with p →·m → T =0\\vec{p} \\cdot \\vec{m}^T=0."
],
[
"Basic Principals of Group Invariance",
"Consider a finite topological group $\\mathcal {G}=\\lbrace G_1,...,G_M\\rbrace $ of $M$ distinct actions on a topological space $\\mathbb {X},G_i : \\mathcal {X}\\rightarrow \\mathcal {X}.$ A real valued function $f\\left(x\\right)$ on $\\mathcal {X}$ is said to be invariant under $\\mathcal {G}$ if $f\\left(Gx\\right)=f\\left(x\\right) \\textrm {for } G \\in \\mathcal {G}$ Regarding a common optical sensor signal $\\lbrace \\vec{p}_i: i=1,...,m\\rbrace $ $\\vec{p} \\in \\mathbb {R}^n = \\lbrace Red,Green,Blue\\rbrace , n=3$ as spatial expectation over a skin operator $s$ and function of time $t$ $\\vec{x}(t)=\\int _{0}^{\\infty } \\mathbb {E}[\\lbrace \\vec{p_i} \\mid s(\\vec{p_i}) \\rbrace ] \\mathrm {d}t$ we assume this multivariate observation is drawn by a normal distribution $\\vec{x}(t) \\sim \\mathcal {N} (\\mu ,\\sigma ^2).$ Local invariance of blood volume changes as function of time for each input feature $\\vec{x}(t)$ under transformations of a differentiable local group of local transformations $\\mathcal {L}_T$ [35] $ \\frac{\\partial }{\\partial T}\\big \\vert _{T=0}=f(\\mathcal {L}_T,\\vec{x}(t))=0$ can be approximately enforced by minimizing the regularizer $ \\frac{1}{l}\\sum _{j=1}^l(\\frac{\\partial }{\\partial T}\\big \\vert _{T=0}f(\\mathcal {L}_T,x_j))^2.$ For the covariance matrix of the observation $\\lbrace x_i: i=1,...,l\\rbrace $ with respect to the transformations $\\mathcal {L}_T$ $C:=\\frac{1}{l}\\sum _{j=1}^l(\\frac{\\partial }{\\partial T}\\big \\vert _{T=0}\\mathcal {L}_T,x_j)(\\frac{\\partial }{\\partial T}\\big \\vert _{T=0}\\mathcal {L}_T,x_j)^\\top $ and the corresponding symmetric eigenvalue problem $CV=V\\Lambda $ we find an operator $P$ with corank $k=1$ for $\\lim \\limits _{l \\rightarrow \\infty }P=I-VV^\\top $ and the corresponding feature vector $\\tilde{x}=P\\cdot x.$ The observation $\\lbrace \\tilde{x_i}: i=1,...,l\\rbrace $ is defining the null space of the projection operator $P$ $H_P=N(P)$"
],
[
"The embedded Riemannian submanifold",
"In general, we're specially interested in the properties of the projection operator $P$ and the resulting linear subspace $H_P$ since the direction of $V$ is assumed to carry most of the PPGIs motion signal component and the subspace $H_P$ the desired quasi periodic perfusion component.",
"It is a well-known from the theory of Banach algebras that the spectral radius $\\rho $ of any $A\\in \\mathcal {M}_n$ is given by Gelfand’s formula $\\rho (A)=\\lim \\limits _{n \\rightarrow \\infty }{\\left\\Vert {A^n}\\right\\Vert ^{1/n}}$ for any matrix norm $\\Vert \\cdot \\Vert $ on $\\mathcal {M}_n$ .",
"For a matrix, the spectrum is just the collection of eigenvalues with $\\rho (A):=max\\lbrace \\left|\\lambda \\right|,\\lambda \\textrm { eigenvalue of } A\\rbrace $ .",
"For the projection operator $P$ , where $V$ is represented by the eigenvector with the largest eigenvalue, this implies a reduction of the spectral radius of the observation $X$ $\\rho (\\tilde{X})<\\rho (X)$ The projection $P$ removes the direction of the largest variance and puts more emphasis on the directions which vary less.",
"It should be clear that the computation of the projection is an explicit operator which has to be estimated on an observation $\\lbrace x_i: i=1,...,l\\rbrace $ .",
"A more convenient way of incorporating invariance to the feature space would be to define an implicit operator.",
"Let $\\lambda $ be an eigenvalue of $A$ , and let $y\\ne 0 $ be a corresponding eigenvector.",
"From $Ay=\\lambda y$ , we have $AY=\\lambda Y$ where $Y:=\\left[ y\\mid ... \\mid y \\right] \\in \\mathcal {M}_n \\setminus \\lbrace 0\\rbrace .$ It follows $\\left|\\lambda \\right| \\Vert X\\Vert = \\Vert \\lambda Y\\Vert = \\Vert AY\\Vert \\le \\Vert A\\Vert \\Vert Y\\Vert $ , and simplifying by $\\Vert Y\\Vert \\left(>0\\right)$ gives $\\left|\\lambda \\right| \\le \\Vert Y\\Vert $ .",
"Taking the maximum over all eigenvalues $\\lambda $ results in $\\rho (A) \\le \\Vert Y\\Vert $ .",
"Now, regarding the optical sensor signal $\\vec{p}_i$ relative to the spectral radius of its elements $\\rho (\\vec{p})$ results in $\\vec{x}_{\\vec{1}}(t)=\\int _{0}^{\\infty } \\mathbb {E}[\\lbrace \\frac{\\vec{p_i}}{\\Vert \\vec{p}\\Vert } \\mid s(\\vec{p_i}) \\rbrace ] \\mathrm {d}t$ with $\\rho (X_{\\vec{1}})<\\rho (X)$ and distributed on $\\mathbb {S}^2=\\lbrace \\vec{x} \\in \\mathbb {R}^3 \\mid \\Vert \\vec{x}\\Vert =1 \\rbrace $ the unit sphere as embedded Riemannian submanifold of the Euclidean space $R^3$ .",
"Intuitively, for real valued observations $\\lbrace y_i: i=1,...,m \\in \\mathbb {R}\\rbrace $ the mean is given by $\\mu =\\frac{1}{m}\\sum _{i=1}^m {y_i}$ .",
"However, for the more general settings $\\lbrace y_i: i=1,...,m \\in \\mathcal {M}\\rbrace $ we do not have the possibility to find such a closed form solution.",
"The solution has to be solved for the optimization problem $\\mu \\in \\underset{x\\in \\mathcal {M}}{\\arg \\min } \\frac{1}{m}\\sum _{i=1}^m d^2_{\\mathcal {M}}\\left(x,x_i \\right)=\\underset{x\\in \\mathcal {M}}{\\arg \\min }F\\left(x\\right).$ by gradient descent algorithm.",
"The resulting $\\mu $ is called Riemannian center of mass or Karcher mean [36].",
"The optimality conditions is give by $0\\overset{!",
"}{=} \\frac{2}{m}\\sum _{i=1}^n log_{x^*}x_{i}=\\nabla F\\left(x^*\\right) \\in T_{x^*}\\cal {M}$ Considering the embedded dimensions of the unit sphere given by its spherical coordinates $r=\\sqrt{x^2+y^2+z^2}=1$ $\\theta =arccos\\frac{z}{\\sqrt{x^2+y^2+z^2}}=arccos \\frac{z}{r}=arccos{z}$ $\\varphi =arctan \\frac{y}{x}$ the directional statistics are represented by the von Mises-Fisher distribution $p\\left(x;\\mu ,\\kappa \\right):=c_{p}\\left(\\kappa \\right)e^{\\kappa \\mu ^{T}x}$ with the normalization constant given by $c_{p}\\left(\\kappa \\right)=\\frac{\\kappa ^{p\\slash 2-1}}{\\left(2\\pi \\right)^{p\\slash 2}I_{p\\slash 2-1}\\left(\\kappa \\right)}$ where $I_s\\left(\\kappa \\right)$ denotes the modified Bessel function of the first kind.",
"The spherical direction $\\varphi $ evolves according to the principals of the stochastic differential equation given by $\\frac{d^2c_n(t)}{dt^2}=-(2\\pi {nf(t)})^2{c_n(t)}+e_n(t)$ whereby $\\theta $ can be expressed as part of a Wiener process given by $\\frac{\\mathrm {d}^2\\theta (t)}{\\mathrm {d}t^2}=w(t).$ For a detailed discussion on the Diffusion process see the previous works of Pilz et al.",
"[5]."
],
[
"Experiments",
"The evaluation of the described feature operator was conducted by experiments against different methods on two public available databases.",
"We decided to compare the operator against the baseline green channel expectation [19], [20], the Spatial Subspace Rotation (SSR) [31], the Projection Orthogonal to Skin (POS) [34] and the Local Group Invariance (LGI) [6] method.",
"All experiments were executed on the French UBFC-RPPG [37] and the German LGI Multi-Session face video database [6].",
"In the following Figures REF and REF some representative images of these two databases are illustrated.",
"The UBFC and the LGI Figure: Example images taken from the UBFC-RPPG and LGI Multi-Session database.database were created using a custom C++ application for video acquisition with a simple low cost Logitech webcam and a CMS50E transmissive pulse oximeter to obtain the ground truth PPG data comprising of the PPG waveform.",
"The total amount of face video recordings yields to 150 sequences containing several couple of minutes respectively.",
"During the recordings, the subjects of the UBFC trials performed moderate face and head motions under indoor environments whereby the LGI recordings contain scenarios from resting and head rotation over sport activities to natural outdoor conversations.",
"Therefore, the evaluation concept ranges from cooperative to challenging scenarios which should be become noticeable in form of the prediction accuracy results.",
"The primary signal processing procedure is selected to be equal for every approach and database.",
"For each video frame a common Viola-Jones face finder was used to pre-select the region of interest.",
"A simple skin operator was applied onto the region by thresholding the blue- and red-difference chroma components.",
"For the set of obtained RGB-pixels the different approach specific operators were computed and stored as time series for further spectral processing and interpretation.",
"Each signal obtained by the different algorithms was band-filtered in the range between 0.5 and 2.5 Hz.",
"All filtered signals were then analyzed by standard Fourier based spectral method with windows size of 256 samples and overlap of 90 percent.",
"A maximum peak energy criterion was applied over the spectral traces to determine the heart rate.",
"All PPG ground truth signals were analyzed in the same way but initially resampled to the camera frame rate.",
"Correlation coefficients were computed against the PPG reference heart rate together with the root-mean-square error (RMSE) for each user and algorithm respectively.",
"We did not perform a Signal-to-noise ratio (SNR) comparison as proposed by De Haan and Jeanne [29] and often used by several other authors [30], [34], [7].",
"This might be useful on short video sequences with a more or less stationary frequency behavior of blood volume changes.",
"A prospective consideration of a SNR metric for system evaluation should at least include information about it's variance computed on short term spectra.",
"Table REF represents the overview of heart rate prediction accuracy of the different operators on the different data sets.",
"Compared to the baseline green channel, SSR and POS approach the LGI and the proposed spherical operator (SPH) are more robust especially under motion scenarios.",
"Although the SPH operator cannot outperform the LGI approach in most cases, it's accuracy is operating in a very similar range.",
"However, in case of fully uncontrolled scenarios with changing illumination as well as different head and face motion, as given in the LGI city talk session, no algorithm is able to perform reasonable well.",
"In Figure 4 and 5 the box plot statistics for the UBFC and the LGI database are visualized.",
"In addition to Table REF the influence of the Diffusion process incorporated into the LGI and SPH approach is constituted.",
"Both approaches benefit from its stochastic interpretation as quasi-periodic nature of blood volume changes.",
"Table: Comparison of heart rate prediction accuracy utilizing different feature operators on diverse face video databases.",
"Each corresponding table entry represents the average Pearson's correlation coefficient together with the average root-mean-square error (RMSE) value in BPM.Figure: UBFC-RPPG: Heart rate prediction accuracy"
],
[
"Discussion",
"We have extended the current knowledge on linear orthogonal operators in the PPGIs feature space.",
"The resulting manifold valued representation is holding implicit properties of invariance with respect to translations of the Group acting onto the set of features.",
"This carries major advantages over the previously prior based assumptions, both POS and LGI, since it comes with a simple change of the topology where the existence of these properties are guaranteed by the fundamental attributes of the space.",
"The computational complexity of the new feature operator is supremely low and it's implementation fairly easy.",
"The comparison against the most popular representatives of this algorithmic family succeeded with a quite promising strength of prediction accuracy.",
"Since the approach is reflecting a fully closed form solution regarding the genuine problem statement no nasty tuning of parameters is necessary, it's operating free of parameters.",
"The major limitation of the operator in its current form is the restriction of invariance with respect to the group of translations.",
"Since we have presented the proof that the intensity of pixels doesn't contribute to the periodic characteristics of skin perfusion, it follows that is indeed a question of the wavelength.",
"Figure: LGI: Heart rate prediction accuracy on different sessions.",
"Session1: Head Resting, Session 2: Head rotation, session 3 : Bicycle ergometer, Session 4: Outdoor city talk"
]
] | 1906.04431 |
[
[
"Reinforcement Learning for Channel Coding: Learned Bit-Flipping Decoding"
],
[
"Abstract In this paper, we use reinforcement learning to find effective decoding strategies for binary linear codes.",
"We start by reviewing several iterative decoding algorithms that involve a decision-making process at each step, including bit-flipping (BF) decoding, residual belief propagation, and anchor decoding.",
"We then illustrate how such algorithms can be mapped to Markov decision processes allowing for data-driven learning of optimal decision strategies, rather than basing decisions on heuristics or intuition.",
"As a case study, we consider BF decoding for both the binary symmetric and additive white Gaussian noise channel.",
"Our results show that learned BF decoders can offer a range of performance-complexity trade-offs for the considered Reed-Muller and BCH codes, and achieve near-optimal performance in some cases.",
"We also demonstrate learning convergence speed-ups when biasing the learning process towards correct decoding decisions, as opposed to relying only on random explorations and past knowledge."
],
[
"Introduction",
"The decoding of error-correcting codes can be cast as a classification problem and solved using supervised machine learning.",
"The general idea is to regard the decoder as a parameterized function (e.g., a neural network) and learn good parameter configurations with data-driven optimization [2], [3], [4], [5], [6], [7].",
"Without further restrictions on the code, this only works well for short codes and typically becomes ineffective for unstructured codes with more than a few hundred codewords.",
"For linear codes, the problem simplifies considerably because one has to learn only a single decision region instead of one region per codeword.",
"One can take advantage of linearity by using message-passing [4] or syndromes [5], [6].",
"Still, the problem remains challenging because good codes typically have complicated decision regions due to the large number of neighboring codewords.",
"Near-optimal performance of learned decoders in practical regimes has been demonstrated, e.g., for convolutional codes [7], which possess even more structure.",
"In this paper, we study the decoding of binary linear block codes from a machine-learning perspective.",
"Rather than learning a direct mapping from observations to estimated codewords (or bits) in a supervised fashion, the decoding is done in steps based on individual bit-flipping (BF) decisions.",
"This allows us to map the problem to a Markov decision process (MDP) and apply reinforcement learning (RL) to find good decision strategies.",
"Following [5], [6], our approach is syndrome-based and the state space of the MDP is formed by all possible binary syndromes, where bit-wise reliability information can be included for general memoryless channels.",
"This effectively decouples the decoding problem from the transmitted codeword.",
"BF decoding has been studied extensively in the literature and is covered in many textbooks on modern coding theory, see, e.g., [8], [9], [10], [11], [12], [13], [14].",
"Despite its ubiquitous use, and to the best of our knowledge, the learning approach to BF decoding presented in this paper is novel.",
"In fact, with the exception of the recent work in [15], we were unable to find references that discuss RL for channel coding.",
"Thus, we briefly review some other iterative decoding algorithms, based on sequential decision-making steps, for which RL is applicable.",
"For a comprehensive survey of RL in the general context of communications, see [16]."
],
[
"Channel Coding Background",
"Let $\\mathcal {C}$ be an $(N,K)$ binary linear code defined by an $ M\\times N $ parity-check (PC) matrix $\\mathbf {H}$ , where $N$ is the code length, $K$ is the code dimension, and $M \\ge N-K$ .",
"The code is used to encode messages into codewords $c=\\left(c_1,...,c_N\\right)^\\intercal $ , which are then transmitted over the additive white Gaussian noise (AWGN) channel according to $y_{n} = (-1)^{c_{n}} + w_{n}$ , where $y_n$ is the $n$ -th component in the received vector $y=\\left(y_1,...,y_N\\right)^\\intercal $ , $w_{n}\\sim \\mathcal {N}(0,(2 R E_\\mathrm {b}/N_0)^{-1})$ , $R \\triangleq K/N$ is the code rate, and we refer to $E_\\mathrm {b}/N_0$ as the signal-to-noise ratio (SNR).",
"The vector of hard-decisions is denoted by $z=\\left(z_1,...,z_N\\right)^\\intercal $ , i.e., $z_n$ is obtained by mapping the sign of $y_n$ according to $+1 \\rightarrow 0$ , $-1 \\rightarrow 1$ .",
"If the decoding is based only on the hard-decisions $z$ , this scenario is equivalent to transmission over the binary symmetric channel (BSC)."
],
[
"Decision Making in Iterative Decoding Algorithms",
"In the following, we briefly review several iterative decoding algorithms that involve a decision-making process at each step.",
"The general idea behind BF decoding is to construct a suitable metric that allows the decoder to rank the bits based on their reliability given the code constraints [14].",
"In its simplest form, BF uses the hard-decision output $z$ and iteratively looks for the bit that, after flipping it, would maximally reduce the number of currently violated PC equations.",
"Pseudocode for standard BF decoding is provided in Alg.",
"REF , where $e_n\\in \\mathbb {F}_2^N$ is a standard basis vector whose $n$ -th component is 1 and all other components are 0, $\\mathbb {F}_2 \\triangleq \\lbrace 0,1\\rbrace $ and $[N] \\triangleq \\lbrace 1,2,\\dots , N\\rbrace $ .",
"BF can be extended to general memoryless channels by including weights and thresholds to decide which bits to flip at each step.",
"This is referred to as weighted BF (WBF) decoding, see, e.g., [8], [9], [10], [11], [12], [13], [14] and references therein."
],
[
"Residual Belief Propagation",
"Belief propagation (BP) is an iterative algorithm where messages are passed along the edges of the Tanner graph representation of the code.",
"In general, it is known that sequential message-passing schedules can lead to faster convergence than standard flooding schedules where multiple messages are updated in parallel.",
"Residual BP (RBP) [17] is a particular instance of a sequential updating approach without a predetermined schedule.",
"Instead, the message order is decided dynamically, where the decisions are based on the residual—defined as the norm of the difference between the current message and the message in the previous iteration.",
"The residual is a measure of importance or “expected progress” associated with sending the message.",
"In the context of decoding, various extensions of this idea have been investigated under the name of informed dynamic scheduling [18]."
],
[
"Anchor Decoding",
"Consider the iterative decoding of product codesGiven a linear code $\\mathcal {C}$ of length $n$ , the product code of $\\mathcal {C}$ is the set of all $n \\times n$ arrays such that each row and column is a codeword in $\\mathcal {C}$ .",
"over the BSC, where the component codes are iteratively decoded in some fixed order.",
"For this algorithm, undetected errors in the component codes, so-called miscorrections, significantly affect the performance by introducing additional errors into the iterative decoding process.",
"To address this problem, anchor decoding (AD) was recently proposed in [19].",
"The AD algorithm exploits conflicts due to miscorrections where two component codes disagree on the value of a bit.",
"After each component decoding, a decision is made based on the number of conflicts whether the decoding outcome is indeed reliable.",
"This can lead to backtracking previous component decoding outcomes and to the designation of reliable component codes as anchors."
],
[
"Decision Making Through Data-Driven Learning",
"While the above decoding algorithms appear in seemingly different contexts, the sequential decision-making strategies in the underlying iterative processes are quite similar.",
"Decisions are typically made in a greedy fashion based on some heuristic metric that assesses the quality of each possible action.",
"As concrete examples for this metric, we have the decrease in the number of violated PC equations in BF decoding, measuring the reliability of bits; the residual in RBP, measuring expected progress and the importance of sending messages; the number of conflicts in AD, measuring the likelihood of being miscorrected.",
"In the next section, we review MDPs which provide a mathematical framework for modeling decision-making in deterministic or random environments.",
"MDPs can be used to obtain optimal decision-making strategies, effectively replacing heuristics with data-driven learning of optimal metrics.",
"mycommfont [t] ShortForfor KwBreakbreak MyWhilewhile MyIfif MySetset MyElseelse MyComputecompute KwEacheach KwAndand hard decisions $z$ , parity-check matrix $\\mathbf {H}$ estimated codeword $\\hat{c}$ $\\hat{c} \\leftarrow z$ $\\mathbf {H} \\hat{c} \\ne 0 $ max.",
"iterations not exceeded $V \\leftarrow \\sum _{m=1}^{M} s_m$ , where $s = \\mathbf {H}\\hat{c}$ *[r]no.",
"unsat checks $n = 1, 2, \\dots , N$ $Q_n \\leftarrow V - \\sum _{m=1}^{M} s_m$ , where $s =\\mathbf {H} (\\hat{c} + e_n)$ update $\\hat{c} \\leftarrow \\hat{c} + e_n $ , where $n = \\arg \\max _{n \\in [N]} Q_n$ Bit-Flipping Decoding"
],
[
"Markov Decision Processes",
"A time-invariant MDP is a Markov random process $S_0$ , $S_1$ , $\\dots $ whose state transition probability $P(s^{\\prime } | s, a) \\triangleq \\mathbb {P}(S_{t+1} = s^{\\prime } | S_t = s , A_t = a)$ is affected by the action $A_t$ taken by an agent based only on knowledge of past events.",
"Here, $s, s^{\\prime } \\in \\mathcal {S}$ and $a \\in \\mathcal {A}$ , where $\\mathcal {S}$ and $\\mathcal {A}$ are finite sets containing all possible states and actions.",
"The agent also receives a reward $R_t =R(S_t, A_t, S_{t+1})$ which depends only on the states $S_t$ , $S_{t+1}$ and the action $A_t$ .",
"The agent's decision-making process is formally described by a policy $\\pi : \\mathcal {S} \\rightarrow \\mathcal {A}$ , mapping observed states to actions.",
"The goal is to find an optimal policy $\\pi ^*$ that returns the best action for each possible state in terms of the total expected discounted reward $\\mathbb {E}\\left[\\sum _{t=0}^\\infty \\gamma ^t R_t\\right]$ , where $0 <\\gamma < 1$ is the discount factor for future rewards.",
"If the transition and reward probabilities are known, dynamic programming can be used to compute optimal policies.",
"If this is not the case, optimal policies can still be discovered through repeated interactions with the environment, assuming that the states and rewards are observable.",
"This is known as RL.",
"In the following, we describe two RL algorithms which will be used in the next sections."
],
[
"Q-learning",
"The most straightforward instance of RL is called Q-learning [20], where the optimal policy is defined in terms of the Q-function $Q : \\mathcal {S} \\times \\mathcal {A} \\rightarrow \\mathbb {R}$ according to $\\pi ^*(s) = \\operatornamewithlimits{arg\\,max}_{a \\in \\mathcal {A}} Q(s,a).$ The Q-function measures the quality of actions and is formally defined as the expected discounted future reward when being in state $s$ , taking action $a$ , and then acting optimally.",
"The key advantage of the Q-function is that it can be iteratively estimated from observations of any “sufficiently-random” agent.",
"Pseudocode for Q-learning is given in Alg.",
"REF , where a popular choice for generating the actions in line 5 is $a = {\\left\\lbrace \\begin{array}{ll}\\text{unif.~random over $\\mathcal {A}$} &\\quad \\text{w.p.~}\\varepsilon \\\\\\operatornamewithlimits{arg\\,max}_{a} Q(s,a) &\\quad \\text{w.p.~} 1- \\varepsilon .\\end{array}\\right.",
"}$ This is referred to as $\\varepsilon $ -greedy exploration.",
"For any $0 < \\varepsilon <1$ , this strategy is sufficient to allow Q-learning to eventually explore the entire state/action space.",
"In the next section, we also describe an alternative exploration strategy for our application that can converge faster than $\\varepsilon $ -greedy exploration.",
"To motivate the update equation in line 7 of Alg.",
"REF , we note that the Q-function can be recursively expressed as $\\!\\!Q(s,a) = \\sum _{s^{\\prime }} P(s^{\\prime }|s,a) \\left( \\!",
"R(s,a,s^{\\prime }) + \\gamma \\max _{a^{\\prime } \\in \\mathcal {A}} Q(s^{\\prime }, a^{\\prime }) \\!",
"\\right) \\!",
".$ This expression forms the theoretical basis for Q-learning which converges to the true Q-function under certain conditionsFor example, if $R(s,a,s^{\\prime })$ depends non-trivially on $s^{\\prime }$ , then $\\alpha $ must decay to zero at sufficiently slow rate.. For a more details, we refer the reader to [20], [21].",
"[t] ShortForfor KwBreakbreak MyWhilewhile MyIfif MySetset MyElseelse MyComputecompute KwEacheach KwAndand learning rate $\\alpha $ , discount factor $\\gamma $ estimated Q-function initialize $Q(s,a) \\leftarrow 0$ for all $s \\in \\mathcal {S}$ , $a \\in \\mathcal {A}$ $i = 1, 2, \\dots $ initialize starting state $s$ *[r]restart the MDP $s$ is not terminal choose action $a$ *[r]$\\varepsilon $ -greedy (REF ) or $(\\varepsilon , \\varepsilon _\\mathrm {g})$ -goal (REF ) execute $a$ , observe reward $r$ and next state $s^{\\prime }$ $Q(s,a) \\leftarrow (1-\\alpha )Q(s,a) + \\alpha (r + \\gamma \\max _{a^{\\prime }\\in \\mathcal {A}} Q(s^{\\prime },a^{\\prime }))$ $s \\leftarrow s^{\\prime }$ Q-learning"
],
[
"Fitted Q-learning with Function Approximators",
"For standard Q-learning, one must store a table of $|\\mathcal {S}|\\times |\\mathcal {A}|$ real values.",
"This will be infeasible if either set is prohibitively large.",
"The idea of fitted Q-learning is to learn a low-complexity approximation of $Q(s,a)$ [21].",
"Let $Q_\\theta (s,a)$ be an approximation of the Q-function, parameterized by $\\theta $ .",
"Fitted Q-learning alternates between simulating the MDP and updating the current parameters to obtain a better estimate of the Q-function.",
"In particular, assume that we have simulated and stored $B$ transition tuples $(s, a, r, s^{\\prime })$ in a set $\\mathcal {D}$ .",
"Then, updating the parameters $\\theta $ is based on reducing the empirical loss $\\!\\!\\!\\mathcal {L}_{\\mathcal {D}}(\\theta ) =\\!\\!",
"\\sum _{(s,a,r,s^{\\prime }) \\in \\mathcal {D}} \\left(r + \\gamma \\max _{a^{\\prime } \\in \\mathcal {A}} Q_\\theta (s^{\\prime }, a^{\\prime }) -Q_\\theta (s,a) \\right)^2 \\!",
"\\!",
".$ Pseudocode for fitted Q-learning is provided in Alg.",
"REF , where gradient descent is used to update the parameters $\\theta $ based on the loss (REF ).",
"It is now common to choose $Q_\\theta (s,a)$ to be a (deep) neural network (NN), in which case $\\theta $ are the network weights and fitted Q-learning is called deep Q-learning.",
"[t] ShortForfor KwBreakbreak MyWhilewhile MyIfif MySetset MyElseelse MyComputecompute KwEacheach KwAndand learning rate $\\alpha $ , batch size $B$ parameterized estimate of the Q-function initialize parameters $\\theta $ and $\\mathcal {D} \\leftarrow \\emptyset $ $i = 1, 2, \\dots $ initialize starting state $s$ *[r]restart the MDP $s$ is not terminal choose action $a$ *[r]$\\varepsilon $ -greedy (REF ) or $(\\varepsilon , \\varepsilon _\\mathrm {g})$ -goal (REF ) execute $a$ , observe reward $r$ and next state $s^{\\prime }$ store transition $(s, a, r, s^{\\prime })$ in $\\mathcal {D}$ $s \\leftarrow s^{\\prime }$ $|\\mathcal {D}| = B$ $\\theta \\leftarrow \\theta - \\alpha \\nabla _\\theta \\mathcal {L}_{\\mathcal {D}}(\\theta ) $ *[r]see (REF ) for def.",
"of $\\mathcal {L}_\\mathcal {D}$ empty $\\mathcal {D}$ Fitted Q-learning"
],
[
"Case Study: Bit-Flipping Decoding",
"In this section, we describe how BF decoding can be mapped to an MDP.",
"In general, this mapping involves multiple design choices that affect the results.",
"We therefore also comment on alternative choices and highlight some potential pitfalls that we encountered during this process."
],
[
"Theoretical Background",
"We start by reviewing the standard maximum-likelihood (ML) decoding problem for a binary linear code $\\mathcal {C} \\subseteq \\mathbb {F}_2^N$ over general discrete memoryless channels.",
"The resulting optimization problem forms the basis for the reward function that is used in the MDP.",
"To that end, consider a collection of $N$ discrete memoryless channels described by conditional probability density functions $\\lbrace P_{Y_n|C_n}(y_n|c_n) \\rbrace _{n \\in [N]}$ , where $c_n \\in \\mathbb {F}_2$ is the $n$ -th code bit and $y_n$ is the $n$ -th channel observation.",
"The ML decoding problem can be written as $\\!\\!\\!",
"\\operatornamewithlimits{arg\\,max}_{c \\in \\mathcal {C}} \\prod _{n=1}^N \\!",
"P_{Y_n|C_n} (y_n | c_n)=\\operatornamewithlimits{arg\\,max}_{c \\in \\mathcal {C}} \\sum _{n=1}^N (-1)^{c_n} \\lambda _n,\\!$ where $\\lambda _n \\triangleq \\ln \\frac{ P_{Y_n|C_n}(y_n|0) }{P_{Y_n|C_n}(y_n|1) }$ is the channel log-likelihood ratio (LLR).",
"Equivalently, one can rewrite the maximization over all possible codewords in terms of error patterns as $&\\operatornamewithlimits{arg\\,max}_{e \\,:\\, z + e \\in \\mathcal {C}} \\sum _{n=1}^N(-1)^{z_n}(-1)^{e_n} \\lambda _n \\\\&=\\operatornamewithlimits{arg\\,max}_{e \\,:\\, z + e \\in \\mathcal {C}} \\sum _{n=1}^N(-1)^{e_n} |\\lambda _n| \\\\&=\\operatornamewithlimits{arg\\,max}_{e \\,:\\, \\mathbf {H}e = s} \\sum _{n=1}^N(-1)^{e_n} |\\lambda _n|\\\\&=\\operatornamewithlimits{arg\\,max}_{e \\,:\\, \\mathbf {H}e = s} \\sum _{n=1}^N-{e_n} |\\lambda _n|$ where $s=\\mathbf {H}z$ is the observed syndrome.",
"Now, consider a multi-stage process where bit $a_t$ is flipped during the $t$ -th stage until the syndrome of the bit-flip pattern matches $s$ .",
"In this case, the optimization becomes $\\operatornamewithlimits{arg\\,max}_{\\tau , a_1, \\dots , a_\\tau \\,:\\, \\sum _{t=1}^\\tau h_{a_t} =s} \\sum _{t = 1}^\\tau - |\\lambda _{a_t}|,$ where $h_{n}$ is the $n$ -th column of the parity-check matrix $\\mathbf {H}$ .",
"By interpreting $-|\\lambda _{a_t}|$ as a reward, one can see that the objective function in (REF ) has the same form as the cumulative reward (without discount) in an MDP.",
"The following points are worth mentioning: For the BSC, all LLRs have the same magnitude and (REF ) returns the shortest flip pattern that matches the observed syndrome.",
"For general channels, (REF ) returns the shortest weighted flip pattern that matches the syndrome, where the weighting is done according to the channel LLRs.",
"In other words, the incurred penality for flipping bit $a_t$ is directly proportional to the reliability of the corresponding received bit.",
"If a bit is flipped multiple times, then there must be a shorter bit-flip sequence with lower cost and the same syndrome.",
"Therefore, it is sufficient to only consider flip patterns that contain distinct bits.",
"We assume that the action $A_t$ encodes which bit is flipped in the received word at time $t$ .",
"Since there are $N$ possible choices, we simply use $\\mathcal {A} = \\lbrace 1,2,\\dots , N\\rbrace \\triangleq [N]$ .",
"The state space $\\mathcal {S}$ is formed by all possible binary syndromes of length $M$ .",
"The initial state $S_0$ is the syndrome $\\mathbf {H}z$ and the next state is formed by adding the $A_t$ -th column of $\\mathbf {H}$ to the current state.",
"The transition probabilities $P(s^{\\prime }|s,a)$ therefore take values in $\\lbrace 0,1\\rbrace $ , i.e., the MDP is deterministic.",
"The all-zero syndrome corresponds to a terminal state.",
"We also enforce a limit of at most $T$ bit-flips per codeword.",
"After this, we exit the current iteration and a new codeword will be decoded.Strictly speaking, the resulting process is not an MDP unless the time $t$ is included in the state space.",
"Remark 1 For the BSC, we also tried (unsuccessfully) to learn BF decoding with fitted Q-learning directly from the channel observations using the state space $\\mathbb {F}_2^N$ .",
"Remark 2 For the AWGN channel, the state space can be extended by including the reliability vector $r = | y |$ , similar to the setup in [6].",
"In this case, each state would correspond to a tuple $(s, r)$ , where $s\\in \\mathbb {F}_2^M$ and $r$ remains constant during decoding.",
"In this paper, we follow a different strategy for BF decoding over the AWGN channel which relies on permuting the bit positions based on their reliability and subsequently discarding the channel LLRs prior to decoding.",
"This approach is described in Sec.",
"and does not require any modifications to the state space."
],
[
"Choosing the Reward Strategy",
"A natural reward function for decoding is to return 1 if the codeword is decoded correctly and 0 otherwise.",
"This would imply that an optimal policy minimizes the codeword error rate.",
"However, the reward is only allowed to depend on the current/next state and the action, whereas the transmitted codeword and its estimate are defined outside the context of the MDP.",
"Based on (REF ) and the discussion in the previous subsection, we instead use the reward function $\\!\\!R(s,a,s^{\\prime }) = {\\left\\lbrace \\begin{array}{ll}-c |\\lambda _a| +1 &\\!",
"\\text{if $s^{\\prime } = 0$}\\\\ -c |\\lambda _a| &\\!",
"\\text{otherwise },\\end{array}\\right.",
"}$ where $c > 0$ is a scaling factor.",
"The additional reward for matching the syndrome is required to prevent the decoder from just flipping the bits where $|\\lambda _a|$ is minimal.",
"For example, it could happen that a single error in position $a$ with large $|\\lambda _a|$ matches the syndrome, but instead one chooses to flip $T$ bits with small absolute LLRs.",
"The scaling factor $c$ is chosen such that the syndrome-matching reward $+1$ always dominates the expected cummulative term $-\\sum _{t=1}^T{c |\\lambda _{a_t}|}$ .",
"As an example, for the BSC, $c$ is chosen such that the reward function becomes $\\!\\!R(s,a,s^{\\prime }) = {\\left\\lbrace \\begin{array}{ll}-\\frac{1}{T}+1 &\\!",
"\\text{if $s^{\\prime } = 0$}\\\\ -\\frac{1}{T} &\\!",
"\\text{otherwise}.\\end{array}\\right.",
"}$ This reward function allows us to interpret optimal BF decoding as a “maze-playing game” in the syndrome domain where the goal is to find the shortest path to the all-zero syndrome.",
"Applying a small negative penalty for each step is a standard technique to encourage short paths.",
"Another alternative in this case is to choose a small discount factor $\\gamma < 1$ ."
],
[
"Choosing the Exploration Strategy",
"Compared to (REF ), we propose another exploration strategy as follows.",
"Let $e$ be the current error pattern, i.e., the channel error pattern plus any bit-flips that have been applied so far.",
"Then, with probability $\\varepsilon _\\mathrm {g}$ , we choose the action randomly from $\\operatorname{supp}(e) \\triangleq \\lbrace i\\in [N] \\,|\\, e_i = 1\\rbrace $ , i.e., we flip one of the incorrect bits.",
"When combined with $\\varepsilon $ -greedy exploration, we refer to this as $(\\varepsilon , \\varepsilon _\\mathrm {g})$ -goal exploration, where $\\varepsilon , \\varepsilon _\\mathrm {g}> 0$ and $0 <\\varepsilon + \\varepsilon _\\mathrm {g}< 1$ : $\\!\\!",
"a = {\\left\\lbrace \\begin{array}{ll}\\text{unif.~random over $\\mathcal {A}$} & \\text{w.p.~}\\varepsilon \\\\\\text{unif.~random over$\\operatorname{supp}(e)$ } & \\text{w.p.~} \\varepsilon _\\mathrm {g}\\\\\\arg \\max _{a} Q(s,a) & \\text{w.p.~} 1 - \\varepsilon - \\varepsilon _\\mathrm {g}.\\end{array}\\right.",
"}$ Remark 3 It may seem that biasing actions towards flipping erroneous bits leads to a form of supervised learning where the learned decisions merely imitate ground-truth decisions.",
"To see that this is not exactly true, consider transmission over the BSC where the error pattern has weight $d_{\\mathrm {min}}- 1$ (where $d_{\\mathrm {min}}$ is the minimum distance of the code) and the observation is at distance 1 from a codeword $\\tilde{c}$ .",
"Then, the optimal decision is to flip the bit that leads to $\\tilde{c}$ , whereas flipping an erroneous bit is suboptimal in terms of expected future reward, even though it moves us closer to the transmitted codeword $c \\ne \\tilde{c}$ ."
],
[
"Choosing the Function Approximator",
"We use fully-connected NNs with one hidden layer to represent $Q_\\theta (s,a)$ in fitted Q-learning.",
"In particular, the NN $f_\\theta $ maps syndromes to length-$N$ vectors $f_\\theta (s) \\in \\mathbb {R}^N$ and the Q-function is given by $Q_\\theta (s,a) = [f_\\theta (s)]_a$ , where $[\\cdot ]_n$ returns the $n$ -th component of a vector and $s$ is the syndrome for state $s$ .",
"The NN parameters are summarized in Tab.",
"REF .",
"In future work, we plan to explore other network architectures, e.g., multi-layer NNs or graph NNs based on the code's Tanner graph.",
"Table: Neural network parameters"
],
[
"Learned Bit-Flipping with Code Automorphisms",
"Let $\\mathcal {S}_N$ be the symmetric group on $N$ elements so that $\\pi \\in \\mathcal {S}_N$ is a bijective mapping (or permutation) from $[N]$ to itself.For a group $(G, \\circ )$ , we also informally refer to the set $G$ as the group.",
"In our context, the group operation $\\circ $ represents function composition defined by $(\\pi \\circ \\sigma )(i)= \\pi (\\sigma (i))$ .",
"The permutation automorphism group of a code $\\mathcal {C}$ is defined as $\\operatornamewithlimits{PAut}(\\mathcal {C}) \\triangleq \\lbrace \\pi \\in \\mathcal {S}_N \\,|\\, x^\\pi \\in \\mathcal {C}, \\forall x \\in \\mathcal {C}\\rbrace $ , where $x^\\pi $ denotes a permuted vector, i.e., $x_i^\\pi =x_{\\pi (i)}$ .",
"The permutation automorphism group can be exploited in various ways to improve the performance of practical decoding algorithms, see, e.g., [22], [23].",
"In the context of learned decoders, the authors in [6] propose to permute the bit positions prior to decoding (and unpermute after) such that the channel reliabilities are approximately sorted.",
"If the applied permutations are from $\\operatornamewithlimits{PAut}(\\mathcal {C})$ , the decoder simply decodes a permuted codeword, rather than the transmitted one.",
"The advantage is that certain bit positions are now more reliable than others due to the (approximate) sorting.",
"This can be advantageous in terms of optimizing parameterized decoders because of the additional structure that the decoder can rely on [6]."
],
[
"A Permutation Strategy for Reed–Muller Codes",
"In [6], the permutation preprocessing approach is applied for Bose–Chaudhuri–Hocquenghem (BCH) codes and permutations are selected from $\\operatornamewithlimits{PAut}(\\mathcal {C})$ such that the total reliabilities of the first $K$ permuted bit positions are maximized, see [6] for details.",
"In the following, we propose a variation of this idea for RM codes.",
"In particular, our goal is to find a permutation that sends as many as possible of the least reliable bits to positions $\\lbrace 0, 1, 2, 4, \\dots ,2^{m-1}\\rbrace \\triangleq \\mathcal {B}$ .",
"Recall that the automorphism group of RM$(r,m)$ is the general affine group of order $m$ over the binary field, denoted by AGL$(m,2)$ [24].",
"The group AGL$(m,2)$ is the set of all operators of the form $T(v) = \\mathbf {A} v + b,$ where $\\mathbf {A} \\in \\mathbb {F}_2^{m \\times m}$ is an invertible binary matrix and $b, v \\in \\mathbb {F}_2^m$ .",
"By interpreting the vector $v$ as the binary representation of a bit position index, (REF ) defines a permutation on the index set $\\lbrace 0,1, \\dots , N-1\\rbrace $ and thus on $[N]$ .",
"A set of vectors $\\lbrace \\mathbf {v}_0,\\mathbf {v}_1,\\ldots ,\\mathbf {v}_{m}\\rbrace $ is called affinely independent if and only if the set $\\lbrace \\mathbf {v}_1-\\mathbf {v}_0,\\ldots ,\\mathbf {v}_{m}-\\mathbf {v}_0\\rbrace $ is linearly independent.",
"The binary representations of the indices in $\\mathcal {B}$ correspond to the all-zero vector and all unit vectors of length $m$ .",
"One can verify that they are affinely independent.",
"The proposed strategy relies on the fact that, for any given set of $m+1$ affinely independent bit positions (in the sense that their binary representation vectors are affinely independent), there always exists a permutation in AGL$(m,2)$ such that the bit positions are mapped to $\\mathcal {B}$ in any desired order.",
"In particular, we perform the following steps to select the permutation prior to decoding: Let $\\pi $ be the permutation that sorts the reliability vector $r = |y|$ , i.e., $r^\\pi $ satisfies $r_i^\\pi < r_j^\\pi $ $\\iff $ $i < j$ .",
"Find the first $m+1$ affinely independent indices for $\\pi $ (e.g., using Gaussian elimination) and denote their binary representations by $v_0, v_1, \\dots , v_m$ .",
"The permutation is then defined by (REF ), where $b = v_0$ and the columns of $\\mathbf {A}$ are $v_1 - v_0, \\dots , v_m - v_0$ ."
],
[
"(Approximate) Sort and Discard",
"For the learned BF decoders over the AWGN channel, our approach is to first apply the permutation strategy described in the previous section and subsequently discard the channel LLRs.",
"From the perspective of the decoder, this scenario can be modeled as $N$ parallel BSCs, where the crossover probabilities for the bit positions in $\\mathcal {B}$ satisfy $p_0 > p_1 > p_2 > p_4 > \\dots > p_{2^{m-1}}$ .",
"This is related to approaches where channel reliabilities are used to mark highly reliable and/or unreliable bit positions, while the actual decoding is performed without knowledge of the reliability values using hard-decision decoding, see, e.g., [25].",
"Figure: BSC crossover probabilities after the proposed permutationstrategy for RM(32,16)(32,16) at E b /N 0 =4E_\\mathrm {b}/N_0=4 dB.The absolute values of the channel LLRs for the parallel BSCs used in the reward function (REF ) are given by $|\\lambda _n| = \\log \\frac{1 - p_n}{p_n},$ where $p_n$ is the crossover probability of the $n$ -th BSC.",
"The individual crossover probabilities can be determined via Monte Carlo estimation before the RL starts.",
"For example, Fig.",
"REF show the expected crossover probabilities after applying the proposed permutation strategy for RM$(32,16)$ assuming transmission at $E_\\mathrm {b}/N_0=4$ dB.",
"Figure: Estimation ofachievable information rates when applying the proposedpermutation strategy for RM codes and subsequently discarding the reliability values.",
"(BI-AWGN: binary-input AWGN, HD: hard decision)Remark 4 One can estimate the capacity of strategies that permute the received bits using the reliabilities and then discard them.",
"Fig.",
"REF shows the estimated information rates for the proposed strategy obtained via Monte Carlo averaging.",
"Our results show that a significant fraction of the achievable information rate is preserved, especially for high-rate codes.",
"For permutations restricted to AGL$(m,2)$ , this is less effective as the blocklength increases because the fraction of sorted channels satisfies $(m+1)/N =(\\log _2(N)+1)/N$ ."
],
[
"Results",
"In this section, numerical results are presented for learned BF (LBF) decoders$\\mathbf {H}$ -matrices and source code for the simulations are available online at https://github.com/fabriziocarpi/RLdecoding.",
"We first used our own Tensorflow RL implementation and later switched to RLlib [27] in order to use multi-core parallelism for training rollouts.",
"for the following RM and BCH codes: RM$(32,16)$ with the standard $16 \\times 32$ PC matrix $\\mathbf {H}_{\\mathrm {std}}$ and overcomplete $620 \\times 32$ PC matrix $\\mathbf {H}_{\\mathrm {oc}}$ whose rows are all minimum-weight dual codewords, see [8], [26] RM$(64, 42)$ with the standard $22 \\times 64$ PC matrix $\\mathbf {H}_{\\mathrm {std}}$ and overcomplete $2604 \\times 64$ PC matrix $\\mathbf {H}_{\\mathrm {oc}}$ BCH$(63, 45)$ with the standard $18\\times 63$ circulant PC matrix $\\mathbf {H}_{\\mathrm {std}}$ and overcomplete $189 \\times 63$ PC matrix $\\mathbf {H}_{\\mathrm {oc}}$ RM$(128,99)$ with the standard $29 \\times 128$ PC matrix $\\mathbf {H}_{\\mathrm {std}}$ and overcomplete $10668 \\times 128$ PC matrix $\\mathbf {H}_{\\mathrm {oc}}$ For some of the considered codes, standard table Q-learning is feasible.",
"For example, RM$(32,16)$ has $|\\mathcal {S}| = 2^{16} =65536$ and $|\\mathcal {A}| = 32$ so the Q-table has $|\\mathcal {S}||\\mathcal {A}| \\approx 2 \\cdot 10^6$ entries."
],
[
"Training Hyperparameters",
"In the following, we set the maximum number of decoding iterations to $T = 10$ and the discount factor to $\\gamma = 0.99$ .",
"For standard table Q-learning, the $(\\varepsilon , \\varepsilon _\\mathrm {g})$ -goal exploration strategy is adopted with fixed $\\varepsilon = 0.6$ , $\\varepsilon _\\mathrm {g}= 0.3$ , and learning rate $\\alpha =0.1$ .",
"For fitted Q-learning based on NNs, we use $\\varepsilon $ -greedy exploration where $\\varepsilon $ is linearly decreased from $0.9$ to 0 over the course of $0.9 K$ learning episodes (i.e., number of decoded codewords), where the total number of episodes $K$ depends on the scenario.",
"For the gradient optimization, the Adam optimizer is used with a batch size of $B=100$ and learning rate $\\alpha = 3 \\cdot 10^{-5}$ .",
"The training SNR for both standard Q-learning and fitted Q-learning is fixed at $E_\\mathrm {b}/N_0= 5\\,$ dB for RM$(128,99)$ and $E_\\mathrm {b}/N_0=4\\,$ dB for all other codes.",
"In general, better performance may be obtained by re-optimizing parameters for each SNR or by adopting parameter adapter networks that dynamically adapt the network parameters to the SNR [28]."
],
[
"Learning Convergence in Q-Learning",
"We start by comparing the learning convergence of the proposed exploration strategy (REF ) to the $\\varepsilon $ -greedy exploration for standard Q-learning assuming RM$(32,16)$ over the BSC.",
"In Fig.",
"REF , the obtained performance in terms of codeword error rate (CER) is shown as a function of the Q-learning iteration.",
"The shown learning curves are generated as follows.",
"During Q-learning, we always decode first the new channel observations (line 3 of Alg.",
"REF ) with the current Q-function without exploration and save the binary outcome (success/failure).",
"Then, we plot a moving average (window size 5000) of the outcomes to approximate the CER.",
"It can be seen that the proposed strategy converges significantly faster than $\\varepsilon $ -greedy exploration.",
"We also show a learning curve for training when a reward of 1 is given only for finding the transmitted codeword; in this case, however, the process is not an MDP (see Sec. )",
"and the performance can become worse during training."
],
[
"Binary Symmetric Channel",
"Fig.",
"REF shows the CER performance for all considered scenarios as a function of $E_\\mathrm {b}/N_0$ .",
"We start by focusing on the “hard-decision” decoding cases, which are equivalent to assuming transmission over the BSC.",
"Supplementary bit error rate (BER) results for the same scenarios are shown in Fig.",
"REF ."
],
[
"Baseline Algorithms",
"As a baseline for the LBF decoders over the BSC, we use BF decoding according to Alg.",
"REF (see also [8] and [14]) applied to both the standard and overcomplete PC matrices $\\mathbf {H}_{\\mathrm {std}}$ and $\\mathbf {H}_{\\mathrm {oc}}$ , respectively.",
"We also implemented optimal syndrome decoding for RM$(32,16)$ and BCH$(63,45)$ .",
"In general, BF decoding shows relatively poor performance when applied to $\\mathbf {H}_{\\mathrm {std}}$ , whereas the performance increases drastically for $\\mathbf {H}_{\\mathrm {oc}}$ (see also [8], [26]).",
"In fact, for RM$(32,16)$ , standard BF for $\\mathbf {H}_{\\mathrm {oc}}$ gives virtually the same performance as optimal decoding and the latter performance curves are omitted from the figure.",
"This performance increase comes at a significant increase in complexity, e.g., for RM$(32,16)$ , the overcomplete PC matrix has 620 rows compared to the standard PC matrix with only 16 rows.",
"For the BCH code, there still exists a visible performance gap between optimal decoding and BF decoding based on $\\mathbf {H}_{\\mathrm {oc}}$ ."
],
[
"Q-learning",
"From Figs.",
"REF (a) and (b), it can be seen that the LBF decoders based on table Q-learning for RM$(32,16)$ and BCH$(63,45)$ converge essentially to the optimal performance.",
"For RM$(64,42)$ in Fig.",
"REF (c), the performance of LBF decoding is virtually the same as for standard BF decoding using $\\mathbf {H}_{\\mathrm {oc}}$ , which leads us to believe that both schemes are optimal in this case.",
"These results show that the proposed RL approach is able to learn close-to-optimal flipping patterns given the received syndromes.",
"Note that for RM$(128,99)$ , Q-learning would require a table with $|\\mathcal {S}| |\\mathcal {A}| \\approx 7\\cdot 10^{10}$ entries which is not feasible to implement on our system."
],
[
"Fitted Q-learning",
"The main disadvantage of the standard Q-learning approach is the large storage requirements of the Q-table.",
"Indeed, the requirements are comparable to optimal syndrome decoding and this approach is therefore only feasible for short or very-high-rate codes.",
"Therefore, we also investigate to what extend the Q-tables can be approximated with NNs and fitted Q-learning.",
"The number of neurons in the hidden layer of the NNs is chosen to be 1500 for RM$(128,99)$ and 500 for all other cases.",
"The achieved performance is shown in Fig.",
"REF , labeled as “LBF-NN”.",
"For the RM codes, it was found that good performance can be obtained using fitted Q-learning using the standard PC matrix $\\mathbf {H}_{\\mathrm {std}}$ .",
"The performance loss compared to table Q-learning is almost negligible for RM$(32,16)$ and increases slightly for the longer RM codes.",
"For the BCH code, we found that fitted Q-learning works better using $\\mathbf {H}_{\\mathrm {oc}}$ compared to $\\mathbf {H}_{\\mathrm {std}}$ .",
"For this case, the gap compared to optimal decoding is less than $0.1\\,$ dB at a CER of $10^{-3}$ ."
],
[
"AWGN Channel",
"Next, we consider the AWGN channel assuming that the reliability information is exploited for decoding.",
"Figure: Bit error rate (BER) results for the samescenarios as considered in Fig.",
"."
],
[
"Baseline Algorithms",
"Ordered statistics decoding (OSD) is used as a benchmark, whose performance is close to ML [29].",
"In this paper, we use order-$\\ell $ processing where $\\ell = 3$ in all cases.",
"Furthermore, we employ WBF decoding according to [14] using $\\mathbf {H}_{\\mathrm {oc}}$ .",
"Similar to BF decoding over the BSC, the performance of WBF is significantly better for overcomplete PC matrices compared to the standard ones (results for WBF on $\\mathbf {H}_{\\mathrm {std}}$ are omitted).",
"From Fig.",
"REF , WBF decoding is within $0.6$ –$1.1\\,$ dB of OSD for the considered codes.",
"We remark that there also exist a number of improved WBF algorithms which may reduce this gap at the expense of additional decoding complexity and the necessity to tune various weight and threshold parameters, see [8], [9], [10], [11], [12], [13].",
"For RM codes of moderate length, ML performance can also be approached using other techniques [30]."
],
[
"Q-Learning",
"As explained in Sec.",
", our approach to LBF decoding over the AWGN channel in this paper consists of permuting the bit positions based on $r$ and subsequently discarding the reliability values.",
"For the RM codes, the particular permutation strategy is described in Sec. .",
"The performance results for standard Q-learning shown in Figs.",
"REF (a) and (c) (denoted as “s+d LBF”) demonstrate that this strategy performs quite close to WBF decoding and closes a significant fraction of the gap to OSD, even though reliability information is only used to select the permutation and not for the actual decoding.",
"For the BCH code, we use the same permutation strategy as described in [6].",
"In this case, however, the performance improvements due to applying the permutations are relatively limited."
],
[
"Fitted Q-Learning",
"For the NN-based approximations of the Q-tables for the sort-and-discard approach, we use the NN sizes from the previous section for the BSC.",
"In this case, fitted Q-learning obtains performance close to the standard Q-learning approach for RM codes.",
"Similar to the BSC, the performance gap is almost negligible for RM$(32,16)$ and increases for the longer RM codes.",
"For RM$(128,99)$ , sort-and-discard LBF decoding with NNs closes roughly half the gap between soft-decision ML (approximated via OSD) and hard-decision ML (approximated via BF on $\\mathbf {H}_{\\mathrm {oc}}$ )."
],
[
"Conclusion",
"In this paper, we have proposed a novel RL framework for BF decoding of binary linear codes.",
"It was shown how BF decoding can be mapped to a Markov decision process by properly choosing the state and action spaces, whereas the reward function can be based on a reformulation of the ML decoding problem.",
"In principle, this allows for data-driven learning of optimal BF decision strategies.",
"Both standard (table-based) and fitted Q-learning with NN function approximators were then used to learn good decision strategies from data.",
"Our results show that the learned BF decoders can offer a range of performance–complexity trade-offs."
]
] | 1906.04448 |
[
[
"EXmatcher: Combining Features Based on Reference Strings and Segments to\n Enhance Citation Matching"
],
[
"Abstract Citation matching is a challenging task due to different problems such as the variety of citation styles, mistakes in reference strings and the quality of identified reference segments.",
"The classic citation matching configuration used in this paper is the combination of blocking technique and a binary classifier.",
"Three different possible inputs (reference strings, reference segments and a combination of reference strings and segments) were tested to find the most efficient strategy for citation matching.",
"In the classification step, we describe the effect which the probabilities of reference segments can have in citation matching.",
"Our evaluation on a manually curated gold standard showed that the input data consisting of the combination of reference segments and reference strings lead to the best result.",
"In addition, the usage of the probabilities of the segmentation slightly improves the result."
],
[
"Introduction",
"The need for data integration from various sources is growing in information systems in order to improve data quality and reusability of the data e.g.",
"for retrieval or data analysis.",
"The procedure of finding records in a database that correspond to the same entity (e.g.",
"files, publications, data sets, ...) across another data set is typically called record linkage.",
"Record linkage has been used in different domains [4], [17].",
"The application of record linkage in the domain of bibliographic data is known as citation matching or reference matching.",
"High quality citation data for research publications is the basis for areas like bibliometrics but also for integrated digital libraries (DL).",
"Citation data are valuable since they show the linkage between publications.",
"The extraction of reference information from full text is called citation extraction.",
"One key challenge for the aforementioned tasks is to match the extracted reference information to given DLs.",
"The process of mapping an extracted reference string to one entity of a given DL is called citation matching [24].",
"Proper citation matching is an essential step for every citation analysis [29] and the improvement of citation matching leads to a higher quality of bibliometric studies.",
"In the DL context, citation data is one important source of effective information retrieval, recommendation systems and knowledge discovery processes [26].",
"Despite the widely acknowledged benefits of citation data, the open access to them is still insufficient.",
"Some commercial companies such as Clarivate Analytics, Elsevier or Google possess citation data in large-scale and use them to provide services for their users.",
"Recently, some initiatives and projects e.g.",
"the \"Open Citations\" project or the \"Initiative for Open Citations\" focus on publishing citation data openlyhttps://i4oc.org/.",
"The \"Extraction of Citations from PDF Documents\" - EXCITEhttp://excite.west.uni-koblenz.de/website/ project is one of these projects.",
"The aim of EXCITE is extracting and matching citations from social science publications [22] and making more citation data available to researchers.",
"With respect to this objective, a set of algorithms for information extraction and matching has been developed focusing on social science publications in the German language.",
"The shortage of citation data for the international and German social sciences is well known to researchers in the field and has itself often been subject to academic studies [29].",
"This paper is dedicated to the step of citation matching in the EXCITE pipeline and the responsible algorithm for this task is called EXmatcherhttps://github.com/exciteproject/EXmatcher.",
"For the matching task in EXCITE, different target databases/DLs are defined a) sowiport [18], b) GESIS Searchhttps://search.gesis.org/ and c) Crossrefhttps://search.crossref.org.",
"The matching target for the study in this paper is solely sowiport.",
"Sowiport contains bibliographic metadata records of more than 9 million references on publications and research projects in the social sciences.",
"This paper makes the following contributions: Introduction of a gold standard for the citation matching task, Evaluate the effect of different inputs in the citation matching steps and Investigate the effect of the utilization of reference segmentation probabilities as features in the citation matching procedure.",
"The remainder of this paper is structured as follows.",
"In section 2, we organize the related work around the concepts of the record linkage pipeline known from [4].",
"Section 3 describes the set-up of citation matching in the EXCITE project.",
"Section 4 is about creating a citation matching gold standard corpus and evaluation of our algorithm with different configurations.",
"Finally, section 5 summarizes the key outcomes of our improvements on citation matching."
],
[
"Related Work",
"Christen et al.",
"[4] suggested general steps for the matching process after reviewing different matching approaches: (1) Input pre-processing, (2) Blocking technique, (3) Feature extraction and classification.",
"EXmatcher also follows these steps for citation matching and considers different input configurations to investigate their affects.",
"In the following, we organize the related work according to these steps."
],
[
"Input pre-processing",
"As the first step, input data need to be pre-processed in a way that it becomes proper for the matching algorithm.",
"To identify similar strings during all parts of the matching process a common method to increase robustness is to normalize the input strings.",
"A simple normalization is to lowercase the input string and remove punctuation and stop words.",
"If an algorithm depends on reference segments for matching, this data need to be extracted from the reference strings.",
"PDFX [7], Exparser [1], GROBID [25], ParsCit [8] are few examples of tools that perform reference segmentation.",
"Wellner et al.",
"[36] investigated the effect of extraction probabilities on citation matching by the consideration of different number of best Viterbi segmentations.",
"EXmatcher considers only the best Viterbi segmentation and uses the probability of each segment in feature vector provided for a binary classifier regarding the citation matching task.",
"Phonetic function is another technique used in this step and the common idea behind all phonetic encoding functions is that they attempt to convert a string into a code based on pronunciation [3].",
"Phonetic algorithms are mainly used for name segments.",
"Pre-processing functions can also be used in other steps.",
"For example, data which has been prepared by phonetic functions can be considered as blocking keys in the indexing step since indexing brings similar values together.",
"These techniques can also be used in the feature extraction step to generate vectors of features for classifiers.",
"This encoding process is often language dependent.",
"Soundex algorithm was developed by Russell and Odell in 1918 [31] for English language pronunciation.",
"Phonex [23], NYSIIS [34], and Cologne functions [33] are some other examples of phonetic functions.",
"The Cologne phonetics is based on the Soundex phonetic algorithm and is optimized to match the German language.",
"We also used Cologne phonetic function in our implementation since our main focus was working on German language papers."
],
[
"Blocking Technique",
"The next step is the blocking technique in order to decrease the number of pairs required to be compared.",
"Imagine, we need to match a set of $n$ references extracted from publications to a bibliographic database with $m$ entries.",
"In a naive way, comparisons of every reference with every entry in the database are required which results in a complexity of $n\\times m$ .",
"Considering a set of 100,000 references and a database with 10 million bibliographic entries this results in $10^{12}$ comparisons.",
"In the blocking approach, we split target and source into blocks of data depending on a common attribute or combination of the attributes.",
"After finding corresponding blocks in source and target we reduce the number of necessary comparisons to the number of combinations between corresponding blocks.",
"For example, we have a reference with \"2001\" as publication year, so in this case, it is not necessary to compare this reference with the entire records in the target database.",
"We only could compare to the entries in the block of records published in 2001.",
"In related works (e.g., [32]), even they use blocks related to one year before and after that year.",
"Several blocking or indexing techniques have been introduced till now [4].",
"As an example, D-Dupe [20] is a tool implemented for data matching and network visualizations.",
"D-Dupe implemented an indexing technique based on standard blocking [12].",
"Hernandez et al.",
"suggested a sorted neighborhood approach [15], [16].",
"This technique, instead of generating a key for each block, sorts the data for matching based on a 'sorting key'.",
"In suffix or q-gram based indexing approaches there is a higher chance to have correct matches in a same block since the idea behind them are for handling different forms of entities and errors.",
"In the citation matching field, Fedoryszak et al.",
"presented a blocking method based on hash functions [10].",
"Another research field deals with the identification of efficient blocking keys.",
"Koo et al.",
"tried to find the best combination of citation record fields [21] that helps increase citation matching performance."
],
[
"Feature Extraction and Classification",
"In the third step of the citation matching process, each candidate record pair (i.e, reference (string and related segments) and each retrieved item by blocking) are compared using a variety of attributes and comparison functions.",
"The output of this step is a feature vector for each pair.",
"In the final step, each compared candidate record pair is classified into one of the classes (i.e., match, non-match) using the related feature vector.",
"Comparison functions such as Jaro-Winkler, Jaccard, or Levenshtein are often used for analyzing textual values.",
"As an example, the D-Dupe tool includes string comparison functions such as Levenshtein distance, Jaro, Jaccard, and Monge-Elkan [20].",
"For the classification step of citation matching the reference can be represented by a reference string or by extracted segments.",
"Also a combination of both is possible as we show in this work.",
"Foufoulas et al.",
"[14] suggested an algorithm which matches reference strings without reference segmentation.",
"Their approach first tries to detect the reference section by some heuristic and then attempts to identify the title of a record in the target repository in the reference section.",
"Finally, it validates this match with more metadata of the record in the target repository.",
"Their title detection and citation validation steps are mostly based on the combination of simple search and comparison functions.",
"One of classification approaches is a threshold-based one.",
"In this type, the similarity between vectors of two items will be calculated (e.g., by using cosine similarity algorithm) and if the similarity score is higher than a predefined threshold, then two items are matched.",
"A rule-based classification employs some rules for classification [6], [16], [30].",
"These rules consist of a combination of smaller parts and the link between these parts are logical \"AND\", \"OR\" and \"NOT\" operands.",
"These rules define the similarity of pairs.",
"In the optimal case, each rule in a set of rules should have a high precision and recall [27].",
"More strict or specific rules usually have high precision, while general or easy rules often have low precision but high recall.",
"The iterative, rule-based citation matching algorithm of CWTS (Center for Science and Technology Studies) [32] relies on a series of matching rules.",
"These rules are applied iterative in decreasing order of strictness.",
"The citation matching algorithm starts with the most restrictive matching rules (e.g., exact match on first author, publication year, publication title, volume number, starting page number, and DOI).",
"Afterward, it proceeds with less restrictive matching rules (e.g.",
"match on Soundex encoding of the last name of the first author, publication year plus or minus one, volume number, and starting page number).",
"The less restrictive matching rules allow for various types of inaccuracies in the bibliographic fields of cited references.",
"In all rules, the Levenshtein distance is used to match the publication name of a cited reference to the publication name of a cited article.",
"Viewing probabilistic record linkage from a Bayesian perspective has also been discussed by Fortini et al.",
"[13] and Herzog et al. [17].",
"If training data are available, then a supervised classification approach can be employed.",
"Many binary classification techniques have been introduced [27], [28], and many of these techniques are used for matching.",
"Decision tree is one of these supervised classification techniques [27].",
"As an example, Cochinwala et al.",
"[5] build a training set and trained a Regression Tree (CART) classifier [2] for data matching.",
"TAILOR tool [9] for data matching uses e.g.",
"a ID3 decision tree.",
"The Support Vector Machine (SVM) classification algorithm [35] is based on the idea of mapping the input data of the classifier into a higher dimensional vector space using a kernel function.",
"This is done to be able to separate samples for the target classes using a hyperplane even if it is not possible in the lower dimension.",
"SVM as a large margin classifier optimizes during training time through maximization of the distance between training samples and hyperplane.",
"Fedoryszak et al.",
"[11] presented a citation matching solution using Apache Hadoop.",
"Their algorithm is based on reference segments and also uses SVM algorithm to confirm the status of items (i.e., match or not match) based on the created features."
],
[
"Input Data for Matching",
"For matching we used two types of information from each reference, the raw reference strings and structured information (i.e.",
"segments).",
"The segmentation is done with Exparserhttps://github.com/exciteproject/Exparser which is a CRF-based algorithm [1].",
"The output of Exparser includes a probability for each predicted segment.",
"This information is taken into account as an additional information to enhance the results of the matching procedure.",
"To enhance the results for the publication year information we extract year mentions from the raw reference strings with a regular expression independently from the parser.",
"We also remove extra characters in the year segment (e.g.",
"b in 1989b).",
"As a last pre-processing step we have combined volume and issue to one segment called number because during parsing the issue was often recognized as a volume and vice versa."
],
[
"Blocking Step",
"We used the search platform Solrhttp://lucene.apache.org/solr/ for blocking.",
"For each reference, EXmatcher retrieves the corresponding block with the help of blocking queries.",
"The whole blocking procedure is described in Algorithm 1.",
"Pre-processed reference $r$ , indexed bibliographic database $D$ , cutoff parameter $c$ A set of suggested matching records $S$ Generate a query set $Q$ based on segments or reference string Initialize an empty suggestion set $S$ query q in query set $Q$ Retrieve ranked result list with query $q$ as $Rl$ size of result list $Rl$ $\\ge $ 0 Cut off ranked result list at position $c$ Join reduced list $Rl$ to $S$ Blocking step for matching in EXmatcher First, queries are formulated with the help of the parsed segments and the raw reference strings.",
"Therefore we used the operators OR and AND from the Solr query syntaxhttps://lucene.apache.org/solr/guide/6_6/query-syntax-and-parsing.html.",
"Additional we use fuzzy search ($\\sim $ -operator) which reflects a fuzzy string similarity search based on the Levenshtein distance.",
"The output of the blocking step is a ranked lists of retrieved items from the target database.",
"The items are ranked by the Lucene score based on tf/idfhttps://lucene.apache.org/core/7_0_0/core/index.html?org/apache/lucene/search/ similarities/TFIDFSimilarity.html .",
"To get the best trade off between retrieving all possible matching items and the reduction of necessary comparisons in the following classification task we identified two opportunities for influence.",
"One is varying the query and select the best query formulation.",
"The other is the selection of a cut off threshold which determines how many of the retrieved items per query are used for further processing.",
"As it is mentioned, firstly, queries out of segment combinations should be generated.",
"For six segments (i.e., 1-Author, 2-Title, 3-Year, 4-Page, 5-Number (Volume/Issue), 6-Source) this results in a maximum of 63 segment combinations.",
"For each query generated based on one of combinations needs to have one correct information about each of the segments queried.",
"For example, if year of publication and authors' names are used, one of the author names has to be correct and also the year of publication.",
"For title and source we used a fuzzy query on the whole segment string.",
"For numbers at least one found number have to be in the volume or the issue field of the record in our database.",
"To exclude not well performing segment combinations for query generation we measure the precision at one of the queries on our gold data.",
"We only select segment combinations where at least 60% of the retrieved items are a correct match.",
"This reduces the number of maximum combinations we consider for query generation by 25% to 48.",
"As an alternative strategy we generated queries only from the reference strings without using information from the segmentation.",
"This strategy tries to deal with the problem that title information is often not correctly identified during segmentation.",
"But since the title is the most effective field for matching, the following approach is used which can act independently from the quality of the segmentation.",
"For this we consider all token of the reference string as potentially including title information.",
"The idea is to formulate a bigram search of the whole reference string.",
"The resulting query leads to results which at least need to include one bigram of the reference string in the title field.",
"But the more bigrams of the reference string are included in the title, the more preferred results are.",
"Therefore a query based on only these bigrams of the reference string will be added to the set of queries.",
"In addition, to increase the precision, a query based on year and bigrams of the reference string will also be considered.",
"For this the year information is taken into account which is extracted with a regular expression.",
"The effect on blocking for the two strategies of query generation and even a comparison with a mixture of both strategies is described in REF ."
],
[
"Classification for citation matching task",
"After retrieving candidates for matches with our blocking procedure we need to decide which of the found candidate our system identify as a match.",
"I.e.",
"the decision if the retrieved item is a match and hence the reference and the entry in the database are representing the same entity.",
"For this we train and evaluate a binary classifier which is able to judge a pair of reference and match candidate as match or non match.",
"It is worth noting, that our approach is able to handle duplicates in our reference database.",
"The crucial step for building this classifier is feature selection.",
"We combine features generated from the raw reference string and from the segmentation.",
"One novelty of our approach is to test the usefulness of utilizing the certainty of our parser for the detected segments as an additional input feature for our classifier.",
"The output of Exparser contains for each token of all segments a probability value reflecting the certainty of the model.",
"If we have a high probability for a segment, the chance of having a wrong predicted label is low.",
"Therefore, we expect that the usage of features reflecting this probabilities will have a noticeable effect on the performance of citation matching.",
"The first group of features is based on the comparison of the reference segments and the retrieved items in the blocking step: Some example of features based on the author segment: Levenshtein score (phono-code and exact), Segmentation probability of first author (surname) Some example of features based on titles and source: Jaccard score (including segmentation probabilities), Levenshtein score (token and letter level) Some example of features based on numbers, pages, and publication year: Jaccard score, and Segmentation probability An example for the usage of the probability is the extended version of the Jaccard score for author names.",
"The Jaccard similarity for the last names is the intersection of last names over the union of the set of last names in two records.",
"If the size of the intersection of the last names of two records is 2 and the size of the union of them is 4, then the Jaccard score would be 0.5 ((1+1)/4).",
"Our enhanced metric uses the extracted probabilities as weights in the intersection.",
"If probabilities of these items in the intersection of lasts names are 0.8 and 0.9, then the new Jaccard score would be 0.42 ((0.8+0.9)/4).",
"For the creation of the features of the second group all information based on segmentation is excluded.",
"These are features based only on the comparison of the raw reference string with the information of the retrieved record.",
"You can find some examples of this group in the following list: Longest common sub-string of title and reference string, and Occurrence of the abbreviation of the source field (e.g., journal abbreviation) in index in reference string."
],
[
"Gold Standard for Matching Algorithm",
"The computation of off-line evaluation metrics such as precision, recall and F-measure need ground truths.",
"A manually checked gold standard was generated to assess the performance of the algorithms.",
"For creating this gold standard, we applied a simple matching algorithm based on blocking on a randomly selected set of reference strings from the EXCITE corpus.",
"The EXCITE corpus contains SSOARhttps://www.gesis.org/ssoar/home/ corpus (about 35k), SOJ:Springer Online Journal Archives 1860-2001 corpus (about 80k), and Sowiport papers (about 116K).",
"We used queries based on different combinations of title, author and publication year segments and considered the top hit in the retrieved blocks based on the Solr score.",
"The result was a document id (from sowiport) for each reference, if the approach could find any match.",
"In the second step, these ids detected by the matching algorithm were completed by duplication information in sowiport to reach a list of all candidates of match items for each references.",
"Afterwards, a trained human assessor checked the results.",
"If the previous step leads to an empty result set the assessor was asked to retrieve a record based on manually curated search queries.",
"These manual queries used the information from the correct segments by manually extracting them from the reference strings.",
"If the corresponding item was found, it was added to the gold standard.",
"It also appeared, that not only one match was found, but also duplicates.",
"In this case the duplicates where also added as matching items.",
"When matching items are found in the previous step, the assessor checked this list to remove wrong items and add missing items.",
"The result of this process is a corpus containing 816 reference strings.",
"517 of these items have at least one matched item in sowiport.",
"We published this corpus and a part of sowiport data (18,590 bibliographic items) openly for interested researchers in our Github repositoryhttps://github.com/exciteproject/EXgoldstandard/tree/master/Goldstandard_EXmatcher."
],
[
"Evaluation of Blocking Step",
"In this evaluation, three different configurations for input of blocking (i.e., 1- using only reference strings, 2- using only reference segments, and 3- the combination of reference segments and strings) were examined.",
"In addition, the effect of the consideration of different numbers of top items from the blocking step was checked.",
"Fig.",
"REF shows that the precision curve of blocking based on reference strings is higher than the two other configurations.",
"This is not a big surprise because using only reference strings in our approach means focusing on the title and year fields (which it is explained in section REF and section REF ) and the usage of these two fields has a high precision score to retrieve items.",
"On the one hand by considering more items of the blocking list the precision is decreasing.",
"On the other hand the recall shown in Fig.",
"REF reach a score higher than 0.9 after the consideration of the 4 top items of blocking.",
"The highest recall has been achieved using the combination of reference strings and segments.",
"Surprisingly, the curve of reference strings become closer to the combination of reference string and segments by consideration of more top items in blocking and almost reach to that in number 14.",
"All these three curves become almost steady after consideration of 11 top retrieved items for each blocking query.",
"Figure: Recall of blockingSince we have another step after blocking which improve the precision, the important point in blocking is keep recall score high and at the same time shrinking the number of items for comparison.",
"The precision of these three curves were not significantly different, therefore, the combination of reference strings and segments is picked in blocking step to generate input for the evaluation of classification step.",
"For the number of top items in blocking, which are used for further processing in our pipeline, five is selected because considering more then five items is not leading to a higher recall value.",
"The selected configuration leads to a number of 1 to 39 retrieved items per reference.",
"The average number was 14 records with a standard deviations of 6.5.",
"For the 816 references of the gold standard 10,997 match candidates are generated with our configuration.",
"For each pair of reference and corresponding match candidate in our reference database sowiport we know if it is a match or not based on our gold standard.",
"In these 10,997 pairs, 1,026 (9.3%) are correct matches and 9,971 (90.7%) are no matches.",
"After blocking, the number of reference strings which have at least one correct match is 507, and 302 references are without any correct pair.",
"It means only ten references (1.2%) which have at least one match in the gold standard could not pass blocking successfully, i.e.",
"blocking step could not suggest any correct match for them."
],
[
"Evaluation of Classification Step",
"In this section we present the results of the classification task.",
"We applied ten-fold cross validation for testing different classifier and feature combinations.",
"Blocking generated results for 809 references were split into ten separated groups and their related pairs placed in the related group to form the ten folds for cross validation.",
"Table REF contains precision, recall and f-measure for our compared configurations.",
"Table: Evaluation macro-metrics of different classifiers including duplicate matches for each reference string - the highest value in each column is marked by * symbol.The results show that the SVM classifier using the combination of reference strings and segment features with considering segmentation probability has the highest F1 and precision scores.",
"The second highest F1 score is related to the SVM classifier which uses the combination input features but this time without the segmentation probabilities.",
"The interpretation of data is that using the combination of inputs (reference segments and strings) has the main impact on the accuracy scores.",
"The average number of references in different folds based on their number of correct predictions of the classifier with the highest F1 score are shown in Fig.",
"REF .",
"Figure: Average number of references in folds with true prediction of match classIn most real world scenarios it is only necessary to find exactly one match in a bibliographic database.",
"Because of this we evaluate our matching algorithm again.",
"For this evaluation only one correct match have to be found for each reference.",
"Regarding this purpose, we pick the highest probability generated by the classifier among match pairs for each reference The threshold for decision between two classes would be the 0.5 - default threshold for the SVM classifier in scikit-learn python package.. For this evaluation, we used the combination of features based on reference strings and segments as the input (including segments probabilities).",
"In this case, average precision and recall scores for SVM algorithm are 0.97 and 0.92.",
"For random forest algorithm, average precision and recall scores are 0.96 and 0.93.",
"To calculate a final score for the complete pipeline, also 10 references which could not pass blocking step should be considered.",
"Since the consideration of these items changes the number of false negatives, we see the effect on the recall score.",
"Consequently, recall in the pipeline with SVM would be 0.913 and the pipeline using the Random Forest classifier would be 0.917.",
"These evaluation scores are included in Table REF .",
"Table: 10-fold cross-validation of SVM classifier regarding finding only one match for each reference"
],
[
"Discussion and Conclusions",
"In this paper, we explained our approach for handling the task of citation matching in the EXCITE project.",
"The implemented algorithm (EXmatcher) follows the classic solution for this task which contains three steps (i.e., 1- data normalization and cleaning, 2- blocking, 3- feature vector creation and classification).",
"We analyzed the impact of different inputs (i.e., reference strings, segments and the combination of both) on the performance of our citation matching algorithm.",
"In addition, we investigated the benefit of using segments probabilities in the citation matching task.",
"The segmentation probabilities are considered directly and as weights for creating specific features for the classifier of EXmatcher.",
"Using the combination of reference strings and segments as input with a SVM classification outperforms the other configurations in terms of F1 and precision scores.",
"Segments probabilities have a good impact on the precision score when the citation matching algorithm uses segments as input.",
"For example, in the configuration of using only segments as the input and using SVM, segments probabilities can improve the precision about 11% (Table REF ).",
"The combination of reference strings and segments can also cover the effect of considering segments probabilities.",
"It means including/excluding segments probability doesn't affect the accuracy when citation matching algorithm uses the combination of two input data.",
"The effect of utilizing different classifiers on the result are very depended on other parameters in the citation matching configuration such as input types (i.e.",
"reference strings, segments or both) and the consideration of segment probabilities.",
"The combination of reference strings and segments as the input for citation matching shows a higher recall than using each of them alone.",
"But still 10 references which have at least one match couldn't pass the blocking step with using the combination the both data.",
"One reason of this incident was that in generating queries, EXmatcher combines the information from reference strings and information from reference segments in one query and links them with OR logical operand.",
"Decreasing the number of failed in blocking step leads to a higher recall.",
"One solution could be to send queries based on reference strings and based on segments in different queries and then the algorithm combines the retrieved items.",
"Also more items can be extracted from reference strings input (such as pages, issue, volume and DOI) with some rule based steps and used in the blocking.",
"The citation matching approach which has been described and evaluated in this paper is implemented in a demonstrator which connects all important steps from reference extraction, reference segmentation and matching in the EXCITE toolchain (see [19] http://excite.west.uni-koblenz.de/excite)."
]
] | 1906.04484 |
[
[
"Derivations on semi-simple Jordan algebras and its applications"
],
[
"Abstract In this paper, we mainly study the derivation algebras of semi-simple Jordan algebras over a field of characteristic $0$ and give sufficient and necessary conditions that the derivation algebras of them are simple.",
"As an application, we prove that for a semi-simple Jordan algebra $J$, $TDer(Der(J)) = Der(Der(J)) = ad(Der(J))$ under some assumptions.",
"Moreover, we also show that for a semi-simple Jordan algebra $J$ which has a finite basis over a field of characteristic $0$, $TDer(J) = Der(J) = Inn(J)$.",
"This is a corollary about our theorem which concerns Jordan algebras with unit."
],
[
"Introduction",
"An algebra $J$ over a field $\\rm {F}$ is called a Jordan algebra if $x \\circ y = y \\circ x,\\quad \\forall x, y \\in J,\\\\(x^{2} \\circ y) \\circ x = x^{2} \\circ (x \\circ y),\\quad x^{2} = x \\circ x ,\\quad \\forall x, y \\in J.$ In 1933, this kind of algebras first appeared in a paper by P. Jordan in his study of quantum mechanics.",
"Subsequently, P. Jordan, J. von Neumann and E. Wigner introduced finite-dimensional formally real Jordan algebras for quantum mechanics formalism in [11].",
"All simple finite-dimensional Jordan algebras over an algebraically closed field $\\rm {F}$ of characteristic different from 2 were classified some years later by A.",
"A. Albert in [1], using the idempotent method.",
"Since then, Jordan algebras were found various applications in mathematics and theoretical physics (see [5], [13], [15], [17] and references therein) and now form an intrinsic part of modern algebra.",
"Latest results on Jordan algebras refer to [12], [16], [6] and references therein.",
"Jordan systems arise naturally as “coordinates\" for Lie algebras having a grading into 3 parts.",
"Over the years, many predecessors have succeeded in generalizing some of the results of Lie algebras to Jordan algebras.",
"In [1], A.",
"A. Albert proved Lie's theorem, Engel's theorem and Cartan's theorem for Jordan algebras.",
"In [7], N. Jacobson successfully showed that $Der(J) = Inn(J)$ where $J$ was a semi-simple Jordan algebra over a field of characteristic zero.",
"In [14], D. J. Meng proved that if a centerless Lie algebra $L$ had the decomposition $L = L_{1} \\oplus L_{2}$ , then the derivation algebra of $L$ also had decomposition $Der(L) = Der(L_{1}) \\oplus Der(L_{2})$ .",
"In this paper, we successfully generalize this result to Jordan algebras(see Theorem REF ).",
"However, there are also some results which couldn't be generalized from Lie algebras to Jordan algebras.",
"For instance, it is well known that a simple Lie algebra $L$ is isomorphic to its derivation algebra $Der(L)$ .",
"The above results doesn't hold in the case of Jordan algebras, since the derivation algebras of Jordan algebras are actually Lie algebras rather than Jordan algebras.",
"Moreover, we also know that the derivation algebras of simple Lie algebras are simple.",
"Whether it is also true in simple Jordan algebras or not?",
"The answer is not, which can be deduced by Theorem REF .",
"Since the result is not necessarily true, the condition for a simple Jordan algebra with simple derivation algebra is given in Theorem REF , REF and REF .",
"In this paper, we mainly study simple Jordan algebras over a field whose characteristic is 0.",
"And we give the sufficient and necessary conditions that the derivation algebras of them are simple.",
"Derivations have been a historic and widely studied subject for many years.",
"Recall that a derivation on an algebra $A$ is a linear map $D : A \\rightarrow A$ satisfying $D(xy) = D(x)y + xD(y),\\quad \\forall x, y \\in A.$ In [8], N. Jacobson showed that every nilpotent Lie algebra had a derivation $D$ which was not inner.",
"Moreover, he also proved that if $L$ was a finite dimensional Lie algebra over the field $\\rm {F}$ such that the killing form of $L$ was non-degenerate, then the derivations of $L$ were all inner in [9].",
"As a generalization, S. Berman proved that if $L$ denoted Lie algebras which generalized the split simple Lie algebras, then the dimension of $Der(L)/ad(L)$ equaled the nullity of the Cartan matrix which defined $L$ in [2].",
"As a nature generalization of derivations, triple derivations appeared in order to study associative algebras(rings).",
"A linear map $D : L \\rightarrow L$ where $L$ is a Lie algebra, is called a triple derivation on $L$ if it satisfies $D([[x, y], z]) = [[D(x), y], z] + [[x, D(y)], z] + [[x, y], D(z)],\\quad \\forall x, y, z \\in L.$ Similarly, one can define triple derivations on Jordan algebras.",
"Obviously, derivations are triple derivations.",
"Naturally, one may has such a question that if triple derivations are all derivations.",
"The answer is not trivial.",
"In [18], J. H. Zhou came to a conclusion that if $L$ denoted a centerless perfect Lie algebra over a field whose characteristic was not 2, then every triple derivation on $L$ was a derivation.",
"What is the answer to this question in Jordan algebras?",
"In this paper, we will study on Jordan algebras with unit and show that triple derivations are all derivations in the case of the characteristic of the basic field is not 2.",
"As an application, this result is valid on semi-simple Jordan algebras over a field of characteristic 0 since a semi-simple Jordan algebra over a field of characteristic 0 has the unique unit.",
"In the following, we'll give the definition of the center of a Jordan algebra, which is denoted by $C(J)$ .",
"The center of a Jordan algebra $J$ is the set $\\lbrace a \\in J | (a \\circ x) \\circ y = (a \\circ y) \\circ x = (x \\circ y) \\circ a, \\forall x, y \\in J\\rbrace .$ The paper is organised as follows: In Section , we'll prove that our main theorem, Theorem REF , which is a generalization of a famous theorem in Lie algebras.",
"Applying this theorem, we can reduce the study of the derivation algebras of semi-simple Jordan algebras to simple Jordan algebras.",
"In the following, we'll study four kinds of simple Jordan algebras over a field of characteristic 0 respectively and give the sufficient and necessary conditions that the derivation algebras of them are simple, Theorem REF and REF , which can be viewed as the most important results in this paper.",
"In Section , we study Jordan algebras with unit over a field of characteristic not 2 and prove that for such Jordan algebra $J$ , $TDer(J) = Der(J)$ (see Theorem REF ).",
"As an application, we get a corollary that for a semi-simple Jordan algebra $J$ over a field of characteristic 0, $TDer(J) = Der(J) = Inn(J)$ (see Corollary REF ).",
"In Section , we also show that under some assumptions, $TDer(Der(J)) = Der(Der(J)) = ad(Der(J))$ where $J$ is a semi-simple algebra, i.e., Theorem REF ."
],
[
"Derivation algebras of semi-simple Jordan algebras",
"Lemma 2.1 [7] Let $J$ be a semi-simple Jordan algebra over a field $\\rm {F}$ .",
"Then $J$ has the decomposition $J = \\oplus ^{s}_{i = 1}J_{i}$ where $J_{i}(1 \\le i \\le s)$ are simple ideals of $J$ .",
"Theorem 2.2 Suppose that $J$ is a Jordan algebra and has a decomposition $J = J_{1} \\oplus J_{2}$ , where $J_{1}$ , $J_{2}$ are ideals of $J$ .",
"Then $C(J) = C(J_{1}) \\dotplus C(J_{2})$ ; If $C(J) = \\lbrace 0\\rbrace $ , then $Der(J) = Der(J_{1}) \\oplus Der(J_{2})$ .",
"(1).",
"Obviously, $C(J_{1}) \\cap C(J_{2}) = \\lbrace 0\\rbrace $ .",
"For all $z_{i} \\in C(J_{i})(i = 1, 2)$ , take $x = x_{1} + x_{2}$ , $y = y_{1} + y_{2} \\in J$ , where $x_{1}, y_{1} \\in J_{1}$ , $x_{2}, y_{2} \\in J_{2}$ .",
"We have $((z_{1} + z_{2}) \\circ x) \\circ y = ((z_{1} + z_{2}) \\circ (x_{1} + x_{2})) \\circ (y_{1} + y_{2}) = (z_{1} \\circ x_{1}) \\circ y_{1} + (z_{2} \\circ x_{2}) \\circ y_{2},$ $((z_{1} + z_{2}) \\circ y) \\circ x = ((z_{1} + z_{2}) \\circ (y_{1} + y_{2})) \\circ (x_{1} + x_{2}) = (z_{1} \\circ y_{1}) \\circ x_{1} + (z_{2} \\circ y_{2}) \\circ x_{2},$ $(z_{1} + z_{2}) \\circ (x \\circ y) = (z_{1} + z_{2}) \\circ ((x_{1} + x_{2}) \\circ (y_{1} + y_{2})) = z_{1} \\circ (x_{1} \\circ y_{1}) + z_{2} \\circ (x_{2} \\circ y_{2}),$ since $z_{i} \\in C(J_{i})(i = 1, 2)$ , we have $(z_{i} \\circ x_{i}) \\circ y_{i} = (z_{i} \\circ y_{i}) \\circ x_{i} = z_{i} \\circ (x_{i} \\circ y_{i})\\;(i = 1, 2).$ Hence, $((z_{1} + z_{2}) \\circ x) \\circ y = ((z_{1} + z_{2}) \\circ y) \\circ x = (z_{1} + z_{2}) \\circ (x \\circ y),$ which implies that $z_{1} + z_{2} \\in C(J)$ , i.e., $C(J_{1}) \\dotplus C(J_{2}) \\subseteq C(J)$ .",
"On the other hand, for all $a \\in C(J)$ , suppose that $a = a_{1} + a_{2}$ where $a_{i} \\in J_{i}(i = 1, 2)$ .",
"Then for all $x_{1}, y_{1} \\in J_{1}$ , $(a_{1} \\circ x_{1}) \\circ y_{1} = ((a - a_{2}) \\circ x_{1}) \\circ y_{1} = (a \\circ x_{1}) \\circ y_{1},$ $(a_{1} \\circ y_{1}) \\circ x_{1} = ((a - a_{2}) \\circ y_{1}) \\circ x_{1} = (a \\circ y_{1}) \\circ x_{1},$ $a_{1} \\circ (x_{1} \\circ y_{1}) = (a - a_{2}) \\circ (x_{1} \\circ y_{1}) = a \\circ (x_{1} \\circ y_{1}),$ since $a \\in C(J)$ , we have $(a \\circ x_{1}) \\circ y_{1} = (a \\circ y_{1}) \\circ x_{1} = a \\circ (x_{1} \\circ y_{1}).$ Hence, $(a_{1} \\circ x_{1}) \\circ y_{1} = (a_{1} \\circ y_{1}) \\circ x_{1} = a_{1} \\circ (x_{1} \\circ y_{1}),$ which implies that $a_{1} \\in C(J_{1})$ .",
"Similarly, we have $a_{2} \\in C(J_{2})$ .",
"Therefore, we have $C(J) = C(J_{1}) \\dotplus C(J_{2})$ .",
"(2).",
"$\\bf {Step 1}$ .",
"We'll show that $\\forall i = 1, 2$ , $D(J_{i}) \\subseteq J_{i},\\quad \\forall D \\in Der(J)$ .",
"Suppose that $x_{i} \\in J_{i}$ , then $D(x_{1}) \\circ x_{2} = D(x_{1} \\circ x_{2}) - x_{1} \\circ D(x_{2}) \\in J_{1} \\cap J_{2} = 0,$ since $J_{1}$ , $J_{2}$ are ideals of $J$ .",
"Suppose that $D(x_{1}) = u_{1} + u_{2}$ where $u_{1} \\in J_{1}$ , $u_{2} \\in J_{2}$ .",
"Then $u_{2} \\circ x_{2} = (u_{1} + u_{2}) \\circ x_{2} = D(x_{1}) \\circ x_{2} = 0,$ which implies that $u_{2} \\in C(J_{2})$ .",
"Note that $C(J) = \\lbrace 0\\rbrace $ , we have $C(J_{i}) = \\lbrace 0\\rbrace (i = 1, 2)$ .",
"Hence, $u_{2} = 0$ .",
"That is to say $D(x_{1}) \\in J_{1}$ .",
"Similarly, we have $D(x_{2}) \\in J_{2}$ .",
"$\\bf {Step 2}$ .",
"We'll show that $Der(J_{1}) \\dotplus Der(J_{2}) \\subseteq Der(J)$ .",
"For any $D \\in Der(J_{1})$ , we extend it to a linear map on $J$ as follow $D(x_{1} + x_{2}) = D(x_{1}),\\quad \\forall x_{1} \\in J_{1}, x_{2} \\in J_{2}.$ Then for any $x, y \\in J$ , suppose that $x = x_{1} + x_{2}$ , $y = y_{1} + y_{2} \\in J$ , where $x_{1}, y_{1} \\in J_{1}$ , $x_{2}, y_{2} \\in J_{2}$ , we have $D(x \\circ y) = D((x_{1} + x_{2}) \\circ (y_{1} + y_{2})) = D(x_{1} \\circ y_{1} + x_{2} \\circ y_{2}) = D(x_{1} \\circ y_{1}),$ $D(x) \\circ y + x \\circ D(y) = D(x_{1} + x_{2}) \\circ (y_{1} + y_{2}) + (x_{1} + x_{2}) \\circ D(y_{1} + y_{2}) = D(x_{1}) \\circ y_{1} + x_{1} \\circ D(y_{1}),$ since $D \\in Der(J_{1})$ , $D(x_{1} \\circ y_{1}) = D(x_{1}) \\circ y_{1} + x_{1} \\circ D(y_{1}),$ we have $D(x \\circ y) = D(x) \\circ y + x \\circ D(y),$ which implies that $D \\in Der(J)$ , i.e., $Der(J_{1}) \\subseteq Der(J)$ .",
"Moreover, $D \\in Der(J_{1})$ if and only if $D(x_{2}) = 0,\\; \\forall x_{2} \\in J_{2}$ .",
"Similarly, we have $Der(J_{2}) \\subseteq Der(J)$ and $D \\in Der(J_{2})$ if and only if $D(x_{1}) = 0,\\; \\forall x_{1} \\in J_{1}$ .",
"Then we have $Der(J_{1}) + Der(J_{2}) \\subseteq Der(J)$ and $Der(J_{1}) \\cap Der(J_{2}) = \\lbrace 0\\rbrace $ .",
"Hence, $Der(J_{1}) \\dotplus Der(J_{2}) \\subseteq Der(J)$ .",
"$\\bf {Step 3}$ .",
"We'll prove that $Der(J_{1}) \\dotplus Der(J_{2}) = Der(J)$ .",
"Suppose that $D \\in Der(J)$ .",
"Set $x = x_{1} + x_{2}, x_{i} \\in J_{i}$ .",
"Define $D_{1}, D_{2}$ as follows $\\left\\lbrace \\begin{aligned}D_{1}(x_{1} + x_{2}) = D(x_{1}),\\\\D_{2}(x_{1} + x_{2}) = D(x_{2}).\\end{aligned}\\right.$ Obviously, $D = D_{1} + D_{2}$ .",
"For any $u_{1}, v_{1} \\in J_{1}$ , $D_{1}(u_{1} \\circ v_{1}) = D(u_{1} \\circ v_{1}) = D(u_{1}) \\circ v_{1} + u_{1} \\circ D(v_{1}) = D_{1}(u_{1}) \\circ v_{1} + u_{1} \\circ D_{1}(v_{1}).$ Hence, $D_{1} \\in Der(J_{1})$ .",
"Similarly, $D_{2} \\in Der(J_{2})$ .",
"Therefore, $Der(J) = Der(J_{1}) \\dotplus Der(J_{2})$ .",
"$\\bf {Step 4}$ .",
"We'll show that $Der(J_{i}) \\lhd Der(J)$ .",
"Suppose that $D_{1} \\in Der(J_{1}), D \\in Der(J), x_{2} \\in J_{2}$ , $[D, D_{1}](x_{2}) = DD_{1}(x_{2}) - D_{1}D(x_{2}) = 0,$ which implies that $[D, D_{1}] \\in Der(J_{1})$ , i.e., $Der(J_{1}) \\lhd Der(J)$ .",
"Similarly, $Der(J_{2}) \\lhd Der(J)$ .",
"Therefore, we have $Der(J) = Der(J_{1}) \\oplus Der(J_{2})$ .",
"According to Lemma REF and Theorem REF , we only need to study the derivation algebras of simple Jordan algebras in order to study the derivation algebras of semi-simple Jordan algebras.",
"Moreover, in [10], we get that the simple Jordan algebras over a field whose characteristic is 0 fall into three “great\" classes and one exceptional class as following: The special Jordan algebras generated by simple associative algebras ${A}$ with the multiplication $a \\circ b = \\frac{1}{2}(ab + ba)$ , denoted by ${A}^{+}$ ; The Jordan algebras $H({A}, P)$ of $P$ -symmetric elements in simple involutorial algebras ${A}$ with an involution $P$ ; The Jordan algebras constructed by the algebras that define Clifford systems.",
"These have a basis $\\lbrace 1, u_{1}, \\cdots , u_{n}\\rbrace $ such that 1 is the unit and $u^{2}_{i} = \\alpha _{i}1$ where $\\alpha _{i} \\ne 0, u_{i} \\circ u_{j} = 0$ if $i \\ne j$ ; The exceptional Jordan algebra corresponding to the system $M^{8}_{3}$ which has 27 dimensions over its center.",
"In the following part, we'll study the derivation algebras of the above four kinds of simple Jordan algebras respectively and give the sufficient and necessary conditions that the derivation algebras of them are simple.",
"Lemma 2.3 [10] Let ${A}$ be a simple associative algebra.",
"Then if $D$ is a derivation on the Jordan algebra ${A}^{+}(i.e., Jordan\\;algebras\\;of\\;type\\;A)$ there exists an element $d$ in ${A}$ such that $D(a) = [a, d]$ for all $a$ .",
"Lemma 2.4 [10] Let ${A}$ be a simple associative algebra that has an involution $P$ (an anti-isomorphism of ${A}$ of period 2).",
"Then if $D$ is a derivation on $H({A}, P)(i.e., Jordan\\;algebras\\\\\\;of\\;type\\;B)$ there exists a $P$ -skew element $d$ in ${A}$ such that $D(a) = [a, d]$ .",
"Lemma 2.5 Let ${A}$ be a simple associative algebra that has an involution $P$ (an anti-isomorphism of ${A}$ of period 2) and $H = \\lbrace a \\mid a \\in {A}, P(a) = -a\\rbrace $ .",
"Then $H$ is closed under the usual Lie bracket.",
"For any $x, y \\in H$ , we have $&P([x, y]) = P(xy - yx) = P(xy) - P(yx) = P(y)P(x) - P(x)P(y)\\\\&= (-y)(-x) - (-x)(-y) = yx - xy = -[x, y],$ which implies that $[x, y]$ is $P$ -skew.",
"Hence, $H$ is closed under the usual Lie bracket.",
"We denote such $D$ in Lemma REF and Lemma REF by $D_{d}$ .",
"Theorem 2.6 Suppose that $J$ is a simple Jordan algebra of type A(respectively, of type B).",
"Let ${A}$ be the associated simple associative algebra and $L$ the Lie algebra generated by ${A}$ (respectively, all $P$ -skew elements in ${A}$ ).",
"Then the following are equivalent $Der(J)$ is simple; $L/Z(L)$ is simple where $Z(L)$ denotes the center of $L$ .",
"(1).",
"Suppose that $L/Z(L)$ is simple.",
"We show that $Der(J)$ is simple by using reduction to absurdity.",
"Otherwise, there exists a non trivial ideal of $Der(J)$ , denoted by $D_{S}$ .",
"According to Lemma REF (respectively, Lemma REF ), there exists an element(respectively, a $P$ -skew element) $d$ in ${A}$ such that $D = D_{d}$ for any $D \\in Der(J)$ .",
"Let $S = \\lbrace \\bar{d} \\mid D_{d} \\in D_{S}\\rbrace $ .",
"It's obvious that $S$ is a subspace of $L/Z(L)$ .",
"Since $D_{S} \\ne \\lbrace 0\\rbrace $ , there exists a nonzero element $D_{d_{0}} \\in D_{S}$ .",
"Then $\\exists 0 \\ne a \\in J$ such that $[a, d_{0}] = D_{d_{0}}(a) \\ne 0,$ which implies that $d_{0} \\notin Z(L)$ .",
"Hence, $\\bar{d_{0}} \\ne \\bar{0}$ , $S \\ne \\lbrace \\bar{0}\\rbrace $ .",
"Since $D_{S} \\ne Der(J)$ , there exists a nonzero element $D_{d_{1}} \\in Der(J)$ and $D_{d_{1}} \\notin D_{S}$ .",
"That is to say $\\bar{0} \\ne \\bar{d_{1}} \\in L/Z(L)$ and $\\bar{d_{1}} \\notin S$ , i.e., $S \\ne L/Z(L)$ .",
"For any $a, b \\in L, c \\in J$ , we have $&[D_{a}, D_{b}](c) = D_{a}D_{b}(c) - D_{b}D_{a}(c) = [[c, b], a] - [[c, a], b] = [[c, b], a] + [[a, c], b]\\\\&= -[[b, a], c] = [c, [b, a]] = -[c, [a, b]] = -D_{[a, b]}(c).$ Hence we have $[D_{a}, D_{b}] = -D_{[a, b]}$ .",
"For all $\\bar{a} \\in L/Z(L)$ and any $\\bar{d} \\in S$ , we have $D_{[a, d]} = -[D_{a}, D_{d}] \\in D_{S},$ which implies that $\\overline{[a, d]} \\in S$ , i.e., $[\\bar{a}, \\bar{d}] \\in S$ .",
"Hence, $S$ is a non trivial ideal of $L/Z(L)$ , contradicting with $L/Z(L)$ is simple.",
"Hence, $Der(J)$ is simple.",
"(2).",
"Suppose that $Der(J)$ is simple.",
"We prove that $L/Z(L)$ is simple by using reduction to absurdity.",
"Otherwise, there exists a non trivial ideal of $L/Z(L)$ , denoted by $S$ .",
"Let $D_{S} = \\lbrace D_{d} \\mid \\bar{d} \\in S\\rbrace $ .",
"It's obvious that $D_{S}$ is a subspace of $Der(J)$ .",
"Since $S \\ne \\lbrace \\bar{0}\\rbrace $ , there exists a nonzero element $\\bar{d_{0}} \\in S$ , i.e., $d_{0} \\notin Z(L)$ .",
"Then there exists $0 \\ne a \\in L$ such that $[a, d_{0}] \\ne 0$ , i.e., $D_{d_{0}}(a) \\ne 0$ .",
"Hence, $D_{d_{0}} \\ne 0$ .",
"Therefore, $D_{S} \\ne \\lbrace 0\\rbrace $ .",
"Assume that $D_{d_{1}} = D_{d_{2}}$ where $d_{1}, d_{2} \\in L$ , then for any $a \\in J$ $[a, d_{1}] = D_{d_{1}}(a) = D_{d_{2}}(a) = [a, d_{2}],$ which implies that $d_{1} - d_{2} \\in Z(L)$ , i.e., $\\bar{d_{1}} = \\bar{d_{2}}$ .",
"Hence, $D_{d_{1}} = D_{d_{2}}$ if and only if $\\bar{d_{1}} = \\bar{d_{2}}$ .",
"Since $S \\ne L/Z(L)$ , there exists a nonzero element $\\bar{d_{1}} \\in L/Z(L)$ and $\\bar{d_{1}} \\notin S$ .",
"That is to say $D_{d_{1}} \\in Der(J)$ and $D_{d_{1}} \\notin D_{S}$ .",
"Hence, $D_{S} \\ne Der(J)$ .",
"For all $D \\in Der(J)$ , according to Lemma REF (respectively, Lemma REF ), there exists an element(respectively, a $P$ -skew element) $d$ in ${A}$ such that $D = D_{d}$ .",
"For all $\\bar{d_{1}} \\in S$ , we have $[D, D_{d_{1}}] = [D_{d}, D_{d_{1}}] = -D_{[d, d_{1}]},$ we have $\\overline{[d, d_{1}]} = [\\bar{d}, \\bar{d_{1}}] \\in S$ since $S$ is an ideal of $L/Z(L)$ , hence $[D, D_{d_{1}}] \\in D_{S}$ .",
"Hence, we have $[Der(J), D_{S}] \\subseteq D_{S}$ , i.e., $D_{S}$ is a non trivial ideal of $Der(J)$ , contradicting with $Der(J)$ is simple.",
"Therefore, $L/Z(L)$ is simple.",
"Lemma 2.7 Let $J$ be a simple Jordan algebra of type $C$ , i.e., $J$ has a basis $\\lbrace 1, u_{1}, \\cdots , u_{n}\\rbrace $ such that 1 is the unit and $u^{2}_{i} = \\alpha _{i}1\\; where\\; \\alpha _{i} \\ne 0, u_{i} \\circ u_{j} = 0 \\;if \\;i \\ne j.$ Then $Der(J)$ is isomorphic to $M$ where $M$ is the Lie algebra generated by all $n \\times n$ -matrices $(v_{ij})_{n \\times n}$ satisfying $\\alpha _{i}v_{ji} + \\alpha _{j}v_{ij} = 0$ .",
"First we'll show that $M$ is closed under the usual Lie bracket.",
"It's obvious that $\\lbrace E_{ij} - \\frac{\\alpha _{j}}{\\alpha _{i}}E_{ji}|1 \\le i < j \\le n\\rbrace $ is a basis of $M$ .",
"We have $[E_{ij} - \\frac{\\alpha _{j}}{\\alpha _{i}}E_{ji}, E_{kl} - \\frac{\\alpha _{l}}{\\alpha _{k}}E_{lk}] =\\left\\lbrace \\begin{aligned}\\frac{\\alpha _{l}}{\\alpha _{i}}(E_{lj} - \\frac{\\alpha _{j}}{\\alpha _{l}}E_{jl}), (i = k, j \\ne l)\\\\\\frac{\\alpha _{j}}{\\alpha _{i}}(E_{ki} - \\frac{\\alpha _{i}}{\\alpha _{k}}E_{ik}), (i \\ne k, j = l)\\\\E_{il} - \\frac{\\alpha _{l}}{\\alpha _{i}}E_{li},(j = k, i \\ne l)\\\\-(E_{kj} - \\frac{\\alpha _{j}}{\\alpha _{k}}E_{jk}),(j \\ne k, i = l)\\end{aligned}\\right.$ others are all zero.",
"Hence, $M$ is a Lie algebra under the usual Lie bracket.",
"Now we will show that $Der(J)$ is isomorphic to $M$ .",
"Define $D : J \\rightarrow J$ to be a linear map by $\\left\\lbrace \\begin{aligned}D(1) = a_{0}1 + \\sum ^{n}_{k = 1}b_{0k}u_{k},\\\\D(u_{i}) = a_{i}1 + \\sum ^{n}_{k = 1}b_{ik}u_{k}.\\end{aligned}\\right.$ Suppose that $D$ is a derivation on $J$ , i.e., $D(x \\circ y) = D(x) \\circ y + x \\circ D(y),\\quad \\forall x, y \\in J$ .",
"It's only need to verify $D$ satisfies the above equation on base elements.",
"Take $x = 1$ , $y = u_{i}(1 \\le i \\le n)$ , we have $D(1 \\circ u_{i}) = D(u_{i}) = a_{i}1 + \\sum ^{n}_{k = 1}b_{ik}u_{k},$ $&D(1) \\circ u_{i} + 1 \\circ D(u_{i}) = (a_{0}1 + \\sum ^{n}_{k = 1}b_{0k}u_{k}) \\circ u_{i} + 1 \\circ (a_{i}1 + \\sum ^{n}_{k = 1}b_{ik}u_{k})\\\\&= (a_{0} + b_{ii})u_{i} + (\\alpha _{i}b_{0i} + a_{i})1 + \\sum ^{n}_{k = 1, k \\ne i}b_{ik}u_{k},$ comparing the coefficients on both sides, we have $\\left\\lbrace \\begin{aligned}a_{i} = \\alpha _{i}b_{0i} + a_{i},\\\\b_{ii} = a_{0} + b_{ii},\\end{aligned}\\right.$ note that $\\alpha _{i} \\ne 0$ , we have $\\left\\lbrace \\begin{aligned}b_{0i} = 0,\\\\a_{0} = 0.\\end{aligned}\\right.$ Similarly, take $x = u_{i}, y = u_{j}, (1 \\le i, j \\le n, i \\ne j)$ , we have $\\left\\lbrace \\begin{aligned}\\alpha _{i}b_{ji} + \\alpha _{j}b_{ij} = 0,\\\\a_{i} = 0.\\end{aligned}\\right.$ Take $x = y = u_{i}, 1 \\le i \\le n$ , we have $\\left\\lbrace \\begin{aligned}b_{ii} = 0,\\\\a_{i} = 0.\\end{aligned}\\right.$ Hence, while $b_{ji} \\ne 0(i \\ne j)$ satisfying $\\alpha _{i}b_{ji} + \\alpha _{j}b_{ij} = 0(i \\ne j)$ and others are all zero, $D$ is a derivation on $J$ .",
"We denote the matrix of $D$ under the basis $\\lbrace 1, u_{1}, \\cdots , u_{n}\\rbrace $ by $N_{D} =\\begin{pmatrix}0 & 0 & 0 & 0 & \\cdots & 0 & 0\\\\0 & 0 & b_{12} & b_{13} & \\cdots & b_{1, n - 1} & b_{1n}\\\\0 & b_{21} & 0 & b_{23} & \\cdots & b_{2, n - 1} & b_{2n}\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\0 & b_{n1} & b_{n2} & b_{n3} & \\cdots & b_{n, n - 1} & 0\\end{pmatrix},$ i.e., $D(1, u_{1}, \\cdots , u_{n}) = N_{D}(1, u_{1}, \\cdots , u_{n})^{T}$ .",
"We set $N$ be the Lie algebra generated the above matrixes.",
"It's obvious that there is a one-to-one correspondence between $Der(J)$ and $N$ by mapping every $D \\in Der(J)$ to $N_{D}$ , i.e., $Der(J)$ is isomorphic to $N$ .",
"Define $f : N \\rightarrow M$ to be a linear map by $f(N_{D}) = M_{D}$ where $M_{D} =\\begin{pmatrix}0 & b_{12} & b_{13} & \\cdots & b_{1, n - 1} & b_{1n}\\\\b_{21} & 0 & b_{23} & \\cdots & b_{2, n - 1} & b_{2n}\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\b_{n1} & b_{n2} & b_{n3} & \\cdots & b_{n, n - 1} & 0\\end{pmatrix}.$ It's easy to verify that $f$ is an isomorphism between $N$ and $M$ .",
"Hence $Der(J)$ is isomorphic to $M$ where $M$ is the Lie algebra generated by all $n \\times n$ -matrices $(v_{ij})_{n \\times n}$ satisfying $\\alpha _{i}v_{ji} + \\alpha _{j}v_{ij} = 0$ .",
"Theorem 2.8 Let $J$ be a simple Jordan algebra of type $C$ .",
"Then $Der(J)$ is simple if and only if $dim(J) \\ne 5$ .",
"We'll prove our conclusion by contradiction.",
"According to Lemma REF , $Der(J)$ has a basis $\\lbrace E_{ij} - \\frac{\\alpha _{j}}{\\alpha _{i}}E_{ji}|1 \\le i < j \\le n\\rbrace $ .",
"Suppose $I$ is a non trivial ideal of $Der(J)$ .",
"Take $0 \\ne a \\in I$ and suppose $a = \\sum ^{n}_{i, j = 1, i < j}a_{ij}(E_{ij} - \\frac{\\alpha _{j}}{\\alpha _{i}}E_{ji})$ .",
"$\\bf Step 1$ .",
"We'll show that there exists $1 \\le i_{0} < j_{0} \\le n$ such that $E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}} \\in I$ .",
"When $n = 2$ , the dimension of $Der(J)$ is 1.",
"The conclusion is clearly true.",
"When $n = 3$ , the dimension of $Der(J)$ is 3.",
"$Der(J) = \\lbrace E_{12} - \\frac{\\alpha _{2}}{\\alpha _{1}}E_{21}, E_{13} - \\frac{\\alpha _{3}}{\\alpha _{1}}E_{31}, E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32}\\rbrace $ .",
"Since $a \\ne 0$ , there exists at least one non zero element in $S = \\lbrace a_{ij}|1 \\le i < j \\le 3\\rbrace $ .",
"If there only exists one non zero element in $S$ , it's obvious that there exists $1 \\le i_{0} < j_{0} \\le 3$ such that $E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}} \\in I$ .",
"If there exist two non zero elements in $S$ , we might as well set $a_{12}, a_{13} \\ne 0$ , i.e., $a = a_{12}(E_{12} - \\frac{\\alpha _{2}}{\\alpha _{1}}E_{21}) + a_{13}(E_{13} - \\frac{\\alpha _{3}}{\\alpha _{1}}E_{31})$ .",
"$[a, E_{13} - \\frac{\\alpha _{3}}{\\alpha _{1}}E_{31}] = -a_{12}\\frac{\\alpha _{2}}{\\alpha _{1}}(E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32}).$ Hence, there exists $1 \\le i_{0} < j_{0} \\le 3$ such that $E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}} \\in I$ .",
"If there exist three non zero elements in $S$ , i.e, $a = a_{12}(E_{12} - \\frac{\\alpha _{2}}{\\alpha _{1}}E_{21}) + a_{13}(E_{13} - \\frac{\\alpha _{3}}{\\alpha _{1}}E_{31}) + a_{23}(E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32})$ .",
"$[a, E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32}] = a_{12}(E_{13} - \\frac{\\alpha _{3}}{\\alpha _{1}}E_{31}) - a_{13}\\frac{\\alpha _{3}}{\\alpha _{2}}(E_{12} - \\frac{\\alpha _{2}}{\\alpha _{1}}E_{21}),$ $[[a, E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32}], E_{12} - \\frac{\\alpha _{2}}{\\alpha _{1}}E_{21}] = a_{12}\\frac{\\alpha _{2}}{\\alpha _{1}}(E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32}).$ Hence, there exists $1 \\le i_{0} < j_{0} \\le 3$ such that $E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}} \\in I$ .",
"When $n \\ge 5$ , we set $S = \\lbrace a_{ij}|1 \\le i < j \\le n\\rbrace $ .",
"We prove the conclusion by induction on the number of nonzero elements in $S$ , denoted by $k$ .",
"Since $a \\ne 0$ , there exists at least one non zero element in $S = \\lbrace a_{ij}|1 \\le i < j \\le n\\rbrace $ .",
"When $k = 1$ , it's obvious that there exists $1 \\le i_{0} < j_{0} \\le n$ such that $E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}} \\in I$ .",
"When $k = 2$ , we set $a_{i_{1}j_{1}}, a_{i_{2}j_{2}} \\ne 0$ , i.e., $a = a_{i_{1}j_{1}}(E_{i_{1}j_{1}} - \\frac{\\alpha _{j_{1}}}{\\alpha _{i_{1}}}E_{j_{1}i_{1}}) + a_{i_{2}j_{2}}(E_{i_{2}j_{2}} - \\frac{\\alpha _{j_{2}}}{\\alpha _{i_{2}}}E_{j_{2}i_{2}})$ .",
"When $i_{1} = i_{2}, j_{1} \\ne j_{2}$ , we have $i_{2} \\ne j_{1}$ .",
"Take $l \\ne i_{1}, i_{2}, j_{1}, j_{2}$ , then $[a, E_{j_{1}l} - \\frac{\\alpha _{l}}{\\alpha _{j_{1}}}E_{lj_{1}}] = a_{i_{1}j_{1}}(E_{i_{1}l} - \\frac{\\alpha _{l}}{\\alpha _{i_{1}}}E_{li_{1}}) \\in I.$ When $i_{1} \\ne i_{2}, j_{1} = j_{2}$ , we have $i_{1} \\ne j_{2}$ .",
"Take $l \\ne i_{1}, i_{2}, j_{1}, j_{2}$ , then $[a, E_{li_{1}} - \\frac{\\alpha _{i_{1}}}{\\alpha _{l}}E_{i_{1}l}] = -a_{i_{1}j_{1}}(E_{lj_{1}} - \\frac{\\alpha _{j_{1}}}{\\alpha _{l}}E_{j_{1}l}) \\in I.$ When $i_{1} \\ne i_{2}, j_{1} \\ne j_{2}$ , take $l \\ne i_{1}, i_{2}, j_{1}, j_{2}$ , then When $i_{2} \\ne j_{1}$ , $[a, E_{j_{1}l} - \\frac{\\alpha _{l}}{\\alpha _{j_{1}}}E_{lj_{1}}] = a_{i_{1}j_{1}}(E_{i_{1}l} - \\frac{\\alpha _{l}}{\\alpha _{i_{1}}}E_{li_{1}}) \\in I$ .",
"When $i_{2} = j_{1}$ , $[a, E_{j_{1}l} - \\frac{\\alpha _{l}}{\\alpha _{j_{1}}}E_{lj_{1}}] = a_{i_{1}j_{1}}(E_{i_{1}l} - \\frac{\\alpha _{l}}{\\alpha _{i_{1}}}E_{li_{1}}) + a_{i_{2}j_{2}}\\frac{\\alpha _{l}}{\\alpha _{i_{2}}}(E_{lj_{2}} - \\frac{\\alpha _{j_{2}}}{\\alpha _{l}}E_{j_{2}l}) \\in I$ ; $[[a, E_{j_{1}l} - \\frac{\\alpha _{l}}{\\alpha _{j_{1}}}E_{lj_{1}}], E_{lj_{2}} - \\frac{\\alpha _{j_{2}}}{\\alpha _{l}}E_{j_{2}l}] = a_{i_{1}j_{1}}(E_{i_{1}j_{2}} - \\frac{\\alpha _{j_{2}}}{\\alpha _{i_{1}}}E_{j_{2}i_{1}}) \\in I$ .",
"Hence, there exists $1 \\le i_{0} < j_{0} \\le n$ such that $E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}} \\in I$ .",
"Assume that the conclusion holds when there are $k(k < \\frac{n(n - 1)}{2})$ nonzero elements in $S$ .",
"While there are $k + 1$ nonzero elements in $S$ , suppose that $a = \\sum ^{k + 1}_{l = 1}a_{i_{l}j_{l}}(E_{i_{l}j_{l}} - \\frac{\\alpha _{j_{l}}}{\\alpha _{i_{l}}}E_{j_{l}i_{l}})$ .",
"Write $b = \\sum ^{k}_{l = 1}a_{i_{l}j_{l}}(E_{i_{l}j_{l}} - \\frac{\\alpha _{j_{l}}}{\\alpha _{i_{l}}}E_{j_{l}i_{l}})$ .",
"Then $a = b + a_{i_{k + 1}j_{k + 1}}(E_{i_{k + 1}j_{k + 1}} - \\frac{\\alpha _{j_{k + 1}}}{\\alpha _{i_{k + 1}}}E_{j_{k + 1}i_{k + 1}})$ .",
"According to the assumption, $b$ becomes $p(E_{i^{^{\\prime }}j^{^{\\prime }}} - \\frac{\\alpha _{j^{^{\\prime }}}}{\\alpha _{i^{^{\\prime }}}}E_{j^{^{\\prime }}i^{^{\\prime }}})$ after finite times of Lie brackets where $p$ is a nonzero coefficient.",
"Meanwhile, $a_{i_{k + 1}j_{k + 1}}(E_{i_{k + 1}j_{k + 1}} - \\frac{\\alpha _{j_{k + 1}}}{\\alpha _{i_{k + 1}}}E_{j_{k + 1}i_{k + 1}})$ vanishes or becomes $q(E_{i^{^{\\prime \\prime }}j^{^{\\prime \\prime }}} - \\frac{\\alpha _{j^{^{\\prime \\prime }}}}{\\alpha _{i^{^{\\prime \\prime }}}}E_{j^{^{\\prime \\prime }}i^{^{\\prime \\prime }}})$ after the same finite times Lie brackets where $q$ is a nonzero coefficient.",
"Hence, $a$ becomes $p(E_{i^{^{\\prime }}j^{^{\\prime }}} - \\frac{\\alpha _{j^{^{\\prime }}}}{\\alpha _{i^{^{\\prime }}}}E_{j^{^{\\prime }}i^{^{\\prime }}})$ or $p(E_{i^{^{\\prime }}j^{^{\\prime }}} - \\frac{\\alpha _{j^{^{\\prime }}}}{\\alpha _{i^{^{\\prime }}}}E_{j^{^{\\prime }}i^{^{\\prime }}}) + q(E_{i^{^{\\prime \\prime }}j^{^{\\prime \\prime }}} - \\frac{\\alpha _{j^{^{\\prime \\prime }}}}{\\alpha _{i^{^{\\prime \\prime }}}}E_{j^{^{\\prime \\prime }}i^{^{\\prime \\prime }}})$ .",
"We have $p(E_{i^{^{\\prime }}j^{^{\\prime }}} - \\frac{\\alpha _{j^{^{\\prime }}}}{\\alpha _{i^{^{\\prime }}}}E_{j^{^{\\prime }}i^{^{\\prime }}}) \\in I$ or $p(E_{i^{^{\\prime }}j^{^{\\prime }}} - \\frac{\\alpha _{j^{^{\\prime }}}}{\\alpha _{i^{^{\\prime }}}}E_{j^{^{\\prime }}i^{^{\\prime }}}) + q(E_{i^{^{\\prime \\prime }}j^{^{\\prime \\prime }}} - \\frac{\\alpha _{j^{^{\\prime \\prime }}}}{\\alpha _{i^{^{\\prime \\prime }}}}E_{j^{^{\\prime \\prime }}i^{^{\\prime \\prime }}}) \\in I$ since $I$ is an ideal of $Der(J)$ .",
"According to the case of $k = 2$ , the conclusion holds.",
"Therefore, there exists $1 \\le i_{0} < j_{0} \\le n$ such that $E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}} \\in I$ .",
"$\\bf Step 2$ .",
"We'll show that $E_{n - 1, n} - \\frac{\\alpha _{n}}{\\alpha _{n - 1}}E_{n, n - 1} \\in I$ .",
"We have $[E_{i_{0}j_{0}} - \\frac{\\alpha _{j_{0}}}{\\alpha _{i_{0}}}E_{j_{0}i_{0}}, E_{j_{0}n} - \\frac{\\alpha _{n}}{\\alpha _{j_{0}}}E_{nj_{0}}] = E_{i_{0}n} - \\frac{\\alpha _{n}}{\\alpha _{i_{0}}}E_{ni_{0}} \\in I,$ if $i_{0} = n - 1$ , the conclusion is proved; if $i_{0} < n - 1$ , then $[E_{i_{0}n} - \\frac{\\alpha _{n}}{\\alpha _{i_{0}}}E_{ni_{0}}, E_{i_{0}n - 1} - \\frac{\\alpha _{n - 1}}{\\alpha _{i_{0}}}E_{n - 1, i_{0}}] = \\frac{\\alpha _{n - 1}}{\\alpha _{i_{0}}}(E_{n - 1, n} - \\frac{\\alpha _{n}}{\\alpha _{n - 1}}E_{n, n - 1}) \\in I$ .",
"$\\bf Step 3$ .",
"We'll show that $I = Der(J)$ .",
"Since $[E_{n - 1, n} - \\frac{\\alpha _{n}}{\\alpha _{n - 1}}E_{n, n - 1}, E_{k, n - 1} - \\frac{\\alpha _{n - 1}}{\\alpha _{k}}E_{n - 1, k}] = -(E_{kn} - \\frac{\\alpha _{n}}{\\alpha _{k}}E_{nk}) \\in I,(1 \\le k < n - 1),$ we have $E_{1n} - \\frac{\\alpha _{n}}{\\alpha _{1}}E_{n1}, \\cdots , E_{n - 1, n} - \\frac{\\alpha _{n}}{\\alpha _{n - 1}}E_{n, n - 1} \\in I$ .",
"Since $[E_{n - 1, n} - \\frac{\\alpha _{n}}{\\alpha _{n - 1}}E_{n, n - 1}, E_{kn} - \\frac{\\alpha _{n}}{\\alpha _{k}}E_{nk}] = \\frac{\\alpha _{n}}{\\alpha _{n - 1}}(E_{k,n - 1} - \\frac{\\alpha _{n - 1}}{\\alpha _{k}}E_{n - 1, k}) \\in I,(1 \\le k < n - 1),$ we have $E_{1, n - 1} - \\frac{\\alpha _{n - 1}}{\\alpha _{1}}E_{n - 1, 1}, \\cdots , E_{n - 2, n - 1} - \\frac{\\alpha _{n - 1}}{\\alpha _{n - 2}}E_{n - 1, n - 2} \\in I$ .",
"Repeat the process above, we can get all $\\lbrace E_{ij} - \\frac{\\alpha _{j}}{\\alpha _{i}}E_{ji}|1 \\le i < j \\le n\\rbrace \\in I$ , i.e., $Der(J) \\subseteq I$ , which implies that $Der(J) = I$ , contradiction.",
"Therefore, $Der(J)$ is simple.",
"When $n = 4$ , then $Der(J)$ has a basis $\\lbrace E_{12} - \\frac{\\alpha _{2}}{\\alpha _{1}}E_{21}, E_{13} - \\frac{\\alpha _{3}}{\\alpha _{1}}E_{31}, E_{14} - \\frac{\\alpha _{4}}{\\alpha _{1}}E_{41}, E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32}, E_{24} - \\frac{\\alpha _{4}}{\\alpha _{2}}E_{42}, E_{34} - \\frac{\\alpha _{4}}{\\alpha _{3}}E_{43}\\rbrace $ .",
"Set $I = \\lbrace -\\sqrt{\\frac{\\alpha _{4}\\alpha _{1}}{\\alpha _{2}\\alpha _{3}}}(E_{12} - \\frac{\\alpha _{2}}{\\alpha _{1}}E_{21}) + (E_{34} - \\frac{\\alpha _{4}}{\\alpha _{3}}E_{43}), \\sqrt{\\frac{\\alpha _{4}\\alpha _{1}}{\\alpha _{2}\\alpha _{3}}}(E_{13} - \\frac{\\alpha _{3}}{\\alpha _{1}}E_{31}) + (E_{24} - \\frac{\\alpha _{4}}{\\alpha _{2}}E_{42}), -\\sqrt{\\frac{\\alpha _{3}\\alpha _{1}}{\\alpha _{2}\\alpha _{4}}}(E_{14} - \\frac{\\alpha _{4}}{\\alpha _{1}}E_{41}) + (E_{23} - \\frac{\\alpha _{3}}{\\alpha _{2}}E_{32})\\rbrace $ .",
"It's easy to verify that $I$ is an ideal of $Der(J)$ .",
"Hence, When $dim(J) = 5$ , $Der(J)$ is not simple.",
"Lemma 2.9 [4] Let $\\rm {F}$ be an algebraically closed field of characteristic 0.",
"Then the derivation algebra of the Jordan algebra of type $D$ is the Lie algebra $F_{4}$ .",
"Theorem 2.10 The derivation algebra of the Jordan algebra of type $D$ over an algebraically closed field of characteristic 0 is simple.",
"According to Lemma REF , the derivation algebra of the Jordan algebra of type $D$ over an algebraically closed field of characteristic 0 is the Lie algebra $F_{4}$ .",
"Note that $F_{4}$ is simple, the proof is completed."
],
[
"Triple derivations of Jordan algebras",
"Definition 3.1 Let $J$ be a Jordan algebra over a field $\\rm {F}$ .",
"A linear map $D : J \\rightarrow J$ is called a triple derivation on $J$ if it satisfies $D((x \\circ y) \\circ z) = (D(x) \\circ y) \\circ z + (x \\circ D(y)) \\circ z + (x \\circ y) \\circ D(z),\\quad \\forall x, y, z \\in J.$ It's easy to verify that derivations are all triple derivations for Jordan algebras.",
"But the reverse is not always true.",
"One can see the following example.",
"Example 3.2 Let $J$ be a Jordan algebra with a basis $\\lbrace e_{1}, e_{2}\\rbrace $ .",
"And the multiplication table is $e_{1} \\circ e_{1} = e_{2} \\circ e_{2} = e_{1} + e_{2}, e_{1} \\circ e_{2} = e_{2} \\circ e_{1} = -e_{1} - e_{2}.$ Then $J$ is a nilpotent Jordan algebra since $J^{3} = (J \\circ J) \\circ J = 0$ .",
"Therefore any linear map on $J$ is a triple derivation on $J$ .",
"Take $D_{0} : J \\rightarrow J$ to be a linear map with $D_{0}(e_{1}) = D_{0}(e_{2}) = e_{1}.$ Then we have $D_{0}(e_{1} \\circ e_{1}) = D_{0}(e_{1} + e_{2}) = D_{0}(e_{1}) + D_{0}(e_{2}) = 2e_{1},$ $D_{0}(e_{1}) \\circ e_{1} + e_{1} \\circ D_{0}(e_{1}) = 2e_{1} \\circ e_{1} = 2(e_{1} + e_{2}),$ since $D_{0}(e_{1} \\circ e_{1}) \\ne D_{0}(e_{1}) \\circ e_{1} + e_{1} \\circ D_{0}(e_{1}),$ $D_{0}$ is not a derivation on $J$ .",
"Hence, $TDer(J) \\ne Der(J)$ .",
"Lemma 3.3 For any Jordan algebra $J$ , $TDer(J)$ is closed under the usual Lie bracket.",
"$\\forall D_{1}, D_{2} \\in TDer(J)$ , $\\forall x, y, z \\in J$ , $&[D_{1}, D_{2}]((x \\circ y) \\circ z) = (D_{1}D_{2} - D_{2}D_{1})((x \\circ y) \\circ z)\\\\&= D_{1}D_{2}((x \\circ y) \\circ z) - D_{2}D_{1}((x \\circ y) \\circ z)\\\\&= D_{1}((D_{2}(x) \\circ y) \\circ z + (x \\circ D_{2}(y)) \\circ z + (x \\circ y) \\circ D_{2}(z)) - D_{2}((D_{1}(x) \\circ y) \\circ z +\\\\&(x \\circ D_{1}(y)) \\circ z + (x \\circ y) \\circ D_{1}(z))\\\\&= (D_{1}D_{2}(x) \\circ y) \\circ z + (D_{2}(x) \\circ D_{1}(y)) \\circ z + (D_{2}(x) \\circ y) \\circ D_{1}(z)\\\\&+ (D_{1}(x) \\circ D_{2}(y)) \\circ z + (x \\circ D_{1}D_{2}(y)) \\circ z + (x \\circ D_{2}(y)) \\circ D_{1}(z)\\\\&+ (D_{1}(x) \\circ y) \\circ D_{2}(z) + (x \\circ D_{1}(y)) \\circ D_{2}(z) + (x \\circ y) \\circ D_{1}D_{2}(z)\\\\&- (D_{2}D_{1}(x) \\circ y) \\circ z - (D_{1}(x) \\circ D_{2}(y)) \\circ z - (D_{1}(x) \\circ y) \\circ D_{2}(z)\\\\&- (D_{2}(x) \\circ D_{1}(y)) \\circ z - (x \\circ D_{2}D_{1}(y)) \\circ z - (x \\circ D_{1}(y)) \\circ D_{2}(z)\\\\&- (D_{2}(x) \\circ y) \\circ D_{1}(z) - (x \\circ D_{2}(y)) \\circ D_{1}(z) - (x \\circ y) \\circ D_{2}D_{1}(z)\\\\&= ([D_{1}, D_{2}](x) \\circ y) \\circ z + (x \\circ [D_{1}, D_{2}](y)) \\circ z + (x \\circ y) \\circ [D_{1}, D_{2}](z).$ Hence, $[D_{1}, D_{2}] \\in TDer(J)$ .",
"The lemma is proved.",
"Theorem 3.4 Suppose that $J$ is a Jordan algebra with unit 1 and $D : J \\rightarrow J$ is a triple derivation.",
"Then $D$ is a derivation on $J$ in the case the characteristic of the field $\\rm {F}$ is not 2.",
"Moreover, we have $TDer(J) = Der(J)$ .",
"$\\forall D \\in TDer(J)$ , we have $D(1) = D((1 \\circ 1) \\circ 1) = (D(1) \\circ 1) \\circ 1 + (1 \\circ D(1)) \\circ 1 + (1 \\circ 1) \\circ D(1) = 3D(1).$ Note that $char \\rm {F} \\ne 2$ , we have $D(1) = 0$ .",
"For all $x, y \\in J$ , $&D(x \\circ y) = D((x \\circ y) \\circ 1) = (D(x) \\circ y) \\circ 1 + (x \\circ D(y)) \\circ 1 + (x \\circ y) \\circ D(1)\\\\&= D(x) \\circ y + x \\circ D(y),$ which implies that $D \\in Der(J)$ .",
"Hence, we have $TDer(J) \\subseteq Der(J)$ .",
"Therefore, we have $TDer(J) = Der(J)$ .",
"Lemma 3.5 [1] Let $J$ be a semi-simple Jordan algebra over a field $\\rm {F}$ of characteristic 0.",
"Then $J$ has the unique unit.",
"Lemma 3.6 [7] Every derivation of a semi-simple Jordan algebra with a finite basis over a field of characteristic 0 is inner.",
"According to Lemma REF and REF , we get a corollary with respect to Theorem REF .",
"Corollary 3.7 Suppose that $J$ is a semi-simple Jordan algebra with a finite basis over a field characteristic 0.",
"Then $TDer(J) = Der(J) = Inn(J)$ ."
],
[
"Triple derivations of the derivation algebras of Jordan algebras",
"Definition 4.1 [18] A linear map $D: L \\rightarrow L$ where $L$ is a Lie algebra, is called a triple derivation on $L$ if it satisfies $D([[x, y], z]) = [[D(x), y], z] + [[x, D(y)], z] + [[x, y], D(z)],\\quad \\forall x, y, z \\in L.$ Lemma 4.2 [18] Let $L$ be a Lie algebra over commutative ring $\\rm {R}$ .",
"If $\\frac{1}{2} \\in \\rm {R}$ , $L$ is perfect and has zero center, then we have that: $TDer(L) = Der(L)$ ; $TDer(Der(L)) = ad(Der(L))$ .",
"Lemma 4.3 [3] Any derivation of a semi-simple Lie algebra $L$ over a field of characteristic of 0 is inner.",
"According to Lemma REF and REF , we have the following theorem.",
"Theorem 4.4 Let $J$ be a simple Jordan algebra of type $A$ or $B$ where $L/Z(L)$ in Theorem REF is simple or a simple Jordan algebra of type $C$ whose dimension isn't 5 or a simple Jordan algebra of type $D$ .",
"Then $TDer(Der(J)) = Der(Der(J)) = ad(Der(J))$ .",
"Lemma 4.5 [14] Suppose that $L$ is a Lie algebra and has a decomposition $L = L_{1} \\oplus L_{2}$ , where $L_{1}$ , $L_{2}$ are ideals of $L$ .",
"Then $C(L) = C(L_{1}) \\oplus C(L_{2})$ ; If $C(L) = \\lbrace 0\\rbrace $ , then $Der(L) = Der(L_{1}) \\oplus Der(L_{2})$ and $ad(L) = ad(L_{1}) \\oplus ad(L_{2})$ .",
"Lemma 4.6 Suppose that $L$ is a Lie algebra and has a decomposition $L = L_{1} \\oplus L_{2}$ , where $L_{1}$ , $L_{2}$ are ideals of $L$ .",
"If $C(L) = \\lbrace 0\\rbrace $ , then $TDer(L) = TDer(L_{1}) \\oplus TDer(L_{2})$ .",
"$\\bf {Step 1}$ .",
"We'll show that for $i = 1, 2$ , $D(L_{i}) \\subseteq L_{i},\\;\\forall D \\in TDer(L)$ .",
"Suppose that $x_{i} \\in L_{i}(i = 1, 2)$ .",
"For any $z \\in L$ , we have $D([[x_{1}, x_{2}], z]) = [[D(x_{1}), x_{2}], z] + [[x_{1}, D(x_{2})], z] + [[x_{1}, x_{2}], D(z)],$ which implies that $[[D(x_{1}), x_{2}], z] + [[x_{1}, D(x_{2})], z] = 0,$ i.e, $[[D(x_{1}), x_{2}], z] = -[[x_{1}, D(x_{2})], z] \\in L_{1} \\cap L_{2} = 0.$ Hence, $[D(x_{1}), x_{2}] \\in C(L), [x_{1}, D(x_{2})] \\in C(L).$ Note that $C(L) = \\lbrace 0\\rbrace $ , we have $[D(x_{1}), x_{2}] = [x_{1}, D(x_{2})] = 0.$ Suppose that $D(x_{1}) = u_{1} + u_{2}$ where $u_{i} \\in L_{i}(i = 1, 2)$ .",
"Then we have $[u_{2}, x_{2}] = [u_{1} + u_{2}, x_{2}] = [D(x_{1}), x_{2}] = 0,$ which implies that $u_{2} \\in C(L_{2})$ .",
"According to Lemma REF , $C(L_{i}) = \\lbrace 0\\rbrace (i = 1, 2)$ .",
"Therefore $u_{2} = 0$ , which is to say $D(L_{1}) \\subseteq L_{1}$ .",
"Similarly, we have $D(L_{2}) \\subseteq L_{2}$ .",
"$\\bf {Step 2}$ .",
"we'll show that $TDer(L_{1}) \\dotplus TDer(L_{2}) \\subseteq TDer(L)$ .",
"For any $D \\in TDer(L_{1})$ , we extend it to a linear map on $L$ as follow $D(x_{1} + x_{2}) = x_{1},\\quad \\forall x_{1} \\in L_{1}, x_{2} \\in L_{2}.$ Then for any $x, y, z \\in L$ , suppose that $x = x_{1} + x_{2}$ , $y = y_{1} + y_{2}$ , $z = z_{1} + z_{2}$ , where $x_{1}, y_{1}, z_{1} \\in L_{1}$ , $x_{2}, y_{2}, z_{2} \\in L_{2}$ , we have $D([[x, y], z]) = D([[x_{1} + x_{2}, y_{1} + y_{2}], z_{1} + z_{2}]) = D([[x_{1}, y_{1}], z_{1}] + [[x_{2}, y_{2}], z_{2}]) = D([[x_{1}, y_{1}], z_{1}]),$ $&[[D(x), y], z] + [[x, D(y)], z] + [[x, y], D(z)]\\\\&= [[D(x_{1} + x_{2}), y_{1} + y_{2}], z_{1} + z_{2}] + [[x_{1} + x_{2}, D(y_{1} + y_{2})], z_{1} + z_{2}] + [[x_{1} + x_{2}, y_{1} + y_{2}],\\\\& D(z_{1} + z_{2})]\\\\&= [[D(x_{1}), y_{1}], z_{1}] + [[x_{1}, D(y_{1})], z_{1}] + [[x_{1}, y_{1}], D(z_{1})].$ Since $D \\in TDer(L_{1})$ , we have $D([[x_{1}, y_{1}], z_{1}]) = [[D(x_{1}), y_{1}], z_{1}] + [[x_{1}, D(y_{1})], z_{1}] + [[x_{1}, y_{1}], D(z_{1})].$ Hence, $D([[x, y], z]) = [[D(x), y], z] + [[x, D(y)], z] + [[x, y], D(z)],$ which implies that $D \\in TDer(L)$ , i.e, $TDer(L_{1}) \\subseteq TDer(L)$ .",
"Moreover, $D \\in TDer(L_{1})$ if and only if $D(x_{2}) = 0,\\; \\forall x_{2} \\in L_{2}$ .",
"Similarly, we have $TDer(L_{2}) \\subseteq TDer(L)$ and $D \\in TDer(L_{2})$ if and only if $D(x_{1}) = 0,\\; \\forall x_{1} \\in L_{1}$ .",
"Then we have $TDer(L_{1}) + TDer(L_{2}) \\subseteq TDer(L)$ and $TDer(L_{1}) \\cap TDer(L_{2}) = \\lbrace 0\\rbrace $ .",
"Hence, $TDer(L_{1}) \\dotplus TDer(L_{2}) \\subseteq TDer(L)$ .",
"$\\bf {Step 3}$ .",
"We'll prove that $TDer(L_{1}) \\dotplus TDer(L_{2}) = TDer(L)$ .",
"Suppose that $D \\in TDer(L)$ .",
"Set $x = x_{1} + x_{2}, x_{i} \\in L_{i}$ .",
"Define $D_{1}, D_{2}$ as follows $\\left\\lbrace \\begin{aligned}D_{1}(x_{1} + x_{2}) = D(x_{1}),\\\\D_{2}(x_{1} + x_{2}) = D(x_{2}).\\end{aligned}\\right.$ Obviously, $D = D_{1} + D_{2}$ .",
"For any $u_{1}, v_{1}, w_{1} \\in L_{1}$ , $&D_{1}([[u_{1}, v_{1}], w_{1}]) = D([[u_{1}, v_{1}], w_{1}]) = [[D(u_{1}), v_{1}], w_{1}] + [[u_{1}, D(v_{1})], w_{1}] + [[u_{1}, v_{1}], D(w_{1})]\\\\&= [[D_{1}(u_{1}), v_{1}], w_{1}] + [[u_{1}, D_{1}(v_{1})], w_{1}] + [[u_{1}, v_{1}], D_{1}(w_{1})].$ Hence, $D_{1} \\in TDer(L_{1})$ .",
"Similarly, $D_{2} \\in TDer(L_{2})$ .",
"Therefore, $TDer(L) = TDer(L_{1}) \\dotplus TDer(L_{2})$ .",
"$\\bf {Step 4}$ .",
"We'll show that $TDer(L_{i}) \\lhd TDer(L)$ .",
"Suppose that $D_{1} \\in TDer(L_{1}), D \\in TDer(L), x_{2} \\in L_{2}$ .",
"$[D, D_{1}](x_{2}) = DD_{1}(x_{2}) - D_{1}D(x_{2}) = 0,$ which implies that $[D, D_{1}] \\in TDer(L_{1})$ , i.e., $TDer(L_{1}) \\lhd TDer(L)$ .",
"Similarly, $TDer(L_{2}) \\lhd TDer(L)$ .",
"Therefore, we have $TDer(L) = TDer(L_{1}) \\oplus TDer(L_{2})$ .",
"Theorem 4.7 Suppose that $J$ is a centerless semi-simple Jordan algebra over a field of characteristic 0 and $J$ has the decomposition $J = \\oplus ^{s}_{i = 1}J_{i}$ where $J_{i}(1 \\le i \\le s)$ are simple Jordan algebras over a field of characteristic 0 satisfying $Der(J_{i})$ is simple.",
"Then $TDer(Der(J)) = Der(Der(J)) = ad(Der(J))$ .",
"According to Theorem REF , we have $Der(J) = \\oplus ^{s}_{i = 1}Der(J_{i})$ and $Der(J_{i}) \\triangleleft Der(J)$ .",
"By Lemma REF , we have $C(Der(J)) = \\oplus ^{s}_{i = 1}C(Der(J_{i}))$ .",
"Since $Der(J_{i})$ is simple, we have $C(Der(J_{i})) = \\lbrace 0\\rbrace $ .",
"Hence, $C(Der(J)) = \\lbrace 0\\rbrace $ .",
"Therefore, we have $TDer(Der(J)) = \\oplus ^{s}_{i = 1}TDer(Der(J_{i})),$ $Der(Der(J)) = \\oplus ^{s}_{i = 1}Der(Der(J_{i})),$ $ad(Der(J)) = \\oplus ^{s}_{i = 1}ad(Der(J_{i})),$ via Lemma REF and Lemma REF .",
"According to Theorem REF , we have $TDer(Der(J_{i})) = Der(Der(J_{i})) = ad(Der(J_{i})),\\quad \\forall 1 \\le i \\le s.$ Hence, we have $TDer(Der(J)) = Der(Der(J)) = ad(Der(J)).$ Remark 4.8 Let $J$ be a simple Jordan algebra of type $C$ .",
"Obviously, $J$ is semi-simple.",
"But $C(J) \\ne \\lbrace 0\\rbrace $ since $1 \\in C(J)$ ."
]
] | 1906.04552 |
[
[
"Exploring universality in neutron star mergers"
],
[
"Abstract We explore the correlation between the pre-merger tidal deformability and the post-merger remnant oscillations seen in numerical simulation of neutron star binaries, with the aim of understanding to what extent the physics support the existence of such a relation.",
"We consider the impact of thermal and rotational effects and argue that the proposed relation does, indeed, make sense and provide simple arguments that help explain the result."
],
[
"Introduction",
"Gravitational waves from the inspiral and merger of binary neutron stars encode the extreme physics of these objects.",
"From the inspiral phase one may extract the faint imprint of the tidal interaction while the violent merger involves the oscillations of the hot remnant that may eventually collapse to form a black hole.",
"The first aspect has already been demonstrated in the case of the spectacular GW170817 signal [2].",
"The contribution from the subsequent merger, which leaves a higher frequency signature, was however buried in the instrument noise in this instance [1], [3].",
"Intuitively, there is no reason why these distinct phases of the gravitational-wave signal should be correlated.",
"Yet, numerical simulations suggest we should expect the unexpected.",
"A typical neutron star merger leads to the formation of a massive, $3-4 M_{\\odot }$ , remnant (often referred to as a Hyper Massive Neutron Star, HMNS), which collapses to a black hole (on a timescale of 100 ms) after shedding the angular momentum that (temporarily) counteracts gravity [22].",
"Numerical simulations demonstrate that the dynamics of the HMNS have robust high-frequency features [8], [21], thought to be associated with the modes of oscillation of the remnant.",
"This would be natural, although the problem is less straightforward than the corresponding one for a cold neutron star (see for example [12] or [11]).",
"In that case one would be considering perturbations with respect to a long-term stable background.",
"Meanwhile, in the case of a HMNS one would have to explore perturbations relative to a background that evolves (and eventually collapses) on a relatively short timescale.",
"The intuitive picture makes sense, but we do not (at least not yet) have precise mode-calculations to test simulations (and eventually observations) against.",
"Numerical simulations have demonstrated the existence of useful phenomenological relations linking the post-merger oscillation frequencies to the matter equation of state [7], [21].",
"This is important as it means that observations could eventually help us get a handle on problematic physics associated with hot high-density matter.",
"This information would complement information gleaned from the inspiral phase (e.g.",
"in terms of the tidal deformability, see [15] and [19]), which relates to cold supranuclear matter.",
"However, it is not clear to what extent the information from inspiral and merger is (at the end of the day) independent.",
"Formally, the underlying equation of state should (obviously) be the same (involving identical many-body interactions etcetera) but one might expect thermal and rotational effects to have decisive impact on the HMNS.",
"Given this, it is interesting to note that the tidal deformability (usually encoded in a mass-weighted combination of the so-called Love numbers of the individual stars, $\\kappa _2^t$ in the following [15]) appears to be linked to the dominant oscillation frequency ($f_2$ ) of the post-merger remnant, see in particular [9].",
"The relation appears to be robust, perhaps hinting at some underlying universality, and may provide a useful constraint on the inferred physics.",
"Of course, before we make use of this information in either a data analysis algorithm or a parameter extraction effort we need to understand why the relation should hold.",
"At first sight it seems peculiar.",
"Why should the properties of the (cold, slowly spinning) inspiralling neutron stars be related to the oscillations of the (hot, differentially rotating) remnant?",
"This is the question we (try to) address in the following."
],
[
"The implied universality",
"In the last few years it has become clear that many neutron star properties are related through universal relations.",
"In fact, since the late 1990s, this observation has provided a foundation for discussions of neutron star asteroseismology [5], which aims to use observed oscillation frequencies to infer mass and radius (and hence constrain the equation of state) for individual stars.",
"More recently, the so-called I-Love-Q relations [24] demonstrate a link between the moment of inertia, the Love number and the quadrupole moment that helps break degeneracies in gravitational-waveform modelling.",
"Finally, one may link the f-mode frequency of a given star to the tidal deformability [10].",
"Since these relations have been demonstrated to be accurate to within a few percent (at least as long as the equation of state does not involve sharp phase transitions, see [16]) it makes sense to take them as our starting point.",
"Labelling the binary partners $a$ and $b$ , with the mass ratio $q = {M_b}/{M_a} \\le 1$ , the effective tidal parameter used by, for example, [9] is given by $\\kappa _2^t = 2\\left[ q \\left(\\frac{X_a}{C_a}\\right)^5 k_2^a+\\frac{1}{q} \\left(\\frac{X_b}{C_b}\\right)^5k_2^b \\right]$ where $C_a=M_a/R_a$ is the compactness of each star, $X_a= {M_a}/{(M_a+M_b)}$ while $k_2^a$ is the quadrupole Love number (and similarly for star $b$ ).",
"For simplicity, let us consider a binary system with two non-spinning equal-mass partners.",
"Then we have $\\kappa _2^t = \\frac{1}{8} \\frac{k_2}{C^5} = {3\\over 16} \\Lambda $ where $\\Lambda $ is commonly used to quantify the tidal deformability.",
"If, in addition, we note that $k_2$ is only weakly dependent on the compactness, we expect the scaling $\\kappa _2^t \\sim C^{-5}$ Meanwhile, we know that (for non-spinning stars) the fundamental mode frequency scales (roughly) as the average density (see for example [5]).",
"That is, we have $Mf_2 \\sim M\\bar{\\rho }^{1/2} = C^{3/2}$ In essence, for a cold neutron star we expect to have $ Mf_2 \\sim C^{3/2} \\sim \\left(\\kappa _2^t\\right)^{-3/10}$ Figure: Power-law fit to the Mf 2 -κ 2 t Mf_2-\\kappa _2^t relation for cold neutron stars from , demonstrating that we can reliably base the discussion on the simpler scaling from ().This is, of course, only a rough indication.",
"A more precise relation between the f-mode frequency and the tidal deformability has been obtained by [10], linking the dimensionless frequency (in geometric units) $\\omega M$ to an expansion in $\\ln \\Lambda $ .",
"This relation is a little bit too complicated for our purposes, but it is easy to demonstrate that it can be replaced by the power-law $Mf_2 \\approx 0.031 \\left(\\kappa _2^t\\right)^{-0.218} \\ .$ Moreover, as is evident from figure REF , this is is close to the $-3/10$ power law suggested by our simple argument.",
"Basically, the origin of the scaling for cold neutron stars is well understood.",
"Figure: The top panel shows the suggested -3/10-3/10 power-law fit to the post-merger f-modes inferred from simulations for a set of equations of state (solid line, data provided by S. Bernuzzi).",
"Also indicated (as a dashed line) is the fit to the f-mode of the individual (cold) pre-merger neutron stars from .",
"The bottom panel illustrates the fractional increase in the f-mode frequency required to explain the scaling of the post-merger remnant oscillations (assuming that no mass is lost during the merger).",
"This factor is seen lie in the range 3-43-4.This is not the scaling relation we are interested in, but it is easy to show that the oscillations of the remnant are well represented by a power law, as well (see [22] and [21]).",
"Using data from a range of simulations (by different groups) we infer the scaling (see figure REF ) $M_tf_2^h \\approx 0.144 (\\kappa _2^t)^{-0.278},$ where $M_t$ is the total mass of the system (assuming that we can neglect mass shedded during the merger) and $f_2^h$ represents the hot post-merger f-mode.",
"Notably, the post-merger f-modes follow almost exactly the same scaling law as the cold f-mode of the individual pre-merger stars.",
"This is the behaviour we are trying to explain.",
"The problem breaks down to two questions.",
"First of all, we need to understand why “the same” scaling with the tidal parameter should apply for the oscillations of cold neutron stars and hot, more massive and differentially rotating, merger remnants.",
"Secondly, we note that the scalings (REF ) and (REF ) imply that $\\left(M_t f_2^h\\right) \\approx \\beta \\left(Mf_2\\right)$ with $\\beta \\approx 3-4$ (see figure REF ).",
"We need (at least at the qualitative level) to understand how the merger physics impacts on this numerical factor."
],
[
"Thermal Effects",
"Just after the merger, the HMNS can reach temperatures as high as 85 MeV [17], [18], making the remnant hotter than the collapsing core of a supernova furnace.",
"Given this, one would expect thermal effect to come into play and we need to explore the implicates for the f-mode oscillations.",
"However, the problem is complicated by the fact that the system is evolving – we are not dealing with a “thermalized” equlibrium background with respect to which we can define a mode perturbation.",
"Still, we can make progress with a simple argument.",
"As we have already pointed out, the f-modes are known to be determined by the average density of the body in question.",
"For the HMNS, this is tricky, as $\\bar{\\rho }$ (or equivalently, as the mass is known, the compactness $C$ ) of the collapsing matter is explicitly dependent on time.",
"However, as the suggested collapse timescale ($\\sim 0.1$ s) is about two orders of magnitude larger than the f-mode oscillation timescale (typically $\\sim 1$ ms) we can expect the HMNS density to be roughly constant on the timescale of a few oscillations.",
"This is all we need to progress, as it allows us to simplify the discussion by considering a single “neutron star”.",
"However, it is not quite enough to make the argument quantitative.",
"After all, stable solutions can never reach masses of $3-4M_{\\odot }$ .",
"This likely requires both support of thermal pressure and differential rotation (see later).",
"Nevertheless, we may be able to gain some insight into the thermal effects from a simple “surrogate” model.",
"Basically, we should be able to estimate the effect on the f-mode as the star becomes bloated due to thermal pressure, causing a relative change in the average density and the compactness.",
"This effect has, in fact, already been studied for proto-neutron stars [14].",
"The main difference here is that we consider a model inspired by the actual temperature distribution inside a HMNS.",
"The thermal profile of a HMNS is not uniform.",
"In essence, the hottest regions arise from heating due to shocks as the stars come into contact.",
"The remnant is not able to to reach thermal equilibrium on the timescale we are interested in.",
"Instead, the heat is advected along with the matter, leading to a nonuniform temperature distribution, see for example the results of [17].",
"In general, the thermal distribution has no obvious symmetry.",
"However, for simplicity we will assume that it has a simple radial profile.",
"Motivated by Figure 6 of [17] we assume the temperature profile to be such that: - thermal effects can be ignored up to about a 5 km distance from centre, simply because the pressure of the cold high-density matter dominates in this region, - the temperature reaches 50 MeV temperature in the region between 5-10 km, - the temperature drops to about 10 MeV temperature in the region between 10-15 km, - we (again) ignore thermal effects beyond a radius of 15 km as the temperature (indicated by simulations) is close to zero (and this matter is likely to be gravitationally unbound, anyway).",
"With this model as our starting point, we can easily calculate the relative compactness of hot and cold models.",
"This, in turn, provides some insight into the thermal impact on the post-merger oscillations.",
"In effect, the post-merger (hot) compactness can be related to the pre-merger (cold) compactness which then relates to the tidal deformability in terms of $\\kappa _2^t$ .",
"In order to estimate the magnitude of this effect, which obviously leads to different fluid configurations, we need some idea of how the compactness of the merger remnant differs from that of the original cold neutron star.",
"This is not at all trivial, but we may try to encode the change in compactness in terms of a nearly constant factor, $F(\\theta ,\\phi )$ .",
"Then we argue that we ought to have $F\\le 2$ , following from taking the (equatorial) radius at the point when the stars first touch.",
"Meanwhile, numerical simulations show that we should have $F \\ge 1$ .",
"For simplicity, we ignore the $\\theta ,\\phi $ dependence which relates to the deviation from spherical symmetry.",
"If we now imagine a hypothetical postmerger compactness, $C_{hyp}$ , we have $C_{hyp} = F(\\theta ,\\phi ) C$ However, we still need to quantify the thermal effects.",
"Figure: Illustrating the relation between the “hot” compactness C h C^h (for our simple thermal surrogate model) and the cold compactness CC for for four different equations of state (EoS).Results from such an exercise are shown in figure REF , which shows the relation between the “hot” compactness and the corresponding cold model, based on the thermal surrogate and four different EoSs.",
"It is notable that the relationship is close to linear for a range of parameter values.",
"This is true for all models we have considered.",
"Since the thermal surrogate ought to capture the relative behaviour, we therefore expect the postmerger compactness $C^h$ to scale linearly with $C_{hyp}$ .",
"That is, we have $C^h = \\alpha C_{hyp} \\quad \\alpha <1$ Combining (REF ) and (REF ), we finally obtain $C^h = (\\alpha F) C ,$ where the pre-factors are expected to be roughly constant.",
"This is an important conclusion.",
"It is now apparent why the thermal pressure would not affect the power-law from (REF ).",
"Moreover, we can estimate the quantitative effect on the f-mode frequency.",
"The results in figure REF suggest that $\\alpha \\approx 0.6-0.9$ , leading to the estimated range $F\\approx 1.5-1.7$ .",
"In essence, we expect the post-merger compactness to be $1 -1.5$ times the pre-merger value.",
"The post-merger frequencies would then increase by factor of $(1-1.5)^{3/2}$ or $\\approx (1 - 2)$ because of the thermal effects, which takes us some way towards explaining the missing factor indicated by the results in figure REF .",
"However, we still seem to be short a factor of 2 or so.",
"Before we move on to consider whether rotation may provide the missing factor, let us consider a related question.",
"In a series of papers, see for example [6] and [8], it has been argued that the f-mode frequency of the HMNS scales with the radius of an isolated neutron star with mass $1.6M_{\\odot }$ .",
"This scaling is interesting because it is not at all obvious why the f-mode of the hot, differentially rotating, star should depend on the radius of an isolated neutron star with a particular mass.",
"Further, the scaling seems to be insensitive to the mass of the HMNS.",
"Motivated by this, we break down the argument for the observed scaling of $M_tf_2^h$ with the isolated neutron star mass into three steps.",
"First of all, given a particular remnant there must exist a variation of the radius of the remnant across different EoS which is unaffected by the mass of the remnant.",
"If this is the case, we can calculate the mass of an isolated NS that reproduces this variation in radius.",
"Finally, since $M_tf_2^h \\sim \\left(C^h\\right)^{-3/2} \\sim \\left(R^h\\right)^{-3/2}$ the f-mode should scale with the radius of the isolated neutron star for which the radius varies with EoS in the same way as $R^h$ .",
"Figure: Variation of the “hot” neutron star radius for different EoS (for the thermal surrogate model) and three different masses.",
"The fits show power laws that best capture the variation, having first of all ordered the equations of state in order of increasing radius (for stars of mass 1.5M ⊙ 1.5M_\\odot ) and then compared the scaling (in terms of a fiducial equation of state parameter xx) for different masses.",
"It is notable these power laws are similar (∼1.7\\sim 1.7) for masses in the range 1.5-1.7M ⊙ 1.5-1.7M_\\odot .Once again we turn to the thermal surrogate model for an answer.",
"Figure REF shows the variation of the radius of the surrogate models for three different masses and different EoS.",
"As before, we emphasize that the surrogate model will not return to “actual” value of the radius, but we expect that with the EoS will be meaningfully captured.",
"It is then clear from figure REF that the variation of radii across EoS closely follows a 1.7 power law.",
"This completes the first item on our list.",
"This moves us on to the issue of determining a mass value for an isolated neutron star showing the same variation.",
"This solution to this problem is provided in figure REF .",
"The results suggest that the f-modes should scale with the radius of a $1.45 M_\\odot $ neutron star, not too different from the $1.6 M_\\odot $ scaling of [8] (especially if we consider that we are using a fairly crude argument).",
"Figure: Variation of radius with EoS for 1.45M ⊙ 1.45 M_{\\odot } isolated neutron star models.",
"The indicated power-law of ≈1.7\\approx 1.7 is that required by the results in figure ."
],
[
"Rotational effects",
"Let us turn to the role of rotation.",
"As in the thermal case, it is easy to argue that the rapid rotation of the HMNS will have decisive impact on the f-mode oscillation frequencies.",
"In fact, in this case we have better quantitative evidence.",
"We may, for example, draw on the results from [13], which show how the f-mode of an isolated NS changes with the angular frequency $\\Omega $ .",
"The key point is that there exists robust phenomenological relations drawn from a collection of EoS.",
"These support the notion that the overall scaling with rotation is likely to be insensitive to the EoS, as required to explain the results in figure REF .",
"However, we have to be a bit careful as the results assume uniform rotation, while we know that the HMNS will rotate differentially.",
"At the same time, the typical rotation profile inferred from simulations [23] has the high-density core rotating close to uniformly.",
"The argument may be strengthened by mode calculations for the appropriate HMNS differential rotation law, but such work has not yet been carried out.",
"With this caveat in mind, let us piece together the argument from the results from [13].",
"In order to do this, we need some rough idea of the spin of the HMNS.",
"This is relatively straightforward.",
"Assuming that the individual stars in the binary are slowly rotating (which makes sense if the system is old enough that the stars have had time to spin down due to dipole emission, which is likely), the angular velocity of the HMNS should arise from the total angular momentum of the system at the innermost stable circular orbit.",
"Conservation of angular momentum then allows us to estimate the HMNS rotation rate.",
"This rough estimate agrees fairly well with the results inferred from simulations, which lead to a dimensionless rotation parameter $\\frac{a}{M} = \\frac{J}{M^2} = \\tilde{I} \\left(\\frac{R}{M}\\right)^2 (M\\Omega ) \\sim (0.75-0.8)$ We can turn this into an estimate for the (here assumed uniform) rotation via the scaling for the dimensionless moment of inertia, $\\bar{I}$ , from [20] $\\tilde{I} = 0.237\\left[ 1 + 4.2\\frac{M/M_{\\odot }}{R/\\mathrm {km}} + 90 \\left(\\frac{M/M_{\\odot }}{R/\\mathrm {km}}\\right)^4 \\right]$ Working out the rotation rate from these relations for (say) a remnant mass, $2.6 M_{\\odot }$ with 15 km radius, we arrive at $\\Omega \\approx 9.5\\times 10^3\\ \\mathrm {s}^{-1}$ .",
"We need to compare this to the expected break-up frequency, $\\Omega _K$ , which is approximated as ${1\\over 2\\pi } \\Omega _K\\ [\\mathrm {kHz}] \\approx 1.716 \\left( {M_0 \\over 1.4M_\\odot } \\right)^{1/2} \\left( {R_0\\over 10\\ \\mathrm {km}}\\right)^{-3/2} - 0.189$ for the models considered in [13] (with $M_0$ and $R_0$ the mass and radius of the corresponding non-rotating model, respectively).",
"Naively using this estimate for our suggested HMNS parameters, we would have $\\Omega _K \\approx 2.6\\times 10^4\\ \\mathrm {s}^{-1}$ .",
"Taking these estimates at face value, the HMNS would rotate at just below half of the Kepler rate.",
"Armed with this (rough) estimate, let us turn to the f-mode frequency.",
"Intuitively one would expect the f-mode that co-rotates with the orbit to be the one that is excited by the merger dynamics simply because this mode most closely resembles the configuration when the two stars come into contact.",
"With the conventions from [13], we are then considering the stable $l=-m=2$ mode and, in a frame rotating with the star, we have $\\sigma _r \\approx \\sigma _0 \\left[ 1 - 0.235 \\left( {\\Omega \\over \\Omega _K}\\right) -0.358 \\left( {\\Omega \\over \\Omega _K}\\right)^2 \\right]$ However, this needs to be translated into the inertial frame (where the gravitational-wave signal is measured).",
"Using $\\sigma _i = \\sigma _r - m\\Omega $ we obtain $\\sigma _i = \\sigma _0 \\left[ 1 - 0.235 \\left( {\\Omega \\over \\Omega _K}\\right)-\\frac{m\\Omega }{\\sigma _0} -0.358 \\left( {\\Omega \\over \\Omega _K}\\right)^2 \\right]$ where $\\sigma _0$ is the f-mode frequency of a non-rotating star.",
"This is estimated as ${1\\over 2\\pi } \\sigma _0 [\\mathrm {kHz}] \\approx 1.562 + 1.151 \\left( {M_0 \\over 1.4M_\\odot } \\right)^{1/2} \\left( {R_0\\over 10\\ \\mathrm {km}}\\right)^{-3/2}$ which for our fiducial parameters returns $\\sigma _0 \\approx \\Omega _K$ .",
"Taking $\\Omega /\\Omega _K\\approx 0.5$ we then have $\\sigma _i \\approx 1.8 \\sigma _0$ We thus arrive at a back-of-the-envelope idea of how much the f-mode changes due to rotation, compared to a rotating model.",
"Basically, we estimate that rotation would take us another factor of almost 2 towards explaining the results in figure REF ."
],
[
"Discussion and Concluding Remarks",
"Using simple approximations, we have tried to explain universal behaviour seen in simulations of neutron star mergers.",
"The basic premise was that we wanted to demonstrate the intuitive association of the dominant oscillation seen in merger simulations with the fundamental oscillation mode of the HMNS.",
"In order to argue the case, we compared the inferred f-modes for hot, differentially rotating post-merger HMNSs to robust scalings relevant for cold NSs.",
"We also made use of phenomenological relations for the oscillations of rapidly (and uniformly) rotating neutron stars.",
"Very roughly, our estimates suggest that thermal effects should increase the f-mode frequency by a factor of (up to) 2, compared to the cold neutron star f-mode.",
"Similarly, rotation would introduce another factor of 2.",
"Combining the effects, the post-merger frequencies should lie a factor 3-4 above the frequency of the individual pre-merger stars.",
"While admittedly simplistic, this estimate allows us to connect the observed frequency HMNS to the (much more easily calculated) f-modes of cold single neutron star.",
"This would “explain” why the features seen in simulations are found to be robust.",
"The arguments we have provided are obviously at the level of back-of-the-envelope estimates.",
"Nevertheless, the exercise opens the door for future possibilities.",
"First, one should be able to extend the rotational estimates to the realistic case of differential rotation (see, for example, [23] for the differential rotation profiles expected for post-merger remnants).",
"This would be an important step as it should quantify the rotational effects.",
"Moreover, given the available computational technology [13], such results should be within reach.",
"The issue of thermal effects (and potential phase transitions, [16]) is more complex, as aspects related to heat (entropy) tend to be treated in a somewhat ad hoc manner in many numerical simulations.",
"Ultimately, one would like demonstrate that the observed relation between tidal deformability and post-merger dynamics is real and not an artefact due to (for example) a “simplified” treatment of the physics.",
"There is scope for improvements in this direction, although it will require some effort as it involves paying detailed attention to the thermodynamics in nonlinear simulations."
],
[
"Acknowledgements",
"We would like to, first of all, thank Sebastiano Bernuzzi for providing the numerical simulation data used in figure REF and acknowledge the use of data from the ComPOSE website for the thermal EoSs.",
"NA gratefully acknowledges support from STFC via grant ST/R00045X/1.",
"Research of K.C.",
"was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program Summer School on Gravitational-Wave Astronomy (Code: ICTS/Prog-gws/2017/07)."
]
] | 1906.04546 |
[
[
"Multiphase modeling of precipitation-induced membrane formation"
],
[
"Abstract We formulate a model for the dynamic growth of a membrane developing in a flow as the result of a precipitation reaction, a situation inspired by recent microfluidic experiments.",
"The precipitating solid introduces additional forces on the fluid and eventually forms a membrane that is fixed in the flow due to adhesion with a substrate.",
"A key challenge is that the location of the immobile membrane is unknown $\\textit{a priori}$.",
"To model this situation, we use a multiphase framework with fluid and membrane phases; the aqueous chemicals exist as scalar fields that react within the fluid to induce phase change.",
"To verify that the model exhibits desired fluid-structure behaviors, we make a few simplifying assumptions to obtain a reduced form of the equations that is amenable to exact solution.",
"This analysis demonstrates no-slip behavior on the developing membrane without $\\textit{a priori}$ assumptions on its location.",
"The model has applications towards precipitate reactions where the precipitate greatly affects the surrounding flow, a situation appearing in many laboratory and geophysical contexts including the hydrothermal vent theory for the origin of life.",
"More generally, this model can be used to address fluid-structure interaction problems that feature the dynamic generation of structures."
],
[
"Introduction",
"One hypothesis for the “origin of life\" is that the first biomolecules were formed in undersea hydrothermal vents.",
"In this theory, passive, anisotropic diffusion across a membrane supports the transmembrane gradients necessary for the first biochemical molecules [30].",
"An experimental approach to study this theory examines simpler systems in microfluidic reactors which allow for the controlled study of the prebiotic chemistry in hydrothermal vent chimneys [3].",
"Microfluidic devices have become an important tool in modern chemistry and biomedical analytics [34].",
"One application is the possibility of a “lab on a chip\", i.e.",
"the miniaturization of chemical separation and analysis procedures onto a disposable device as small as a few square centimeters.",
"The devices are typically made from etched glass or lithographically-processed elastomers and the fluid flow is usually controlled mechanically by external pumps or electrically via electro-osmotic flows [29].",
"Recent studies have used microfluidic methods to form inorganic membranes within Y-shaped devices [4], [50].",
"The membranes result from chemical reactions between two different solutions that are injected separately but later merge in a long reaction channel that brings the reactants into direct contact.",
"This merging is usually performed under low Reynolds number (Re) conditions and for miscible liquids, such as aqueous solutions of NaOH, and a dissolved metal salt, such as NiCl$_2$ .",
"At the reactive interface between these liquids, a precipitate, such as Ni(OH)$_2$ , swiftly forms a thin porous membrane (Figure 1).",
"This precipitate reaction typically involves the formation of microscopically small colloid particles and their aggregation or addition to the membrane.",
"This phenomenon is related to so-called “chemical gardens\" which consist of thin cylindrical precipitate membranes separating a metal salt solution from silicate or hydroxide solutions [41], [28].",
"Figure: Inorganic precipitate membrane formed in a microfluidic channel.",
"(a) The photograph of the resulting membrane after 2 h with 0.5 M NaOH and 0.5 M NiCl 2 _2 solutions being injected simultaneously into the microfluidic device.",
"The mixing part of the channel is 50 mm long, 2 mm wide, and approximately 130 μ\\mu m high.",
"(b) A magnified view of a selected area from (a).",
"(c-g) A sequence of micrographs showing the unidirectional thickening process after (c) 1 min, (d) 15 min, (e) 30 min, (f) 60 min, and (g) 120 min.",
"Scale bars correspond to (a) 1 cm, (b) 5 mm, and (c) 200 μ\\mu m.[16] showed experimentally that membrane thickness increases with the square root of time, indicating diffusion-controlled growth.",
"The membrane thickening occurs only in the direction of the metal salt solution (e.g.",
"NiCl$_2$ ) and not in the direction of the anionic precipitation partner (e.g.",
"OH$^-$ ), indicating that the membrane is more permeable to anions than cations.",
"This phenomenon has been qualitatively explained by the charged nature of the membrane that suppresses the transmembrane transport of the positive metal ion [4], [50].",
"The modeling challenges presented by this experiment involve a confluence of topics that have been studied before, namely ionic reactions [43], [51], precipitation [52], [1], passive diffusion through a membrane [33], [12], [16], and fluid-structure interaction [49], [11], [40], [47], [46], [32], [38].",
"The particular combination of these aspects provides the opportunity for a new model that captures them all.",
"One key challenge is that the “structure” in this problem is generated dynamically according to equations governing the chemistry.",
"We choose to model the fluid-structure combination as a single multiphase material: one component “fluid” or solvent and one component “structure” or precipitate membrane.",
"Such multiphase models have proven useful in a variety of complex-fluid applications, such as bacterial biofilms [14], [13], tumor-growth [9], [37], [21], [44], and biological membranes [27]; their formulation is based on averaging momentum and stresses in separated, multi-component fluids [18], [17].",
"The multiphase framework developed here builds on previous ones [13], [52], [25], but with some keys differences that are guided by a combination of physical principles, model simplicity, and the micro-fluidic experiments mentioned above.",
"First, our formulation conserves the total mass of the components — solvent, dissolved species, and precipitate membrane — throughout evolution.",
"In particular, the model accounts for changes in solute concentrations that result from the formation of new membrane and the associated exclusion of solvent volume.",
"This effect is neglected in previous models that treat reaction chemicals as scalar fields distinct from the multiphase material, but is essential for overall mass conservation.",
"The treatment of reaction chemicals as additional components of a multiphase material has been successfully modeled by many [35], [51] but greatly complicates the analysis, interpretation, and simulation of the governing equations.",
"Second, by making certain choices in the averaging procedure for the multicomponent stress, our formulation becomes equivalent to an incompressible Brinkman system with variable permeability.",
"This equivalence is important for a few reasons.",
"First, it guarantees that the model reduces to the Stokes equations in the fluid limit and to Darcy’s equation in the porous-medium limit [8], [20], [23].",
"In particular, it guarantees that when membrane is fully developed, the interface behaves as an impermeable surface with a no-slip condition on fluid velocity.",
"As demonstrated in section REF , many existing multiphase models fail to exhibit this behavior [7], [15], [13], [44], as they were developed primarily for highly permeable systems.",
"Second, the equivalence to Brinkman significantly simplifies the overall structure of the partial differential equation (PDE) system by eliminating certain cross-terms in the stress divergence that arise in other models.",
"This simplification is one key that will allow reduction to a non-trivial case where chemical and phase dynamics can be solved exactly.",
"To demonstrate that the new framework possesses the desired properties listed above, we consider a simplified system in which the incoming reactant concentrations are held fixed via chemostat [42].",
"By assuming parallel flow and neglecting solute diffusion, the governing equations reduce to a planar system of ordinary differential equations (ODEs).",
"This nonlinear system can be linearized around the fixed points, and eigenvalue analysis provides an estimate for the rate at which new membrane forms.",
"Moreover, we find that the equation for the aqueous product is a second-order nonlinear ODE known as the Ricatti equation [39], [48].",
"Exact solutions to the Ricatti equation give explicit formulas for the time dependence of the chemical product and, consequently, the formation of new membrane.",
"Once the membrane dynamics are known exactly, the flow profile can be obtained through the numerical solution of a simple boundary value problem (BVP).",
"Access to the resulting flow profile allows a careful comparison between variants of the multiphase framework.",
"In particular, we demonstrate that the framework developed here properly captures the transition from one-channel to two-channel flow as membrane develops.",
"The paper proceeds as follows: in section we develop the governing equations for both the reacting chemicals and multiphase material such that the total mass is conserved.",
"Section contains analysis and results based on simplifying assumptions.",
"These assumptions generate a reduced form for an idealized scenario which can be solved with a combination of analytic and simple numerical methods.",
"Finally in section the predictive power of the model and further applications are discussed."
],
[
"Mathematical Model",
"The model requires the accurate description of several aspects of the experiment, including the flow transport of the two ionic species and their reaction to form a product, the precipitation of the product out of solution, and finally the response of the bulk fluid motion to the dynamically-generated precipitate membrane.",
"Advection-Diffusion-Reaction (ADR) equations are derived for the aqueous chemical concentrations, while the fluid and membrane dynamics are described by multiphase mass and momentum balance equations.",
"In many multiphase models, either constituent can be viscous, viscoelastic, poroelastic, or otherwise.",
"Here, since the membrane adheres to the substrate, it can be treated as an immobile solid, leading to considerable simplifications.",
"We assume that aqueous reactants and product contribute mass, but not volume, to the fluid phase.",
"The solvent and membrane each have their own distinct mass densities, and any arbitrary control volume can be divided into solvent and membrane volume fractions.",
"The formation of new membrane involves the precipitation of product out of solution and the sequestration of solvent.",
"A key modeling assumption is that the volume of fluid sequestered equals the volume of the resulting membrane.",
"As shown in section REF , this assumption ensures incompressibility of the phase-averaged velocity field, i.e.",
"the so-called Darcy velocity.",
"We now detail the model equations.",
"First we derive evolution equations for the reaction of aqueous ionic species, then we list mass balances for all chemical species as well as solvent and membrane phases, and finally we describe the momentum equation for the fluid.",
"In the end we obtain a closed, coupled PDE system governing the chemistry and physics of the system, where total mass is conserved throughout aqueous reactions and phase transitions."
],
[
"Model for Chemical Reactions",
"In this section we derive equations for the chemical reactions.",
"We follow the “nucleation and growth\" model of precipitation [31] and separate the reaction into two sequential parts: in the first, two reactants come together to form an aqueous product, and the second describes the aggregation of the aqueous product into a solid precipitate.",
"While many aqueous chemical reactions do not alter the solution volume significantly, the formation of a membrane excludes fluid volume and therefore can alter the local concentration of the dissolved species.",
"Accordingly, our model neglects the volume occupied by the aqueous species but does account for changes in species concentration that are due to the precipitated solid excluding fluid volume.",
"This effect introduces additional terms in the aqueous reaction equations that are required for mass conservation.",
"To our knowledge these additional terms are not accounted for in the multiphase precipitation literature that treats aqueous chemicals as scalar fields distinct from of the multiphase material.",
"The aqueous reaction is written as a generic net ionic equation $aA\\text{(aq)} \\,+\\,bB\\text{(aq)} \\rightarrow cC\\text{(aq)} $ where $A\\text{(aq)}$ , $B\\text{(aq)}$ , and $C\\text{(aq)}$ are chemicals in the aqueous phase and $a$ , $b$ , and $c$ are their respective stoichiometric coefficients; the precipitation reaction is written simply as $C\\text{(aq)} \\rightarrow C\\text{(s)}\\,.$ As a concrete example consider the reaction described in the introduction, $\\text{Ni}^{2+}\\text{(aq)} + 2\\text{(OH)}^-\\text{(aq)} \\rightarrow \\text{Ni(OH)}_2\\text{(aq)} \\\\[1em]\\text{Ni(OH)}_2\\text{(aq)} \\rightarrow \\text{Ni(OH)}_2\\text{(s)}\\,.$ Then $A=\\text{Ni}^{2+}$ , $B=\\text{(OH)}^-$ and $C=\\text{Ni(OH)}_2$ and $a=c=1$ , $b=2$ .",
"The aqueous chemicals will be measured with a variable for the number of chemicals per unit solvent volume, i.e.",
"molarity, which we will call $\\psi _i$ for chemical species $i\\in \\lbrace A,B,C\\rbrace $.",
"Reaction rates depend on a reactants' molarity, and molarity can change due to two independent factors: either the number of molecules changes due to the aqueous reaction, or the solvent volume changes due to precipitation.",
"Because either one can occur in a precipitation reaction, these two competing effects must be carefully considered when formulating the reaction equations.",
"We begin by deriving equations for how the aqueous reaction proceeds in a spatially homogeneous environment; later in section REF the effects of advective and diffusive spatial fluxes will be added.",
"Suppose the chemicals exist in some aqueous solution of fixed control volume $V_0$ .",
"The chemicals undergo both the aqueous and precipitation reactions which results in fluid mass and solvent volume being converted to membrane mass and volume (see figure REF ).",
"The fluid component has mass $\\mathcal {M}_f = \\rho _f\\theta _sV_0 = \\left(\\rho _s + \\sum M_i\\psi _i\\right)\\theta _sV_0 $ where $\\rho _s$ is the constant solvent mass density (without any reactants or products present), $\\theta _s$ is the solvent volume fraction, and $M_i$ is the molar mass of chemical species $i\\in \\lbrace A,B,C \\rbrace $ .",
"The summation represents the contribution of the chemical species to fluid mass, so that the fluid mass density $\\rho _f$ is not constant.",
"The membrane component has mass $\\mathcal {M}_m = \\rho _m\\theta _m V_0$ where $\\rho _m$ is the constant membrane mass density and $\\theta _m$ is the membrane volume fraction.",
"Physically, membrane mass is composed of both precipitated chemical $C$ and sequestered solvent mass.",
"The change in $\\psi _i$ purely due to aqueous reaction, i.e.",
"no precipitation, can be modeled as a second-order kinetics reaction $\\dot{\\psi }^{(aq)}_A = -ar\\psi _A\\psi _B,\\qquad \\dot{\\psi }^{(aq)}_B = -br\\psi _A\\psi _B,\\qquad \\dot{\\psi }^{(aq)}_C = cr\\psi _A\\psi _B\\,.",
"$ where the dot indicates a derivative with respect to time, and $r$ is the rate of aqueous reaction per chemical concentration.",
"More general power laws are sometimes used to model chemical kinetics, but here we use purely second-order kinetics for simplicity [10].",
"None of the analysis, however, depends specifically on this choice and the results could be carried forward for other kinetics.",
"To derive equations for the change in $\\psi _i$ purely due to precipitation we appeal to ideas from continuum mechanics.",
"The concentration of ions $A$ in the control volume is written as $\\psi _A=n_A/(\\theta _sV_0)$ where $n_A$ is the number of $A$ ions in $V_0$ .",
"Note that this formulation makes explicit the dependence of $\\psi _A$ on both $n_A$ and $\\theta _s$ .",
"Consider the change in a small increment of time $\\Delta t$ .",
"Then the time-dependent variables are updated so that $\\psi _A + \\Delta \\psi _A = \\frac{n_A}{(\\theta _s+\\Delta \\theta _s)V_0}\\,.$ Recall that $n_A$ is constant during precipitation as only $C\\text{(aq)}$ precipitates.",
"Approximating for small $\\Delta \\theta _s$ and neglecting higher-order terms gives $\\psi _A + \\Delta \\psi _A = \\frac{n_A}{\\theta _sV_0}\\left(1 - \\frac{\\Delta \\theta _s}{\\theta _s}\\right) = \\psi _A\\left(1 - \\frac{\\Delta \\theta _s}{\\theta _s}\\right) \\, .$ Then, cancelling the $\\psi _A$ , dividing both sides by $\\Delta t$ , and letting $\\Delta t \\rightarrow 0$ gives the change in $\\psi _A$ purely due to precipitate reaction as $\\dot{\\psi }^{(p)}_A=-\\psi _A\\dot{\\theta }_s/\\theta _s$ .",
"By symmetry, a similar formula holds for $\\dot{\\psi }^{(p)}_B$ .",
"Note that both of these are essentially applications of the product rule for $\\partial _t(\\psi _i\\theta _s)=0$ , which physically means that the total number of ions of $i\\in \\lbrace A,B\\rbrace $ in the control volume does not change in time due to precipitation.",
"Figure: Schematic of precipitate reaction in control volume.",
"Precipitation causes solution (white) to transform into membrane (shaded) after a certain concentration threshold is reached of aqueous product CC.",
"Aqueous chemicals AA, BB and CC are volumeless scalar fields while the solvent and membrane is treated as a multiphase material.",
"The volume of membrane gained is exactly equal to the volume of solvent lost.A similar procedure can be followed for $\\psi _C$ , except now the number of aqueous chemicals $n_C$ changes as $C\\text{(aq)}$ precipitates into membrane, $\\psi _C+\\Delta \\psi _C = \\frac{n_C+\\Delta n_C}{(\\theta _s+\\Delta \\theta _s)V_0} \\, .$ Above, both $n_C$ and $\\theta _s$ change in time.",
"Expanding both expressions while linearizing for small $\\Delta \\theta _s$ , dividing by $\\Delta t$ , and taking the limit as $\\Delta t\\rightarrow 0$ one obtains $\\dot{\\psi }^{(p)}_C = - \\psi _C\\dot{\\theta }_s/\\theta _s + \\dot{n}_C/\\theta _s$.",
"The first term in this expression is analogous to those obtained for reactants $A$ and $B$ , and simply describes the effect on concentration when solvent volume is changing.",
"The second term, however, is new and describes the effect on $\\psi _C$ as aqueous $C$ molecules are converted into membrane.",
"We write $\\dot{n}_C = \\alpha \\dot{\\theta }_s$ where the specific value of $\\alpha $ will be found shortly to guarantee conservation of mass throughout the entire reaction.",
"The expressions for the rate of change of aqueous chemical concentrations due to precipitation are thus: $\\dot{\\psi }^{(p)}_A = -\\psi _A\\dot{\\theta }_s/\\theta _s,\\qquad \\dot{\\psi }^{(p)}_B = -\\psi _B\\dot{\\theta }_s/\\theta _s, \\qquad \\dot{\\psi }^{(p)}_C = -\\psi _C\\dot{\\theta }_s/\\theta _s + \\alpha \\dot{\\theta }_s/\\theta _s\\,.",
"$ Assuming that the aqueous and precipitate reactions act independently, $\\dot{\\psi }_i = \\dot{\\psi }^{(aq)}_i + \\dot{\\psi }^{(p)}_i$ , gives $\\dot{\\psi }_A & = -ar\\psi _A\\psi _B -\\psi _A\\dot{\\theta }_s/\\theta _s \\\\\\dot{\\psi }_B & = -br\\psi _A\\psi _B -\\psi _B\\dot{\\theta }_s/\\theta _s \\\\\\dot{\\psi }_C & = cr\\psi _A\\psi _B -\\psi _C\\dot{\\theta }_s/\\theta _s + \\alpha \\dot{\\theta }_s/\\theta _s $ These equations describe the dynamics of aqueous species concentrations in the absence of spatial fluxes.",
"To obtain the value of $\\alpha $ that guarantees conservation of mass, we again apply a continuum mechanics argument.",
"The change in membrane mass after a small time step is $\\Delta \\mathcal {M}_m = \\rho _m\\Delta \\theta _m V_0$ .",
"To simplify the expression for change in fluid mass, we expand $\\Delta \\mathcal {M}_f$ while neglecting second order terms to get $\\Delta \\mathcal {M}_f = V_0\\rho _s\\Delta \\theta _s + V_0\\theta _s\\sum _i M_i\\Delta \\psi _i + V_0\\Delta \\theta _s\\sum _i M_i\\psi _i \\, .$ Replacing the $\\Delta \\psi _i$ with their respective differential terms in equations (REF ) and performing some algebraic manipulation produces $\\Delta \\mathcal {M}_f = V_0\\rho _s\\Delta \\theta _s + V_0\\theta _s\\psi _A\\psi _B(cM_c - aM_A - bM_B) + V_0M_C\\Delta n_C \\, .$ Conservation of mass during the aqueous reaction (REF ) implies $aM_A+bM_B=cM_C\\, .$ Thus, the term in parenthesis in (REF ) vanishes.",
"Meanwhile, conservation of mass of the entire system implies $\\Delta \\mathcal {M}_m = -\\Delta \\mathcal {M}_f$ , i.e.",
"the mass lost by the fluid equals the mass gained by membrane.",
"Additionally, the assumption that fluid volume is converted perfectly to membrane volume implies $\\Delta \\theta _m = -\\Delta \\theta _s$ .",
"Using the respective definitions of $\\mathcal {M}_i$ and solving for $\\Delta n_C$ gives $\\Delta n_C = \\left(\\frac{\\rho _m-\\rho _s}{M_C}\\right)\\Delta \\theta _s \\, .$ Diving by $\\Delta t$ and taking the limit $\\Delta t\\rightarrow 0$ gives $\\dot{n}_C = \\alpha \\dot{\\theta }_s$ where $\\alpha = (\\rho _m - \\rho _s)/M_C$ .",
"Physically, this value of $\\alpha $ corresponds to the concentration of $C\\text{(aq)}$ that must leave the fluid phase during precipitation in order for mass to be conserved.",
"The reaction equations derived in this section, along with the specific $\\alpha $ term, will be used to provide reaction terms for the chemistry mass balance equations, as described in the next section."
],
[
"Mass Balance Equations",
"In the experiments, the aqueous reaction occurs within the flow of a microfluidic device and therefore spatial fluxes must be considered.",
"To describe these fluxes, consider the general conservation law for the chemical mass per unit control volume $\\phi =M_i\\psi _i\\theta _s$ , $\\frac{\\partial \\phi }{\\partial t} + \\nabla \\cdot \\mathbf {J} = \\Gamma $ where $\\mathbf {J}$ is the flux of $\\phi $ and $\\Gamma $ is a transfer term for the rate that $\\phi $ enters the system.",
"We choose $\\mathbf {J}$ to account for advection and diffusion of the chemical concentrations, $\\frac{\\partial (M_A\\psi _A\\theta _s)}{\\partial t} + \\nabla \\cdot \\big (M_A\\psi _A\\theta _s\\mathbf {v}_s - \\kappa _A\\nabla (M_A\\psi _A)\\big ) & = \\Gamma _A \\\\\\frac{\\partial (M_B\\psi _B\\theta _s)}{\\partial t} + \\nabla \\cdot \\big (M_B\\psi _B\\theta _s\\mathbf {v}_s - \\kappa _B\\nabla (M_B\\psi _B)\\big ) & = \\Gamma _B \\\\\\frac{\\partial (M_C\\psi _C\\theta _s)}{\\partial t} + \\nabla \\cdot \\big (M_C\\psi _C\\theta _s\\mathbf {v}_s - \\kappa _C\\nabla (M_C\\psi _C)\\big ) & = \\Gamma _C $ where $\\mathbf {v}_s$ is the (tracer) velocity of the solvent and $\\kappa _i$ are diffusion coefficients which possibly depend on the solvent volume fraction.",
"Note that the diffusive flux used above transports mass according to gradients in molarity $\\psi _i$ , not gradients in $\\phi _i$ .",
"This choice produces the physically realistic steady state of uniform molarity in a quiescent, non-reacting fluid that has inhomogeneous volume fraction.",
"Assuming that the reactions and spatial fluxes act independently, the $\\Gamma _i$ correspond to the rates given in equations (REF ).",
"Rearranging and multiplying each equation by its respective molar mass $M_i$ gives $\\Gamma _A = -arM_A\\theta _s\\psi _A\\psi _B \\\\\\Gamma _B = -brM_B\\theta _s\\psi _A\\psi _B \\\\\\Gamma _C = crM_C\\theta _s\\psi _A\\psi _B + \\alpha M_C\\dot{\\theta }_s$ where $\\alpha =(\\rho _m-\\rho _s)/M_C$ .",
"Now that mass balance equations for the chemistry are established, mass balance equations for the multiphase solvent-membrane system are needed.",
"A simple but necessary assumption is that our volume is occupied by only solvent and membrane, i.e.",
"there are no “voids\".",
"This no-void assumption implies $\\theta _s + \\theta _m = 1 $ everywhere.",
"Mass balances for the solvent and membrane phases provide $& \\frac{\\partial (\\rho _s\\theta _s)}{\\partial t} + \\nabla \\cdot (\\rho _s\\theta _s\\mathbf {v}_s)= R_s \\\\& \\frac{\\partial (\\rho _m\\theta _m)}{\\partial t} = R_m $ where $R_i$ denotes the rate of mass added to phase $i$ .",
"Equation () has no advective term since the membrane is assumed to be immobile.",
"To ensure conservation of total mass, the rates $R_m$ and $R_s$ must be related.",
"To derive this relationship, let $V_0$ be an arbitrary control volume.",
"The total mass (of all components) in $V_0$ is $\\mathcal {M}(V_0) = \\int _{V_0} {\\rho _s\\theta _s + \\rho _m\\theta _m + \\sum M_i\\psi _i\\theta _s}\\,\\, dV$ Summing the five mass balance equations, (REF )–() and (REF )–(), integrating over $V_0$ , and applying the divergence theorem gives $\\frac{d }{d t}\\mathcal {M}(V_0)+ \\int _{\\partial V_0} \\underbrace{\\left( \\rho _s \\theta _s \\mathbf {v}_s + \\sum \\mathbf {J}_i \\right) \\cdot \\hat{\\bf {n}} }_{\\text{boundary flux}} \\,\\, dS= \\int _{V_0} \\underbrace{R_s + R_m + \\sum \\Gamma _i}_{\\text{transfer \\& reaction}}\\,\\, dV \\, .$ where $ \\hat{\\bf {n}} $ is the outward unit normal vector.",
"Summing equations (REF ) and applying (REF ) gives $ \\sum \\Gamma _i = \\alpha M_C\\dot{\\theta }_s$ .",
"For the sake of obtaining a relationship between $R_s$ and $R_m$ , briefly consider the case of zero boundary flux.",
"Then to conserve total mass within any control volume, we must have $R_s + R_m + \\alpha M_C\\dot{\\theta }_s = 0$ holding point-wise.",
"Substituting the value of $\\alpha $ , using the no-void assumption (REF ) and the definition of $R_m$ in () gives $R_s = -\\frac{\\rho _s}{\\rho _m} \\, R_m$ This relationship is required for overall mass conservation.",
"Now consider the so-called Darcy velocity field $\\mathbf {q}_s = \\theta _s\\mathbf {v}_s$ .",
"Substituting the value of $R_s$ from equation (REF ) into equation (REF ), using a consequence of the no-void assumption ($\\dot{\\theta }_s=-\\dot{\\theta }_m$ ) and substituting $R_m$ by its value in equation (REF ) implies that the Darcy velocity field is in fact incompressible $\\nabla \\cdot \\mathbf {q}_s = 0 \\, .$ Now return to (REF ) and consider the entire domain $\\Omega $ with total mass $\\mathcal {M}=\\mathcal {M}(\\Omega )$ .",
"From the above argument, the right-hand-side of this equation vanishes as a necessary condition on mass conservation.",
"Then applying incompressibility (REF ), gives the total mass balance $\\frac{d \\mathcal {M}}{d t} =- \\int _{\\partial \\Omega } \\sum M_i \\big ( \\mathbf {q}_s \\psi _i - \\kappa _i \\nabla \\psi _i \\big ) \\cdot \\hat{\\bf {n}} \\,\\, dS \\, .$ As expressed in this equation, the total mass of the system is conserved as long as the chemical flux at the boundary vanishes.",
"More generally, the total mass of the system can change according to how much chemical mass is being injected or removed via the boundary flux terms.",
"We now specify our choice for the form for the precipitation term $R_m$ .",
"Although complicated models of precipitation exist [31], [36], we employ a simple model in which the rate of membrane mass growth is proportional to the amount of product, provided that the product concentration exceeds some precipitation threshold, i.e.",
"$R_m = \\beta \\psi _C\\theta _s\\mathcal {H}(\\psi _C - \\psi _C^*) $ where $\\mathcal {H}$ is the standard Heaviside function and $\\psi _C^*$ is the concentration threshold for precipitation to occur."
],
[
"Momentum Balance Equations",
"Since inertial effects are assumed negligible ($\\text{Re}\\ll 1$ ), the solvent momentum balance can be written as $\\nabla \\cdot \\mathbf {T} - \\theta _s\\nabla P - \\xi \\mathbf {v}_s = \\mathbf {0} $ where $\\mathbf {T}$ is the multiphase stress tensor, $P$ is a so-called common pressure that is shared by the fluid and membrane phases [13], and $\\xi $ is a friction coefficient.",
"Deviating slightly from the predominant multiphase literature, we chose the form of the stress tensor as $\\mathbf {T} = \\eta \\big (\\nabla \\mathbf {q}_s + \\nabla \\mathbf {q}_s^\\top \\big )$ where $\\eta $ is the fluid viscosity.",
"In particular, since $\\mathbf {q}_s = \\theta _s \\mathbf {v}_s$ , we have placed the fluid volume fraction $\\theta _s$ inside the gradient, whereas most multiphase models place the $\\theta _s$ outside of the gradient but inside the divergence [9], [15], [13].",
"Such a choice must be made for model closure, and neither is fully justified by first principles.",
"We make the above choice to obtain equivalence to the Brinkman system, which offers considerable gains in model tractability.",
"The incompressibility of the Darcy velocity, while being notable in itself, allows the momentum equation to be transformed into something more familiar.",
"Applying the divergence-free property to equation (REF ) reduces it to a Brinkman equation with variable coefficients $\\eta \\nabla ^2 \\mathbf {q}_s - \\frac{\\xi }{\\theta _s} \\mathbf {q}_s = \\theta _s \\nabla P\\,.",
"$ We can now see that equation (REF ), equivalently equation (REF ), does not have any cross-derivative terms that appear in [13], [24].",
"Because the membrane is assumed immobile ($\\mathbf {v}_m\\equiv \\mathbf {0}$ ), no momentum equation is needed for it.",
"The friction coefficient $\\xi $ should be chosen in such a way that, at high membrane volume fraction, friction becomes the dominant effect in equation (REF ).",
"The choice made here, and mentioned briefly in [26], is to use the Kozeny-Carman (KC) formula for permeability as it depends on porosity [19].",
"In the present notation, the KC relationship gives the friction coefficient as $\\xi _{KC}(\\theta _s) = h\\frac{(1-\\theta _s)^2}{\\theta _s}$ where $h$ is an arbitrary constant.",
"This friction coefficient will provide the desired no-slip behavior in the membrane limit $\\theta _m\\rightarrow 1$ .",
"[2] discusses the implications of a similar singular friction term in a Brinkman system, although their model does not include the solvent volume fraction term in front of the pressure gradient and only applies to domains with spatially discontinuous volume fractions; the current fluid-membrane model generalizes this notion by being able to account for smooth spatial and temporal gradients of the volume fraction.",
"As an alternative to the singular $\\xi _{KC}$ , the friction coefficient can be chosen to be a (non-singular) Hill function, as used in [25], [26], $\\xi _H(\\theta _s) = h\\frac{(1-\\theta _s)^n}{K^n + (1-\\theta _s)^n}\\,.$ The use of Hill functions is largely empirical, although it has significant advantages in that it is finite in the membrane limit and therefore more numerically stable.",
"Additionally, $K$ determines the half-saturation point and $n$ indicates the qualitative manner at which this saturation is achieved.",
"These parameters allow for fine-tuning to specific experimental observations.",
"Finally, a friction term employed in many biofilm multiphase models is $\\xi _B(\\theta _s) = h\\theta _s(1-\\theta _s)\\,.$ This choice has become popular in the literature [9], [14], [15], [13], [44], and is often justified by the idea that friction should vanish if either phase, $\\theta _s$ or $\\theta _m$ , is absent.",
"While it is an intuitive notion, we will show in section REF that this friction coefficient does not produce the physically realistic behavior of no-slip velocity on fully developed solid surfaces.",
"To produce this behavior, it is necessary that friction dominates, not vanishes, in the limit $\\theta _m \\rightarrow 1$ .",
"A visual comparison of these three friction coefficients is shown in figure REF .",
"For the sake of comparison, we have chosen the constant $h$ so that the three curves intersect at a reference porosity $\\theta _s^*$ , i.e.",
"$\\xi (\\theta _s^*) = \\xi ^*$ in each case.",
"For $\\xi _{KC}$ and $\\xi _H$ , the value $\\theta _s^*$ can be interpreted as the percolation threshold, i.e.",
"the critical porosity below which the medium essentially behaves as impermeable to flow [22].",
"Figure: Comparison of friction terms ξ\\xi .",
"ξ KC \\xi _{KC} (solid) is singular in the limit θ s →0\\theta _s\\rightarrow 0, ξ H \\xi _H (dash) is non-singular in the porous limit (K=0.5K=0.5, n=2n=2) and ξ B \\xi _B (dot) yields maximum friction when both phases are present in equal amounts.",
"All terms have been normalized by choosing hh such that ξ(θ S * )=ξ * \\xi (\\theta _S^*)=\\xi ^*."
],
[
"Model Summary",
"The governing equations are now summarized for the reader.",
"Rearranging equations (REF )–() and applying incompressibility of $\\mathbf {q}_s = \\theta _s \\mathbf {v}_s$ gives the following ADR evolution equations for aqueous chemical concentration: $&\\frac{\\partial \\psi _A}{\\partial t} = \\overbrace{\\frac{1}{\\theta _s}\\nabla \\cdot \\big (\\kappa _A\\nabla \\psi _A\\big )}^{\\text{\\small diffusion}} - \\overbrace{\\nabla \\psi _A\\cdot \\mathbf {v}_s}^{\\text{\\small advection}} - \\overbrace{a r \\psi _A\\psi _B}^{\\begin{array}{c}\\text{\\small aqueous} \\\\ \\text{\\small reaction}\\end{array}} - \\overbrace{\\psi _A\\dot{\\theta }_s/\\theta _s}^{\\begin{array}{c}\\text{\\small precipitate} \\\\ \\text{\\small reaction}\\end{array}} \\\\&\\frac{\\partial \\psi _B}{\\partial t} = \\frac{1}{\\theta _s}\\nabla \\cdot \\big (\\kappa _B\\nabla \\psi _B\\big ) - \\nabla \\psi _B\\cdot \\mathbf {v}_s - b r \\psi _A\\psi _B - \\psi _B\\dot{\\theta }_s/\\theta _s \\\\&\\frac{\\partial \\psi _C}{\\partial t} =\\frac{1}{\\theta _s}\\nabla \\cdot \\big (\\kappa _C\\nabla \\psi _C\\big ) - \\nabla \\psi _C\\cdot \\mathbf {v}_s + c r \\psi _A\\psi _B - (\\psi _C-\\alpha )\\dot{\\theta }_s/\\theta _s $ where $\\alpha =(\\rho _m-\\rho _s)/M_C$ .",
"Meanwhile, the evolution equations for the fluid and solid phases can be summarized as $& \\theta _s + \\theta _m = 1\\,, \\\\& \\rho _m \\frac{\\partial \\theta _m}{\\partial t} = R_m= \\beta \\psi _C\\theta _s\\mathcal {H}(\\psi _C - \\psi _C^*) \\, .$ The momentum equation is a Brinkman equation with variable permeability $&\\eta \\nabla ^2\\mathbf {q}_s - h\\frac{\\theta _m^2}{\\theta _s^2}\\mathbf {q}_s = \\theta _s \\nabla P \\\\&\\nabla \\cdot \\mathbf {q}_s = 0 $ where we have employed the $\\xi _{KC}$ friction term.",
"We will now consider these coupled PDEs in a simplified setting that will permit exact solutions."
],
[
"Analysis of Reduced Model",
"Analysis of any complicated system is aided by reduction into a form that is analytically tractable.",
"Inspired by microfluidic experiments, we assume that some chemostat controls the influx of reactants' molarity far upstream.",
"All variables are kept constant along the longitudinal axis by neglecting diffusion and assuming parallel flow.",
"This requirement on parallel flow also means that the reaction takes place everywhere along the longitudinal axis simultaneously.",
"Finally, we assume that the precipitation threshold is negligible.",
"Applying these assumptions to the governing equations reduces the system considerably so that it becomes a Poiseuille analysis; these assumptions generate the following reduced system $&\\dot{\\psi }_C = c r\\psi _A\\psi _B - (\\psi _C - \\alpha )\\dot{\\theta }_s/\\theta _s \\\\&\\theta _s + \\theta _m = 1 \\\\&\\dot{\\theta }_s = -\\beta \\psi _C \\theta _s/\\rho _m \\\\&\\eta \\frac{\\partial ^2 q_y}{\\partial x ^2} - h\\frac{\\theta _m^2}{\\theta _s^2}q_y = \\theta _s G\\,.",
"$ where only $\\psi _C$ , $\\theta _s$ , $\\theta _m$ and $q_y$ are unknown.",
"Note that the longitudinal axis is chosen to be $y$ such that only this component of the Darcy velocity $\\mathbf {q}_s = q_x\\mathbf {\\hat{x}} + q_y\\mathbf {\\hat{y}}$ remains.",
"In section REF we obtain an analytic estimate on the rate of membrane formation by treating $\\psi _C$ and $\\theta _m$ as a planar dynamical system.",
"Then in section REF we solve equations () with a combination of analytic and numerical methods to visualize how the membrane affects the flow profile in time."
],
[
"Fixed Point Analysis",
"Our approach is to linearize the reduced model system, then perform an eigenvalue analysis about a steady state fixed point.",
"The benefit of this is the eigenvalue will correspond to the rate that membrane develops, a quantity that is possible to measure experimentally.",
"Before doing a fixed point analysis, it is helpful to understand the conditions on which the existence and stability of fixed points depends.",
"To do so, eliminate the explicit dependence of $\\dot{\\psi }_C$ on volume fraction and replace all $\\dot{\\theta }_s$ in equation (REF ) with equation () to obtain a quadratic ODE of the form $\\dot{\\psi }_C = \\frac{\\beta }{\\rho _m}\\psi _C^2 - \\frac{\\alpha \\beta }{\\rho _m}\\psi _C + cr\\psi _A\\psi _B\\,.",
"$ which is an ODE in time alone, as the $x\\text{-dependence}$ of $\\psi _A$ and $\\psi _B$ are determined by the initial conditions.",
"Examining the qualitative behavior of this ODE by considering $\\dot{\\psi }_C=\\dot{\\psi }_C (\\psi _C)$ , it is quadratic in $\\psi _C$ , intercepts the $\\dot{\\psi }_C$ axis at $cr\\psi _A\\psi _B\\ge 0$ , is concave up, and has equilibria at $\\psi _C^{\\pm } = \\frac{1}{2}\\left(\\alpha \\pm \\frac{1}{\\beta }\\sqrt{\\alpha ^2\\beta ^2 - 4c r\\rho _m \\beta \\psi _A\\psi _B} \\right)$ whose existence depends on the sign of $\\chi = \\alpha ^2\\beta ^2 - 4 c r \\rho _m \\beta \\psi _A\\psi _B\\,.",
"$ If $\\chi >0$ , equation (REF ) will have two fixed points, for $\\chi =0$ these fixed points coalesce, and for $\\chi <0$ there are no fixed points and $\\dot{\\psi }_C$ will grow without bound; see figure REF (a).",
"We now ask the question of whether fixed points exist in the reduced system, i.e.",
"$\\chi \\ge 0$ or $\\alpha ^2\\beta \\ge 4cr\\rho _m\\psi _A\\psi _B$ ?",
"We interpret this condition based on the physical meaning of the parameters: $\\alpha =(\\rho _m-\\rho _s)/M_C$ has dimension of molarity and is $O(10 \\,\\text{M})$ where $\\text{M}$ refers to molar units, $1 \\,\\text{M}=\\text{1 mol}/\\text{liter}$ ; for example, using the reaction system in the introduction gives $\\alpha \\approx 30 \\, \\text{M}$ .",
"We note that a similar analysis also justifies neglecting the precipitation threshold $\\psi _C^*$ , as $\\psi _C^*\\approx 0.001 \\text{ M}$ .",
"Although both $r$ and $\\beta $ scale the rates of the aqueous and precipitate reactions, respectively, they have different units.",
"$r$ has units of volume per time while $\\beta $ has units of mass per time.",
"Because experimental values of $r$ and $\\beta $ are expensive to acquire, for the sake of this simplified analysis we will assume that $\\beta \\approx r\\rho _m$ such that their effects don't impact the sign $\\chi $ .",
"The stoichiometric coefficient $c$ for $C\\text{(aq)}$ can be assumed $O(1)$ .",
"Finally, examine the reactants $\\psi _A$ and $\\psi _B$ .",
"Most experiments in microfluidic chambers use molar concentrations with an upper bound of $O(1 \\,\\text{M})$ ; for example, in [16] the maximum concentration of reactants was $0.5\\, \\text{M}$ .",
"Therefore, using parameter values taken from experiments, $\\chi > 0$ and fixed points exist for the reduced system.",
"Figure: Dynamical system for ψ C \\psi _C and θ m \\theta _m.",
"(a) Qualitative stability of ψ ˙ C (ψ C )\\dot{\\psi }_C(\\psi _C) ODE.",
"The left equilibrium ψ C - \\psi _C^- is stable and exists for χ≥0\\chi \\ge 0.",
"(b) Visualization of planar dynamical system.",
"The thicker line corresponds to homogeneous initial conditions for ψ C \\psi _C and θ m \\theta _m.",
"Shaded region is outside of the domain of (ψ C ,θ m )∈[0,∞)×[0,1](\\psi _C,\\theta _m)\\in [0,\\infty )\\times [0,1].We now consider the planar dynamical system in phase space $(\\psi _C,\\theta _m)\\in [0,\\infty )\\times [0,1]$ with fixed points $(\\Psi _C,\\Theta _m)=(\\psi _C^-,1)$ .",
"The dynamical system is $\\dot{\\psi }_C & = f(\\psi _C,\\theta _m) = \\frac{\\beta }{\\rho _m}\\psi _C^2 - \\frac{\\alpha \\beta }{\\rho _m}\\psi _C + cr\\psi _A\\psi _B \\\\\\dot{\\theta }_m & = g(\\psi _C,\\theta _m) = \\beta \\psi _C(1-\\theta _m)/\\rho _m $ where $r$ , $c$ , $\\rho _m$ , $\\alpha $ , $\\beta $ , $\\psi _A$ , and $\\psi _B$ are assumed to be known and constant.",
"The eigenvalues of the Jacobian generated by equations (REF ) evaluated at the fixed points provides information about the rate of growth of $\\psi _C$ and $\\theta _m$ .",
"This particular eigen-system is simple to interpret because the eigenvectors align with the coordinate axes and therefore the eigenvalues correspond to the rates that the physical variables $(\\psi _C,\\theta _m)$ approach their equilibria.",
"These rates are given by: $\\lambda _{\\psi _C} = -\\frac{1}{\\rho _m}\\sqrt{\\chi }, \\qquad \\lambda _{\\theta _m} = -\\frac{1}{2}\\left(\\frac{\\alpha \\beta }{\\rho _m}+\\lambda _{\\psi _C}\\right)\\, .",
"$ Both eigenvalues are negative, and therefore the fixed point is stable, because $\\alpha \\beta /\\rho _m +\\lambda _{\\psi _C} = \\alpha \\beta /\\rho _m - \\sqrt{\\chi }/\\rho _m > 0$ always.",
"For this same reason, it is true that $|\\lambda _{\\theta _m}| < |\\lambda _{\\psi _C}|$ , meaning the membrane volume fraction approaches its fixed point at a slower rate than the aqueous product.",
"This agrees with our intuition, as the conversion of $C\\text{(aq)}$ to $C\\text{(s)}$ means we would expect $\\theta _m$ production to lag behind $\\psi _C$ ."
],
[
"Poiseuille Analysis",
"We now solve the equations to see how the solvent velocity transitions from one-channel to two-channel flow.",
"An exact solution for $\\psi _C(x,t)$ can be found by solving a Ricatti equation with constant coefficients.",
"This solution $\\psi _C$ can be integrated using elementary functions, so $\\theta _s(x,t)$ can be found exactly because equation is separable; $\\theta _m(x,t)$ can then be found using the no-void assumption.",
"Finally, we solve the variable-coefficient BVP for $q_y$ numerically using finite differences.",
"For initial conditions, let $\\psi _C(x,t) = 0$ and $\\theta _s(x,t)=1$ .",
"Fix $\\psi _A^0(x)$ and $\\psi _B^0(x)$ to be piecewise-constant values in $x$ such that there is only a middle region in $x\\in (0,L)$ in which the reactants $A$ and $B$ are simultaneously present.",
"Given $\\psi _A^0(x)$ and $\\psi _B^0(x)$ , the Ricatti equation (REF ) can be solved exactly for $\\psi _C(x,t)$ (see appendix ): $\\psi _C(x,t) = \\gamma _1\\gamma _2\\frac{\\rho _m}{\\beta }\\left(\\frac{e^{\\gamma _2 t} - e^{\\gamma _1 t}}{\\gamma _2e^{\\gamma _1 t} - \\gamma _1e^{\\gamma _2 t}}\\right) $ where $\\gamma _{1,2}(x) = \\frac{1}{2}\\left(-\\alpha \\beta \\pm \\frac{\\sqrt{\\chi }}{\\rho _m}\\right)\\, .$ Because both the antiderivative of $\\psi _C$ can be given in terms of elementary functions (see appendix ) and equation is separable, we can also solve for solvent volume fraction $\\theta _s(x,t)$ exactly: $\\theta _s(x,t) = \\frac{\\gamma _1e^{\\gamma _2 t} - \\gamma _2 e^{\\gamma _1 t}}{\\gamma _1-\\gamma _2}$ where have implemented the initial condition $\\theta _s(x,0) = 1$ .",
"Then, the membrane volume fraction $\\theta _m(x,t)$ can be computed easily using the no-void assumption.",
"Until this point, all solutions in space have been treated independently.",
"The effect of variations in space is taken into account when solving for the longitudinal component of the Darcy velocity, $q_y$ .",
"We numerically solve the momentum equation () for $q_y$ using a finite difference method.",
"The $x$ domain is discretized into $N$ intervals of equal width $\\Delta x=1/N$ such that $x_j=j\\Delta x$ , $j=1,\\dots ,N-1$ , and use centered-difference approximations to the derivatives.",
"Note that $q_y(x_0)=q_y(x_N)=0$ due to the no-slip boundary conditions.",
"This discretization results in a tridiagonal linear system which can be solved in $O(N)$ complexity by using the Thomas algorithm [45].",
"The velocity is constrained to satisfy constant flux in accordance with experiments, which mathematically is represented by $\\int _0^Lq_y\\,dx = \\text{constant}\\,.$ This constant-flux condition allows the computation of $G(t_n)$ at each time step.",
"The authors use the Julia programming language to solve the BVP [5].",
"Figure: Developing membrane affects fluid flow.",
"(a) Flow profile in a 1D channel transitions from one-channel to two-channel flow.",
"The reaction region is shaded.",
"(b) Relevant variables evaluated in the reaction region at x=L/2x=L/2, normalized for legibility; ψ C \\psi _C develops first, followed by θ m \\theta _m, which when large enough triggers the transition from one- to two-channel flow.",
"The percolation threshold θ s * \\theta _s^* is set to θ s * =0.3\\theta _s^*=0.3, which is why negligible change in fluid velocity is seen until θ m ≈0.7\\theta _m\\approx 0.7.The developing membrane for the 1D reduced model geometry is shown in figure REF (a).",
"Membrane develops within the shaded region, which in this example is 10% of the domain.",
"Because the membrane has finite width, the constant-flux condition causes the pressure gradient $G(t)$ to increase with the developing membrane, causing the maximum speed for the two-channel flow to be slightly higher than the maximum speed for the one-channel flow.",
"In this sense, the developing membrane splits the domain into two symmetric one-channel flows.",
"Figure REF (b) shows the three main variables of the reduced model as functions of time, evaluated in the middle of the reaction region ($x=L/2$ ).",
"The $\\psi _C$ variable increases immediately due to the presence of $\\psi _A$ and $\\psi _B$ .",
"The membrane initially has zero growth rate due to the absence of $\\psi _C$ , and grows at a slower rate than $\\psi _C$ .",
"This ordering on the growth rates matches our expectations from the eigenvalue analysis of section REF .",
"The $q_y$ curve demonstrates the transition from one-channel to two-channel pipe flow by measuring the normalized value in the middle of the pipe as a function of time.",
"By comparing $q_y$ with $\\theta _m$ , one can see the effect of the percolation threshold $\\theta _s^*=0.3$ .",
"After the solvent volume fraction declines past this value, the fluid velocity begins to react more strongly to precipitating membrane.",
"Figure: Effect of singular friction term.",
"(a) Kozeny-Carman friction term, (b) Hill friction term and (c) generic friction term ξ B =hθ s θ m \\xi _B=h\\theta _s\\theta _m common in biofilm models.",
"The coefficients hh are scaled so as to make the three coefficients comparable in strength.",
"The first two friction terms produce the desired no-slip on the membrane interface and the third, while affecting the fluid flow, does not generate the desired no-slip boundary condition.Figure REF demonstrates the effect of using different friction coefficients.",
"Both figures REF (a,b) demonstrate the desired no-slip behavior in the membrane limit by using $\\xi _{KC}$ and $\\xi _H$ , respectively.",
"In figure REF (c) the effect of the friction coefficient $\\xi _B$ is shown.",
"While there is some effect on the flow profile, $\\xi _B$ does not demonstrate the no-slip condition on the membrane.",
"While the $\\xi _B$ term was developed primarily for high permeability applications, our framework was developed to capture the transition from purely fluid behavior, to partially permeable, to a fully-developed impermeable solid.",
"As demonstrated, this full transition requires either the $\\xi _{KC}$ or $\\xi _H$ friction coefficient.",
"Figure REF shows three flows with reaction regions of various sizes, and therefore different width of membranes.",
"The initial flow profiles of all are equivalent, as the reaction has not yet occurred and no membrane is present.",
"However, as membrane develops, the constant-flux condition requires that for regions with thicker membranes, the flow velocity must increase in the non-reacting regions to compensate for the loss of flux in the membrane region.",
"These results demonstrate that, once the membrane is fully developed, the flow domain treats the membrane portion as a no-slip boundary and the prescribed constant-flux conditions lead to the expected results from single-phase fluids.",
"Figure: Effect of increasing membrane thickness.",
"As a percentage of the domain length, membrane width is (a) 5%5\\%, (b) 15%15\\%, and (c) 33%33\\%.",
"The increasing maximum flow speed is due to the constant-flux constraint, and is analogous to that what would occur if the membrane boundaries were prescribed a priori in a single-phase flow."
],
[
"Conclusion",
"Typically, no-slip conditions on boundaries must be prescribed a priori when modeling fluid flows.",
"However, motivated by microfluidic experiments of a precipitate reaction, there exist situations where solid materials develop dynamically and so the exact location of such a boundary is not known a priori.",
"This situation exposes a deficiency in modeling techniques that rely on knowledge of where no-slip boundaries exist.",
"To formulate the model, we assumed conservation of mass for the entire chemistry-fluid-membrane system.",
"The reacting chemical species were assumed to exist in the fluid phase as volumeless scalar fields subject to mass flux determined by a combination of advection with the fluid flow and diffusion down gradients in molarity.",
"The careful consideration of changing solvent volume inside the domain led to extra terms being included in the reaction equations, and in particular conservation of mass for the entire system was used to determine the closure for the aqueous product's reaction equation.",
"A momentum equation for the fluid velocity followed the usual Cauchy stress formulation with the slight modification that all tensor quantities depend on the Darcy velocity $\\mathbf {q}_s=\\theta _s\\mathbf {v}_s$ , not simply the tracer velocity $\\mathbf {v}_s$ as is usual in the single- multi-phase fluids literature.",
"This choice, paired with incompressibility of the fluid Darcy velocity, led to a simplification of the momentum balance to a variable-coefficient Brinkman equation.",
"In order to demonstrate that the model reflects the expected behaviors, we first sought growth rate estimates from a linearized form of reduced equations.",
"Then, to show that the dynamically-generated no-slip boundary corresponded to development of the membrane, we performed what was essentially a Poisseuille analysis on the reduced model to show that the no-slip behavior agreed qualitatively with expected behaviors, specifically the recovery of one-to-two channel transitions and the effect of membrane width paired with a constant-flux condition.",
"We examined three potential terms to be used in the variable-coefficient Brinkman friction term, both by a direct comparison as functions of fluid volume fraction and by examining their effect on fluid flow from the perspective of how well they demonstrated no-slip behavior.",
"These results demonstrated that, qualitatively, the friction terms derived from Karman-Cozeny relationship and using Hill functions gave no-slip flow behavior on the developed membrane.",
"Additionally, the percolation threshold $\\theta _s^*$ can be chosen to reflect specific permeability properties of the structure under investigation.",
"Both of these friction coefficients were preferable to the term commonly employed in biofilm models because $\\xi _B$ does not recover the no-slip behavior in low-permeability regions.",
"To our knowledge, this is the first model that qualitatively captures the fluid-structure dynamics of a precipitate reaction in a low-Re environment where the dynamically-developing precipitate significantly affects the surrounding fluid flow.",
"Future work will focus on numerical simulation of the full model in various geometries to be used as a predictive tool for experimentalists.",
"In particular, exploring sufficient modeling conditions to generate the asymmetric growth in membrane in a 2D setting analogous to the experimental microfluidic domains was something that we were not able to explore in this paper, as diffusion was ignored to reduce the model equations.",
"A model capable of accurately capturing microfluidic experiments would be valuable to researchers using these devices to study precipitate reactions at the microscale and ultimately useful in examining origin of life theories."
],
[
"Acknowledgements",
"P.S.E.",
"is supported by the National Science Foundation (NSF) Graduate Research Fellowship under Grant 1449440.",
"M.N.J.M.",
"is supported by Simons Collaboration Grant 524259.",
"N.G.C.",
"is supported by NSF-CBET 1510743.",
"Q.W.",
"and O.S.",
"are supported by NSF Grants 1609495 and 1565734."
],
[
"Solving Ricatti's Differential Equation",
"The solution technique to Ricatti's differential equation [48] is not very well known, so the derivation is stated here for the interested reader.",
"A homogeneous 1st order ODE is called a Ricatti equation if it is quadratic in the unknown, i.e.",
"$y^{\\prime }(x) = q_0(x) + q_1(x)y(x) + q_2(x)y^2(x)\\,.",
"$ If $q_0(x) \\equiv 0$ , this reduces to Bernoulli's equation [48].",
"In general, one can transform Ricatti's equation to an equivalent 2nd order linear differential equation.",
"In this appendix, we detail this transformation and explicitly solve for the case of constant coefficients $q_0$ , $q_1$ and $q_2$ .",
"First, define a new variable $v$ by $v = q_2y $ so that equation (REF ) becomes $v^{\\prime } = v^2 + Rv + S $ where $R = q_1 + q_2^{\\prime }/q_2$ and $S = q_0q_2$ .",
"Then, introduce another variable, $u$ , related to $v$ via a Cole-Hopf transform: $v = -\\frac{u^{\\prime }}{u} $ and now the original equation, in terms of $u$ , becomes $u^{\\prime \\prime }-Ru^{\\prime }+Su = 0\\,.",
"$ For constant coefficients $R$ and $S$ , equation (REF ) can be solved exactly.",
"Using terminology from the present paper, the constant coefficient Ricatti equation is $\\dot{\\psi }_C = c r \\psi _A\\psi _B - \\frac{\\alpha \\beta }{\\rho _m} \\psi _C + \\frac{\\beta }{\\rho _m}\\psi _C^2$ where $\\psi _C=\\psi _C(t)$ is the unknown variable and $r$ , $c$ , $\\rho _m$ , $\\alpha $ , $\\beta $ , $\\psi _A$ , and $\\psi _B$ are fixed parameters.",
"Relating this to equation (REF ), let $y=\\psi _C$ , $q_0 = c r \\psi _A\\psi _B$ , $q_1=-\\alpha \\beta /\\rho _m$ , and $q_2=\\beta /\\rho _m$ .",
"Therefore, in the final equation, $R = q_1 + q_2^{\\prime }/q_2 = -\\alpha \\beta /\\rho _m$ and $S = q_0q_2 = c r \\beta \\psi _A\\psi _B/\\rho _m$ .",
"The solution to equation (REF ) when $R$ and $S$ are constant depends on the eigenvalues of its corresponding characteristic equation.",
"More specifically, it depends on the sign of the determinant of the root of the characteristic equation $R^2-4S=\\chi /\\rho _m^2$ , where $\\chi $ , defined in section REF , was determined to be positive for physically realistic parameter values.",
"The solution to this case – where the characteristic equation to equation (REF ) has two real, distinct roots – can be found in any introductory book on ODEs [6], but for completeness the solution is detailed here with homogeneous initial conditions $y(0) = 0$ : $y(t) = \\frac{\\gamma _1\\gamma _2}{q_2}\\left[\\frac{e^{\\gamma _2 t} - e^{\\gamma _1 t}}{\\gamma _2 e^{\\gamma _1 t} - \\gamma _1e^{\\gamma _2 t}}\\right]\\,.",
"$ where $\\gamma _{1} & = \\frac{R + \\sqrt{R^2-4S}}{2}, & \\gamma _{2} & = \\frac{R - \\sqrt{R^2-4S}}{2} \\\\R & = q_1, & S & = q_0q_2.$ and the antiderivative of this solution is given by $\\int y(t) \\,\\, dt = -\\frac{1}{q_2}\\log (\\gamma _1 e^{\\gamma _2 t} - \\gamma _2 e^{\\gamma _1 t}) + C$ where $C$ is an arbitrary constant of integration."
]
] | 1906.04216 |
[
[
"Vi\\`ete's fractal distributions and their momenta"
],
[
"Abstract Solutions of Schr\\\"oder-Poincar\\'e's polynomial equations $f(az)=P(f(z))$ usually do not admit a simple closed-form representation in terms of known standard functions.",
"We show that there is a one-to-one correspondence between zeros of $f$ and a set of discrete functions stable at infinity.",
"The corresponding Vi\\`ete-type infinite products for zeros of $f$ are also provided.",
"This allows us to obtain a special kind of closed-form representation for $f$ based on the Weierstrass-Hadamard factorization.",
"From this representation, it is possible to derive explicit momenta formulas for zeros.",
"We discuss also the rate of convergence of WH-factorization and momenta formulas.",
"Obtaining explicit closed-form expressions is the main motivation for this work.",
"Finally, all the branches of the multi-valued function $f^{-1}$ are computed explicitly."
],
[
"Introduction and main results",
"The classical Viète's formula $\\frac{2}{\\pi }=\\frac{\\sqrt{2}}{2}\\cdot \\frac{\\sqrt{2+\\sqrt{2}}}{2}\\cdot \\frac{\\sqrt{2+\\sqrt{2+\\sqrt{2}}}}{2}...$ uses nested square root radicals to represent the constant $\\pi $ .",
"Wiki says \"By now many formulas similar to Viète's involving either nested radicals or infinite products of trigonometric functions are known for $\\pi $ , as well as for other constants such as the golden ratio\", see, e.g., [1], [2], [4], [3].",
"In this note, we derive formulas for zeros of functions satisfying Schröder-Poincaré's polynomial equations.",
"In general, the formulas for zeros will involve various nested-radicals products similar to Viète's.",
"These formulas can be used in Weierstrass-Hadamard factorization to obtain various closed-form expressions.",
"Finally, looking through \"A chronology of continued square roots and other continued compositions\" [11], I found paper [12], where a detailed analysis of real roots of $f$ , satisfying $f(az)=f(z)^2+c$ , is provided.",
"Many interesting facts are presented in [11], e.g., an interesting story of the famous formula $\\varepsilon _0\\sqrt{2+\\varepsilon _1\\sqrt{2+\\varepsilon _2\\sqrt{2+...}}}=2\\sin \\biggl (\\frac{\\pi }{4}\\sum _{n=0}^{+\\infty }\\frac{\\varepsilon _0\\varepsilon _1\\varepsilon _2...\\varepsilon _n}{2^n}\\biggr ),$ where $\\varepsilon _i=-1,0,1$ .",
"We assume facts about existence of entire solutions of SP-equation to be known, see, e.g., [6], [9].",
"Let $P$ be some polynomial of degree $d\\geqslant 2$ .",
"Let $b$ be some its repelling point $P(b)=b$ , with $|a|>1$ for $a:=P^{\\prime }(b)$ .",
"Consider the entire solution $f$ of SP-equation $f(az)=P(f(z))$ satisfying $f(0)=b$ , $f^{\\prime }(0)=1$ .",
"This solution can be taken as $f(z)=\\lim _{n\\rightarrow \\infty }\\underbrace{P\\circ ...\\circ P}_n(b+a^{-n}z),\\ \\ z\\in {\\mathbb {C}},$ see, e.g., [14].",
"Composition (REF ) converges uniformly in any compact subset of ${\\mathbb {C}}$ .",
"For simplicity, let us assume $b\\ne 0$ .",
"This is not a restriction, since $\\widetilde{f}:=f+c$ , $c\\in {\\mathbb {C}}$ , also satisfies some polynomial SP-equation.",
"Let $P_0^{-1}(w)$ , $w\\in {\\mathbb {C}}$ be the principal branch of $P^{-1}$ analytic in some open domain containing $b$ , where $P_0^{-1}(b)=b$ .",
"We assume also that Hypothesis 1.",
"For any $w\\in {\\mathbb {C}}$ the orbit $(P_0^{-1})^{\\circ n}(w):=\\underbrace{P_0^{-1}\\circ ...\\circ P_{0}^{-1}}_n(w)\\rightarrow b.$ This assumption means that point $b$ repelling for $P$ is attracting for $P_0^{-1}$ .",
"Note that once $(P_0^{-1})^{\\circ k}(w)\\in \\lbrace |w-b|<\\delta \\rbrace $ for some small $\\delta > 0$ and some $k\\in {\\mathbb {N}}$ , then $(P_0^{-1})^{\\circ n}(w)$ stays in $\\lbrace |w-b|<\\delta \\rbrace $ for $n>k$ and $(P_0^{-1})^{\\circ n}(w)\\rightarrow b$ , since $|(P_0^{-1})^{\\prime }(b)|=|a^{-1}|<1.$ Let $P_j^{-1}$ , $1\\leqslant j\\leqslant d-1$ be other branches of $P^{-1}$ so that $\\lbrace z_j(w)\\rbrace _{j=0}^{d-1}=\\lbrace P_j^{-1}(w)\\rbrace _{j=0}^{d-1}$ is a complete set of solutions of $P(z)=w$ , defined for all $w\\in {\\mathbb {C}}$ .",
"For our research, it does not matter how the branches of $P^{-1}$ are numbered.",
"There are only two things that we should pay close attention to: 1) analyticity of the principal branch $P_0^{-1}$ at an open neighbourhood of its attracting point $b$ ; 2) Hypothesis 1.",
"Introduce the polynomial $Q(z):=\\frac{P(z)-P(b)}{z-b}=\\frac{P(z)-b}{z-b}$ and the set of discrete functions stable at infinity $\\Sigma =\\lbrace \\sigma :{\\mathbb {N}}\\rightarrow \\lbrace 0,...,d-1\\rbrace ,\\ \\lim _{n\\rightarrow \\infty }\\sigma _n=0\\rbrace .$ Theorem 1.1 The set of zeros of $f$ coincides with $\\lbrace z(\\sigma )\\rbrace _{\\sigma \\in \\Sigma }$ , where $z(\\sigma )=-b\\prod _{n=1}^{\\infty }\\frac{a}{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0))}.$ Each zero is counted according to its multiplicity.",
"In other words, the multiplicity of $z_0$ as zero of $f$ is $\\#\\lbrace \\sigma \\in \\Sigma :\\ z(\\sigma )=z_0\\rbrace $ .",
"We may apply Theorem REF to the function $F(z):=f(z)-w$ with some constant $w\\in {\\mathbb {C}}$ , because $F$ also satisfies SP-equation $F(az)=P(F(z)+w)-w$ similar to that for $f$ .",
"We only should care about the assumption $f(0)-w=b-w\\ne 0$ , see after (REF ).",
"Corollary 1.2 All the solutions of $f(z)=w$ , where $w\\in {\\mathbb {C}}\\setminus \\lbrace b\\rbrace $ , have the form $z=g_{\\sigma }(w)=(w-b)\\prod _{n=1}^{\\infty }\\frac{a}{Q(P^{-1}_{\\sigma _n}\\circ ...\\circ P^{-1}_{\\sigma _1}(w))},\\ \\ \\sigma \\in \\Sigma .$ Each solution is counted according to its multiplicity.",
"The case $w=b$ is special, because formal setting $w=b$ in (REF ) leads to $g_{\\sigma }(b)=0$ , $\\forall \\sigma \\in \\Sigma $ which seems doubt.",
"The next theorem is devoted to the case $w=b$ .",
"Theorem 1.3 All the solutions of $f(z)=b$ have the form $z=g_{0,0}=0$ or $z=g_{\\sigma ,m}=a^m(P_{\\sigma _1}^{-1}(b)-b)\\prod _{n=2}^{\\infty }\\frac{a}{Q(P^{-1}_{\\sigma _n}\\circ ...\\circ P^{-1}_{\\sigma _1}(b))},\\ \\ m\\in {\\mathbb {N}}\\cup \\lbrace 0\\rbrace ,\\ \\ \\sigma \\in \\Sigma ^{\\prime },$ where $\\Sigma ^{\\prime }=\\lbrace \\sigma \\in \\Sigma :\\ \\sigma _1\\ne 0\\rbrace $ .",
"In fact, $\\lbrace g_{\\sigma }\\rbrace _{\\sigma \\in \\Sigma }$ are all the branches of super-multi-valued function $f^{-1}$ .",
"Depending on the choice of the branches $P^{-1}$ , the functions $g_{\\sigma }(w)$ may or may not be analytic.",
"We can only state that the branch $g_0(w)$ ($\\sigma =0$ ) is analytic in $\\Omega \\setminus \\lbrace b\\rbrace $ , where $\\Omega $ is some small neighborhood of $b$ , e.g., considered in the remark after Hypothesis 1.",
"Due to (REF ) and arguments presented before (REF ), we have $g_0(w)=\\lim _{n\\rightarrow \\infty }a^n(\\underbrace{P_0^{-1}\\circ ...\\circ P_0^{-1}}_n(w)-b).$ Differentiating (REF ) and using $(P_0^{-1})^{\\prime }(w)=1/P^{\\prime }(P_0^{-1}(w))$ , $w\\in \\Omega $ we obtain $g_0^{\\prime }(w)=\\prod _{n=1}^{\\infty }\\frac{a}{P^{\\prime }(\\underbrace{P_0^{-1}\\circ ...\\circ P_0^{-1}}_n(w))}.$ The convergence of the product (REF ), as well as (REF ), and (REF ), (REF ) is exponentially fast, as discussed in the beginning of Section .",
"The order of the entire function $f$ can be computed explicitly by substituting $\\alpha e^{A|z|^{\\rho }}$ into SP-equation $f(az)=P(f(z))$ , see, e.g., [13].",
"Extracting leading terms after the substitution, we obtain $\\rho =\\frac{\\ln d}{\\ln |a|}$ .",
"If $\\rho <1$ then the Weierstrass-Hadamard (WH) factorization for $f$ does not contain exponential factors, see [15].",
"Corollary 1.4 Suppose that $d<|a|$ .",
"If $w\\ne b$ then WH-factorization for $f$ is $f(z)=w+(b-w)\\prod _{\\sigma \\in \\Sigma }\\biggl (1-\\frac{z}{g_{\\sigma }(w)}\\biggr ),\\ \\ z\\in {\\mathbb {C}}.$ In particular, $f(z)=b\\prod _{\\sigma \\in \\Sigma }\\biggl (1+\\frac{z}{b}\\prod _{n=1}^{\\infty }\\frac{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0))}{a}\\biggr ),\\ \\ z\\in {\\mathbb {C}}.$ If $w=b$ then $f(z)=b+z\\prod _{m\\geqslant 0}\\prod _{\\sigma \\in \\Sigma ^{\\prime }}\\biggl (1-\\frac{z}{a^m g_{\\sigma ,0}}\\biggr )=b+\\biggl (\\biggl \\lbrace \\frac{z}{g_{\\sigma ,0}}\\biggr \\rbrace _{\\sigma \\in \\Sigma ^{\\prime }};\\frac{1}{a}\\biggr )_{\\infty }z,\\ \\ z\\in {\\mathbb {C}},$ where $(\\lbrace \\alpha \\rbrace _{i\\in R};\\beta )_{\\infty }:=\\prod _{i\\in R}\\prod _{n=0}^{\\infty }(1-\\alpha _i\\beta ^n)$ is the q-Pochhammer symbol.",
"Equation (REF ) allows us to compute explicitly momentum formulas for zeros of $f(z)-w$ , for any fixed $w\\ne b$ .",
"This new type of formulas will include both: infinite products and infinite sums.",
"The first (negative) momentum formula for zeros follows from (REF ) immediately $\\sum _{\\sigma \\in \\Sigma }\\prod _{n=1}^{\\infty }\\frac{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w))}{a}=f^{\\prime }(0)=1,\\ \\ \\forall w\\in {\\mathbb {C}}\\setminus \\lbrace b\\rbrace .$ Let us note how to compute explicitly other momenta of zeros.",
"First, differentiating $f(az)=P(f(z))$ at $z=0$ and using $f(0)=b$ , $f^{\\prime }(0)=1$ , $P^{\\prime }(b)=a$ , we obtain recurrent formulas to determine all the derivatives: $f^{\\prime \\prime }(0)&=(a^2-a)^{-1}P^{\\prime \\prime }(b), \\\\f^{(m)}(0)&=(a^m-a)^{-1}\\sum _{j=2}^{m}P^{(j)}(b)B_{m,j}(f^{\\prime }(0),...,f^{(m-j+1)}(0)),\\ \\ m\\geqslant 2,$ where $B_{m,j}$ are Bell polynomials.",
"They are given by $B_{m,j}(x_1,...,x_{m-j+1})=\\sum \\frac{m!}{k_1!...k_{m-j+1}!",
"}\\biggl (\\frac{x_1}{1!",
"}\\biggr )^{k_1}...\\biggl (\\frac{x_{m-j+1}}{(m-j+1)!",
"}\\biggr )^{k_{m-j+1}},$ where the sum is taken over all sequences $k_1$ , $k_2$ , ..., $k_{m-j+1}$ of non-negative integers such that the two conditions are satisfied: $\\sum _{i=1}^{m-j+1}k_i=j,\\ \\ \\sum _{i=1}^{m-j+1}ik_i=m,$ see more about Faà di Bruno's formula for high order derivatives of compositions in, e.g., wiki.",
"Now, differentiating $\\ln (f(z)-w)$ at $z=0$ and using (REF ), we obtain the momenta formulas of high orders $m\\geqslant 2$ : $\\sum _{\\sigma \\in \\Sigma }\\prod _{n=1}^{\\infty }\\frac{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w))^2}{a^2}&=f^{\\prime }(0)^2-bf^{\\prime \\prime }(0)=1-\\frac{b P^{\\prime \\prime }(b)}{a^2-a} \\\\\\sum _{\\sigma \\in \\Sigma }\\prod _{n=1}^{\\infty }\\frac{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w))^m}{a^m}&=\\sum _{j=1}^{m}\\frac{(-b)^{m-j}(j-1)!}{(m-1)!",
"}B_{m,j}(f^{\\prime }(0),...,f^{(m-j+1)}(0)).$ Another type of Vieta formulas also follows from (REF ): $\\sum _{\\sigma \\in \\Sigma }\\frac{w-b}{g_{\\sigma }(w)}=f^{\\prime }(0)=1,\\ \\ \\ \\sum _{\\sigma \\ne \\tau }\\frac{(w-b)^2}{g_{\\sigma }(w)g_{\\tau }(w)}=\\frac{f^{\\prime \\prime }(0)}{2!",
"}=\\frac{P^{\\prime \\prime }(b)}{2(a^2-a)}$ and so on.",
"There is a natural extension of $g_{\\sigma }(w)$ to the case $w=b$ : $g_{\\sigma }(b)={\\left\\lbrace \\begin{array}{ll}0, & \\sigma =0,\\\\g_{\\sigma ^{\\prime },m}, & \\sigma ^{\\prime }=\\sigma |_{\\lbrace n:\\ n\\geqslant m\\rbrace },\\ {\\rm where}\\ \\sigma _m\\ne 0\\ {\\rm and}\\ \\sigma _n=0\\ {\\rm for}\\ n<m,\\end{array}\\right.",
"}\\ \\ \\sigma \\in \\Sigma .$ Then all the solutions $f(z)=w$ , where $w\\in {\\mathbb {C}}$ , have the form $z=g_{\\sigma }(w)$ , $\\sigma \\in \\Sigma $.",
"It is possible to estimate the remainder of series (), (REF ), and products (REF )-(REF ).",
"They converge exponentially fast, regarding the length of the support of functions $\\sigma \\in \\Sigma $ .",
"Theorem 1.5 There is $C>0$ , depending on the polynomial $P$ only, such that $|g_{\\sigma }(w)|\\geqslant C|a|^{|\\mathop {\\mathrm {supp}}\\nolimits \\sigma |},\\ \\sigma \\in \\Sigma ,\\ \\ \\sigma \\ne 0.$ Moreover, if $d<|a|$ then $\\sum _{|\\mathop {\\mathrm {supp}}\\nolimits \\sigma |\\geqslant N}|g_{\\sigma }(w)|^{-m}=O((d|a|^{-m})^N),\\ \\ \\prod _{|\\mathop {\\mathrm {supp}}\\nolimits \\sigma |\\geqslant N}\\biggl |1-\\frac{z}{g_{\\sigma }(w)}\\biggr |=1+O(|z|(d|a|^{-1})^N),$ for $N\\rightarrow \\infty $ , $m\\geqslant 1$ , and any fixed $z\\in {\\mathbb {C}}$ .",
"We will begin the next section with examples.",
"The proof of the main result is placed in the final section."
],
[
"Examples",
"1.",
"Consider the case $P(z)=2z^2-1$ .",
"SP-equation is $f(az)=2f(z)^2-1$ .",
"We take $f(0)=b=1$ , $f^{\\prime }(0)=1$ .",
"Then $a=(2z^2-1)^{\\prime }|_{z=b}=4$ .",
"Polynomial (REF ) is $Q(z)=\\frac{2z^2-1-1}{z-1}=2z+2.$ There are two branches of $P^{-1}$ : $P_{1}^{-1}(w)=\\sqrt{\\frac{1+w}{2}},\\ \\ P_{-1}^{-1}(w)=-\\sqrt{\\frac{1+w}{2}}.$ We assume that $\\sqrt{z}=r^{\\frac{1}{2}}e^{\\frac{i\\vartheta }{2}}\\ \\ {\\rm for}\\ \\ z=r e^{i\\vartheta },\\ r\\geqslant 0,\\ \\vartheta \\in (-\\pi ,\\pi ].$ The branch $P_1^{-1}$ is principal.",
"It is analytically defined near the attracting (for $P_1^{-1}$ ) point $b$ .",
"Moreover, $(P_1^{-1})^{\\circ n}(w)$ converges to its fixed point $b$ for any $w\\in {\\mathbb {C}}$ , since $\\sqrt{z}$ is a contraction mapping in the closed domain $D=\\lbrace z:\\ \\mathop {\\mathrm {Re}}\\nolimits z\\geqslant 1/\\sqrt{2}\\rbrace $ : $|\\sqrt{z_1}-\\sqrt{z_2}|=\\frac{|z_1-z_2|}{|\\sqrt{z_1}+\\sqrt{z_2}|}\\leqslant \\frac{|z_1-z_2|}{\\sqrt{2}},\\ \\ z_1,z_2\\in D,$ and $P_1^{-1}\\circ P_1^{-1}({\\mathbb {C}})\\subset D$ .",
"Thus, Hypothesis 1 is satisfied and we can use Theorem REF and its Corollaries.",
"To parameterize zeros of $f$ , we should use the set $\\Sigma =\\lbrace \\sigma :{\\mathbb {N}}\\rightarrow \\lbrace \\pm 1\\rbrace ,\\ \\lim _{n\\rightarrow \\infty }\\sigma _n=1\\rbrace .$ Then zeros of $f$ have form (REF ) $z(\\sigma )=-\\prod _{n=1}^{\\infty }\\frac{4}{2+2\\sigma _n\\sqrt{\\frac{1}{2}+...+\\frac{\\sigma _1}{2}\\sqrt{\\frac{1}{2}}}}=-\\prod _{n=1}^{\\infty }\\frac{1}{\\frac{1}{2}+\\frac{\\sigma _n}{2}\\sqrt{\\frac{1}{2}+...+\\frac{\\sigma _1}{2}\\sqrt{\\frac{1}{2}}}}.$ Computations show $z(1,1,1,...)=-\\frac{\\pi ^2}{8},\\ \\ z(-1,1,1,...)=-\\frac{9\\pi ^2}{8},\\ \\ z(-1,-1,1,...)=-\\frac{25\\pi ^2}{8},\\ \\ z(1,-1,1,...)=-\\frac{49\\pi ^2}{8}$ and so on.",
"This is in full agreement with expected values, since $f(z)=\\cos \\sqrt{-2z}$ .",
"In this case, the formulas for zeros are, in fact, modified Viète's formulas, see also [1], [2].",
"The order of entire function $f$ is $1/2$ .",
"WH-factorization is $\\cos \\sqrt{-2z}=\\prod _{n=1}^{\\infty }\\biggl (1+\\frac{8z}{(2n-1)^2\\pi ^2}\\biggr )=\\prod _{\\sigma \\in \\Sigma }\\biggl (1+z\\prod _{n=1}^{\\infty }\\biggl (\\frac{1}{2}+\\frac{\\sigma _n}{2}\\sqrt{\\frac{1}{2}+...+\\frac{\\sigma _1}{2}\\sqrt{\\frac{1}{2}}}\\biggr )\\biggr ).$ 2.",
"Consider the case $P(z)=z^2-1$ .",
"SP-equation is $f(az)=f(z)^2-1$ .",
"We take $f(0)=b=\\frac{\\sqrt{5}+1}{2}$ , $f^{\\prime }(0)=1$ .",
"Then $a=(z^2-1)^{\\prime }|_{z=b}=2b$ .",
"Polynomial (REF ) is $Q(z)=\\frac{z^2-1-b}{z-b}=z+b.$ There are two branches of $P^{-1}$ : $P_{1}^{-1}(w)=\\sqrt{1+w},\\ \\ P_{-1}^{-1}(w)=-\\sqrt{1+w}.$ Again, using the arguments from Example 1, we can state that Hypothesis 1 is satisfied.",
"To parametrize zeros of $f$ , we should use the same set as in the previous example $\\Sigma =\\lbrace \\sigma :{\\mathbb {N}}\\rightarrow \\lbrace \\pm 1\\rbrace ,\\ \\lim _{n\\rightarrow \\infty }\\sigma _n=1\\rbrace .$ Then zeros of $f$ have the form $z(\\sigma )=-b\\prod _{n=1}^{\\infty }\\frac{2b}{b+\\sigma _n\\sqrt{1+...+\\sigma _1\\sqrt{1}}}.$ Figure: (a) zeros of f(z)f(z) in the complex plane, |z|⩽5·10 5 |z|\\leqslant 5\\cdot 10^5; (b) images f -1 (w)(=g σ (w))f^{-1}(w)(=g_{\\sigma }(w)), see (), of circles |w|=r 3 125.05|w|=\\frac{r^3}{125.05}, r=1,...,10r=1,...,10 for the first 10 3 10^3 values σ∈Σ\\sigma \\in \\Sigma depicted in different colors; (c) real and imaginary parts of f(z)f(z), where min{| Re f(z)|,| Im f(z)|}⩽2\\min \\lbrace |\\mathop {\\mathrm {Re}}\\nolimits f(z)|,|\\mathop {\\mathrm {Im}}\\nolimits f(z)|\\rbrace \\leqslant 2 and 10 -3 z∈[-2.5,0]×[-1.25,1.25]10^{-3}z\\in [-2.5,0]\\times [-1.25,1.25].The first negative zero $z(1,1,1,...)=-2C$ relates to the so-called Paris constant $C$ appearing in the approximation of the golden ratio by nested square root radicals, see [4], [5], [10].",
"Zeros of $f$ are also related to the polynomial dynamics generated by $P=z^2-1$ and, hence, approximate the corresponding Julia set growing up to infinity, see more in [7], [8], [9].",
"The zeros form impressive fractal structures, see Fig.",
"REF .",
"The order of entire function $f$ is $\\ln 2/\\ln a<1$ .",
"Hence, there is WH-factorization $f(z)=b\\prod _{\\sigma \\in \\Sigma }\\biggl (1+\\frac{z}{b}\\prod _{n=1}^{\\infty }\\frac{b+\\sigma _n\\sqrt{1+...+\\sigma _1\\sqrt{1}}}{2b}\\biggr ).$ There are infinitely many complex zeros of multiplicities $2^n$ for any $n\\geqslant 0$ , see [10].",
"All the multiplicities are taken into account in WH-factorization mentioned above.",
"The first, second and third momentum formulas for zeros, see (REF ), (REF ) and (), are $\\sum _{\\sigma \\in \\Sigma }\\prod _{n=1}^{\\infty }\\frac{(b+\\sigma _n\\sqrt{1+...+\\sigma _1\\sqrt{1}})^m}{(2b)^m}={\\left\\lbrace \\begin{array}{ll}1,& m=1, \\\\ 1-\\frac{1}{\\sqrt{5}},& m=2,\\\\ \\frac{2}{5},& m=3.",
"\\end{array}\\right.",
"}$ 3.",
"Let us consider the cubic SP-equation $f(az)=f(z)^3-6$ , $f(0)=b=2$ , $f^{\\prime }(0)=1$ .",
"Then $a=3b^2=12$ .",
"The order of the entire function $f(z)$ is $\\ln 3/\\ln 12<1$ .",
"Let us skip the similar arguments as in the previous examples that show that the principal branch of $P^{-1}$ for $P(z)=z^3-6$ satisfies Hypothesis 1.",
"So, we can use (REF ), (REF ) to obtain explicit momentum formulas $\\sum _{k_n\\in \\lbrace 0,1,2\\rbrace ; \\ \\lim k_n=0}\\ \\prod _{n=1}^{\\infty }\\frac{(e^{\\frac{2\\pi i k_n}{3}}\\@root 3 \\of {6+...+e^{\\frac{2\\pi i k_1}{3}}\\@root 3 \\of {6}})^2+2e^{\\frac{2\\pi i k_n}{3}}\\@root 3 \\of {6+...+e^{\\frac{2\\pi i k_1}{3}}\\@root 3 \\of {6}}+4}{12}=1,$ $\\sum _{k_n\\in \\lbrace 0,1,2\\rbrace ; \\ \\lim k_n=0}\\biggl (\\prod _{n=1}^{\\infty }\\frac{(e^{\\frac{2\\pi i k_n}{3}}\\@root 3 \\of {6+...+e^{\\frac{2\\pi i k_1}{3}}\\@root 3 \\of {6}})^2+2e^{\\frac{2\\pi i k_n}{3}}\\@root 3 \\of {6+...+e^{\\frac{2\\pi i k_1}{3}}\\@root 3 \\of {6}}+4}{12}\\biggr )^2=\\frac{9}{11}$ and so on.",
"All of the momenta are rational numbers."
],
[
"Proof of Theorems ",
"First of all let us show that infinite products (REF ) are well defined.",
"Suppose that $Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w))=0$ for some $w\\in {\\mathbb {C}}$ and $n\\in {\\mathbb {N}}$ .",
"If $n=1$ then (REF ) gives $w=b$ .",
"Consider the case $n>1$ .",
"We have that $P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w)\\ne b$ , since $Q(b)=P^{\\prime }(b)=a\\ne 0$ .",
"Next, if $P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w)\\ne b$ then (REF ) and (REF ) give us $P(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w))=P_{\\sigma _{n-1}}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w)=b,$ which leads to $w=b$ , since $P(b)=b$ .",
"Hence any denominator in (REF ) is non-zero, since $b\\ne 0$ by the assumption from the beginning of the article.",
"Due to analyticity of $P_0^{-1}$ in some open neighbourhood of its attracting point $b$ , where $|(P_0^{-1})^{\\prime }(b)|=|a^{-1}|<1$ , we have that for any $\\varepsilon >0$ there is $\\delta >0$ such that for any $w\\in \\lbrace |w-b|<\\delta \\rbrace $ : $P_0^{-1}(b+w)=b+R(w),\\ \\ |R(w)|<(|a^{-1}|+\\varepsilon )|w|.$ Identity and inequality (REF ) along with (REF ) and the stability condition $\\lim \\sigma _n=0$ in (REF ) lead to $P_{\\sigma _{n}}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)=b+O(c^{n}),\\ \\ n\\rightarrow \\infty ,$ where small $\\varepsilon >0$ is taken such that $c:=|a^{-1}|+\\varepsilon <1$ .",
"Hence $Q(P_{\\sigma _{n}}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0))=Q(b)+O(c^{n})=P^{\\prime }(b)+O(c^{n})=a+O(c^{n}).$ This guaranties the convergence of infinite products (REF ) for any $\\sigma \\in \\Sigma $ .",
"The convergence of the products is exponentially fast, since $c<1$ .",
"Note that the same arguments give also the exponential rate of convergence of (REF ).",
"Let $\\widetilde{z}$ be some zero of $f(z)$ of multiplicity $\\widetilde{m}\\in {\\mathbb {N}}$ , i.e.",
"$f^{(j)}(\\widetilde{z})=0\\ \\ {\\rm for}\\ \\ j=0,...,\\widetilde{m}-1.$ SP-equation gives $f(z)=P^{\\circ n}(f(a^{-n}z)),\\ \\ n\\in {\\mathbb {N}}.$ Taking $n>\\widetilde{m}$ such that $f^{\\prime }(a^{-n}\\widetilde{z})\\ne 0$ (recall that $f^{\\prime }(0)=1\\ne 0$ and $|a|>1$ ), differentiating (REF ) at $z=\\widetilde{z}$ and using (REF ), we obtain that $(P^{\\circ n})^{(j)}(f(a^{-n}\\widetilde{z}))=0,\\ \\ j=0,...,\\widetilde{m}-1.$ Thus, $f(a^{-n}\\widetilde{z})$ is a root of $P^{\\circ n}$ of a multiplicity at least $\\widetilde{m}$ .",
"This means that $f(a^{-n}\\widetilde{z})=P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)$ for at least $\\widetilde{m}$ different $\\sigma :\\lbrace 1,...,n\\rbrace \\rightarrow \\lbrace 0,...,d\\rbrace $ .",
"SP-equation can be written in the form $f(a^{-1}z)=P^{-1}(f(z))$ .",
"Since $f(0)=b$ , the branch $P^{-1}$ should coincide with the principal branch $P_0^{-1}$ in a small neighbourhood of $b$ , i.e.",
"$f(a^{-1}z)=P_0^{-1}(f(z))$ for all sufficiently small $z$ (see also the remark before Hypothesis 1).",
"Let $\\widetilde{n}$ be such that $f(a^{-\\widetilde{n}}\\widetilde{z})$ belongs to this small neighbourhood of $b$ .",
"We assume also that $\\widetilde{n}$ is large enough to satisfy (REF ) with at least $\\widetilde{m}$ different $\\sigma $ .",
"Then, by (REF ), we have $f(a^{-\\widetilde{n}-k}\\widetilde{z})=\\underbrace{P_0^{-1}\\circ ...\\circ P_0^{-1}}_k\\circ P_{\\sigma _{\\widetilde{n}}}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0),\\ \\ k\\geqslant 0.$ Denote $\\sigma _n=0$ for $n>\\widetilde{n}$ .",
"Thus, using $f(0)=b$ , $f^{\\prime }(0)=1$ , we get $\\widetilde{z}=\\lim _{n\\rightarrow \\infty }a^n(f(a^{-n}\\widetilde{z})-b)=\\lim _{n\\rightarrow \\infty }a^n(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)-b)=\\\\\\lim _{n\\rightarrow \\infty }\\frac{a(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)-b)}{P_{\\sigma _{n-1}}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)-b}a^{n-1}(P_{\\sigma _{n-1}}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)-b)=\\\\\\lim _{n\\rightarrow \\infty }\\frac{a}{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0))}(P_{\\sigma _{n-1}}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)-b)=-b\\prod _{n=1}^{\\infty }\\frac{a}{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0))}.$ Like (REF ), identity $\\widetilde{z}=-b\\prod _{n=1}^{\\infty }\\frac{a}{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0))}$ holds for at least $\\widetilde{m}$ different $\\sigma \\in \\Sigma $ .",
"Conversely, suppose that (REF ) holds for $\\widetilde{m}$ different $\\sigma \\in \\Sigma $ .",
"To finish the proof we need to show that $\\widetilde{z}$ is a zero of $f$ of a multiplicity at least $\\widetilde{m}$ .",
"Using (REF ) and the second identity in (REF ), we obtain $f(\\widetilde{z})=\\lim _{n\\rightarrow \\infty }f(a^n(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)-b))=\\\\\\lim _{n\\rightarrow \\infty }\\underbrace{P\\circ ...\\circ P}_n(b+a^{-n}(a^n(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)-b)))=\\\\\\lim _{n\\rightarrow \\infty }\\underbrace{P\\circ ...\\circ P}_n\\circ P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)=0,$ since the convergence of (REF ) is uniform in any bounded domain.",
"Hence $\\widetilde{z}$ is a zero of $f$ .",
"Now, let $N$ be such that $P_{\\sigma _N}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)$ is sufficiently close to $b$ , where $f^{-1}$ is defined, so that $P_{\\sigma _N}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0)=f(t_{\\sigma }).$ We can do this because $P_{\\sigma _n}^{-1}=P_0^{-1}$ for large $n$ by definition (REF ), and we are under Hypothesis 1.",
"We also assume that $N$ is so large that (REF ) is valid for at least $\\widetilde{m}$ different $\\sigma :\\lbrace 1,...,N\\rbrace \\rightarrow \\lbrace 0,...,d\\rbrace $ coinciding with the segments of those $\\sigma \\in \\Sigma $ mentioned in (REF ), and, also, all $\\sigma _n=0$ for $n>N$ .",
"For simplicity, in the previous sentence we use the same symbol for $\\sigma \\in \\Sigma $ and for its segment $\\sigma |_{\\lbrace 1,...,N\\rbrace }$ .",
"Finally, it is assumed that $N$ is so large that $\\underbrace{P_0^{-1}\\circ ...\\circ P_0^{-1}}_n\\circ P_{\\sigma _N}^{-1}\\circ ...\\circ P_{\\sigma _1}(0)=f(a^{-n}t_{\\sigma }),$ see comments before (REF ).",
"Using (REF ), (REF ), the assumption $\\sigma _n=0$ , $n>N$ , and the arguments similar to (REF ), we obtain $t_{\\sigma }=\\lim _{n\\rightarrow \\infty }a^n(f(a^{-n}t_{\\sigma })-b)=-a^{-N}b\\prod _{n=1}^{\\infty }\\frac{a}{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(0))}=a^{-N}\\widetilde{z}.$ Hence, all $t_{\\sigma }$ are equal to each other.",
"Using (REF ), the remark after (REF ) about $\\widetilde{m}$ different $\\sigma $ , and (REF ), we conclude that $f(a^{-N}\\widetilde{z})$ is a zero of $P^{\\circ N}$ of a multiplicity at least $\\widetilde{m}$ .",
"Thus, differentiating $f(z)=P^{\\circ N}(f(a^{-N}z))$ at $z=\\widetilde{z}$ , we obtain that $f^{(j)}(\\widetilde{z})=0$ , $j=0,...,\\widetilde{m}-1$ .",
"Hence, $\\widetilde{z}$ is a zero of $f$ of a multiplicity at least $\\widetilde{m}$ .",
"The proof of Theorem REF is finished.",
"To prove Theorem REF let us note that $g_{\\sigma }(w)=a^m(P_{\\sigma _m}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w)-b)\\prod _{n=m+1}^{\\infty }\\frac{a}{Q(P_{\\sigma _n}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w))},\\ \\ m\\geqslant 0,\\ \\ w\\ne b,$ where $P_{\\sigma _0}^{-1}(w):=w$ and $g_{\\sigma }(w)$ are defined in (REF ).",
"Moreover, using the similar arguments as in the proof of Theorem REF , we can state that if $\\sigma _m\\ne 0$ then (REF ) is true for $w=b$ .",
"Now, choosing the first $m$ such that $\\sigma _m\\ne 0$ , we finish the proof of Theorem REF .",
"Recall that $P_0^{-1}$ is the unique branch of $P^{-1}$ such that $P_0^{-1}(b)=b$ and $P_0^{-1}$ is analytic in the neighbourhood of $b$ .",
"Let $\\delta >0$ be so small that $P_{j}^{-1}({\\mathbb {C}})\\cap \\lbrace w:\\ |w-b|<\\delta \\rbrace =\\emptyset ,\\ \\ \\forall j\\ne 0.$ Identities (REF ) and (REF ) give $f(a^{-m}g_{\\sigma }(w))=P_{\\sigma _m}^{-1}\\circ ...\\circ P_{\\sigma _1}^{-1}(w).$ If $\\sigma _m\\ne 0$ then (REF ) leads to $|f(a^{-m}g_{\\sigma }(w))-b|\\geqslant \\delta .$ Since $f$ is continuous and $f(0)=b$ , there is $C>0$ such that $\\lbrace z:\\ |z|<C\\rbrace \\subset f^{-1}(\\lbrace w:\\ |w-b|<\\delta \\rbrace ).$ Thus, by (REF ) and (REF ), we have $|a^{-m}g_{\\sigma }(w)|\\geqslant C$ , which gives (REF ).",
"Using (REF ), we get $\\sum _{|\\mathop {\\mathrm {supp}}\\nolimits \\sigma |\\geqslant N}|g_{\\sigma }(w)|^{-m}=\\sum _{n\\geqslant N}\\sum _{|\\mathop {\\mathrm {supp}}\\nolimits \\sigma |=n}|g_{\\sigma }(w)|^{-m}\\leqslant \\frac{1}{C^m}\\sum _{n\\geqslant N}d^n|a|^{-mn}=O((d|a|^{-m})^N).$ By analogy, we can estimate the product in (REF ).",
"The proof of Theorem REF is finished."
]
] | 1906.04579 |
[
[
"An Unsupervised Framework for Comparing Graph Embeddings"
],
[
"Abstract Graph embedding is a transformation of vertices of a graph into set of vectors.",
"Good embeddings should capture the graph topology, vertex-to-vertex relationship, and other relevant information about graphs, subgraphs, and vertices.",
"If these objectives are achieved, they are meaningful, understandable, and compressed representations of networks.",
"They also provide more options and tools for data scientists as machine learning on graphs is still quite limited.",
"Finally, vector operations are simpler and faster than comparable operations on graphs.",
"The main challenge is that one needs to make sure that embeddings well describe the properties of the graphs.",
"In particular, the decision has to be made on the embedding dimensionality which highly impacts the quality of an embedding.",
"As a result, selecting the best embedding is a challenging task and very often requires domain experts.",
"In this paper, we propose a ``divergence score'' that can be assign to various embeddings to distinguish good ones from bad ones.",
"This general framework provides a tool for an unsupervised graph embedding comparison.",
"In order to achieve it, we needed to generalize the well-known Chung-Lu model to incorporate geometry which is interesting on its own rights.",
"In order to test our framework, we did a number of experiments with synthetic networks as well as real-world networks, and various embedding algorithms."
],
[
"Introduction",
"The study of networks has emerged in diverse disciplines as a means of analyzing complex relational data.",
"Indeed, capturing aspects of a complex system as a graph can bring physical insights and predictive power [1].",
"Network Geometry is a rapidly developing approach in Network Science [2] which further abstracts the system by modelling the vertices of the network as points in a geometric space.",
"There are many successful examples of this approach that include latent space models [3], and connections between geometry and network clustering and community structure [4], [5].",
"Very often, these geometric embeddings naturally correspond to physical space, such as when modelling wireless networks or when networks are embedded in some geographic space [6], [7].",
"See [8] for more details about applying spatial graphs to model complex networks.",
"Another important application of geometric graphs is in graph embedding.",
"The idea here is that, for a given network, one tries to embed it in a geometric space by assigning coordinates to each vertex such that nearby vertices are more likely to share an edge than those far from each other.",
"In a good embedding most of the network's edges can be predicted from the coordinates of the vertices.",
"For example, in [9] protein interaction networks are embedded in low-dimension Euclidean space.",
"Unfortunately, in the absence of a general-purpose representation for graphs, very often graph embedding requires domain experts to craft features or to use specialized feature selection algorithms.",
"Having said that, there are some graph embedding algorithms that work without any prior or additional information other than graph structure.",
"However, these are randomized algorithms that are usually not so stable; that is, the outcome is often drastically different despite the fact that all the algorithm parameters remain the same.",
"Consider a graph $G=(V,E)$ on $n$ vertices, and several embeddings of its vertices to some multidimensional spaces (possibly in different dimensions).",
"The main question we try to answer in this paper is: how do we evaluate these embeddings?",
"Which one is the best and should be used?",
"In order to answer these questions, we propose a general framework that assigns the divergence score to each embedding which, in an unsupervised learning fashion, distinguishes good from bad embeddings.",
"In order to benchmark embeddings, we generalize well-known Chung-Lu random graph model to incorporate geometry.",
"The model is interesting on its own and should be useful for many other problems and tools.",
"In order to test our algorithm, we experiment with synthetic networks as well as real-world networks, and various embedding algorithms.",
"The paper is structured as follows.",
"In Section , we describe our algorithm for comparing graph embeddings, and we illustrate our approach on one simple graph.",
"The Chung-Lu model is generalized in Section .",
"Some theoretical results that justify the model can be found in Appendix .",
"In Section , we describe the experiments and present results.",
"In particular, the datasets and embedding algorithms used are outlined respectively in subsections REF and REF .",
"We conclude with a discussion on some future directions in Section ."
],
[
"General Framework",
"Suppose that we are given a graph $G=(V,E)$ on $n$ vertices with the degree distribution $\\textbf {w}=(w_1, \\ldots , w_n)$ and an embedding of its vertices to $k$ -dimensional space, $\\mathcal {E}: V \\rightarrow {\\mathbb {R}}^k$ .",
"Our goal is to assign a “divergence score” to this embedding.",
"The lower the score, the better the embedding is.",
"This will allow us to compare several embeddings, possibly in different dimensions."
],
[
"Intuition Behind the Algorithm",
"What do we expect from a good embedding?",
"As already mentioned, in a good embedding, one should be able to predict most of the network's edges from the coordinates of the vertices.",
"Formally, it is natural to expect that if two vertices, say $u$ and $v$ , are far away from each other (that is, $\\mathrm {dist}(\\mathcal {E}(u),\\mathcal {E}(v))$ is relatively large), then the chance they are adjacent is smaller compared to another pair of vertices that are close to each other.",
"But, of course, in any real-world network there are some sporadic long edges and some vertices that are close to each other are not adjacent.",
"In other words, we do not want to pay attention to local properties such as existence of particular edges (microscopic point of view) but rather evaluate some global properties such as density of some relatively large subsets of vertices (macroscopic point of view).",
"So, how can we evaluate if the global structure is consistent with our expectations and intuition without considering individual pairs?",
"The idea is as follows.",
"We identify dense parts of the graph by running some good graph clustering algorithm.",
"As we will illustrate in Section , the choice of graph clustering algorithm is flexible so long as the vertex set is partitioned into clusters such that there are substantially more edges captured within clusters than between them.",
"The clusters that are found will provide the desired macroscopic point of view of the graph.",
"Note that for this task we only use information about graph $G$ ; in particular, we do not use the embedding $\\mathcal {E}$ at all.",
"We then consider the graph $G$ from a different point of view.",
"Using the Geometric Chung-Lu (GCL) model that we introduce in this paper especially for this purpose, based on the degree distribution $\\textbf {w}$ and the embedding $\\mathcal {E}$ , we compute the expected number of edges within each cluster found earlier, as well as between them.",
"The embedding is scored by computing a divergence score between those expected number of edges, and the actual number of edges present in $G$ .",
"Our approach falls into a general and commonly used method of statistical inference, in our case applied to the Geometric Chung-Lu model.",
"With these methods, one fits a generative model of a network to observed network data, and the parameters of the fit tell us about the structure of the network in much the same way that fitting a straight line through a set of data points tells us about their slope."
],
[
"Algorithm",
"Given a graph $G=(V,E)$ , its degree distribution $\\textbf {w}$ on $V$ , and an embedding $\\mathcal {E}: V \\rightarrow {\\mathbb {R}}^k$ of its vertices in $k$ -dimensional space, we perform the five steps detailed below to obtain $\\Delta _\\mathcal {E}(G)$ , a divergence score for the embedding.",
"We can apply this algorithm to compare several embeddings $\\mathcal {E}_1,\\dots ,\\mathcal {E}_m$ , and select the best one via $\\operatornamewithlimits{argmin}_{i\\in [m]} \\Delta _{\\mathcal {E}_i}(G)$ (here and later in the paper, we use $[n]$ to denote the set of natural numbers less than or equal to $n$ ; that is, $[n] := \\lbrace 1, \\ldots , n\\rbrace $ ).",
"Note that our algorithm is a general framework and some parts have flexibility.",
"We clearly identify these below.",
"Step 1: Run some stable graph clustering algorithm on $G$ to obtain a partition $\\textbf {C}$ of the vertex set $V$ into $\\ell $ communities $C_1, \\ldots , C_\\ell $ .",
"Note: In our implementation, we used the ensemble clustering algorithm for graphs (ECG) which is based on the Louvain algorithm and the concept of consensus clustering [10], and is shown to have good stability.",
"We experiment with other algorithms in Section .",
"Step 2: For each $i \\in [\\ell ]$ , let $c_{i}$ be the proportion of edges of $G$ with both endpoints in $C_i$ .",
"Similarly, for each $1 \\le i < j \\le \\ell $ , let $c_{i,j}$ be the proportion of edges of $G$ with one endpoint in $C_i$ and the other one in $C_j$ .",
"Let $\\bar{\\textbf {c}} = (c_{1,2},\\ldots , c_{1,\\ell }, c_{2,3}, \\ldots , c_{2,\\ell }, \\ldots , c_{\\ell -1,\\ell } )\\quad \\text{and }\\hat{\\textbf {c}} = (c_1, \\ldots , c_\\ell )$ be two vectors with a total of $\\binom{\\ell }{2} + \\ell = \\binom{\\ell +1}{2}$ entries that sum to one.",
"These graph vectors characterize partition $\\textbf {C}$ from the perspective of $G$ .",
"Note: The embedding $\\mathcal {E}$ does not affect the vectors $\\bar{\\textbf {c}}$ and $\\hat{\\textbf {c}}$ .",
"It is calculated purely based on $G$ and the partition $\\textbf {C}$ .",
"Step 3: For a given parameter $\\alpha \\in {\\mathbb {R}}_+$ and the same vertex partition $\\textbf {C}$ , we consider $\\mathcal {G}(\\textbf {w}, \\mathcal {E}, \\alpha )$ , the GCL Model presented in Section .",
"For each $1 \\le i < j \\le \\ell $ , we compute $b_{i,j}$ , the expected proportion of edges of $\\mathcal {G}(\\textbf {w}, \\mathcal {E}, \\alpha )$ with one endpoint in $C_i$ and the other one in $C_j$ .",
"Similarly, for each $i \\in [\\ell ]$ , let $b_i$ be the expected proportion of edges within $C_i$ .",
"That gives us another two vectors $\\bar{\\textbf {b}}_\\mathcal {E}(\\alpha ) = (b_{1,2},\\ldots , b_{1,\\ell }, b_{2,3}, \\ldots , b_{2,\\ell }, \\ldots , b_{\\ell -1,\\ell } ) \\qquad \\text{ and } \\qquad \\hat{\\textbf {b}}_\\mathcal {E}(\\alpha ) = (b_{1},\\ldots , b_{\\ell } )$ with a total of $\\binom{\\ell +1}{2}$ entries that sum to one.",
"These vectors characterizes partition $\\textbf {C}$ from the perspective of the embedding $\\mathcal {E}$ .",
"Note: The structure of graph $G$ does not affect the vectors $\\bar{\\textbf {b}}_\\mathcal {E}(\\alpha )$ and $\\hat{\\textbf {b}}_\\mathcal {E}(\\alpha )$ ; only its degree distribution $\\textbf {w}$ and embedding $\\mathcal {E}$ are used.",
"Note: We used the Geometric Chung-Lu Model but the framework is flexible.",
"If, for any reason (perhaps there are some restrictions for the maximum edge length; such restrictions are often present in, for example, wireless networks) it makes more sense to use some other model of random geometric graphs, it can be easily implemented here.",
"If the model is too complicated and computing the expected number of edges between two parts is challenging, then it can be approximated easily via simulations.",
"Step 4: Compute the distances between the two pairs of vectors, between $\\bar{\\textbf {c}}$ and $\\bar{\\textbf {b}}_\\mathcal {E}(\\alpha )$ , and between $\\hat{\\textbf {c}}$ and $\\hat{\\textbf {b}}_\\mathcal {E}(\\alpha )$ , in order to measure how well the graph $G$ fits the model $\\mathcal {G}(\\textbf {w}, \\mathcal {E}, \\alpha )$ .",
"Let $\\Delta _\\alpha $ be a weighted average of the two distances.",
"Note: We used the well-known and widely used Jensen–Shannon divergence (JSD) to measure the dissimilarity between two probability distributions.",
"The JSD was originally proposed in [11] and can be viewed as a smoothed version of the Kullback-Leibler divergence.",
"In our implementation, we used simple average, that is, $\\Delta _\\alpha = \\frac{1}{2} \\cdot \\left(JSD(\\bar{\\bf c},\\bar{\\bf b}(\\alpha )) + JSD(\\hat{\\bf c},\\hat{\\bf b}(\\alpha ))\\right).$ Other weighted averages can be used if more weight needs to be put on internal or external edges.",
"Step 5: Select $\\hat{\\alpha } = \\operatornamewithlimits{argmin}_{\\alpha } \\Delta _\\alpha $ , and define the divergence score for embedding $\\mathcal {E}$ on $G$ as: $\\Delta _\\mathcal {E}(G) = \\Delta _{\\hat{\\alpha }}$ .",
"Note: The parameter $\\alpha $ is used to define a distance in the embedding space, as we detail in Section .",
"In our implementation we simply checked values of $\\alpha $ on a grid between 0 and 10.",
"There are clearly better ways to search the space of possible values of $\\alpha $ but, since the algorithm worked very fast on our graphs, we did not optimize this part.",
"In order to compare several embeddings for the same graph $G$ , we repeat steps 3-5 above and compare the divergence scores (the lower score, the better).",
"Let us stress again that steps 1-2 are done only once, so we use the same partition into $\\ell $ communities for each embedding."
],
[
"Illustration",
"We illustrate our framework on the well-known Zachary's Karate Club graph (see Subsection REF for the description of this and other datasets).",
"Illustrations with other datasets are shown in Appendix .",
"The parameter $\\alpha \\ge 0$ in the GCL model controls the distance used in the embedding space.",
"With $\\alpha =0$ , the embedding is not taken into account and the classic Chung-Lu model is obtained, so only the degree distribution is accounted for.",
"As $\\alpha $ gets larger, long edges in the embedding space are penalized more severely.",
"In the left plot of Figure REF , we show the impact of varying $\\alpha $ on the two components of equation (REF ) which respectively consider pairs of vertices that are internal (to some cluster) or external (between clusters).",
"Recall that the divergence score for a given embedding is obtained by choosing $\\hat{\\alpha } = \\operatornamewithlimits{argmin}_{\\alpha } \\Delta _\\alpha $ .",
"In the right plot of Figure REF , we show a 2-dimensional projection of the best embedding as obtained by our framework.",
"The vertices are coloured according to the two known communities.",
"Figure: The Karate Club Graph.",
"We illustrate the divergence score as a function of α\\alpha (left) for the best embedding found by our framework (right).",
"The colors represent the two ground-truth communities.We can use the GCL model to generate edges, as with the standard Chung-Lu model.",
"In Figure REF , we generate 3 such graphs using the best embedding shown in Figure REF .",
"The left plot uses $\\alpha =0$ , which ignores the embedding and clearly generates too many long edges between the clusters.",
"The center plot uses the optimal value ($\\hat{\\alpha }=2.75$ in this case), generating a graph that resembles the true one.",
"The rightmost plot uses the larger value $\\alpha = 7$ , which penalizes long edges more severely, yielding a graph with less edges between the two communities.",
"Figure: Zachary's Karate Club Graph.",
"We generate random edges following the Geometric Chung-Lu Model with the same expected degree distribution and with the highest scoring embedding.",
"We look at three cases: α=0\\alpha =0 which ignores the embedding (left), α=7\\alpha =7 which penalizes long edges too severely (right), and the best α ^=4\\hat{\\alpha }=4 (center)."
],
[
"Geometric Chung-Lu Model",
"It is known that classical Erdős-Rényi (binomial) random graphs $G(n,p)$ can be generalized to $G(\\textbf {w})$ , the random graph with a given expected degree distribution $\\textbf {w}=(w_1, \\ldots , w_n)$ .",
"Because of our application, we will define it as a function of a given graph $G$ on $n$ vertices but, in fact, it is only a function of its degree sequence.",
"Since our goal is to compare different embeddings of the same graph, we will generalize the Chung-Lu model further, including geometry coming from graph embedding.",
"In such models vertices are embedded in some metric space and link formation is influenced by the metric distance between vertices.",
"Such models are called spatial models or geometric graphs.",
"The main principle of spatial models is that vertices that are metrically close are more likely to link to each other.",
"This is a formal expression of the intuitive notion we have about virtual networks: Web links are likely to point to similar pages, people that share similar interests are more likely to become friends on Facebook, and scientific papers mostly refer to papers on similar topics.",
"See [8] for more details about applying spatial graphs to model complex networks."
],
[
"Original Model",
"Let $G=(V,E)$ be a graph, where $V = \\lbrace v_1,\\ldots ,v_n\\rbrace $ are the vertices, the edges $E$ are multisets of $V$ of cardinality 2 (loops are allowed), and $\\deg _G(v)$ is the degree of $v$ in $G$ (with a loop at $v$ contributing 2 to the degree of $v$ ).",
"We define $\\mathcal {G}(G)$ to be the probability distribution on graphs on the vertex set $V$ following the well-known Chung-Lu model [12], [13], [14], [15].",
"In this model, each set $e=\\lbrace v_i,v_j\\rbrace $ , $v_i,v_j \\in V$ , is independently sampled as an edge with probability given by: $P(v_i,v_j) ={\\left\\lbrace \\begin{array}{ll}\\frac{\\deg _G(v_i) \\deg _G(v_j)}{2|E|}, & i \\ne j \\\\\\frac{\\deg _G^2(v_i)}{4|E|}, & i = j.\\end{array}\\right.",
"}$ (Let us mention about one technical assumption.",
"Note that it might happen that $P(v_i,v_j)$ is greater than one and so it should really be regarded as the expected number of edges between $i$ and $j$ ; for example, as suggested in [1], one can introduce a Poisson-distributed number of edges with mean $P(v_i,v_j)$ between each pair of vertices $i$ , $j$ .",
"However, since typically the maximum degree $D$ satisfies $D^2 \\le 2 |E|$ it rarely creates a problem and so we may assume that $P(v_i,v_j) \\le 1$ for all pairs.)",
"As already mentioned, this model is a function of the degree sequence of $G$ .",
"One desired property of this random model is that it yields a distribution that preserves the expected degree for each vertex, namely: for any $i \\in [n]$ , $\\mathbb {E}_{G^{\\prime } \\sim \\mathcal {G}(G)}[\\deg _{G^{\\prime }}(v_i)] &=& \\sum _{j \\in [n] \\setminus \\lbrace i\\rbrace } \\frac{\\deg _G(v_i)\\deg _G(v_j)}{2|E|} + 2 \\cdot \\frac{\\deg _G^2(v_i)}{4|E|} \\\\&=& \\frac{\\deg _G(v_i)}{2|E|} \\sum _{j \\in [n]} \\deg _G(v_j) ~~=~~ \\deg _G(v_i).$"
],
[
"Geometric Model",
"This time we are not only given the expected degree distribution $\\textbf {w}=(w_1, \\ldots , w_n) = (\\deg _G(v_1), \\ldots , \\deg _G(v_n))$ but also an embedding of its vertices in some $k$ -dimensional space, function $\\mathcal {E}: V \\rightarrow {\\mathbb {R}}^k$ .",
"In particular, for each pair of vertices, $v_i$ , $v_j$ , we know the distance between them: $d_{i,j} = \\mathrm {dist}( \\mathcal {E}(v_i), \\mathcal {E}(v_j)).$ It is desired for the probability that $v_i$ and $v_j$ are adjacent to be a function of $d_{i,j}$ , that is, to be proportional to $g(d_{i,j})$ for some function $g$ .",
"Function $g$ should be a decreasing function as long edges should occur less frequently than short ones.",
"There are many natural choices such as $g(d) = d^{-\\beta }$ for some $\\beta \\in [0, \\infty )$ or $g(d) = \\exp (-\\gamma d)$ for some $\\gamma \\in [0, \\infty )$ .",
"We use the following, normalized function $g:[0,\\infty ) \\rightarrow [0,1]$ : for a fixed $\\alpha \\in [0,\\infty )$ , let $g(d) := \\left( 1 - \\frac{d - d_{\\min }}{d_{\\max } - d_{\\min }} \\right)^{\\alpha },$ where $d_{\\min } &=& \\min \\lbrace \\mathrm {dist}(\\mathcal {E}(v), \\mathcal {E}(w)): v,w \\in V \\rbrace \\\\d_{\\max } &=& \\max \\lbrace \\mathrm {dist}(\\mathcal {E}(v), \\mathcal {E}(w)): v,w \\in V \\rbrace $ are the minimum, respectively maximum, distance between vertices in embedding $\\mathcal {E}$ .",
"One convenient and desired property of this function is that it is invariant with respect to an affine transformation of the distance measure.",
"Clearly, $g(d_{\\min })=1$ and $g(d_{\\max })=0$ ; in the computations, we can use clipping to force $g(d_{\\min })<1$ and/or $g(d_{\\max })>0$ if required.",
"Let us also note that if $\\alpha = 0$ (that is, $g(d)=1$ for any $d \\in [0,\\infty )$ with $g(d_{\\max })=0^0=1$ ), then we recover the original Chung-Lu model as the pairwise distances are neglected.",
"Moreover, the larger parameter $\\alpha $ is, the larger aversion for long edges is.",
"Since this family of functions (for various values of the parameter $\\alpha $ ) captures a wide spectrum of behaviours, it should be enough to concentrate on this choice but one can easily experiment with other functions.",
"So, for now we may assume that the only parameter of the model is $\\alpha \\in [0,\\infty )$ .",
"The Geometric Chung-Lu (GCL) model is the random graph $G(\\textbf {w}, \\mathcal {E}, \\alpha )$ on the vertex set $V = \\lbrace v_1, \\ldots , v_n \\rbrace $ in which each pair of vertices $v_i, v_j$ , independently of other pairs, forms an edge with probability $p_{i,j}$ , where $p_{i,j} = x_i x_j g(d_{i,j}) $ for some carefully tuned weights $x_i \\in {\\mathbb {R}}_+$ .",
"The weights are selected such that the expected degree of $v_i$ is $w_i$ ; that is, for all $i \\in [n]$ $w_i = \\sum _{j \\in [n]} p_{i,j} = x_i \\sum _{j \\in [n]} x_j g(d_{i,j}).$ Indeed, we prove in Appendix that there exists the unique selection of weights, provided that the maximum degree of $G$ is less than the sum of degrees of all other vertices.",
"Since each connected component of $G$ can be embedded independently, we may assume that $G$ is connected and so the minimum degree of $G$ is at least 1.",
"As a result, this very mild condition is trivially satisfied unless $G$ is a star on $n$ vertices.",
"Finally, let us mention that in Appendix it is assumed that $g(d_{i,j}) > 0$ for all pairs $i,j$ .",
"In our case, $g(d_{i,j}) = 0$ for a pair of vertices that are at the maximum distance.",
"It causes no problems in practice but, as mentioned earlier, one can easily scale the outcome of function $g(\\cdot )$ to move away from zero without affecting the divergence score in any non-negligible way.",
"It is not clear how to find weights explicitly but they can be easily (and efficiently) approximated numerically to any desired precision.",
"In Appendix , we prove that, if the solution exists, which is easy to check, then the set of right hand sides of the equations considered as a function from $\\mathbb {R}^n$ to $\\mathbb {R}^n$ is a local diffeomorphism everywhere in its domain.",
"As a result, standard gradient root-finding algorithms should be quite effective in finding the desired weights.",
"We discuss one simple numerical approximation procedure the next subsection."
],
[
"Numerical Approximation",
"Let us start with an arbitrary vector $\\textbf {t}^0=(t^0_1, \\ldots , t^0_n)$ (say, $\\textbf {t}^0=(1,\\ldots ,1)$ ) that we will carefully tune by iteratively constructing a sequence $(\\textbf {t}^s)_{s \\ge 0}$ .",
"Suppose that we are given a vector $\\textbf {t}^s=(t^s_1, \\ldots , t^s_n)$ .",
"If we introduce an edge between $v_i$ and $v_j$ with probability $p^s_{i,j} = t^s_i t^s_j g(d_{i,j}),$ then the expected degree of $v_i$ would be $s^s_i = \\sum _j p^s_{i,j} = t^s_i \\sum _j t^s_j g(d_{i,j}).$ Note that, for a given vertex $v_i$ , it easy to adjust the weights so that $s^s_i$ matches $w_i$ , the desired expected degree of $v_i$ ; indeed, one can replace $t^s_i$ with $t^s_i (w_i/s^s_i)$ .",
"However, unfortunately, it will also affect other values of $\\textbf {s}^s$ and vice versa, changes in other parts of $\\textbf {t}$ affect $s^s_i$ too.",
"Hence, instead of doing it, each vertex should take a small step into the right direction and this process should quickly converge to the desired state: $s^s_i$ being very close to $w_i$ for all $i$ .",
"Let us note that, for our application, we do not need to get close to the desired expected degree sequence.",
"The divergence score we defined earlier is not too sensitive.",
"Fix some small constants $\\varepsilon , \\delta > 0$ .",
"For example, in our experiments we used $\\varepsilon = 0.1$ and $\\delta = 0.001$ .",
"For each $i$ , $1 \\le i \\le n$ , we define $t^{s+1}_i = (1-\\varepsilon ) t^s_i + \\varepsilon t^s_i (w_i/s^s_i) = t^s_i + \\varepsilon t^s_i (w_i/s^s_i-1).$ We repeat the tuning process until $\\max _i |\\textbf {w}_i - s^s_i| < \\delta $ ."
],
[
"Datasets",
"In order to test our algorithm, we benchmark it against synthetic graphs with communities as well as some real-world graphs with known community structures.",
"This is a natural and often used approach (see, for example, [16]).",
"In all examples, we colour the vertices with respect to the known, ground-truth communities, but in the algorithm, we use the partition obtained via a graph clustering algorithm, as the algorithm is unsupervised."
],
[
"LFR",
"As common in the literature, we analyze how the algorithm performs on artificially constructed networks with communities.",
"A popular example is the LFR benchmark by Lancichinetti, Fortunato, and Radicchi [17] that generates synthetic graphs that resemble real world graphs.",
"In this benchmark, the size of each community is drawn from a power-law distribution, as is the degree of each vertex.",
"The LFR model has a number of parameters.",
"The most important one is the mixing parameter $\\mu $ that controls the fraction of external edges (that is, edges between communities).",
"Essentially this can be viewed as the amount of noise in the graph.",
"In one extreme case, if $\\mu =0$ , then all the edges are within communities.",
"On the other hand, if $\\mu =1$ , then all edges are between different communities.",
"Other parameters control the number of vertices ($n$ ), the negative exponent of the power-law degree distribution ($\\gamma _1$ ), the negative exponent of the distribution of community sizes ($\\gamma _2$ ), and the average ($d$ ) and maximum degree ($d_{\\max }$ ).",
"We generated small LFR graphs with $n=100$ for visualization purpose.",
"For the mixing parameter $\\mu $ , we used $\\mu =0.15$ (strong communities, low noise), $\\mu =0.35$ (intermediate case), and $\\mu =0.55$ (noisy graph, weaker communities).",
"For the other parameters, we used, $\\gamma _1=2$ , $\\gamma _2=1$ , $d=8$ , and $d_{\\max }=20$ ."
],
[
"Zachary's Karate Club",
"This well-known social network consists of 34 members of a karate club and 78 pairwise links observed over a period of three years.",
"During the famous study of Zachary [18], a political conflict arose between the club president and the instructor caused the club to split into two parts (the communities), each with half of the members.",
"Zachary recorded a network of friendships among members of the club shortly before the fission."
],
[
"College Football",
"This graph represents the schedule of United States football games between Division IA colleges during the regular season in Fall 2000 [19].",
"This is another well-studied, more complex, real-world network with known community structures.",
"The data consists of 115 teams (vertices) and 613 games (edges).",
"The teams are divided into conferences containing around 8–12 teams each.",
"In general, games are more frequent between members of the same conference than between members of different conferences, with teams playing an average of about seven intra-conference games and four inter-conference games in the 2000 season.",
"There are a few exceptions to this rule, as detailed in [16]: one of the conferences is really a group of independent teams, one conference is really broken into two groups, and 3 other teams play mainly against teams from other conferences.",
"We refer to those as outlying vertices, which we represent with a distinctive triangular shape."
],
[
"Email-Eu-core Network",
"This network was generated using email data from a large European research institution and is available as one of the SNAP Datasets [20].",
"There is a directed edge $(u, v)$ in the network if person $u$ sent person $v$ at least one email.",
"The e-mails only represent communication between institution members (the core), and the dataset does not contain incoming messages from or outgoing messages to the rest of the world.",
"The dataset also contains community memberships of the vertices: each individual belongs to exactly one of 42 departments at the research institute.",
"There are 1005 vertices and 25571 edges in the original directed graph.",
"After making it undirected and keeping only the largest connected component, we ended up with a graph on 986 vertices and 16064 edges.",
"Most of the communities are very weak; in fact, only 1 community qualifies as a weak community as defined in equation (9.2) in [21].",
"Those are communities for which the ratio between the total internal degree and the total degree is above 0.5.",
"In our study, we looked at the communities for which this ratio was the highest."
],
[
"The Graph Embeddings",
"We considered three popular graph embedding algorithms in our examples: node2vec, VERSE and LINE.",
"Recall that our goal is to experiment with our framework and not to do extensive comparison of graph embeddings.",
"Comparisons are done in the original dimension of the embedding.",
"For visualization purposes, we project the embeddings into 2-dimensional space."
],
[
"node2vec",
"node2vec [22] is a semi-supervised algorithm for scalable feature learning in networks.",
"Intuitively, the goal is to find feature representations that maximize the likelihood of preserving network neighbourhoods of vertices in a $d$ -dimensional feature space.",
"By choosing an appropriate notion of a neighborhood, node2vec can learn representations that organize vertices based on their network roles and/or communities they belong to.",
"It is achieved by developing a family of biased random walks, which efficiently explore diverse neighborhoods of a given vertex.",
"There are two main parameters in this algorithm.",
"The “return parameter” $p$ controls the likelihood of immediately revisiting a vertex in the random walk.",
"Setting it to a high value ensures that we are less likely to sample an already-visited vertex in the following two steps (unless, of course, the next vertex in the walk had no other neighbour).",
"The “in-out parameter” $q$ allows the search to differentiate between “inward” and “outward” vertices.",
"Loosely speaking, this parameter guides the neighborhood sampling strategy which allows us to smoothly interpolate between BFS and DFS.",
"For node2vec, we fixed parameter $q=1$ and we varied parameter $0.5 \\le p \\le 2$ .",
"We considered embedding dimensions $2 \\le D \\le 128$ .",
"For the other parameters we used the default values."
],
[
"VERSE",
"VERtex Similarity Embeddings (VERSE) [23] is a simple, versatile, and memory-efficient method that derives graph embeddings explicitly calibrated to preserve the distributions of a selected vertex-to-vertex similarity measure.",
"It is a general framework that learns any similarity measures among vertices via training a simple, yet expressive, single-layer neural network.",
"This includes popular similarity measures such as Personalized PageRank (PPR), SimRank, and adjacency similarity.",
"We used the default recommended parameters for this algorithm and we considered embedding dimensions $2 \\le D \\le 128$ ."
],
[
"LINE",
"Large-scale Information Network Embedding (LINE) [24] is an efficient method for vertex embedding that uses an approximate factorization of the adjacency matrix, trying to preserve first as well as second order proximities.",
"We used the default recommended parameters for this algorithm, but we varied the number of threads used from 1 (default) to 10.",
"We considered embedding dimensions $2 \\le D \\le 128$ ."
],
[
"Visualization",
"Dimension reduction seeks to produce a low dimensional representation of high dimensional data that preserves relevant structure.",
"As a result, it is an important problem in data science as a potential pre-processing step for many machine learning algorithms.",
"Here, we use it for visualization of tested graph embeddings that are done in high dimensions.",
"We used UMAP (Uniform Manifold Approximation and Projection) [28], a novel manifold learning technique for dimension reduction.",
"UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology.",
"It provides a practical scalable algorithm that applies to real world datasets."
],
[
"Artificial Graphs",
"For the three LFR graphs described earlier, we generated 5 embeddings for every choice of parameter(s) and for each algorithm considered, for a total of 315 embeddings for each graph.",
"In Figure REF , we plot a 2-dimension projection of the best and worst results as identified by our divergence score.",
"The colors correspond to the ground-truth communities generated by the LFR benchmark.",
"For the best embeddings, we clearly see that the ground-truth communities are grouped together, even (but to a lesser extent) for the noisy graph with $\\mu =0.55$ .",
"This is not the case for the worst embeddings.",
"Figure: Best (left column) and worst (right column) scoring embeddings for 3 different LFR graphs.In the top row, we show the results for the graph with 7 strong communities (μ=0.15\\mu =0.15).In the middle row, we show the results for the graph with μ=0.35\\mu =0.35 and 9 communities.Finally in the bottom row, we show the results for the noisy graph (μ=0.55\\mu =0.55) with 13 communities."
],
[
"Zachary's Karate Club Network",
"We already considered this dataset in Section .",
"In Figure REF , we show the best and worst scoring embeddings we found using our algorithm.",
"The colors correspond to the 2 real communities.",
"Figure: Zachary's Karate Club.",
"We show the best (left) and the worst (right) embedding according to our algorithm.",
"The left plot is clearly a better choice."
],
[
"The College Football Graph",
"This graph consists of 115 teams spread between 12 conferences.",
"In Figure REF , we show the best and worst scoring embeddings.",
"The colors of vertices correspond to the conferences, white triangular shaped vertices correspond to outlying vertices as explained earlier.",
"Figure: The College Football Graph.",
"We show the best scoring embedding (left) and the worst one (right).",
"The communities are very clear in the left plot.For each choice of embedding model, parameters and dimension, we obtained 5 different embeddings.",
"In Figure REF , we summarize those results by showing the mean and standard deviation of the divergence score $\\Delta _\\mathcal {E}(G)$ .",
"This allows view some overall properties: for example, embedding in 2-dimensions is not enough in this case; we obtained the best result with LINE with 32 dimensions or more.",
"Recall that we are not performing a overall study of graph embedding algorithms, and we are not looking for a winning model.",
"In fact, both VERSE and node2vec gave the best results for other graphs, which indicates the need for a comparison framework.",
"Figure: For the College Football Graph, for each choice of algorithm, parameters and dimension, we generated 5 embeddings.",
"We show the mean and standard deviation of the divergence score Δ ℰ (G)\\Delta _\\mathcal {E}(G) for each of those."
],
[
"The email-Eu-core Graph",
"In Figure REF , we show the highest scoring embedding we obtained for the email graph described earlier.",
"Since most communities are very weak, we highlight only the 3 strongest communities for which the ratio of internal to total degree are respectively 0.54, 0.42 and 0.39.",
"Figure: The best embedding we obtained for the email-Eu-core graph with the 3 strongest communities respectively shown in yellow, red and blue, which are clearly separated."
],
[
"Graph Clustering",
"An important part of our algorithm consists of building a reasonable partition of the vertices, over which we build the representative vectors (REF ) and (REF ).",
"The ECG algorithm was shown to generally yield good and stable clusters [25], and we used it throughout.",
"We re-ran all embeddings on the College Football graph using respectively the Louvain [26] and InfoMap [27] clustering algorithms, and in each case, we produced a ranking of the embeddings with respect to our method.",
"We compared those rankings as well as the ranking obtained with the ECG clustering algorithm by computing the Kendall-tau correlation between those.",
"The results, which are summarized in Table REF , show very high correlation.",
"Table: Kendall-tau correlation between all ranked embeddings on the College Football graph using 3 different graph clustering algorithms."
],
[
"Future Directions",
"In this paper, our aim was to introduce a general framework for evaluating embeddings.",
"This exploratory research showed that our algorithm is very promising.",
"The next natural step is to do extensive experiments of various embedding algorithms on large real-world datasets in order to evaluate and compare them.",
"Hypergraphs are natural generalizations of graphs in which a single (hyper)edge can connect any number of vertices.",
"As a consequence, hypergraphs are often more suitable and useful to model many important networks and processes.",
"Typical applications are related to social data analysis and include situations such as exchanging emails with several recipients, co-authorship of research papers, or analyzing security vulnerabilities of information networks.",
"In many situations, using hypergraphs instead of classical graphs allows us to better capture and analyze dependencies within the network.",
"We are interested in generalizing classic notions to hypergraphs, such as clustering via modularity [29], as well as developing new algorithms for them [30].",
"Hence, a natural line of development of the proposed embedding algorithm is to generalize it to allow for evaluation of embeddings of hypergraphs.",
"As a side effect of our research on graph embeddings, we have introduced the Geometric Chung-Lu model that is interesting on its own rights and potentially applicable in other problems.",
"As it is not the main focus of this paper, we did not analyze its graph-theoretic properties in detail.",
"Their study remains as a subject for further research.",
"The properties, that are standard in the analysis of other models of complex networks, include clustering coefficient, connectivity, average path length, centrality measures."
],
[
"Appendix — Geometric Chung-Lu Model is Well-defined",
"Let us state the problem in a slightly more general situation.",
"Suppose that for each pair $i, j \\in [n]$ we have $a_{ij}=a_{ji} \\in {\\mathbb {R}}_+$ , and for each $i \\in [n]$ we have $a_{ii}=0$ and $b_i \\in {\\mathbb {R}}_+$ .",
"In our application, elements of vector $\\mathbf {b} = (b_i)_{i\\in [n]}$ satisfy $b_i = w_i \\in [n-1]$ and correspond to the degree distribution, and elements of matrix $A$ satisfy $a_{i,j} = g(d_{i,j}) \\in (0,1]$ and correspond to the embedding distance between vertices.",
"Our goal is to investigate if there is a solution, $x_i \\in {\\mathbb {R}}_+$ for $i \\in [n]$ , of the following system of equations: $b_i = x_i \\sum _{j=1}^na_{ij}x_j \\quad \\text{ for all } i\\in [n].$ If there is one, then is this solution unique?",
"The $n=2$ case a degenerate case that exhibit a different behaviour but is easy to investigate: $b_1 = x_1 x_2 a_{12}$ and $b_2 = x_2 x_1 a_{21}$ .",
"It follows that the solution exists if and only if $b_1 = b_2$ .",
"If this is the case, then there are infinite number of solutions, each of them of the form $(x_1, x_2) = (t,b_1/(a_{12}t))$ for some $t \\in {\\mathbb {R}}_+$ .",
"Suppose now that $n \\ge 3$ .",
"We will show that the desired solution of (REF ) exists if and only if $\\sum _{i=1}^nb_i > 2\\max _{i\\in [n]}b_i.$ In other words, the condition is that the maximum element in vector $\\mathbf {b}$ is smaller than the sum of the other elements.",
"This is a very mild condition that holds in our application.",
"More importantly, this solution is unique.",
"We will start with proving uniqueness.",
"After that, we will show that (REF ) is indeed an obvious, necessary condition before proving that it is also a sufficient one."
],
[
"Uniqueness",
"Let us assume that $n \\ge 3$ .",
"For a contradiction, suppose that we have two different solutions: $\\mathbf {x} = (x_i)_{i\\in [n]}$ ($x_i \\in {\\mathbb {R}}_+$ , $i \\in [n]$ ) and $\\mathbf {y} = (y_i)_{i \\in [n]}$ ($y_i \\in {\\mathbb {R}}_+$ , $i \\in [n]$ ).",
"It follows that for all $i \\in [n]$ we have $b_i = f_i(\\mathbf {x}) = f_i(\\mathbf {y}), \\quad \\text{ where } f_i(\\mathbf {x}) = x_i \\sum _{j=1}^n a_{ij} x_j.$ Let us analyze what happens at point $\\mathbf {z} = t\\mathbf {x} + (1-t)\\mathbf {y}$ for some $t \\in [0,1]$ (that is, $z_i = tx_i+(1-t)y_i$ , $i \\in [n]$ ).",
"For each $i \\in [n]$ we get $f_i(\\mathbf {z}) &=& (tx_i+(1-t)y_i) \\sum _{j=1}^n a_{ij} (tx_j+(1-t)y_j) \\\\&=& \\sum _{j=1}^n a_{ij} \\left( t^2 x_i x_j+ t(1-t)(x_i y_j + x_j y_i) + (1-t)^2 y_i y_j \\right) \\\\&=& f(\\textbf {x}) t^2 + \\frac{t(1-t)}{x_i y_i} (f(\\textbf {y}) x_i^2 + f(\\textbf {y}) y_i^2) + f(\\textbf {y}) (1-t)^2 \\\\&=& b_i \\left( t^2 + \\frac{t(1-t)}{x_i y_i} (x_i^2 + y_i^2) + (1-t)^2 \\right) \\\\&=& b_i \\left( 1 - 2t(1-t) + \\frac{t(1-t)}{x_i y_i} (x_i^2 + y_i^2) \\right) \\\\&=& b_i \\left( 1 + \\frac{t(1-t)}{x_i y_i} (x_i^2 - 2x_iy_i + y_i^2) \\right) \\\\&=& b_i \\left( 1+t(1-t)\\frac{(x_i-y_i)^2}{x_iy_i} \\right) =: h_i(t).$ Note that $h_i^{\\prime }(1/2)=0$ for all $i$ (as either $x_i-y_i$ vanishes and so $h_i(t)$ is a constant function or it does not vanish but then $h_i(t)$ is a parabola with a maximum at $t=1/2$ ).",
"For convenience, let $\\mathbf {v} = (\\mathbf {x} + \\mathbf {y})/2$ and $\\mathbf {s} = (\\mathbf {x} - \\mathbf {y})/2$ (that is, $v_i=(x_i+y_i)/2$ and $s_i=(x_i-y_i)/2$ for all $i \\in [n]$ ).",
"It follows that $\\frac{df_i}{dh} \\Big ( \\mathbf {v}+h\\mathbf {s} ~|~ h=0 \\Big )=0.$ On the other hand, $f_i(\\mathbf {v}+h\\mathbf {s}) = \\sum _{j=1}^na_{ij}(v_i+hs_i)(v_j+hs_j)$ so $\\frac{df_i}{dh}(\\mathbf {v}+h\\mathbf {s})=\\sum _{j=1}^na_{ij}(s_j(v_i+hs_i)+s_i(v_j+hs_j)).$ Combining the two observations, we get that $0=\\frac{df_i}{dh} \\Big ( \\mathbf {v}+h\\mathbf {s} ~|~ h=0 \\Big ) =\\sum _{j=1}^na_{ij}(s_jv_i+s_iv_j)=s_i\\sum _{j=1}^na_{ij}v_j+\\sum _{j=1}^ns_ja_{ij}v_i.$ Since $a_{ii}=0$ , it is equivalent to $\\mathbf {g}_i \\ \\mathbf {s}= 0,$ where $\\mathbf {g}_i$ is a row vector such that $g_{ij}=a_{ij}v_i$ if $i\\ne j$ and $g_{ii}=\\sum _{j=1}^na_{ij}v_j$ .",
"Since this is true for all $i \\in [n]$ , we get $G\\mathbf {s}=\\mathbf {0},$ where $g_{ij}$ 's, the elements of matrix $G$ , are defined as above.",
"Since our assumption is that $\\mathbf {x} \\ne \\mathbf {y}$ , we have that $\\Vert s \\Vert >0$ and so we get that $\\det (G)=0$ .",
"Now, as multiplying a column of a matrix by a positive constant does not change the sign of its determinant, we may multiply column $j$ by $v_j > 0$ to conclude that $\\det (C)=0,$ where $c_{ii}=\\sum _{j=1}^na_{ij}v_iv_j$ and $c_{ij}=a_{ij}v_iv_j$ for $i\\ne j$ .",
"Thus, $C$ is a symmetric matrix whose diagonal is a sum of all its entries in a row off diagonal.",
"Now, since $\\det (C)=0$ , we get that there exists a non-zero vector $\\mathbf {u}$ such that $C\\mathbf {u}=\\mathbf {0}$ .",
"Without loss of generality, we may assume that the first entry of this vector is a largest one in absolute value.",
"Since we may scale this vector, without loss of generality, we may also assume that $\\Vert \\mathbf {u} \\Vert _\\infty =1$ ; that is, $u_1=1$ and $|u_i|\\le 1$ for all $i$ .",
"Let us consider the product of the 1st row of $C$ and vector $\\mathbf {u}$ .",
"We have that $0 = 1\\cdot \\sum _{j=2}^na_{1j}v_1v_j + \\sum _{j=2}^n u_ja_{1j}v_1v_j= \\sum _{j=2}^n (1+u_j)a_{1j}v_1v_j.$ But this means that $u_j=-1$ for $j>1$ (since all other numbers involved are positive).",
"Finally, by considering the 2nd row of the product $C\\mathbf {u}=\\mathbf {0}$ , we get that $0 = a_{21}v_2v_1 - \\sum _{j \\ne 2} a_{2j}v_2v_j - \\sum _{j=3}^n a_{2j}v_2v_j = -2 \\sum _{j=3}^n a_{2j}v_2v_j.$ This is possible only if $n=2$ ; that is, when the sum is trivial.",
"Since $n\\ge 3$ , we get the desired contradiction.",
"This finishes the proof of the uniqueness of the solution of (REF ) for $n \\ge 3$ .",
"Also note that $G$ is the Jacobian matrix of the function $[f_i(\\mathbf {x})]$ and from the proof it follows that $\\det (G)\\ne 0$ in all admissible points where the system of equations $b_i=f_i(\\mathbf {x})$ has a solution."
],
[
"Necessity",
"Without loss of generality, we may assume that the sequence of $b_i$ 's is non-increasing.",
"In particular, $b_1$ is a largest value.",
"Suppose that $x_i \\in {\\mathbb {R}}_+$ , $i \\in [n]$ , is a solution of the system (REF ).",
"Since $a_{ij}=a_{ji}>0$ for $i\\ne j$ and $a_{ii}=0$ , we observe that for $n\\ge 3$ $\\sum _{i=2}^n b_i= \\sum _{i=2}^n \\left( x_i \\sum _{j=1}^na_{ij}x_j \\right)> \\sum _{i=2}^n \\left( x_i a_{i1} x_1 \\right)= x_1 \\sum _{i=1}^n a_{1i} x_i = b_1.$ Hence, we get that (REF ) is a necessary condition for the existence of the solution."
],
[
"Sufficiency",
"In this subsection, we will still assume that $n\\ge 3$ .",
"For a contradiction, suppose that there exists a vector $\\mathbf {b} = (b_i)_{i \\in [n]}$ , with $b_i>0$ for all $i$ , that satisfies (REF ) but for which there is no solution to the system (REF ).",
"We will call such vectors infeasible; on the other hand, vectors that yield a solution $\\mathbf {x} = (x_i)_{i \\in [n]}$ , with $x_i>0$ for all $i$ , will be called feasible (as proved above, the solution must be unique).",
"Without loss of generality, we may assume that $b_1$ is a largest value in vector $\\mathbf {b}$ .",
"Let us now construct another vector $\\mathbf {b}^{\\prime } = (b^{\\prime }_i)_{i \\in [n]}$ for which there is a solution to (REF ) (that is, $\\mathbf {b}^{\\prime }$ is feasible) but also $b^{\\prime }_1=b_1$ is a largest element in $\\mathbf {b}^{\\prime }$ .",
"Indeed, it can be done easily by, for example, taking $x^{\\prime }_1=s$ for some large enough $s \\in {\\mathbb {R}}_+$ and $x^{\\prime }_2 = \\ldots = x^{\\prime }_n = b_1 / (s \\sum _{j \\in [n]} a_{1j})$ .",
"We immediately get that $b^{\\prime }_1 = x^{\\prime }_1 \\sum _{j \\in [n]} a_{1j} x^{\\prime }_j = b_1$ and for $i \\ge 2$ we have $b^{\\prime }_i = x^{\\prime }_i \\sum _{j \\in [n]} a_{ij} x^{\\prime }_j = x^{\\prime }_i a_{i1}x^{\\prime }_1 + x^{\\prime }_i \\sum _{j=2}^n a_{ij} x^{\\prime }_j = b_1\\frac{a_{1i}}{\\sum _{j \\in [n]} a_{1j}} + \\left( \\frac{b_1}{s \\sum _{j \\in [n]} a_{1j}}\\right)^2 \\sum _{j=2}^n a_{ij} < b_1,$ provided that $s \\in {\\mathbb {R}}_+$ is sufficiently large (since $n\\ge 3$ , we get that $a_{1i}/\\sum _{j \\in [n]} a_{1j} < 1$ , and the second term above tends to zero as $s \\rightarrow \\infty $ ).",
"We will consider points along the line segment between $\\mathbf {b}^{\\prime }$ and $\\mathbf {b}$ , namely, $\\mathbf {b}(t) = (b_i(t))_{i \\in [n]} = (1-t)\\mathbf {b}^{\\prime }+t\\mathbf {b}, \\qquad \\text{ for } t\\in [0,1].$ Since $\\mathbf {b}^{\\prime }$ is feasible and we already proved that (REF ) is a necessary condition, $\\mathbf {b}^{\\prime }$ satisfies (REF ).",
"But, not only $\\mathbf {b}$ and $\\mathbf {b}^{\\prime }$ satisfy it but also $\\mathbf {b}(t)$ satisfies it for any $t \\in [0,1]$ .",
"Let us fix any $t \\in (0,1)$ .",
"Clearly, $b_1(t)=b_1=b^{\\prime }_1$ is a largest value in $\\mathbf {b}(t)$ .",
"More importantly, $\\sum _{i \\in [n]} b_i(t) = (1-t) \\sum _{i \\in [n]} b^{\\prime }_i + t \\sum _{i \\in [n]} b_i > (1-t)\\cdot 2 \\max _{i \\in [n]} b^{\\prime }_i + t \\cdot 2 \\max _{i \\in [n]} b_i = 2 b_1(t) = 2 \\max _{i\\in [n]} b_i(t)$ and so, indeed, (REF ) holds for $\\mathbf {b}(t)$ .",
"It follows that there exists a universal $\\varepsilon > 0$ such that for any $t \\in [0,1]$ we have $2 \\max _{i \\in [n]} b_i(t) < (1-\\varepsilon ) \\sum _{i \\in [n]} b_i(i).$ Fix $t \\in (0,1)$ and suppose that $\\mathbf {b}(t)$ is feasible.",
"Let $\\mathbf {x}(t) = (x_i(t))_{i\\in [n]}$ be the (unique) solution for $\\mathbf {b}(t)$ .",
"Let $G(t) = \\left( \\frac{\\partial f_i}{\\partial x_j} \\right)_{ij}$ be the Jacobian matrix of our transformation function at point $\\mathbf {x}(t)$ ; that is, $g_{ij}(t) = a_{ij} x_i(t)$ if $i \\ne j$ and $g_{ii}(t) = \\sum _{j \\in [n]} a_{ij} x_j(t)$ .",
"From the analysis performed in the proof of uniqueness of the solution it follows that $\\det (G(t))\\ne 0$ and so our transformation is a local diffeomorphism.",
"As a result, any open set in ${\\mathbb {R}}^n$ containing $\\mathbf {x}(t)$ is mapped to an open set in ${\\mathbb {R}}^n$ containing $\\mathbf {b}(t)$ .",
"In particular, there exists $\\delta > 0$ such that $\\mathbf {b}(s)$ is feasible for any $t - \\delta \\le s \\le t + \\delta $ .",
"Combining this observation with the fact that $\\mathbf {b}$ is not feasible we get that there exists $T \\in (0,1]$ such that $\\mathbf {b}(T)$ is not feasible but $\\mathbf {b}(t)$ is feasible for any $t \\in [0,T)$ .",
"Consider then any sequence $(t_i)_{i \\in {\\mathbb {N}}}$ of real numbers $t_i \\in [0,T)$ such that $t_i\\rightarrow T$ as $i \\rightarrow \\infty $ ; for example, $t_i = T(1-1/i)$ .",
"All limits from now on will be for $i \\rightarrow \\infty $ .",
"Recall that $\\mathbf {b}(t_i)$ is feasible and so $\\mathbf {x}(t_i)$ is well-defined.",
"Let us first note that it is impossible that $\\Vert \\mathbf {x}(t_i) \\Vert $ is bounded for infinitely many $i$ .",
"Indeed, if it were the case, then by Bolzano-Weierstrass theorem it would have a subsequence $(\\mathbf {x}(t_{s_i}))_{i \\in [n]}$ such that $\\Vert \\mathbf {x}(t_{s_i}) \\Vert \\rightarrow c \\in {\\mathbb {R}}$ .",
"However, then, by continuity of our transformation, the limit value $\\mathbf {b}(T)$ would be feasible.",
"Note, in particular, that it is impossible that $x_j(t_{s_i}) \\rightarrow 0$ for some $j$ as then the corresponding $b_j(t_{s_i})$ would also tend to zero which implies that $b_j(T) = 0$ ; this is not possible as both $b_j$ and $b^{\\prime }_j$ are bounded away from zero.",
"It follows that $\\Vert \\mathbf {x}(t_i) \\Vert \\rightarrow \\infty $ .",
"This means that there exist $p \\in [n]$ and subsequence $(t_{s_i})_{i \\in {\\mathbb {N}}}$ such that $x_p(t_{s_i})\\rightarrow \\infty $ .",
"Let us now observe that it is not possible that $x_p(t_{s_i})\\rightarrow \\infty $ and $x_q(t_{s_i})$ does not tend to zero for some $q\\ne p$ .",
"Indeed, since $b_p(t_{s_i}) \\ge a_{pq} x_q(t_{s_i}) x_p(t_{s_i})$ , this would imply that for any constant $c$ , $b_p(t_{s_i}) > c$ for an infinite number of $i$ 's.",
"This is impossible as $b_p(t_{s_i}) \\rightarrow b_p(T) \\le b_1(T)$ .",
"We get that $x_p(t_{s_i})\\rightarrow \\infty $ for precisely one index $p$ and $x_q(t_{s_i})\\rightarrow 0$ for all $q\\ne p$ .",
"Now, for a given $q \\ne p$ , note that $\\sum _{j\\ne p} a_{qj} x_q(t_{s_i}) x_j(t_{s_i}) \\rightarrow 0$ and $a_{qp} x_q(t_{s_i}) x_p(t_{s_i}) + \\sum _{j\\ne p} a_{qj} x_q(t_{s_i}) x_j(t_{s_i}) = \\sum _{j \\in [n]} a_{qj} x_q(t_{s_i}) x_j(t_{s_i}) = b_q(t_{s_i}) \\rightarrow b_q(T).$ We get that for sufficiently large values of $i$ , $a_{qp} x_q(t_{s_i}) x_p(t_{s_i}) > (1-\\varepsilon /2) b_q(T)$ , where $\\varepsilon > 0$ is the same as in (REF ).",
"It follows that for all sufficiently large $i$ , $b_p(t_{s_i}) = \\sum _{q \\ne p} a_{pq} x_p(t_{s_i}) x_q(t_{s_i}) \\ge \\left( 1 - \\frac{\\varepsilon }{2} \\right) \\sum _{q \\ne p} b_q(T)$ and so also in the limit $b_p(T) \\ge \\left( 1 - \\frac{\\varepsilon }{2} \\right) \\sum _{q \\ne p} b_q(T).$ But this means that $2 \\max _{i \\in [n]} b_i(T) \\ge 2 b_p(T) \\ge b_p(T) + \\left( 1 - \\frac{\\varepsilon }{2} \\right) \\sum _{q \\ne p} b_q(T) \\ge \\left( 1 - \\frac{\\varepsilon }{2} \\right) \\sum _{q \\in [n]} b_q(T)$ that contradicts (REF ).",
"Hence, we get that (REF ) is also a sufficient condition for the existence of the solution."
],
[
"More Illustrations of the GCL Model",
"In Section , we illustrated our framework with the small Zachary's Karate Club graph.",
"In this section, we provide the same illustration for all the other graphs we considered except for the larger email graph which is less amenable to nice visualization.",
"For each graph considered, we select the best scoring embedding we found, and we plot the divergence score as a function of the $\\alpha $ -parameter that we use in our framework.",
"Recall that we search over a grid of values for $\\alpha $ and we select the one for which the divergence score in (REF ) is minimized.",
"We also plot the graph itself using this best embedding.",
"The Geometric Chung-Lu model computes probabilities of seeing edges between vertices as a function of the degrees of the corresponding vertices and their distance in embedded space, the latter being a function of $\\alpha $ .",
"Thus, we can use this model to generate random graphs given the degrees of vertices and some embedding.",
"With small values of $\\alpha $ , the importance of the distances is small (at the extreme case, when $\\alpha =0$ , it is completely neglected), and several long edges (between communities) are generated.",
"With large values of $\\alpha $ , it is the opposite and we expect much fewer edges between communities.",
"The optimal value, which we denote $\\hat{\\alpha }$ , aims at preserving the right balance between long and short edges (respectively between and within communities).",
"For each graph considered, we show three generated random graph respectively using $\\alpha =0, \\alpha = \\hat{\\alpha }$ , and some value of $\\alpha > \\hat{\\alpha }$ ."
],
[
"The College Football Graph",
"For this graph, most communities are very strong, and the best embedding puts their vertices quite close, as we see in Figure REF .",
"The impact of the choice of $\\alpha $ for the distance function in embedded space is clearly seen in the bottom row of Figure REF .",
"With $\\alpha =0$ , there are many edges between communities, as only the expected degrees are preserved.",
"For $\\alpha $ larger than the optimal value, we see the opposite behaviour.",
"The random graph generated with the optimal $\\alpha $ value is visually very similar to the real graph shown in the top row.",
"Figure: The College Football Graph.",
"On the top row, we illustrate the divergence score as a function of α\\alpha (left) for the best embedding found by our framework (right).In the bottom row, we generate edges using the Geometric Chung-Lu Model respectively with α=0\\alpha =0 (left), the optimal value α ^=5.25\\hat{\\alpha }=5.25 (center) and α=7\\alpha =7 (right)."
],
[
"LFR Graph with Strong Communities",
"The communities are very strong given the choice of value $\\mu =0.15$ , so good embeddings should put their vertices quite close, as we see in Figure REF .",
"The impact of the choice of $\\alpha $ for the distance function in embedded space is clearly seen in the bottom row of Figure REF , with the same conclusion as for the College Football graph.",
"Figure: For the LFR graph with mixing parameter μ=0.15\\mu =0.15, on the top row, we illustrate the divergence score as a function of α\\alpha (left) for the best embedding found by our framework (right).On the bottom row, we generate edges via theGeometric Chung-Lu Model with distance parameters α=0\\alpha =0 (left),the optimal value α ^=4.75\\hat{\\alpha }=4.75 (center) and α=7\\alpha =7 (right)."
],
[
"LFR Graph with Weaker Communities",
"Results for the LFR graph with mixing parameter $\\mu =0.35$ are shown in Figure REF .",
"For the best embedding, vertices within communities are not as close as in the previous LFR graph, as expected.",
"Conclusions regarding the values of $\\alpha $ are the same as for the previous examples.",
"Figure: For the LFR graph with mixing parameter μ=.35\\mu =.35, on the top row, we illustrate the divergence score as a function of α\\alpha (left) for the best embedding found by our framework (right).On the bottom row, we generate edges via theGeometric Chung-Lu Model with distance parameters α=0\\alpha =0 (left),the optimal value α ^=6\\hat{\\alpha }=6 (center) and α=10\\alpha =10 (right)."
],
[
"Noisy LFR Graph",
"In this graph, LFR graph with mixing parameter $\\mu =0.55$ , for each community, there are more expected external edges than internal edges.",
"Nevertheless, in the best embedding shown in Figure REF , we still see some spatial grouping of the communities.",
"Figure: For the LFR graph with mixing parameter μ=0.55\\mu =0.55, on the top row, we illustrate the divergence score as a function of α\\alpha (left) for the best embedding found by our framework (right).On the bottom row, we generate edges via theGeometric Chung-Lu Model with distance parameters α=0\\alpha =0 (left),the optimal value α ^=4.5\\hat{\\alpha }=4.5 (center), and α=7\\alpha =7 (right)."
]
] | 1906.04562 |
[
[
"Classifying galaxies according to their HI content"
],
[
"Abstract We use machine learning to classify galaxies according to their HI content, based on both their optical photometry and environmental properties.",
"The data used for our analyses are the outputs in the range $z = 0-1$ from MUFASA cosmological hydrodynamic simulation.",
"In our previous paper, where we predicted the galaxy HI content using the same input features, HI rich galaxies were only selected for the training.",
"In order for the predictions on real observation data to be more accurate, the classifiers built in this study will first establish if a galaxy is HI rich ($\\rm{log(M_{HI}/M_{*})} > -2 $) before estimating its neutral hydrogen content using the regressors developed in the first paper.",
"We resort to various machine learning algorithms and assess their performance with various metrics such as accuracy for instance.",
"The performance of the classifiers gets better with increasing redshift and reaches their peak performance around $z = 1$.",
"Random Forest method, the most robust among the classifiers when considering only the mock data for both training and test in this study, reaches an accuracy above $98.6 \\%$ at $z = 0$ and above $99.0 \\%$ at $z = 1$.",
"We test our algorithms, trained with simulation data, on classification of the galaxies in RESOLVE, ALFALFA and GASS surveys.",
"Interestingly, SVM algorithm, the best classifier for the tests, achieves a precision, the relevant metric for the tests, above $87.60\\%$ and a specificity above $71.4\\%$ with all the tests, indicating that the classifier is capable of learning from the simulated data to classify HI rich/HI poor galaxies from the real observation data.",
"With the advent of large HI 21 cm surveys such as the SKA, this set of classifiers, together with the regressors developed in the first paper, will be part of a pipeline, a very useful tool, which is aimed at predicting HI content of galaxies."
],
[
"Introduction",
"Much effort has been put into understanding the role of neutral hydrogen in galaxy formation and evolution.",
"In the canonical picture based on the Hubble Sequence, the spiral galaxies are rich in cold gas and star forming, whereas the ellipticals are red and quiescent.",
"However, an increasing number of observational evidence shows that these correlations are not always true.",
"Local early-type galaxies from the ATLAS$^\\mathrm {3D}$ survey were shown to contain significant cold gaseous components [7].",
"They found that the relative angles between the gaseous and stellar planes show a bimodal distribution, but found no plausible explanation for such difference.",
"This indicates that the gas distribution of a galaxy does not necessarily follow that of the stellar component.",
"Therefore, direct inference of the gas content of galaxy based on its optical content is inaccurate.",
"Elliptical galaxies are observed to form stars in cool core massive clusters [8] that is suggestive of the presence of cold gas in those objects.",
"The amount of gas components in massive ellipticals is crucial to understanding the evolution and growth of galaxies at the massive end, but the presence of kinematic abnormalities in their gas content as well as the uncertain effects of the Active Galactic Nuclei (AGN) feedback can affect the surface density of the gas content to pull the galaxies below the Hi detection limit, especially at higher redshifts.",
"Spiral galaxies are gas rich, but the limitations of observing the neutral gas at intermediate redshift prevent a robust study of the evolution of their gas content.",
"Low redshift ($z\\lesssim 0.4$ ) Hi can be observed with the 21cm emission line to provide the neutral hydrogen mass distribution of nearby galaxies.",
"For instance, the Arecibo Legacy Fast ALFA [16] observed $\\sim 30000$ galaxy Hi fluxes.",
"The highest redshift galaxy ($z=0.376$ ) detected in 21 cm emission was observed with the COSMOS Hi Large Extragalactic Survey (CHILES) [9].",
"At any substantially higher redshift, the Hi content of galaxies is inferred from Damped Lyman Alpha systems (DLAs) in the spectra of background quasars, but it is difficult to measure the Hi mass from DLAs, and the relationship between galaxies and DLAs is not completely clear.",
"The upcoming blind surveys such as Looking At the Distant Universe with the MeerKAT Array (LADUMA) on MeerKAT and eventually follow-up surveys on the SKA aim to measure the Hi content of galaxies at intermediate redshifts, to $z\\sim 1$ and beyond.",
"The gas content of satellite galaxies are substantially impacted by environmental effects.",
"Observationally, only $25\\%$ of $\\alpha .40$ [15] galaxies were found to be in groups or clusters [17], which is lower than for the overall galaxy population.",
"They found that in contrast to increasing optical sources towards to the center of groups or clusters, the number of Hi sources decreases.",
"This is also supported from theoretical views.",
"Using hydrodynamical simulation, showed that the fraction of Hi deficient galaxies increases towards higher halo masses.",
"This is related to the star formation quenching timescale decrease towards higher halo mass: from $>3$ Gyr for $\\mathrm {M_{halo}} <10^{12}\\;{\\rm M}_{\\odot }$ to $<1$ Gyr for $\\mathrm {M_{halo}} > 10^{13}\\;{\\rm M}_{\\odot }$ .",
"Recent observational work by [11] agrees with this prediction, but in contrast [12] argues for no relationship between galaxy quenching timescales and halo mass.",
"Simulations also suggest that the presence of Hi is strongly correlated with star formation, even if the star formation is physically occurring in molecular gas [6].",
"Therefore, the Hi content appears to have a complex relationship with respect to stellar mass, star formation rate, morphology, and environment.",
"This makes it challenging to predict what the Hi content of any given galaxy will be without accounting for the full range of its properties.",
"In order to better design and interpret upcoming Hi surveys, it is useful to be able to estimate the expected Hi content of galaxies that will be observed based on their already-measured multi-wavelength properties.",
"To do so, here we develop and employ galaxy classification tools using machine learning.",
"Galaxy classification is a very useful approach as it can provide insights into the physical processes by which galaxies evolve over cosmic time.",
"There exist different and complementary ways to classify galaxies depending on the availability of the data, for instance morphological classification or spectral classification.",
"The Hubble Sequence focuses on morphological classification, while spectral classification via absorption and emission lines provides more information about the chemical composition and stellar populations of galaxies [21].",
"developed a $\\chi ^2$ -fitting approach to identify the best linear combination of template spectra that matches the observed spectrum in order to classify galaxies spectroscopically with low signal to noise ratio (S/N), and found good correlations of $\\ge 80\\%$ between spectra and morphology from Hubble classification.",
"presented a novel information bottleneck (IB) approach, improving on the then-standard geometrical and statistical approaches, to classify galaxy spectra using 2dF Galaxy Survey [5], [10].",
"In a seminal work, [13] conducted morphological classification of galaxies which was achieved by simple visual inspection where volunteers catalogued thousands of objects from Sloan Digital Sky Survey Data Release 3 in order to obtain the rate of interacting galaxies.",
"The need for automated classification arose with the increasing amount of available survey data, and it was demonstrated by [22] and [20] that accuracy achieved by a trained Artificial Neural Network in classifying galaxies is comparable to that of a human expert.",
"In a morphological classification of high redshift galaxies that [18] conducted using Support Vector Machines, they argued that at $z > 1$ early type galaxies were underestimated in the classifications using sample from COSMOS HST/ACS [19] owing to the effects of morphological k-correction.",
"In galaxy morphological classification, tree-based algorithms have also proved to be relatively robust classifier compared to other machine learning algorithms, as reported by [14].",
"Hence there is a long history of using sophisticated galaxy classification methods in astronomy, but so far this has not been extensively applied to studying Hi.",
"In our previous work in ([rad18]rad18 hereafter), we investigated the possibility of estimating the Hi content of galaxies using a variety of machine learning algorithms.",
"Considering both the optical and environmental properties of the galaxies as input features, the algorithms were trained using large subsets of data from Mufasa simulation and tested on different subsets.",
"They found that the performance of all regressors – assessed by using root mean squared error (rmse) and Pearson's correlation coefficient (r) as metrics – degraded at higher redshift.",
"Despite the tendency of all learners to under-predict the high Hi richness and over-predict the low one, random forest method – followed tightly by deep neural network – exhibited an overall best performance; achieving an rmse $\\sim 0.25$ (corresponding to $\\textbf {r} \\sim 0.9$ ) at $z = 0$ .",
"They then applied the regressors to real data from two different surveys, RESOLVE and ALFALFA.",
"To this end, they trained the algorithms with an output from Mufasa at $z = 0$ and used them to predict the Hi content of galaxies from real observations.",
"Their results proved that the learners which they built can be potentially used for Hi study with the upcoming large Hi surveys like the SKA.",
"Prior to this work, related study by also investigated the estimation of Hi content of galaxies based on the SDSS and ALFALFA data using 15 derived galaxy parameters.",
"However, in [rad18]rad18 we only considered Hi rich galaxies ($\\rm {log(M_{{\\sc Hi}}/M_{*})} \\ge -2 $ ), hence the machine learning methods were trained to predict the gas content of Hi rich galaxies only.",
"Therefore, at this stage, those algorithms on their own can't be deployed in real world application where not all galaxies will be Hi rich.",
"Models generally predict that galaxies are bimodal in their Hi content, particularly since satellite galaxies lose their Hi quite rapidly, after a delay period, once they enter another halo .",
"To extend our work to be more generally applicable, we therefore need a way to classify galaxies as Hi rich or Hi poor based on available photometric data.",
"In this follow-up paper, we address this issue by building a set of learners that filter out the Hi poor galaxies in real survey, such that the regressors built in [rad18]rad18 only predict galaxy gas content known to be above a certain threshold.",
"Together with the classifiers, the regressors will form a pipeline which will be used to estimate Hi gas of galaxies in real observation.",
"The approach is to use the same set of input features as in [rad18]rad18 for the classification.",
"This paper thus extends our approach to be more generally applicable to any galaxy survey that contains the requisite input features, which are chosen to be typically observationally accessible in present and upcoming multi-wavelength surveys.",
"We present our machine learning setup for our analyses in § and list all the algorithms we consider in §.",
"The results are shown in § and we demonstrate how the methods can be applied to data from real surveys in §.",
"We finally conclude in §."
],
[
"Setups",
"It is first noted that we make use of the same outputs ($z = 0 - 1$ ) from Mufasa simulation to build our classifiers.",
"Considering the Planck cosmological parameters $\\Omega _m =0.3$ , $\\Omega _\\Lambda = 0.7$ , $\\Omega _b = 0.048$ , $H_0 = 68 \\;{\\rm km}\\,{\\rm s}^{-1}\\;{\\rm Mpc}^{-1}$ , $\\sigma _8 = 0.82$ and $n_s = 0.97$ [23], each snapshot results from simulating a comoving box of 50$h^{-1}$ Mpc with a resolution of N = $512^3$ for each species (dark matter and gas).",
"For the training, the features $\\lbrace u,g,r,i,z,U,V,J,H,K_s, \\Sigma _3, v_{gal}\\rbrace $ are considered whereas our target – as in the case of a binary classification – is one of the two classes; 0 to denote Hi depleted galaxies ($\\rm {log(M_{{\\sc Hi}}/M_{*})} < v_{thresh} $ ) and 1 for Hi rich galaxies ($\\rm {log(M_{{\\sc Hi}}/M_{*})} \\ge v_{thresh} $ ).",
"To split the galaxies into two classes, one simply needs to run through all galaxies in the data and assign 0 or 1 to it if its gas content is below or above the threshold value v$_\\mathrm {thresh}$ .",
"In our case, we adopt v$_\\mathrm {thresh} = -2$ , i.e.",
"the Hi content is 2 orders of magnitude fewer than the stellar content.",
"Table: List of all the setups that are considered in the analysis.For easy reference, each setup has been given a name.As in [rad18]rad18, we adopt different setups both in terms of features and type of training which we present again in Table REF for reference.",
"For “$z-$ training”, a classifier is built at each redshift bin whereas for “$f-$ training” we make use of all data available in the range $z = 0 - 0.5$ .",
"In contrast with the $f-$ training in [rad18]rad18, we do not go to higher $z$ to train the learner.",
"In all cases, $75\\%$ of the data is used for training and the remaining is used for testing."
],
[
"Algorithms",
"We used a rather wide variety of machine learning algorithms in [rad18]rad18 to see which one captures best the features from the data in order to make good predictions.",
"Having gained a better understanding about how the methods dealt with information from the data, we consider most of them for this classification problem.",
"It is worth reiterating that as opposed to regression task where the label is a numerical variable, the label for a classification task is a class – represented by integers mainlyCategorical variable.. k-Nearest Neighbour (kNN) - Classification: the principle remains the same as in regression but instead of averaging the targets of $k-$ closest neighbours to make prediction, the predicted class $y_{new}$ of a new instance $\\textbf {x}_{new}$ is simply the majority of the classes of $k-$ neighbours of $\\textbf {x}_{new}$ .",
"Random forest (RF) and Gradient boosting (GRAD) - Classification: decision tree is still the base estimator of both RF and GRAD.",
"In contrast with its regressor counterpart, the decision tree classifier splits the training set at a split point $s_{i}$ using a feature $i$ .",
"The splitting is done in such a way as to minimize the objective function $\\mathcal {F} = \\frac{n_{R_{1}}}{n}G_{R_{1}} + \\frac{n_{R_{2}}}{n}G_{R_{2}},$ where $n_{R_{1}}$ is the number of examples in region $R_{1}$ and $n_{R_{2}}$ the number of examples in $R_{2}$ .",
"The total number of instances $n$ before the split is simply $n = n_{R_{1}} + n_{R_{2}}$ .",
"The Gini impurityAlso called Gini index.",
"$G$ of each region is given by $G = 1 - \\sum _{i = 1}^{k}p^{2}_{i},$ where $p_{i}$ is the probability of an instance to belong to a class $i$ in the region.",
"This can be computed by the ratio between the number of intances belonging to a class $i$ and the number of all instances in the region.",
"The splitting can be done recursively on the resulting nodes depending on the required size of the tree.",
"The RF method predicts the class of a new instance $\\textbf {x}_{new}$ by aggregating the predictions of all its decision trees.",
"The expression of the GRAD classifier is quite similar to Eq.",
"6 in [rad18]rad18.",
"Deep neural network (DNN) - Classification: In contrast with the DNN regressor, the activation function of the output layer is a sigmoid functionAlso named logit.",
"$\\sigma (x) = \\frac{1}{1 + e^{-x}},$ which computes the probabibility $p_{i}$ that an instance belongs to class $i$ .",
"In this case specifically, if $p \\ge 0.5$ , $y_{new}$ is $\\textbf {1}$ (positive class) whereas for $p < 0.5$ $y_{new}$ is $\\textbf {0}$ (negative class).",
"The objective function, known as log loss, is defined as $\\mathcal {F} = -\\frac{1}{N}\\sum _{i=1}^{N}y_{i}{\\rm {log}}(p_{i})+(1 - y_{i}){\\rm {log}}(1-p_{i}).$ The weights and biases are updated via backpropagation as usual.",
"The cost function in Eq.",
"REF can be generalised for multiclass case by using what is called cross entropy defined as $\\mathcal {F} = -\\frac{1}{N}\\sum _{i=1}^{N}\\sum _{k=1}^{\\mathcal {K}}y_{k}^{i}{\\rm {log}}(p_{k}^{i}),$ where $\\mathcal {K}$ is number of classes."
],
[
"Galaxy classification",
"The objective in this work is to be able to establish whether a galaxy is Hi rich or Hi poor by exploiting both its optical and environmental data.",
"To do so, we build various classifiers (see §) and compare their performance qualitatively using various metrics which we present now along with some useful terminology in machine learning.",
"Accuracy: In binary classificationAnd even in mutliclass case., it measures the ratio of the correct predictions on a test sample, i.e.",
"$accuracy = \\frac{TP + TN}{FN + FP + TP + TN},$ where $TP$ and $TN$ are True Positive – number of instances that are correctly predicted by the classifier to belong to 1 – and True negative – number of instances that are correctly predicted by the classifier to belong to 0 – respectively.",
"FN or False Negative denotes the number of instances that belong to 1 but are classified as 0 and FP or False Positive indicates the number of instances that belong to 0 but are predicted as 1.",
"A confusion matrix, which is represented in Figure.",
"REF , is $2 \\times 2$$n \\times n$ in multiclass case.",
"matrix which summarizes the predictions of a classifier on a test set.",
"Figure: Confusion matrix 2×22 \\times 2 for a binary classification.",
"Negative class is Hi poor, Positive class is Hi rich.Precision: It indicates how well the algorithm minimizes the number of instances incorrectly identified as a Positive class (FP) and is given by $precision = \\frac{TP}{TP + FP}.$ A good precision (high value close to one) translates to low FP.",
"Recall: Also called sensitivity, it characterizes the ability of the method to minimize the number of instances wrongly identified as a Negative class ($FN$ ).",
"It is given by $recall = \\frac{TP}{TP + FN}.$ It is worth noting that, provided a classifier, if FP increases then FN decreases and vice versa.",
"In other words, an increase in precision implies a decrease in recall – the so called precision-recall tradeoff.",
"In our case, since we are mainly interested in identifying Hi rich galaxies whose gas content is to be predicted by our regressors built in [rad18]rad18, we require our classifier to have good precision, as having a learner with a lower FP (hence higher it FN) – lower number of Hi poor galaxies predicted to belong to class of Hi rich galaxies – is in our case more preferable than a learner with a lower FN, hence higher FP.",
"$F_{1}$ score: This metric which combines $precision$ and $recall$ is their harmonic mean, given by $F_{1} = \\frac{TP}{TP + \\frac{FN + FP}{2}}.$ High $F_{1}$ score simply means that both $precision$ and $recall$ are also high, which is the ideal case.",
"Log Loss: This quantity, given by Eq.",
"REF , is also used as a metric.",
"The lower its value, the better the classifier is.",
"Receiving Operating Characteristic - Area Under the Curve (ROC AUC): It is also possible to plot recall against FP rate which is given by $1 - specificity$ where $specificity = \\frac{TN}{TN + FP}.$ As can be seen from Eqs.REF -REF , $FP$ follows the increase of recall as a consequence of the precision-recall trade-off.",
"Another measure of the performance of a classifier is then to compute the area under the curve (recall vs FP rate).",
"A perfect learner would have ROC AUC = 1.",
"A binary classifier uses a threshold parameter such that a new instance will be classified as positive or negative if the predicted probability is above or below the threshold respectively.",
"A precision-recall (alternatively recall-FP rate) pair corresponds to a single value of a threshold parameter of a classifier and the idea behind the ROC curve is to find the best pair values precision-recall (alternatively recall-FP rate) in order to mitigate the trade-off between them, i.e.",
"finding a threshold parameter value of the classifier such that both precision and recall are high.",
"The results are now presented in the following."
],
[
"Dependence on redshift",
"Table REF lists the various setups that we feed to our machine learning algorithms.",
"The name specifies whether it is uses $f-$ training or $z-$ training, whether we use SDSS data only (S) or all data including near-IR (A), and whether we use magnitudes (Mg) or colors (Clr) or combine them (Cmb).",
"In all cases we use environment as measured by the third nearest neighbor ($\\Sigma _3$ ), as well as the galaxy peculiar velocity ($v_{gal}$ ).",
"In Figure.",
"REF , we show the results corresponding to each classifier selected in our investigation, considering only two metrics here, accuracy and $f_{1}$ , for illustration purpose.",
"The first column shows the accuracy achieved by each method with different input features for “$f-$ training”, the second column is the resulting accuracy for “$z-$ training”, the third column presents the $f_{1}$ score for “$f-$ training” and finally the fourth one is the $f_{1}$ score for “$z-$ training”.",
"Most classifiers attain accuracy and $f_{1}$ scores exceeding 0.9, which indicates that it is robustly possible to classify galaxies into Hi rich vs. Hi poor based on observable properties, at least in the idealised case of training and testing on simulated data alone.",
"Still, there are clear differences among the classifiers.",
"Random forests (RF; green) clearly exhibits the best performance whereas GRAD (purple) is relatively the weakest.",
"For instance, RF (“$z-$ training”) $accuracy$ and $f_{1}$ both reach $\\sim 0.98$ at $z = 0$ and $\\sim 0.99$ at $z = 1$ , with similar values when combining data from $z=0-0.5$ (“$f-$ training\").",
"kNN shows values $\\sim 0.95$ , while DNN's performance is consistently poorer.",
"Figure: Accuracy and f 1 f_{1} are shown on the 2 columns from the left and right,respectively.",
"Good performance means high values of both accuracy and f 1 f_{1}.",
"The dots, color codedby the training models we use, represent the performance (accuracy and f 1 f_{1}) of each classifier trained on all the data available between z=0-0.5z=0-0.5, “f-f-training”.",
"In the same way, the lines denote the value of the two metrics of each learner as a function of redshift “z-z-training”.",
"Each row shows different results for differentsetups.",
"The accuracy values are shown on the left y-axes and the f 1 f_{1} valueson the right y-axes.The dependences of both accuracy and $f_{1}$ on redshift follow similar trend; they both increase as we go to higher $z$ .",
"This indeed looks very promising, since the improving performance of the classifier with increasing $z$ may compensate for the decreasing performance of the regressor built in [rad18]rad18 at higher $z$ , although this is only valid up to $z \\sim 1$ since the performance of the classifier reaches their peak around that redshift then starts to degrade.",
"That limitation is the reason we only show the results up to $z\\sim 1$ .",
"In other words, most of the Hi poor galaxies can be filtered out by the classifier such that the regressor will only estimate the gas content of the Hi rich galaxies.",
"As expected, the value of the accuracy and that of $f_{1}$ when training the learners with all the data available between $z = 0-0.5$ is approximately the average of accuracy's and that of $f_{1}$ 's within that $z$ -bin.",
"As already mentioned in [rad18]rad18 the main idea behind the “$f-$ training” is to anticipate the fact that in real observations, retrieving redshift information is not an easy task.",
"Therefore we make an attempt at also building a classifier without relying on redshift information.",
"The high values of both accuracy and $f_{1}$ $\\sim 0.9$ for all learners with any setup except fSMg demonstrate that it is indeed possible to build a relatively good classifier without taking into account redshift information."
],
[
"Dependence on input features",
"We now look in more detail at how the classification is affected by the selected input features, i.e.",
"comparing the rows in Figure.",
"REF .",
"In realistic scenarios, it is not always possible to have all the features available.",
"This leads us to investigate different scenarios by considering different combinations of features.",
"The best classifier (RF) does appear to be insensitive to the choice of input features with values of accuracy and $f_{1}$ $\\ge 0.98$ at all redshift bins, which is good news.",
"However, for the learner with the worst performance (GRAD), it does not seem to be the case as its performance measures fluctuate with respect to the setup considered and are at their lowest values with zSMg setup (at $z = 0$ , $f_{1}$ and accuracy are both $\\sim 0.87$ ; $z = 1$ , $f_{1}$ $\\sim 0.91$ and $accuracy \\sim 0.88$ ) for “$z-$ training” and $accuracy \\sim 0.857$ and $f_{1}$ $\\sim 0.877$ for “$f-$ training” fSMg.",
"In Figure.",
"REF , we show other metrics of the RF, namely ROC AUC and log loss, as function of redshift for zSCmb.",
"As expected, the better performance at higher redshift bin corresponds to a lower log loss.",
"A ROC AUC $\\ge 0.99$ at all redshifts corroborates the fact that RF is our best classifier for this ideal scenario where classifiers are both trained and tested with mock data.",
"It is noted that the effects of the class imbalance – potential issue owing to a big difference between the number of instances of each class in the training set which might cause a classifier to fail to label new instances in the test set properly – have been checked by compensating the imbalance using imblearn.",
"No noticeable differenceIf not the exact same results.",
"has been found between the two cases – with and without compensation – by comparing their resulting metrics.",
"Figure: Four metrics as a function of zz of both RF and kNN methods for zSCmb setup.",
"Top left: log loss, top right: ROC AUC, bottom left: accuracy and bottom right: f 1 f_{1}."
],
[
"Effects of setting up the classes",
"In our main analyses, the Hi galaxies are split into two distinct classes according to whether their Hi gas masses are above or below a threshold of 0.01 times their stellar masses.",
"The threshold value is broadly in accordance with observational Hi fraction limits.",
"However, other classifications are possible.",
"Here we explore the impact of changing the classification metric.",
"We consider three new classification schemes.",
"The Galex Arecibo SDSS Survey [3] set a threshold limit of $\\log (\\mathrm {M_{HI}}) = 8.7$ for galaxies with $\\mathrm {M_*}<10^{10.5}\\;{\\rm M}_{\\odot }$ and $\\log (\\mathrm {M_{HI}/M_*}) = -1.8$ otherwise.",
"However, in order to be consistent with the threshold value of gas fraction used in [rad18]rad18 to denote Hi depleted galaxies, we set it to be $\\log (\\mathrm {M_{HI}/M_*}) = -2$ .",
"We call this type of splitting BIN.",
"Another potential classification may be on whether a galaxy has higher Hi mass than stellar mass.",
"In this case, the classes are given by $\\Big \\lbrace \\rm {log(M_{{\\sc Hi}}/M_{*})} < 0 \\rightarrow \\textbf {0};\\:\\rm {log(M_{{\\sc Hi}}/M_{*})} \\ge 0 \\rightarrow \\textbf {1} \\Big \\rbrace $ .",
"We name this type of splitting LOW.",
"Finally, we attempt splitting into three classes, as follows: $\\Big \\lbrace \\rm {log(M_{{\\sc Hi}}/M_{*})} < -2 \\rightarrow \\textbf {0};\\: -2 \\le \\rm {log(M_{{\\sc Hi}}/M_{*})} < 0 \\rightarrow \\textbf {1};\\: \\rm {log(M_{{\\sc Hi}}/M_{*})} \\ge 0 \\rightarrow \\textbf {2} \\Big \\rbrace $ , which we call MULTI.",
"In Figure.",
"REF , we compare the results corresponding to the RF method when considering three types of splitting, namely BIN (blue), LOW (orange) and MULTI (green).",
"For brevity we only consider RF, since it is our best classifier, and $z-$ training since the $f-$ training values are expected to be similar.",
"Overall, both accuracy and $f_{1}$ are $\\ge 0.80$ for all three types of splitting at all $z$ bins and it is quite clear that the algorithm performs best with our main type of splitting Hi poor/Hi rich, namely BIN.",
"It is also interesting to see that the accuracy decreases with increasing redshift for both LOW and MULTI whereas $f_{1}$ increases as we go at higher redshift for LOW.",
"Based on $accuracy$ , the method performs similarly for LOW and MULTI splittings, but the difference in performance of the algorithm is striking when considering $f_{1}$ as a metric.",
"This indicates that the classifier performance does depend on the classes chosen, but for our purposes of separating Hi rich and Hi poor galaxies, it performs very well even with minor changes to the schemeSlight change to the gas fraction limit..",
"It is worth noting that in this idealised case and in the light of the results in [rad18]rad18 we did not include SVM method.",
"However, as will be shown later, we include it for the different tests on real observation data.",
"Figure: Left panel: accuracy as function of zz, right panel: f 1 f_{1} as a function of zz.",
"Blue is for BIN, orange for LOW and green for MULTI.",
"The results are all related to RF algorithm."
],
[
"Application to observational data",
"The lack of available data is one of the drawbacks of using machine learning when solving a problem, be it regression or classification.",
"To mitigate that issue, in the context of Hi study, we aim at building classifiers trained with mock data from simulation and using them to identify Hi rich galaxies in real surveys.",
"As already demonstrated in [rad18]rad18 the regressors that they built were able to learn from the mock data in order to predict the Hi content of the galaxies from both RESOLVE and ALFALFA.",
"Our approach here is to redo the same exercise but for a classification task, i.e.",
"training some classifiers with Mufasa data and utilising them to identify Hi rich galaxies from the same surveys, RESOLVE and ALFALFA.",
"In this study, we also consider another survey, GASS, in which both Hi poor and Hi rich galaxies are better represented for our tests.",
"For the description of the first two surveys, we refer the interested reader to [rad18]rad18, and will now give a brief description of GASS."
],
[
"GASS was aimed at investigating Hi properties of a selected sample of galaxies ($\\sim $ 1000) with available optical properties.",
"The last data release (DR3) [4], which we use in our analyses, was built upon the first two data releases [1], [2].",
"Within a relatively large volume survey of 200 Mpc already probed by SDSS primary spectroscopy survey, the GALEX Medium Imaging Survey and ALFALFA, galaxies have stellar masses of $10 < \\rm {log}(M_{*}/\\rm {M_{\\odot }}) < 11.5$ that encompasses the transition mass.",
"The targets have Hi richness above the detection limit of $0.015$ for $10.5 < \\rm {log}(M_{*}/\\rm {M_{\\odot }})$ and a fix Hi mass of $10^{8.7} \\rm {M_{\\odot }}$ for lower stellar masses.",
"The targets are designed to fall within $0.025<z<0.05$ .",
"Using the Arecibo radio telescope, [4] compiled a sample which has a fairly good representationGASS representative sample as they call it.",
"in which $62\\%$ are referred to as detections and the remaining $38\\%$ as non detections.",
"The latter represent galaxies in which a relatively small gas mass fraction was observed hence required a longer integration time (but not more than 3h), whereas the former was found to have relatively large amount of gas mass fraction.",
"For our analyses, we retrieved all the optical properties of each galaxy in the sample from SDSS database using their SDSS-ID.",
"In order to have a more balanced test sample, we then split the sample into two classes: Hi poor galaxies (class 0) are those with $\\rm {log}(M_{HI}/\\rm {M_{*}}) < -1.55$ and the remaining are Hi rich galaxies (class 0).",
"With this type of splitting, we have $56.8\\%$ of the sample Hi rich and the remaining Hi poor."
],
[
"Testing the built classifiers",
"We consider four different tests according to both the survey and input features TEST 1: RESOLVE DATA, color indices from all the band magnitudes available; SDSS (u,g,r,i,z), 2MASS (J,H,K), GALEX (NUV) and UKIDSS (Y,H,K) TEST 2: RESOLVE DATA, color indices from only SDSS (u,g,r,i,z) photometric data.",
"TEST 3: ALFALFA, color indices from only SDSS (u,g,r,i,z) photometric data.",
"TEST 4: GASS data, color indices from only SDSS (u,g,r,i,z) photometric data.",
"In all cases we split the simulated data for training and the considered test set into two categories, Hi poor (class 0) and Hi rich (class 1).",
"Our results are summarised in Table REF and shown in Figure REF ."
],
[
"TEST 1",
"The training set is composed of the data of snapshot at $z = 0$ from Mufasa since the galaxies to be classified in RESOLVE survey are all at present epoch.",
"We make use of all the photometric data available in RESOLVE, i.e.",
"$\\lbrace u, g, r, i, z, J, H, Ks, NUV\\rbrace $ .",
"We consider 5 metrics – accuracy, $f_{1}$ , ROC AUC, precision and specificity.",
"The results of the classification from the learners selected in this work are presented in Table REF and similarly shown in Figure REF .",
"DNN has the highest accuracy amongst the algorithms followed by RF.",
"This is reminiscent to the results found in the regression problem in [rad18]rad18.",
"Despite the weaker performance of DNN compared to RF when testing on the simulated data (see Figure REF ), testing on observational data really show the power of the algorithm.",
"Nonetheless, all algorithms agree within $<10\\%$ .",
"Based on the $f_{1}$ score and precision the methods are all comparable as well.",
"Interestingly, DNN's ROC AUC = 0.589 is the worst among all the methods, just above that of a classifier with a random guess.",
"Judging by the values of the precision which are $\\ge 0.95$ for all methods, they satisfy what we require; classifiers that minimise the number of Hi poor galaxies incorrectly classified as Hi rich (FP) or in other words with high precision.",
"However, a specificity equal to zero implies that all the negative class instances in the data are incorrectly classified (FP), bearing in mind that only 2$\\%$ of this test sample are Hi-poor galaxies.",
"Along with its high precision, SVM exhibits the highest specificity = 0.714, indicating its robustness, hence the best choice among the algorithms for this test.",
"We finally note that although $\\sim 98\\%$ for the RESOLVE galaxies are Hi-rich, Mufasa sample contains a balanced proportion of $\\sim 52\\%$ positive class making the training robust against class imbalance effect.",
"The most important thing is the training part which is achieved using a well balanced sample (50/50 poor-rich), therefore the algorithms are not biased toward any class.",
"In Test 4 we will consider a testing set that is more balanced (albeit smaller), which allows us to test our algorithm more fully.",
"Table: Summary of the results when using the simulation trained methods to classify Hi galaxies in the three different test tests."
],
[
"TEST 2",
"For this second test, we still use the RESOLVE data but consider color indices formed out of SDSS photometric data only, i.e.",
"$\\lbrace u,g,r,i,z\\rbrace $ .",
"In contrast with TEST 1, the results in Table.",
"REF (Figure REF ) suggest that, with the selected inputs features, the methods are capable of better identifying the gas poor galaxies with specificity all above $0.5$ , except for DNN with $0.286$ .",
"In terms of Accuracy and $f_{1}$ scores, DNN is remarkably better and GRAD noticeably worse compared to RF and $k$ NN.",
"Based on ROC AUC, RF and $k$ NN score the best and worst respectively.",
"Based on the value of its precision = 0.998, it is tempting to say that GRAD is the best method for this test, however the results suggest that SVM generalises better than GRAD, as indicated by its $f_{1}$ score and accuracy.",
"It is quite surprising to notice that with the same data (training/test), decreasing the number of selected features provide better information to the algorithms such that they get better at classifying the instances properly i.e precision (TEST 2) > precision (TEST 1); specificity (TEST 2) > specificity (TEST 1)."
],
[
"TEST 3",
"In this test, we use ALFALFA data and only consider SDSS photometric data for the input features as in TEST 2.",
"Overall, all the methods perform much better as suggested by the high values of the metrics considered (see Table.",
"REF ).",
"We note that the training set is the same as the one used for RESOLVE, hence class imbalance is not an issue that requires to be alleviated during training.",
"The precision and specificity which are both equal to 1 clearly imply that FP is zero, hence class 0 instances, despite their relatively low number, are all correctly classified.",
"This applies to all classifiers with the exception of DNN which has a specificity = 0.",
"The results for this test then suggest that our classifiers are capable of recognizing Hi rich and Hi poor galaxies to a very good precision.",
"The $f_{1}$ scores (all $> 0.9$ ) of all the learners show that their recall's are optimised, which also means that FN (Hi rich galaxies that incorrectly classified as Hi poor) is minimised.",
"The relatively higher average precision (ROC AUC) of all classifiers (> 0.9) can indeed be used as an indicator that on average both FP and FN are minimised, this is not the case for DNN.",
"All the trained non-neural network algorithms appear to meet our requirements but for the sake of comparison, RF method seems to be the best in this test, with the highest accuracy and $f_{1}$ values despite its ROC AUC is only second best.",
"Conversly with the RESOLVE data, the DNN is definitely not favoured in properly classifying Hi-poor and Hi-rich galaxies when tests are done with blind survey data such as ALFALFA."
],
[
"TEST 4",
"We use GASS data [4] for this test, considering SDSS photometric data as input features.",
"Unlike the other samples used for testing so far, all classes (0, 1) are well represented in this dataset, with $56\\% $ of this test set are Hi rich.",
"Although kNN exhibits the highest precision and specificity, it does not generalise well, given its relatively low values of both accuracy and $f_{1}$ .",
"Results suggest that our best classifier for this test is SVM which has a relatively high precision (second best after kNN) and its tendency to generalise well as justified by its overall scores.",
"In general, the other classifiers (RF, GRAD and DNN) are also capable of learning the features from the mock data in order to classify the real data, however the neural network model classifies poorly the Hi poor galaxies (i.e.",
"low specifity values), as can be noticed in all the tests conducted.",
"Figure: Summary of the results when using the simulation trained methodsto classify the Hi content of galaxies from three different testsets from observational data.",
"The y-axes are in exponential scale to preventfor data point cluttering.We have demonstrated in this work that it is possible to classify Hi galaxies based on their gas content using both their photometric and environmental data.",
"We have built various algorithms by training them using large subset of the mock data ($75\\%$ ) from Mufasa simulation.",
"While being sensitive to: the inputs features, type of training (f-training or z-training), type of class splitting.",
"the test results, using smaller subset of Mufasa mock data (different from the subset on which they have been trained), look very promising.",
"For instance, both Accuracy and $f_{1}$ score $> 0.9$ .",
"We have shown the good performance of the built classifiers when being tested on real observation data – RESOLVE, ALFALFA and GASS surveys – after training them on the mock data from Mufasa .",
"Our findings can be summarized as follows: On using Mufasa to both train and test the learners, RF shows the best performance amongst the learners with an Accuracy of $99.00 \\%$ ROC AUC above $99.96\\%$ , $f_{1}$ score $99.4\\%$ at $z = 1$ .",
"Other classifiers like k-NN and DNN also perform similarly well in general, however GRAD method shows poor performance when considering zSMg and fSMg setups.",
"For z- training, Accuracy and $f_{1}$ score increase from present to higher redshift.",
"The increase is steeper at $z<0.5$ and flattens out at higher redshift.",
"This indeed compensates the fact that regressors built in [rad18]rad18 perform best at low redshift and more poorly with increasing $z$ .",
"The performances of the regressors appear to be insensitive to the selected input features for the training except with the case of GRAD method which struggles to properly classify the galaxies in the test set when only considering SDSS magnitudes and environmental information as input features (zSMg and fSMg).",
"The results are affected by the definition of the class of galaxies(BIN, LOW and MULTI).",
"BIN, which is the type of splitting behind the motivation for this work, corresponds to better results compared to the other two types of splittings.",
"Comparing the results corresponding to four different tests using real observational data from RESOLVE, ALFALFA and GASS surveys, with the exception of DNN as suggested by its low value of ROC AUC and zero specificity, the classifiers perform best on TEST 3 in which the test set is ALFALFA data and the input features considered are color indices formed out of SDSS magnitudes only.",
"All learners correctly classified the Hi poor galaxies with a specificity = 1.0 and their precision is also maximised (precision = 1.0), which is what we really aim for.",
"For TEST 3, it is quite clear that most of the errors (if not all) come from FN, i.e.",
"Hi rich galaxies misclassified as Hi poor, although this quantity is already minimised given the rather high $f_{1}$ score of all the learners.",
"By comparing TEST 1 and 2, it is clear that using color indices from SDSS data only is the optimal option to better identify the Hi poor galaxies given the higher precision in TEST 2.",
"DNN has the highest Accuracy and $f_{1}$ for TEST 1 and TEST 2, indicative of being robust in classifying the Hi-rich galaxies.",
"However, DNN fails to achieve a resonable classification of the Hi-poor galaxies as shown by the low values of Specificity ($<0.3$ ) for all tests.",
"The relatively poor performance of DNNCompared to other classifiers in this test.",
"quantified by the slightly lower values of Accuracy and $f_{1}$ for TEST 3 compared to TEST 1 and TEST 2 might be due to the nature of the test samples.",
"We speculate that the neural network is able to achieve higher performance in a cleaner set of data such as from the RESOLVE survey but under-perform in a sample from blind survey data such as ALFALFA.",
"This does not mean the learner itself is not performing well, it only means that the data to test on are prone to higher systematic errors.",
"In TEST 4 we use a test sample from GASS, which unlike the other samples used in the first three tests, has a fairly good representation of the two classes (i.e.Hi rich-Hi poor).",
"This makes it a good dataset for assessing how well the classifiers are able to apply the learned features from the mock data.",
"Based on the most important performance metric in this study, k-NN is the best classifier for TEST 4 with a precision = 0.985.",
"It also classifies the Hi poor galaxies properly as demonstrated by its high specificity (0.995).",
"However, even though our purpose is to build a classifier that has a very good precision which translates to its ability to correctly classify Hi rich galaxies, in all kinds of machine learning tasks, the algorithm that can minimise the generalisation errors well is the more preferable.",
"In this case specifically, as the results suggest, SVM proves to be able to generalise well as shown by its accuracy (0.717), $f_{1}$ score (0.702), ROC AUC (0.809) and both precision and specificity are the second best.",
"Overall in terms of performance, based on the scores in all tests on real data, we find that SVM is the best classifier as it demonstrates quite well its generalisation ability, learning from simulated data in order to classify real data.",
"With the advent of large Hi surveys like LADUMA and MIGHTEE, we have presented the possibility of properly classifying galaxies according to their gas content, using machine learning.",
"The robustness of our methods lie in the fact that the trained algorithms can learn from mock data in order to classify galaxies in real surveys, which is indeed a strong asset in the sense that in reality the lack of enough data to train the methods turns out to be an issue that requires to be mitigated.",
"Together with the regressors built in [rad18]rad18, the classifiers in this work will form a useful pipeline to create mock Hi surveys for assisting with survey design, and eventually, will enable more detailed tests of the input model by comparing observed Hi to that predicted from the regressor on a case-by-case basis.",
"We only analysed the performance of single models in both this work and [rad18]rad18.",
"However, the use of more complex models using ensemble or stacking techniques are increasingly favoured in the literatures.",
"We will explore such methods in future work despite their level of complexity as well as their interpretability."
],
[
"Acknowledgements",
"SA acknowledges financial support from the South African Radio Astronomy Observatory (SARAO).",
"MR and RD acknowledge support from the South African Research Chairs Initiative and the South African National Research Foundation.",
"Support for MR was also provided by the Square Kilometre Array post-graduate bursary program.",
"The Mufasa simulations were run on the Pumbaa astrophysics computing cluster hosted at the University of the Western Cape, which was generously funded by UWC's Office of the Deputy Vice Chancellor.",
"Additional computing resources are obtained from the Max Planck Computing & Data Facility (http://www.mpcdf.mpg.de) and SARAO."
]
] | 1906.04198 |
[
[
"Evaluating two-electron-repulsion integrals over arbitrary orbitals\n using Zero Variance Monte Carlo: Application to Full Configuration\n Interaction calculations with Slater-type orbitals"
],
[
"Abstract A Monte Carlo method for evaluating multi-center two-electron-repulsion integrals over any type of orbitals (Slater, Sturmian, finite-range, numerical, etc.)",
"is presented.",
"The approach is based on a simple and universal (orbital-independent) gaussian sampling of the two-electron configuration space and on the use of efficient zero-variance Monte Carlo estimators.",
"Quite remarkably, it is shown that the high level of accuracy required on two-electron integrals to make Hartree-Fock (HF) and configuration interaction (CI) calculations feasible can be achieved.",
"A first zero-variance estimator is built by introducing a gaussian approximation of the orbitals and by evaluating the two-electron integrals using a correlated sampling scheme for the difference between exact and approximate orbitals.",
"A second one is based on the introduction of a general coordinate transformation.",
"The price to pay for this simple and general Monte Carlo scheme is the high computational cost required.",
"However, we argue that the great simplicity of the algorithm, its embarrassingly parallel nature, its ideal adaptation to modern computational platforms and, most importantly, the possibility of using more compact and physically meaningful basis sets make nevertheless the method attractive.",
"HF and near full CI (FCI) calculations using Slater-type orbitals (STO) are reported for Be, CH4 and [H2N(CH)NH2]^+ (a simple model of cyanine).",
"To the best of our knowledge, our largest FCI calculation involving 18 active electrons distributed among 90 orbitals for the cyanine molecule, is the most extensive molecular calculation performed so far using pure STO orbitals (no gaussian approximation, even for the challenging four-center two-electron integrals)."
],
[
"Introduction",
"In recent years most of the standard methods of quantum chemistry have been revisited within the framework of stochastic processes.",
"In short, the very same equations and quantities are considered but, instead of solving the equations using standard linear algebra techniques (diagonalization) or explicit calculations of very large sums (perturbational quantities), stochastic implementations are employed.",
"Let us cite the stochastic versions of the second-order Møller-Plesset (MP2),[1] coupled-cluster with single and double and perturbative triple excitations (CCSDT),[2] complete active space self-consistent field (CASSCF),[3] multi-reference with second-order perturbation (MRPT2), [4], [5], random phase approximation (RPA)[6], GW[7], and FCI[8] approaches.",
"In practice, by avoiding the practical limitations in terms of memory (no storage of very large vectors and matrices) and number of determinants to consider (only a small subspace consisting of the determinants contributing the most to the averages is sampled) calculations beyond the limits of the standard ”deterministic” versions can be performed.",
"As a representative example, let us mention the recent stochastic CASSCF calculation of Smith et al.",
"involving 44 electrons distributed among 44 active orbitals for a model complex of Fe-porphyrin.",
"[9] From a general perspective, the major driving force behind the active developement of stochastic techniques is their very good adaptation to massive parallelism and to modern computational platforms (simplicity of the algorithm, low-memory fingerprint, easy implementation on graphics processing units (GPU) and efficient arithmetic co-processors, cache optimization, etc.).",
"In the same spirit, we propose here to calculate the two-electron-repulsion integrals of quantum chemistry using a stochastic approach.",
"As well-known, the choice of the basis functions (orbitals) in electronic structure wave-function calculations is one of the critical aspects.",
"Ideally, orbitals should obey the electron-nucleus cusp condition removing the divergence of the one-electron component of the (local) energy at short electron-nucleus distances; they should also display the physically correct exponential-like decay at large distances, and be flexible enough to reproduce any type of behavior at intermediate distances.",
"Unfortunately, the high computational cost required to evaluate the very large numbers of integrals involved in calculations limits in practice the type of orbitals that can be employed.",
"As well-known, the compromise between cost and efficiency adopted in virtually all calculations for molecular systems consists in using Gaussian-type basis functions.",
"Although fast and efficient algorithms have been developed over the years to calcutate gaussian integrals, the price to pay is the need of using large sets of basis functions, larger than those based on more physical representations (for example, Slater-type orbitals with the correct cusp and long-range behavior).",
"Considering the sharp increase of the computational cost of accurate post-HF methods with the number of basis functions [e.g., $N_b^7$ scaling for the ”gold standard” CCSD(T), $N_b$ number of basis functions], to have the possibility of using more compact basis set is important, particularly for large systems.",
"Here, we present a Monte Carlo approach to calculate two-electron integrals for arbitrary orbitals (STO, numerical, finite-range, etc.).",
"In this approach no analytic integration is performed and only the values of the orbitals at each Monte Carlo configuration are to be calculated, making the approach particularly simple and general.",
"However, at first sight using a Monte Carlo approach to calculate accurately low (six)-dimensional integrals may appear unrealistic.",
"Indeed, the statistical error is usually large and its very slow decay with the number $N$ of drawings -$\\sim 1/\\sqrt{N}$ - precludes any brute force approach (i.e.",
"increasing $N$ indefinitely) to improve the accuracy.",
"Here, this problem is particularly acute since a high accuracy on the two-electron integrals is known to be needed to get stabilized and unbiaised HF or post-HF calculations.",
"For example, the use of single precision floating point representation is in general not sufficient and an absolute error at least smaller than $10^{-8}$ is necessary.",
"[10] In this work it is shown that by resorting to a zero-variance strategy the statistical error on two-electron integrals can be tremendously reduced and the targeted accuracy can be attained.",
"For example, in the case of the biggest system treated here (the cyanine molecule) an average absolute error of about $\\sim 2 \\times 10^{-9}$ on the two-electron integrals is achieved.",
"From a general perspective, a zero-variance strategy is based on the introduction of improved estimators having the same average as the standard estimator but a (much) smaller variance.",
"[11] In this way, for a given number of Monte Carlo configurations (much) more accurate averages (smaller statistical error) can be obtained at essentially the same computational cost.",
"When building up such improved estimators it is usually possible to define the ideal zero-variance limit where statistical fluctuations entirely vanish.",
"In practice, approaching this limit is a guarantee of decreasing the statistical error.",
"In this work on computing two-electron integrals, we define a first zero-variance estimator based on a gaussian approximation of the orbitals and on the evaluation of the exact integrals using a correlated sampling scheme for the difference between exact and approximate orbitals.",
"It is most important to emphasize that, although a gaussian approximation for the orbitals is introduced, the calculated integrals are independent of this approximation, only the magnitude of the statistical error is affected.",
"The zero-variance limit is attained in the limit of an exact representation (infinite number of gaussian functions).",
"A second zero-variance estimator defined here is obtained by introducing a coordinate transformation.",
"In this case it is possible to write down a so-called zero-variance equation defining the best transformation.",
"In practice, searching for good approximations of this equation is a precious guide to build efficient improved estimators.",
"However, once again, we note that the results are independent of the quality of the approximation made for the transformation.",
"The introduction of zero-variance estimators being instrumental to the success of the method, the approach will be referred to as zero-variance Monte Carlo (ZVMC).",
"In this work ZVMC is applied to the calculation of two-electron integrals over Slater-type orbitals.",
"The problem of computing such integrals has a long history from the very start of quantum chemistry and has given rise to numerous works (for references see, e.g., [hoggan,smiles]).",
"Here, it is shown that STO integrals can be computed with sufficient accuracy to allow converged HF and FCI-type calculations for Be, CH$_4$ , and [H$_2$ N(CH)NH$_2$ ]$^+$ (a simple model of cyanine).",
"To the best of our knowledge, the FCI calculation presented here for the cyanine molecule involving 18 active electrons (and 6 frozen core electrons) distributed among 90 orbitals is the most extensive molecular calculation performed so far using pure STO orbitals (no approximate gaussian expansion for two-electron integrals, even for the challenging four-center integrals).",
"However, the price to pay for this simple and general approach is the need of using (very) large Monte Carlo statistics.",
"It is clearly the major drawback of the approach.",
"However, as illustrated and discussed in this work we believe that the unique features of the approach nevertheless make the method attractive.",
"Finally, let us note that we shall here restrict ourselves to atomic orbitals.",
"However, there is no fundamental difficulty to consider molecular orbitals; this will be presented in a forthcoming work.",
"The paper is organized as follows.",
"In Sec.",
"the basic theory is presented.",
"Some illustrative numerical applications are discussed in section .",
"We first present the main aspects of the method with calculations of several representative four-center two-electron integrals over Slater-type orbitals.",
"Then, HF and near-FCI calculations are presented for Be, CH$_4$ and [H$_2$ N(CH)NH$_2$ ]$^+$ (a simple model of cyanine).",
"Finally, a summary and discussion are given in Sec."
],
[
"General theory",
"In this work we are concerned with the calculation of general two-electron-repulsion integrals of the form $I= (ab|cd)=\\int d{\\bf r}_1 d{\\bf r}_2{\\phi _a({\\bf r}_1) \\phi _b({\\bf r}_1) \\frac{1}{r_{12}} \\phi _c({\\bf r}_2) \\phi _d({\\bf r}_2)}$ where the $\\phi $ 's are real atomic orbitals written under the general unnormalized cartesian form $\\phi _a({\\bf r})= (x-A_x)^{a_x} (y-A_y)^{a_y} (z-A_z)^{a_z} u_a(|{\\bf r}-{\\bf A}|).$ Here, ${\\bf A}=(A_x,A_y,A_z)$ is the center, ${\\bf a}=(a_x,a_y,a_z)$ a triplet of non-negative integers (angular momentum vector), and $u_a(r)$ some general radial part.",
"Standard examples of radial parts are, e.g., $u_a(r) = e^{-\\alpha r^2}$ for Gaussian-type orbitals (GTO), $u_a(r) = r^{n-l-1}e^{-\\alpha r}$ for Slater-type orbitals (STO), or $u_a(r) =F(r) e^{-\\alpha r}$ for Sturmian orbitals (where $F$ is the confluent hypergeometric function).",
"Alternatively, the radial part may be defined on a one-dimensional grid or using a spline representation.",
"Note that the expression for the radial function does not need to be the same among orbitals, so mixed basis sets (e.g.",
"STO-GTO) can also be used.",
"In what follows, we will characterize the radial extension of a general orbital $\\phi _a$ by introducing an effective exponent $\\alpha > 0$ equal to the inverse of the average radial width of the orbital, that is $\\alpha \\sim \\frac{1}{\\langle r \\rangle }_{\\phi _a},$ where ${\\langle r \\rangle }_{\\phi _a}= \\frac{\\int r^2 dr \\;r \\;{\\phi ^2_a}}{\\int r^2 dr {\\phi ^2_a}}$ .",
"The exponents associated with $\\phi _b$ , $\\phi _c$ , and $\\phi _d$ will be denoted as $\\beta $ ,$\\gamma $ , and $\\delta $ , respectively.",
"The densities $\\rho _{ab}({\\bf r})$ are defined as $\\rho _{ab}({\\bf r}) =\\phi _a({\\bf r})\\phi _b({\\bf r}),$ and the integral writes $I=\\int d{\\bf r}_1 d{\\bf r}_2\\frac{1}{r_{12}} \\rho _{ab}({\\bf r}_1)\\rho _{cd}({\\bf r}_2).$ Remark on notation: For simplicity the $abdc$ -dependency of the integral has not been indicated here; in what follows it will be the case for all quantities for which this dependency is obvious, except when some confusion is possible."
],
[
"Simple Monte Carlo estimator",
"The two-electron-repulsion integral is expressed as $I= \\int d{\\bf r}_1 d{\\bf r}_2\\pi _{ab|cd}({\\bf r}_1,{\\bf r}_2)\\Big [ \\frac{1}{r_{12}}\\frac{ \\rho _{ab}({\\bf r}_1)\\rho _{cd}({\\bf r}_2)}{\\pi _{ab|cd}({\\bf r}_1,{\\bf r}_2)}\\Big ],$ where $\\pi _{ab|cd}$ is some arbitrary probability density ($\\pi _{ab|cd} \\ge 0$ and $\\int d{\\bf r}_1 d{\\bf r}_2 \\pi _{ab|cd}=1$ ).",
"Here, we use a simple factorized gaussian density reproducing the overall shape of the one-electron distributions, for example $\\pi _{ab|cd}( {\\bf r}_1,{\\bf r}_2)= {\\Big (\\frac{\\sqrt{\\zeta \\eta }}{2\\pi }\\Big )}^3e^{-\\frac{\\zeta }{2} ({\\bf r}_1-{\\bf P} )^2}e^{-\\frac{\\eta }{2} ({\\bf r}_2-{\\bf Q} )^2},$ with $\\zeta = \\alpha +\\beta \\;\\;\\;\\;\\;\\;\\;\\;\\; \\;\\;\\; \\eta = \\gamma +\\delta $ and ${\\bf P}=\\frac{\\alpha {\\bf A} + \\beta {\\bf B}}{\\alpha +\\beta } \\;\\;\\;\\;\\;\\;\\;\\;\\; \\;\\;\\;{\\bf Q}=\\frac{\\gamma {\\bf C} + \\delta {\\bf D}}{\\gamma +\\delta }$ After simple changes of variables and relabelling, the integral can be written under the form $I= \\int d{\\bf r}_1 d{\\bf r}_2\\pi _{0}({\\bf r}_1,{\\bf r}_2) F({\\bf r}_1,{\\bf r}_2)$ with $F({\\bf r}_1,{\\bf r}_2) =(\\zeta \\eta )^{-\\frac{3}{2}}\\frac{1}{|{\\bf u}_1-{\\bf u}_2|}\\frac{ \\rho _{ab}({\\bf u}_1)\\rho _{cd}({\\bf u}_2)}{\\pi _0({\\bf r}_1,{\\bf r}_2)},$ and $\\left\\lbrace \\begin{array}{l}{\\bf u}_1 = {\\zeta }^{-\\frac{1}{2}}{\\bf r}_1+ {\\bf P}\\\\{\\bf u}_2 = {\\eta }^{-\\frac{1}{2}} {\\bf r}_2+ {\\bf Q}\\\\\\end{array}\\right.$ Here, $\\pi _0$ is the product of the normal distribution for each electron coordinate $\\pi _0({\\bf r}_1,{\\bf r}_2)=\\frac{1}{(2\\pi )^3}e^{-\\frac{1}{2} ( {\\bf r}^2_1 + {\\bf r}^2_2 )}.$ To apply Monte Carlo techniques, the integral is rewritten as $I = \\langle F \\rangle _{\\pi _0},$ where $\\langle ...\\rangle _{\\pi _0}$ denotes the average over the probability distribution $\\pi _0$ .",
"In practice, the integral is evaluated from a finite random sample of $N$ configurations $({\\bf r}^i_1,{\\bf r}^i_2)$ drawn with $\\pi _0$ , $I \\simeq I_N=\\frac{1}{N} \\sum _{i=1}^N F({\\bf r}^i_1,{\\bf r}^i_2),$ the exact value being obtained as $N$ goes to infinity.",
"At finite $N$ , the statistical error bar on $I_N$ is calculated using elementary statistical techniques."
],
[
"Zero Variance Monte Carlo estimators",
"The general idea of variance reduction techniques[11] is to replace the initial estimator $F$ , by a new “improved” one, denoted here as $\\tilde{F}$ , having the same average but a smaller variance, $\\sigma ^2(\\tilde{F})$ $\\langle \\tilde{F} \\rangle = \\langle {F} \\rangle \\;\\;\\; {\\rm with}\\;\\;\\; \\sigma ^2(\\tilde{F}) <\\sigma ^2(F).$ In this formula the average is defined over some general probability density and the variance is given by $\\sigma ^2(F)= \\langle F^2 \\rangle -\\langle F \\rangle ^2.$ As long as the calculation of $\\tilde{F}$ is not too expensive, calculating the average using $\\tilde{F}$ instead of $F$ leads to a decrease of the statistical error, the gain in computational cost being essentially proportional to the reduction in variance.",
"The ideal zero-variance limit where the statistical fluctuations vanish is reached when $\\tilde{F}$ can be made constant for all configurations, more precisely $\\tilde{F}= \\langle \\tilde{F} \\rangle .$"
],
[
"Zero Variance using control variates",
"The first zero-variance (ZV) estimator introduced here is based on the use of the so-called control variate method, e.g.",
"[control].",
"Denoting $F_0$ some approximation of $F$ whose average, $\\langle F_0 \\rangle $ , is known the following improved estimator is considered $\\tilde{F}= F + \\lambda (F_0-\\langle F_0 \\rangle )$ where $\\lambda $ is some real parameter.",
"By construction, $\\langle \\tilde{F} \\rangle = \\langle {F} \\rangle $ , for all $\\lambda $ .",
"Minimizing the variance with respect to $\\lambda $ , the variance of the optimized estimator is found to be $\\sigma ^2(\\tilde{F})= \\sigma ^2({F})-\\frac{ {\\langle (F-\\langle F \\rangle )(F_0-\\langle F_0 \\rangle ) \\rangle }^2}{ \\sigma ^2(F_0) }.$ As seen, by using the control variate $F_0$ a systematic decrease of the variance is obtained, whatever the choice of $F_0$ .",
"However, a significant variance reduction is possible in practice only if the fluctuations of $F_0$ are correlated enough to those of $F$ , that is, if the correlator $\\langle (F-\\langle F \\rangle )(F_0-\\langle F_0 \\rangle ) \\rangle $ is large enough.",
"Here, the control variate $F_0$ is chosen by using some gaussian approximation $\\rho ^G$ of the exact density $\\rho $ , more precisely $F_0({\\bf r}_1,{\\bf r}_2) =({\\zeta \\eta })^{-\\frac{3}{2}}\\frac{1}{|{\\bf u}_1-{\\bf u}_2|}\\frac{ \\rho ^G_{ab}({\\bf u}_1)\\rho ^G_{cd}({\\bf u}_2)}{\\pi _0({\\bf r}_1,{\\bf r}_2)}.$ The average of $F_0$ given by $\\langle F_0 \\rangle = I^G=\\int d{\\bf r}_1 d{\\bf r}_2\\frac{1}{r_{12}} \\rho ^G_{ab}({\\bf r}_1)\\rho ^G_{cd}({\\bf r}_2)$ can be efficiently evaluated using standard algorithms for gaussian integrals.",
"We now decompose $I$ as $I= I^G + \\Delta I$ where $\\Delta I$ is a residual integral given as $\\Delta I =\\int d{\\bf r}_1 d{\\bf r}_2{\\pi _0({\\bf r}_1,{\\bf r}_2)}\\Delta F ({\\bf r}_1,{\\bf r}_2)$ where $\\Delta F({\\bf r}_1,{\\bf r}_2)= { ({\\zeta \\eta })^{-\\frac{3}{2}} }\\frac{1}{|{\\bf u}_1-{\\bf u}_2|}$ $\\times \\frac{\\rho _{ab}({\\bf u}_1)\\rho _{cd}({\\bf u}_2)-\\rho ^G_{ab}({\\bf u}_1)\\rho ^G_{cd}({\\bf u}_2)}{\\pi _0({\\bf r}_1,{\\bf r}_2)}.$ The formula for the integral becomes $I= I^G + \\langle \\Delta F \\rangle _{\\pi _0}$ where the first contribution, $I^G$ , is calculated deterministically and the residual integral, $\\Delta I$ , computed with Monte Carlo.",
"While $\\rho ^G$ is approaching $\\rho $ , $\\Delta I$ becomes smaller and smaller and the same for the statistical error.",
"In the zero-variance limit where $\\rho =\\rho ^G$ , the error entirely vanishes.",
"In practice, using accurate gaussian approximation leads to (very) important reduction in statistical fluctuations.",
"Now, since the integrals are independent of $\\rho ^G$ -whatever the quality of the approximation- we have a great freedom in choosing the way the densities $\\rho _{ab}$ are approximated.",
"For example, it can be done by using density fitting or related techniques where auxiliary basis sets are introduced to approximate products of one-electron functions.",
"Here, we shall not elaborate on this aspect (this is let for future work) but use instead the simple procedure consisting in building $\\rho ^G$ as the product of some gaussian approximation $\\phi ^G_a$ for the orbitals $\\phi ^G_a({\\bf r})= (x-A_x)^{a_x} (y-A_y)^{a_y} (z-A_z)^{a_z} u^G_a(|{\\bf r}-{\\bf A}|).$ with $u^G_a(r) = \\sum _{i=1}^{n_g} c^a_i r^{2n_i}e^{-\\gamma ^a_i r^2}.$ Here, $n_g$ is the number of elementary gaussian functions used, $\\lbrace n_i\\rbrace $ a fixed set of positive integers, and ($c^a_i$ , $\\gamma ^a_i$ ) the parameters resulting from some fitting process, for example by minimizing the $\\chi ^2$ quantity $\\chi ^2= \\int _{0}^{+\\infty } r ^2 dr\\;\\; \\frac{1}{r} {[u_a(r)-u^G_a(r)]}^2.$"
],
[
"Zero Variance using a coordinate transformation",
"Our second zero-variance estimator is based on the fact that a coordinate transformation can be used to reduce the statistical error.",
"Let us note $[\\tilde{\\bf r}_1({\\bf r}_1,{\\bf r}_2),\\tilde{\\bf r}_2({\\bf r}_1,{\\bf r}_2)]$ such a one-to-one correspondance.",
"The residual integral, $\\Delta I$ , computed with Monte Carlo , Eq.",
"(REF ) writes $\\Delta I =\\int d{\\tilde{\\bf r}_1} d{\\tilde{\\bf r}_2}\\pi _0(\\tilde{\\bf r}_1,\\tilde{\\bf r}_2)\\Delta F({\\tilde{\\bf r}_1},{\\tilde{\\bf r}_2})$ $=\\int d{\\bf r}_1 d{\\bf r}_2 \\pi _0({\\bf r}_1,{\\bf r}_2) \\widetilde{\\Delta F}({\\bf r}_1,{\\bf r}_2)$ with $\\widetilde{\\Delta F}({\\bf r}_1,{\\bf r}_2)=\\frac{\\pi _0(\\tilde{\\bf r}_1,\\tilde{\\bf r}_2) }{\\pi _0({\\bf r}_1,{\\bf r}_2)} J({\\bf r}_1,{\\bf r}_2)\\Delta F(\\tilde{\\bf r}_1,\\tilde{\\bf r}_2),$ where $J$ is the Jacobian of the transformation $J({\\bf r}_1,{\\bf r}_2) = \\left|{\\rm det} \\frac{\\partial {\\tilde{\\bf r}}_\\mu }{\\partial {\\bf r}_\\nu } \\right| \\;\\;\\;\\mu ,\\nu =1,2$ The expression for the complete two-electron integral thus writes $I = I^G + \\int d{\\bf r}_1 d{\\bf r}_2 \\pi _0( {\\bf r}_1,{\\bf r}_2 ) \\widetilde{\\Delta F}({\\bf r}_1,{\\bf r}_2).$ or, equivalently $I = I^G + \\langle \\widetilde{\\Delta F} \\rangle _{\\pi _0}.$ Although the Monte Carlo average is independent on the transformation, $\\langle \\widetilde{\\Delta F} \\rangle _{\\pi _0} =\\langle \\widetilde{\\Delta F} \\rangle _{\\pi _0}$ , it is not at all true for its variance.",
"Now, a precious guide to construct a coordinate-transformation leading to a large reduction in variance consists in invoking the zero-variance equation that the ideal transformation (no statistical fluctuations) obeys.",
"This equation is obtained by equating the quantity to be averaged to its average,[11] that is, here $\\widetilde{\\Delta F}({\\bf r}_1,{\\bf r}_2)=\\frac{\\pi _0(\\tilde{\\bf r}_1,\\tilde{\\bf r}_2) }{\\pi _0({\\bf r}_1,{\\bf r}_2)} J({\\bf r}_1,{\\bf r}_2)\\Delta F(\\tilde{\\bf r}_1,\\tilde{\\bf r}_2)= \\Delta I$ In the next section it will be illustrated how this ZV equation can be exploited in the particular case of STO integrals.",
"Note that, although $\\Delta I$ -the unknown quantity to be computed- is present in the equation, it is not a problem in practice.",
"Indeed, a simple solution consists in replacing the exact value $\\Delta I$ by some approximate one.",
"It is legitimate as long as the variation of $\\widetilde{\\Delta F}$ in configuration space -measured for example by its variance, is larger than the error made for $\\Delta I$ , which is always the case except for very simple cases.",
"Once a functional form for the coordinate transformation has been chosen, its parameters can be optimized by minimizing the variance of $\\widetilde{\\Delta F}$ evaluated over a fixed set of configurations drawn according to $\\pi _0$ ."
],
[
"The case of Slater-type orbitals",
"In this section, we make more explicit the general approach just described for the important case of STO atomic orbitals.",
"The real cartesian unnormalized STO orbitals $\\phi _a({\\bf r})$ , Eq.",
"(REF ), are defined by choosing the radial part as $u_a(r) = r^{n_a-l_a-1} e^{-\\alpha r}$ where $n_a=1,2,...$ is the principal quantum number and $l_a$ the total angular momentum $l_a=a_x+a_y+a_z.$ In what follows, we shall employ the usual notation for STO orbitals, namely $1s=e^{-\\alpha r}, 2s=r e^{-\\alpha r},3s=r^2 e^{-\\alpha r}, 2p_x= x e^{-\\alpha r}, 3p_x=x r e^{-\\alpha r}, 3d_{xx}= x^2 e^{-\\alpha r}$ , and so on.",
"For the particular case of STO orbitals, we have not built the gaussian approximations of the radial part, Eq.",
"(REF ) by minimization of the $\\chi ^2$ , Eq.",
"(REF ).",
"Instead, we have preferred to use the accurate representations of the exponential $e^{-r}=\\sum _{i=1}^{n_g} c_i e^{-\\gamma _i r^2}$ given by Lopez et al.",
"[15] for a number of gaussian functions ranging from $n_g=1$ to $n_g=30$ .",
"For $n=2$ , $r^{n-1} e^{-r}$ is expanded as $r e^{-r}=2\\sum _{i=1}^{n_g} c_i \\gamma _i r^2 e^{-\\gamma _i r^2},$ an expression obtained by considering the derivative $\\frac{\\partial }{\\partial a} e^{-ar}\\Bigr \\vert _{a = 1}$ where the gaussian expansion of $e^{-ar}$ , Eq.",
"(REF ), is used.",
"For $n=3$ , the polynomial $r^2=x^2+y^2+z^2$ is withdrawn from the radial part and transferred to the polynomial part of the orbital.",
"Several choices of the functional form for the coordinate transformation have been investigated.",
"The following simple form is proposed $\\tilde{\\bf r}_i({\\bf r}_1,{\\bf r}_2) =f(|{\\bf r}_i|){\\bf r}_i \\;\\;\\; i=1,2.$ where $f$ is a general (smooth enough) function.",
"In this case, the Jacobian is given by $J({\\bf r}_1,{\\bf r}_2)=j(r_1)j(r_2)$ where $j(r) = f^2(r) |f(r)+r f^{\\prime }(r)|.$ Let us now use the ZV equation to get information on $f$ .",
"Fixing electron 2 at some position ${\\bf r}_2$ , the ZV equation writes $f^2(r_1) |f(r_1)+r_1 f^{\\prime }(r_1)| \\frac{1}{|{\\bf u}_1(\\tilde{\\bf r}_1)-{\\bf u}_2(\\tilde{\\bf r}_2)|}$ $\\times \\frac{\\rho _{ab}[{\\bf u}_1(\\tilde{\\bf r}_1)]\\rho _{cd}[{\\bf u}_2(\\tilde{\\bf r}_2)]-\\rho ^G_{ab}[{\\bf u}_1(\\tilde{\\bf r}_1))\\rho _{cd}({\\bf u}_2(\\tilde{\\bf r}_2)]}{\\pi _0({\\bf r}_1,{\\bf r}_2)}= C$ where $C$ is a constant collecting the terms independent of ${\\bf r}_1$ .",
"Without the coordinate transformation ($f=1$ ) the left-hand-side diverges in the large $r_1$ -limit as the ratio $\\sim \\frac{e^{-\\frac{(\\alpha +\\beta )}{\\sqrt{\\zeta }} r_1} }{ e^{-\\frac{ {r_1}^2 }{2} } }.$ Here, we have used the fact that in the large-distance limit $\\rho ^G_{ab}\\rho ^G_{cd}$ becomes negligible with respect to $\\rho _{ab}\\rho _{cd}$ .",
"The divergence makes the variance of the estimator infinite and the Monte Carlo estimators do not converge (see, Fig.1 below).",
"Now, by using the coordinate transformation the divergence can be removed, for example by using the simplest form $f(r) = \\mu r^{\\nu }$ where $\\mu = \\frac{\\kappa \\sqrt{\\zeta } }{2 (\\alpha +\\beta )},$ $\\kappa $ a positive constant and $\\nu $ some real exponent.",
"Now, taking expression (REF ) for $\\zeta $ , the preceding ratio becomes $\\sim \\frac{ e^{ -\\frac{\\kappa }{2} r_1^{\\nu +1} } }{ e^{-\\frac{{r_1}^2}{2} } }$ The divergence is removed when ($\\nu = 1$ and $\\kappa \\ge 1$ ) or ($\\nu > 1$ and $\\kappa \\ge 0$ ).",
"In applications both parameters can be optimized by minimization of the statistical fluctuations.",
"Of course, more elaborate forms for $f$ can be used, this is let for future work."
],
[
"Removing the infinite variance",
"In the absence of the coordinate transformation we have seen that the Monte Carlo estimators of the STO integrals have an infinite variance.",
"This point is illustrated in Figure REF where the Monte Carlo average as a function of the exponent $\\nu $ of the function $f$ involved in the coordinate transformation [Eqs.",
"(REF ) and (REF )] is shown.",
"The results are presented for the simplest possible STO integral, namely $(1s1s|1s1s)= \\frac{1}{(4\\pi )^2}\\int d{\\bf r}_1 d{\\bf r}_2e^{-r_1} \\frac{1}{r_{12}} e^{-r_2}= \\frac{5}{4}$ but a similar behavior is obtained for all STO integrals considered here.",
"The calculation is performed by approximating the $1s$ orbital with five gaussian functions, $n_g=5$ , Eq.",
"(REF ), and by drawing $N= 10^6$ Monte Carlo configurations.",
"The constant $\\kappa $ in $f$ , Eqs.",
"[REF ], is taken to be equal to 1.",
"As expected, for small values of $\\nu $ uncontrolled fluctuations resulting from the infinite variance are present.",
"For large enough $\\nu $ the wild fluctuations disappear and the Monte Carlo average becomes very close to the exact value of 1.25.",
"The critical value of $\\nu $ corresponding to the change of regime is compatible with $\\nu 1$ ."
],
[
" $(1s_A 1s_B|1s_C 1s_D)$ for STO atomic orbitals",
"As a first application, we consider the calculation of four-center two-electron integrals over $1s$ Slater-type orbitals.",
"For quantitative comparison we calculate the four integrals introduced by Pérez et al.",
"[15] and presented in Tables I-IV of their work.",
"Following their convention for the normalization constant, the $1s$ orbital is written as $1s_A({\\bf r}) =\\mathcal {N}_{\\alpha } e^{-\\alpha |{\\bf r}-{\\bf A}|}.$ with $\\mathcal {N}_{\\alpha }=\\sqrt{\\frac{\\alpha ^3}{\\pi }}.$ The four two-electron integrals are denoted here as $I_k$ with $k$ ranging from 1 to 4; the exponents and nuclei positions are given in Tables I-IV of [integrals].",
"In Table REF the convergence of the two-electron integral $I_1$ as a function of the number of Monte Carlo drawings $N$ and number of gaussian functions $n_g$ is presented.",
"The parameters of the coordinate transformation are taken to be $\\kappa =1$ and $\\nu =1$ .",
"All computed values are in agreement with the exact value within the 2-$\\sigma $ limits of the confidence interval.",
"Here, the exact value is evaluated by using the approximate gaussian integral with the most accurate $n_g$ =30-representation of the exponential function to our disposal, that is $I^G(n_g=30)=0.1426742806$ .",
"All given digits are converged as a function of $n_g$ and the value is in very close agreement with that given in [integrals], $I=0.14267429$ (difference of about $10^{-8}$ ).",
"As it should be, the statistical error decreases both as a function of $N$ at fixed $n_g$ and of $n_g$ at fixed $N$ .",
"At fixed $n_g$ , the error decreases as $\\sim \\frac{1}{\\sqrt{N}}$ as expected in a Monte Carlo calculation.",
"When passing from $n_g$ to $n_g+1$ at fixed $N$ , an average reduction of the statistical error between 2 and 3 is obtained, except for $n_g=1$ where the factor is about 10, a larger reduction resulting from the very poor representation of the radial part $u(r)$ using only a single gaussian function.",
"The gain in accuracy when increasing $n_g$ being directly related to the quality of the fit, no general rule is expected for it as a function of $n_g$ .",
"For each $n_g$ the value of the approximate gaussian integral $I^G_1=(1s^G_A 1s^G_B|1s^G_C 1s^G_D)$ is also reported.",
"These values allow to quantify the magnitude of the bias $\\epsilon =|I_1-I^G_1|$ recovered by the Monte Carlo part.",
"Of course, the approach is of interest only if the statistical error on the unbiased ZVMC integral is smaller than $\\epsilon $ .",
"Table REF shows that it is always the case, except for the smallest number of Monte Carlo steps $N=10^3$ (for almost all $n_g$ ) and also for $N=10^5$ with $n_g=7$ .",
"The most accurate value of the integral is obtained for the largest value of $n_g$ and $N$ and is only in error of about $9 \\times 10^{-10}$ .",
"As we shall see below, such a typical accuracy will be sufficient to perform molecular calculations.",
"In Table REF the results for the three other two-electron integrals, $I_{k=2-4}$ as a function of $n_g$ and for $N=10^{11}$ are reported.",
"For $n_g$ =7 the absolute errors on the integrals $I_2$ , $I_3$ , and $I_4$ are comparable to those obtained for $I_1$ .",
"The biases $\\epsilon $ associated with the $n_g$ =7-gaussian approximation are about $3 \\times 10^{-7}$ , $2 \\times 10^{-5}$ , and $3 \\times 10^{-8}$ for $I_2$ , $I_3$ and $I_4$ , respectively.",
"These biases are much larger than the corresponding statistical errors on the Monte Carlo values which are $4 \\times 10^{-10}$ , $2 \\times 10^{-10}$ , and $2 \\times 10^{-11}$ , respectively.",
"It illustrates the effectiveness of the Monte Carlo approach for recovering the exact STO values, starting from the approximate gaussian ones.",
"Table: Convergence of the two-electron integral I 1 I_1 as a function of the numberof Monte Carlo drawings NN and gaussian functions n g n_g.",
"Error bars on the last digit (one-sigma confidence intervals) given in parentheses.",
"I 1 G I^G_1 value of the approximate reference gaussian integral.",
"Parametersof the coordinate transformation (κ=1\\kappa =1, ν=1\\nu =1).",
"Exact value obtained with n g =30n_g=30, see text.Table: *Figure: Value of I=(1s1s|1s1s)I=(1s 1s|1s 1s) for N=10 6 N=10^6 and n g =5n_g=5as a function of ν\\nu .",
"Coordinate transformation paramter, κ=1\\kappa =1."
],
[
"$(n_A l_A n_B l_B|n_C l_C n_D l_D)$ for STO atomic orbitals",
"In Table REF some results for four-center two-electron integrals over STO orbitals with non-zero angular momenta are presented.",
"The atomic orbitals considered are $1s_A=\\mathcal {N}_{\\alpha } e^{-\\alpha |{\\bf r}-{\\bf A}|}$ , $2p_A=\\mathcal {N}_{\\alpha } (x-x_A)e^{-\\alpha |{\\bf r}-{\\bf A}|}$ , and $3d_A= \\mathcal {N}_{\\alpha }(x-x_A)^2 e^{-\\alpha |{\\bf r}-{\\bf A}|}$ , with the same choice for the other nuclei.",
"Ten particular integrals combining these atomic orbitals have been selected.",
"The nucleus centers and exponents have been chosen to avoid any particular spatial symmetry.",
"$I^G(n_g=30)$ are taken as exact values, all reported digits being converged as a function of $n_g$ .",
"For a given value of $n_g$ , all ten integrals are calculated simultaneously over the same Monte Carlo configurations.",
"Most of the computational effort is spent in computing quantities independent of the polynomial part of the orbitals, Eq.",
"(REF ).",
"Thus, the additional cost for calculating all integrals compared to that needed for the $(1s_A 1s_B |1s_C 1s_D)$ integral alone is marginal.",
"It is one of the attractive properties of the approach.",
"For $n_g=7$ the absolute errors obtained for the ten integrals range from $7 \\times 10^{-9}$ to $5 \\times 10^{-8}$ .",
"For a fixed number of drawings, the statistical error is proportional to the square root of the variance of the estimator.",
"As the total angular momentum $L=l_A+l_B+l_C+l_D$ is increased, the variance is also expected to increase (higher and higher moments of the probability distribution are calculated) and so the error.",
"It is indeed what is observed in Table REF where the error increases continuously when going from $L=0$ to $L=5$ .",
"However, the absolute errors obtained in the less favorable case ($L=5$ ) are still very small.",
"Table: Two-electron integrals for STO orbitals with non-zero momenta.α=1,β=1.2,γ=1.6,δ=2.1\\alpha =1,\\beta =1.2,\\gamma =1.6, \\delta =2.1;𝐀{\\bf A}=(0.4,-0.2,0.5), 𝐁{\\bf B}=(-0.5, 0.3,-0.4), 𝐂{\\bf C}=(0.5,-0.6,0.6), 𝐃{\\bf D}=(-0.4,0.5,-0.4), N=10 11 N=10^{11}.",
"Coordinate transformation parameters(κ=1\\kappa =1, ν=1\\nu =1).",
"Exact values of the integrals obtained with n g =30n_g=30 (all digits converged)."
],
[
"Application to atomic and molecular systems",
"In this section Hartree-Fock and near full CI calculations using Slater-type atomic orbitals for Be, CH$_4$ , and [H$_2$ N(CH)NH$_2$ ]$^+$ are presented.",
"For that, the full set of two-electron integrals is to be computed.",
"After removal of the redundancy among orbital indices the number of integrals is about $\\frac{N_b^4}{8}$ , where $N$ is the number of orbitals (basis functions).",
"The sampling distribution being independent of the orbitals, all integrals are computed over the same Monte Carlo realization.",
"In this way, the approach is embarrassingly parallel not only under splitting of the full set of integrals into independent blocks as in any approach but also with respect to the Monte Carlo sampling that can be performed on independent blocks over an arbitrary number of compute cores.",
"The CI calculations are performed using the CIPSI algorithm[16](Configuration Interaction using a Perturbative Selection made Iteratively) as implemented in the freely available electronic structure software QUANTUM PACKAGE.",
"[17] CIPSI combines a selected CI (sCI) step based on a second-order energetic criterion to select perturbatively the most important determinants and on a perturbative step where the second-order Epstein-Nesbet pertubative estimate $E_{PT2}$ of the difference between the FCI and the variational reference energy is evaluated.",
"$E_{PT2}$ is efficiently computed with a recently proposed hybrid stochastic-deterministic algorithm.",
"[18] In order to extrapolate the sCI results to the FCI limit, the method recently proposed by Holmes, Umrigar and Sharma in the context of the HBCI method[5] is employed.",
"It consists in extrapolating the sCI energy, $E_{sCI}$ , as a function of $E_{PT2}$ , i.e $E_{sCI} \\simeq E_{FCI}-E_{PT2}$ .",
"When $E_{PT2}=0$ , the FCI limit has effectively been reached.",
"This extrapolation procedure has been shown to be robust, even for challenging chemical situations.",
"In the calculations presented here the number of selected determinants is about a few millions and E$_{PT2}$ is small enough to enter the quasi-linear regime of the difference $E_{FCI}-E_{sCI}$ as a function of the number of selected determinants.",
"Our estimate of FCI is denoted as exFCI (extrapolated FCI).",
"For the various aspects of the CIPSI implementation and several examples molecular applications the interested reader is referred to [qp2] and references therein.",
"Very few STO basis sets adapted to post-HF calculations (that is, including optimized polarization functions to describe the virtual space) have been proposed in the literature.",
"Here, in all applications we employ the Slater-type atomic orbital (STO) valence basis set VB1 developed by Ema et al.",
"[19] for the first and second row atoms."
],
[
"Beryllium atom",
"For Be the VB1 basis set consists of two $1s$ , three $2s$ and one $2p$ , for a total of 8 atomic STO basis functions and about 700 two-electron integrals to evaluate.",
"The second column of table REF gives for increasing values of $n_g$ the Hartree-Fock energies obtained with the gaussian basis sets used in the deterministic part of the calculation.",
"Denoted here as $\\lbrace n_g\\rbrace $ , these GTO basis sets are made of the approximate gaussian orbitals $\\phi ^G_a$ , as expressed in Eq.",
"(REF ).",
"By definition, they have the same size as the STO basis set, they only differ by the quality of the approximation made for representing the STO orbitals.",
"As it should be, as $n_g$ increases the Hartree-Fock energies converge to the exact Slater Hartree-Fock energy of -$14.572976251$ .",
"This latter value has been computed using the exact expressions for the one- and two-electron STO integrals that are known in the case of a single nucleus center.",
"Note that this value is in perfect agreement with that given by Ema et al.",
"[19] The third column gives the Hartree-Fock energies obtained with the STO integrals computed with ZVMC and using the $\\lbrace n_g\\rbrace $ gaussian basis set for the deterministic part.",
"The number of Monte Carlo drawings is $N=10^7$ and the coordinate transformation parameters ($\\kappa =3.2$ , $\\nu =1$ ).",
"The value of $\\kappa $ has been optimized by minimization of the statistical error.",
"As it should be, the HF energies obtained with the STO integrals computed by ZVMC are independent of the $n_g$ -approximation, only the magnitude of the statistical error is affected.",
"This error decreases very rapidly as a function of $n_g$ ranging from $10^{-5}$ a.u.",
"to less than $10^{-8}$ .",
"For $n_g=14$ the value of -14.57297625(1) is in perfect agreement with the exact STO Hartree-Fock energy.",
"In Table REF the exFCI values for Be are presented.",
"For comparison the exFCI value computed with the exact one- and two-electron STO integrals and the FCI value of Ema et al.",
"are given.",
"Similarly to the Hartree-Fock results, i.)",
"the exFCI values obtained with the $\\lbrace n_g\\rbrace $ gaussian basis sets converge to the exact ones as $n_g$ increases, ii.)",
"the exFCI values obtained with the ZVMC STO integrals are independent of $n_g$ , and iii.)",
"the statistical error decreases rapidly as the gaussian approximation is improved.",
"For $n_g=10$ our exFCI energy is converged with 7 decimal places and is in full agreement with the exact value.",
"Table: Be atom.",
"Hartree-Fock (HF) energies for the approximate gaussian basis sets{n g }\\lbrace n_g\\rbrace (see, text) and for the exact STO basis set with Monte Carlo integrals computedusing the {n g }\\lbrace n_g\\rbrace basis set for the deterministic part.Total statistics: N=10 7 N=10^7, coordinate transformation parameters (κ=3.2\\kappa =3.2, ν=1\\nu =1).Energies in atomic units.Table: Be atom.",
"exFCI energies for the approximate gaussian basis sets{n g }\\lbrace n_g\\rbrace (see, text) and for the exact STO basis set with Monte Carlo integrals computedusing the {n g }\\lbrace n_g\\rbrace basis set for the deterministic part.Total statistics: N=10 7 N=10^7, coordinate transformation parameters (κ=3.2\\kappa =3.2, ν=1\\nu =1).Energies inatomic units."
],
[
"CH$_4$",
"Table REF presents the HF and $1s^2$ -frozen-core exFCI energies of CH$_4$ .",
"The geometry of the molecule -close to the experimental one- is given in the Supporting Information.",
"Results using the $\\lbrace $ 6-9$\\rbrace $ and VB1 STO basis sets are shown.",
"For comparison, we also report those obtained with the cc-pVDZ basis set.",
"Here, the notation $\\lbrace $ 6-9$\\rbrace $ refers to the gaussian basis set defined above, except that a different number of gaussian functions is used to approximate the various STO orbital, the motivation being to get a more uniform quality among orbital approximations.",
"To be more precise, the orbitals are generically expanded with $n_g=6$ .",
"For 2p and 3d orbitals, $n_g$ is increased by one unit ($n_g=7$ ) or two ($n_g=8$ ), respectively.",
"In addition, when the exponent is too large (here, greater than 4) $n_g$ is increased by 3 ($n_g=9$ ).",
"The VB1 STO basis set is made of three 1s and one 2p for H and two 1s, four 2s, three 3p and one 3d, for a total of 44 cartesian STO orbitals.",
"The cc-pVDZ basis is made of 35 cartesian orbitals.",
"The number of two-electron integrals calculated with ZVMC is about 500 000.",
"All mono-center integrals have been computed with the exact expressions for these STO integrals.",
"The total number of Monte Carlo drawings is about $3 \\times 10^{11}$ The HF and FCI results are presented in table REF .",
"Quite remarkably, both the STO VB1 Hartree-Fock and exFCI energies are obtained with a very good accuracy, that is 6 and 5 decimal places, respectively.",
"The statistical errors have been obtained by running ten statistically independent calculations.",
"Figure REF presents the convergence of the CIPSI energies as a function of the number of selected determinants up to $N_{det}=8 \\times 10^{6}$ by plotting the ten curves obtained for each independent Monte Carlo run.",
"The upper curve shows the convergence of the CIPSI variational energy and the lower one the variational + $E_{PT2}$ energy.",
"As seen the dispersion of the curves decreases as a function of the determinant and converge to the FCI energy.",
"Figure: CH 4 _4 molecule.",
"Convergence of the CIPSI variational energy E var E_{var} (upper curve) andE var E_{var} + E PT2 E_{PT2} energy (lower curve) as a function of the number of selected determinants.Convergence curves realized for 10 statistically independent ZVMC calculations of thetwo-electron STO integrals.",
"Energies in atomic units.Table: CH 4 _4 molecule.",
"Hartree-Fock and 1s 2 1s^2-frozen-core exFCI energies for the cc-pVDZ,{\\lbrace 6-9}\\rbrace (see, text), and Slater VB1 basis sets.Total statistics: N∼3×10 11 N \\sim 3 \\times 10^{11}.",
"Coordinate transformation parameters (κ=3.2\\kappa =3.2, ν=1\\nu =1).Energies in atomic units."
],
[
"A simple cyanine: [H$_2$ N(CH)NH{{formula:f351fded-1d06-4cd1-a35e-6f09570fea96}} ]{{formula:63a13a74-5cac-4045-9a70-65d3c6eb8978}}",
"In the last application the results obtained for a simple model of cyanine molecule, [H$_2$ N(CH)NH$_2$ ]$^+$ are presented.",
"The geometry of the molecule is available in the Supporting Information.",
"The VB1 basis set consists of (three 1s, one 2p) for H and (two 1s, three 2s, three 3p and one 3d) for C and N, for a total of 90 cartesian STO orbitals.",
"The total number of two-electron integrals is about 8.3 $10^6$ .",
"Table REF presents the Hartree-Fock and $1s^2$ -frozen core exFCI results.",
"As seen the accuracy reached is lower than in the case of CH$_4$ but still very good.",
"The statistical error on the exFCI energy is $2 \\times 10^{-4}$ a.u.",
"($\\sim $ 0.1 kcal/mole), that is, the sub-chemical accuracy is reached.",
"Table: Cyanine molecule.",
"Hartree-Fock and 1s 2 1s^2-frozen core exFCI resultsfor the cc-pVDZ,{\\lbrace 6-9}\\rbrace (see, text), and Slater VB1 basis sets.Total statistics: N∼10 10 N \\sim 10^{10}.",
"Coordinate transformation parameters (κ=3.6\\kappa =3.6, ν=1\\nu =1).Energies in atomic units."
],
[
"Summary and discussion",
"In this work an efficient Monte Carlo approach to calculate general two-electron integrals has been presented.",
"Using variance reduction techniques it has been shown that the very high level of precision required on two-electron integrals by Hartree-Fock and post-HF calculations can be achieved.",
"The major advantage of the approach is its great generality and flexibility.",
"It can be used with any type of orbitals provided that sufficiently accurate gaussian approximations are available for them.",
"Various schemes can be used to construct such approximations, so it is not a severe practical limitation.",
"Actually, the key point is that ZVMC results do not depend on this approximation (whatever its quality), only the magnitude of the statistical error is affected.",
"We also note that the approach can be generalized without difficulty to various situations, for example, in the case of an arbitrary two-body interaction or for the calculation of three-particle integrals, the sole condition being that the approximate gaussian integrals involved can be efficiently evaluated.",
"Now, it is clear that the major drawback of the approach is its (very) high computational cost.",
"In the applications presented in this work, the number of Monte Carlo drawings required to make molecular calculations possible ranges from 10$^9$ to 10$^{11}$ .",
"In terms of computational burden, the most extensive simulation realized here (90 orbitals and about 8 millions STO-type two-electron integrals for the cyanine molecule) has been performed using 4800 compute cores running in parallel during a few hours.",
"Although such a cost may appear very high, we emphasize that using the approach in the present form Monte Carlo calculations are feasible with the required accuracy and that we have already been able to realize for a system of the size of the cyanine molecule (24 electrons) a FCI calculation involving 18 active electrons distributed among 90 orbitals, which is, to the best of our knowledge, the most extensive molecular calculation performed so far using pure STO orbitals (no gaussian approximation, even for the challenging four-center two-electron integrals).",
"However, there is clearly much room for the improvement of the method and (much) smaller timings should be easily reachable.",
"Indeed, no particular attention has been paid here to the algorithmic implementation, our objective being mainly to demonstrate the feasibility of the method.",
"No doubt that, in view of the simplicity of the approach and the very repetitive character of the basic floating point operations to be performed, more efficient implementations taking full advantage of the most advanced capabilities of modern processors should be possible.",
"Furthermore, a lot remains to be done to improve the approach itself, particularly in the way the correlated part is performed and in the choice of the coordinate transformation.",
"Finally, we would like to insist on the fact that the most interesting source of (indirect) computational savings is the possibility of using more compact and physically meaningful basis sets, a key aspect considering the sharp increase of the cost of post-Hartree-Fock methods with the number of orbitals.",
"The author would like to thank P.F.",
"Loos and A. Scemama for helpful discussions and a careful reading of the manuscript.",
"We also thank the Centre National de la Recherche Scientifique (CNRS) for funding.",
"This work was performed using HPC resources from i) GENCI-TGCC (Grant No.",
"2018-A0040801738), and ii) CALMIP (Toulouse) under allocations 2018-0510, 2018-18005 and 2019-18005."
]
] | 1906.04515 |
[
[
"Deep learning analysis of coronary arteries in cardiac CT angiography\n for detection of patients requiring invasive coronary angiography"
],
[
"Abstract In patients with obstructive coronary artery disease, the functional significance of a coronary artery stenosis needs to be determined to guide treatment.",
"This is typically established through fractional flow reserve (FFR) measurement, performed during invasive coronary angiography (ICA).",
"We present a method for automatic and non-invasive detection of patients requiring ICA, employing deep unsupervised analysis of complete coronary arteries in cardiac CT angiography (CCTA) images.",
"We retrospectively collected CCTA scans of 187 patients, 137 of them underwent invasive FFR measurement in 192 different coronary arteries.",
"These FFR measurements served as a reference standard for the functional significance of the coronary stenosis.",
"The centerlines of the coronary arteries were extracted and used to reconstruct straightened multi-planar reformatted (MPR) volumes.",
"To automatically identify arteries with functionally significant stenosis that require ICA, each MPR volume was encoded into a fixed number of encodings using two disjoint 3D and 1D convolutional autoencoders performing spatial and sequential encodings, respectively.",
"Thereafter, these encodings were employed to classify arteries using a support vector machine classifier.",
"The detection of coronary arteries requiring invasive evaluation, evaluated using repeated cross-validation experiments, resulted in an area under the receiver operating characteristic curve of $0.81 \\pm 0.02$ on the artery-level, and $0.87 \\pm 0.02$ on the patient-level.",
"The results demonstrate the feasibility of automatic non-invasive detection of patients that require ICA and possibly subsequent coronary artery intervention.",
"This could potentially reduce the number of patients that unnecessarily undergo ICA."
],
[
"Introduction",
"Obstructive coronary artery disease (CAD) is the most common type of cardiovascular disease [1].",
"Obstructive CAD develops when atherosclerotic plaque builds up in the wall of the coronary arteries, narrowing the coronary artery lumen [2].",
"This is defined as coronary stenosis, which can potentially limit blood supply to the myocardium, and could lead to ischemia and irreversible damage [3].",
"Only functionally significant stenoses, i.e.",
"those stenoses which significantly limit blood flow, need to be invasively treated in order to reduce CAD morbidity [3], [4], [5], [6].",
"Contrarily, invasively treating a functionally non-significant stenosis may lead to harmful output [5], [7].",
"Therefore, it is crucial to assess the functional significance of a coronary stenosis to guide treatment.",
"Cardiac CT angiography (CCTA) is typically used to noninvasively identify patients with suspected CAD and visually detect coronary artery stenosis [8].",
"Although CCTA has high sensitivity in determining the functional significance of the stenosis, its specificity for this task is low [9], [10], [11].",
"Therefore, to determine whether a coronary artery stenosis is functionally significant, patients with obstructive CAD typically undergo invasive coronary angiography (ICA) to measure the fractional flow reserve (FFR) in the coronary arteries.",
"FFR is currently the reference standard for establishing the functional significance of a coronary stenosis and it is used to guide treatment [3], [4].",
"However, because of the low specificity of CCTA, up to 50% of patients undergo invasive FFR measurement unnecessarily [11].",
"To reduce the number of unnecessary invasive procedures, noninvasive determination of the functional significance of stenoses based on CCTA images has been intensively investigated.",
"Several automatic methods for determination the functional significance of coronary artery stenosis in CCTA have been proposed [12].",
"These methods can be divided into those that simulate and analyze blood flow in the coronary arteries [13], [14], [15], [16], and those that analyze and characterize the left ventricle (LV) myocardium [12], [17].",
"Methods that simulate and analyze the blood flow in the coronary arteries in CCTA images estimate FFR values along the coronary artery, which can be used to determine the functional significance of coronary artery stenosis.",
"Taylor et al.",
"[13] were the first to propose noninvasive flow-based FFR estimation from CCTA images, which was later validated in multiple clinical studies[18], [19].",
"To determine FFR values along the coronary artery, computational fluid dynamics, coupled with assumptions of physiological boundary conditions, were used.",
"Itu et al.",
"[14] also presented a method to estimate FFR in the coronary artery tree in CCTA images by simulating blood flow.",
"This method uses a parametric lumped heart model, while modeling the patient-specific hemodynamics in both healthy and diseased coronary arteries.",
"Nickisch et al.",
"[15] determined FFR values along the coronary artery by simulating blood flow and pressure along the coronary artery arteries using an electrical patient-specific parametric lumped model.",
"Moreover, Itu et al.",
"[16] presented a machine-learning-based model for estimating FFR along the coronary artery.",
"The model is trained on a large number of synthetically generated coronary anatomies, where the target values are computed using a blood flow-based model [14].",
"This method was further evaluated in [20].",
"While these techniques [13], [14], [15], [16] achieved high accuracy, they are remarkably dependent on the accuracy of coronary artery lumen segmentation [21].",
"Manual annotation of the coronary artery lumen is a time consuming and a complex task, where commercially available automatic software tools typically require substantial manual interaction and correction, especially in CCTA scans with excessive atherosclerotic calcifications or imaging artefacts due to stents and cardiac motion [22].",
"Recently, methods that do not model the blood flow in the coronary arteries but employ characteristics extracted from the myocardium in CCTA scans, have shown to be feasible.",
"Our recent work [12], [23] presented a deep learning approach to automatically identify patients with a functionally significant coronary artery stenosis using analysis of the LV myocardium in CCTA.",
"The method first characterizes the LV myocardium using a convolutional autoencoder (CAE).",
"Thereafter, using the extracted characteristics, patients are classified according to the presence of functionally significant stenosis using an SVM classifier.",
"Previously, Xiong et al.",
"[17] presented a machine learning based approach to detect patients with anatomically significant stenosis using characteristics of the LV myocardium derived from a CCTA scan.",
"In this method, the LV myocardium is aligned with the standard 17-segments model [24] to relate each myocardial segment to its perfusing coronary artery.",
"Then, hand-crafted features, describing each myocardial segment, are extracted and used for supervised classification of patients according to the presence of anatomical significant stenosis.",
"Thereafter, Han et al.",
"[25] employed the technique described in [17] to detect patients with functionally significant stenosis, as defined by the invasively measured FFR.",
"Although these new methods [12], [17] have presented promising results without the need for accurate coronary artery lumen segmentation, they still need to be validated in large and diverse patients cohorts.",
"Moreover, in our recent work [26], we have analyzed the coronary arteries employing a recurrent convolutional neural network (RCNN) for detecting and classifying the anatomical significance of the coronary artery stenosis.",
"The RCNN employs a 3D convolutional neural network to extract local features along the coronary artery.",
"Subsequently, a recurrent neural network aggregates the features to perform the classification tasks.",
"However, such an approach cannot be directly employed for the detection of the functional significance of a coronary stenosis for two reasons.",
"First, such RCNN only performs a local analysis of the artery, were the complete artery is not taken into account.",
"Second, to train such an RCNN, local reference labels are required.",
"Such a requirement is not practical in the case of the functional significance of a coronary stenosis, where FFR is used as the reference and is usually provided on the artery level only.",
"Here, we present a method to automatically and non-invasively identify coronary arteries and patients requiring further invasive evaluation, i.e.",
"ICA, as determined by the invasively measured FFR.",
"Blood flow in the coronary artery may be affected by multiple coronary artery stenoses and arterial plaques [3], [27].",
"Therefore, to classify an artery according to the functional significance of the coronary artery stenosis, local analysis of a single stenosis may be insufficient.",
"Hence analysis of the complete artery should be performed.",
"Moreover, in clinical practice, usually a single, i.e.",
"lowest, FFR value per coronary artery is reported.",
"Consequently, employing supervised machine learning methods to directly analyze a whole volume of an artery (e.g.",
"with 3D-CNN or RCNN [26]) to detect the functional significance of each stenosis or estimating the invasively measured FFR values at every point along the coronary artery is unfeasible.",
"Therefore, in the proposed work, a complete artery is analyzed in an unsupervised manner to extract lower-dimensional encoding, and thereafter to determine the presence of abnormal FFR.",
"First, using the extracted coronary artery centerline [28], the straightened 3D multi-planar reformatted (MPR) volume is reconstructed.",
"Then, an MPR volume of a complete artery is characterized with a fixed number of encodings using convolutional autoencoders (CAEs) [29], [30], [31], which serve as unsupervised feature extractors.",
"As MPR volumes of complete coronary arteries have large volumetric sizes and variable lengths and shapes, a single traditional CAE cannot be successfully and directly applied to efficiently encode a complete artery.",
"Therefore, in the here proposed work, two disjoint CAEs are employed.",
"The first CAE performs spatial encoding of local sub-volumes along the artery.",
"Then, a second CAE encodes the output of the first CAE - which depends on the artery length - into a fixed-length encoding.",
"Finally, a support vector machine (SVM) [32] classifies arteries based on these encodings according to presence of functionally significant stenosis, as defined by the invasively measured FFR.",
"The proposed approach is illustrated in Fig.",
"REF .",
"Our contributions are twofold.",
"Firstly, we propose to jointly employ two disjoint CAEs that perform spatial and sequential encoding of large volumes with varying lengths.",
"Secondly, in contrast to previous methods that detect the presence of functionally significant stenosis or determine FFR values non-invasively, our method does not require accurate and difficult to obtain segmentation of the coronary artery lumen or LV myocardium.",
"Instead, it only requires the coronary artery centerline, which can be obtained automatically or semi-automatically [33].",
"The remainder of the manuscript is organized as follows.",
"Section describes the data and reference standard.",
"Section describes the method.",
"Section reports our experimental results, which are then discussed in Section .",
"This study includes retrospectively collected CCTA scans of 187 patients (age: $58.6 \\pm 8.7$ years, 145 males) acquired between 2012 and 2016.",
"The Institutional Ethical Review Board waived the need for informed consent.",
"All CCTA scans were acquired using an ECG-triggered step-and-shoot protocol on a 256-detector row scanner (Philips Brilliance iCT, Philips Medical, Best, The Netherlands).",
"A tube voltage of 120 kVp and tube current between 210 and 300 mAs were used.",
"For patients $\\le 80$ kg contrast medium was injected using a flow rate of 6 mL/s for a total of 70 mL iopromide (Ultravist 300 mg I/mL, Bayer Healthcare, Berlin, Germany), followed by a 50 mL mixed contrast medium and saline (50:50) flush, and next a 30 mL saline flush.",
"For patients $>80$ kg the flow rate was 6.7 mL/s and the volumes of the boluses were 80, 67 and 40 mL, respectively.",
"Images were reconstructed to an in-plane resolution ranging from 0.38 to 0.56 mm, and 0.9 mm thick slices with 0.45 mm spacing.",
"In each CCTA scan, coronary arteries were tracked and their centerlines were extracted using the method previously described by Wolterink et al.",
"[28].",
"The method tracks the visible coronary arteries, where the arterial centerlines are extracted between the ostia and the most distal visible locations.",
"Using the extracted centerlines, a 3D straightened MPR volumes with 0.3 $mm^3$ isotropic resolution were reconstructed for all coronary arteries and used for further analysis.",
"Note that we define an artery as the vessel starting from the ostium until the most distal location visible in the CCTA."
],
[
"FFR Measurements",
"Out of the 187 patients, 137 patients suspected of obstructive CAD underwent invasive FFR measurements ($0.81 \\pm 0.10$ , interquartile range: 0.74-0.89), up to one year after the acquisition of the CCTA scan.",
"In these patients, FFR was measured in 192 different arteries.",
"FFR was recorded with a coronary pressure guidewire (Certus Pressure Wire, St. Jude Medical, St. Paul, Minnesota) at maximal hyperemia conditions.",
"Maximal hyperemia was induced by administration of intravenous adenosine (at a rate of 140 $\\mu $ g/kg per minute) through a central vein.",
"The FFR wire was placed at the most distal part possible in the target artery.",
"Using manual pullback, a single minimal FFR value was assessed and recorded for each artery."
],
[
"Methods",
"Blood flow in the coronary artery may be affected by a single or multiple coronary artery stenoses [3], [27], located anywhere along the coronary artery; starting from the ostium until the most distal location visible in the CCTA.",
"Therefore, to classify an artery according to the functional significance of a stenosis, local analysis of a single stenosis may be insufficient, but the analysis of the complete artery is needed.",
"Moreover, in clinical practice, determining invasive FFR values for each voxel within the artery lumen, or recording an FFR value for each point on the coronary artery centerline, is impractical and typically not performed.",
"Instead, the minimal single FFR value per coronary artery is recorded, resulting in a single reference label per artery.",
"Hence, given the sparsity of reference labels along the artery, the large input dimensions and the limited dataset size, employing a supervised end-to-end machine learning methods (e.g.",
"with 3D-CNN or RCNN [26]) to directly detect the functional significance of each stenosis or estimating FFR values at every point along the coronary artery would be prone to overfitting.",
"Therefore, in the proposed work, an MPR of a complete artery is analyzed to determine the presence of abnormal FFR.",
"First, to extract robust features of complete arteries, MPR volumes are characterized by a fixed number of encodings using convolutional autoencoders (CAEs) [29], [30], [31], regardless of the artery length.",
"Then, the extracted encodings are used as input to an SVM classifier that determines whether the artery needs further invasive evaluation, in the form of ICA, to establish the need of intervention.",
"Figure: Illustration of the proposed workflow.",
"In a CCTA scan, the centerlines of the coronary arteries are extracted and used to reconstruct straightened multi-planar reformatted (MPR) images of the coronary arteries.",
"Then, an unsupervised analysis is performed, where the MPR volume of a complete artery is encoded into a fixed number of encodings (features) using two disjoint convolutional autoencoders, applied sequentially: a 3D variational convolutional autoencoder (3D-VCAE), that spatially encodes local sub-volumes of the coronary artery, and a 1D convolutional autoencoder (1D-CAE), that sequentially encodes the encodings of the complete artery.",
"Then the final extracted encodings are employed in a supervised fashion to classify arteries according to the need of further invasive evaluation, in the form of ICA, to establish the need of intervention, using a support vector machine (SVM) classifier."
],
[
"Encoding the artery",
"The main purpose of a convolutional autoencoder (CAE) is to extract robust compact features from unlabeled data, while removing input redundancies and preserving essential aspects of the data [29], [30], [34].",
"A CAE consists of two main parts, an encoder and a decoder [29], [30].",
"The encoder compresses the data to a lower dimensional latent space by convolutions and down-sampling.",
"The decoder expands the compressed form to reconstruct the input data by deconvolutions and upsampling.",
"A CAE is trained to minimize a difference loss between the encoder input and decoder output.",
"This ensures that the encodings contain sufficient information to reconstruct inputs with low error [30].",
"Once the CAE is trained, the decoder is removed and the encoder is used to generate encodings for unseen data.",
"Coronary arteries are complex anatomical 3D structures, with varying lengths and anomalies across patients [35].",
"The resolution of modern CT scanners is high and a large number of voxels (millions) is contained in an MPR volume of a single artery.",
"Therefore, following the straightforward approach of training a single CAE, applied directly to the complete artery volume without a large reconstruction error, is infeasible.",
"Therefore, in this work, we propose a two-stage encoding approach to encode a complete MPR volume of the coronary artery, regardless of its length.",
"Fig.",
"REF illustrates the proposed encoding flow.",
"First, a 3D variational convolutional autoencoder (3D-VCAE) is applied to local sub-volumes extracted from the MPR along the artery centerline.",
"As the 3D-VCAE is only applied to small input volumes, the number of its trainable parameters is relatively low.",
"The 3D-VCAE encodes each sub-volume into a set of small number of encodings.",
"When applied to all sequential sub-volumes along the artery, the result is a feature map of the same height as the number of encodings and the same length as the artery length.",
"This feature map is then represented as a set of individual 1D sequences of encodings.",
"Each sequence contains an individual encoding out of the set of encodings, running along the artery (colored signals in Fig.",
"REF ).",
"This allows the analysis of complete arteries with varying length by a 1D convolutional autoencoder (1D-CAE), with low number of trainable parameters which decreases the chance of overfitting.",
"Hence, the 1D-CAE encodes the varying length sequences of encodings further into a fixed number of encodings, that represent the complete artery, regardless of its length.",
"Figure: Illustration of the proposed encoding approach.",
"To encode an MPR volume of a complete artery into a fixed number of encodings, a two stage encoding approach is applied.",
"First, a 3D variational convolutional autoencoder (3D-VCAE) is applied to local 40x40x5 voxel sub-volumes extracted from the MPR along the artery.",
"The 3D-VCAE encodes each volume into an encoding in a R 16 R^{16} latent space.",
"When applied to all sequential sub-volumes along the artery, the result is a feature map of the same height as the number of encodings and the same length as the artery length (L).",
"This feature map is then represented as a set of individual 1D sequences of encodings.",
"Each sequence contains an individual value in the R 16 R^{16} latent space encoding, running along the artery.Then, a 1D convolutional autoencoder (1D-CAE) is applied separately to each of the 16 sequences of encodings and encodes each further into a second latent space with 64 dimensions (R 64 R^{64}).",
"This results in a fixed number of encodings (1024) per artery, that represent the complete artery volume, regardless of its length and shape.VAEs are generative models, which approximate data generating distributions [31].",
"Through approximation and compression, the resulting models have been shown to capture the underlying data manifold; a constrained, smooth, continuous, lower dimensional latent (feature) space where data is distributed [36], [37].",
"Having in mind possible reconstruction errors of the encoding sequences by the second CAE, a variational CAE is chosen as the first CAE for the ability of its decoder to handle small variations in the encodings [36].",
"Inspired by these advantageous properties of the latent space, a VCAE is employed to compress and encode local volumes along the artery.",
"To capture local volumetric characteristics of the artery, the input to the 3D-VCAE is set to a volume of 40x40x5 voxels, centered around a coronary artery centerline point.",
"The size of the input is chosen so that it contains the whole arterial lumen and the vicinity of the artery [2].",
"The output of the encoder in the 3D-VCAE is set to 16; i.e.",
"an encoding in a $R^{16}$ latent space.",
"The dimension of the input volume and encoding size are determined in preliminary experiments to balance between the compactness (i.e.",
"size of the encoding) and the expressiveness of the encodings (i.e.",
"reconstruction error).",
"Table REF lists these findings.",
"To encode the complete artery, overlapping volumes with stride of 1 are extracted and encoded with 3D-VCAE (Fig.",
"REF ).",
"This results in 16xL encodings, where L is the length of the artery.",
"To reconstruct a complete artery, the middle slices of each overlapping reconstructed volume are used.",
"The 3D-VCAE architecture used in this work is shown in Fig.",
"REF (a).",
"In the 3D-VCAE , batch normalization [38] layers and rectified linear units (ReLUs) are used after all convolution layers except the encoder and decoder output layers.",
"Figure: Architectures of autoencoders.",
"(a) A 3D variational convolutional autoencoder (3D-VCAE), its input and outputs are volumes of 40x40x5 voxels, where the input is encoded into an encoding of size 16.Key: N kernel N_{kernel}@size kernel size_{kernel} is a convolutional layer with N kernel N_{kernel} kernels of size size kernel size_{kernel}.",
"MP@size kernel size_{kernel} is a max-pooling layer with kernel size size kernel size_{kernel}.",
"US@size kernel size_{kernel} is an upsampling layer with kernel size size kernel size_{kernel}.",
"FC@N units N_{units} is a fully connected layer with N units N_{units} units.",
"Once the 3D-VCAE is trained, the output of the μ\\mu layer is used to generate encodings for the input.",
"(b) A 1D convolutional autoencoder (1D-CAE), its input and outputs are 16xL sequences of encodings.",
"Each sequence is padded into a maximal length of 800, and is encoded into an encoding of size 64.",
"Key: N kernel N_{kernel}@(size kernel size_{kernel}:size stride size_{stride}) is a 1D convolutional layer with N kernel N_{kernel} kernels of size size kernel size_{kernel} and stride of size stride size_{stride}.",
"US@size kernel size_{kernel} is a 1D upsampling layer with kernel size size kernel size_{kernel}.",
"The 1D-CAE is applied separately, but with shared weights, to each of the 16 1D-sequences.",
"Once the 1D-CAE is trained, the output of the ee layer is used to generate encodings for each input sequence.Descending and ascending arrows represent the encoder and the decoder in each autoencoder, respectively.Table: Average reconstruction MAPE, (within lumen HU range), total size of the final encoding used for classification and the achieved AUC for artery-level classification across a range of different input sizes of first CAE (CAE 1), different encoding sizes of first and second CAE (CAE 2) or different encoding strategies: PCA: using two consecutive principle component analyses; 1D: using a 1D-CAE (as CAE 2); 2D: using a 2D-CAE (as CAE 2); Global: using a single 3D-VCAE applied to the complete artery.",
"Please note that the proposed configuration (input size of 40x40x5 with 16/64 encoding sizes) is listed multiple times for easy comparison.",
"* ^*Number of principle components."
],
[
"Sequential encoding with 1D convolutional autoencoder",
"When representing the coronary artery to determine the functionally significant stenosis according to FFR, characteristics along the artery, starting from the ostium to the most distal part of the artery, need to be taken into account [13], [14], [15].",
"Therefore, to analyze the complete artery at once, the local encodings extracted previously by the 3D-VCAE along the length of the artery need to be merged.",
"To accomplish this, the feature map, consisting of L sets of 16 values of encoding generated by 3D-VAE at each coronary artery center point, is represented as L 1D sequences.",
"As in the 3D-VCAE design, the size of the 1D-CAE encoding is determined in preliminary experiments to balance between the compactness (i.e.",
"size of the encoding), the expressiveness of the encodings (i.e.",
"reconstruction error) and the classification performance (AUC).",
"Table REF lists these findings.",
"Each sequence consists of 1xL values, where L represents the length of the artery, i.e.",
"number of coronary artery centerline points.",
"To encode arteries with different lengths, sequences of encodings of short arteries were padded into a maximum length of 800, which corresponds to the number of centerline points in the longest artery in the dataset.",
"This representation leads each sequence to represent a specific member of the encoding in the $R^{16}$ latent space along the artery (colored 1D signals in Fig.",
"REF ).",
"This, consequently, allows us to apply a 1D-CAE to each of the 16 sequences separately.",
"The weights of the 16 1D-CAEs are shared, where each 1D-CAE encodes one of the 16 sequences into an encoding of a second latent space of 64 dimensions ($R^{64}$ ).",
"This results in 1024 (16x64) features that represent the complete artery.",
"The 1D-CAE architecture used in this work is shown in Fig.",
"REF (b).",
"In the 1D-CAE, the exponential linear units (ELUs) are used after all convolutions layers except the encoder and decoder output layers.",
"Based on the extracted encodings from the encoding stage, arteries are classified according to the need of further invasive evaluation, in the form of ICA.",
"This was defined by the invasively measured FFR.",
"As the standard deviation of the differences in repeated FFR measurements can reach up to $5\\%$ [39], [40], [41], [42], especially in the so called \"gray-zone\" [41], where the measured FFR is between 0.75 and 0.85, in our experiments, the threshold on FFR value was set to 0.9.",
"This results in a positive class with $FFR\\le 0.9$ representing arteries requiring ICA to establish the need of intervention, and a negative class with $FFR>0.9$ representing absence of functionally significant stenosis, where ICA is not necessary.",
"The classification is performed using an SVM classifier with a linear kernel and an $L_1$ regularization.",
"For each classified artery, the continuous output of the trained SVM is used to assign a predicted class.",
"As patients with suspected obstructive CAD undergo ICA to measure the FFR in all diseased coronary arteries, in this work, classification of patients is also performed.",
"To classify patients, the highest output value of all classified arteries in a patient is used to assign a predicted class to the patient.",
"The minimal FFR across the arteries of a patient is taken as a reference.",
"Classification performance is evaluated using a receiver operating characteristic (ROC) curve and the corresponding area under the ROC curve (AUC)."
],
[
"Encoding the artery",
"To train the 3D-VCAE and the 1D-CAE, a set of CCTA images of 50 patients, who did not undergo ICA and hence had no FFR measurements, were used.",
"From these, 38 CCTA images were randomly selected for training, and the remaining 12 images were used for validation.",
"In both sets, MPR volumes of the extracted arteries were used to train and validate the autoencoders.",
"Both autoencoders' hyperparameters were determined in preliminary experiments using the validation set.",
"Please note that no data augmentation was performed for training the autoencoders.",
"As a baseline reconstruction, principal component analysis (PCA) was employed in a similar way as the two disjoint autoencoders: A first PCA was applied to all 40x40x5 voxels volumes along the artery to reduce each into 16 principal components.",
"Then, a second PCA was applied to all outputs of the first PCA to reduce the dimensionality of the outputs into 1024 components.",
"Table REF lists the findings.",
"To train and validate the 3D-VCAE, 40x40x5 voxels volumes were randomly extracted along centerlines of arteries in the training and validation sets, respectively.",
"Mini-batches of 32 volumes were used to minimize the loss function with Adam optimizer [43] with a learning rate $0.001$ .",
"The mean squared error between the input and the reconstructed volumes, and the Kullback-Leibler (KL) divergence with the reparameterization trick [31] were employed as a loss function for the variational autoencoder.",
"L2 regularization was used with $\\gamma =0.001$ for all layers.",
"Training was performed until convergence.",
"Fig.",
"REF (a)-(b) show an example of a complete artery which was encoded and reconstructed with the trained 3D-VCAE.",
"This was performed by extracting, encoding and reconstructing input volumes around each point along the MPR centerline.",
"Fig.",
"REF (c) shows the absolute reconstruction error, i.e.",
"the absolute difference between the input and the reconstructed artery.",
"Fig.",
"REF (d) shows all 16 sets of encodings, presented as continuous sequences running along the artery.",
"These sequences are to be encoded further in a later stage using the 1D-CAE into a fixed number of encodings.",
"Figure: Examples of outputs from different stages in encoding and reconstructing a complete artery.",
"(a) Original MPR of a complete artery; (b) The reconstructed MPR by only the 3D-VCAE.",
"This was performed by extracting, encoding and reconstructing 40x40x5 voxels volumes around each point along the MPR centerline; (c) The absolute error between (a) and (b); (d) Encodings extracted by the 3D-VCAE, presented as continuous sequences running along the artery; (e) Three randomly selected encodings sequences of a complete artery (dashed line) which are encoded and reconstructed (solid line) with the 1D-CAE.",
"(f) The reconstructed artery that was encoded and reconstructed back with the 3D-VCAE and 1D-CAE combined.",
"(g) The absolute error between (a) and (f).To train and validate the 1D-CAE, arteries with the corresponding sets of 16 encodings sequences, obtained by the 3D-VCAE, were randomly chosen from the training and validation sets, respectively.",
"Sequences of encodings of short arteries were padded into a maximum length 800, which corresponds to the longest artery in the dataset.",
"Mini-batches of 32 sets of encodings sequences were used to minimize the loss function with Adam optimizer with a learning rate $0.001$ .",
"The masked mean squared error was employed as a loss function, where padded values in the input sequences did not contribute to the loss value or its gradients and were therefore ignored.",
"L2 regularization was used with $\\gamma =0.001$ for all layers.",
"Training was performed until convergence.",
"Fig.",
"REF (e) shows an example of 3 randomly chosen encoding sequences of a complete artery which were encoded and reconstructed with the trained 1D-CAE.",
"To demonstrate the effectiveness of the proposed combined two-stage encoding approach in preserving the original shape and appearance of the artery, both disjoint trained autoencoders were combined and tested on complete arteries.",
"To accomplish this, an inference with four steps was performed.",
"First, the encoder of the 3D-VCAE was applied to local volumes along the MPR volume of a complete artery, resulting in 16 sequences of encodings with L values each.",
"Second, the encoder of the 1D-CAE encoded the sequences into a single encoding vector of 1024 values.",
"Third, the decoder of the 1D-CAE decoded the encoding vector back to 16 encodings sequences.",
"Last, the decoder of the 3D-VCAE reconstructed those reconstructed encodings sequences to the original MPR volume size.",
"Fig.",
"REF (a),(f),(g) show an example of a complete artery that was encoded with the combined strategy, reconstructed back to the original volume dimensions, and the corresponding reconstruction error.",
"Fig.",
"REF compares the average mean absolute reconstruction percentage errors (MAPE) between the local reconstructions made by only the 3D-VCAE, the baseline PCA reconstruction approach and the combined approach reconstruction, across a range of CT Hounsfield units (HU).",
"As the MAPE might be misleading around small image values or image values equal to zero, the range of intensity values characteristic for coronary artery lumen (250-450 HU) is highlighted.",
"Fig.",
"REF and Fig.",
"REF demonstrate the high resemblance and the low reconstruction error between the results of the local and the combined approaches compared to the original volume.",
"Figure: The average and standard deviation of the mean absolute reconstruction percentage errors (MAPE) obtained by the different reconstruction strategies, across a range of CT Hounsfield units (HU).",
"Artery lumen: indicates the typical range (250-450 HU) of the CT values of the coronary artery lumen.",
"Local: local reconstructions by the 3D-VCAE only (as in Fig.",
"(b)); PCA: baseline reconstruction strategy, using 2 consecutive PCAs; 1D: combined reconstruction strategy, using the 1D-CAE (as in Fig.",
"(f)); 2D: combined reconstruction strategy, using the 2D-CAE (as in Fig.",
"(d)); Global: global reconstruction strategy, using a 3D-VCAE (refer to Fig. )",
"applied to the complete artery (as in Fig.",
"(f))."
],
[
"Evaluation of alternative encoding strategies",
"To demonstrate that the proposed combined encoding strategy is advantageous, two additional encoding strategies were evaluated and compared to the proposed sequential disjoint autoencoders.",
"First, the most straightforward approach was evaluated, where a single autoencoder analyzes the complete MPR volume, encodes it to a fixed number of encodings (1024), and reconstructs it back to the input size.",
"To handle arteries with different lengths, shorter arteries were padded to the maximal artery length in our dataset (800).",
"Hence, the input of the autoencoder was defined as 40x40x800.",
"The architecture of the evaluated VCAE is shown in Fig.",
"REF .",
"Figure: The 3D-VCAE architecture used to encode a complete artery, where its input and outputs are MPR volumes of a complete artery of size 40x40x800 voxels.",
"Keys are the same as in Fig.",
".Second, the 1D-CAE used in the combined encoding strategy (Fig.",
"REF ) was replaced by a 2D-CAE to jointly process the encoding sequences.",
"While the proposed 1D-CAE encoded each sequence of encodings separately, the here evaluated 2D-CAE mutually encoded all sequences of encodings.",
"This was performed by representing the encodings map as a 2D image and applying two-dimensional convolutional kernels.",
"The architecture of the evaluated 2D-CAE is identical to the 1D-CAE (Fig.",
"REF (b)), but the convolutions were performed by applying two-dimensional kernels of the same size (3x3).",
"Additionally, dropout of 0.1 was applied between fully connected layers to avoid overfitting.",
"The two additional autoencoders were trained and validated in a similar manner as in the combined encoding strategy.",
"In the case of the 2D-CAE, the trained decoder of the 3D-VCAE was used to reconstruct the MPR volume.",
"Fig.",
"REF shows an example of an artery that was encoded and reconstructed using the two evaluated auto-encoding strategies, and compared with the reconstruction of the proposed combined encoding strategy.",
"Fig.",
"REF also shows the average MAPE across a range of different HUs.",
"Both figures demonstrate a clear advantage for the reconstruction of the combined strategy over the two additionally evaluated approaches.",
"Figure: Examples of different reconstructions of a complete artery by different reconstruction strategies.",
"(a) Original MPR of a complete artery; (b) The reconstructed MPR by the proposed combined 3D-VCAE and 1D-CAE; (c) The absolute error between (a) and (b); (d) The reconstructed MPR by the combined 3D-VCAE and 2D-CAE; (e) The absolute error between (a) and (d); (f) The reconstructed MPR by the global 3D-VCAE (refer to Fig.",
"), applied to the complete artery; (g) The absolute error between (a) and (f)."
],
[
"Classification of arteries and patients",
"Classification of arteries was performed using the arteries' encodings, extracted by the two disjoint autoencoders (Section REF ), and an SVM classifier.",
"In preliminary experiments, different classifiers (logistic regression, random forest) and various SVM configurations, including different regularizations and kernels types, were tested.",
"The best performance was achieved using a linear $L_1$ -regularized SVM.",
"All 50 CCTA images of patients used in training and validation of the autoencoders were excluded.",
"Thus, CCTA images of 137 patients and the corresponding reference FFR measurements in 192 different arteries were used for this analysis.",
"To assess the performance and the robustness of the classification, 1000 stratified Monte-Carlo cross-validation experiments were performed.",
"In each experiment, 10 arteries were used as a test set, and the remaining arteries were used as a training set.",
"The assignment of arteries to test or training sets was random, however it insured that arteries from the same patient were included either in the training or the test set.",
"Optimal SVM parameters were selected in every experiment using a grid search on the training set only.",
"The obtained results are shown in Fig.",
"REF .",
"On the artery-level, an average AUC of $0.81\\pm 0.02$ was achieved, while on the patient-level, an average AUC of $0.87\\pm 0.02$ was achieved.",
"Table REF lists the average diagnostic accuracy on the artery- and patient-levels across four different ranges of FFR measurements, and Table REF lists the achieved performance in the three main coronary arteries.",
"Moreover, implemented in Keras with TensorFlow, the runtime of encoding and classifying a single MPR volume of a complete coronary artery was on average 11 seconds, while using a single NVIDIA TITAN X (Pascal) GPU with an Intel Xeon machine with 256 GB RAM.",
"Figure: Average ROC curves, and corresponding area under curve (AUC), for classification of (a) arteries and (b) patients requiring ICA.",
"The classification was performed using the extracted encodings of the arteries and a support vector machine classifier.",
"The shaded area represents the standard deviation of the sensitivity across the cross validation experiments.Table: Average diagnostic accuracy for the detection of arteries and patients requiring ICA on the artery- and patient-levels shown in four different subgroups corresponding to four ranges of FFR measurements.",
"N indicates the number of arteries and patients in each subgroup.Table: Average diagnostic accuracy for the detection of arteries requiring ICA shown in three subgroups, corresponding to the three main coronary arteries: Left circumflex artery (LCX), right coronary artery (RCA) and left anterior descending (LAD).",
"N indicates the number of arteries in the data set."
],
[
"Evaluation of alternative classification strategies",
"To demonstrate the effectiveness of the proposed classification scheme, we have performed several additional classification experiments that can be divided into two categories.",
"First, the influence of the FFR threshold was investigated.",
"As in Section REF , arteries were classified into positive or negative class with a binary SVM classifier using the extracted encodings.",
"Different thresholds on reference FFR values were applied, resulting in different class interpretations: When a threshold of 0.7 was applied, positive classes represented arteries in need for an invasive intervention without the need of establishing the FFR value first.",
"When a 0.8 threshold was applied, positive class represented arteries with a functionally significant stenosis.",
"Second, the influence of regression vs. classification was investigated.",
"In contrast to the former classification scenarios where retraining the SVM classifier was needed for each different FFR threshold, here, a single SVM regressor was trained to estimate continuous values of FFR (i.e.",
"regression) and then an FFR threshold (0,7, 0.8 or 0.9) was applied to output the binary classes.",
"The ROC curves showing the results are given in Fig.",
"REF .",
"The results show that binary classifications outperform the corresponding regression experiments, regardless of FFR threshold.",
"Moreover, when an FFR threshold of 0.7 was used, performance of the both classification and regression approaches were moderate, while the experiments using threshold of 0.8 on FFR values showed lowest areas under the ROC curves.",
"Figure: Average ROC curves, and corresponding area under curve (AUC), for classification of arteries under different classification strategies and various FFR thresholds (0,7, 0.8, 0.9).",
"The classification was performed using the extracted encodings of the arteries and a support vector machine classifier to: (1) binary classify arteries (C.) or (2) regress continuous FFR value and subsequently use the FFR threshold to produce the binary output (R.).",
"The shaded areas represent the standard deviation of the sensitivity across the cross validation experiments."
],
[
"Comparison with other FFR classification methods",
"We compare our classification performance with the reported results of previous methods.",
"These methods either analyzed the blood flow in the coronary arteries [18], [44], [20], requiring a highly accurate segmentation of the arterial lumen, or analyzed the LV myocardium [25], [12], requiring a segmentation of the LV myocardium.",
"Table REF lists the results as originally reported.",
"However, all of the compared methods were evaluated on different datasets that included different patients cohorts.",
"Moreover, the compared methods employed an FFR cut-off value of 0.8 to define the functional significance of a stenosis, while in this study, a 0.9 cut-off point was used.",
"Therefore, these results only indicate the differences in approaches and should not be directly compared.",
"Table: Performance comparison with previous work.",
"Table lists number of evaluated patients and arteries, achieved accuracy and the area under the ROC curve (AUC) per-patient and per-artery for classification according to FFR as reported in the original studies.",
"Please note that these methods use different FFR thresholds and perform different analyses: either analyzing of the blood flow in the coronary arteries (Flow), detecting ischemic changes directly in LV myocardium (Myo.",
"), or, as proposed in this work; classifying coronary arteries with features extracted by CAEs.A method for automatic and non-invasive identification of patients requiring evaluation with invasive coronary angiography has been presented.",
"The method analyzes complete coronary arteries with two convolutional autoencoders that characterize the MPR volume of each artery with general robust features, and encode the complete artery into a fixed number of encodings to reduce the dimensions of the input.",
"Then, these encodings are used with an SVM classifier to identify arteries with functionally significant stenosis in an supervised manner.",
"Unlike previous methods that detect functionally significant stenosis by relying either on the coronary artery lumen segmentation [13], [14], [15], [16] or the left ventricle myocardium segmentation [12], [17], the proposed method requires only the coronary artery centerline as an input along with the CCTA scan.",
"Artery centerline extraction is a simplified task compared to myocardium segmentation and to the arterial lumen segmentation, where the latter occasionally requires substantial manual interaction, especially in diseased population with heavily calcified arteries.",
"In this work, to extract the coronary artery centerlines, we have employed our previously designed method for artery centerline extraction [28].",
"However, any other manual, semi-automatic or automatic method could be employed instead.",
"As the dimensions of MPR volumes of complete arteries are large, and the reference labels are only provided on the artery level, employing a straight-forward supervised 3D-CNN to detect the functional significance of a stenosis is far from feasible.",
"Therefore, here, we have used unsupervised learning to characterize and encode each artery before employing a supervised classifier to detected abnormal FFR.",
"To do so, two disjoint CAEs, that were applied sequentially, were employed.",
"This is contrary to the more common approach of using a single CAE that encodes the complete artery volume at once.",
"The results show that the learned encodings were able to represent the artery shape and appearance accurately, as was demonstrated qualitatively (Fig.",
"REF ) and quantitatively by the relatively small mean absolute error between the input and the reconstructed volumes (Fig.",
"REF ).",
"The output of the combined encoding strategy was examined and compared to the output of the 3D-VCAE on local volumes.",
"Fig.",
"REF and Fig.",
"REF show that, in the range of CT values of the artery lumen (250-450 Hounsfield units), both the local and the combined encoders (with 1D-CAE) achieved satisfactory results, where the local approach was slightly advantageous.",
"However, the lower error achieved with the local approach could be explained by the larger number of encodings used per artery compared to the combined approach.",
"It can be noticed (Fig.",
"REF (b) and (f)) that the reconstructions of both approaches preserved the shape and the morphology of the artery, while not being able to preserve the texture of neither the lumen nor the background.",
"Although the lumen of the coronary artery was accurately reconstructed by the proposed encoding method, some small calcifications within the artery were entirely or partially lost (Fig.",
"REF and Fig.",
"REF , respectively).",
"This might be due to the low number of encodings used in the 3D-VCAE (16).",
"Beside increasing the size of the encoding, future work could address this by modifying the loss function of the 3D-VCAE to penalize such errors in reconstruction, or by over-sampling such calcifications in the training process.",
"In the proposed combined encoding strategy, both autoencoders were disjoint during training and were combined only during inference, which could lead to error propagation.",
"To overcome this, one alternative would be training both autoencoders simultaneously and end-to-end.",
"However, in preliminary experiments, this was proven difficult, mainly due to hardware limitations.",
"Another alternative would be training the 3D-VCAE separately, and then, using its trained decoder during training the 1D-CAE.",
"This could be done by directly minimizing the mean squared error between the original and the reconstructed MPR volumes of the complete artery instead of between its original and reconstructed encodings sequences.",
"Such training might potentially compensate for errors or prevent error propagation between the two disjoint training processes.",
"Future work might address this.",
"Additional encoding strategies were performed and compared with the proposed one.",
"Although other studies showed that a single 3D-CAE might be successfully used on large volumetric input [45], in the proposed work training a single 3D-VCAE (Fig.",
"REF ), applied directly to the complete artery volume without a large reconstruction error, was proven infeasible (Fig.",
"REF and Fig.",
"REF ).",
"This could be due to a number of reasons: The very large number of trainable parameters ($\\sim 65\\times 10^6$ ) of the CAE, the high variability among the shapes and lengths of the arteries, the atypical aspect ratio of the input (40x40x800), or the lack of large set of training data.",
"Moreover, treating all encoding sequences as a single 2D image and encoding this image with a 2D-CAE proved inferior when compared with the proposed 1D-CAE (Fig.",
"REF and Fig.",
"REF ).",
"This might be explained by the lack of local spatial relations between the different encodings ($\\mu _0-\\mu _{15}$ ) in Fig.",
"REF at a given location along the coronary artery.",
"These local spatial relations among the encodings motivate the use of 2D kernels in a typical 2D-CAE, when analyzing natural or medical images.",
"The artery was represented by multiple sequences of encodings, obtained after applying the 3D-VCAE to local sub-volumes along the artery.",
"To further encode the artery to a fixed number of encodings, a 1D-CAE was applied to each sequence of encodings separately.",
"As the fully connected layer in 1D-CAE (layer $e$ in Fig.",
"REF (b)) expects a fixed number of inputs, the input sequences were padded to the maximal length of an artery in the dataset.",
"A masked loss function was employed during training the 1D-CAE to minimize the effect of such padding on the reconstruction error.",
"Despite this masking, the proposed padding could affect the classification performance as a function of the artery length.",
"To enable the autoencoder to handle variable length sequences, without the need of padding it, a recurrent autoencoder could be employed [46].",
"In such a recurrent autoencoder, known as sequence-to-sequence autoencoders, a recurrent layer, with Gated Recurrent Units (GRUs) [47] or long short-term memory (LSTM) units [48], replaces the fully connected layer in the proposed 1D-CAE, to recursively process and encode a sequential varying length input.",
"Future work might investigate such a recurrent autoencoder and its affect on the classification performance.",
"Our experiments show that moderate classification performance on both the artery- and patient-levels was achieved, while using only features derived in an unsupervised manner from a CCTA of coronary arteries (Fig.",
"REF ).",
"These results show that the proposed approach could potentially lead to a reduction in the number of patients that unnecessarily undergo invasive coronary angiography.",
"For example, as seen in Fig.",
"REF (b), at the sensitivity of 80% or 90% in detecting patients requiring ICA, i.e.",
"those having $FFR\\le 0.9$ , unnecessary ICA could have been prevented in 76% or 53% of the negative patients, i.e.",
"those having $FFR>0.9$ , respectively.",
"Moreover, we have compared the classification results across different ranges of FFR measurements (Table REF ).",
"The comparison shows slightly higher accuracies for $FFR>0.8$ , on both the artery- and patient-levels.",
"The reason might be that arteries with $FFR>0.8$ typically contain less plaque and therefore were better characterized by the autoencoders.",
"A comparison of the diagnostic accuracy in the three main coronary arteries (Table REF ) shows that the highest accuracy was achieved for the LAD.",
"This might be due the small number of available training examples for the LCX and RCA.",
"Unlike this study, most previous methods that analyze blood flow [13], [14], [15], [16] for detection of functionally significant stenosis as determined by invasive FFR estimate continuous values of FFR along the coronary artery and localize the functionally significant stenoses using a threshold of 0.8 on the determined FFR [41].",
"However, in preliminary experiments for estimation of continuous FFR values (i.e.",
"with regression) or applying such a threshold, the proposed method has not achieved satisfactory results (refer to Section REF and Fig.",
"REF ).",
"The reason may be threefold.",
"First, unlike other methods [13], [14], [15], [16], the here proposed method analyzes only a single coronary artery at once, while not taking into account the other arteries in the complete coronary artery tree.",
"Analyzing the entire coronary tree might be crucial to differentiate between arteries with functionally significant or non-significant stenoses.",
"Second, as the complete coronary artery is characterized as a whole using the CAEs, spatial information about a specific stenosis can not be retained.",
"Third, the small number of encodings used in this work preserves the coarse shape and morphology of the analyzed artery (see Fig.",
"REF (f)).",
"However, fine morphology might be crucial for differentiating the functionally significant stenoses with FFR measurements in the most difficult range around the FFR of 0.8 [41].",
"Although these methods that analyze blood flow reported better results [13], [14], [15], [16], they are heavily dependent on the accuracy of coronary artery lumen segmentation [21].",
"Highly accurate lumen segmentation is extremely challenging task especially in patients with excessive atherosclerotic calcifications or imaging artefacts or stents [22].",
"As a result, these patients are typically not eligible for such analysis and are excluded[19], [49], [50].",
"In contrast, our method does not require lumen segmentation and therefore heavily diseased patients were not excluded.",
"With a larger data set, future work may further investigate estimation of FFR on the continuous scale, possible performance enhancement when complete coronary artery tree is taken into analysis, and investigate different encoding approaches that may preserve fine morphology of the arteries.",
"To conclude, this study presented an automatic and non-invasive analysis of the coronary arteries in CCTA for detection of patients requiring invasive coronary angiography to establish the need of coronary intervention.",
"The method is based on two disjoint convolutional autoencoders that characterize and encode volumes of complete coronary arteries into a set of encodings.",
"Thereafter, a support vector machine classifier classifies arteries, employing these encodings, according to the presence of abnormal invasively measured FFR.",
"The achieved moderate classification performance shows the feasibility of reducing the number of patients that unnecessarily undergo invasive FFR measurements."
]
] | 1906.04419 |
[
[
"Parallel Scheduled Sampling"
],
[
"Abstract Auto-regressive models are widely used in sequence generation problems.",
"The output sequence is typically generated in a predetermined order, one discrete unit (pixel or word or character) at a time.",
"The models are trained by teacher-forcing where ground-truth history is fed to the model as input, which at test time is replaced by the model prediction.",
"Scheduled Sampling aims to mitigate this discrepancy between train and test time by randomly replacing some discrete units in the history with the model's prediction.",
"While teacher-forced training works well with ML accelerators as the computation can be parallelized across time, Scheduled Sampling involves undesirable sequential processing.",
"In this paper, we introduce a simple technique to parallelize Scheduled Sampling across time.",
"Experimentally, we find the proposed technique leads to equivalent or better performance on image generation, summarization, dialog generation, and translation compared to teacher-forced training.",
"In dialog response generation task, Parallel Scheduled Sampling achieves 1.6 BLEU score (11.5%) improvement over teacher-forcing while in image generation it achieves 20% and 13.8% improvement in Frechet Inception Distance (FID) and Inception Score (IS) respectively.",
"Further, we discuss the effects of different hyper-parameters associated with Scheduled Sampling on the model performance."
],
[
"Introduction",
"Auto-regressive models are a popular choice for generating sequences of any kind including audio [25], images [24], and text [23], [3].",
"Here, the joint probability of the sequence is factorized in a pre-determined order during train and test time.",
"For example, auto-regressive models for text generation factorize the joint probability left-to-right.",
"The text sequence is generated by a decoder network left-to-right, one token (word or word-piece or character) at a time and are widely used in text generation tasks such as summarization [13], machine translation [23] and dialog response generation [2] in the encoder-decoder [3], [23] setting.",
"Such models are typically trained by teacher-forcing [28] where ground-truth history is fed to the model as input, which at test time is replaced by the model prediction.",
"Auto-regressive models applied in any domain suffer from this train-test time discrepancy.",
"Scheduled Sampling [1] aims to mitigate the discrepancy between train and test time in teacher-forcing by randomly replacing some tokens in the history with the model's prediction.",
"More concretely, at a given time step in generating the output sequence, the model is conditioned either on ground-truth or model prediction from the previous time-step with some probability.",
"The probability of selecting model predicted token is gradually increased as training progresses.",
"This procedure potentially allows the model to recover from its own errors, and [1] observe better empirical performance in natural language parsing, image captioning, and speech recognition compared to teacher-forced training.",
"Scheduled Sampling has also been used to get better performance in other tasks such as video prediction [6], knowledge base completion [16] and piano music transcription [22].",
"A key bottleneck in training models with Scheduled Sampling is its inherently sequential nature.",
"Unlike teacher-forcing, tokens must be processed one time-step at a time.",
"The sequential procedure makes Scheduled Sampling impractical for training neural networks, particularly on problems involving long sequence generation.",
"In this work, we describe a simple technique to parallelize Scheduled Sampling.",
"Given an input example, we first generate an entire model prediction sequence in parallel by conditioning on ground-truth history (equivalent to forward-pass of teacher-forcing).",
"Then, we employ a (parallel) mixing step where we generate a new sequence whose token at every time step is either from the model prediction or the ground-truth.",
"Finally, we perform training as in teacher-forcing by conditioning on the sequence obtained from mixing.",
"In Section REF we show that by performing multiple passes of parallel prediction and mixing, we obtain a conditioning sequence that converges to a sample decode from the model.",
"Our key contributions are, A novel, fully parallelizable approach to reducing train-test discrepency of auto-regressive models.",
"We find the method produces the same empirical benefits of Scheduled Sampling while using as little as 0.3% of the training time.",
"We prove equivalence of the proposed approach to Scheduled Sampling under certain choices of hyperparameters.",
"This recovers the clear interpolation between training-time teacher-forcing and test-time decoding described in [1].",
"We extensively evaluate our approach on four auto-regressive tasks in text and image domains.",
"We find that Parallel Scheduled Sampling matches Scheduled Sampling's benefits, takes massively less compute time, and significantly improves performance compared to teacher-forcing on most tasks.",
"In dialog response generation task, Parallel Scheduled Sampling achieves 1.6 BLEU score (11.5%) improvement over teacher-forcing while in image generation it achieves 20% and 13.8% improvement in Frechet Inception Distance (FID) and Inception Score (IS) respectively."
],
[
"Method",
"Our proposed technique can be applied to both conditional and unconditional auto-regressive generative models.",
"For notational simplicity, we consider the task of conditional sequence generation.",
"The training set is given in terms of $N$ input-output sequences $\\lbrace (x^i, y^i) \\rbrace _{i=1}^n$ , where $x^i$ is the input and target $y^i$ is the desired output.",
"The target $y^i$ is a variable-length sequence of $T_i$ tokens (or pixels), $( y^i_1, y^i_2, \\ldots , y^i_{T_i})$ , whereas $x^i$ may be variable-length (as in translation) or fixed-length (as in image captioning).",
"The goal is to learn a model that accurately predicts $y^i$ given $x^i$ .",
"We use $y_{1:t}$ to denote the sequence of tokens $(y_1, y_2, \\ldots , y_t)$ ."
],
[
"Teacher-Forcing and Decoding",
"Given an input $x$ and a target $y$ , the log-probability of the target can be decomposed autoregressively: $P(y | x) &= \\prod _{t=1}^{T} P(y_t | y_{1:t-1}, x)$ Auto-regressive sequence generation models learn to assign high likelihood to token $y_t$ given previous target tokens $y_{1:t-1} = (y_1, \\ldots , y_{t-1})$ and inputs $x$ via a learned likelihood model $p_{\\theta }$ .",
"Neural language models such as RNNs [15] and Transformer [26] adopt left-to-right decomposition while image generation models [24], [18] adopt a raster-scan order.",
"Such models are typically trained with teacher-forcing [28].",
"In teacher-forcing, the log likelihood of the training set is directly maximized, $\\theta ^{*}= \\operatornamewithlimits{argmax}_{\\theta } \\mathcal {L}_{\\text{tf}}(\\theta )= \\operatornamewithlimits{argmax}_{\\theta } \\sum _{i=1}^N \\sum _{t=1}^{T_i} \\log p_{\\theta }(y^i_t | y_{1:t-1}^i, x^i)$ Importantly, teacher-forcing conditions on gold target prefixes $y_{1:t-1}$ , enabling backpropagation through all timesteps with a single pass of inference.",
"At inference time, beam or sample decoding is often used to generate a candidate target $\\hat{y}^i$ .",
"In this regime, target tokens are generated one at a time while conditioning on previously-generated tokens.",
"$\\hat{y}_t \\sim p_{\\theta }(\\hat{y}_t | \\hat{y}_{1:t-1}, x)\\quad \\text{ or } \\quad \\hat{y}_t = \\operatornamewithlimits{argmax}_{y} p_{\\theta }(y | \\hat{y}_{1:t-1}, x)$ A potential failure mode for teacher-forcing-trained models is in conditioning on previously unobserved target prefixes $\\hat{y}_{1:t-1}$ .",
"As the model has not conditioned on these prefixes at training time, it may generate bland, repetitive, or nonsensical candidate targets [9]."
],
[
"Scheduled Sampling",
"Scheduled Sampling [1], hereafter Sequential Scheduled Sampling is a training technique designed to bridge the gap between teacher-forcing and sample decoding.",
"In its simplest form, Sequential Scheduled Sampling generates tokens $\\tilde{y}_{1:t}$ and conditions on these target prefixes during training.",
"Sequential Scheduled Sampling uses the same objective function as teacher-forcing (Equation REF ) except the conditioning tokens $\\tilde{y}_{1:t}$ are a random mixture of gold tokens $y_{1:t}$ and sampled tokens $\\hat{y}_{1:t}$ instead of gold tokens $y_{1:t}$ .",
"See Algorithm REF for implementation.",
"Sequential Scheduled Sampling (single example) timesteps $t = 1, \\ldots , T_i$ Sample $\\hat{y}_t \\sim p_{\\theta }(\\hat{y}_t | \\tilde{y}_{1:t-1}, x)$ Choose next conditioning token, $\\tilde{y}_t = {\\left\\lbrace \\begin{array}{ll}y_t & \\text{with probability $1-p$} \\\\\\hat{y}_t & \\text{with probability $p$} \\\\\\end{array}\\right.",
"}$ Accumulate loss $\\sum _{t} \\log p_{\\theta }(y_t | \\tilde{y}_{1:t-1}, x)$ As $p \\rightarrow 0$ , we condition on $y_{1:t-1}$ as in teacher-forcing, and as $p \\rightarrow 1$ , we condition on $\\hat{y}_{t:t-1}$ as in sample decoding.",
"Typically a schedule will be used to gradually increase $p$ over the course of training.",
"As illustrated in [1], Scheduled Sampling leads to a performance improvement in a variety of language generation tasks.",
"In spite of its benefits, Sequential Scheduled Sampling is inherently a sequential algorithm: choosing conditioning token $\\tilde{y}_t$ requires conditioning autoregressively on tokens $\\tilde{y}_{1:t-1}$ .",
"While this is natural for sequential architectures such as RNNs and LSTMs, it is poorly suited to self-attending feed-forward models such as Transformer where inference for multiple timesteps can be carried out simultaneously."
],
[
"Parallel Scheduled Sampling",
"We propose a natural extension to Sequential Scheduled Sampling called Parallel Scheduled Sampling.",
"Whereas Sequential Scheduled Sampling selects conditioning tokens one after another, we propose generating conditioning tokens for all timesteps in parallel over the course of one or more passes.",
"While this technique requires strictly more operations than Sequential Scheduled Sampling, it is better suited to hardware accelerators such as GPUs and TPUs [10].",
"Moreover, we find in our experiments that only a modest number of passes is necessary for improving model performance.",
"Parallel Scheduled Sampling generates conditioning tokens for all timesteps simultaneously.",
"The procedure consists of multiple passes, each pass consisting of parallel sampling and mixing steps (Figure REF ).",
"In the first pass, the algorithm conditions on gold tokens $y_{1:t}$ , generating tokens $\\hat{y}_{1:t}$ i.i.d.",
"according to $p_{\\theta }(\\hat{y}_{t} | y_{1:t-1}, x)$ .",
"Sampling tokens in the first pass is equivalent to the forward-pass of teacher-forcing.",
"The sampled tokens, $\\hat{y}_{1:t}$ , are mixed (in parallel) with gold tokens, $y_{1:t}$ , to produce conditioning tokens for the next pass, $\\tilde{y}_{1:t}$ .",
"We now describe the multiple-pass procedure.",
"Let $\\hat{y}_{k,1:t}$ , and $\\tilde{y}_{k,1:t}$ denote sampled and mixed tokens respectively on pass $k$ .",
"The mixed tokens from pass $k$ , $\\tilde{y}_{k,1:t}$ , are used for conditioning on pass $k+1$ in place of gold tokens $y_{1:t}$ .",
"Finally, the loss is calculated as before, conditioning on the final mixture of gold and sampled tokens $\\tilde{y}_{K,1:t}$ .",
"See Algorithm REF for implementation.",
"Parallel Scheduled Sampling (single example) Set $\\tilde{y}_{0, t} = y_t$ .",
"passes $k = 1, \\ldots , K$ Sample $\\hat{y}_{k,t} \\sim p_{\\theta }(\\hat{y}_{k,t} | \\tilde{y}_{k-1,1:t-1}, x)$ for all timesteps $t$ in parallel.",
"timesteps $t$ in parallel $t < k$ Copy $\\tilde{y}_{k,t} = \\tilde{y}_{k-1,t}$ Sample $\\tilde{y}_{k,t} = {\\left\\lbrace \\begin{array}{ll}y_t & \\text{with probability $1-p$} \\\\\\hat{y}_{k,t} & \\text{with probability $p$} \\\\\\end{array}\\right.",
"}$ Accumulate loss $\\sum _{t} \\log p_{\\theta }(y_t | \\tilde{y}_{K,1:t-1}, x)$ Finally, we prove that by running the sampling and mixing steps for multiple passes as described in Algorithm REF , the final sample from Parallel Scheduled Sampling converges to a random sample decode from the model when $p=1$ and $K \\ge T$ .",
"Theorem 2.1 Consider a sequence of tokens $z = (z_1, z_2, \\ldots , z_{T})$ of length $T$ .",
"Let $p = 1$ and $K \\ge T$ be fixed.",
"Then the likelihood of $z_{1:T}$ under Parallel Scheduled Sampling's proposal distributionWe drop conditioning on $x$ in the following for conciseness over conditioning tokens on pass $K$ , $q_{\\theta }^{K}(z_{1:T})$ , is identical to random sample decoding's, $p_{\\theta }(z_{1:T})$ , $q_{\\theta }^{K}(z_{1:T}) = p_{\\theta }(z_{1:T})$ We begin by establishing notation.",
"Let $p_{\\theta }(z_{1:T})$ be the likelihood of a sequence $z_{1:T}$ according to random sample decoding.",
"Let $q_{\\theta }^{K}(z_{1:t})$ be the likelihood of the same according to Parallel Scheduled Sampling's proposal distribution on pass $K$ .",
"The proof proceeds by induction.",
"First we show that the proposal distribution for the first token matches random sampling's on the first pass, $q_{\\theta }^{1}(z_{1}) = p_{\\theta }(z_{1})$ .",
"Then we show that if $q_{\\theta }^{K}(z_{1:t}) = p_{\\theta }(z_{1:t})$ holds for some $K$ , it also holds for all $K^{\\prime } > K$ .",
"Finally, we show that if the previous statement holds, it also holds for tokens $z_{ 1:t+1}$ on pass $K+1$ .",
"Thus, it follows that the proposal distribution matches random sampling's for all $T$ tokens so long as $K \\ge T$ .",
"Base Case: Consider the proposal distribution for the first token on the first pass, $z_{1}$ .",
"As $p = 1$ , the first token is sampled from $p_{\\theta }(z_{1})$ by construction.",
"Thus, $q_{\\theta }^{1}(z_{1}) = p_{\\theta }(z_{1})$ Induction over $K$: Suppose that the proposal distribution for tokens $q_{\\theta }^{K}(z_{1:t}) = p_{\\theta }(z_{1:t})$ some $K \\ge t$ .",
"Then the equality also hold for the proposal distribution on pass $K+1$ .",
"This follows trivially as tokens $z_{1:t}$ are “copied” from pass $K$ to $K+1$ and thus their likelihood is unchanged, $q_{\\theta }^{K+1}(z_{1:t})= q_{\\theta }^{K}(z_{1:t})= p_{\\theta }(z_{1:t})$ Induction over $t$: Suppose that the proposal distribution matches random sample decoding's for the first $t$ tokens for $t = K$ ; that is, $q_{\\theta }^{K}(z_{1:t}) = p_{\\theta }(z_{1:t})$ .",
"We show that the statement holds for pass $K+1$ for tokens $z_{1:t+1}$ .",
"First, recall that by construction the proposal distribution for token $z_{t+1}$ given previous tokens $z_{1:t}$ is the same as random sampling's when $t \\ge K$ , $q_{\\theta }^{K+1}(z_{t+1} | z_{1:t}) = p_{\\theta }(z_{t+1} | z_{1:t})$ Note that this only holds when $p = 1$ .",
"Then, $q_{\\theta }^{K+1}(z_{1:t+1})&= q_{\\theta }^{K+1}(z_{t+1} | z_{1:t})q_{\\theta }^{K+1}(z_{1:t}) \\\\&= q_{\\theta }^{K+1}(z_{t+1} | z_{1:t})q_{\\theta }^{K}(z_{1:t}) \\\\&= p_{\\theta }(z_{t+1} | z_{1:t})p_{\\theta }(z_{1:t}) \\\\&= p_{\\theta }(z_{1:t+1})$ Where we use the chain rule, induction over $K$ for $z_{1:t}$ , the inductive assumption for $q_{\\theta }^{K}(z_{1:t})$ , and the definition of $q_{\\theta }^{K+1}(z_{t+1} | z_{1:t})$ when $t \\ge K$ ."
],
[
"Related Work",
"Professor forcing [12] has a similar motivation as Scheduled Sampling, where a discriminator network is trained jointly with the generator to distinguish between generator's hidden states produced by conditioning on ground-truth and model prediction sample.",
"The generator apart from maximizing the likelihood of the data is also trained to fool the discriminator [7].",
"With this new objective, the dynamics of the generator would be the same for conditioning on both ground-truth and model prediction.",
"Our parallel sampling contribution is orthogonal to professor forcing and can be potentially applied in their framework.",
"[4] use beam search which is a sequential search procedure during both during training and testing time, and update the weights of the model using a variant of the Perceptron algorithm [19].",
"Methods with similar motivation of mitigating the discrepancy between train and test time behavior have also been studied in the sequential decision making, and reinforcement learning setting [5], [20]."
],
[
"Experiments",
"We evaluate our proposed technique on image and text domains.",
"In the text domain, we evaluate Parallel Scheduled Sampling on text summarization [13], task-oriented dialog response generation [2], and machine translation [23], [26] and compare it to teacher-forced training.",
"We further evaluate the proposed technique on image generation on CIFAR-10.",
"Since our procedure is intended to generate better sequences, we evaluate it at the sequence level and not at the word token or pixel level.",
"We compare our method to Sequential Scheduled Sampling only on the dialog task [2] as we find runtime infeasible on all other tasks.",
"We also conduct ablation studies on the dialog task.",
"We use the Tensor2Tensor framework for all experiments [27]."
],
[
"Dialog Response Generation",
"We evaluate our method on text response generation task using MultiWOZ [2], a task-oriented dialog dataset.",
"Here, we consider the problem of mapping conversation history consisting of alternating user and assistant turns to a single turn of assistant response.",
"We use a Transformer model containing approximately one million parameters for this study as the dataset is much smaller (approximately 100k training examples) than those in other experiments.",
"We truncate the length of the input and output to 512, and train all the models for 50k steps.",
"As both model and dataset are small, we are able to empirically compare our method to Sequential Scheduled Sampling (such experiments are infeasible in larger models).",
"Table REF summarizes results for all experiments on the MultiWOZ dataset.",
"Both Sequential Scheduled Sampling and Parallel Scheduled Sampling (with just one pass) achieve better results than teacher-forced trained models.",
"However, as can be seen in Table REF , Parallel Scheduled Sampling and teacher-forcing are both two orders of magnitude faster to train than Sequential Scheduled Sampling.",
"A single pass of Parallel Scheduled Sampling is approximately 25% slower than teacher-forced training while producing the benefits of Sequential Scheduled Sampling.",
"Table REF also shows the impact of mixing probability, number of passes, warm-up steps, and the mixing probability schedule [1] on model performance.",
"Overall, we find a single pass with 50% gold/sampled mixing probability sufficient for improving performance.",
"In the best setting, Parallel Scheduled Sampling achieves 1.6 BLEU score (11.5%) improvement over teacher-forcing."
],
[
"Summarization",
"[13] propose a multi-document summarization task, where the task is to generate the text of a Wikipedia article given its references and other related documents.",
"The dataset has close to 1.9 million training examples, and 230,000 test examples.",
"We use a Transformer seq2seq model for this task in two hyper-parameter settings: a base model with 60 million parameters and a large model with 210 million parameters.",
"For the base model, we restrict the maximum length of input and output to be 500, while for the large model the maximum length is set to 1500.",
"Table REF shows the results of training base and large Transformer models for the summarization task.",
"The base and large models were trained for 250k steps and 500k steps respectively.",
"We use teacher-forcing for the first 50% of training steps in Parallel Scheduled Sampling as warm-up steps.",
"The mixing probability is set to 50% and we perform a single pass of sampling and mixing (Algorithm REF ).",
"With the base model, Parallel Scheduled Sampling obtains better performance than teacher-forcing with both beam search and greedy decoding while it performs better only with greedy decoding when the large model is used.",
"Since we use models that are much bigger than the ones in the dialog task discussed before, it is runtime infeasible to apply Sequential Schedule Sampling here."
],
[
"Image Generation",
"In the image domain, we evaluate our method for image generation on the CIFAR-10 dataset.",
"We compare class-conditional Image Transformer [18] trained with teacher-forcing and Parallel Scheduled Sampling.",
"After training, we decode a total of 50,000 randomly-sampled images conditioned on classes drawn from the training set.",
"In additional to a baseline model, we compare all metrics to ground truth samples from the CIFAR-10 training set.",
"We evaluate the image samples using Frechet Inception Distance (FID) [8] and Inception Score (IS) [21] metrics.",
"These metrics have been widely used to evaluate the quality of image samples from GANs [14], [17], [11], [29].",
"Table REF compares teacher-forcing with Parallel Scheduled Sampling on image generation.",
"We train the 50 million parameter Image Transformer model for 200K steps in both the cases.",
"We find that Parallel Scheduled Sampling with a single pass and a mixing probability of 50% significantly decreases FID by 20% and increases IS by 13.8% compared to the baseline.",
"Similarly to our summarization experiment, it is runtime infeasible to apply Sequential Schedule Sampling here."
],
[
"Machine Translation",
"We evaluate our method on the WMT 2014 English-German task which consists of approximately 4.5 million training sentences.",
"We experiment with the large Transformer model that contains approximately 210 million parameters.",
"We did not see performance improvements by using Parallel Scheduled Sampling.",
"The model trained with teacher-forcing for 500k steps gets 28.74 BLEU.",
"The same model trained with 250k warm-up steps using teacher-forcing and the next 250k steps trained with Parallel Scheduled Sampling with mixing probability set to 50% and a single pass of sampling and mixing (Algorithm REF ) obtains 28.57 BLEU.",
"Hyper-parameter tuning of warm-up steps and mixing probability did not improve performance.",
"We hypothesize the lack of performance improvement may be due to the fact that the summarization, dialog response generation and image generation tasks have much longer output sequences than in machine translation, though further investigation is required."
],
[
"Conclusion",
"We introduce a simple technique to parallelize Scheduled Sampling that allows Schedule Sampling to be applied for training models with hundreds of millions of parameters on large datasets.",
"The technique potentially mitigates discrepancy between train and test time in autoregressive sequence generation models.",
"We find that in most cases our technique leads to better empirical performance on summarization, dialog generation, and image generation compared to teacher-forced training.",
"Our empirical results indicate that Parallel Scheduled Sampling can potentially improve the performance of autoregressive sequence generation models particularly on tasks containing long sequences."
]
] | 1906.04331 |
[
[
"Polaron-transformed dissipative Lipkin-Meshkov-Glick Model"
],
[
"Abstract We investigate the Lipkin-Meshkov-Glick model coupled to a thermal bath.",
"Since the isolated model itself exhibits a quantum phase transition, we explore the critical signatures of the open system.",
"Starting from a system-reservoir interaction written in positive definite form, we find that the position of the critical point remains unchanged, in contrast to the popular mean-field prediction.",
"Technically, we employ the polaron transform to be able to study the full crossover regime from the normal to the symmetry-broken phase, which allows us to investigate the fate of quantum-critical points subject to dissipative environments.",
"The signatures of the phase transition are reflected in observables like stationary mode occupation or waiting-time distributions."
],
[
"Introduction",
"In closed systems, Quantum Phase Transitions (QPTs) are defined as non-analytic changes of the ground state energy when a control parameter other than temperature is varied across a critical point [1].",
"They are accompanied by non-analytic changes in observables or correlation functions [2], [3], [4] and form a fascinating research area on their own.",
"Nowadays, it is possible to study such QPTs in experimental setups with cold atoms [5], [6], [7], [8], [9], which provide high degree of control and allow to test theoretical predictions.",
"However, each experimental set-up is an open system, such that the impact of the reservoir on the QPT should not be neglected.",
"To the contrary, the presence of a reservoir can fundamentally change the nature of the QPT.",
"For example, in the famous Dicke phase transition, it is the presence of the reservoir that actually creates a QPT via the environmental coupling of a collective spin [10].",
"With the renewed interest in quantum thermodynamics, it has become a relevant question whether QPTs can be put to use e.g.",
"as working fluids of quantum heat engines [11], [12], [13], [14].",
"This opens another broad research area of dissipative QPTs in non-equilibrium setups.",
"Here, the non-equilibrium configuration can be implemented in different ways, e.g.",
"by periodic driving [15], [16], [17], by quenching [18], [19], [20], by coupling to reservoirs [21], [22], [23] or by a combination of these approaches [24], [25].",
"One has even considered feedback control of such quantum-critical systems [26], [27], [28], [29], [30].",
"All these extensions should however be applied in combination with a reliable microscopic description of the system-reservoir interaction.",
"For example, in the usual derivation of Lindblad master equations one assumes that the system-reservoir interaction is weak compared to the splitting of the system energy levels [21], [31].",
"In particular in the vicinity of a QPT – where the energy gap above the ground state vanishes – this condition cannot be maintained.",
"Therefore, while in particular the application of the secular approximation leads to a Lindblad-type master equation preserving the density matrix properties, it has the disadvantage that its range of validity is typically limited to non-critical points or to finite-size scaling investigations [32], [33].",
"In principle, the weak-coupling restriction can be overcome with different methods such as e.g.",
"reaction-coordinate mappings [34], [35], [36].",
"These however come at the price of increasing the dimension of the system, which renders analytic treatments of already complex systems difficult.",
"In this paper, we are going to study at the example of the Lipkin-Meshkov-Glick (LMG) model how a QPT is turned dissipative by coupling the LMG system [37] to a large environment.",
"To avoid the aforementioned problems, we use a polaron [38], [39], [40], [41], [42] method, which allows to address the strong coupling regime [43], [44], [45], [46], [34], [47], [48], [49] without increasing the number of degrees of freedom that need explicit treatment.",
"In particular, we show that for our model the position of the QPT is robust in presence of dissipation.",
"We emphasize that the absence of a reservoir-induced shift – in contrast to mean-field-predictions [50], [51], [23], [52], [53], [54], [55] – is connected with starting from a Hamiltonian with a lower spectral bound and holds without additional approximation.",
"Our work is structured as follows.",
"In Sec.",
"we introduce the dissipative LMG model, in Sec.",
"we show how to diagonalize it globally using the Holstein-Primakoff transformation.",
"There, we also derive a master equation in both, original and polaron, frames and show that the QPT cannot be modeled within the first and that the QPT position is not shifted within the latter approach.",
"Finally, we discuss the effects near the QPT by investigating the excitations in the LMG system and the waiting time distribution of emitted bosons in Sec. .",
"The isolated LMG model describes the collective interaction of $N$ two-level systems with an external field and among themselves.",
"In terms of the collective spin operators $J_\\nu = \\frac{1}{2}\\sum _{m=1}^N \\sigma _\\nu ^{(m)}\\,,\\qquad \\nu \\in \\lbrace x,y,z\\rbrace $ and $J_\\pm = J_x \\pm {\\rm i} \\cdot J_y$ with $\\sigma _\\nu ^{(m)}$ denoting the Pauli matrix of the $m$ th spin, the anisotropic LMG Hamiltonian reads [56] $H_{\\rm LMG}(h,\\gamma _x) = -h J_z - \\frac{\\gamma _x}{N} J_x^2\\,,$ where $h$ is the strength of a magnetic field in $z$ direction and $\\gamma _x$ is the coupling strength between each pair of two-level systems.",
"As such, it can be considered a quantum generalization of the Curie-Weiss model [57].",
"Throughout this paper, we consider only the subspace with the maximum angular momentum $j=\\frac{N}{2}$ , where the eigenvalues of the angular momentum operator $J^2 = J_x^2 + J_y^2 + J_z^2$ are given by $j(j+1)$ .",
"Studies of the LMG model are interesting not only due to its origin in the nuclear context [58], [37], [59], but also due to its experimental realization with cold atoms and high possibility of control [8].",
"In particular the existence of a QPT at $\\gamma _x^{\\rm cr} = h$ with a non-analytic ground-state energy density has raised the interest in the community [60], [61], [62], [63]: For $\\gamma _x < \\gamma _x^{\\rm cr}$ , the system has a unique ground state, which we denote as the normal phase further-on.",
"In contrast, for $\\gamma _x > \\gamma _x^{\\rm cr}$ it exhibits a symmetry-broken phase [2], [64], where e.g.",
"the eigenvalues become pairwise degenerate and the $J_z$ -expectation exhibits a bifurcation [65], [19].",
"Strictly speaking, the QPT is found only in the thermodynamic limit (for $N \\rightarrow \\infty $ ), for finite sizes $N$ smoothing effects in the QPT signatures will appear [66], [67], [68].",
"Here, we want to investigate the LMG model embedded in an environment of bosonic oscillators $c_k$ with frequencies $\\nu _k$ .",
"The simplest nontrivial embedding preserves the conservation of the total angular momentum and allows for energy exchange between system and reservoir.",
"Here, we constrain ourselves for simplicity to the case of a $J^x$ coupling.",
"Furthermore, to ensure that the Hamiltonian has a lower spectral bound for all values of the system-reservoir coupling strength, we write the interaction in terms of a positive operator $H_{\\rm tot} &= H_{\\rm LMG}(h,\\gamma _x)\\\\&\\;+\\sum _k \\nu _k \\left( c_k^\\dag + \\frac{g_k}{\\sqrt{N} \\nu _k}J_x \\right)\\left( c_k + \\frac{g_k}{\\sqrt{N} \\nu _k}J_x \\right)\\,.$ Here, $g_k > 0$ represent emission/absorption amplitudes (a possible phase can be absorbed in the bosonic operators), and the factor $N^{-1/2}$ needs to be included to obtain a meaningful thermodynamic limit $N \\rightarrow \\infty $ , but can also be motivated from the scaling of the quantization volume $V \\propto N$ .",
"Since the LMG Hamiltonian has a lower bound, the spectrum of this Hamiltonian $H_{\\rm tot}$ is (for finite $N$ ) then bounded from below for all values of the coupling strength $g_k$ .",
"Upon expansion and sorting spin and bosonic operators, this form implicates an effective rescaling of the system Hamiltonian $H_{\\rm LMG}(h,\\tilde{\\gamma }_x)$ with a renormalized spin-spin interaction $\\tilde{\\gamma }_x = \\gamma _x - \\sum _k \\frac{g_k^2}{\\nu _k}\\,,$ which indeed leads to a shift of the critical point within a naive treatment."
],
[
"Local LMG diagonalization",
"In the thermodynamic limit Eq.",
"(REF ) can be diagonalized using the Holstein-Primakoff transform which maps collective spins to bosonic operators $b$ [69], [70], [23] $J_+ &= \\sqrt{N - b^\\dag b} b\\,, \\qquad J_- = b^\\dag \\sqrt{N - b^\\dag b}\\,,\\\\J_z &= \\frac{N}{2} - b^\\dag b\\,.$ However, to capture both phases of the LMG Hamiltonian, one has to account for the macroscopically populated ground state in the symmetry-broken phase.",
"This can be included with the displacement $b = \\sqrt{N}\\alpha + a$ with complex $\\alpha $ in Eq.",
"(REF ), where $N\\left| \\alpha \\right|^2$ is the classical mean-field population of the mode [70], [23], [62] and $a$ is another bosonic annihilation operator.",
"The next step is then to expand for either phase Eq.",
"(REF ) with the inserted transformation (REF ) in terms of $1/\\sqrt{N}$ for $N\\gg 1$ – see App.",
"– which yields a decomposition of the Hamiltonian $H_{\\rm LMG}^{\\rm HP}(h,\\gamma _x) &= N \\cdot H_0^{\\rm HP} + \\sqrt{N} H_1^{\\rm HP} + H_2^{\\rm HP}\\\\&\\qquad + {\\mathcal {O}}\\left(\\frac{1}{\\sqrt{N}}\\right)\\,,$ with individual terms depending on the phase $H_0^{\\rm HP} &= {\\left\\lbrace \\begin{array}{ll}-\\frac{h}{2} &: \\gamma _x < \\gamma _x^{\\rm cr}\\\\-\\frac{h^2 + \\gamma _x^2}{4\\gamma _x} &: \\gamma _x > \\gamma _x^{\\rm cr}\\end{array}\\right.",
"}\\,,\\\\H_1^{\\rm HP} &\\stackrel{!",
"}{=} {\\left\\lbrace \\begin{array}{ll}0 &: \\gamma _x < \\gamma _x^{\\rm cr}\\\\0 &: \\gamma _x > \\gamma _x^{\\rm cr}\\end{array}\\right.",
"}\\,,\\\\H_2^{\\rm HP} &={\\left\\lbrace \\begin{array}{ll}(h - \\frac{\\gamma _x}{2})a^\\dag a - \\frac{\\gamma _x}{4} (a^2 + {a^\\dag }^2)-\\frac{\\gamma _x}{4} &: \\gamma _x < \\gamma _x^{\\rm cr}\\\\+\\frac{5 \\gamma _x - 3 h}{4} a^\\dagger a+\\frac{3 \\gamma _x - 5 h}{8} \\left(a^2 + {a^\\dag }^2\\right) &: \\gamma _x > \\gamma _x^{\\rm cr}\\\\\\qquad {+\\frac{\\gamma _x - 3 h}{8}}\\end{array}\\right.",
"}\\,.$ We demand in both phases that $H_1^{\\rm HP}$ is always zero.",
"Technically, this enforces that only terms quadratic in the creation and annihilation operators occur in the Hamiltonian.",
"Physically, this enforces that we expand around the correct ground state, i.e., in the final basis, the ground state is the state with a vanishing quasiparticle number.",
"This requirement is trivially fulfilled in the normal phase with $\\alpha =0$ but requires a finite real value of the mean-field $\\alpha $ in the symmetry-broken phase [70], [23], [62], altogether leading to a phase-dependent displacement $\\alpha (h,\\gamma _x) = \\sqrt{\\frac{1}{2} \\left(1 - \\frac{h}{\\gamma _x}\\right)} \\Theta (\\gamma _x - h)\\,,$ which approximates $H_{\\rm LMG}^{\\rm HP}$ by a harmonic oscillator near its ground state.",
"Here we note that $-\\alpha (h,\\gamma _x)$ is also a solution.",
"The mean-field expectation value already allows to see the signature of the phase transition in the closed LMG model at $\\gamma _x = h$ , since $\\alpha $ is only finite for $\\gamma _x > h$ and is zero elsewhere.",
"Since up to corrections that vanish in the thermodynamic limit, the Hamiltonian defined by Eq.",
"(REF ) is quadratic in $a$ , it can in either phase be diagonalized by a rotation of the old operators $a=\\cosh (\\varphi ) d+\\sinh (\\varphi ) d^\\dagger $ with $\\varphi \\in \\mathbb {R}$ to new bosonic operators $d$ .",
"The system Hamiltonian $H_{\\rm LMG}^{\\rm HP}$ then transforms into a single harmonic oscillator, where the frequency $\\omega $ and ground state energy are functions of $h$ and $\\gamma _x$ $H_{\\rm LMG}^{\\rm HP}(h,\\gamma _x) &= \\omega (h,\\gamma _x) d^\\dag d + C_2(h,\\gamma _x)\\\\&\\qquad - N \\cdot C_1(h,\\gamma _x)+ {\\mathcal {O}}\\left(\\frac{1}{\\sqrt{N}}\\right)\\,.$ The actual values of the excitation energies $\\omega (h,\\gamma _x)$ and the constants $C_i(h,\\gamma _x)$ are summarized in table REF .",
"Table: Parameters of the diagonalization procedure of the LMG model H LMG (h,γ x )H_{\\rm LMG}(h,\\gamma _x) for the normal phase (γ x <h\\gamma _x<h, second column) and for the symmetry-broken phase (γ x >h\\gamma _x>h, last column).In both phases, the dd operators correspond to fluctuations around the mean-field value α\\alpha , which is zero only in the normal phase.Fig.",
"REF confirms that the thus obtained spectra from the bosonic representation agree well with finite-size numerical diagonalization when $N$ is large enough.",
"Figure: Lower part of the isolated LMG model spectrum for finite-size numerical diagonalization of Eq.",
"() (thin curves) and using the bosonic representation (bold curves) based on Eq.",
"() for the three lowest energies.For large NN, the spectra are nearly indistinguishable.In the symmetry-broken phase (right), two numerical eigenvalues approach the same oscillator solution.These correspond to the two different parity sectors, formally represented by two possible displacement solutions ±α(h,γ x )\\pm \\alpha (h,\\gamma _x) in Eq.",
"().First, one observes for consistency that the trivial spectra deeply in the normal phase ($\\gamma _x \\approx 0$ ) or deeply in the symmetry-broken phase ($h \\approx 0$ ) are reproduced.",
"In addition, we see that at the QPT $\\gamma _x = \\gamma _x^{\\rm cr}=h$ , the excitation frequency $\\omega $ vanishes as expected, which is also reflected e.g.",
"in the dashed curve in Fig.",
"REF (a).",
"For consistency, we also mention that all oscillator energies $E_n$ are continuous at the critical point $\\gamma =h$ .",
"Furthermore, the second derivative with respect to $\\gamma _x$ of the continuum ground state energy per spin $\\lim _{N\\rightarrow \\infty } E_0/N$ is discontinuous at the critical point, classifying the phase transition as second order.",
"Finally, we note that this treatment does not capture the excited state quantum phase transitions present in the LMG model as we are only concerned with the lower part of the spectrum."
],
[
"Master Equation",
"We first perform the derivation of the conventional Born-Markov-secular (BMS) master equation in the usual way, starting directly with Eq.",
"(REF ) [23], [22], [71].",
"Afterwards, we show that a polaron transform also allows to treat regions near the critical point."
],
[
"Conventional BMS master equation",
"The conventional BMS master equation is derived in the energy eigenbasis of the system, i.e., the LMG model with renormalized spin-spin interaction $\\tilde{\\gamma }_x$ , in order to facilitate the secular approximation.",
"In this eigenbasis the master equation has a particularly simple form.",
"Applying the very same transformations (that diagonalize the closed LMG model) to its open version (REF ), we arrive at the generic form $H_{\\rm tot}^{\\rm HP} &= H_{\\rm LMG}^{\\rm HP}(h,\\tilde{\\gamma }_x) + \\sum \\nu _k c_k^\\dag c_k\\\\&\\qquad + \\left[A(h,\\tilde{\\gamma }_x) (d + d^\\dag ) + \\sqrt{N} Q(h,\\tilde{\\gamma }_x)\\right] \\times \\\\&\\qquad \\qquad \\times \\sum _k g_k (c_k + c_k^\\dagger )\\,,$ where we note that the LMG Hamiltonian is now evaluated at the shifted interaction (REF ).",
"The phase-dependent numbers $A$ and $Q$ are defined in Table REF .",
"Table: Additional parameters of the diagonalization procedure for the derivation of the master equationin the original frame for the normal phase (γ ˜ x <h\\tilde{\\gamma }_x < h, second column) and for the symmetry-broken phase (γ ˜ x >h\\tilde{\\gamma }_x > h, last column).Note that as compared to the closed model in Tab.",
", functions are evaluated at the shifted interaction ().In particular, in the normal phase we have $Q=0$ , and we recover the standard problem of a harmonic oscillator weakly coupled to a thermal reservoir.",
"In the symmetry-broken phase we have $Q \\ne 0$ , such that the shift term in the interaction Hamiltonian formally diverges as $N\\rightarrow \\infty $ , and a naive perturbative treatment does not apply.",
"Some thought however shows, that this term can be transformed away by applying yet another displacement for both system and reservoir modes $d \\rightarrow d + \\sigma $ and $c_k \\rightarrow c_k + \\sigma _k$ with $\\sigma ,\\sigma _k\\in \\mathbb {C}$ chosen such that all terms linear in creation and annihilation operators vanish in the total Hamiltonian.",
"This procedure does not change the energies of neither system nor bath operators, such that eventually, the master equation in the symmetry-broken phase is formally equivalent to the one in the normal phase, and the interaction proportional to $Q$ is not problematic.",
"Still, when one approaches the critical point from either side, the system spacing $\\omega $ closes in the thermodynamic limit, which makes the interaction Hamiltonian at some point equivalent or even stronger than the system Hamiltonian.",
"Even worse, one can see that simultaneously, the factor $A \\sim e^{+\\varphi }$ in the interaction Hamiltonian diverges at the critical point, such that a perturbative treatment is not applicable there.",
"Therefore, one should consider the results of the naive master equation in the thermodynamic limit $N\\rightarrow \\infty $ with caution.",
"The absence of a microscopically derived master equation near the critical point is a major obstacle in understanding the fate of quantum criticality in open systems.",
"Ignoring these problems, one obtains a master equation having the standard form for a harmonic oscillator coupled to a thermal reservoir $\\dot{\\rho }(t) &= - {\\rm i}\\left[H_{\\rm LMG}^{\\rm HP}(h,\\tilde{\\gamma }_x),\\rho \\right]+ F_e \\mathcal {D}(d)\\rho + F_a \\mathcal {D}(d^\\dag )\\rho \\,,\\\\F_e &= A^2(h,\\tilde{\\gamma }_x) \\Gamma (\\omega (h,\\tilde{\\gamma }_x)) [1 + n_B(\\omega (h,\\tilde{\\gamma }_x)]\\,,\\\\F_a &= A^2(h,\\tilde{\\gamma }_x) \\Gamma (\\omega (h,\\tilde{\\gamma }_x)) n_B(\\omega (h,\\tilde{\\gamma }_x))\\,.$ Here, we have used the superoperator notation $\\mathcal {D}(O)\\rho \\hat{=} O \\rho O^\\dag - \\frac{1}{2} \\rho O^\\dag O -\\frac{1}{2} O^\\dag O \\rho $ for any operator $O$ and $\\Gamma (\\omega ) = 2 \\pi \\sum _k g_k^2 \\delta (\\omega - \\nu _k)$ is the original spectral density of the reservoir, and $n_B(\\omega )=[e^{\\beta \\omega }-1]^{-1}$ is the Bose distribution with inverse reservoir temperature $\\beta $ .",
"These functions are evaluated at the system transition frequency $\\omega (h,\\tilde{\\gamma }_x)$ .",
"The master equation has the spontaneous and stimulated emission terms in $F_e$ and the absorption term in $F_a$ , and due to the balanced Bose-Einstein function these will at steady state just thermalize the system at the reservoir temperature, as is generically found for such BMS master equations.",
"Note that $H_{\\rm LMG}^{\\rm HP}$ from Eq.",
"(REF ) is evaluated at the rescaled coupling $\\tilde{\\gamma }_x$ .",
"Therefore, the position of the QPT is at $\\tilde{\\gamma }_x^{\\rm cr} = h$ and shifted to higher $\\gamma _x$ couplings, see (REF ).",
"Similar shifts of the QPT position in dissipative quantum optical models are known e.g.",
"from mean-field treatments [50], [72].",
"However, here we emphasize that we observe them as a direct consequence of ignoring the divergence of interaction around the phase transition in combination with positive-definite form of the initial total Hamiltonian Eq.",
"(REF )."
],
[
"Polaron master equation",
"In this section, we apply a unitary polaron transform to the complete model, which has for other (non-critical) models been used to investigate the full regime of system-reservoir coupling strengths [73], [74].",
"We will see that for a critical model, it can – while still bounded in the total coupling strength – be used to explore the systems behaviour at the QPT position."
],
[
"Polaron transform",
"We choose the following polaron transform $U_p$ $U_p = e^{-J_x \\hat{B}}\\,,\\qquad \\hat{B} = \\frac{1}{\\sqrt{N}} \\sum _{k} \\frac{g_k}{\\nu _k} \\left(c_k^\\dag - c_k\\right)\\,.$ The total Hamiltonian (REF ) in the polaron frame then becomes $\\bar{H}_{\\rm tot} &= U_p^\\dag H_{\\rm tot} U_p\\\\&= - h D \\cdot J_z - \\frac{\\gamma _x}{N} J_x^2+ \\sum _k \\nu _k c_k^\\dag c_k\\\\&\\qquad - h \\cdot \\left[J_z \\cdot \\left(\\cosh (\\hat{B}) - D\\right) - {\\rm i}J_y \\sinh (\\hat{B})\\right]\\,.",
"$ Here, $\\gamma _x$ is the original interaction of the local LMG model, and the renormalization of the external field $D$ is defined via $D &= \\left< \\cosh (\\hat{B}) \\right> = {\\rm Tr}\\left\\lbrace \\cosh (\\hat{B}) \\frac{e^{-\\beta \\sum _k \\nu _k c_k^\\dagger c_k}}{{\\rm Tr}\\left\\lbrace e^{-\\beta \\sum _k \\nu _k c_k^\\dagger c_k} \\right\\rbrace } \\right\\rbrace \\\\&= \\exp \\left[-\\frac{1}{N}\\sum _k \\left(\\frac{g_k}{\\nu _k}\\right)^2 \\left(n_k + \\frac{1}{2}\\right) \\right] > 0\\,,\\\\n_k &= \\frac{1}{e^{\\beta \\nu _k} - 1}\\,.$ It has been introduced to enforce that the expectation value of the system-bath coupling vanishes for the thermal reservoir state.",
"More details on the derivation of Eq.",
"(REF ) are presented in App .",
"The operator $\\hat{B}\\propto \\frac{1}{\\sqrt{N}}$ decays in the thermodynamic limit, such that for these studies, only the first few terms in the expansions of the $\\sinh (\\hat{B})$ and $\\cosh (\\hat{B})$ terms need to be considered.",
"Accordingly, the position of the QPT in the polaron frame is now found at the QPT of the closed model $\\gamma _x^{\\rm cr} = h D\\stackrel{N\\rightarrow \\infty }{\\rightarrow } h\\,.$ Here, we have with $D\\rightarrow 1$ implicitly assumed that the thermodynamic limit is performed in the system first.",
"If a spectral density is chosen that vanishes faster than quadratically for small frequencies, the above replacement holds unconditionally (see below).",
"We emphasize again we observe the absence of a QPT shift as a result of a proper system-reservoir interaction with a lower spectral bound.",
"Without such an initial Hamiltonian, the reservoir back-action would shift the dissipative QPT [50], [72].",
"For the study of strong coupling regimes, polaron transforms have also been applied e.g.",
"to single spin systems [73] and collective non-critical spin systems [74].",
"Treatments without a polaron transformation should be possible in our case too, by rewriting Eq.",
"(REF ) in terms of reaction coordinates [75], [36], [35], leading to an open Dicke-type model.",
"In the thermodynamic limit, we can use that the spin operators $J_\\nu $ scale at worst linearly in $N$ to expand the interaction and $D$ , yielding $\\bar{H}_{\\rm tot} &\\approx - h \\left[1-\\frac{1}{N} \\delta \\right] \\cdot J_z - \\frac{\\gamma _x}{N} J_x^2+ \\sum _k \\nu _k c_k^\\dag c_k\\\\&\\qquad - h \\cdot \\left[\\frac{J_z}{N} \\left(\\frac{1}{2} \\bar{B}^2 + \\delta \\right) - {\\rm i}\\frac{J_y}{\\sqrt{N}} \\bar{B}\\right]\\\\&= -h J_z - \\frac{\\gamma _x}{N} J_x^2 + \\sum _k \\nu _k c_k^\\dag c_k\\\\&\\qquad - h \\cdot \\left[\\frac{J_z}{N} \\frac{1}{2} \\bar{B}^2 - {\\rm i}\\frac{J_y}{\\sqrt{N}} \\bar{B}\\right]\\,,$ where $\\bar{B} = \\sqrt{N} \\hat{B}$ and $D \\equiv e^{-\\frac{\\delta }{N}}$ has been used.",
"As in the thermodynamic limit, $J_z/N$ just yields a constant, the first term in the last row can be seen as an all-to-all interaction between the environmental oscillators, which only depends in a bounded fashion on the LMG parameters $h$ and $\\gamma _x$ .",
"Since it is quadratic, it can be formally transformed away by a suitable global Bogoliubov transform $c_k = \\sum _q (u_{kq} b_q + v_{kq} b_q^\\dagger )$ of all reservoir oscillators, which results in $\\bar{H}_{\\rm tot} &\\approx -h J_z - \\frac{\\gamma _x}{N} J_x^2 + \\sum _k \\tilde{\\nu }_k b_k^\\dag b_k\\\\&\\qquad + h \\frac{{\\rm i}J_y}{\\sqrt{N}} \\sum _k \\left(h_k b_k - h_k^* b_k^\\dagger \\right)\\,,$ and where $h_k \\in \\mathbb {C}$ are the transformed reservoir couplings and the $\\tilde{\\nu }_k$ the transformed reservoir energies.",
"In case of weak coupling to the reservoir which is assumed here however, we will simply neglect the $\\bar{B}^2$ -term since it is then much smaller than the linear $\\bar{B}$ term."
],
[
"System Hamiltonian diagonalization",
"To proceed, we first consider the normal phase $\\gamma _x < h$ .",
"We first apply the Holstein-Primakoff transformation to the total Hamiltonian, compare appendix .",
"Since in the normal phase the vanishing displacement implies $a=b$ , this yields $\\bar{H}_{\\rm tot, N}^{\\rm (HP)} &= - \\frac{h}{2} N + \\left(h-\\frac{\\gamma _x}{2}\\right) a^\\dag a -\\frac{\\gamma _x}{4} ({a^\\dag }^2 + a^2+1)\\\\&\\quad + \\sum _k \\tilde{\\nu }_k b_k^\\dag b_k + \\frac{h}{2}(a - a^\\dag ) \\sum _k \\left(h_k b_k - h_k^* b_k^\\dagger \\right)\\,.$ Here, the main difference is that the system-reservoir interaction now couples to the momentum of the LMG oscillator mode and not the position.",
"Applying yet another Bogoliubov transform $a = \\cosh (\\varphi (h,\\gamma _x)) d + \\sinh (\\varphi (h,\\gamma _x)) d^\\dagger $ with the same parameters as in table REF eventually yields a Hamiltonian of a single diagonalized oscillator coupled via its momentum to a reservoir.",
"Analogously, the symmetry-broken phase $\\gamma _x > h$ is treated with a finite displacement as outlined in App. .",
"The requirement, that in the system Hamiltonian all terms proportional to $\\sqrt{N}$ should vanish, yields the same known displacement (REF ).",
"One arrives at a Hamiltonian of the form $\\bar{H}_{\\rm tot, S}^{\\rm (HP)} &= -\\frac{h^2+\\gamma _x^2}{4\\gamma _x} N +\\frac{5 \\gamma _x - 3 h}{4} a^\\dagger a\\\\&\\qquad +\\frac{3 \\gamma _x - 5 h}{8} \\left(a^2 + {a^\\dag }^2\\right)+\\frac{\\gamma _x - 3 h}{8}+ \\sum _k \\tilde{\\nu }_k b_k^\\dag b_k\\\\&\\qquad + \\frac{h}{2} \\sqrt{1-\\left| \\alpha (h,\\gamma _x) \\right|^2} (a-a^\\dagger ) \\sum _k (h_k b_k - h_k^* b_k^\\dagger )\\,.$ Using a Bogoliubov transformation to new bosonic operators $d$ the system part in the above equation can be diagonalized again.",
"Thus, in both phases the Hamiltonian acquires the generic form $\\bar{H}_{\\rm tot}^{\\rm (HP)} &= \\omega (h,\\gamma _x) d^\\dagger d - N C_1(h,\\gamma _x) + C_2(h,\\gamma _x)\\\\&\\qquad + \\bar{A}(h,\\gamma _x) (d-d^\\dagger ) \\sum _k \\left(h_k b_k - h_k^* b_k^\\dagger \\right)\\\\&\\qquad + \\sum _k \\tilde{\\nu }_k b_k^\\dag b_k\\,,$ where the system-reservoir coupling modification $\\bar{A}(h,\\gamma _x)$ is found in Tab.",
"REF .",
"Table: Additional parameters of the diagonalization procedure of H LMG H_{\\rm LMG} in the polaron frame for the normal phase (γ x <h\\gamma _x < h, second column) and symmetry broken phase (γ x >h\\gamma _x > h, last column).Note that ϕ(h,γ x )\\varphi (h,\\gamma _x) – see Tab.",
"– is evaluated at the original spin-spin coupling γ x \\gamma _x.To this form, we can directly apply the derivation of the standard quantum-optical master equation."
],
[
"Master Equation",
"In the polaron-transformed interaction Hamiltonian, we do now observe the factor $\\bar{A}(h,\\gamma _x)$ , which depends on $h$ and $\\gamma _x$ , see tables REF and REF .",
"This factor is suppressed as one approaches the shifted critical point, it vanishes there identically.",
"Near the shifted QPT, its square $\\bar{A}^2(h,\\gamma _x)$ shows the same scaling behaviour as the system gap $\\omega (h,\\gamma _x)$ , such that in the polaron frame, the system-reservoir interaction strength is adaptively scaled down with the system Hamiltonian, and a naive master equation approach can be applied in this frame.",
"From either the normal phase or the symmetry-broken phase we arrive at the following generic form of the system density matrix master equation $\\dot{\\rho }(t) &= - {\\rm i}\\left[H_{\\rm LMG}^{\\rm HP}(h,\\gamma _x),\\rho \\right]+ \\bar{F}_e \\mathcal {D}(d)\\rho + \\bar{F}_a \\mathcal {D}(d^\\dag )\\rho \\,,\\\\\\bar{F}_e &= \\bar{A}^2(h,\\gamma _x) \\bar{\\Gamma }(\\omega (h,\\gamma _x)) [1 + n_B(\\omega (h,\\gamma _x))]\\,,\\\\\\bar{F}_a &= \\bar{A}^2(h,\\gamma _x) \\bar{\\Gamma }(\\omega (h,\\gamma _x)) n_B(\\omega (h,\\gamma _x))\\,.$ Here, $\\bar{\\Gamma }(\\omega ) = 2 \\pi \\sum _k \\left| h_k \\right|^2 \\delta (\\omega - \\tilde{\\nu }_k)$ denotes the transformed spectral density, which is related to the original spectral density via the Bogoliubov transform that expresses the $c_k$ operators in terms of the $b_k$ operators, and $n_B(\\omega )$ again denotes the Bose distribution.",
"The mapping from the reservoir modes $c_k$ to the new reservoir modes $b_k$ has been represented in an implicit form, but in general it will be a general multi-mode Bogoliubov transformation [76], [77] with a sophisticated solution.",
"However, if $h g_k/\\nu _k$ is small in comparison to the reservoir frequencies $\\nu _k$ , the Bogoliubov transform will hardly change the reservoir oscillators and thereby be close to the identity.",
"Then, one will approximately recover $\\bar{\\Gamma }(\\omega ) \\approx \\Gamma (\\omega )$ .",
"Even if this assumption is not fulfilled, we note from the general form of the master equation that the steady state will just be the thermalized system – with renormalized parameters depending on $\\Gamma (\\omega )$ , $h$ , and $\\gamma _x$ .",
"Therefore, it will not depend on the structure of $\\bar{\\Gamma }(\\omega )$ – although transient observables may depend on this transformed spectral density as well.",
"In our results, we will therefore concentrate on a particular form of $\\Gamma (\\omega )$ only and neglect the implications for $\\bar{\\Gamma }(\\omega )$ ."
],
[
"Results",
"To apply the polaron transform method, we require that all involved limits converge.",
"All reasonable choices for a spectral density (REF ) will lead to convergence of the renormalized spin-spin interaction (REF ).",
"However, convergence of the external field renormalization (REF ) may require subtle discussions on the order of the thermodynamic limits in system ($N\\rightarrow \\infty $ ) and reservoir ($\\sum _k g_k^2 [\\ldots ] \\rightarrow \\frac{1}{2\\pi }\\int \\Gamma (\\omega )[\\ldots ]d\\omega $ ), respectively.",
"These discussions can be avoided if the spectral density grows faster than quadratically for small energies, e.g.",
"$\\Gamma (\\omega ) =\\eta \\frac{\\omega ^3}{\\omega _c^2} \\cdot \\exp (-\\omega /\\omega _c)\\,,$ where $\\omega _c$ is a cutoff frequency and $\\eta $ is a dimensionless coupling strength.",
"With this choice, the renormalized all-to-all interaction (REF ) becomes $\\tilde{\\gamma }_x = \\gamma _x - \\frac{\\eta \\cdot \\omega _c}{\\pi }\\,,$ such that the QPT position Eq.",
"(REF ) is shifted to $\\gamma _x^{\\rm cr} \\rightarrow h + \\frac{\\eta \\cdot \\omega _c}{\\pi }$ .",
"We emphasize again that – independent of the spectral density – both derived master equations Eq.",
"(REF ) and (REF ) let the system evolve towards the respective thermal state $\\rho = \\frac{\\exp (-\\beta H_{\\rm LMG}^{\\rm HP}(h,\\tilde{\\gamma }_x))}{Z}\\,,\\;\\;\\bar{\\rho }= \\frac{\\exp (-\\beta H_{\\rm LMG}^{\\rm HP}(h,\\gamma _x))}{\\bar{Z}}\\,,$ in the original and polaron frame, respectively, where $\\beta $ is the inverse temperature of the bath and $Z/\\bar{Z}$ are the respective normalization constants.",
"The difference between the treatments is therefore that within the BMS treatment (REF ) the rates may diverge and that the system parameters are renormalized.",
"The divergence of rates within the BMS treatment would also occur for a standard initial Hamiltonian.",
"To illustrate this main result, we discuss a number of conclusions can be derived from it below."
],
[
"Magnetization",
"In general, the role of temperature in connection with the thermal phase transition in models like LMG or Dicke has been widely studied using partition sums or by using naive BMS master equations [78], [79], [80], [81].",
"Since in our case the stationary system state is just the thermalized one, standard methods (compare Appendix ) just analyzing the canonical Gibbs state of the isolated LMG model can be used to obtain stationary expectation values such as e.g.",
"the magnetization.",
"For the polaron approach we obtain $\\left< J^z \\right> = -\\frac{\\partial E_0(h,\\gamma _x)}{\\partial h} - \\frac{1}{e^{\\beta \\omega (h,\\gamma _x)}-1} \\frac{\\partial \\omega (h,\\gamma _x)}{\\partial h}\\,,$ where $E_0(h,\\gamma _x)=C_2(h,\\gamma _x)-N C_1(h,\\gamma _x)$ is the ground state energy and $\\omega (h,\\gamma _x)$ the energy splitting, compare Tab.",
"REF .",
"The quantum-critical nature is demonstrated by the first (ground state) contribution, where the nonanalytic dependence of the ground state energy on the external field strength will map to the magnetization.",
"The second contribution is temperature-dependent.",
"In particular, in the thermodynamic limit $N\\rightarrow \\infty $ , only a part of the ground state contribution remains and we obtain $\\lim _{N\\rightarrow \\infty }\\frac{\\left< J^z \\right>}{N} \\rightarrow \\frac{1}{2}\\left\\lbrace \\begin{array}{ccc}1 &:& h > \\gamma _x\\\\\\frac{h}{\\gamma _x} &:& \\gamma _x>h\\end{array}\\right.\\,.$ For finite system sizes however, finite temperature corrections exist.",
"In Fig.",
"REF , we show a contour plot of the magnetization density $\\left< J^z \\right>/N$ from the exact numerical calculation of the partition function (dashed contours) and compare with the results from the bosonic representation (solid green contours).",
"Figure: Contour plot of the magnetization density J z /N\\left< J^z \\right>/N versus spin-spin interaction γ x \\gamma _x and temperature k B Tk_B T for N=1000N=1000.At the critical point γ x =h\\gamma _x = h, the magnetization density at low temperatures (bottom) suddenly starts to drop from a constant value in the normal phase (left) to a decaying curve in the symmetry-broken phase (right) as predicted by ().At higher temperatures, the transition is smoother and the predictions from the bosonic representation (solid green contours, based on Eq.",
"())and the finite-size numerical calculation of the partition function (dashed contours, based on the Gibbs state with Eq.",
"()) disagree for γ x ≈h\\gamma _x \\approx h.For the finite-size calculation, weak coupling has been assumed k B T≪Nω c /ηk_B T \\ll N \\omega _c/\\eta , such that U p † J z U p ≈J z U_p^\\dag J_z U_p \\approx J_z instead of ().We see in the contour lines of the magnetization convincing agreement between the curves of the bosonic representation (solid green) and the finite-size calculation (dashed black) only for very low temperatures or away from the critical point.",
"The disagreement for $\\gamma _x \\approx h$ and $T>0$ can be attributed to the fact that the bosonization for finite sizes only captures the lowest energy eigenstates well, whereas in this region also the higher eigenstates become occupied.",
"However, it is clearly visible that in the low temperature regime, the magnetization density will drop suddenly when $\\gamma _x \\ge h$ , such that the QPT can be detected at correspondingly low temperatures.",
"At high temperatures, the magnetization density falls of smoothly with increasing spin-spin interaction."
],
[
"Mode Occupation",
"The master equations appear simple only in a displaced and rotated frame.",
"When transformed back, the steady-state populations $\\left< d^\\dagger d \\right> = {\\rm Tr}\\left\\lbrace d^\\dagger d \\rho \\right\\rbrace $ and $\\overline{\\left< d^\\dagger d \\right>} = {\\rm Tr}\\left\\lbrace d^\\dagger d \\bar{\\rho } \\right\\rbrace $ actually measure displacements around the mean-field.",
"Fig.",
"REF compares the occupation number and system frequency with (solid) and without (dashed) polaron transform.",
"Panel (a) demonstrates that the LMG energy gap is in the BMS treatment strongly modified by dissipation, such that in the vicinity of the closed QPT the non-polaron and polaron treatments lead to very different results.",
"Panel (b) shows the fluctuations in the diagonal basis $\\overline{\\left< d^\\dag d \\right>}$ ($\\left< d^\\dag d \\right>$ ) around the mean-field $\\alpha (h,\\gamma _x)$ (or $\\alpha (h,\\tilde{\\gamma }_x)$ ) in the polaron (or non-polaron) frame.",
"Finally, panel (c) shows the mode occupation $\\left< a^\\dagger a \\right> = \\sinh ^2(\\varphi (h,\\gamma _x)) + 2 \\cosh ^2(\\varphi (h,\\gamma _x)) \\left< d^\\dagger d \\right>$ (and analogous in the symmetry-broken phase) in the non-diagonal basis.",
"These are directly related to the deviations of the $J_z$ -expectation value from its mean-field solution, compare App. .",
"Since the frequency $\\omega (h,\\tilde{\\gamma }_x)$ (Tab.",
"REF ) vanishes at $\\gamma _x = h + \\frac{\\eta \\cdot \\omega _c}{\\pi }$ in the non-polaron frame, the BMS approximations break down around the original QPT position, see dashed line in Fig.",
"REF (a).",
"Mode occupations in both the diagonal and non-diagonal bases diverge at the QPT point, see the dashed lines in Fig.",
"REF (b-c).",
"In particular, in the polaron frame the fluctuation divergence occurs around the original quantum critical point at $\\gamma _x = h$ , see the solid lines in Fig.",
"REF .",
"Figure: (a) LMG oscillator frequency ω(h,γ x )\\omega (h,\\gamma _x) or ω(h,γ ˜ x )\\omega (h,\\tilde{\\gamma }_x), (b) diagonal frame steady-state mode occupations d † d ¯\\overline{\\left< d^\\dag d \\right>} (d † d\\left< d^\\dag d \\right>), (c) non-diagonal frame steady-state mode occupations a † a ¯(a † a)\\overline{\\left< a^\\dag a \\right>} (\\left< a^\\dag a \\right>) for the polaron (solid) and non-polaron (dashed) master equations.Divergent mode occupations indicate the position of the QPT where the excitation frequency vanishes.For the polaron treatment, the QPT position stays at γ x /h=1\\gamma _x/h = 1 just as in the isolated LMG modelin contrast to the shift predicted by the BMS master equation.Parameters: η=2π·0.1,ω c =0.5h,β=1.79/h\\eta =2\\pi \\cdot 0.1, \\omega _c = 0.5 h, \\beta = 1.79/h."
],
[
"Waiting times",
"The coupling to the reservoir does not only modify the system properties but may also lead to the emission or absorption of reservoir excitations (i.e., photons or phonons depending on the model implementation), which can in principle be measured independently.",
"Classifying these events into classes $\\nu $ describing e.g.",
"emissions or absorptions, the waiting-time distribution between two such system-bath exchange processes of type $\\mu $ after $\\nu $ is characterized by [82] $\\mathcal {w}_{\\mu \\nu }(\\tau ) = \\frac{{\\rm Tr}\\left(\\mathcal {J}_\\mu \\exp (\\mathcal {L}_0 \\tau )\\mathcal {J}_\\nu \\rho \\right)}{{\\rm Tr} \\left(\\mathcal {J}_\\nu \\rho \\right)}\\,.$ Here $\\mathcal {J}_\\mu , \\mathcal {L}_0$ are super operators describing the jump $\\mu $ and the no-jump evolution $\\mathcal {L}_0$ .",
"For example, in master equation (REF ), there are only two distinct types of jumps, emission `e' and absorption `a'.",
"Their corresponding super-operators are then acting as $\\mathcal {J}_{e} \\rho &= F_e d \\rho d^\\dag \\,,\\qquad \\mathcal {J}_{a} \\rho = F_a d^\\dag \\rho d\\,, \\\\\\mathcal {L}_0 \\rho &= -{\\rm i}\\left[\\omega d^\\dagger d, \\rho \\right] - \\frac{F_e}{2} \\left\\lbrace d^\\dagger d, \\rho \\right\\rbrace - \\frac{F_a}{2} \\left\\lbrace d d^\\dagger , \\rho \\right\\rbrace \\,,$ such that the total Liouvillian is decomposable as $\\mathcal {L} = \\mathcal {L}_0 + \\mathcal {J}_e + \\mathcal {J}_a$ .",
"The same equations are valid in the polaron frame (REF ), just with the corresponding overbar on the variables.",
"It is straightforward to go to a frame where the Hamiltonian dynamics is absorbed $\\tilde{\\rho } = e^{+{\\rm i}\\omega t d^\\dagger d} \\rho e^{-{\\rm i}\\omega t d^\\dagger d}$ , we see that the whole Liouvillian in this frame $\\tilde{\\mathcal {L}}$ is just proportional to the spectral density, evaluated at the system transition frequency $\\omega $ .",
"Thereby, it enters as a single parameter, a different spectral density could be interpreted as a rescaling $\\Gamma (\\omega ) \\rightarrow \\alpha \\Gamma (\\omega )$ , which would imply ${\\cal L}_0 \\rightarrow \\alpha {\\cal L}_0$ and ${\\cal J}_\\mu \\rightarrow \\alpha {\\cal J}_\\mu $ .",
"These transformations would only lead to a trivial stretching of the waiting time distribution $\\mathcal {w}_{\\mu \\nu }(\\tau ) \\rightarrow \\alpha \\mathcal {w}_{\\mu \\nu }(\\alpha \\tau )$ , compare also Eq.",
"(REF ).",
"Since the LMG Hamiltonian and the steady state (REF ) are diagonal, analytic expressions for the waiting time distributions can be derived, see App. .",
"Figure: Waiting time distributions (WTD) between two emission (absorption and emission) events 𝓌 ¯ ee(ae) \\bar{\\mathcal {w}}_{ee(ae)} (solid, dot-dashed) calculated in the polaron frame as a function of τ\\tau (a) for a fixedγ x \\gamma _x value and (b) distribution 𝓌 ¯ ee \\bar{\\mathcal {w}}_{ee} as a function of γ x \\gamma _x for two different fixed τ\\tau values (b).Additionally, the WTD in the non-polaron frame is shown in (b) for τ=0\\tau = 0 case (dashed), which wrongly diverges around the shifted critical point.At the true critical point a non-analytic dependence of the distribution on the intra-spin coupling strength γ x \\gamma _x is clearly visible, within the polaron treatment however all WTDs remain finite.Parameters: η=2π·0.1,ω c =0.5h,β=1.79/h\\eta =2\\pi \\cdot 0.1, \\omega _c = 0.5 h, \\beta = 1.79/h, (a) γ x =0.5h\\gamma _x = 0.5 h.In Fig.",
"REF we show two waiting-time distributions $\\bar{\\mathcal {w}}_{ee(ae)}$ as a function of time $\\tau $ for fixed coupling strength $\\gamma _x$ (a) and the repeated-emission waiting-time distribution $\\bar{\\mathcal {w}}_{ee}(\\tau )$ as a function of $\\gamma _x$ for two fixed waiting times $\\tau $ (b).",
"A typical feature of a thermal state is bunching of emitted photons, which we see in Fig.",
"REF (a): After an emission event the same event has the highest probability for $\\tau \\rightarrow 0$ , thus immediately.",
"When looking at waiting time distributions of different phases, like in panel (a), a significant difference is not visible.",
"However, fixing the waiting time $\\tau $ and varying $\\gamma _x$ we find, that the waiting times have their maximum at the position of QPT, see Fig.",
"REF (b).",
"Essentially, this is related to the divergence of $n_B(\\omega )$ when the energy gap vanishes.",
"Whereas the non-polaron treatment predicts a divergence of waiting times around the critical point $\\tilde{\\gamma }_x^{\\rm cr}$ , see the dashed curve in Fig.",
"REF (b), the waiting times within the polaron approach remain finite but depend non-analytically on the Hamiltonian parameters.",
"Therefore, the quantum-critical behaviour is not only reflected in system-intrinsic observables like mode occupations but also in reservoir observables like the statistics of photoemission events."
],
[
"Summary",
"We have investigated the open LMG model by using a polaron transform technique that also allows us to address the vicinity of the critical point.",
"First, within the polaron treatment, we have found that the position of the QPT is robust when starting from an initial Hamiltonian with a lower spectral bound.",
"This shows that the choice of the starting Hamiltonian should be discussed with care for critical models, even when treated as weakly coupled.",
"Second, whereas far from the QPT, the approach presented here reproduces naive master equation treatments, it remains also valid in the vicinity of the QPT.",
"In the transformed frame, the effective interaction scaled with the energy gap of the system Hamiltonian, which admits a perturbative treatment at the critical point.",
"We therefore expect that the polaron-master equation approach is also applicable to other models that bilinearly couple to bosonic reservoirs via position operators.",
"Interestingly, we obtained that for a single reservoir the stationary properties are determined by those of the isolated system alone, such that a standard analysis applies.",
"The critical behaviour (and its possible renormalization) can be detected with system observables like magnetization or mode occupations but is also visible in reservoir observables like waiting-time distributions, which remain finite in the polaron frame.",
"We hope that our study of the LMG model paves the way for further quantitative investigations of dissipative quantum-critical systems, e.g.",
"by capturing higher eigenstates by augmented variational polaron treatments [83] or by investigating the non-equilibrium dynamics of critical setups."
],
[
"Acknowledgements",
" The authors gratefully acknowledge financial support from the DFG (grants BR 1528/9-1, BR 1528/8-2, and SFB 910) as well as fruitful discussions with M. Kloc, A. Knorr, and C. Wächtler."
],
[
"Thermodynamic limit of large spin operators",
"Without any displacement, the Holstein-Primakoff representation leads to a simple large-$N$ expansion $J_- &\\approx \\sqrt{N} b^\\dag \\,,\\qquad J_+ \\approx \\sqrt{N} b\\,,\\\\J_z &= \\frac{N}{2} - b^\\dag b\\,,$ where we have neglected terms that vanish in the thermodynamic limit.",
"Insertion of these approximations lead to the Hamiltonians for the normal phase, and in effect, no term of order $\\sqrt{N}$ occurs in the Hamiltonian.",
"In the symmetry-broken phase, one allows for a displacement $b = a + \\alpha \\sqrt{N}$ with bosonic operators $a$ and in general complex number $\\alpha $ .",
"Then, the large-$N$ expansion of the large spin operators is more complicated $J_- &\\approx N \\alpha ^* \\sqrt{1-\\left| \\alpha \\right|^2}\\\\&\\qquad + \\sqrt{N} \\sqrt{1-\\left| \\alpha \\right|^2} \\left[a^\\dagger - \\frac{1}{2} \\frac{(\\alpha ^*)^2 a + \\left| \\alpha \\right|^2 a^\\dagger }{1-\\left| \\alpha \\right|^2}\\right]\\\\&\\qquad - \\frac{\\sqrt{1-\\left| \\alpha \\right|^2}}{2\\left(1-\\left| \\alpha \\right|^2\\right)}\\Big [\\alpha (a^\\dagger )^2 + 2 \\alpha ^* a^\\dagger a\\\\&\\qquad \\qquad \\qquad +\\frac{\\alpha ^* \\left(\\alpha ^* a + \\alpha a^\\dagger \\right)^2}{4 \\left(1-\\left| \\alpha \\right|^2\\right)}\\Big ]\\,,\\\\J_+ &\\approx N \\alpha \\sqrt{1-\\left| \\alpha \\right|^2}\\\\&\\qquad + \\sqrt{N} \\sqrt{1-\\left| \\alpha \\right|^2} \\left[a - \\frac{1}{2} \\frac{\\alpha ^2 a^\\dagger + \\left| \\alpha \\right|^2 a}{1-\\left| \\alpha \\right|^2}\\right]\\\\&\\qquad - \\frac{\\sqrt{1-\\left| \\alpha \\right|^2}}{2\\left(1-\\left| \\alpha \\right|^2\\right)}\\Big [\\alpha ^* a^2 + 2 \\alpha a^\\dagger a\\\\&\\qquad \\qquad \\qquad +\\frac{\\alpha \\left(\\alpha ^* a + \\alpha a^\\dagger \\right)^2}{4 \\left(1-\\left| \\alpha \\right|^2\\right)}\\Big ]\\,,\\\\J_z &= N \\left(\\frac{1}{2}-\\left| \\alpha \\right|^2\\right) - \\sqrt{N} \\left(\\alpha ^* a + \\alpha a^\\dag \\right) - a^\\dag a\\,.$ For consistency, one can check that by setting $\\alpha \\rightarrow 0$ , the previous representation is reproduced.",
"Insertion of this expansion leads to the Hamiltonians for the symmetry-broken phase, and the displacement $\\alpha $ is chosen such that the $\\sqrt{N}$ terms in the LMG Hamiltonian vanish.",
"One might be tempted to neglect the last expansion terms in $J_\\pm $ from the beginning, as these operators enter the Hamiltonian always with a factor of $1/\\sqrt{N}$ .",
"However, we stress that in terms like $J_x^2/N$ they will yield a non-vanishing contribution and thus need to be considered to obtain the correct spectra of the LMG model."
],
[
"Polaron transform",
"Here we provide more details how to derive Eq.",
"(REF ) in the main text.",
"Using the Hadamard lemma $e^{+X} Y e^{-X} &= \\sum _{m=0}^\\infty \\frac{1}{m!}",
"\\left[X,Y\\right]_m\\,,\\\\\\left[X,Y\\right]_m &= \\left[X,\\left[X,Y\\right]_{m-1}\\right]\\,,\\qquad [X,Y]_0=Y\\,,$ one can see that the polaron transform (REF ) leads to $U_p^\\dag c_k U_p &= c_k - \\frac{J_x}{\\sqrt{N}} \\frac{g_k}{\\nu _k}\\,,$ and analogous for the transformation of the creation operator.",
"Furthermore, it is trivial to see that $U_p^\\dag J_x U_p = J_x$ .",
"From this, it directly follows that the polaron-transform of the interaction and reservoir Hamiltonian becomes $U_p^\\dag \\left( c_k^\\dag + \\frac{g_k}{\\sqrt{N} \\nu _k}J_x \\right)\\left( c_k + \\frac{g_k}{\\sqrt{N} \\nu _k}J_x \\right) U_p&= c_k^\\dag c_k\\,.$ In addition, the polaron transform of $J_z$ has to be calculated, which yields via the commutation relations $[J_x,J_y]={\\rm i}J_z$ the relation $U^\\dag _p J_z U_p &= J_z \\cosh (\\hat{B}) - {\\rm i}J_y \\sinh (\\hat{B})\\,,$ where $\\hat{B}$ is defined in (REF ) in the main text.",
"Therefore, the full polaron-transformed Hamiltonian $H_{\\rm tot}$ becomes $U_p^\\dag H_{tot} U_p&= -h D J_z - \\frac{\\gamma _x}{N}J_x^2 + \\sum _k \\nu _k b_k^\\dag b_k\\\\&\\quad - h \\cdot \\left[J_z \\cdot \\left(\\cosh (\\hat{B})-D\\right) - {\\rm i}J_y \\sinh (\\hat{B})\\right]\\,,$ such that there is no rescaling of the spin-spin interaction $\\gamma _x$ .",
"We have also already inserted the temperature-dependent shift $D$ , which is necessary in order to ensure that the first order expectation values of the system-reservoir coupling operators vanish, eventually yielding Eq.",
"(REF ) in the main text.",
"For the $\\sinh $ -term this is not necessary as its expectation value vanishes anyhow."
],
[
"Magnetization",
"It is well known that for a Hamiltonian depending on an external parameter $\\lambda $ (which for your model could be $h$ or $\\gamma _x$ ), the canonical partition function $Z={\\rm Tr}\\left\\lbrace e^{-\\beta H(\\lambda )} \\right\\rbrace $ allows to evaluate the thermal expectation value of particular operators $\\frac{-1}{\\beta } \\frac{\\partial \\ln Z}{\\partial \\lambda } &= \\frac{-1}{Z \\beta } {\\rm Tr}\\left\\lbrace \\frac{\\partial }{\\partial \\lambda } e^{-\\beta H(\\lambda )} \\right\\rbrace \\\\&= \\frac{1}{Z} \\sum _{n=1}^\\infty \\frac{(-\\beta )^{n-1}}{(n-1)!}",
"{\\rm Tr}\\left\\lbrace \\frac{\\partial H(\\lambda )}{\\partial \\lambda } H^{n-1}(\\lambda ) \\right\\rbrace \\\\&= \\frac{1}{Z} {\\rm Tr}\\left\\lbrace \\frac{\\partial H(\\lambda )}{\\partial \\lambda } e^{-\\beta H(\\lambda )} \\right\\rbrace = \\left< \\frac{\\partial H(\\lambda )}{\\partial \\lambda } \\right>\\,,$ where we have used the invariance of the trace under cyclic permutations to sort all derivatives of $H(\\lambda )$ to the left.",
"In particular, for a harmonic oscillator $H=E_0(\\lambda )+\\omega (\\lambda ) a^\\dagger (\\lambda ) a(\\lambda )$ with bosonic operators $a(\\lambda )$ , the partition function becomes $Z = \\frac{e^{-\\beta E_0(\\lambda )}}{1-e^{-\\beta \\omega (\\lambda )}}\\,.$ With $\\lambda \\rightarrow -h$ , this eventually leads to Eq.",
"(REF ) in the main text."
],
[
"Waiting time distribution",
"Starting from the spectral decomposition of a thermal state in terms of Fock states $\\rho &= \\frac{e^{-\\beta \\omega d^\\dagger d}}{{\\rm Tr}\\left\\lbrace e^{-\\beta \\omega d^\\dagger d} \\right\\rbrace } = \\sum _{n=0}^\\infty P_n \\mathinner {|{n}\\rangle }\\mathinner {\\langle {n}|}\\\\P_n &= \\left(\\frac{n_B}{1+n_B}\\right)^n \\frac{1}{1+n_B}\\,,$ with the shorthand notation $n_B = [e^{\\beta \\omega }-1]^{-1}$ , it is straightforward to compute the action of the emission or absorption jump superoperators $\\mathcal {J}_e \\rho &= F_e \\sum _{n=0}^\\infty P_{n+1} (n+1) \\mathinner {|{n}\\rangle }\\mathinner {\\langle {n}|}\\,,\\\\\\mathcal {J}_a \\rho &= F_a \\sum _{n=1}^\\infty P_{n-1} n \\mathinner {|{n}\\rangle }\\mathinner {\\langle {n}|}\\,,$ which also implies ${\\rm Tr}\\left\\lbrace \\mathcal {J}_e \\rho \\right\\rbrace = {\\rm Tr}\\left\\lbrace \\mathcal {J}_a \\rho \\right\\rbrace = \\Gamma n_B (1+n_B)\\,,$ where $\\Gamma =A^2(h,\\tilde{\\gamma }_x) \\Gamma (\\omega (h,\\tilde{\\gamma }_x))$ or $\\Gamma = \\bar{A}^2(h,\\gamma _x) \\bar{\\Gamma }(\\omega (h,\\gamma _x))$ in the main text.",
"Since $\\mathcal {L}_0$ does not induce transitions between different Fock states, its action on a diagonal density matrix can be computed via $e^{\\mathcal {L}_0 t} \\mathinner {|{n}\\rangle }\\mathinner {\\langle {n}|} = e^{-\\left[(1+n_B) n + n_B (1+n)\\right] \\Gamma t} \\mathinner {|{n}\\rangle }\\mathinner {\\langle {n}|}\\,,$ which implies for the relevant terms $\\mathcal {w}_{ee}(\\tau ) &= \\frac{2 \\Gamma n_B (1+n_B) e^{(2+3 n_B) \\Gamma \\tau }}{\\left[(1+n_B) e^{(1+2 n_B)\\Gamma \\tau } - n_B\\right]^3}\\,,\\\\\\mathcal {w}_{ae}(\\tau ) &= \\frac{\\Gamma n_B e^{(2+3 n_B) \\Gamma \\tau } \\left[n_B+ (1+n_B) e^{(1+2 n_B) \\Gamma \\tau }\\right]}{\\left[(1+n_B) e^{(1+2 n_B)\\Gamma \\tau } - n_B\\right]^3}\\,,\\\\\\mathcal {w}_{ea}(\\tau ) &= \\frac{\\Gamma (1+n_B) e^{(1+n_B) \\Gamma \\tau } \\left[n_B+ (1+n_B) e^{(1+2 n_B) \\Gamma \\tau }\\right]}{\\left[(1+n_B) e^{(1+2 n_B)\\Gamma \\tau } - n_B\\right]^3}\\,,\\\\\\mathcal {w}_{aa}(\\tau ) &= \\frac{2 \\Gamma n_B (1+n_B) e^{(2+3 n_B) \\Gamma \\tau }}{\\left[(1+n_B) e^{(1+2 n_B)\\Gamma \\tau } - n_B\\right]^3}\\,.$ For consistency, we note that the normalization conditions $\\int \\left(\\mathcal {w}_{ae}(\\tau ) + \\mathcal {w}_{ee}(\\tau )\\right)d\\tau = 1$ and $\\int \\left(\\mathcal {w}_{aa}(\\tau ) + \\mathcal {w}_{ea}(\\tau )\\right)d\\tau = 1$ always hold, which simply reflects the fact that only emission or absorption processes can occur.",
"Furthermore, in the low-temperature limit $n_B \\rightarrow 0$ , only the conditional waiting time distribution for emission after absorption can survive $\\mathcal {w}_{ea} \\rightarrow \\Gamma e^{-\\Gamma \\tau }$ : Once a photon has been absorbed from the reservoir, it must be emitted again since no further absorption is likely to occur.",
"For $\\tau \\gg 1$ all waiting time distributions $\\bar{\\mathcal {w}}_{\\mu \\nu }$ decay to zero."
]
] | 1906.04260 |
[
[
"Associative Convolutional Layers"
],
[
"Abstract Motivated by the necessity for parameter efficiency in distributed machine learning and AI-enabled edge devices, we provide a general and easy to implement method for significantly reducing the number of parameters of Convolutional Neural Networks (CNNs), during both the training and inference phases.",
"We introduce a simple auxiliary neural network which can generate the convolutional filters of any CNN architecture from a low dimensional latent space.",
"This auxiliary neural network, which we call \"Convolutional Slice Generator\" (CSG), is unique to the network and provides the association between its convolutional layers.",
"During the training of the CNN, instead of training the filters of the convolutional layers, only the parameters of the CSG and their corresponding \"code vectors\" are trained.",
"This results in a significant reduction of the number of parameters due to the fact that the CNN can be fully represented using only the parameters of the CSG, the code vectors, the fully connected layers, and the architecture of the CNN.",
"We evaluate our approach by applying it to ResNet and DenseNet models when trained on CIFAR-10 and ImageNet datasets.",
"While reducing the number of parameters by $\\approx 2 \\times$ on average, the accuracies of these networks remain within 1$\\%$ of their original counterparts and in some cases there is an increase in the accuracy."
],
[
"Introduction",
"Current state-of-the-art Convolutional Neural Networks (CNNs) consist of hundreds or even thousands of convolutional layers [12], [35] and the resulting large number of parameters presents a limit to their wider application.",
"More efficient implementations are desired, for training and inference phases of large CNNs running on the cloud.",
"Also, on the other end of the spectrum, as these networks proliferate to small embedded devices at the edge of the internet, and closer to observation and control in real-life applications, their size and implementation efficiency becomes critical.",
"For the training of very large CNNs, for instance those running on cloud computing resources, distributed machine learning approaches are typically used.",
"In this case, communication constraints present a key challenge, as gradients of the network parameters need to be communicated among different nodes [30].",
"Federated learning is an example of distributed machine learning where a neural network is optimized and customized in a distributed manner using numerous users' edge devices [17].",
"Recent works have focused on coding, quantization, and compression techniques to reduce the amount of data that needs to be communicated among the different nodes [5], [30], [22].",
"To improve the inference time of CNN models, especially on edge devices, recent studies propose various techniques such as pruning the network parameters and connections [10], [20], [3].",
"Although these techniques overcome limited storage capacities in edge devices and reduce the number and cost of operations, they are only applicable to the models after the training phase.",
"In addition, to recover the accuracy degradation resulting from these methods, extra training and fine-tuning is required.",
"Motivated by the challenges described above, in this paper we focus on reducing the redundancy in the parameters describing the convolutional layers of CNNs and provide an approach which can be used in conjunction with all of the aforementioned solutions and is applicable during both the training and inference phases.",
"Our contribution stems from the observation that although there has been considerable progress in more efficient implementation of neural networks, large convolutional filters are always needed.",
"Such convolutional filters are inherently redundant, to a point that pruning [20], quantization [14], [18], [4], and low rank approximations on these filters [16] can be performed.",
"This redundancy suggests that these layers could be represented in a much smaller space than their natural tensors space.",
"Our approach is particularly relevant in view of the recent trend of adding additional convolutional layers [12], [35] or adding additional filters to the convolutional layers [37] to achieve higher accuracy.",
"We provide a method to obtain a low-dimensional representation of the parameter space of the set of filters of convolutional layers during both the training and inference phases by introducing an auxiliary neural network that can be used alongside any CNN architecture.",
"This auxiliary neural network generates slices of sets of convolutional filters and is called Convolutional Slice Generator (CSG).",
"The CSG takes as input a set of code vectors corresponding to a partition of a set of convolutional filters of each layer in a latent, low-dimensional space, and produces these slices, which are then combined and used as the set of convolutional filters of that layer.",
"The code vectors, which lie in a space of cardinality $\\approx 20 \\times $ smaller than the cardinality of the corresponding slice of the convolutional filter, are optimized during the training of the CNN instead of the set of filters.",
"The auxiliary neural network can either be trained alongside the main network or be provided to the network in advance with pre-trained and fixed parameters.",
"In our experiments on classification tasks, we show that while this approach significantly reduces the cardinality of the parameter space of the CNN, the resulting networks, except in extreme compression cases, still achieve top-1 accuracies that are within one percent of the original CNNs, or even achieve improved accuracies.",
"Finally, one could argue that in this work we trade computation efficiency for parameter efficiency, and hence communication and storage efficiency.",
"However, we also show that the added computational cost is negligible in practice, and with customized hardware for edge devices, our approach is also expected to improve timing performance."
],
[
"Related Works",
"There are several works, mostly in the intersection of signal processing and computer vision, that focus on the design of convolutional filters.",
"For instance, in [15] the authors, inspired by scattering networks [27], [4], [23], introduce a structured method based on the family of Gaussian filters and its smooth derivatives, to produce the CNN filters from some basis functions that are also learned during the training phase.",
"Steerable filter design is another approach that has also been studied for about two decades [9].",
"Closer to our approach is the design of low rank and separable filters.",
"In this case, the main goal has been of achieving better computing performance [28], [24], [16].",
"For example, the work in [26] shows that multiple image filters can be approximated by a shared set of separable (rank-1) filters, allowing large speedups with minimal loss in accuracy.",
"Other works have exploited the computing efficiency of Fast-Fourier-Transform (FFT) based multiplications [1], [8].",
"These schemes require complex multiplications and efficient implementations of FFT.",
"There are also methods based on the Winograd algorithm [34] for performing efficient convolutions in the real domain [19].",
"All of these methods can be applied in conjunction with our approach to compress and accelerate the operations in the fully connected layer(s) or to accelerate the convolution operations.",
"Additional works are concerned with methods to perform different stages of the training in parallel, or to reduce the amount of information that needs to be communicated between different nodes of the distributed computation network using compression, or quantizing the gradients [30], [22], [36], [21], [25], [31].",
"However, these works are not concerned with the architecture of the network or on how the filters are designed, and can be applied to any architecture including our CSG-augmented CNNs."
],
[
"Our Contribution",
"We present three distinct contributions: We provide a novel and general method for reducing the number of parameters that are needed to represent the sets of filters of convolutional layers during both the training and inference phases, through the use of an auxiliary neural network which transforms a set of code vectors in a low dimensional space to slices of sets of convolutional filters.",
"The software implementation of our method is straightforward and it can be done by adding only a few lines of codes to the implementation of any CNN architecture.",
"We provide an example of a simple CSG-augmented CNN and show that the training time for this network is polynomial in the number of data points, number of input features (e.g., pixels), and inverse of the minimum distance between data points.",
"In addition to this analysis, inspired by Discrete Cosine Transform (DCT)-based compression techniques for images, we provide an estimate on the relationship between the size of the slices and the cardinality of the code vector space which eliminates the need for tuning for these parameters.",
"This analysis also suggests that our approach can be applied to at least a large set of architectures.",
"We experimentally investigate the performance of our method by applying it to ResNet and DenseNet architectures, and show that significant parameter reductions, without compromising the accuracy, are possible.",
"Furthermore, when running on a single GPU, we observe that the training time and the inference time of the augmented networks remain almost unaltered.",
"The paper is organized as follows.",
"In Section we provide the preliminaries and set the stage for introducing our method.",
"In Section we formally introduce the CSG, provide a rough estimate on the cardinality of the code vector space, and theoretically investigate its effect on the convergence of the training phase.",
"In Section we provide the results of our experiments on ResNet and DenseNet architectures.",
"Finally, Section includes our concluding remarks and future directions."
],
[
"Convolutional Neural Network (CNN)",
"In a typical classification task, a CNN is composed of several convolutional layers and one or more fully connected layers, at the very end of the network, responsible for the classification.",
"Each convolutional layer consists of a set of filters (i.e., kernels) and perhaps some batch normalization layers and ReLu activations.",
"Here, we focus on the sets of filters of the convolutional layers.",
"Let $k\\in \\mathbb {R}^{s_1s_2s_3s_4}$ , for $s_1,s_2,s_3,s_4 \\in \\mathbb {N}$ , denote a set of $s_1$ filters in the CNN, where $s_2$ denotes the number of channels and $s_3$ and $s_4$ denote the height and width of the kernel, respectively.",
"We denote the collection of all the sets of filters in a CNN by $\\mathcal {K}$ .",
"Let $\\mathcal {O}$ denote the set of all the other parameters in the CNN.",
"We denote the set of all the parameters by $\\mathcal {P} := \\mathcal {K} \\cup \\mathcal {O}$ ."
],
[
"Slices",
"We define a slice as a tensor $\\hat{k} \\in \\mathbb {R}^{\\hat{s}_1\\hat{s}_2\\hat{s}_3\\hat{s}_4}$ , for $\\hat{s}_1,\\hat{s}_2,\\hat{s}_3,\\hat{s}_4\\in \\mathbb {N}$ .",
"We partition each set of filters $k\\in \\mathcal {K}\\setminus \\lbrace k_0\\rbrace $ , where $k_0$ denotes the set of filters of the first convolutional layer, into $\\lceil s_1/\\hat{s}_1 \\rceil \\lceil s_2/\\hat{s}_2 \\rceil \\lceil s_3/\\hat{s}_3 \\rceil \\lceil s_4/\\hat{s}_4 \\rceil $ slices and denote the set of all these slices by $\\hat{\\mathcal {K}}$ .",
"For simplicity we assume that this partitioning is possible.",
"In practice we consider additional slices for fractional partitions and only use part of the final slice(s) to reconstruct the set of convolutional filters."
],
[
"Code Vectors",
"Let $c\\in \\mathbb {R}^{n_c}$ , where $n_c\\in \\mathbb {N}$ , denotes a vector of $n_c$ elements.",
"We refer to $c$ as a code vector.",
"Each slice of each filter $\\hat{k} \\in \\hat{\\mathcal {K}}$ in the CNN corresponds to one code vector and their relationship is illustrated in the following section."
],
[
"The Convolutional Slice Generator",
"The Convolutional Slice Generator (CSG) is the core element of our approach.",
"The CSG provides a linear approximation for slices of a convolutional filter."
],
[
"The CSG Network",
"Let $vec(k)$ denote the vectorized version of a tensor $k$ .",
"Let $k_{i}$ , for $i\\in \\lbrace 1,...,|\\hat{\\mathcal {K}}| \\rbrace $ denote a slice in $\\hat{\\mathcal {K}}$ and $c_i$ for $i\\in \\lbrace 1,...,|\\hat{\\mathcal {K}}| \\rbrace $ denote the code vector corresponding to the $i$ 'th slice in $\\hat{\\mathcal {K}}$ $vec(\\hat{k}_{i}) = A_{CSG}c_{i} \\mbox{, \\ \\ \\ for \\ \\ \\ } i\\in \\lbrace 1,...,|\\hat{\\mathcal {K}}|\\rbrace ,$ where $A_{CSG}$ denotes an $\\hat{s}_1\\hat{s}_2\\hat{s}_3\\hat{s}_4$ by $n_c$ matrix representing the weights of the CSG network, $c_i$ denotes the code vector corresponding to the $i$ 'th slice where $i \\in \\lbrace 1,2,...,|\\hat{\\mathcal {K}}|\\rbrace $ .",
"See Fig.",
"REF for an example of how a single slice of a set of filters for a single convolutional layer is generated.",
"Let $\\hat{\\mathcal {G}}$ denote all the parameters of the CSG, i.e., the elements of the matrix $A_{CSG}$ , $\\hat{\\mathcal {C}}$ denote the set of all the code vectors, and let $\\hat{\\mathcal {O}}$ denote all the parameters of the CNN except for the parameters in $\\hat{\\mathcal {G}}$ or $\\hat{\\mathcal {C}}$ , e.g., biases, batch normalization parameters, fully connected layer(s), and the first convolutional filter.",
"Hence, we can denote the set of all the parameters of the network by $\\hat{\\mathcal {P}} := \\hat{\\mathcal {C}} \\cup \\hat{\\mathcal {G}} \\cup \\hat{\\mathcal {O}}$ .",
"Figure: Generation of a single slice of a set of convolutional filters of a convolutional layer.",
"Our method can be applied to any filter shape.",
"In this example there are 32 filters, kernels are 6×\\times 6 and have 32 channels.",
"Each slice, generated by the CSG, is assumed to be 16×16×3×316\\times 16 \\times 3\\times 3.",
"The figure shows one slice, that spans across multiple channels and multiple filters, and its corresponding code vector."
],
[
"Estimating the Cardinality of the Code Vector Space",
"In this section, we discuss our method for having a very rough estimate on the cardinality of the code vector space $n_c$ .",
"First, we need to choose a shape for the slices.",
"In order to decide about this shape, we considered several widely used CNNs including VGG16, VGG19, ResNet, etc., and concluded that a $3\\times 3$ filter size is the most common size for the filters.",
"Also, these architectures suggests that a slice with channel size of 16 and the depth of 16 would divide most of these filters.",
"Hence, we chose $\\hat{s}_1 = 16, \\hat{s}_2 = 16, \\hat{s}_3 = 3, \\hat{s}_4 = 3$ for this part of our work.",
"In order to decide about the cardinality of the vector space, we need an estimate on the number of the elements of the slice in its possible latent domain, namely an estimate for $n_c$ .",
"Inspired by the fact that these filters are responsible for detecting visual features and knowing that usage of DCT leads to a very good encoding of visual representations [32], we looked at the four-dimensional Type-II DCTs (4-D DCT-II) of about 29000 slices of pre-trained filters extracted from VGG-16, VGG-19, ResNet-50, InceptionV3, DenseNet-169, DenseNet-201, InceptionResNetV2 (available in Tensorflow).",
"We then computed the 4-D DCT-II representation of these slices and removed the elements of this representation in such a way that the remaining elements would result in an inverse transform which is not very different from the original slice.",
"Our analysis, presented in Appendix , suggests that a code vector that has close to $20\\times $ fewer number of elements would be sufficient.",
"In our experiments, we chose code vectors that have $18\\times $ fewer elements than the slices, and our experiments on the neural networks confirm this choice."
],
[
"Training Convergence",
"While convergence is always observed in all our experiments, in this section, we provide a proof of convergence for a simple CNN with only one convolutional layer based on the recent work [2].",
"Let $m$ denote the number of channels of the input, and $d$ denote the number of its features (e.g., pixels).",
"For simplicity, let us assume that the number of channels remains $m$ after the convolutional layer.",
"Let $n$ denote the number of data points, and $d^{\\prime }$ denote the number of labels.",
"We assume that the data-set is non-degenerate meaning that there does not exist similar inputs with dissimilar labels.",
"We denote by $\\delta $ the minimum distance between two training points.",
"We restate the following theorem from [2] for the CNN defined in Appendix B of this reference.",
"Theorem 1 (CNN [2]) As long as $m\\ge \\tilde{\\Omega }(poly(n,d,\\delta ^{-1})d^{\\prime })$ , with a probability that approaches one as $m\\rightarrow {\\infty }$ , Stochastic Gradient Decent (SGD) finds an $\\epsilon $ -error solution for $l_2$ regression in $T=\\tilde{\\Omega }\\left( \\frac{poly(n,d)}{\\delta ^2}\\log \\epsilon ^{-1} \\right)$ iterations for a CNN.",
"The above theorem as discussed in [2] can be easily extended for other convergence criteria including the cross-entropy.",
"Now let us consider our CSG-augmented CNN which we denote by CNN-CSG.",
"For simplicity, in the following theorem, we consider the case when only a single layer convolutional layer is present.",
"Theorem 2 (CNN-CSG) If $|\\hat{\\mathcal {C}}| \\ge \\tilde{\\Omega }(poly(n,d,\\delta ^{-1})d^{\\prime })$ , with a probability that approaches one as $|\\hat{\\mathcal {C}}|\\rightarrow {\\infty }$ , then SGD finds an $\\epsilon $ -error solution for $l_2$ regression in $T=\\tilde{\\Omega }\\left( \\frac{poly(n,d)}{\\delta ^2}\\log \\epsilon ^{-1} \\right)$ iterations for a CNN-CSG.",
"The proof of the above theorem, which follows from the fact that the code vectors following the CSG layer can simply be viewed as an additional fully connected layer, can be found in Appendix .",
"Similar to Theorem REF , Theorem REF can be easily extended for other convergence criteria including the cross-entropy."
],
[
"Setup",
"We evaluated our approach on three different CNN models (ResNet56, DenseNet-BC-40-48, DenseNet-BC-40-36) on CIFAR-10 dataset.",
"CIFAR-10 includes 50K training images and 10K test images from 10 different classes.",
"The CSGs are integrated into the models implemented in Pytorch.",
"Our implementations along with detailed documentations of our codes are available in the supplementary materials.",
"For training the models, we used a machine with a single GPU (Nvidia Geforce 2080 Ti).",
"It is worth mentioning that we did not do any parameter tuning for our CSG-augmented networks and the experiments are all done using the same settings that we used for the original networks.",
"Also, as it is clear from the previous sections, we did not apply our method to the very first convolutional layer of any network."
],
[
"Training CSG alongside the CNN",
"In this set of experiments we train all the models from scratch.",
"We initialize the parameters of CSG $\\hat{\\mathcal {G}}$ with random initial values and train it alongside the code vectors $\\hat{\\mathcal {C}}$ as well as other parameters of the network $\\hat{\\mathcal {O}}$ ."
],
[
"CIFAR-10 Dataset",
"See Table REF for a summary of the results.",
"As we can see, when we used $[16,16,3,3]$ slices and code vectors of size 128 for ResNet-56 [12], we achieved $\\approx 2.5\\times $ reduction with less than $1\\%$ increase in top-1 error.",
"If we allow a higher accuracy degradation of $\\approx 1.5\\%$ , we can achieve over $5.3\\times $ parameter reduction by using $[12,12,3,3]$ slices and code vectors of size 72.",
"In case of DenseNet [13], we considered the most challenging cases, namely, when bottlenecks are used and the network has a $50\\%$ compression factor (i.e., $\\theta =0.5$ ), which is abbreviated as DenseNet-BC.",
"We only considered $3\\times 3$ kernels and did not compress the bottleneck or transition layers in these implementations.",
"Since the number of filters is a multiple of 12, we chose slices of shape $[12,12,3,3]$ and code size of 72 to keep the ratio between the number of elements in the slice and $n_c$ the same.",
"We considered two cases when $L=40, K=48$ , and $L=40, K=36$ , where $L$ is the number of layers and $K$ is the growth rate.",
"For the first case, we could achieve $\\approx 2\\times $ reduction with a slight improvement in accuracy.",
"For the second case, the use of CSG had little effect on the accuracy of the network while reducing its parameters by over $1.8\\times $ .",
"Table: Training results on CIFAR-10 dataset.",
"When CSG is used, the slice shape and the code vector size are indicated as CSG-[s ^ 1 ,s ^ 2 ,s ^ 3 ,s ^ 4 ][\\hat{s}_1,\\hat{s}_2,\\hat{s}_3,\\hat{s}_4]-n c n_c following the name of the original network.",
"In the “Top-1 Err.” column the average and standard deviations of test errors at the last epoch for three non-selective trainings and on the “Ratio” column the compression ratios with respect to the original networks are reported.Figure: Training and test error for DenseNet-BC-40-48 and ResNet-56 and their CSG-augmented versions on CIFAR-10 dataset over the course of 200 epochs.",
"For the first 100 epochs the learning rate was set to 0.050.05 and for the two final 50 epochs it was set to 5×10 -3 5 \\times 10^{-3} and 5×10 -4 5\\times 10^{-4} respectively, and the batch size was 128 for all DenseNet models and 192 for all ResNet models.Table: Training results on ImageNet-1000 (ILSVRC2012) dataset.",
"When CSG is used, the slice shape and the code vector size are indicated as CSG-[s ^ 1 ,s ^ 2 ,s ^ 3 ,s ^ 4 ][\\hat{s}_1,\\hat{s}_2,\\hat{s}_3,\\hat{s}_4]-n c n_c following the name of the original network.",
"In the “Top-1 Error” column the validation error for the center cropped images at the last epoch for the training and on the “Ratio” column the compression ratios with respect to the original networks are reported.",
"The results indicated with a \"*\" are reported from .Figure: Train and validation errors during the training of ResNet-18-CSG-[16,16,3,3]-128, and ResNet-50-CSG-[16,16,3,3]-128 on ImageNet dataset."
],
[
"ImageNet-1000 (ILSVRC2012) Dataset",
"We have also trained the CSG-augmented versions of ResNet-18 and ResNet-50 on the ImageNet-1000 (ILSVRC2012) dataset.",
"We used the same hyperparameters as the ones mentioned in the original paper [11], namely we used batch sizes of 256 images, and started from the learning rate of 0.1 and divided the learning rate by 10 every 30 epochs.",
"We continued the training for 100 epochs which is 20 epochs fewer than the original paper.",
"While ResNet-18-CSG-[16,16,3,3]-128 has a compression ratio of $1.54\\times $ , it achieves a top-1 error of $28.5 \\%$ which is $1.7\\%$ better than the implementation of the original ResNet-18 as reported in [11].",
"ResNet-50-CSG-[16,16,3,3]-128 which has almost the same number of parameters as ResNet-18, achieves $24.9 \\%$ top-1 error with a compression ration of $1.68\\times $ .",
"The results are summarized in Table REF .",
"More details of training and validation errors over the course of 100 epochs are brough in Figure REF ."
],
[
"Using Pre-Trained CSG",
"When using pre-trained CSG parameters during the training of the CSG-augmented CNNs, the number of parameters to be trained reduces to $|\\hat{\\mathcal {C}}|+|\\hat{\\mathcal {O}}|$ .",
"This can result in significant reduction in the number of the parameters of the network depending on its architecture.",
"For ResNet-56, in the case of using fixed pre-trained parameters for the CSG that was trained alongside ResNet-20 architecture (in ResNet-20, due to the small size of the network, use of the CSG-augmented network does not result in parameter reduction, i.e., $\\hat{\\mathcal {G}}$ is larger than the number of parameters - training and test details of ResNet-20 are available in supplementary materials), the number of parameters reduces from about $850K$ to merely $50K$ , a reduction of more than $16\\times $ but at the cost of higher accuracy loss.",
"For DenseNet-BC when $L=40$ ,$K = 48$ , our approach of using pre-trained CSG that was trained alongside DenseNet-BC with $L=40$ , $K = 36$ reduces the number of parameters from $\\approx 2.7$ million to $\\approx 1.3$ million (i.e., $2.06\\times $ ) while also improving the accuracy.",
"For DenseNet-BC when $L=40$ , $K = 36$ , this approach (using a pre-trained CSG obtained from DenseNet-BC with $L=40$ , $K = 48$ ) reduces the number of parameters from $\\approx 1.5$ million to $\\approx 0.75$ million while having a degradation of less than $0.5\\%$ in accuracy."
],
[
"End-to-End Timings",
"We evaluated the training and inference time of CSG-augmented CNNs on a single GPU for different CNN models, and compared it with the baseline ones.",
"We measured the execution time of training and inference stages on the whole train and test datasets.",
"The average epoch time for each network is summarized in Table REF .",
"The results show that for DenseNet, both inference time and training time are slightly improved in the CSG-augmented models compared to the baseline.",
"For ResNet-56, execution times reported for the CSG-augmented model are slightly higher than the baseline model.",
"The results indicate that although CSG is added to each convolutional layer in CSG-augmented CNNs, the execution time on a single GPU remains almost the same.",
"The reason is that in CSG-augmented networks, due to the reduced number of parameters, costly memory accesses like DRAM accesses across memory hierarchy in a computing system are decreased; thus, the communication and memory accesses cost are reduced.",
"Therefore, the cost of more computation performed in CSG-augmented CNN models (i.e., additional operations related to matrix multiplications in CSG) compared to the baseline models do not increase the execution time of the baseline model.",
"This results in a slight improvement of timing in DenseNet models, however, in case of ResNet, due to its smaller size, the lower memory access cost does not fully compensate the additional cost due to the use of CSG.",
"Table: Training time, inference time and model sizes (Mega Bytes (MB)) and host to device model data transfer.",
"Training and inference time for each epoch are reported in the following format: “mean” ±\\pm “standard deviation” over all 200 epochs.",
"Model size reports the size of the Pytorch model in MB.",
"“H to D” column reports the amount of model related data transmitted from the host (main memory) to the device (GPU memory) collected using the `nvprof' tool.We expect the execution time to be improved considerably on specialized hardware architectures such as Application Specific Integrated Circuits (ASICs) and Field Programmable Gate Arrays (FPGAs), mostly used in edge devices, for the following reasons.",
"First, in these hardware architectures, as shown in [6], the required memory bandwidth for the model parameters are relatively high compared to other data such as input/output feature maps.",
"In contrast, on GPU, because of data parallelization, the intermediate results such as feature maps for different input images consume a more significant part of the on-chip memories and off-chip memory bandwidth.",
"Second, in specialized architectures designed for CNNs such as the ones introduced in [7], [6], the proposed mapping of the operations and data on the processing elements helps to increase the reusability of data and parameters loaded into the on-chip memories, which reduces the number of accesses to costly memories (i.e., off-chip memories).",
"In these cases, with CSG-augmented CNNs, fewer number of parameters are loaded onto the on-chip memories such as global buffers in Eyeriss architecture [7], and their reuse distance is increased due to the fact that fewer number of parameters can be loaded onto on-chip memories to generate the same number of weights that must be loaded onto the on-chip memories in baseline models.",
"In addition to evaluating the timing, we reported the size of the model related parameters that are transferred from the main memory of a computing machine (Host) to DRAM of the GPU (Device) (i.e., Host to Device (“H to D” column in Table REF )) for an inference task.",
"The numbers, extracted from an Nvidia profiling tool (nvprof) show that with CSG-augmented CNNs, the communication between host and device memories to transfer the model is reduced by, on average, 2.03$\\times $ , compared to the baseline models."
],
[
"Comparison with Other Methods",
"As mentioned before, our approach can be used on top of most of the available approaches.",
"We have implemented two of the other compression methods mentioned in the above for DenseNet-BC-40-48: Separable filters [26] and low rank filters using singular value decomposition (SVD) [28].",
"Separable convolutions approach reduced the accuracy by $\\approx 1\\%$ with $\\approx 2 \\times $ compression with roughly similar timings for training and inference on the GPU, while our approach slightly increases the accuracy with $\\approx 2\\times $ compression.",
"Parameter tuning for the low rank filters method (SVD-based) is needed and with moderate tuning, we have not been able to achieve better than $\\approx 11\\%$ top-1 error.",
"Also, the training is $\\approx 3\\times $ slower due to need for computing the SVDs.The number of trainable parameters is similar to the original model ; however, after training, the decomposed parameters (reduced by $2\\times $ ) are used for the inference.",
"We will also provide a qualitative comparison between major approaches in a table in the final version which we could not include due to limited space.",
"Table: Comparison of compression methods."
],
[
"Conclusion and Future Directions",
"Although several methods for making the convolutional layers of CNNs more efficient are used, the number of parameters of these layers constitute the most significant portion of the model parameters.",
"In this work we focused on reducing the number of unnecessary parameters of convolutional layers by representing them in a low dimensional space through the use of a simple auxiliary neural network without significantly compromising the accuracy or tangibly adding to the processing burden.",
"There are still several directions that can be pursued.",
"The use of this method for other tasks, especially other than vision related tasks, such as natural language processing, etc.",
"needs to be assessed.",
"The extension of the theoretical analysis to other more complicated architectures is an attractive future direction.",
"The combination of this method with efficient computation, compression, and quantization methods mentioned in this paper for distributed machine learning and machine learning acceleration for edge devices are all worthwhile studies.",
"Also the use of more than one CSG for different classes of filters or the use of non-linear and/or multi-layer CSGs should be investigated."
],
[
"Estimating the Cardinality of the Code Vector Space",
"In this appendix we make the statements in Section REF more precise.",
"We first take the 4-D DCT-II of each slice defined in Section REF .",
"The 4-D DCT-II that we use, after removing the scaling factors, is stated as follows.",
"$&K[u,v,w,t] := \\sum _{i=0}^{\\hat{s}_1-1}\\sum _{j=0}^{\\hat{s}_2-1}\\sum _{k=0}^{\\hat{s}_3-1}\\sum _{l=0}^{\\hat{s}_4-1} k[i,j,k,l] \\nonumber \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\cos {\\left(\\frac{\\pi }{\\hat{s}_1}\\left(i+\\frac{1}{2}\\right)u\\right)}\\cos {\\left(\\frac{\\pi }{\\hat{s}_2}\\left(j+\\frac{1}{2}\\right)v\\right)}\\cos {\\left(\\frac{\\pi }{\\hat{s}_3}\\left(k+\\frac{1}{2}\\right)w\\right)}\\cos {\\left(\\frac{\\pi }{\\hat{s}_4}\\left(l+\\frac{1}{2}\\right)t\\right)}$ After taking the 4-D DCT-II, we then remov the elements of the slice in the transformed domain that were smaller than a threshold.",
"We then took the inverse transform.",
"The inverse 4-D DCT transform, after neglecting its scaling factors, can be stated as follows.",
"$&k[i,j,k,l] := \\sum _{u=0}^{\\hat{s}_1-1}\\sum _{v=0}^{\\hat{s}_2-1}\\sum _{w=0}^{\\hat{s}_3-1}\\sum _{t=0}^{\\hat{s}_4-1} K[u,v,w,t] \\nonumber \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\cos {\\left(\\frac{\\pi }{\\hat{s}_1}\\left(u+\\frac{1}{2}\\right)i\\right)}\\cos {\\left(\\frac{\\pi }{\\hat{s}_2}\\left(v+\\frac{1}{2}\\right)j\\right)}\\cos {\\left(\\frac{\\pi }{\\hat{s}_3}\\left(w+\\frac{1}{2}\\right)k\\right)}\\cos {\\left(\\frac{\\pi }{\\hat{s}_4}\\left(t+\\frac{1}{2}\\right)l\\right)}$ In order to measure the similarity between the inverse transformed version of the slice and the original slice, inspired by image compression similarity measures, we use a variation of a known measure called PSNR [33] which we define as follows.",
"Let $\\hat{k^*}$ denote the inverse DCT of the pruned DCT of the slice $\\hat{k}$ .",
"We re-scale the elements of the slices and their corresponding approximate version to $[0,1]$ with a bit of abuse of notation we represent the re-scaled versions with the same notations.",
"$PSNR^* = 10\\log \\frac{1^2}{MSE},$ where $MSE = \\frac{1}{\\hat{s}_1\\hat{s}_2\\hat{s}_3\\hat{s}_4}||\\hat{k}-\\hat{k}^*||^2_2.$ We chose the threshold for keeping the elements in the DCT domain such that the average PSNR* is above 20dB which from image compression literature is expected to result in images that are still recognizable (see, for instance [29]).",
"We then calculated the mean of the number of remaining elements in the DCT domain after the pruning step.",
"This suggests that a code size of 20 times fewer elements than its corresponding slice would be sufficient.",
"Based on these estimates, in most of our experiments we choose code vectors whose number of elements is 18 times smaller than that of their corresponding slices."
],
[
"Training on CIFAR-10 Dataset",
"The training and test error of different models for CIFAR-10 dataset and their CSG-augmented versions reported in Table REF , but not included in Figure REF are provided in figures REF -REF .",
"Figure: Training and test error for the CSG-augmented versions of DenseNet-BC-40-48 reported in Table not included in Figure over the course of 200 epochs.Figure: Training and validation error for DenseNet-BC-40-36 and its CSG-augmented versions over the course of 200 epochs.Figure: Training and test error for the CSG-augmented versions of DenseNet-BC-40-48 reported in Table not included in Figure over the course of 200 epochs."
],
[
"Training Convergence",
"In this section of the appendix we provide the proof of Theorem REF .",
"[Proof of Theorem REF ] First of all we note that since the number of weights in the convolutional layer is a polynomial function of $|\\hat{\\mathcal {C}}|$ , it has replaced the $m$ in Theorem REF .",
"Now, let $C = \\left[c_1,...,c_{\\hat{|\\mathcal {C}|}}\\right],$ denote a matrix whose columns are the code vectors corresponding to the slices of the convolutional layer.",
"Now, instead of assuming that the convolutional filter is first generated and then it is used for the convolution operation, equivalently, using associativity, we can assume that each column of the matrix $A_{CSG}$ denotes a vectorized version of a slice of a convolutional filter.",
"It means that, for each column of $A_{CSG}$ , we need to calculate the convolution of a slice for its $|\\hat{\\mathcal {C}}|$ possible locations in the filter.",
"But each of these would be an ordinary convolution with appropriate zero-paddings.",
"Now, the matrix $C$ can be viewed as an additional fully connected layer before the final classification layers.",
"Hence we are dealing with a CNN with an additional fully connected layer at the final stage for which the results in [2] and specially Theorem REF holds."
]
] | 1906.04309 |
[
[
"Classification of Radio Signals and HF Transmission Modes with Deep\n Learning"
],
[
"Abstract This paper investigates deep neural networks for radio signal classification.",
"Instead of performing modulation recognition and combining it with further analysis methods, the classifier operates directly on the IQ data of the signals and outputs the transmission mode.",
"A data set of radio signals of 18 different modes, that commonly occur in the HF radio band, is presented and used as a showcase example.",
"The data set considers HF channel properties and is used to train four different deep neural network architectures.",
"The results of the best networks show an excellent accuracy of up to 98%."
],
[
"Introduction",
"Classification of radio signals is an essential task in signal intelligence and surveillance applications and is recently adopted in applications like cognitive radio and dynamic spectrum access to continuously monitor the spectrum and its occupancy by various different radio signals, modes and services.",
"Traditional approaches to radio signal classification rely on signal features based on probabilistic methods, statistics or cyclostationarity.",
"These features need to be carefully designed by expert developers and thus depend on their experience and knowledge on the problem structure [1].",
"Recently advanced machine learning techniques, like deep neural networks, gained huge interest and showed extremely good performance for classification tasks in various applications.",
"Instead of formulating hand-crafted features by a designer, a training algorithm uses large amounts of labelled example data to learn and extract good features from the data that are discriminative for the classification task.",
"While this data-driven approach first proved to be very successful for object classification in image processing [2], it quickly advanced to numerous further applications, such as different challenges in radio communications [3].",
"For the task of signal classification, the neural network trains on large amounts of raw data of radio signals (e.g.",
"IQ data samples), such that it is enabled to distinguish the different signal classes.",
"Especially for the task of modulation classification, neural networks have shown to exhibit very good performance [4], [5].",
"Although recent work is based on IQ data samples as input, the output classes are restricted to modulation types (e.g.",
"FSK, PSK, AM).",
"To determine the actual transmission mode or service (Wi-fi, mobile telephony (GSM, LTE), digital radio (DAB), radioteletype (RTTY), AM broadcasting, etc.)",
"further processing is required, like the additional measurement of baud rate, pulse shape or bit pattern structures.",
"These additional features are not learnt by the neural network trained on modulation types and therefore this second step follows the classical way of designing expert features by hand.",
"This paper investigates how neural networks can be used to classify signals by their transmission mode directly, instead of classifying only modulation types.",
"The approach purely follows the data driven paradigm by mapping input IQ data to the output modes directly.",
"As an example application this paper considers radio signals typically present in the HF band (3-30 MHz), because this wireless band contains many different modes that coexist closely spaced in the frequency spectrum.",
"In total, 18 different HF transmission modes are considered for classification.",
"Four different types of neural networks are trained on a synthetically generated data set, considering a noisy HF channel environment under imperfect receiver conditions.",
"The investigation shows, that neural networks are very powerful in classifying signals into their transmission modes even if they exhibit very similar properties.",
"An accuracy of 98% for moderate SNRs can be obtained with reasonable training effort.",
"The remainder of the paper is structured as follows: Section describes the training data in more detail, Section introduces the models and the training process, followed by the results in Section .",
"The training data consists of the 18 different transmission modes shown in Table REF , of which many are commonly found in the HF bands.",
"This includes AM broadcasting, single-sideband (SSB) audio, radioteletype (RTTY, Navtex) [6], morse code [7], facsimile and further modes for digital data transmission like PSK31 [8], Olivia [9] and others.",
"The selected 18 modes cover very different modulation techniques, including analog (AM, facsimilie and SSB) and various digital modulation types (PSK, FSK, M-FSK, OOK, multicarrier).",
"However, there are also very similar modes, such as RTTY45 and RTTY50, that differ only by their baud rate of 45 and 50 Bd and are expected to be especially hard to classify.",
"Other similar modes are PSK31 and PSK63, as well as SSB in upper (USB) and lower (LSB) sideband.",
"Table: Transmission ModesThe training data is generated synthetically by simulation.",
"The raw data used to generate the training signals is plain text for the digital modes and speech and music (various genres) for the analog modes.",
"Fax data is generated by modulating different black & white images, such as e.g.",
"transmitted by weather services.",
"Then standard software modulates the raw signals in order to obtain baseband signals, which are then artificially distorted by a HF channel model, described in detail below.",
"Transmission in the HF band is mostly characterized by comparably small bandwidths (often less than 3 kHz) and therefore low data rates.",
"The data set contains vectors of complex IQ data of length 2048 with a sample rate of 6 kHz.",
"Thus a data vector corresponds to a time of approximately 0.3 s. In total, the data set consists of 120,000 training vectors and another 30,000 vectors for validation."
],
[
"HF Channel Properties",
"Since signals that are transmitted via the HF band exhibit special distortions, these effects needs to be reflected by the training data.",
"The HF band ranges from 3 to 30 MHz and is characterized by sky wave propagation, i.e.",
"radio waves do not travel by line-of-sight, but are reflected by the earth's ionosphere, which enables intercontinental communication with little technical effort.",
"In fact, different ionospheric layers, that may be located in different heights, contribute to the propagation of radio waves.",
"This complex propagation behaviour presents a multi path propagation environment resulting in fading effects.",
"Moreover, these ionospheric layers are not stationary, but are in continuous movement, which introduces varying doppler shifts.",
"For modelling radio wave propagation in the HF band, the Watterson model is commonly applied [10], that covers fading and doppler shift introduced by multipath propagation effects.",
"The ITU has defined several channel models based on the Watterson model, called CCIR 520 [11].",
"CCIR 520 includes different scenarios for complex wave propagation, that differ in the amount of distortion introduced, i.e.",
"frequency spread, frequency offset and differential time delay.",
"The scenarios employed in the training data sets are good conditions, moderate conditions, bad conditions, flutter fading and doppler fading, plus data vectors without any fading and doppler distortion.",
"In addition, all data vectors are distorted by Gaussian noise (AWGN) of different strength, such that the SNR of the data is evenly distributed from -10 to +25 dB.",
"To account for non-coherent reception and receiver frequency mismatch also random phase and frequency offsets are applied.",
"In summary, the training data set incorporates the following types of distortion to provide robust training results that perform well in real-world scenarios: CCIR 520 channel models AWGN noise (from -10 to +25 dB) random frequency offset (+- 250 Hz) random phase offset"
],
[
"Models and Training",
"This paper investigates four neural networks or models for signal classification, that range from ordinary convolutional neural nets (CNN) [12] to advanced models, like residual nets [13].",
"The structure of the models is depicted in Fig.",
"REF .",
"Classical CNN: It consists of six convolutional layers each followed by max pooling.",
"At the output two large dense layers are located.",
"All Convolutional Net: It is only composed of convolutional layers with stride 2 (instead of max pooling layers) and it has a global average pooling layer at its output [14].",
"Deep CNN: It uses 16 convolutional layers, with max pooling after every second convolutional layer and a single small dense layer at the output, roughly following the concept of the VGG net [15].",
"Residual Net: It uses a large number of layers enhanced by residual connections.",
"While residual nets as described in [13] (including the bottleneck architecture) did not provide convincing results, an arrangement of eight 5-layer residual stacks of [4] proved to be more appropriate for the task of signal classification and is applied here (Fig.",
"REF ).",
"The nets are chosen to have a similar number of parameters around 1.4M.",
"They adopt ReLU activation functions and softmax for the output layers.",
"The nets use dropout layers for regularization and batch normalization.",
"Following the ideas of [15] the convolutional layers mostly use a filter size of 3.",
"Figure: Different neural network architecturesFigure: The residual stack as used in the residualnet.",
"The parameter N provides the number of filters for the stack.Training is done with adam optimization [16] enhanced by a learning rate scheduler, that reduces the learning rate as soon as the training process plateaus.",
"The batch size is 128 and the number of training epochs is 30 to keep the training time moderate while achieving very good training accuracy."
],
[
"Results",
"Table REF shows a comparison of the results for the four models that have been trained with the previously described data set.",
"The validation data set to measure accuracy also contains signal data evenly distributed over the whole SNR range from -10 to 25 dB.",
"Therefore the provided accuracy values in Table REF are an average over all SNR values.",
"Best performance show the deep CNN (17 layers) with 93.7% and the residual net (41 layers) with 94.1%.",
"Although the residual net has much more layers, it provides only a minor improvement in accuracy.",
"Larger improvement in accuracy by the much deeper residual net may be obtained when using much more training time than the 30 epochs used for economic training in this paper.",
"Table: Overview of the different modelsand their performanceFig.",
"REF shows the accuracy of the different models over SNR.",
"Even for small SNR values of -5 dB, the accuracy for the best models is above 90%.",
"When SNR is above 5 dB the accuracy increases to an excellent value of approximately 98%.",
"Figure: Comparison of the results of the differentmodelsFig.",
"REF shows the confusion matrix for the residual net, that provides the best results.",
"Since the overall performance of the net is very good, confusions are very rare.",
"Minor confusions occur between QPSK and BPSK modes.",
"Also RTTY45 and RTTY50 tend to rarely be confused, because the only difference between these modes is the slightly deviating baud rate of 45 and 50.",
"Although LSB und USB are quite similar modes, they are classified with very high reliability.",
"Figure: Confusion matrix for the residual net(all SNR values)"
],
[
"Conclusion",
"This paper presented four different neural networks for the classification of radio signals.",
"Instead of focusing on modulation recognition, the models learn to classify different transmission modes directly.",
"This saves additional post-processing required to determine the modes from modulation and other signal parameters.",
"An exemplary data set with 18 different transmission modes that occur the HF band has been utilized, with an excellent accuracy of up to 98%."
]
] | 1906.04459 |
[
[
"Competing (Semi)-Selfish Miners in Bitcoin"
],
[
"Abstract The Bitcoin protocol prescribes certain behavior by the miners who are responsible for maintaining and extending the underlying blockchain; in particular, miners who successfully solve a puzzle, and hence can extend the chain by a block, are supposed to release that block immediately.",
"Eyal and Sirer showed, however, that a selfish miner is incentivized to deviate from the protocol and withhold its blocks under certain conditions.",
"The analysis by Eyal and Sirer, as well as in followup work, considers a \\emph{single} deviating miner (who may control a large fraction of the hashing power in the network) interacting with a remaining pool of honest miners.",
"Here, we extend this analysis to the case where there are \\emph{multiple} (non-colluding) selfish miners.",
"We find that with multiple strategic miners, specific deviations from honest mining by multiple strategic agents can outperform honest mining, even if individually miners would not be incentivised to be dishonest.",
"This previous point effectively renders the Bitcoin protocol to be less secure than previously thought."
],
[
"Introduction",
"One of the key innovations in the Nakamoto protocol behind Bitcoin [1] is the assumption that agents involved in the upkeep of the digital ledger, so called miners, are strategic rather than adversarial, which invites a game-theoretic analysis of the underlying protocol.",
"Under this relaxed assumption, Bitcoin enjoys more robust guarantees on its security: the adage being “it is in a miner's best interest to be honest when there is an honest majority of miners”.",
"This adage however was famously proven to be incorrect by Eyal and Sirer [2], when they first described “Selfish Mining”, a non-honest miner strategy that gives more returns to miners than honest mining, even if a majority of other agents are honest.",
"Subsequently, there has been much work exploring the extensions and limitations of selfish mining, but most of this work is limited to the case in which there is a single selfish miner and the rest of the network acts honestly.",
"In this paper we study scenarios where more than one miner deviates from the honest mining protocol.",
"We show that there are substantial game-theoretic differences when multiple miners can be strategic with implications to Bitcoin's security.",
"First of all, there are hash rates where a miner is incentivised to be honest if mining is treated as a one-shot game, yet where the miner is incentivised to be strategic if he is the leader in sequential (Stackleberg) game.",
"Second of all, we show that with multiple strategic miners, specific deviations from honest mining by multiple strategic agents can outperform honest mining, even if individually miners would not be incentivised to be dishonest.",
"These two previous points effectively render the Bitcoin protocol to be less secure than previously thought.",
"We study miner incentives when multiple miners employ variants of selfish mining strategies.",
"Original selfish mining (SM) consists of secretly withholding mined blocks and judiciously publishing private blocks in an attempt to increase stale block rates of other miners.",
"Though such an attack is not immediately profitable, as the block rate of all miners decreases, it can be profitable in a longer time horizon as block difficulty rates decrease.",
"In SM, miners may keep an arbitrarily long private chain, which makes it difficult to analytically solve for relative revenues when more than one miner employs SM.",
"For this reason, we study a truncation of this strategy, semi-selfish mining (SSM), where miners keep a private chain of length at most 2.",
"SSM falls within the family of generalised selfish mining strategies of [3] and [4], and our paper begins by studying analytic properties of SSM's performance against honest mining.",
"In Section , we show that although SSM achieves less relative revenue than SM against honest mining, it is always a more profitable strategy for a strategic miner than honest mining if the miner has a hash rate larger than $38\\%$ of the total system hash rate, and if the strategic miner is able to propagate blocks to other miners quickly, this threshold lowers to around $26.8\\%$ .",
"In fact, we show the relative revenue of SSM is an asymptotically tight lower bound to the relative revenue of SM as a strategic miner's hash power tends to 0.",
"As mentioned before, the benefit of SSM is that it can be represented with a reduced state space, and hence we can explicitly solve for relative revenues in the case where multiple strategic miners employ SSM.",
"In Section we focus on systems with two strategic miners and describe the Markov chain that governs block publishing dynamics.",
"This allows us to explicitly solve for relative revenues of all miners in the steady state.",
"With the steady state solutions in hand, we are able to study the incentives that govern the decision whether a miner uses SSM against another strategic miner.",
"To do so, we define a binary action two-player game amongst both strategic miners which we call the SSM game.",
"In the SSM game both miners are denoted by $m_1$ and $m_2$ and they have corresponding utility functions $U_1$ and $U_2$ .",
"In addition, each miner has the action set $\\lbrace H,S\\rbrace $ representing honest mining and SSM mining.",
"Interestingly, we find multiple scenarios at different hash rates: For all pure strategy profiles, $s \\in \\lbrace (H,H),(S,H),(H,S),(S,S)\\rbrace $ , there exist hash rates of both strategic miners such that $s$ is a unique pure Nash equilibrium.",
"When both strategic miners have roughly around 0.2 to 0.27 of the system's hash power, both $(H,H)$ and $(S,S)$ are simultaneously pure Nash equilibria of the SSM game.",
"There exist hash rates where a specific miner is not unilaterally incentivised to employ SSM, yet $(S,S)$ is the only pure Nash equilibrium of the game.",
"This effectively lowers the minimum hash rate required for SSM to be profitable by virtue of the existence of another strategic miner.",
"There exist hash rates where $U_1(S,S) < U_1(H,H) < U_1(S,H)$ (once again, an identical result holds with the roles of miners reversed).",
"This is interesting because although SSM is individually rational for the first strategic miner, the second (larger) strategic miner has the ability to “penalise” the first miner were they to retaliate by using SSM.",
"We also consider a richer action space for miners: we allow them to partition their hash power into an honest portion and an SSM portion.",
"The game specified by these utilities is called the partition game, and when treated as a one-shot game, it yields the same pure Nash equilibria as the SSM game.",
"The more interesting result stems from treating this game as a Stackelberg game and understanding optimal commitments a miner may make to elicit a desired behaviour in the other miner.",
"It turns out that in the partition game, there exist hash rates with non-trivial Stackelberg equilibria that can result in large gains for leader miners.",
"In fact, there are even hash rates where a miner is honest in the one-shot SSM game, yet strategic in the sequential partition game's Stackelberg equilibrium.",
"This has important consequences for the security of Bitcoin, as miners with smaller hash rates than what was known before may be incentivised to be strategic in a sequential setting.",
"In Section we consider the scenario where $M > 2$ miners are strategic.",
"For $1 \\le M \\le 8$ , we compute bounds on the minimal $\\alpha \\in [0,1]$ such that if the $M$ strategic miners each with hash power $\\alpha $ have to decide between employing honest mining and SSM, the strategy profile where all such miners employ SSM Pareto-dominates honest mining.",
"For each $M$ , we call $\\alpha $ the uniform profitability threshold for SSM, and we show that not only is it a decreasing function in $M$ , but that already for $M = 8$ , $\\alpha $ is as low as 0.11.",
"This is striking, because at such hash rates, miners are far from being individually incentivised to employ SSM, implying that the existence of other strategic miners can effectively hurt the stability of Bitcoin.",
"As an aside, we also note that in Appendix we explicitly extend our game-theoretic formalism from Section and Section to the multi-player setting, and we specify how to compute utilities in these games.",
"Furthermore, in Appendix we extensively map incentives of 3 strategic miners akin to Section and Section .",
"We find that the game-theoretic observations of the two-player setting generalise appropriately."
],
[
"Related Work",
"Selfish mining was originally introduced by Eyal and Sirer in [2].",
"In this work, the authors describe Selfish Mining (SM), a specific mining strategy that deviates from the prescribed honest mining strategy of the Bitcoin network with the key property that it is more profitable than honest mining for miners with over $1/3$ of the hash power of the entire Bitcoin network.",
"Subsequently, [4] and [3] identify a generalised class of selfish mining strategies to which SM belongs and show that in general there are more aggressive and profitable strategies than SM within this family of strategies.",
"In a similar vein, [5] uses game theory to formalise the decision a single strategic miner may take to employ different strategies from the generalised family of selfish mining strategies.",
"In particular, they define analogous complete information games to real-life mining and show that for these games, if no miner has a large enough hash power, honest mining is a Nash equilibrium.",
"Perhaps most similar to our work is [6], where the authors simulate multiple strategic miners employing strategies other than honest mining.",
"Their results are simulation-based, whereas we provide closed-form results for the specific SSM strategy.",
"In fact, our model can be seen as a variant of the model used in [7], which we developed concurrently to allow for an arbitrary number of strategic agents employing SSM.",
"Furthermore we focus on the game-theoretic considerations miners may take in deciding whether to employ SSM in varying degrees.",
"Subversive mining strategies can also be combined with network level attacks to exacerbate undue profits.",
"This is discussed in [4] where the authors combine selfish mining strategies with eclipse attacks; an eclipse attack is when an entity holds all connections with a subset of the mining swarm and can thus control all communication between them and the rest of the miners.",
"The authors show that no combination of a selfish mining strategy and eclipse attack is optimal at all times.",
"The choice of what selfish mining strategy to adopt as well as how to eclipse a victim is highly dependent on the network parameters in which one is operating.",
"These parameters include computational power, percentage of the network that can be eclipsed, and the percentage of remaining miners that can be influenced.",
"There are additional attacks miners can wage outside the family of selfish mining.",
"At the pool level, managers can wage withholding attacks as per [8] [9], where a malicious pool infiltrates a victim pool, submitting shares and withholding full solutions.",
"Indeed this notion of “partitioning” one's pool is similar to our partition games from section .",
"[9] shows that this can be profitable for a single malicious pool, but when multiple pools engage in block withholding attacks, this results in a situation akin to the prisoner's dilemma, where the equilibrium of all malicious pools is to infiltrate and thus reduce the overall profit of every pool in the network.",
"Withholding attacks are further refined in [10], where a malicious pool still withholds full solutions from a victim pool, but may share said full solutions when it hears of a full solution being found by a miner outside of the malicious and victim pool.",
"The intent of this strategy is to incentivise the victim pool manager to cause a fork, and this behaviour does away with the prisoner's dilemma of [9], as there are equilibria where larger pools are strictly better off than honest mining.",
"Furthermore, there is some evidence showing that this family of pool-level attacks can be difficult to detect for victim pools [8].",
"Along with work covering subversive mining attacks and which strategies miners should adopt based on network parameters, there have also been efforts to defeat these attacks.",
"In [11] the authors outline a new blockchain protocol, Bitcoin Next Generation, which decouples leader election and transaction serialization for better scalability.",
"In addition to this they also modify which chain honest miners adopt as the one they will mine on.",
"Currently, when honest miners are presented with two chains of the same length, they will opt to accept the older one.",
"This fact gives selfish miners an advantage in that they become more powerful the more connected they are to the rest of the network and can lower the necessary computational power needed to selfish mine successfully.",
"In their new protocol, they propose that when an honest miner is presented with two chains of the same length, they choose which one to mine on uniformly at random.",
"With this change, the lower bound on computational power needed to selfish mine increases, thus making it harder to act subversively.",
"While this was conjectured to be true and showed to be so with simulation, there are contradictory results.",
"In [12] the authors show that while this change does limit the strength of large selfish miners, it enhances the strength of medium sized selfish miners and that selfish miners with computational power less than 25% can still gain from acting subversively.",
"The decentralised design of Bitcoin consists of clients: users of Bitcoin, who own accounts designated by addresses.",
"A client can send Bitcoin from an address he owns to an arbitrary address by broadcasting a transaction to the Bitcoin P2P network.",
"This transaction will eventually be appended to a a global ledger called the Blockchain.",
"The upkeep of the Blockchain is performed by miners, who collect transactions in blocks and append these blocks to the chain.",
"For this task, miners are rewarded with Bitcoin, either in the form of a block reward or transaction fees.",
"We model the Blockchain system as a set of $M$ strategic miners, $m_1,...,m_M$ , and an implicit honest miner $m_{M+1}$ .",
"Each strategic miner $m_i$ , controls an $\\alpha _i \\in (0,0.5]$ portion of the system hash power (we don't consider strategic miners strong enough to perform a 51 percent attack), and the honest miner $m_{M+1}$ controls a $\\beta = 1 - \\sum _{i=1}^M \\alpha _i > 0$ portion of the system hash power.",
"The implicit honest miner is without loss of generality for if any number of miners (beyond the strategic miners $m_1,...,m_M$ ) employ honest mining, this is equivalent to one miner of their combined hash power employing honest mining.",
"For convenience we denote the set of valid strategic miner hash rates by $\\mathcal {H}^M = \\lbrace \\alpha \\in (0,0.5]^M \\ | \\ \\sum _{i=1}^M \\alpha _i < 1 \\rbrace $ .",
"Given strategic miner hash rates $\\alpha \\in \\mathcal {H}^M$ , any found block has an $\\alpha _i$ probability of being found by the $i$ -th strategic miner $m_i$ , and a $\\beta $ probability of being found by $m_{M+1}$ .",
"We also assume that the system overall finds blocks at a rate of $\\lambda $ according to a Poisson process.",
"In terms of the actual implementation of the Bitcoin protocol, $\\lambda $ is roughly one block every 10 minutes, which is ensured by dynamically adjusting the difficulty of the block hash target.",
"The append-only nature of the block renders the Blockchain into a tree with a root at the genesis block.",
"Since the longest path of the tree is the agreed-upon transaction history, a miner's revenue consists of his block rewards and transaction fees arising from blocks that eventually become a part of the longest path in the Blockchain.",
"In this paper we focus on block rewards and normalise such rewards to unit value, hence the revenue of a miner is the number of his blocks that are accepted in the longest path of the blockchain.",
"Indeed it could be the case that a longest path in the blockchain is eventually surpassed by a competing path: this is a key aspect to selfish mining strategies.",
"This of course makes it difficult to ascertain revenues when miners are arbitrary agents.",
"In our paper however we pit specific mining strategies against each other and hence obtain a well-defined block creation rates for all agents involved.",
"Furthermore, we assume that agents are rational and that the utility they wish to maximise is their relative revenue: which is for a miner $m_i$ is the expected number of blocks $m_i$ publishes in the blockchain normalised by the expected number of blocks produced by all miners $m_1,...,m_{M+1}$ .",
"The justification behind this utility function comes from the fact that Bitcoin dynamically adjusts its difficulty, hence relative revenue in the long-term corresponds to overall revenue."
],
[
"Miner Strategies",
"Mining strategies are often defined with an implicit assumption that a miner following the strategy will be pitted against miners employing a specific strategy (i.e.",
"honest mining).",
"Since our paper focuses on miner incentives when multiple miners deviate from honest mining, we find ourselves in need of rigorously defining miner strategies with respect to all possible changes in the blockchain, not just those changes that can occur against a specific kind of miner.",
"In this vein, we formally describe three specific mining strategies: honest mining, selfish mining, and semi-selfish mining.",
"We describe the strategies for an arbitrary miner denoted by $m$ .",
"To execute these strategies, $m$ must keep track of their private chains, the public chain, a block upon which to mine and an internal state $\\ell \\in \\lbrace 0,0^{\\prime }\\rbrace \\cup \\mathbb {N}$ .",
"As for additional notation, $priv$ denotes the private chain of $m$ , $pub$ denotes the public chain and $F$ (frontier) denotes the set of blocks at the ends of the longest paths of the public chain.",
"Arbitrary blocks are usually denoted by $B$ .",
"We also let $len(priv)$ and $len(pub)$ denote the length of the longest path in the miner's private chain and the length of the longest path of the public chain respectively.",
"For a given set of a blocks $S$ , we let $oldest(S)$ denote the oldest block in $S$ of which $m$ was aware.",
"Finally, we let $end(priv)$ be the block at the end of the miner's private chain and $p(m)$ be the block upon which $m$ is mining.",
"The integer of the internal state, $\\ell $ represents a miner's “lead”: how much longer the miner's private chain is than the public chain.",
"For all three mining strategies states 0 and 0' will not only mean that the miner has no lead with respect to the public chain, but that the miner's private chain is in fact the public chain (a fact which follows from the rules governing the strategies).",
"Finally, the difference between 0 and 0' is that the latter state occurs when there is a tie on the public chain, i.e.",
"$|F|>1$ .",
"The choices available to miners are where to mine, $p(m)$ , and whether to reveal parts of their private chain."
],
[
"Honest Mining",
"Honest miners are those who follow the prescribed Bitcoin mining protocol faithfully.",
"We describe the strategy in terms of what actions $m$ takes when in states $\\ell \\in \\lbrace 0, 0^{\\prime }\\rbrace $ : Case 1: $m$ finds a block, $B$ .",
"$m$ publishes $B$ .",
"$\\ell \\leftarrow 0$ .",
"$p(m) \\leftarrow B$ .",
"Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ .",
"If $|F^{\\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime })$ .",
"It is straightforward to check that if all miners mine honestly, their expected relative revenue is precisely their hash rate: For any $\\alpha \\in \\mathcal {H}^M$ , if all strategic miners are honest, the expected (block) reward of any strategic miner $m_i$ is $\\alpha _i$ (and $\\beta $ for the extra honest miner $m_M$ )."
],
[
"Selfish Mining",
"Eyal and Sirer introduced Selfish Mining (SM) in [9] as a specific strategy that outperforms honest mining when a rational agent has sufficient computational resources.",
"SM can be described by the actions $m$ takes in the following states:"
],
[
"$\\ell = 0$ and {{formula:f7ac43da-7573-4fac-b84a-fc4f5ae7d8f6}}",
" Case 1: $m$ finds a block: $B$ .",
"$m$ keeps $B$ private.",
"$\\ell \\leftarrow 1$ .",
"$p(m) \\leftarrow B$ .",
"Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ If $|F^{\\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime })$ Case 1: $m$ finds a block: $B$ .",
"$m$ publishes $B$ .",
"$\\ell \\leftarrow 0$ .",
"$p(m) \\leftarrow B$ Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ .",
"If $|F^{\\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime })$ Case 1: $m$ finds a block: $B$ .",
"$m$ keeps $B$ private.",
"$\\ell \\leftarrow \\ell + 1$ .",
"$p(m) \\leftarrow B$ .",
"Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ and $k = len(pub^{\\prime }) - len(pub) \\le max(\\ell -2, 0)$ .",
"$m$ publishes $k$ -prefix of $priv$ .",
"$\\ell \\leftarrow \\ell - k$ .",
"$p(m) \\leftarrow p(m)$ .",
"Case 3: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ and $k = len(pub^{\\prime }) - len(pub) > max(\\ell -2, 0)$ .",
"$m$ publishes $priv$ , resulting in $pub^{\\prime \\prime }$ with frontier $F^{\\prime \\prime }$ If $|F^{\\prime \\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime \\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime \\prime })$ .",
"At a glance, this characterisation of SM may look different to how it is usually described.",
"Upon closer inspection however, one can see that this is equivalent to what was presented in [9].",
"In particular, the fact that in state $\\ell = 0^{\\prime }$ , $p(m) = oldest(F)$ , means that when a tie involves a block mined by $m$ (as would be the case if they had published a previously private block), they will indeed continue mining upon it, as they will have necessarily seen it first amongst blocks in $F$ .",
"Figure: SM Dynamics.",
"A square with MM is a block mined by mm.",
"The circle with mm represents the value of p(m)p(m).",
"A solid line means that portion of the chain is public, and a dashed line means that portion of the chain is private.",
"If a miner employing SM has a lead of ℓ=1\\ell = 1 that is diminished, he publishes his private chain and hopes to win the tie (Top).",
"If the miner has a larger lead that is partially encroached, he publishes a prefix of his private chain to push other miners into a race (Middle).",
"If a miner of lead ℓ>1\\ell > 1 sees his lead encroached to ℓ=1\\ell = 1, he publishes all blocks to overtake (Bottom)."
],
[
"Semi-Selfish Mining",
"SM can be generalised to a class of strategies where a miner maintains a private chain and has the following actions at hand: publishing a portion of his private chain, mining upon his private chain, and foregoing his private chain to mine upon the public chain.",
"Indeed, this general class of selfish mining strategies is studied in [13] and [3].",
"We focus on the simplest selfish mining strategies from this family by looking at strategies where the selfish miner never maintains a private chain of length greater than 2.",
"Notice that this is necessary if the selfish miner is to gain any benefit from selfish mining, for if the miner only maintains at most one private block, he can only hurt his chances of having this block (and hence any block) published when facing honest miners.",
"On the other hand, for private chains of length 2, we exhibit a specific strategy Semi-Selfish Mining (SSM) that much like the original SM strategy, leads to increased revenue ratios for the selfish miner if they have sufficient hash power.",
"The reason we study such a simple strategy from the rich space of selfish mining strategies is that it still obtains higher relative revenues than honest mining in certain parameter regimes, yet it has a much simpler state space than most selfish mining strategies.",
"This reduced state space will eventually allow us to explicitly solve for expected relative revenues when two selfish miners play against each other.",
"SSM can be described by the actions $m$ takes in the following states:"
],
[
"$\\ell = 0$ and {{formula:fc10d0f1-187c-41e3-910b-832b7771acf3}}",
" Case 1: $m$ finds a block: $B$ .",
"$m$ keeps $B$ private.",
"$\\ell \\leftarrow 1$ .",
"$p(m) \\leftarrow B$ .",
"Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ If $|F^{\\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime })$ Case 1: $m$ finds a block: $B$ .",
"$m$ publishes $B$ .",
"$\\ell \\leftarrow 0$ .",
"$p(m) \\leftarrow B$ Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ .",
"If $|F^{\\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime })$ Case 1: $m$ finds a block: $B$ .",
"$m$ keeps $B$ private.",
"$\\ell \\leftarrow 2$ .",
"$p(m) \\leftarrow B$ .",
"Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ and $k = len(pub^{\\prime }) - len(pub) = 0$ .",
"$m$ does nothing.",
"Case 3: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ and $k = len(pub^{\\prime }) - len(pub) > 0$ .",
"$m$ publishes $priv$ , resulting in $pub^{\\prime \\prime }$ with frontier $F^{\\prime \\prime }$ If $|F^{\\prime \\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime \\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime \\prime })$ .",
"Case 1: $m$ finds a block: $B$ .",
"$m$ publishes $oldest(priv \\setminus pub)$ .",
"$\\ell \\leftarrow 2$ .",
"$p(m) \\leftarrow B$ .",
"Case 2: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ and $k = len(pub^{\\prime }) - len(pub) = 0$ .",
"$m$ does nothing.",
"Case 3: $pub$ changes to $pub^{\\prime }$ with frontier $F^{\\prime }$ and $k = len(pub^{\\prime }) - len(pub) > 0$ .",
"$m$ publishes $priv$ , resulting in $pub^{\\prime \\prime }$ with frontier $F^{\\prime \\prime }$ If $|F^{\\prime \\prime }| = 1$ , then $\\ell \\leftarrow 0$ .",
"If $|F^{\\prime \\prime }| > 1$ , then $\\ell \\leftarrow 0^{\\prime }$ .",
"$p(m) \\leftarrow oldest(F^{\\prime \\prime })$ .",
"Figure: SSM as a truncation of SM.",
"Once again, a square with MM is a block mined by mm.",
"The circle with mm represents the value of p(m)p(m).",
"A solid line means that portion of the chain is public, and a dashed line means that portion of the chain is private.",
"Here mm has a lead of ℓ=2\\ell = 2 and upon mining a block, publishes his oldest private block."
],
[
"One Strategic Miner",
"We begin by studying how one strategic miner of hash power $\\alpha \\in \\mathcal {H}^1 = (0,0.5]$ performs against honest miners of hash power $\\beta = 1 - \\alpha $ in terms of relative revenue.",
"As in [9], we let $\\gamma $ be the proportion of honest miners who mine upon an SSM chain in the case of a tie, a parameter which we call the propagation of the strategic miner.",
"In what follows, we let $r_{SM}$ and $r_{SSM}$ be the expected block creation rate of a single miner using SM and SSM respectively against honest miners.",
"Consequently, we let $r_{others}$ be the block creation of other honest miners in the system (this is dependant upon whether SM or SSM is used, but we use the same term for the sake of simplicity).",
"Finally, we let $R_{SM}$ and $R_{SSM}$ denote the relative revenues of a single miner using SM and SSM respectively against honest miners.",
"[Selfish Mining Relative Revenue [9]] A single strategic miner of hash power $\\alpha $ and propagation $\\gamma $ , attains the following revenue ratio using SM against honest miners: $R_{SM} = \\frac{r_{SM}}{r_{SM} + r_{others} } =\\frac{\\alpha (1-\\alpha )^2(4\\alpha + \\gamma (1-2\\alpha )) - \\alpha ^3}{1 - \\alpha (1 + (2-\\alpha )\\alpha )}$ Asymptotically around $\\alpha = 0$ the expression is the following: $R_{SM} = \\alpha \\gamma + \\alpha ^2(4-3\\gamma ) + \\alpha ^3(4\\gamma - 5) + \\alpha ^4 (7 - 5\\gamma ) + \\alpha ^5(6\\gamma - 7) + O(\\alpha ^6)$ We can use similar Markov chain methods to derive the revenue ratio of SSM against honest miners.",
"The details of the analysis can be found in Appendix .",
"A strategic miner of hash power $\\alpha $ and propagation $\\gamma $ , attains the following revenue ratio when using SSM against honest miners: $R_{SSM} = \\frac{r_{SSM}}{r_{SSM} + r_{others}} =\\frac{\\alpha (\\alpha (\\alpha (2\\alpha -5) + 4) - (\\alpha -1)^3\\gamma )}{(\\alpha -1)\\alpha ^2 + 1}$ Asymptotically around $\\alpha = 0$ , the expression is the following: $R_{SSM} = \\alpha \\gamma + \\alpha ^2(4 - 3\\gamma ) + \\alpha ^3(4\\gamma - 5) - \\alpha ^4(6 - 5\\gamma ) + \\alpha ^5(7\\gamma - 9) + O(\\alpha ^6)$"
],
[
"Comparing Performance of SM and SSM",
"Asymptotically SM and SSM have the same performance as $\\alpha \\rightarrow 0$ .",
"In fact $R_{SM} - R_{SSM} = O(\\alpha ^4)$ .",
"For all parameter settings SM outperforms SSM, as evidenced in the graphs in Figure REF .",
"At $\\gamma = 0$ SM becomes profitable at $\\alpha = 1/3$ and SSM becomes profitable at $\\alpha = 0.38$ .",
"At $\\gamma = 0.25$ SM becomes profitable at $\\alpha = 0.3$ and SSM becomes profitable at $\\alpha = 1/3$ .",
"Finally, at $\\gamma = 0.5$ SM becomes profitable at $\\alpha = 1/4$ and SSM becomes profitable at $\\alpha = 0.26795$ Figure: R SM R_{SM} and R SSM R_{SSM} against honest miners at γ=0\\gamma = 0, 0.250.25 and 0.50.5"
],
[
"Two Strategic Miners",
"The benefit of SSM lies in the fact that it can be a rational strategy distinct from honest mining and more importantly, describing it in terms of a Markov chain does not require many states.",
"The simplicity of the state space allows us to explore the scenario where two agents of different hash rates employ SSM and analytically solve for relative revenues."
],
[
"Markov Chain Analysis",
"Suppose that $\\alpha = (\\alpha _1,\\alpha _2) \\in \\mathcal {H}^2$ is the strategic hash rate of the system.",
"Since we have two strategic miners, our state space, $S$ , consists of nine states of the form $S_{i,j}$ where $0 \\le i,j \\le 2$ .",
"These represent the relative lead SSM miners 1 and 2 have with respect to the public chain.",
"Given our description of SSM we can describe the state transitions in the same way as we did for the single SSM case.",
"Both of these can be found in Appendix ."
],
[
"Transition Matrix and Steady State",
"The above state space gives rise to an ergodic Markov chain, so there is a unique stationary distribution we can solve for.",
"In order to do so, we define the following transition matrix, $P$ , on $\\mathbb {R}^9$ , where the coordinate axes of $\\mathbb {R}^9$ (in ascending order) represent probability mass in states $S_{0,0}, S_{0,1}, S_{1,0}, S_{0,2}, S_{1,1}, S_{2,0}, S_{1,2}, S_{2,1}$ , and $S_{2,2}$ respectively.",
"Each $P_{x,y}$ is the probability of transitioning to state $x$ from state $y$ in the Markov chain.",
"$P=\\begin{bmatrix}\\beta & \\beta & \\beta & \\beta & \\beta & \\beta & \\alpha _2(1-\\alpha _2) + \\beta \\ & \\alpha _1(1-\\alpha _1) + \\beta \\ & 1 \\\\\\alpha _2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\alpha _1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & \\alpha _2 & 0 & \\alpha _2 & 0 & 0 & \\alpha _2^2 & 0 & 0 \\\\0 & \\alpha _1 & \\alpha _2 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & \\alpha _1 & 0 & 0 & \\alpha _1 & 0 & \\alpha _1^2 & 0 \\\\0 & 0 & 0 & \\alpha _1 & \\alpha _2 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & \\alpha _1 & \\alpha _2 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & \\alpha _1 & \\alpha _2 & 0\\end{bmatrix}$ Since this is an ergodic chain, there is a unique steady state distribution, $\\pi $ , such that $P \\pi = \\pi $ , which we can solve for with Gaussian elimination."
],
[
"Propagation and Revenues",
"In the original selfish mining paper, much attention was given to the propagation parameter $\\gamma $ .",
"Indeed block dissemination is important because it allows an attacker to persuade other miners to work on their chain in the case of a tie.",
"We also note from the previous section that the steady state distribution $\\pi $ is independent of the propagation of the system.",
"The expected number of blocks published per state however, crucially depends on the propagation of the system, and these two objects specify the relative revenue of agents.",
"In our work, when there is a single strategic miner employing SSM, the ability to propogate blocks is parametrised by $\\gamma $ as in the original analysis of SM.",
"When there are strategic miners employing SSM however, how propagation is modelled becomes more complicated, since different strategic miners may have a different influence on the P2P network topology.",
"For the rest of the paper we assume that propagation is uniform.",
"In other words, whenever there is a tie in the public chain (of arbitrary size), all miners not involved in the tie are assumed to have a uniformly random chance of contributing their hash power to any element of the tie.",
"Under the assumption of uniform propagation, we can compute the expected block rate per state of the Markov chain for both strategic miners and honest miners.",
"The following matrix $R$ encodes this information: the first and second row are expected block rates per state for the first and second strategic miners respectively, the third column is the block creation rate for honest miners.",
"If $\\pi $ is a steady state vector for $M$ above, then $R^T \\pi \\in \\mathbb {R}^3$ gives steady state expected block creation rates for all miners.",
"$R=\\begin{bmatrix}0 & 0 & \\beta \\\\\\beta \\alpha _1 & 2\\beta \\alpha _2 + \\frac{1}{2}\\beta (1- \\alpha _2) & \\frac{1}{2} \\beta \\alpha _1 + \\frac{3}{2}\\beta ^2 \\\\2\\beta \\alpha _1 + \\frac{1}{2}\\beta (1- \\alpha _1) \\ & \\beta \\alpha _2 & \\frac{1}{2} \\beta \\alpha _2 + \\frac{3}{2}\\beta ^2 \\\\0 & \\alpha _2 + 2\\beta & 0 \\\\2\\beta \\alpha _1 + \\frac{1}{3}\\beta ^2 & 2\\beta \\alpha _2 + \\frac{1}{3}\\beta ^2 & \\frac{4}{3}\\beta ^2 \\\\\\alpha _1 + 2\\beta & 0 & 0 \\\\0 & 2\\beta + 2\\alpha _2^2 + 3\\alpha _2(1-\\alpha _2) & 0 \\\\2\\beta + 2\\alpha _1^2 + 3\\alpha _1(1-\\alpha _1) & 0 & 0 \\\\3\\alpha _1 + 3\\beta \\alpha _1 + \\frac{1}{2}\\beta ^2 & 3\\alpha _2 + 3\\beta \\alpha _2 + \\frac{1}{2}\\beta ^2 & \\beta ^2\\end{bmatrix}$ For the sake of completeness, in Appendix we include a model for different propagation rates when two strategic miners are involved as well as their effects on relative revenues of all miners."
],
[
"To SSM or not to SSM? A Revenue Analysis",
"Although our Markov chain analysis gives us a closed-form solution for the relative revenue of both strategic miners when using SSM, the expression is unwieldy.",
"We can however explicitly solve the expression for specific hash values, $\\alpha _1$ and $\\alpha _2$ and use these values to describe a two-player, binary action game governing the decision as to whether a player employs SSM or not.",
"Suppose that $\\alpha = (\\alpha _1, \\alpha _2) \\in \\mathcal {H}^2$ describes the hash rates of both strategic miners.",
"We let $R_{SSM}(\\alpha ) = R_{SSM}((\\alpha _1,\\alpha _2)) \\in [0,1]^3$ be the revenue ratios of all miners (including the honest miner $m_3$ ) when both strategic miners employ SSM.",
"Specifically, $R_{SSM}(\\alpha )_i$ is the revenue ratio of $m_i$ for $i = 1,2,3$ .",
"With this in place we can define a two-player binary action game governing the incentives behind employing SSM or not for $m_1$ and $m_2$ .",
"[Two-player SSM Games] Suppose that $\\alpha = (\\alpha _1,\\alpha _2) \\in \\mathcal {H}^2$ is a strategic hash distribution.",
"We define the SSM Game, $G_\\alpha $ as a two-player binary action game.",
"In $G_\\alpha $ each strategic miner has a binary action set $\\lbrace H,S\\rbrace \\cong \\lbrace 0,1\\rbrace $ , where $H \\cong 0$ represents mining honestly and $S \\cong 1$ represents employing SSM.",
"We define the utilities of all pure strategy profiles as follows: $U_1(H,H) = \\alpha _1$ , $U_2(H,H) = \\alpha _2$ $U_1(H,S) = \\frac{\\alpha _1}{1- \\alpha _2}R_{SSM}((0,\\alpha _2))_3$ , $U_2(H,S) = R_{SSM}((0,\\alpha _2))_2$ $U_1(S,H) = R_{SSM}((\\alpha _1,0))_1$ , $U_2(S,H) = \\frac{\\alpha _2}{1 - \\alpha _1}R_{SSM}((\\alpha _1,0))_3$ $U_1(S,S) = R_{SSM}(\\alpha )_1$ , $U_2(S,S) = R_{SSM}(\\alpha )_2$ For notational convenience, we interchangeably denote a pure strategy profile of all players by either a tuple, as in $(H,H)$ for both miners employing honest mining, or a string, as in $HH$ As a first region of interest, in Figure REF we display hash rates where $U_1(S,S) < U_1(H,H) < U_1(S,H)$ .",
"For such $\\alpha $ , although SSM may be unilaterally rational for the first strategic miner, a larger miner can penalise the first strategic miner for deviating from the honest protocol by retaliating with SSM.",
"As a specific example of this phenomenon, let us consider the hash distribution $\\alpha = (0.33, 0.48)$ which leads to $G_\\alpha $ with utilities summarised in Table REF .",
"The second, larger, strategic miner $m_2$ can retaliate from $SH$ by deviating to $SS$ , in which case $m_1$ is worse off by approximately $0.04$ in utility than if he had mined honestly at the outset.",
"Table: Example of G α G_\\alpha where m 2 m_2 can retaliate against SHFigure: Hash rates where U 1 (S,S)<U 1 (H,H)<U 1 (S,H)U_1(S,S) < U_1(H,H) < U_1(S,H) and subsequent penalty values given by U 1 (S,S)-U 1 (H,H)U_1(S,S) - U_1(H,H).Now that we have defined the game $G_\\alpha $ , it is natural to ask about what equilibria it has.",
"Our results suggest that for all values of $\\alpha \\in \\mathcal {H}^2$ , $G_\\alpha $ has at least one pure Nash equilibrium (PNE), so that if we let PNE$(G)$ denote the PNE of a given game $G$ , PNE$(G_\\alpha ) \\ne \\emptyset $ for $\\alpha \\in \\mathcal {H}^2$ .",
"In the first image of Figure REF we show which regions of $\\mathcal {H}^2$ demonstrate different combinations of PNE.",
"For the most part, hash rates lead to a single PNE in $G_\\alpha $ , with distinct regions where each pure strategy profile $(HH, HS, SH$ , and $SS)$ occurs as a sole equilibrium.",
"The most interesting observation however, is that for $\\alpha $ roughly in the region $[0.2,0.27]^2$ , $\\text{PNE}(G_\\alpha ) = \\lbrace HH \\text{ and } SS \\rbrace $ .",
"For all of these hash rates, $SS$ Pareto dominates $HH$ as it results in more utility for both agents involved.",
"As a concrete example, consider $G_\\alpha $ for $\\alpha = (0.24, 0.24)$ with utilities in Table REF .",
"Clearly $HH$ and $SS$ are PNE in $G_\\alpha $ , and the utility surplus between $SS$ and $HH$ is approximately $0.02$ for $m_1$ and $m_2$ .",
"Table: Example of G α G_\\alpha with HH and SS as PNEThe second image in Figure REF focuses on $[0.2,0.27]^2 \\subset \\mathcal {H}^2$ and visualises the difference in utility between $SS$ and $HH$ for $m_1$ .",
"The difference in utility for $m_2$ is symmetric since $G_\\alpha $ is an anonymous game, meaning the role $m_1$ and $m_2$ can be interchanged.",
"Figure: PNE types and the welfare surplus of SS over HH for m 1 m_1 when both are PNE.Interestingly, there are hash rates $\\alpha \\in \\mathcal {H}^2$ where $SS$ is an equilibrium, yet $SH$ is not profitable relative to $HH$ for $m_1$ .",
"This means that the existence of another strategic miner can make mining with SSM profitable and stable for $m_1$ whereas this is not the case when $m_1$ with hash power $\\alpha _1$ is the only strategic miner in the system.",
"For these $\\alpha $ we say the profitability threshold of SSM has decreased.",
"The set of $\\alpha \\in \\mathcal {H}^2$ such that the profitability threshold of SSM decreases is graphed in Figure REF .",
"Furthermore, there are hash rates in this region where $SS$ is the only PNE, such as $\\alpha = (0.235, 0.345)$ which leads to $G_\\alpha $ with utilities in Table REF .",
"Table: Example G α G_\\alpha where SSM Profitability Threshold DecreasesFigure: Hash rates where the profitability threshold of SSM is reduced.The logical next step is to ask about mixed Nash equilibria in $G_\\alpha $ , however the meaning of mixed strategies is not well-suited for selfish mining attacks.",
"For example, what would the mixed strategy $0.2H + 0.8S$ represent?",
"One interpretation could be a randomised commitment, where with probability $0.2$ a miner commits to $H$ and with probability $0.8$ a miner commits to SSM.",
"This however does not make much sense for selfish mining attacks, since their profitability takes time (due to adjustments in the block difficulty of the system), meaning that an opposing agent would have ample time to perform a best response to the realised commitment over the initial randomisation.",
"Another approach is to have $0.2H + 0.8S$ mean that a miner partitions his hash power into honest mining and SSM mining and commits to this partition henceforth.",
"Although utilities of mixed strategies do not directly correspond to convex combinations of utilities, we use this approach to study an extended action space for miners."
],
[
"Partition Games and Strong Stackelberg Equilibria",
"As mentioned at the end of the previous section, we also study incentives when miners are given a richer set of pure strategies beyond that of choosing between honest mining and SSM.",
"In particular, we now allow a given miner with hash power $\\alpha _i$ to partition his computational power into a portion following SSM and a portion using honest mining.",
"Before continuing we also clarify notation: for $x,y \\in \\mathbb {R}^n$ , we use $x \\circ y$ to denote the Hadamard product of $x$ and $y$ .",
"[Two-player Partition Games] Suppose that $\\alpha = (\\alpha _1,\\alpha _2) \\in \\mathcal {H}^2$ is a strategic hash distribution.",
"We define the Partition Game, $G^P_\\alpha $ , as a two-player game, where each player has the same action set $[0,1]$ , representing the proportion of their hash power dedicated to employing SSM.",
"For a given pure strategy profile $s = (s_1, s_2) \\in [0,1]^2$ , we define the utilities of $G^P_\\alpha $ as follows: $U_1(s_1,s_2) = s_1 R_{SSM}(s \\circ \\alpha )_1 + (1 - s_1)\\frac{(1-s_1)\\alpha _1}{1 - s \\cdot \\alpha } R_{SSM}(s\\circ \\alpha )_3$ $U_2(s_1,s_2) = s_2 R_{SSM}(s \\circ \\alpha )_2 + (1 - s_2)\\frac{(1-s_2)\\alpha _2}{1 - s \\cdot \\alpha } R_{SSM}(s \\circ \\alpha )_3$ In Figure REF , for $\\alpha = (0.46,0.25)$ we graph the pure strategy utilities of $m_1$ and $m_2$ as a function of $s \\in [0,1]^2$ .",
"The most glaring observation is that for fixed $s_{-i}$ , $U_i(s_i,s_{-i})$ is a convex function of $s_i$ , attaining local maxima at $s_i = 0$ and $s_i = 1$ .",
"This is clear from the fact that the blockchain eventually has one common history, so both sides of a miner's partition inherently compete with one another.",
"Figure: Utilities in G α P G_\\alpha ^P for α=(0.46,0.25)\\alpha = (0.46,0.25).Game theoretically, this means best responses for any $m_i$ are always from the set $\\lbrace 0,1\\rbrace $ .",
"Immediately, this tells us that the set of pure Nash equilibria of $G^P_\\alpha $ are the same as those in $G_\\alpha $ , since $G^P_\\alpha $ restricted to pure strategy profiles in $\\lbrace 0,1\\rbrace ^2$ is isomorphic to $G_\\alpha $ (recall that $H \\cong 0$ and $S \\cong 1$ in $G_\\alpha $ ).",
"It may thus seem the augmented strategy space of $G^P_\\alpha $ buys us nothing, however if we treat $G^P_\\alpha $ as a leadership game, where $m_1$ gets to commit to a pure strategy, $s_1$ , to which $m_2$ retaliates, then we get a different story.",
"To formally treat $G^P_\\alpha $ as a leadership game, we let $m_1$ be the leader and $m_2$ the follower.",
"For a given pure strategy $s_1 \\in [0,1]$ of $m_1$ , we let $BR(s_1)$ denote the best response $m_2$ has to $s_1$ .",
"Since we have observed that best responses for any $m_i$ are always from the set $\\lbrace 0,1\\rbrace $ , it follows that $BR(s_1) = \\text{argmax}_{x \\in \\lbrace 0,1\\rbrace }U_2(s_1,x)$ .",
"If $U_2(s_1,0) = U_2(s_1,1)$ , then we let $BR(s_1) = \\text{argmax}_{x \\in \\lbrace 0,1\\rbrace }U_1(s_1,x)$ , so that $m_2$ breaks ties in favour of $m_1$ .",
"The value of commitment $s_1$ for $m_1$ is denoted by $v_1(s_1) = U_1(s_1, BR(s_1))$ and for the value of commitment $s_1$ for miner 2 is denoted by $v_2(s_1) = U_2(s_1, BR(s_1))$ .",
"In a leadership game, a common solution concept is that of a Strong Stackelberg Equilibrium (SSE), which is a strategy pair $(s_1^*,s_2^*)$ such that $s_1^* \\in \\text{argmax}_{x} v_1(x)$ and $s_2^* = BR(s_1^*)$ .",
"This can be seen as a subgame perfect equilibrium of $G^P_\\alpha $ , or the optimal commitment under $v_1$ .",
"Furthermore, we let $\\text{SSE}(G)$ denote the SSE of an arbitrary game $G$ .",
"In Figure REF we graph (optimal) commitment values for $m_1$ at the SSE of $G^P_\\alpha $ for different values of $\\alpha \\in \\mathcal {H}^2$ .",
"Furthermore, we graph the value of these optimal commitments when compared to utility players obtain at their respectively optimal PNE of $G_\\alpha $ at the given hash rate."
],
[
"Non-trivial SSE",
"Since $G^P_\\alpha $ can be seen as an augmented action space to $G_\\alpha $ , we categorise $\\alpha \\in \\mathcal {H}^2$ depending on how the sets $\\text{PNE}(G_\\alpha ) = \\text{PNE}(G^P_\\alpha )$ and $\\text{SSE}(G^P_\\alpha )$ compare.",
"[Commitment/SSE Types] For every $\\alpha \\in \\mathcal {H}^2$ we associate a commitment type denoted $com(\\alpha ) \\in \\lbrace 0,1,2,3\\rbrace $ defined as follows: If $\\text{SSE}(G^P_\\alpha ) = \\text{PNE}(G_\\alpha )$ , then $com(\\alpha ) = 0$ .",
"If $\\text{SSE}(G^P_\\alpha ) \\subset \\text{PNE}(G_\\alpha )$ , then $com(\\alpha ) = 1$ .",
"If $\\text{SSE}(G^P_\\alpha ) \\lnot \\subset \\text{PNE}(G_\\alpha )$ , and $\\exists s^* \\in \\text{SSE}(G^P_\\alpha )$ such that $s^*_1 \\in \\lbrace 0,1\\rbrace $ , then $com(\\alpha ) = 2$ .",
"If $\\text{SSE}(G^P_\\alpha ) \\lnot \\subset \\text{PNE}(G_\\alpha )$ , and $\\lnot \\exists s^* \\in \\text{SSE}(G^P_\\alpha )$ such that $s^*_1 \\in \\lbrace 0,1\\rbrace $ , then $com(\\alpha ) = 3$ .",
"If $com(\\alpha ) = 0$ we say $\\alpha \\in \\mathcal {H}^2$ gives rise to a trivial commitment and that the collection of SSE in $G^P_\\alpha $ are trivial.",
"Accordingly, if $com(\\alpha ) \\ne 0$ , we say $\\alpha $ gives rise to a non-trivial commitment and the collection of SSE in $G^P_\\alpha $ is non-trivial.",
"Furthermore, we also say that if $\\alpha \\in \\mathcal {H}^2$ is such that $com(\\alpha ) = i$ , then all $s^* \\in \\text{SSE}(G^P_\\alpha )$ are of type $i$ as well.",
"In the two-miner scenario, we make the following observations about $\\alpha \\in \\mathcal {H}^2$ with non-trivial commitment types: $com(\\alpha ) = 1$ occurs at hash values such that the PNE of $G_\\alpha $ are $HH$ and $SS$ .",
"$m_1$ commits to $S$ to nudge the system to converge to the $SS$ equilibrium which Pareto-dominates $HH$ in $G_\\alpha $ .",
"$com(\\alpha ) = 2$ occurs at hash rates where there is one SSE of $G^P_\\alpha $ , $s^* = (s_1^*,s_2^*) \\in \\lbrace 0,1\\rbrace ^2$ , yet $s^*$ does not correspond to a PNE of $G_\\alpha $ .",
"$s^*$ is unstable in $G_\\alpha $ from the perspective of $m_1$ , who would prefer deviating from $s_1$ when pitted against $s_2$ .",
"These SSE make use of the sequentiality of $G^P_\\alpha $ but not of the extended action space given by partitioning.",
"$com(\\alpha ) = 3$ occurs at hash rates such that $HS$ is the only PNE of $G_\\alpha $ , but where $\\alpha $ is close to the region in $\\mathcal {H}^2$ where $SS$ arises as the sole PNE of $G_\\alpha $ .",
"At these values, $m_2$ only slightly prefers $SH$ to $SS$ , hence $m_1$ can bait $m_2$ into playing $S$ by reserving a small portion of hash power to mine honestly.",
"For any non-trivial SSE, $v_1(s^*_1)$ is lower bounded by the lowest-utility $m_1$ obtains amongst PNE in $G_\\alpha $ .",
"On the other hand, if $com(\\alpha ) = 1,3$ , the SSE of $G^P_\\alpha $ are such that $v_1(s^*_1)$ is strictly greater than the highest utility $m_1$ obtains amongst PNE in $G_\\alpha $ .",
"This strict surplus in utility is visible in the latter graphs of Figure REF , and we can see that these non-trivial commitments also benefit $m_2$ in spite of being the follower.",
"Figure: SSE types for m 1 m_1, optimal commitments for m 1 m_1, and relative surplus of SSE against best PNE for m 1 m_1 and for m 2 m_2 respectively."
],
[
"Plots of Non-Trivial SSE by Type",
"We now focus on plotting optimal $\\mathcal {H}^2$ that exhibit SSE of types 1, 2 and 3.",
"For SSE of type 1, it suffices to look at Figure REF and Table REF for visualisation of the benefit of SSE over PNE (Since it is just the difference in welfare between PNE in this case).",
"As for SSE of type 2, these are plotted in more detail in Figure REF .",
"For these values of $\\alpha $ , we can see that $HH$ is the only PNE in $G_\\alpha $ , but $SS$ is the SSE of $G^P_\\alpha $ , which is forcibly unstable in the one shot game, $G_\\alpha $ , as $m_1$ prefers $HS$ to $SS$ .",
"Table REF shows the utilities for $G_\\alpha $ at a specific value of $\\alpha $ exhibiting this behaviour.",
"Note that in this example, the leader, $m_1$ , has a hash rate of $\\alpha _1 = 0.2$ , at which normally they would not be incentivised to unilaterally employ SSM in the one-shot SSM game.",
"The power to commit makes SSM viable at smaller hash rates than in the one-shot game.",
"Table: Example G α G_\\alpha where the SSE in G α P G^P_\\alpha is of type 2Figure: Optimal Commitments for m 1 m_1, as well as SSE surplus against best PNE for m 1 m_1, and for m 2 m_2 respectively in the region [0.185,0.2]×[0.21,0.27][0.185, 0.2] \\times [0.21, 0.27].",
"This region exhibits SSE of type 2.Figure REF focuses on hash rates where SSE are of type 3.",
"Furthermore, Figure REF looks specifically at $\\alpha = (0.431,0.239)$ , which is a hash rate such that $G^P_\\alpha $ has an SSE of type 3, and graphs utilities and best responses as a function of the leader commitment in $G^P_\\alpha $ .",
"This gives a better way of visualising how $s_1 = 0.98$ is an optimal commitment where $m_2$ is rendered indifferent between $S$ and $H$ .",
"Figure: Optimal Commitments for m 1 m_1, as well as SSE surplus against best PNE for m 1 m_1, and for m 2 m_2 respectively in the region [0.2,0.5]×[0.19,0.28][0.2, 0.5] \\times [0.19, 0.28].",
"This region exhibits SSE of type 2.Figure: Partition Game Analysis for α=(0.431,0.239)\\alpha = (0.431,0.239).",
"The top left image plots follower utilities when playing HH or SS against a leader commitment partition.",
"The bottom left image plots follower utility when best responding to a leader commitment.",
"The best response at a given commitment dictates which of the two utilities the leader obtains in the top right plot.",
"Putting everything together, the bottom right plot gives the value of a leader commitment (for the leader) as a function of their commitment.",
"Note how this function is maximised at approximately 0.98, where the follower is indifferent between HH and SS."
],
[
"$M > 2$ Strategic Miners",
"Our analysis from Section extends in a straightforward fashion to when there are $M > 2$ strategic miners.",
"Consequently, for any hash distribution $\\alpha \\in \\mathcal {H}^M$ , we can compute $R_{SSM}(\\alpha ) \\in [0,1]^{M+1}$ , the revenue ratio of all $M$ strategic miners and all other honest miners, when all strategic miners of hash power $\\alpha _i$ employ SSM.",
"The full details of the corresponding Markov chain and reward vectors can be found in Appendix .",
"It is also straightforward to extend the game-theoretic formalism of Section to study incentives when $M > 2$ strategic miners interact.",
"This formalism can also be found in Appendix .",
"In Appendix we also plot similar graphs to Section for $M = 3$ at different hash rates to visualise strategic miner behaviour.",
"When $M > 3$ however, it becomes difficult to visualise how aspects of $G_\\alpha $ and $G^P_\\alpha $ precisely vary with $\\alpha $ .",
"That being said, we do find very similar structures as in the $M = 2$ and $M = 3$ case, such as: penalising coalitions, existence of PNE, and for some regions multiple PNE, in $G_\\alpha $ , non trivial commitments in $G^P_\\alpha $ , and finally, hash rates $\\alpha $ where the SSM profitability threshold decreases with the existence of other strategic miners.",
"We expand upon this final point to specifically see how the number of strategic miners $M$ affects the profitability threshold of SSM."
],
[
"Decreasing SSM Profitability Threshold",
"To study the effect of the number of miners on the profitability threshold of SSM, we define the following: [Uniform Profitability Threshold for SSM] For $M \\ge 1$ miners we say the uniform profitability threshold for SSM is the smallest $\\eta \\in [0,1]$ such that $\\alpha = \\eta \\vec{1} \\in \\mathcal {H}^M$ and $\\vec{1} \\in \\text{PNE}(G_\\alpha )$ (all players employing SSM is a PNE in $G_\\alpha )$ .",
"With our methods from Appendix , we can approximate the uniform SSM profitability threshold for various values of $M$ .",
"In particular, Figure REF shows these threshold values for $M = 1,...,8$ .",
"Furthermore, the second plot takes the uniform SSM profitability threshold, $\\eta $ , and for $\\alpha = \\eta \\vec{1}$ , computes the utilities of both $\\vec{0}$ and $\\vec{1}$ which are both PNE in $G_\\alpha $ .",
"Interestingly, for $M = 1,..,8$ , not only does the uniform profitability threshold decrease as a function of $M$ , but all miners employing SSM is a PNE that Pareto dominates all miners being honest.",
"These results thus show that the presence of multiple strategic miners may have more of an impact on the stability of Bitcoin than previously thought.",
"Figure: Upper bounds on the uniform profitability threshold for SSM as a function of the number of strategic miners.",
"We also plot the welfare of 1 →\\vec{1} (all SSM) versus 0 →\\vec{0} (all honest)."
],
[
"Conclusion and Further Work",
"In this paper we have described a specific miner strategy, semi-selfish mining (SSM) that is a truncated variant of Selfish Mining (SM).",
"SSM has the benefit of being a profitable strategy for large enough miners (in the same way as SM), and also structured enough for us to explicitly solve for relative revenues when more than one strategic miner employs SSM.",
"With this in hand, we have been able to use a game-theoretic lense to glean some information on miner incentives when more than one miner is strategic within the bitocin system.",
"In particular, for any $\\alpha \\in \\mathcal {H}^M$ , we define the SSM game $G_\\alpha $ which governs strategic miner incentives in choosing to employ SSM or mine honestly, and the partition game $G^P_\\alpha $ , which extends the action space of $G_\\alpha $ to allow miners to partition their hash power between honest mining and SSM.",
"For $M > 1$ strategic miners we find the following main takeaways from studying $G_\\alpha $ and $G^P_\\alpha $ : All $\\alpha \\in \\mathcal {H}^M$ seem to lead to $G_\\alpha $ with pure Nash equilibria.",
"Furthermore, there are regions in $\\mathcal {H}^M$ such that $G_\\alpha $ has multiple PNE.",
"A single miner might prefer to use SSM over honest mining in $G_\\alpha $ , but there can exist a coalition of miners who may retaliate against this action and punish the original SSM miner into receiving less utility than their hash power.",
"Though the set of PNE in $G^P_\\alpha $ is identical to those of $G_\\alpha $ , when treating $G^P_\\alpha $ as a sequential game leads to non-trivial commitments, some of which involve a miner employing SSM even though SSM is not rational in the one-shot SSM game.",
"Finally, there exist hash rates, $\\alpha \\in \\mathcal {H}^M$ such that $m_1$ does not unilaterally prefer to employ SSM, but some PNE of $G_\\alpha $ includes $m_1$ employing SSM, effectively reducing the profitability threshold of SSM and consequently affecting the stability of Bitcoin.",
"The action spaces in $G_\\alpha $ and $G^P_\\alpha $ may seem limited due to the fact that they only interpolate between honest mining and SSM, but there is nothing barring a variant $G_\\alpha $ and $G^P_\\alpha $ from studying the choice of employing other subversive mining strategies over honest mining.",
"In fact, $G_\\alpha $ and $G^P_\\alpha $ can be defined by using empirical estimates to steady state payoffs instead of closed form solutions, which could glean some information into how mining dynamics change when a larger palette of subversive strategies is available to interdependent strategic miners.",
"In fact, $G^P_\\alpha $ could be extended so that the action space of miners is no longer simply partitioning mining power between honest mining and SSM, but any partition of mining power amongst a given list of subversive mining strategies.",
"In addition, the fact that penalising coalitions exist hints at the possibility of modelling such structures in a repeated game framework.",
"The issue of course comes in modelling how much utility a penalising coalition gains in maintaining everyone honest, but there could be interesting subgame perfect Nash equilibria in an appropriate model.",
"Finally, along the same vein of penalising coalitions, there is also scope for a more fine-grained cooperative game theoretic analysis of SSM and Partition games."
],
[
"SSM vs. Honest Mining",
"We can use a similar Markov chain analysis to derive the revenue ratio of SSM against honest miners.",
"We recall that the strategic miner, $m_1$ , has hash power $\\alpha \\in \\mathcal {H}^1 = (0,0.5]$ and the honest miner $m_2$ has hash power $\\beta = 1 - \\alpha $ .",
"Let us define the state space $S = \\lbrace S_0,S_1,S_2\\rbrace $ corresponding to the number of private blocks belonging to the miner employing SSM.",
"We can now describe the transitions and their corresponding revenues (expected block creation rate per state):"
],
[
"Transitions from state $S_0$",
" $S_0 \\rightarrow S_0$ occurs if $m_2$ find a block.",
"The probability of this transition is $\\beta $ and $m_2$ wins a block.",
"$S_0 \\rightarrow S_1$ occurs if $m_1$ finds a block.",
"The probability of this transition is $\\alpha $ and no players win a block."
],
[
"Transitions from state $S_1$",
" $S_1 \\rightarrow S_0$ occurs if $m_2$ finds and publishes a block, which occurs with probability $\\beta $ .",
"A fork is created when $m_1$ subsequently publishes his hidden block and from here three events can occur: A first scenario occurs when $m_1$ finds another block to resolve the tie in his favour, resulting in two blocks for $m_1$ .",
"This occurs with probability $\\alpha $ .",
"A second scenario occurs when an honest miner finds a block that resolves the tie in favour of $m_1$ , resulting in one block for $m_1$ and one block for $m_2$ .",
"This occurs with probability $\\gamma \\beta $ .",
"A final scenario occurs when an honest miner finds a block that resolves the tie in favour of $m_2$ which results in two blocks for $m_2$ .",
"This final event occurs with probability $(1-\\gamma )\\beta $ .",
"In all aforementioned scenarios the resulting state is $S_0$ , thus the probability of the transition to state $S_0$ is $\\beta $ .",
"$S_1 \\rightarrow S_2$ occurs if $m_1$ finds a block and keeps it private as per SSM.",
"This event occurs with probability $\\alpha $ and no blocks are awarded to any agent."
],
[
"Transitions from state $S_2$",
" $S_2 \\rightarrow S_0$ occurs when $m_2$ finds a block.",
"The probability of this transition is $\\beta $ and $m_1$ wins two blocks.",
"$S_2 \\rightarrow S_2$ occurs if $m_1$ finds a block.",
"The probability of this transition is $\\alpha $ and $m_1$ wins a block.",
"Figure: States and Transitions for SSM vs.",
"Honest MinersThe transitions are visualised in Figure REF .",
"Furthermore, we can fully express the transition matrix of the Markov chain as follows: $M=\\begin{bmatrix}1- \\alpha & 1-\\alpha & 1-\\alpha \\\\\\alpha & 0 & 0\\\\0 & \\alpha & \\alpha \\\\\\end{bmatrix}$ For a given probability distribution $x \\in \\mathbb {R}^3$ over the state space $S$ , $Mx$ gives the resulting probability distribution over $S$ after one transition under the Markov chain above.",
"Since the chain is easily seen to be ergodic, there exists a unique steady state distribution, $\\pi $ , such that $M \\pi = \\pi $ .",
"Using Gaussian elimination we obtain $ \\pi = (1-\\alpha , \\alpha (1-\\alpha ), \\alpha ^2)^T$ as the unique steady state.",
"Furthermore, from the transitions mentioned above we obtain the following expected block creation rates (denoted by $r_{SSM}$ and $r_{others}$ ) per state: Table: Expected revenue per stateWe let $r_{SSM}$ and $r_{others}$ denote the expected revenue per round at steady state $\\pi $ for $m_1$ and $m_2$ .",
"We also let $R_{SSM}$ and $R_{others}$ denote the revenue ratios of $m_1$ and $m_2$ at steady state.",
"Given our expected revenues per state, we obtain $r_{SSM} = (2-\\gamma )\\alpha ^4 + (3\\gamma - 5)\\alpha ^3 + (4-3\\gamma )\\alpha ^2 + \\gamma \\alpha $ and $r_{others} = (1-\\alpha )^2 \\left( (\\gamma - 2)\\alpha ^2 + (2 - \\gamma )\\alpha + 1 \\right)$ .",
"A strategic miner of hash power $\\alpha $ attains the following revenue ratio when playing against honest miners: $R_{SSM} = \\frac{r_{SSM}}{r_{SSM} + r_{others}} =\\frac{\\alpha (\\alpha (\\alpha (2\\alpha -5) + 4) - (\\alpha -1)^3\\gamma )}{(\\alpha -1)\\alpha ^2 + 1}$ Asymptotically around $\\alpha = 0$ , the expression is the following: $R_{SSM} = \\alpha \\gamma + \\alpha ^2(4 - 3\\gamma ) + \\alpha ^3(4\\gamma - 5) - \\alpha ^4(6 - 5\\gamma ) + \\alpha ^5(7\\gamma - 9) + O(\\alpha ^6)$"
],
[
"Markov Chain Formalism for $M\\ge 2$ Strategic Miners and Arbitrary Propagation",
"In this section we delve into the Markov chain governing revenues (block creation rates) when multiple strategic miners employ SSM.",
"In what follows we assume that $\\alpha \\in \\mathcal {H}^M$ .",
"This implies that $m_1,...,m_M$ are strategic miners with hash power $\\alpha _1,...,\\alpha _m$ , and $m_{M+1}$ is an honest miner with hash power $\\beta = 1 - \\sum _{i=1}^M \\alpha _i$ .",
"As in the one miner case, we let $S = \\lbrace 0,1,2\\rbrace ^M$ be the state space of all possible private leads held by $m_1,...,m_M$ employing SSM.",
"For a given $x \\in S$ , $x_i$ denotes the private lead of $m_i$ .",
"In addition, for a given $x \\in S$ , we let $A_x = \\lbrace i \\in [M] \\ | \\ x_i = 1\\rbrace $ and $B_x = \\lbrace i \\in [M] \\ | \\ x_i = 2\\rbrace $ .",
"Clearly $A_x \\cap B_x = \\emptyset $ , furthermore, we can completely establish transition probabilities from $x$ by looking at $A_x$ and $B_x$ ."
],
[
"State Transitions",
"Let us suppose $x \\in S$ is arbitrary.",
"In what follows we let $e_i \\in \\lbrace 0,1\\rbrace ^M$ be the unit vector with 1 in the $i$ -th coordinate.",
"Furthermore, we let $P_{x \\rightarrow y}$ denote the probability of transitioning from $x$ to $y$ .",
"To fully describe all transitions for any $x \\in S$ , we look at four different cases depending on $A_x$ and $B_x$ .",
"If any strategic miner $m_i$ obtains a block, they keep it private as per SSM extending their private chain by 1 (which they forcibly have a margin to do so).",
"This results in state $x + e_i$ and occurs with probability $\\alpha _i$ .",
"If $m_{M+1}$ finds a block, they publish it as per the honest mining protocol, which occurs with probability $\\beta $ .",
"All $m_i$ such that $x_i \\ne 0$ then publish their private chains as per SSM and a race ensues.",
"The conditions of SSM and honest mining dictate that the race is settled in the following turn, and hence we return to state 0.",
"In summary: $P_{x \\rightarrow x + e_i} = \\alpha _i$ for all $i \\le M$ $P_{x \\rightarrow 0} = \\beta $"
],
[
"$|A_x| = 0$ , {{formula:55ed3461-ff1e-44c7-87a7-52c3560b4362}}",
"In this case a single miner has a private lead of 2 and all other miners have no private lead.",
"If any strategic miner $m_i$ such that $i \\ne j$ finds a block, SSM dictates that they keep this block private and proceed to having a private chain of length 1.",
"This corresponds to transitioning from $x$ to $x + e_i$ , which occurs with probability $\\alpha _i$ .",
"If the $m_j$ finds a block, an event which happens with probability $\\alpha _j$ , SSM dictates he publish his oldest private block.",
"Since $A_x = \\emptyset $ , this block will be the longest public chain, and the resulting state will be $x$ again.",
"Finally, if $m_{M+1}$ finds a block, honest mining dictates he publish it.",
"$m_j$ in turn sees his private lead decrease to 1 and hence publishes his entire private chain.",
"As a consequence state 0 ensues, and this transition occurs with probability $\\beta $ .",
"In summary we have the following transitions: $P_{x \\rightarrow x + e_i} = \\alpha _i$ for all $i \\ne j$ $P_{x \\rightarrow x} = \\alpha _j$ $P_{x \\rightarrow 0} = \\beta $"
],
[
"$|A_x| \\ge 0$ , {{formula:ca4fd030-4266-4ffe-b07b-91a8ad6ae647}}",
"Suppose that $m_i$ such that $i \\notin B_x$ finds a block, which occurs with probability $\\alpha _i$ .",
"As per SSM $m_i$ has a margin to keep this block private, hence state $x + e_i$ ensues.",
"On the other hand, if $m_i$ is such that $i \\in B_x$ , then by SSM, $m_i$ publishes their oldest private block.",
"As a result, all miners in $A_x$ publish their private leads to start a race, and all miners in $B_x$ publish their private leads to overtake.",
"$m_i$ thus sees his private lead diminish to 1, hence by SSM he publishes his entire private chain.",
"This chain is the longest of all miners, hence we return to state 0.",
"Finally, if $m_{M+1}$ finds a block, which occurs with prability $\\beta $ , he publishes it as per honest mining, all strategic miners with hidden chains once again publish their hidden chains.",
"There is a multi-way race amongst all miners in $B_x$ , but as per SSM and honest mining, this race is decided in the following turn and we return to state 0.",
"In summary we have the following transitions: $P_{x \\rightarrow x+e_i} = \\alpha _i$ for $i \\notin B_x$ $P_{x \\rightarrow 0} = \\beta + \\sum _{i \\in B_x} \\alpha _i$"
],
[
"$|A_x| > 1$ , {{formula:4049db0a-2b0c-4628-bca1-16880afba6a8}}",
"If any $m_i$ such that $i \\ne j$ finds a block, an event which occurs with probability $\\alpha _i$ , then SSM dictates they keep this block private and the resulting state is $x + e_i$ .",
"If $m_{M+1}$ finds a block, which occurs with probability $\\beta $ , they publish it as per honest mining, and $m_j$ sees his lead diminished and by the rules of SSM, publishes his private chain to create the longest public chain.",
"The resulting state is thus 0.",
"Finally, if $m_j$ finds the following block, he publishes his oldest private block as per SSM, and consequently the public tie is amongst a prefix of the chain of $m_j$ the chains of all $m_i$ such that $i \\in A_x$ , since they also publish their private chains.",
"At this point $m_j$ is mining upon his private chain whereas all other miners, including $m_{M+1}$ mine upon some of the chains partaking in the public tie.",
"From here there are two scenarios.",
"Either $m_j$ also finds the following block, in which case SSM dictates he publish it, and the new public prefix of his chain is the longest public chain and the ensuing state is $2e_j$ , or any miner other than $m_j$ finds the next block, in which case $m_j$ sees his lead diminished and publishies his entire private chain resulting in state 0.",
"The overall probability of the first scenario is $\\alpha _2^2$ and the overall probability of the second scenario is $\\alpha _2(1 - \\alpha _2)$ .",
"In summary we have the following transitions: $P_{x \\rightarrow x+e_i} = \\alpha _i$ for $i \\ne j$ $P_{x \\rightarrow 2e_j} = \\alpha _j^2$ $P_{x \\rightarrow 0} = \\beta + \\alpha _j(1- \\alpha _j)$"
],
[
"Propagation Formalism ",
"In the original analysis of selfish mining, much attention was given to a data propagation parameter $\\gamma $ .",
"Propagation is important because it allows an attacker to persuade honest miners to work on their end the public chain when forks occur.",
"When there are $M \\ge 2$ strategic miners however, propagation intricacies cannot be captured by a single parameter, as different strategic agents have different abilities to convinces other miners of their own chains.",
"To encompass this generality, let us suppose that $D \\subseteq [M+1]$ is a subset of miners engaged in a tie (we recall that $m_{M+1}$ is the implicit honest miner in the system).",
"For $j \\in D$ and $i \\in [M+1]$ we let $\\gamma ^D_{i,j}$ be the probability that $m_i$ mines upon the chain of $m_j$ in the tie composed of all $D$ miners.",
"The only restriction we place on these parameters is that $\\gamma ^D_{i,i} = 1$ for $i \\ne M+1$ .",
"The reason for this is that a strategic miner will mine upon their public chain in case of a tie.",
"Finally, we note that in the uniform propagation model we use throughout the paper, we simply let $\\gamma ^D_{i,j} = \\frac{1}{|D|}$ if $i \\notin D$ and $i \\ne j$ or if $M+1 \\in D$ and $i = j = M+1$ ."
],
[
"Expected Revenue per State with Arbitrary Propagation",
"For a given state $x \\in S$ , we compute the expected revenue per agent under the underlying Markov chain governing SSM dynamics.",
"We denote this quantity by $rev(x) \\in \\mathbb {R}^{M+1}$ , where $rev(x)_i$ denotes the expected revenue of $m_i$ when the system is in state $x \\in S$ .",
"In order to compute these quantities, it will be useful to define the expected revenue all agents obtain when there is an arbitrary tie involving a set $D \\subseteq [M]$ of miners.",
"As we have seen in state transitions, for a given $x \\in S$ , ties can involve either agents with a private lead of 1 or agents with a private lead of 2.",
"We denote the expected revenue all agents receive when a tie of $D \\subseteq [M]$ miners with private lead of $i = 1,2$ occurs by $T_i(D) \\in \\mathbb {R}^{M+1}$ : $T_1(D) = \\sum _{i= 1}^{M+1} \\alpha _i \\left( \\sum _{j \\in D} \\gamma _{i,j}^D (e_i + e_j) \\right)$ $T_2(D) = \\sum _{i= 1}^{M+1} \\alpha _i \\left( \\sum _{j \\in D} \\gamma _{i,j}^D (e_i + 2e_j) \\right)$ As with state transitions, for a given $x \\in S$ , we can characterise $rev(x)$ by looking at $A_x$ and $B_x$ , the indices of strategic miners with a private lead of 1 and 2 respectively."
],
[
"$|A_x| = |B_x| = 0$",
"In this case, only $m_{M+1}$ revceives a block if he finds one, which occurs with probability $\\beta $ .",
"$rev(x) = \\beta e_{M+1}$"
],
[
"$|A_x| > 0$ , {{formula:e222bfdf-20de-4741-9fac-7d58fbc677dc}}",
"In this case, blocks are only won in the event of a tie, which in turn only happens if $m_{M+1}$ originally finds a block with probability $\\beta $ .",
"In such a case, there is a tie amongst the indices $A_x \\cup \\lbrace M+1\\rbrace $ .",
"$rev(x) = \\beta \\left( (T_1(A_x \\cup \\lbrace M+1\\rbrace ) \\right)$"
],
[
"$|A_x| = 0$ , {{formula:563a1883-5f4c-4dd4-a2d3-afca0586ffe8}}",
"In this case, if $m_j$ finds a block, he publishes his oldest block as per SSM and thus wins a block in the turn.",
"If $m_{M+1}$ finds and publishes a block as per honest mining, $m_j$ publishes his entire chain as per SSM and wins two blocks in the turn.",
"$rev(x) = \\alpha _j (e_j) + \\beta (2e_j)$"
],
[
"$|A_x| \\ge 0$ , {{formula:3a3a8663-6ff7-4fec-a987-5515283ee7b8}}",
"If any $m_j$ such that $j \\in B_x$ finds a block, by SSM they publish their oldest private block.",
"Other miners with indices in $B_x$ thus publish their entire private chains of length 2, and consequently $m_j$ publishes his entire private chain of length 2 (relative to the original fork so still longer than all other private chains), winning 3 blocks overall.",
"If any $m_i$ such that $i \\notin B_x$ finds a block, they simply keep it private as per SSM and no blocks are definitively won.",
"Finally, if $m_{M+1}$ finds a block with probability $\\beta $ , then all $m_j$ such that $j \\in B_x$ publish their private chains and a tie ensues amongst these agents, which results in $T_2(B_x)$ expected revenue for all miners.",
"$rev(x) = \\sum _{j \\in B} \\alpha _j (3 e_j) + \\beta T_2(B_x)$"
],
[
"$|A_x| > 1$ , {{formula:1223876c-167d-4203-b73f-e1c707174707}}",
"If $m_{M+1}$ finds a block with probability $\\beta $ , $m_j$ sees his lead diminished and publishes his entire private chain, thus winning two blocks.",
"If any $m_i$ such that $i \\notin B_x$ finds a block, they simply keep it private as per SSM and no one immediately wins blocks.",
"Finally, if $m_j$ finds two blocks in a row he publishes a prefix of his private chain and wins two blocks (transitioning to state $2e_j$ in the process).",
"If $m_j$ finds a block (thus leading him to publish his oldest private block as per SSM), and subsequently any other miner finds the next one, $m_j$ sees his lead diminished and publishes his entire private chain, winning 3 blocks overall.",
"$\\beta (2e_j) + \\alpha _j^2(2e_j) + \\alpha _j(1 - \\alpha _j)(3e_j)$"
],
[
"Game Theoretic Formalism for $M > 2$ Strategic Miners",
"Our analysis from Section extends in a straightforward fashion to when there are $M > 2$ strategic miners.",
"Consequently, for any hash distribution $\\alpha \\in \\mathcal {H}^M$ , we can compute $R_{SSM}(\\alpha ) \\in [0,1]^{M+1}$ , the revenue ratio of all $M$ strategic miners and all other honest miners, when all strategic miners of hash power $\\alpha _i$ employ SSM.",
"The full details of the corresponding Markov chain and reward vectors can be found in Appendix .",
"In this section, we extend the game-theoretic formalism of Section to to study incentives when $M > 2$ strategic miners interact."
],
[
"To SSM or not to SSM in the Multiplayer Setting",
"We recall that Section introduced SSM games, a family of binary action games $G_\\alpha $ that governed the incentives behind choosing to employ SSM or honest mining.",
"We extend this game to the multiplayer setting in a natural way.",
"Suppose that $\\alpha \\in \\mathcal {H}^M$ is a hash rate of all strategic miners.",
"Once again, we let $R_{SSM}(\\alpha ) \\in [0,1]^{M+1}$ be the revenue ratios of all miners.",
"Just as before, $R_{SSM}(\\alpha )_i$ is the relative revenue of $m_i$ .",
"[Multi-player SSM Games] For every $\\alpha \\in \\mathcal {H}^M$ , we define the SSM Game, $G_\\alpha $ as a $M$ -player binary action game.",
"Each strategic miner has a binary action set $\\lbrace H,S\\rbrace $ , where $H$ represents mining honestly and $S$ represents employing SSM.",
"For convenience, we associate this action space with $\\lbrace 0,1\\rbrace ^M$ , where action 0 denotes honest mining and action 1 denotes employing SSM.",
"In order to specify utilities, we suppose that $x \\in \\lbrace 0,1\\rbrace ^M$ is a pure action profile such that $x_i = 1$ and $x_j = 0$ : $U_i(x) = R_{SSM}(\\alpha \\circ x)_i$ $U_j(x) = \\frac{\\alpha _j}{1 - \\alpha \\cdot x} R_{SSM}(\\alpha \\circ x)_{M+1}$ In Section we explored scenarios where an agent might be unilaterally incentivised to use SSM, but a second larger agent can retaliate by employing SSM to make the original agent worse off than when everyone mines honestly.",
"In the multiplayer setting, any subset of agents can retaliate in a similar fashion, thus we formally define what constitutes a penalising coalition.",
"In what follows, we use the notation $\\vec{\\chi }_C \\in \\lbrace 0,1\\rbrace ^M$ to denote an indicator vector for a subset $C \\subset [M]$ .",
"[Penalising Coalition] Suppose that $\\alpha \\in \\mathcal {H}^M$ is a distribution of hash power amongst $M$ strategic miners.",
"We say that $C \\subset \\lbrace 2,...,M\\rbrace $ is a penalising coalition for miner 1 if the following hold: $U_1(\\vec{\\chi }_1) > U_1(\\vec{0})$ $U_i(\\vec{\\chi }_{ 1 \\cup C}) > U_i(\\vec{\\chi }_{1 \\cup C \\setminus i})$ for all $i \\in C$ $U_1(\\vec{\\chi }_{1 \\cup C}) < U_1 (\\vec{0})$ Furthermore, we say that $C$ incurs a penalty of $U_1 (\\vec{0}) - U_1(\\vec{\\chi }_{1 \\cup C})$ on miner 1 when retaliating The first condition ensures that miner 1 has a unilateral incentive to deviate and employ SSM.",
"The second condition ensures that each miner in the penalising coalition is better off retaliating than defecting from the retaliation (ensuring retaliation is in a loose sense a credible threat), and finally the third condition ensures that miner 1 is worse off when being retaliated against than when everyone is honest."
],
[
"Partition Games in the Multiplayer Setting",
"In Section , we introduced the notion of a partition game, $G^P_\\alpha $ , which extended the action space of $G_\\alpha $ to allow miners to partition their hash power into honest mining and employing SSM.",
"This definition extends naturally to the $M$ -player case.",
"[Multi-player Partition Games] Suppose that $\\alpha \\in \\mathcal {H}^M$ is a hash distribution for $M$ strategic miners.",
"We define the Partition Game, $G^P_\\alpha $ , as a $M$ -player game, where each player has the same action set $[0,1]$ , representing the proportion of their hash power dedicated to employing SSM.",
"For a given pure strategy profile $s \\in [0,1]^M$ , we define the utility of the $i$ -th player in $G^P_\\alpha $ as follows: $U_i (s) = s_i R_{SSM}(s \\circ \\alpha )_i + (1 - s_i) \\frac{\\alpha _i (1 - s_i)}{1 - \\sum \\alpha _i s_i} R_{SSM}(s \\circ \\alpha )_{M+1}$"
],
[
"Optimal Commitments in $G^P_\\alpha $",
"As in the $M = 2$ miner case, for any miner $i$ , every action $s_i \\in (0,1)$ is dominated by either $s_i = 0$ or $s_i = 1$ if $G^P_\\alpha $ is treated as a one shot game.",
"The reason for this is that partitioning hash power results in unnecessary self competition, hence it will never be a best response to fixed opponent strategies.",
"Consequently, the PNE of $G^P_\\alpha $ as a one shot game are identical to the PNE of $G_\\alpha $ .",
"On the other hand, we can once again treat $G^P_\\alpha $ as a full information sequential game where $m_1$ commits to a strategy and all other $M-1$ players react.",
"The subgame perfect Nash equilibria (SGPNE) of this game are generalisations of the Stackelberg equilbria of Section .",
"The most subtle issue with generalising SSE however arises in tie-breaking.",
"The assumption in SSE for two player games is that the follower will break ties in favour of the leader.",
"This is a fair assumtion in the two-player setting, because it is often the case that commitments that lead to indifference in responses are of lower measure than those that invoke unique best responses.",
"For this reason a leader can commit to strategies in an arbitrarily small neighbourhood of an SSE to ellicit the desired best response in the case of a tie for the follower.",
"In the multi-player setting however, it can be the case that a non-trivial neighbourhood of leader commitments give rise to subgames with multiple PNE.",
"For this reason it may be unfeasible to assume that follower agents converge to a PNE that maximises the welfare of the leader, as there is nothing in the power of the leader to even approximately guarantee this behaviour.",
"For this reason, we take a pessimistic approach to SGPNE of $G^P_\\alpha $ .",
"In particular, we assume that for a leader commitment, all other agents will settle on a PNE that minimises welfare for the leader.",
"To be precise, for a given pure strategy $s_1 \\in [0,1]$ , we let $G^P_\\alpha (s_1,-)$ denote the $(M-1)$ -player subgame for miners $2,...,M$ conditioned on miner 1 committing to $s_1$ .",
"Furthermore, we let $WSN(s_1)$ (Worst sub-Nash) be the lowest utility pure Nash equilibrium of $G^P_\\alpha (s_1,-)$ for miner 1.",
"The value of commitment $s_1$ in the leadership game $G^P_\\alpha $ for miner 1 is $v_1(s_1) = U_1(s_1,WSN(s_1))$ , and for any other miner $i = 2,...,M$ , $v_i(s_1) = U_i(s_1,WSN(s_1))$ .",
"We call the family of all such pure strategy profile the collection of Pessimistic Sub-game Perfect Nash Equilibria, (P-SGPNE).",
"In particular, we are interested in values of $\\alpha $ where the set of P-SGPNE of $G^P_\\alpha $ result in strictly larger welfare for $m_1$ , implying that either the possiblity of commitment or partitioining strictly benefits $m_1$ in the worst case.",
"In a similar fashion to the two-player case, we study how different values of $\\alpha \\in \\mathcal {H}^M$ give rise to different P-SGPNE($G^P_\\alpha )$ vs PNE($G_\\alpha $ ) = PNE($G^P_\\alpha $ ).",
"[Multiplayer Commitment/ SGPNE Types] Suppose that $\\alpha \\in \\mathcal {H}^M$ , we classify its commitment type, $com(\\alpha )$ , depending on the relationship between the sets P-SGPNE($G^P_\\alpha )$ and PNE$(G_\\alpha )$ If $\\text{P-SGPNE}(G^P_\\alpha ) = \\text{PNE}(G_\\alpha )$ , then $com(\\alpha ) = 0$ .",
"If $\\text{P-SGPNE}(G^P_\\alpha ) \\subset \\text{PNE}(G_\\alpha )$ , then $com(\\alpha ) = 1$ .",
"If $\\text{P-SGPNE}(G^P_\\alpha ) \\lnot \\subset \\text{PNE}(G_\\alpha )$ and $\\exists s^* \\in \\text{P-SGPNE}$ such that $s^*_1 \\in \\lbrace 0,1\\rbrace $ , then $com(\\alpha ) = 2$ If $\\text{P-SGPNE}(G^P_\\alpha ) \\lnot \\subset \\text{PNE}(G_\\alpha )$ and $\\lnot \\exists s^* \\in \\text{P-SGPNE}$ such that $s^*_1 \\in \\lbrace 0,1\\rbrace $ , then $com(\\alpha ) = 3$ .",
"As in the two-player case, if $com(\\alpha ) = 0$ we say $\\alpha \\in \\mathcal {H}^M$ gives rise to a trivial commitment and that the collection of P-SGPNE in $G^P_\\alpha $ are trivial.",
"Accordingly, if $com(\\alpha ) \\ne 0$ , we say $\\alpha $ gives rise to a non-trivial commitment and the collection of P-SGPNE in $G^P_\\alpha $ is non-trivial.",
"Furthermore, we also say that if $\\alpha \\in \\mathcal {H}^M$ is such that $com(\\alpha ) = i$ , then all $s^* \\in \\text{P-SGPNE}(G^P_\\alpha )$ are of type $i$ as well."
],
[
"Results for $M = 3$ Miners",
"In order to visualise results for $M = 3$ miners, we fix the hash rate of the third player, $\\alpha _3$ and repeat our analysis for Section when $\\alpha _1$ and $\\alpha _2$ are allowed to vary.",
"We observe qualitative difference in the family of games $G_\\alpha $ for four different regions of $\\alpha _3$ values: $R_1 = [0,0.17]$ , $R_2 = [0.175,0.203]$ , $R_3 = [0.208, 0.27]$ and $R_4 = [0.274, 0.5]$ ."
],
[
"Pure Nash Equilibria in $G_\\alpha $",
"As mentioned in the previous section, our results show four main regimes of results as a function of $\\alpha _3$ .",
"In terms of PNE, When $\\alpha _3 \\in R_1$ , $H$ strictly dominates $S$ for miner 3, hence the three player games $G_\\alpha $ and $G^P_\\alpha $ reduce to a two player game conditioned on player 3 playing $H$ .",
"For $\\alpha _3 \\in R_2$ , we see the emergence of $SSS$ as a PNE near the centre of the hash space, and the size of this region grows as a function of $\\alpha _3$ .",
"For $\\alpha _3 \\in R_3$ , $SSS$ is still a PNE for central values of $\\alpha $ , however we see the emergence of distinct regions where $SHS$ and $HSS$ are PNE.",
"Finally, when $\\alpha _3 \\in R_4$ , $S$ strictly dominates $H$ for player 3, and once again $G_\\alpha $ and $G^P_\\alpha $ reduce to subgames conditioned on miner 3 playing $S$ .",
"In Figure REF , we visualise this phenomenon by picking representative values of $\\alpha _3$ in $R_1,R_2,R_3$ and $R_4$ and graphing regions where distinct PNE occur in $G_\\alpha $ as well as the difference in welfare between the best and worst PNE for each player respectively.",
"Figure: PNE for α 3 ∈{0.14,0.2,0.22,0.36}\\alpha _3 \\in \\lbrace 0.14, 0.2, 0.22, 0.36\\rbrace .",
"For the areas that have multiple PNE, the difference in welfare at the best PNE and worst PNE for miner 1 and miner 3 are mapped in the second and third rows respectively.",
"Since G α G_\\alpha is an anonymous game, the difference in welfare for miner 2 is the same as that of miner 1 reflected about the axis y=xy = x."
],
[
"SSM Profitability Threshold Diminished",
"As in the two-player case, we find that there are hash rates where $m_1$ is not unilaterally incentivised to employ SSM, yet there exist equilibria where $m_1$ employs SSM.",
"In Figure REF we visualise the hash rates where this happens for all $R_i$ relevant regions of $\\alpha _3$ values.",
"In particular, for $\\alpha _3$ values in $R_2, R_3$ and $R_4$ , we see that the emergence of SSS as a PNE can occur when $m_1$ has much smaller hash power than the 0.26795 necessary to make SSM profitable unilaterally.",
"Figure: Profitability Threshold Diminished for α 3 ∈{0.14,0.2,0.22,0.36}\\alpha _3 \\in \\lbrace 0.14, 0.2, 0.22, 0.36\\rbrace"
],
[
"Optimal Commitments",
"As mentioned in Section , we treat $G^P_\\alpha $ as a full information sequential game where $m_1$ commits to a strategy and all other miners subsequently act.",
"We recall that for a given pure strategy commitment $s_1 \\in [0,1]$ for $m_1$ , the worst Nash equilibrium of the resulting subgame $G_\\alpha (s_1,-)$ for $m_1$ (so in terms of $U_1$ ) is denoted by $WSN(s_1)$ .",
"In addition, $v_i(s_1) = U_i(s_1,WSN(s_1))$ for $i = 1,...,M$ , denotes the utility obtained by each miner at $(s_1,WSN(s_1))$ .",
"For $\\alpha _3 \\in R_1,R_2,R_3,R_4$ , we look at what values of $s_1$ optimise $v_1(s_1)$ as optimal commitments from the leader of $G_\\alpha ^P$ , $m_1$ .",
"As mentioned in Section , any $s^* = (s_1^*, WSN(s_1^*))$ that optimises $v_1$ is necessarily a (pessimistic) subgame perfect Nash equilibrium.",
"In Figure REF we plot these optimal commitments with fixed $\\alpha _3$ as a function of $(\\alpha _1,\\alpha _2)$ .",
"Furthermore, we also plot $com(\\alpha )$ as per Definition REF , and the subsequent surplus between $v_1$ at the aforementioned pessimistic SGPNE and the worst lowest utility PNE for $m_1$ .",
"Similar observations can be made as in the two-player case of Section : When $com(\\alpha ) = 0$ , pessimistic SGPNE of $G^P_\\alpha $ are identical to $PNE$ of $G_\\alpha $ , so the ability to partition and the ability to commit to strategies do not give $m_1$ an undue advantage in the worst case.",
"When $com(\\alpha ) = 1$ it is generally the case that $G_\\alpha $ has multiple PNE, and the commitment of $m_1$ “nudges” other players to a PNE that Pareto-dominates the worst PNE in $G_\\alpha $ .",
"The only exception to the previous observation is the left-most region of $com(\\alpha ) = 1$ when $\\alpha _3 = 0.2$ is fixed (second column of Figure REF ).",
"In this area, the optimal commitment for $m_1$ is $s_1 = 0$ .",
"In response to this, the subgame $G_\\alpha (0,-)$ only has $HH$ as a PNE.",
"As a consquence, the only pessimistic SGPNE at these $\\alpha $ values is $(0,0,0)$ , yet both $(0,0,0)$ and $(1,1,1)$ are PNE in $G_\\alpha $ .",
"The reason for this however, is that if we consider the commitment $s_1 = 1$ (i.e.",
"$m_1$ employing SSM), then $G_\\alpha (1,-)$ actually has two PNE: $HH$ and $SS$ .",
"The worst of these two equilibria however is $HH$ , and thus the strategy profile $(1, WSN(1)) = (1,0,0)$ , which is strictly worse than $(0,0,0)$ fpr $m_1.$ When $com(\\alpha ) = 2$ there exist pessimistic SGPNE, $s^* = (s_1^*, WSN(s_1^*)) \\notin \\text{PNE}(G_\\alpha )$ such that $s_1 \\in \\lbrace 0,1\\rbrace $ and $s^*_1$ is not a best response to $WSN(s_1^*)$ for $m_1$ .These non-trivial commitments make use of sequentiality of $G^P_\\alpha $ but not of the augmented action space granted by partitioning.",
"When $com(\\alpha ) = 3$ , $m_1$ has enough hash power that PNE($G_\\alpha $ ) only has strategy profiles that exemplify $m_2$ and $m_3$ being disincentivised to use SSM.",
"That being said, at these values of $\\alpha $ , $m_2$ and $m_3$ are almost indifferent between employing SSM and honest mining (hence the reason $com(\\alpha ) = 3$ occurs along boundaries of where PNE($G_\\alpha $ ) changes values), hence $m_1$ can bait them into employing SSM by judiciously giving away some hash power to honest mining in a partition.",
"Figure: Optimal m 1 m_1 commitment for α 3 ∈{0.14,0.2,0.22,0.36}\\alpha _3 \\in \\lbrace 0.14, 0.2, 0.22, 0.36\\rbrace , commitment types, and utility surplus in P-SGPNE vs. worst PNE for m 1 m_1."
],
[
"Penalising Coalitions",
"In Figure REF we plot hash rates where there exist penalising coalitions against miner 1 along with the smallest given penalty they can incur on miner 1.",
"Furthermore, in the top row of the plot, we specify precisely which coalitions $C \\subset [2,3]$ satisfy the conditions of Definition REF .",
"The plots show that for $\\alpha _3 \\in \\lbrace 0.14,0.36\\rbrace $ there is only one kind of penalising coalition ($C = \\lbrace 2\\rbrace $ or $C = \\lbrace 3\\rbrace $ respectively), but for $\\alpha _3 \\in \\lbrace 0.2,0.22\\rbrace $ , $C = \\lbrace 2\\rbrace , \\lbrace 3\\rbrace $ and $\\lbrace 2,3\\rbrace $ are all penalising coalitions at different hash rates and for some values of $\\alpha $ .",
"Figure: All Penalising Coalitions for miner 1 when α 3 ∈{0.14,0.2,0.22,0.36}\\alpha _3 \\in \\lbrace 0.14, 0.2, 0.22, 0.36\\rbrace , and the smallest penalty they incur."
]
] | 1906.04502 |
[
[
"Super-biderivations of the contact Lie superalgebra\n $K(m,n;\\underline{t})$"
],
[
"Abstract Let $K$ denote the contact Lie superalgebra $K(m,n;\\underline{t})$ over a field of characteristic $p>3$, which has a finite $\\mathbb{Z}$-graded structure.",
"Let $T_K$ be the canonical torus of $K$, which is an abelian subalgebra of $K_{0}$ and operates on $K_{-1}$ by semisimple endomorphisms.",
"Utilizing the weight space decomposition of $K$ with respect to $T_K$, %we show the action of the skew-symmetric super-biderivation on the elements of $T$ and the contact of $K$.",
"%Moreover, we prove that each skew-symmetric super-biderivation of $K$ is inner."
],
[
"Introduction",
"Let $L$ be a Lie algebra over an arbitrary field $\\mathbb {F}$ .",
"An $\\mathbb {F}$ -linear map $D:L\\rightarrow L$ is a derivation satisfying $D([x,y])=[D(x),y]+[x,D(y)],$ for all $x,y\\in L$ .",
"A bilinear map $\\psi :L\\times L\\rightarrow L$ is called a biderivation if it is a derivation with respect to both components, meaning that $\\psi (x,[y,z])&=[\\psi (x,y),z]+[y,\\psi (x,z)],\\\\\\psi ([x,y],z)&=[\\psi (x,z),y]+[x,\\psi (y,z)],$ for all $x,~y,~z\\in L$ .",
"A biderivation $\\psi $ is called skew-symmetric if $\\psi (x,y)=-\\psi (y,x)$ for all $x,~y\\in L$ .",
"Obviously, if a biderivation $\\psi $ is skew-symmetric, we can omit one of the equations (1.1) and (1.2).",
"Meanwhile, we can view $\\psi (x,\\cdot )$ or $\\psi (\\cdot ,x)$ as a derivation of $L$ .",
"The study of biderivations traces back to the research on the commuting map in the associative ring [1], where the author showed that all biderivations on associative prime rings are inner.",
"The notation of biderivations of Lie algebras was introduced in [2].",
"In recent years, there exist a lot of interests in studying biderivations and commuting maps on Lie algebras[3], [4], [5], [6], [7], [8], [9].",
"Moreover, the authors gave the notion of the skew-symmetric super-biderivation in [11].",
"So the results about the skew-symmetric super-biderivation of Lie superalgebras arise in [11], [12], [13].",
"The Cartan modular Lie superalgebra is an important branch of the modular Lie superalgebra, which is a Lie superalgebra over an algebraically closed field of characteristic $p>0$ .",
"And the contact Lie superalgebra $K(m,n;\\underline{t})$ is an important class of Cartan modular Lie superalgebras.",
"There are many research results about the contact Lie superalgebra $K(m,n;\\underline{t})$ , such as, derivation superalgebras[14], [15], [16], noncontractible filtrations[17], nondegenerate associative bilinear forms[18].",
"In this paper, we prove that each skew-symmetric super-biderivation of $K(m,n;\\underline{t})$ is inner.",
"The paper is organized as follows.",
"In Section 2, we recall the basic notation.",
"In Section 3, we use the weight space decomposition of $K(m,n;\\underline{t})$ with respect to the canonical torus $T_K$ to prove that all skew-symmetric super-biderivation of $K$ is inner(Theorem 3.14)."
],
[
"Preliminaries",
"Let $\\mathbb {F}$ denote the prime field of the characteristic $p>2$ and $\\mathbb {Z}_{2}=\\lbrace \\overline{0},\\overline{1}\\rbrace $ the additive group of two elements.",
"For a vector superspace $V=V_{\\overline{0}}\\oplus V_{\\overline{1}}$ , we use $\\mathrm {p}(x)$ for the parity of $x\\in V_{\\alpha }$ , $\\alpha \\in \\mathbb {Z}_{2}$ .",
"If $V=\\oplus _{i\\in \\mathbb {Z}}V_i$ is a $\\mathbb {Z}$ -graded vector space and $x\\in V$ is a $\\mathbb {Z}$ -homogeneous element, write $|x|$ for the $\\mathbb {Z}$ -degree of $x$ .",
"Once the symbol $\\mathrm {p}(x)$ or $|x|$ appears in this paper, it implies that $x$ is a $\\mathbb {Z}_{2}$ -homogeneous element or that $x$ is a $\\mathbb {Z}$ -homogeneous element.",
"Throughout this paper all vector spaces or algebras are over $\\mathbb {F}$ ."
],
[
"Skew-symmetric super-biderivations of a Lie superalgebra",
"Let us recall some facts related to the superderivation and skew-symmetric super-biderivation of Lie superalgebras.",
"A Lie superalgebra is a vector superspace $L=L_{\\overline{0}}\\oplus L_{\\overline{1}}$ with an even bilinear mapping $[\\cdot ,\\cdot ]:L\\times L\\rightarrow L$ satisfying the following axioms: $[x,y]&=-(-1)^{\\mathrm {p}(x)\\mathrm {p}(y)}[y,x],\\\\[x,[y,z]]&=[[x,y],z]+(-1)^{\\mathrm {p}(x)\\mathrm {p}(y)}[y,[x,z]],$ for all $x,y,z\\in L$ .",
"We call a linear mapping $D:L\\times L\\rightarrow L$ a superderivation of $L$ if it satisfies the following axiom: $D([x,y])=[D(x),y]+(-1)^{\\mathrm {p}(D)\\mathrm {p}(x)}[x,D(y)],$ for all $x,y\\in L$ , where $\\mathrm {p}(D)$ denotes the $\\mathbb {Z}_{2}$ -degree of $D$ .",
"Write $\\mathrm {Der}_{\\overline{0}}(L)$ (resp.",
"$\\mathrm {Der}_{\\overline{1}}(L)$ ) for the set of all superderivations of $\\mathbb {Z}_2$ -degree $\\overline{0}$ (resp.",
"$\\overline{1}$ ) of $L$ .",
"We call a bilinear mapping $\\phi :L\\times L\\rightarrow L$ a skew-symmetric super-biderivation of $L$ if it satisfies the following axioms: $&skew-symmetry:~~\\phi (x,y)=-(-1)^{\\mathrm {p}(x)\\mathrm {p}(y)}\\phi (y,x),\\\\&\\phi ([x,y],z)=(-1)^{\\mathrm {p}(\\phi )\\mathrm {p}(x)}[x,\\phi (y,z)]+(-1)^{\\mathrm {p}(y)\\mathrm {p}(z)}[\\phi (x,z),y],\\\\&\\phi (x,[y,z])=[\\phi (x,y),z]+(-1)^{(\\mathrm {p}(\\phi )+\\mathrm {p}(x))\\mathrm {p}(y)}[y,\\phi (x,z)],$ for all $\\mathbb {Z}_2$ -homogeneous elements $x,y,z\\in L$ .",
"A super-biderivation $\\phi $ of $\\mathbb {Z}_{2}$ -degree $\\gamma $ of $L$ is a super-biderivation such that $\\phi (L_{\\alpha },L_{\\beta })\\subset L_{\\alpha +\\beta +\\gamma }$ for any $\\alpha ,~\\beta \\in \\mathbb {Z}_{2} $ .",
"Denote by $\\mathrm {BDer}_{\\gamma }(L)$ the set of all skew-symmetric super-biderivations of $\\mathbb {Z}_2$ -degree $\\gamma $ .",
"Obviously, $\\mathrm {BDer}(L)=\\mathrm {BDer}_{\\overline{0}}(L)\\oplus \\mathrm {BDer}_{\\overline{1}}(L).$ Specially, if the bilinear map $\\phi _{\\lambda }:L\\times L\\rightarrow L$ is defined by $\\phi _{\\lambda }(x,y)=\\lambda [x,y]$ for all $x,y\\in L$ , where $\\lambda \\in \\mathbb {F}$ , then it is easy to check that $\\phi _{\\lambda }$ is a super-biderivation of $L$ .",
"This class of super-biderivations is called inner.",
"Denote by $\\mathrm {IBDer}(L)$ the set of all inner super-biderivations."
],
[
"Contact Lie superalgebras $K(m,n;\\underline{t})$",
"We propose to construct a $\\mathbb {Z}_2$ -gradation tensor algebra via a divided power algebra and a exterior superalgebra.",
"In the follow, we introduce the divided power algebra $\\mathcal {O}(m)$ and the exterior superalgebra $\\Lambda (n)$ .",
"Fix two positive integers $m>1$ and $n>1$ .",
"For $\\alpha =(\\alpha _1,\\ldots ,\\alpha _m)\\in \\mathbb {N}^{m}$ , where $ \\mathbb {N}$ denote the set of natural numbers, put $|\\alpha |=\\Sigma _{i=1}^{m}\\alpha _i$ .",
"For two $m$ -tuples $\\alpha =(\\alpha _1,\\ldots ,\\alpha _m)$ and $\\beta =(\\beta _1,\\ldots ,\\beta _m)\\in \\mathbb {N}^m$ , we write ${\\alpha \\atopwithdelims ()\\beta }=\\prod _{i=1}^{m}{\\alpha _i\\atopwithdelims ()\\beta _i}$ and define $\\beta \\le \\alpha \\Longleftrightarrow \\beta _i\\le \\alpha _i$ , $1\\le i\\le m$ .",
"Let $\\mathcal {O}(m)$ denote the $\\mathbb {F}$ -algebra of divided power series in the variable $x_1,\\ldots ,x_m$ , which is called a $divided~power~algebra$ .",
"For convenience, we replace $x_1^{\\alpha _1}x_2^{\\alpha _2}\\cdots x_m^{\\alpha _m}$ by $x^{(\\alpha )}$ , $\\alpha =(\\alpha _1,\\alpha _2,\\ldots ,\\alpha _m)$ .",
"Obviously, $\\mathcal {O}(m)$ has an $\\mathbb {F}$ -basis $\\lbrace x^{(\\alpha )}|\\alpha \\in \\mathbb {N}^{m}\\rbrace $ and satisfies the formula: $x^{(\\alpha )}x^{(\\beta )}={\\alpha +\\beta \\atopwithdelims ()\\alpha }x^{(\\alpha +\\beta )},~\\forall ~\\alpha ,\\beta \\in \\mathbb {N}^{m}.$ Let $\\Lambda (n)$ denote the $exterior~superalgebra$ over $\\mathbb {F}$ with $n$ variables $x_{m+1},\\ldots ,x_{s}$ , where $s=m+n$ .",
"The tensor product $\\mathcal {O}(m,n)=\\mathcal {O}(m)\\otimes _{\\mathbb {F}}\\Lambda (n)$ is an associative superalgebra with a $\\mathbb {Z}_2$ -gradation induced by the trivial $\\mathbb {Z}_2$ -gradation of $\\mathcal {O}(m)$ and the natural $\\mathbb {Z}_2$ -gradation of $\\Lambda (n)$ .",
"Obviously, $\\mathcal {O}(m,n)$ is super-commutative.",
"For $g\\in \\mathcal {O}(m)$ , $f\\in \\Lambda (n)$ , it is customary to write $gf$ instead of $g\\otimes f$ .",
"Including the formula (REF ), the following formulas also hold in $\\mathcal {O}(m,n)$ : $x_kx_l=-x_lx_k,~\\forall ~k,l\\in \\lbrace m+1,\\ldots ,s\\rbrace ;$ $x^{(\\alpha )}x_k=x_kx^{(\\alpha )},~\\forall ~\\alpha \\in \\mathbb {N}^{m}, k\\in \\lbrace m+1,\\ldots ,s\\rbrace .$ For $k=1,\\ldots ,n$ , set $\\mathbb {B}_k:=\\lbrace \\langle i_1,i_2,\\ldots ,i_k\\rangle \\mid m+1\\le i_1<i_2<\\ldots <i_k\\le s\\rbrace $ and $\\mathbb {B}:=\\cup _{k=0}^{n}\\mathbb {B}_k$ , where $\\mathbb {B}_0=\\emptyset $ .",
"For $u=\\langle i_1,i_2,\\ldots ,i_k\\rangle \\in \\mathbb {B}_k$ , set $|u|:=k$ , $x^u=x_{i_1}\\cdots x_{i_k}$ .",
"Specially, we define $|\\emptyset |=0$ , $x^{\\emptyset }=1$ , $|\\omega |=n$ and $x^{\\omega }=x_{m+1}\\cdots x_{m+n}$ .",
"Clearly, the set $\\lbrace x^{(\\alpha )}x^u|~\\alpha \\in \\mathbb {N}^{m},u\\in \\mathbb {B}\\rbrace $ constitutes an $\\mathbb {F}$ -basis of $\\mathcal {O}(m,n)$ .",
"Put $\\mathrm {I}_0:=\\lbrace 1,\\ldots ,m\\rbrace $ , $\\mathrm {I}_1:=\\lbrace m+1,\\ldots ,m+n\\rbrace $ and $\\mathrm {I}:=\\mathrm {I}_0\\cup \\mathrm {I}_1$ .",
"Let $\\partial _1,\\partial _2,\\ldots ,\\partial _{s}$ be the linear transformations of $\\mathcal {O}(m,n)$ such that $\\partial _i(x^{(\\alpha )})=x^{(\\alpha -\\varepsilon _i)}$ for $i\\in \\mathrm {I}_0$ , and $\\partial _i(x_k)=\\delta _{ik}$ , $k\\in \\mathrm {I}_1$ , for $i\\in \\mathrm {I}_1$ , where $\\delta _{ij}$ is denoted the Kronecker symbol.",
"Obviously, $\\mathrm {p}(\\partial _i)=\\overline{0}$ if $i\\in \\mathrm {I}_0$ and $\\mathrm {p}(\\partial _i)=\\overline{1}$ if $i\\in \\mathrm {I}_1$ .",
"Then $\\partial _1,\\partial _2,\\ldots ,\\partial _{s}$ are superderivations of the superalgebra $\\mathcal {O}(m,n)$ .",
"Let $W(m,n):=\\left\\lbrace \\sum f_r\\partial _r|~f_r\\in \\mathcal {O}(m,n),r\\in \\mathrm {I}\\right\\rbrace .$ Then $W(m,n)$ is an infinite-dimensional Lie superalgebra contained in $\\mathrm {Der}(\\mathcal {O}(m,n))$ .",
"One can verify that $[f\\partial _i,g\\partial _j]=f\\partial _i(g)\\partial _j-(-1)^{\\mathrm {p}(f\\partial _i)\\mathrm {p}(g\\partial _j)}g\\partial _j(f)\\partial _i,$ for all $f,g\\in \\mathcal {O}(m,n)$ and $i,j\\in \\mathrm {I}$ .",
"Fix two $m$ -tuples of positive integers $\\underline{t}=(t_1,t_2,\\ldots ,t_m)$ and $\\pi =(\\pi _1,\\pi _2,\\ldots ,\\pi _m)$ , where $\\pi _i=p^{t_i}-1$ for all $i\\in \\mathrm {I}_0$ and $p$ is denoted the characteristic of the basic field $\\mathbb {F}$ .",
"For two $m$ -tuples $\\alpha =(\\alpha _1,\\ldots ,\\alpha _m)$ and $\\beta =(\\beta _1,\\ldots ,\\beta _m)\\in \\mathbb {N}^m$ , we have ${\\alpha +\\beta \\atopwithdelims ()\\alpha }=0$ if there is some $i\\in \\lbrace 1,\\ldots ,m\\rbrace $ satisfying $\\alpha _i+\\beta _i\\ge p^{t_i}$ .",
"Thence the set $O(m,n;\\underline{t})=\\lbrace x^{(\\alpha )}x^u~|~0\\le \\alpha \\le \\pi , u\\in \\mathbb {B}\\rbrace $ is a subalgebra of $O(m,n)$ and the set $W(m,n;\\underline{t})=\\mathrm {span}_{\\mathbb {F}}\\lbrace x^{(\\alpha )}x^u\\partial _r~|~0\\le \\alpha \\le \\pi , u\\in \\mathbb {B}, r\\in \\mathrm {I}\\rbrace $ is a finite-dimensional simple subalgebra of $W(m,n)$ , which is called the generalized Witt Lie superalgebra.",
"$W(m,n;\\underline{t})$ possesses a $\\mathbb {Z}$ -graded structure: $W(m,n;\\underline{t})=\\bigoplus _{r=-1}^{\\xi -1}W(m,n;\\underline{t})_r,$ where $W(m,n;\\underline{t})_r:=\\mathrm {span}_{\\mathbb {F}}\\lbrace x^{(\\alpha )}x^u\\partial _j|~|\\alpha |+|u|=r+1,~j\\in \\mathrm {I}\\rbrace $ and $\\xi :=|\\pi |+n$ .",
"For $i\\in \\mathrm {I}_0$ , we abbreviate $x^{(\\varepsilon _i)}$ to $x_i$ , where $\\varepsilon _i$ is denoted the $m$ -tuple with 1 as the i-th entry and 0 elsewhere.",
"Hereafter, suppose $m=2r+1$ is odd and $n=2t$ is even.",
"Let $\\mathrm {J}=\\mathrm {I} \\setminus \\lbrace m\\rbrace $ and $\\mathrm {J}_0=\\mathrm {I}_0 \\setminus \\lbrace m\\rbrace $ .",
"For $i\\in \\mathrm {J}$ , put $i^{\\prime }:={\\left\\lbrace \\begin{array}{ll}i+r,&1\\le i\\le r,\\\\i-r, &r<i\\le 2r,\\\\i, &i=m,\\\\i+t, &m<i\\le m+t,\\\\i-t, &m+t<i\\le s;\\end{array}\\right.",
"}~~~~\\sigma (i):={\\left\\lbrace \\begin{array}{ll}1, &1\\le i\\le r\\\\-1, &r<i\\le 2r\\\\1, &2r<i\\le s.\\end{array}\\right.",
"}$ Define a linear mapping $D_{K}:\\mathcal {O}(m,n)\\rightarrow W(m,n)$ by means of $D_{K}(f)=\\sum \\limits _{i\\in \\mathrm {J}}(-1)^{\\mathrm {p}(\\partial _i)\\mathrm {p}(f)}(x_{i}\\partial _{m}(f)+\\sigma (i^{\\prime })\\partial _{i^{\\prime }}(f))\\partial _{i}+(2f-\\sum \\limits _{i\\in \\mathrm {J}}x_{i}\\partial _{m}(f))\\partial _{m}.$ The restricted linear mapping of $D_{K}$ on $\\mathcal {O}(m,n;\\underline{t})$ still is denoted by $D_{K}$ , that is $D_{K}:\\mathcal {O}(m,n;\\underline{t})\\rightarrow K(m,n;\\underline{t}).$ Let $\\widetilde{K}(m,n;\\underline{t})$ denote the image of $\\mathcal {O}(m,n;\\underline{t})$ under $D_{K}$ .",
"Consider the derived algebra of $\\widetilde{K}(m,n;\\underline{t})$ : $K(m,n;\\underline{t})=[\\widetilde{K}(m,n;\\underline{t}),\\widetilde{K}(m,n;\\underline{t})].$ The derived algebra $K(m,n;\\underline{t})$ is a finite dimensional simple Lie superalgebra, which is called the contact Lie superalgebra.",
"We define a Lie bracket $\\langle \\cdot ,\\cdot \\rangle $ on the tensor superalgebra $\\mathcal {O}(m,n;\\underline{t})$ by $ \\langle f, g \\rangle :=D_{K}(f)(g)-2\\partial _{m}(f)(g),$ for all $f,g\\in \\mathcal {O}(m,n;\\underline{t})$ .",
"Since $D_{K}$ is injective and $D_{K}(\\langle f,g\\rangle )=[ D_{K}(f),D_{K}(g)]$ , there exists an isomorphism, that is, $({K}(m,n;\\underline{t}),[\\cdot ,\\cdot ])\\cong (\\mathcal {O}(m,n;\\underline{t}),\\langle \\cdot ,\\cdot \\rangle ).$ For convenience, we use $W$ and $W_r$ denote $W(m,n;\\underline{t})$ and its $\\mathbb {Z}$ -graded subspace $W(m,n;\\underline{t})_r$ , respectively, ${K}(m,n;\\underline{t})$ is denoted by $K$ ."
],
[
"Skew-symmetry Super-biderivation of $K(m,n;\\underline{t})$",
"Lemma 3.1 [10] Let $L$ be a Lie superalgebra.",
"Suppose that $\\phi $ is a skew-symmetric super-biderivation on $L$ , then $[\\phi (x,y),[u,v]]=(-1)^{\\mathrm {p}(\\phi )(\\mathrm {p}(x)+\\mathrm {p}(y))}[[x,y] ,\\phi (u,v)] $ for any homogenous element $x,y,u,v\\in L$ .",
"Lemma 3.2 [10] Let $L$ be a Lie superalgebra.",
"Suppose that $\\phi $ is a skew-symmetric super-biderivation on $L$ .",
"If $\\mathrm {p}(x)+\\mathrm {p}(y)=\\overline{0}$ , then $[\\phi (x,y),[x,y]]=0$ for any homogenous element $x,y\\in L$ .",
"Lemma 3.3 [10] Let $L$ be a Lie superalgebra.",
"Suppose that $\\phi $ is a skew-symmetric super-biderivation on $L$ .",
"If $[x,y]=0$ , then $\\phi (x,y)\\in C_L([L,L])$ , where $C_L([L,L])$ is the centralizer of $[L,L]$ .",
"Lemma 3.4 Let $K$ denote the contact Lie superalgebra.",
"Suppose $\\phi $ is a skew-symmetric super-biderivation on $K$ .",
"If $\\langle x,y \\rangle =0$ for $x, y\\in K$ , then $\\phi (x,y)=0 .$ Since $K$ is a simple Lie superalgebra, it is obvious that $K=\\langle K,K \\rangle $ and $C(K)=0$ .",
"if $\\langle x,y \\rangle =0$ for $x, y\\in K$ , by Lemma REF , we obtain $\\phi (x,y)\\in C_K(\\langle K,K \\rangle )=C(K)=0.$ Set $T_K=\\mathrm {span}_{\\mathbb {F}}\\lbrace x_{i}x_{i^{\\prime }}~|~i\\in \\mathrm {J}\\rbrace $ .",
"Obviously, $T_{K}\\subseteq K(m,n;\\underline{t})_{0}\\bigcap K(m,n;\\underline{t})_{\\bar{0}}$ .",
"$T_K$ is an abelian subalgebra of $K$ .",
"For any $x^{(\\alpha )}x^u\\in K$ , we have $\\langle x_{i}x_{i^{\\prime }},x^{(\\alpha )}x^u \\rangle =(\\alpha _{i^{\\prime }}-\\alpha _{i}+\\delta _{(i^{\\prime }\\in u)}-\\delta _{(i\\in u)})x^{(\\alpha )}x^u,$ where $\\delta _{(\\mathrm {P})}=1$ if the proposition $\\mathrm {P}$ is true, $=0$ if the proposition $\\mathrm {P}$ is false.",
"Fixed an $m$ -tuple $\\alpha $ , where $\\alpha \\in \\mathbb {N}^m$ , $0\\le \\alpha \\le \\pi $ and $u\\in \\mathbb {B}$ , we define a linear function $(\\alpha +\\langle u\\rangle ): T\\rightarrow \\mathbb {F}$ such that $(\\alpha +\\langle u\\rangle )(x_{i}x_{i^{\\prime }})= \\alpha _{i^{\\prime }}-\\alpha _{i}+\\delta _{(i^{\\prime }\\in u)}-\\delta _{(i\\in u)}.$ Further, $K$ has a weight space decomposition with respect to $T_K$ : $K=\\bigoplus _{(\\alpha +\\langle u\\rangle )}K_{(\\alpha +\\langle u\\rangle )}.$ Lemma 3.5 Suppose that $\\phi $ is a $\\mathbb {Z}_2$ -homogeneous skew-symmetric super-biderivation on $K$ .",
"Let $x^{(\\alpha )}x^u\\in K$ such that $\\phi (x_{i}x_{i^{\\prime }},x^{(\\alpha )}x^u)\\in K_{(\\alpha +\\langle u\\rangle )},$ for any $x_{i}x_{i^{\\prime }}\\in T_K$ .",
"The equation by Lemma REF , it follows that $\\phi (x_{i}x_{i^{\\prime }},x_{j}x_{j^{\\prime }})=0$ for any $i,~j\\in \\mathrm {J}$ from $[x_{i}x_{i^{\\prime }},x_{j}x_{j^{\\prime }}]=0$ .",
"Note that $\\mathrm {p}(x_{l}x_{l^{\\prime }})=\\overline{0}$ for all $l\\in \\mathrm {J}$ , then all $x^{(\\alpha )}x^u\\in K$ , it is clear that $&\\langle x_{l}x_{l^{\\prime }},\\phi (x_{i}x_{i^{\\prime }},x^{(\\alpha )}x^u) \\rangle \\\\=&(-1)^{(\\mathrm {p}(\\phi )+\\mathrm {p}(x_{i}x_{i^{\\prime }}))\\mathrm {p}(x_{l}x_{l^{\\prime }})}(\\phi (x_{i}x_{i^{\\prime }},\\langle x_{l}x_{l^{\\prime }},x^{(\\alpha )}x^u \\rangle )-\\langle \\phi (x_{i}x_{i^{\\prime }},x_{l}x_{l^{\\prime }}),x^{(\\alpha )}x^u \\rangle )\\\\=&(\\alpha _{i^{\\prime }}-\\alpha _{i}+\\delta _{(i^{\\prime }\\in u)}-\\delta _{(i\\in u)})\\phi (x_{i}x_{i^{\\prime }},x^{(\\alpha )}x^u).$ The proof is completed.",
"Remark 3.6 Due to Lemma REF , we can find that any $\\mathbb {Z}_2$ -homogeneous skew-symmetric super-biderivation on $K$ is an even bilinear map.",
"Since $\\phi (x_{i}x_{i^{\\prime }},x^{(\\alpha )}x^u)$ and $x^{(\\alpha )}x^u$ have the same $\\mathbb {Z}_2$ -degree.",
"Then the $\\mathbb {Z}_2$ -degree of $\\phi $ is even.",
"Lemma 3.7 [17] Let $M=\\lbrace x^{(\\kappa _i\\varepsilon _i)}|0\\le \\kappa _i\\le \\pi _i, i\\in \\mathrm {I}_0\\rbrace $ and $N=\\lbrace x_i|i\\in \\mathrm {I}_1\\rbrace $ .",
"Then $K$ is generated by $M\\cup N$ .",
"Lemma 3.8 Let $i\\in \\mathrm {J}_0$ , $j \\in \\mathrm {I}_1$ and $q_i\\in \\mathbb {N}$ , $1\\le q_i\\le \\pi _i$ .",
"Then the following statements hold: (1) $K(m,n;\\underline{t})_{(0)}=\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}\\mathbb {F}(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_{0}}} \\limits _{\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _{l})})x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}};$ (2) $K(m,n;\\underline{t})_{(q_i\\varepsilon _i)}=\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}\\mathbb {F}(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0\\backslash \\lbrace i,{i^{\\prime }}\\rbrace }} \\limits _{\\alpha _{{l^{\\prime }}}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{(\\alpha _i\\varepsilon _i)}x^{(\\overline{(\\alpha _i-q_{i})}\\varepsilon _{i^{\\prime }})}x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}};$ (3) $K(m,n;\\underline{t})_{(\\langle j\\rangle )}=\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}\\mathbb {F}(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_{0}}} \\limits _{\\alpha _{{l^{\\prime }}}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _{l})})x^{(\\alpha _m\\varepsilon _m)}x_{j}x^{\\bar{u}};$ where $i$ and ${i^{\\prime }}$ are both in $\\bar{u}$ for $i\\in \\mathrm {J}$ , and $\\alpha _{l}^{\\overline{q}}$ is denoted some integer and $\\alpha _{l}^{\\overline{q}}\\equiv q~(\\mathrm {mod}~p)$ .",
"(1) We first discuss the vector of the same weight with 1 in $K$ with respect to $T_K$ .",
"Since we have the equation $\\left\\langle {x_{l}x_{l^{\\prime }}},{1} \\right\\rangle =D_{K}(x_{l}x_{l^{\\prime }})(1)-2\\partial _{m}(x_{l}x_{l^{\\prime }})(1)=0.$ For any $l \\in \\mathrm {J}$ , in contrast with equation (3.1), we get that $\\alpha _{l^{\\prime }}-\\alpha _{l}+\\delta _{(l^{\\prime }\\in u)}-\\delta _{(l\\in u)}=0.$ Then if $l \\in \\mathrm {J_{0}}, $ it is obvious that is $\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)$ .",
"If $l\\in \\mathrm {I_{1}}$ , it is obvious that $l$ and $l^{^{\\prime }}$ are both in $\\bar{u}$ .",
"It proves that $K(m,n;\\underline{t})_{(0)}=\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}\\mathbb {F}(\\mathop {\\prod \\limits _{l\\in \\mathrm {J_{0}}}} \\limits _{\\alpha _{{l^{\\prime }}}-\\alpha _{l}\\equiv 0(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _{l})})x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}}.$ (2) Without loss of generality, we choose a fixed element $i\\in \\mathrm {J}$ .",
"For any $l \\in \\mathrm {J}$ , we have the equation $\\left\\langle {x_{l}x_{l^{\\prime }}},{x^{(q_{i}\\varepsilon _{i})}} \\right\\rangle =D_{K}(x_{l}x_{l^{\\prime }})(x_{i})-2\\partial _{m}(x_{l}x_{l^{\\prime }})(x_{i})=-q_{l}x_{l}\\delta _{(li)}.$ For any $l \\in \\mathrm {J}$ , by equation(3.1) we have that $\\alpha _{l^{\\prime }}-\\alpha _{l}+\\delta _{(l^{\\prime }\\in u)}-\\delta _{(l\\in u)}=-q_{l}\\delta _{(li)}.$ Then we try to discuss the choice of $l\\in \\mathrm {J}.$ If $l\\in \\mathrm {J_{0}}\\setminus \\lbrace i,i^{^{\\prime }}\\rbrace $ , it is obvious that $\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)$ .",
"If $ l=i$ , it is obvious that $\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv -q_{i}~(\\mathrm {mod}~p)$ .",
"If $l \\in \\mathrm {I_{1}}$ , we have that $l$ and $l^{\\prime }$ are both in $\\bar{u}$ .",
"So we proves that $K(m,n;\\underline{t})_{(q_i\\varepsilon _i)}=\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}\\mathbb {F}(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0\\backslash \\lbrace i,{i^{\\prime }}\\rbrace }} \\limits _{\\alpha _{{l^{\\prime }}}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{(\\alpha _i\\varepsilon _i)}x^{(\\overline{(\\alpha _i-q_{i})}\\varepsilon _{i^{\\prime }})}x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}}.$ (3) Without loss of generality, we choose a fixed element $i\\in \\mathrm {J}$ .",
"For any $l \\in \\mathrm {J}$ , we have the equation $\\left\\langle {x_{l}x_{l^{\\prime }}},{x_{j}} \\right\\rangle =D_{K}(x_{l}x_{l^{\\prime }})(x_{j})-2\\partial _{m}(x_{l}x_{l^{\\prime }})(x_{j})=-x_{l}\\delta _{(lj)}.$ By equation (3.1), for any $l \\in \\mathrm {J}$ , we have that $\\alpha _{l^{\\prime }}-\\alpha _{l}+\\delta _{(l^{\\prime }\\in u)}-\\delta _{(l\\in u)}=-\\delta _{(lj)}.$ Then we try to discuss the choice of $l\\in \\mathrm {J}.$ If $l \\in \\mathrm {J_{0}}$ , it is obvious that $\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)$ .",
"If $l \\in \\mathrm {I_{1}}$ , we have that $l$ and $l^{\\prime }$ are both in $\\bar{u}$ .",
"It proves that $K(m,n;\\underline{t})_{(\\langle j\\rangle )}=\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}\\mathbb {F}(\\mathop {\\prod \\limits _{l\\in \\mathrm {J_{0}}}} \\limits _{\\alpha _{{l^{\\prime }}}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _{l})})x^{(\\alpha _m\\varepsilon _m)}x_{j}x^{\\bar{u}}.$ Lemma 3.9 Suppose that $\\phi $ is a $\\mathbb {Z}_2$ -homogeneous skew-symmetric super-biderivation on $K$ .",
"For any $x_{l}x_{l^{\\prime }}\\in T_K$ and $x^{(q_{m}\\varepsilon _{m})}\\in M$ , where $0\\le q_m\\le \\pi _m$ , we have $\\phi (x_{l}x_{l^{\\prime }},x^{(q_{m}\\varepsilon _{m})})=0.$ When $q_{m}=0$ , by Lemma REF , it is obvious that $\\phi (x_{l}x_{l^{\\prime }},1)=0$ for $l\\in \\mathrm {I} $ from the equation $\\langle x_{l}x_{l^{\\prime }},1\\rangle =0$ .",
"When $q_{m}\\ne 0$ , we have that $&\\langle x_{l}x_{l^{\\prime }},x^{(q_m\\varepsilon _m)} \\rangle \\\\=&D_{K}(x_{l}x_{l^{\\prime }})(x^{(q_m\\varepsilon _m)})-2\\partial _{m}(x_{l}x_{l^{\\prime }})(x^{(q_m\\varepsilon _m)})\\\\=&(2(x_{l}x_{l^{\\prime }})-\\sum \\limits _{i \\in \\mathrm {J}}x_{i}\\partial _{i}(x_{l}x_{l^{\\prime }}))\\partial _{m}(x^{(q_m\\varepsilon _m)})\\\\=&(2(x_{l}x_{l^{\\prime }})-2(x_{l}x_{l^{\\prime }}))\\partial _{m}(x^{(q_m\\varepsilon _m)})\\\\=&0.$ Hence, we have that $\\phi (x_{l}x_{l^{\\prime }},x^{(q_{m}\\varepsilon _{m})})=0.$ The proof is completed.",
"Lemma 3.10 Suppose that $\\phi $ is a $\\mathbb {Z}_2$ -homogeneous skew-symmetric super-biderivation on $K$ .",
"For any $i\\in \\mathrm {J} $ and $x_{l}x_{l^{\\prime }}\\in T_K$ , there is an element $ \\lambda _i\\in \\mathbb {F}$ such that $\\phi (x_{l}x_{l^{\\prime }},x_{i})=\\lambda _i\\langle x_{l}x_{l^{\\prime }},x_{i} \\rangle ,$ where $\\lambda _i$ is dependent on the second component.",
"Without loss of generality, we choose a fixed element $i\\in \\mathrm {J}$ .",
"By Lemma REF , it is obvious that $\\phi (x_{l}x_{l^{\\prime }},x_{i})=0$ for $l\\in \\mathrm {J}\\setminus \\lbrace i,{i^{\\prime }}\\rbrace $ from $\\langle x_{l}x_{l^{\\prime }},x_{i}\\rangle =0$ .",
"So we only need to discuss the case with the condition $l=i$ .",
"When $q_{i}=1$ , by Lemma REF (2), we can suppose that $\\phi (x_{l}x_{l^{\\prime }},x_{i})=&\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\overline{u}\\in \\mathbb {B}}a(\\alpha ,\\bar{u})(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0\\backslash \\lbrace i,{i^{\\prime }}\\rbrace }} \\limits _{\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{(\\alpha _i\\varepsilon _i)}x^{((\\overline{\\alpha _i-1})\\varepsilon _{i^{\\prime }})}x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}}.$ It is obvious that $0=&\\phi (x_{i}x_{i^{\\prime }},\\langle 1,x_{i} \\rangle )-\\langle \\phi (x_{i}x_{i^{\\prime }},1),x_{i}\\rangle \\\\=&(-1)^{(\\mathrm {p}(\\phi )+\\mathrm {p}(x_{i}x_{i^{\\prime }}))\\mathrm {p}(1)}\\langle 1,\\phi (x_{i}x_{i^{\\prime }},x_{i}) \\rangle \\\\=&D_{K}(1)( \\phi (x_{i}x_{i^{\\prime }},x_{i}))-2\\partial _{m}(1)( \\phi (x_{i}x_{i^{\\prime }},x_{i}))\\\\=&\\partial _{m}( \\phi (x_{i}x_{i^{\\prime }},x_{i}))\\\\=&\\partial _{m}(\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\overline{u}\\in \\mathbb {B}}a(\\alpha ,\\bar{u})(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0\\backslash \\lbrace i,{i^{\\prime }}\\rbrace }} \\limits _{\\alpha _{{l^{\\prime }}}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{(\\alpha _i\\varepsilon _i)}x^{((\\overline{\\alpha _i-1})\\varepsilon _{i^{\\prime }})}x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}}).$ By computing the equation, we find that $a(\\alpha ,\\bar{u})=0$ if $\\alpha _{m}>0$ .",
"Putting $l\\in \\mathrm {J}\\backslash \\lbrace i,i^{\\prime }\\rbrace $ , we have that $0=&(-1)^{(\\mathrm {p}(\\phi )+\\mathrm {p}(x_{i}x_{i^{\\prime }}))\\mathrm {p}(x_{l})}(\\phi (x_{i}x_{i^{\\prime }}, \\langle x_{l},x_{i} \\rangle )-\\langle \\phi (x_{i}x_{i^{\\prime }},x_{l}),x_{i} \\rangle )\\\\=&\\langle x_{l},\\phi (x_{i}x_{i^{\\prime }},x_{i}) \\rangle \\\\=&D_{K}(x_{l})( \\phi (x_{i}x_{i^{\\prime }},x_{i}))-2\\partial _{m}(x_{l})( \\phi (x_{i}x_{i^{\\prime }},x_{i}))\\\\=&\\partial _{l^{\\prime }}( \\phi (x_{i}x_{i^{\\prime }},x_{i}))\\\\=&\\partial _{l^{\\prime }}(\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\overline{u}\\in \\mathbb {B}} a(\\alpha ,i)(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0\\backslash \\lbrace i,{i^{\\prime }}\\rbrace }} \\limits _{\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)}x^{(\\alpha _i\\varepsilon _i)} x^{(\\alpha _{^{l}}\\varepsilon _l)})x^{((\\overline{\\alpha _i-1})\\varepsilon _{i^{\\prime }})}x^{\\bar{u}}).$ By computing the equation, we find that $a(\\alpha ,\\bar{u})=0$ if $\\alpha _{l}>0$ for $l\\in \\mathrm {J}_0\\backslash \\lbrace i,i^{\\prime }\\rbrace $ or $|\\bar{u}|>0$ .",
"Then we can suppose that $\\phi (x_{l}x_{l^{\\prime }},x_{i})=\\sum \\limits _{0\\le \\alpha \\le \\pi }a(\\alpha )x^{((\\overline{\\alpha _i-1})\\varepsilon _{i^{\\prime }})}x^{(\\alpha _i\\varepsilon _i)}.$ Since $\\mathrm {p}(x_{i}x_{i^{\\prime }})+\\mathrm {p}(x_{i})=\\overline{0}$ for any $ i\\in \\mathrm {I}_0$ , by Lemma REF , we have $0=&\\langle \\phi (x_{i}x_{i^{\\prime }},x_{i}),\\langle x_{i}x_{i^{\\prime }},x_{i}\\rangle \\rangle \\\\=&\\langle \\phi (x_{i}x_{i^{\\prime }},x_{i}),-x_{i}\\rangle \\\\=&D_{K}(x_{i})( \\phi (x_{i}x_{i^{\\prime }},x_{i}))-2\\partial _{m}(x_{i})( \\phi (x_{i}x_{i^{\\prime }},x_{i}))\\\\=&\\partial _{i^{\\prime }}( \\phi (x_{i}x_{i^{\\prime }},x_{i}))\\\\=&\\partial _{i^{\\prime }}(\\sum \\limits _{0\\le \\alpha \\le \\pi } a(\\alpha )x^{((\\overline{\\alpha _i-1})\\varepsilon _{i^{\\prime }})}x^{(\\alpha _i\\varepsilon _i)}).$ By computing the equation, we find that $a(\\alpha )=0$ if $\\overline{\\alpha _i-1}>0$ .",
"Hence we get that $\\alpha _i=1$ .",
"Let $\\lambda _i=a(\\varepsilon _i)$ .",
"From what has been discussed above, for any $i\\in \\mathrm {J}_0$ we have that $\\phi (x_{l}x_{l^{\\prime }},x_{i})=-x_{i}=\\lambda _{i}\\langle x_{l}x_{l^{\\prime }},x_{i} \\rangle ,$ where $\\lambda _i$ is dependent on the second component.",
"Similarly, we choose a fixed element $j\\in \\mathrm {I}_1$ .",
"By Lemma REF (3), we can suppose that $\\phi (x_{j}x_{j^{\\prime }},x_{j})=&\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}a(\\alpha ,\\bar{u},j)(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0}} \\limits _{\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}}x_{j}$ where $a(\\alpha ,\\bar{u},j)\\in \\mathbb {F}$ .",
"By the definition of the shew-symmetric super-biderivation, we have $0=&\\phi (x_{j}x_{j^{\\prime }},\\langle 1,x_{j} \\rangle )-\\langle \\phi (x_{j}x_{j^{\\prime }},1),x_{j}\\rangle \\\\=&(-1)^{(\\mathrm {p}(\\phi )+\\mathrm {p}(x_{j}x_{j^{\\prime }}))\\mathrm {p}(1)}\\langle 1,\\phi (x_{j}x_{j^{\\prime }},x_{j}) \\rangle \\\\=&D_{K}(1)( \\phi (x_{j}x_{j^{\\prime }},x_{j}))-2\\partial _{m}(1)( \\phi (x_{j}x_{j^{\\prime }},x_{j}))\\\\=&2\\partial _{m}( \\phi (x_{j}x_{j^{\\prime }},x_{j}))\\\\=&2\\partial _{m}(\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}a(\\alpha ,\\bar{u},j)(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0}} \\limits _{\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{(\\alpha _m\\varepsilon _m)}x^{\\bar{u}}x_{j})\\\\=&\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\bar{u}\\in \\mathbb {B}}a(\\alpha ,\\bar{u},j)(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0}} \\limits _{\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{((\\alpha _m-1)\\varepsilon _m)}x^{\\bar{u}}x_{j}.$ By computing the equation, we find that $a(\\alpha ,\\bar{u},j)=0$ if $\\alpha _{m}>0$ .",
"For $k\\in \\mathrm {J}\\backslash \\lbrace j,j^{\\prime }\\rbrace $ , it is obvious that $0=&(-1)^{(\\mathrm {p}(\\phi )+\\mathrm {p}(x_{l}x_{l^{\\prime }}))\\mathrm {p}(x_{k})}(\\phi (x_{l}x_{l^{\\prime }},\\langle x_{k},x_{j} \\rangle )-\\langle \\phi (x_{l}x_{l^{\\prime }},x_{k}),x_{j} \\rangle )\\\\=&\\langle x_{k},\\phi (x_{l}x_{l^{\\prime }},x_{j}) \\rangle \\\\=&\\langle x_{k},\\sum \\limits _{0\\le \\alpha \\le \\pi ,~\\overline{u}\\in \\mathbb {B}} a(\\alpha ,\\bar{u},j)(\\mathop {\\prod \\limits _{l\\in \\mathrm {J}_0}} \\limits _{\\alpha _{l^{\\prime }}-\\alpha _{l}\\equiv 0~(\\mathrm {mod}~p)} x^{(\\alpha _{l}\\varepsilon _l)})x^{\\bar{u}}x_{j} \\rangle .$ Putting $k\\in \\mathrm {I}_1$ , we can deduce $a(\\alpha ,\\bar{u},j)=0$ if $|\\bar{u}|>0$ .",
"Putting $k\\in \\mathrm {J}_0$ , we have that $a(\\alpha ,\\bar{u},j)=0$ if $\\alpha _{k}>0$ .",
"Let $a(0,0,j)=\\lambda _j$ .",
"Hence, for any $j\\in \\mathrm {I}_1$ , we have that $\\phi (x_{l}x_{l^{\\prime }},x_{j})=-a(0,0,j)x_{j}=\\lambda _{j}\\langle x_{l}x_{l^{\\prime }},x_{j} \\rangle .$ The proof is completed.",
"Lemma 3.11 Suppose that $\\phi $ is a $\\mathbb {Z}_{2}$ -homogenous skew-symmetric super-biderivation on $K$ .",
"For any $x^{(q_i\\varepsilon _i)}\\in M$ , where $1\\le q_i\\le \\pi _i$ , $i\\in \\mathrm {J}_0 $ , there is an element $\\lambda _{i}\\in \\mathbb {F}$ such that $\\phi (x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _i)})=\\lambda _{i}\\langle x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _i)} \\rangle .$ Without loss of generality, we choose a fixed element $i\\in \\mathrm {I}_0$ .",
"By Lemma REF , it is obvious that $\\phi (x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _i)})=0$ for $l\\in \\mathrm {J}\\setminus \\lbrace i,i^{\\prime }\\rbrace $ from $\\langle x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _i)} \\rangle =0$ .",
"So we only need to consider the condition that $l=i$ , it is clear that $\\langle x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)} \\rangle =-q_ix^{(q_i\\varepsilon _i)}$ .",
"If $q_i>1$ , by Lemma REF and REF , we have $\\begin{split}0=&\\langle \\phi (x_{k}x_{k^{\\prime }},x_{k}),\\langle x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)} \\rangle \\rangle -(-1)^{\\mathrm {p}(\\phi )(\\mathrm {p}(x_{k}x_{k^{\\prime }})+\\mathrm {p}(x_{k}))}\\langle \\langle x_{k}x_{k^{\\prime }},x_{k} \\rangle ,\\phi (x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)}) \\rangle \\\\=&\\langle \\lambda _k\\langle x_{k}x_{k^{\\prime }},x_{k} \\rangle ,(-q_{i})x^{(q_i\\varepsilon _i)} \\rangle -\\langle -x_{k},\\phi (x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)}) \\rangle \\\\=&\\langle x_{k},-\\lambda _kq_{i}x^{(q_i\\varepsilon _i)}+\\phi (x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)}) \\rangle .\\end{split}$ Because of $\\mathcal {C}_{K_{-1}}(K)=\\lbrace f\\in K|\\langle f,x_{i} \\rangle =0, \\forall ~i\\in \\mathrm {I} \\rbrace = K_{-2}=\\mathbb {F}1$ , the equation (REF ) implies that $\\phi (x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)})=\\lambda _i\\langle x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)} \\rangle +b,$ where $\\lambda _i$ is denoted in Lemma REF and $b\\in \\mathbb {F}$ .",
"Since $\\phi (x_{i}x_{i^{\\prime }},x^{(q_i\\varepsilon _i)})\\in K(m,n;\\underline{t})_{(q_i\\varepsilon _i)}$ by Lemma REF .",
"It is easily seen from Lemma REF (2) that $K_{-2}\\cap K_{(q_i\\varepsilon _i)}=\\emptyset $ for $q_i>1$ .",
"So $b=0$ and $\\phi (x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _i)})=\\lambda _i\\langle x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _i)} \\rangle ,$ where $\\lambda _{i}$ is dependent on the second component.",
"Lemma 3.12 Suppose that $\\phi $ is a $\\mathbb {Z}_{2}$ -homogenous skew-symmetric super-biderivation on $K$ .",
"For any $x^{(q_m\\varepsilon _m)}\\in M$ , where $0\\le q_{m}\\le \\pi _{m}$ , there is an element $\\lambda \\in \\mathbb {F}$ such that $\\phi (1,x^{(q_m\\varepsilon _m)})=\\lambda \\langle 1,x^{(q_m\\varepsilon _m)} \\rangle .$ When $q_{m}=1$ , we suppose that $\\phi (1,x_m)=\\sum _{0\\le \\alpha \\le \\pi ,~u\\in \\mathbb {B}}c_{(\\alpha ,u)}x^{(\\alpha )}x^{u},$ where $c_{(\\alpha ,u)}\\in \\mathbb {F}$ .",
"For $k\\in \\mathrm {J_{0}}$ , by the definition of the skew-symmetric super-biderivation, we have the equation $0=&(-1)^{(p(\\phi )+p(1))p(1)}(\\phi (1,\\langle 1,x_m\\rangle )-\\langle \\phi (1,1),x_m\\rangle )\\\\=&\\langle 1,\\phi (1,x_m)\\rangle \\\\=&\\langle 1,\\sum _{0\\le \\alpha \\le \\tau ,~u\\in \\mathbb {B}}c_{\\alpha }x^{(\\alpha )}x^{u}\\rangle \\\\=&2\\sum _{0<\\alpha \\le \\tau ,~u\\in \\mathbb {B}}c_{\\alpha }x^{(\\alpha -\\varepsilon _{m})}x^{u}.$ By computing the equation, we find that $c_{(\\alpha ,u)}=0$ if $\\alpha _{m}-\\varepsilon _{m}\\ge 0$ .",
"We suppose that $\\phi (1,x_m)=\\sum _{0\\le \\alpha _{\\widehat{m}}\\le \\tau _{\\widehat{m}},~u\\in \\mathbb {B}}c_{\\alpha _{(\\widehat{m},u)}}x^{(\\alpha _{\\widehat{m}})}x^{u},$ where $\\widehat{m}$ represents an m-tuple with 0 as the m-th entry.",
"For $k\\in \\mathrm {J}$ , by the definition of the skew-symmetric biderivation, we have the equation $0=&(-1)^{(p(\\phi )+p(1))p(x_{m})}(\\phi (x_{k},\\langle 1,x_m\\rangle )-\\langle \\phi (x_k,1),x_m\\rangle )\\\\=&\\langle x_{k},\\phi (1,x_m)\\rangle \\\\=&\\langle x_{k},\\sum _{0\\le \\alpha _{\\widehat{m}}\\le \\tau _{\\widehat{m}},~u\\in \\mathbb {B}}c_{\\alpha _{(\\widehat{m},u)}}x^{(\\alpha _{\\widehat{m}})}x^{u}\\rangle \\\\=&\\sum _{0\\le \\alpha _{\\widehat{m},~u\\in \\mathbb {B}}\\le \\tau _{\\widehat{m}}}c_{\\alpha _{(\\widehat{m},u)}}x^{(\\alpha _{(\\widehat{m},u)}-\\varepsilon _{k^{\\prime }})}x^{u}.$ By computing the equation, we find that $c_{\\alpha _{(\\widehat{m},u)}}=0$ if $\\alpha _{\\widehat{m}}-\\varepsilon _{k^{\\prime }}>0$ or $|u|>0$ .",
"Then we can suppose that $\\phi (1,x_{m})=c_{0}.$ Set $\\lambda =\\frac{c_0}{2}$ , then we can get that $\\phi (1,x_m)=\\lambda \\langle 1,x_m\\rangle .$ When $q_{m}\\ge 2$ , we suppose that $\\phi (1,x^{(q_{m}\\varepsilon _m)})=\\sum _{0\\le \\alpha \\le \\tau ,~u\\in \\mathbb {B}}c_{(\\alpha ,u)}x^{(\\alpha )}x^{u}.$ By Lemma REF and the conclusion of the case $q_{m}=1$ , we have the equation $0=&\\langle \\phi (1,x_{m}),\\langle 1,x^{(q_{m}\\varepsilon _{m})}\\rangle \\rangle -\\langle \\langle 1,x_{m}\\rangle ,\\phi (1,x^{(q_{m}\\varepsilon _{m})})\\rangle \\\\=&\\langle \\lambda \\langle 1,x_{m}\\rangle ,\\langle 1,x^{(q_{m}\\varepsilon _{m})}\\rangle \\rangle -\\langle \\langle 1,x_{m}\\rangle ,\\sum _{0\\le \\alpha \\le \\tau ,~u\\in \\mathbb {B}}c_{\\alpha }x^{(\\alpha )}x^{u}\\rangle \\\\=&4\\lambda \\langle 1,x^{((q_{m-1})\\varepsilon _{m})}\\rangle -2\\langle 1,\\sum _{0\\le \\alpha \\le \\tau ,~u\\in \\mathbb {B}}c_{(\\alpha ,u)}x^{(\\alpha )}x^{u}\\rangle \\\\=&8\\lambda x^{((q_{m}-2)\\varepsilon _{m})}-4\\sum _{0\\le \\alpha \\le \\tau ,~u\\in \\mathbb {B}}c_{(\\alpha ,u)}x^{(\\alpha -\\varepsilon _{m})}x^{u}.$ By computing the equation, we find that $c_{(\\alpha ,u)}=0$ if $\\alpha -(q_{m}-1)\\varepsilon _{m}\\ne 0$ or $|u|>0$ .",
"And $c_{(q_{m}-1)}=2\\lambda $ .",
"We suppose that $\\phi (1,x^{(q_{m}\\varepsilon _m)})=\\sum _{0\\le \\alpha _{\\widehat{m}}\\le \\tau _{\\widehat{m}}}c_{\\alpha _{\\widehat{m}}}x^{(\\alpha _{\\widehat{m}})}+2\\lambda x^{((q_{m}-1)\\varepsilon _m)}.$ For any $i \\in \\mathrm {J_{0}}$ , by Lemma REF , we have the equation $\\begin{split}0=&\\langle \\phi (1,x^{(q_{m}\\varepsilon _{m})}),\\langle x_m,x_i\\rangle \\rangle -\\langle \\langle 1,x^{q_{m\\varepsilon _{m}}}\\rangle ,\\phi (x_m,x_i)\\rangle \\\\=&\\langle \\sum _{0\\le \\alpha _{\\widehat{m}}\\le \\tau _{\\widehat{m}}}c_{\\alpha _{\\widehat{m}}}x^{(\\alpha _{\\widehat{m}})}+2\\lambda x^{((q_{m}-1)\\varepsilon _m)},-x_i\\rangle -\\langle 2x^{((q_{m}-1)\\varepsilon _n)},\\lambda _i\\langle x_m,x_i \\rangle \\rangle \\\\=&\\langle x_i,\\sum _{0\\le \\alpha _{\\widehat{m}}\\le \\tau _{\\widehat{m}}}c_{\\alpha _{\\widehat{m}}}x^{(\\alpha _{\\widehat{m}})}+2\\lambda x^{((q_{m}-1)\\varepsilon _m)}-2\\lambda _ix^{((q_{m}-1)\\varepsilon _m)} \\rangle .\\end{split}$ Since $\\mathcal {C}_{K_{-1}}(K)= K_{-2}$ , we have that $\\sum _{0\\le \\alpha _{\\widehat{m}}\\le \\tau _{\\widehat{m}}}c_{\\alpha _{\\widehat{m}}}x^{(\\alpha _{\\widehat{m}})}+2\\lambda x^{((q_{m}-1)\\varepsilon _m)}-2\\lambda _ix^{((q_{m}-1)\\varepsilon _m)}\\in \\mathbb {F}.$ Then we have $c_{\\alpha _{\\widehat{m}}}=0$ for $\\alpha _{\\widehat{m}}>0$ and $\\lambda =\\lambda _i$ for $i \\in \\mathrm {J_{0}}$ .",
"Then we can get that $\\phi (1,x^{(q_{m}\\varepsilon _m)})=c_0+2\\lambda x^{((q_{m}-1)\\varepsilon _m)}.$ Utilizing the definition of the skew-symmetry biderivation, by Lemma REF , we have that $0=&\\langle \\phi (1,x^{q_{m}\\varepsilon _{m}}),\\langle 1,x^{q_{m}\\varepsilon _{m}}\\rangle \\rangle \\\\=&\\langle c_0+2\\lambda x^{((q_{m}-1)\\varepsilon _m)},x^{(q_{m}-1)\\varepsilon _{m}}\\rangle \\\\=&2c_0x^{(q_{m}-2)\\varepsilon _{m}}.$ It is obvious that $c_0=0$ for $p>2$ .",
"So we can get that $\\phi (1,x^{(q_{m}\\varepsilon _m)})=\\lambda \\langle 1,x^{(q_{m}\\varepsilon _m)}\\rangle .$ The proof is complete.",
"Remark 3.13 We claim that $\\lambda _1=\\cdots =\\lambda _m=\\cdots =\\lambda _{m+n}$ .",
"Choose two mutually different elements $i,j\\in \\mathrm {J}$ .",
"Since the characteristic $p>3$ , there are two positive integers $q_i$ and $q_m$ , which are greater than 1 and are neither congruent to 0 modulo $p$ , such that we have $0=&\\langle \\phi (x_{i}x_{i^{\\prime }},x_{i}),\\langle 1,x^{(q_m\\varepsilon _m)}\\rangle \\rangle -\\langle \\langle x_{i}x_{i^{\\prime }},x_{i}\\rangle ,\\phi (1,x^{(q_m\\varepsilon _m)})\\rangle \\\\=&\\langle \\lambda _{i}\\langle x_{i}x_{i^{\\prime }},x_{i}\\rangle ,2x^{((q_m-1)\\varepsilon _m)}\\rangle -\\langle -x_{i},\\lambda \\langle 1,x^{(q_m\\varepsilon _m)}\\rangle \\rangle \\\\=&\\langle -\\lambda _{i}x_{i},2x^{((q_m-1)\\varepsilon _m)}\\rangle -\\langle -x_{i},2\\lambda x^{((q_m-1)\\varepsilon _m)}\\rangle \\\\=&-2\\lambda _{i}\\langle x_{i},x^{((q_m-1)\\varepsilon _m)}\\rangle +2\\lambda \\langle x_{i},x^{((q_m-1)\\varepsilon _m)}\\rangle \\\\=&2(\\lambda -\\lambda _{i})\\langle x_{i},x^{((q_m-1)\\varepsilon _m)}\\rangle \\\\=&2(\\lambda -\\lambda _{i})x^{(\\varepsilon _{i}+(q_m-1)\\varepsilon _m)}.$ By direct calculation, it is easily seen that $\\lambda _i=\\lambda $ for any $i\\in \\mathrm { I }$ .",
"Set $\\lambda :=\\lambda _1=\\cdots =\\lambda _m=\\lambda _{m+1}=\\cdots =\\lambda _{m+n}$ .",
"Then we can conclude that for any $x^{(q_i\\varepsilon _{i})}\\in M$ , $1\\le q_i\\le \\pi _i$ and $x_{l}x_{l^{\\prime }}\\in T$ , there is an element $\\lambda \\in \\mathbb {F}$ such that $\\phi (x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _{i})})=\\lambda \\langle x_{l}x_{l^{\\prime }},x^{(q_i\\varepsilon _i)}\\rangle ,$ where $\\lambda $ depends on neither $x^{(q_i\\varepsilon _i)}$ nor $x_{l}x_{l^{\\prime }}$ .",
"Theorem 3.14 Let $K$ be the contact Lie superalgebra $K(m,n;\\underline{t})$ over the prime field $\\mathbb {F}$ of the characteristic $p>3$ , where $m,~n\\in \\mathbb {N}+1$ and $\\underline{t}=(t_1,t_2,\\ldots ,t_m)$ is an $m$ -tuple of positive integers.",
"Then $\\mathrm {BDer}(K)=\\mathrm {IBDer}(K).$ Suppose that $\\phi $ is a skew-symmetric super-biderivation on $K$ .",
"By Lemmas REF and REF , there is an element $\\lambda \\in \\mathbb {F}$ such that $\\phi (x_{i}x_{i^{\\prime }},x_{i})=\\lambda \\langle x_{i}x_{i^{\\prime }},x_{i}\\rangle $ for all $i\\in \\mathrm {J}$ .",
"For any $x^{(\\alpha )}x^u, x^{(\\beta )}x^v\\in K$ and $x_{l}x_{l^{\\prime }}\\in T_{k}$ , by Lemma REF and Remark REF , we have the equation $\\begin{split}0&=\\langle \\phi (x_{l}x_{l^{\\prime }},x_{l}),\\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle \\rangle -\\langle \\langle x_{l}x_{l^{\\prime }},x_{l} \\rangle ,\\phi (x^{(\\alpha )}x^u,x^{(\\beta )}x^v) \\rangle \\\\&=\\langle \\langle x_{l}x_{l^{\\prime }},x_{l} \\rangle ,\\lambda \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle \\rangle -\\langle \\langle x_{l}x_{l^{\\prime }},x_{l} \\rangle ,\\phi (x^{(\\alpha )}x^u,x^{(\\beta )}x^v) \\rangle \\\\&=\\langle x_{l},\\phi (x^{(\\alpha )}x^u,x^{(\\beta )}x^v)-\\lambda \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle \\rangle .\\end{split}$ Since $\\mathcal {C}_{K_{-1}}(K)= K_{-2}$ , we have that $\\phi (x^{(\\alpha )}x^u,x^{(\\beta )}x^v)=\\lambda \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle +b,$ where $\\lambda $ is denoted in Remark REF and $b\\in \\mathbb {F}$ .",
"By Lemma REF and Remark REF , we have $\\begin{split}0&=\\langle \\phi (x^{(\\alpha )}x^u,x^{(\\beta )}x^v),\\langle 1,x^{(2\\varepsilon _{m})}\\rangle \\rangle -\\langle \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle ,\\phi (1,x^{(2\\varepsilon _{m})}) \\rangle \\\\&=\\langle \\lambda \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle +b,2x_{m}\\rangle -\\langle \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle ,\\lambda \\langle 1,x^{(2\\varepsilon _{m})} \\rangle \\rangle \\\\&=\\langle \\lambda \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle +b,2x_{m}\\rangle -\\langle \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle ,\\lambda 2x_{m}\\rangle \\\\&=\\langle b,2x_{m}\\rangle \\\\&=4b.\\end{split}$ Then $b=0$ .",
"Hence, $\\phi (x^{(\\alpha )}x^u,x^{(\\beta )}x^v)=\\lambda \\langle x^{(\\alpha )}x^u,x^{(\\beta )}x^v \\rangle $ for any $x^{(\\alpha )}x^u,x^{(\\beta )}x^v\\in K $ and $\\phi $ is an inner super-biderivation.",
"Acknowledgements The authors would like to thank the referee for valuable comments and suggestions on this article."
]
] | 1906.04549 |
[
[
"Label-Agnostic Sequence Labeling by Copying Nearest Neighbors"
],
[
"Abstract Retrieve-and-edit based approaches to structured prediction, where structures associated with retrieved neighbors are edited to form new structures, have recently attracted increased interest.",
"However, much recent work merely conditions on retrieved structures (e.g., in a sequence-to-sequence framework), rather than explicitly manipulating them.",
"We show we can perform accurate sequence labeling by explicitly (and only) copying labels from retrieved neighbors.",
"Moreover, because this copying is label-agnostic, we can achieve impressive performance when transferring to new sequence-labeling tasks without retraining.",
"We additionally consider a dynamic programming approach to sequence labeling in the presence of retrieved neighbors, which allows for controlling the number of distinct (copied) segments used to form a prediction, and leads to both more interpretable and accurate predictions."
],
[
"Introduction",
"Retrieve-and-edit style structured prediction, where a model retrieves a set of labeled nearest neighbors from the training data and conditions on them to generate the target structure, is a promising approach that has recently received renewed interest [9], [8], [7], [32].",
"This approach captures the intuition that while generating a highly complex structure from scratch may be difficult, editing a sufficiently similar structure or set of structures may be easier.",
"Recent work in this area primarily uses the nearest neighbors and their labels simply as an additional context for a sequence-to-sequence style model to condition on.",
"While effective, these models may not explicitly capture the discrete operations (like copying) that allow for the neighbors to be edited into the target structure, making interpreting the behavior of the model difficult.",
"Moreover, since many retrieve-and-edit style models condition on dataset-specific labels directly, they may not easily allow for transfer learning and in particular to porting a trained model to a new task with different labels.",
"We address these limitations in the context of sequence labeling by developing a simple label-agnostic model that explicitly models copying token-level labels from retrieved neighbors.",
"Since the model is not a function of the labels themselves but only of a learned notion of similarity between an input and retrieved neighbor inputs, it can be effortlessly ported (zero shot) to a task with different labels, without any retraining.",
"Such a model can also take advantage of recent advances in representation learning, such as BERT [6], in defining this similarity.",
"We evaluate the proposed approach on standard sequence labeling tasks, and show it is competitive with label-dependent approaches when trained on the same data, but substantially outperforms strong baselines when it comes to zero-shot transfer applications, such as when training with coarse labels and testing with fine-grained labels.",
"Finally, we propose a dynamic programming based approach to sequence labeling in the presence of retrieved neighbors, which allows for trading off token-level prediction confidence with trying to minimize the number of distinct segments in the overall prediction that are taken from neighbors.",
"We find that such an approach allows us to both increase the interpretability of our predictions as well as their accuracy."
],
[
"Related Work",
"Nearest neighbor based structured prediction (also referred to as instance- or memory-based learning) has a long history in machine learning and NLP, with early successes dating back at least to the taggers of Daelemans [3], [4] and the syntactic disambiguation system of [2].",
"Similarly motivated approaches remain popular for computer vision tasks, especially when it is impractical to learn a parametric labeling function [28], [27].",
"More recently, there has been renewed interest in explicitly conditioning structured predictions on retrieved neighbors, especially in the context of language generation [9], [8], [7], [32], although much of this work uses neighbors as extra conditioning information within a sequence-to-sequence framework [30], rather than making discrete edits to neighbors in forming new predictions.",
"Retrieval-based approaches to structured prediction appear particularly compelling now with the recent successes in contextualized word embedding [16], [19], [21], [6], which should allow for expressive representations of sentences and phrases, which in turn allow for better retrieval of neighbors for structured prediction.",
"Finally, we note that there is a long history of transfer-learning based approaches to sequence labeling [1], [5], [26], [34], [18], [33], [23], though it is generally not zero-shot.",
"There has, however, been recent work in zero-shot transfer for sequence labeling problems with binary token-labels [22].",
"Figure: A visualization of POS tagging an input sentence xx (bottom) by copying token-labels from M=3M=3 retrieved sentences x '(m) x^{\\prime (m)} for which we know the true corresponding label sequences y '(m) y^{\\prime (m)}; see the text for details."
],
[
"Nearest Neighbor Based Sequence Labeling",
"While nearest-neighbor style approaches are compelling for many structured prediction problems, we will limit ourselves here to sequence-labeling problems, such as part-of-speech (POS) tagging or named-entity recognition (NER), where we are given a $T$ -length sequence $x = x_{1:T}$ (which we will assume to be a sentence), and we must predict a corresponding $T$ -length sequence of labels $\\hat{y} = \\hat{y}_{1:T}$ for $x$ .",
"We will assume that for any given task there are $Z$ distinct labels, and denote $x$ 's true but unknown labeling as $y \\, {=} \\, y_{1:T} \\, {\\in } \\, \\lbrace 1, \\ldots , Z\\rbrace ^T$ .",
"Sequence-labeling is particularly convenient for nearest-neighbor based approaches, since a prediction $\\hat{y}$ can be formed by simply concatenating labels extracted from the label-sequences associated with neighbors.",
"In particular, we will assume we have access to a database $= \\lbrace x^{\\prime (m)}, y^{\\prime (m)}\\rbrace _{m=1}^M$ of $M$ retrieved sentences $x^{\\prime (m)}$ and their corresponding true label-sequences $y^{\\prime (m)}$ .",
"We will predict a labeling $\\hat{y}$ for $x$ by considering each token $x_t$ , selecting a labeled token $x^{\\prime (m)}_k$ from $$ , and then setting $\\hat{y}_t = y^{\\prime (m)}_k$ .More precisely, we will set $\\hat{y}_t$ to be an instance of the label type of which $y^{\\prime (m)}_k$ is a label token; this distinction between label types and tokens can make the exposition unnecessarily obscure, and so we avoid it when possible."
],
[
"A Token-Level Model",
"We consider a very simple token-level model for this label-agnostic copying, where the probability that $x$ 's $t$ 'th label $y_t$ is equal to $y^{\\prime (m)}_k$ — the $k$ 'th label token of sequence $x^{\\prime (m)}$ — simply depends on the similarity between $x_t$ and $x^{\\prime (m)}_k$ , and is independent of the surrounding labels, conditioned on $x$ and $$ .While recent sequence labeling models [14], [13], often model inter-label dependence with a first-order CRF [12], [6] have recently shown that excellent performance can be obtained by modeling labels as conditionally independent given a sufficiently expressive representation of $x$ .",
"In particular, we define p(yt = y'(m)k x, ) (t '(m)k), where the above probability is normalized over all label tokens of all label-sequences in $$ .",
"Above, $_t$ and $^{\\prime (m)}_k$ (both in $^D$ ) represent the contextual word embeddings of the $t$ 'th token in $x$ and the $k$ 'th token in $x^{\\prime (m)}$ , respectively, as obtained by running a deep sequence-model over $x$ and over $x^{\\prime (m)}$ .",
"In all experiments we use BERT [6], a model based on the Transformer architecture [31], to obtain contextual word embeddings.",
"We fine-tune these contextual word embeddings by maximizing a latent-variable style probabilistic objective t=1T m=1M k: y'(m)k = yt p(yt = y'(m)k x, ), where we sum over all individual label tokens in $$ that match $y_t$ .",
"At test time, we predict $\\hat{y}_t$ to be the label type with maximal marginal probability.",
"That is, we set $\\hat{y}_t$ to be $_z \\sum _{m=1}^M \\sum _{k: \\, y^{\\prime (m)}_k = z} p(y_t \\, {=} \\, y^{\\prime (m)}_k x, )$ , where $z$ ranges over the label types (e.g., POS or named entity tags) present in $$ .",
"As noted in the introduction, predicting labels in this way allows for the prediction of any label type present in the database $$ used at test time, and so we can easily predict label types unseen at training time without any additional retraining."
],
[
"Data and Methods",
"Our main experiments seek to determine both whether the label-agnostic copy-based approach introduced above results in competitive sequence-labeling performance on standard metrics, as well as whether this approach gives rise to better zero-shot transfer.",
"Accordingly, our first set of experiments consider several standard sequence-labeling tasks and datasets, namely, POS tagging the Penn Treebank [15] with both the standard Penn Treebank POS tags and Universal POS tags [20], [17], and the CoNLL 2003 NER task [24], [25].",
"We compare with the sequence-labeling performance of BERT [6], which we take to be the current state of the art.",
"We use the standard dataset-splits and evaluations for all tasks, and BIO encoding for all segment-level tagging tasks.",
"We evaluate zero-shot transfer performance by training on one dataset and evaluating on another, without any retraining.",
"In particular, we consider three zero-shot transfer scenarios: training with Universal POS Tags on the Penn Treebank and then predicting the standard, fine-grained POS tags, training on the CoNLL 2003 NER data and predicting on the fine-grained OntoNotes NER data [10] using the setup of [29], and finally training on the CoNLL 2003 chunking data and predicting on the CoNLL 2003 NER data.",
"We again compare with a BERT baseline, where labels from the original task are deterministically mapped to the most frequent label on the new task with which they coincide.For the Chunk $\\rightarrow $ NER task, this results in mapping all tags to `O', so we instead use the more favorable mapping of NPs to PERSON tags.",
"Our nearest-neighbor based models were fine-tuned by retrieving the 50 nearest neighbors of each sentence in a mini-batch of either size 16 or 20, and maximizing the objective (REF ) above.",
"For training, nearest neighbors were determined based on cosine-similarity between the averaged top-level (non-fine-tuned) BERT token embeddings of each sentence.",
"In order to make training more efficient, gradients were calculated only with respect to the input sentence embeddings (i.e., the $_t$ in (REF )) and not the embeddings $^{\\prime (m)}_k$ of the tokens in $$ .",
"At test time, 100 nearest neighbors were retrieved for each sentence to be labeled using the fine-tuned embeddings.",
"The baseline BERT models were fine-tuned using the publicly available huggingface BERT implementation,https://github.com/huggingface/pytorch-pretrained-BERT and the “base” weights made available by the BERT authors [6].",
"We made word-level predictions based on the embedding of the first tokenized word-piece associated with a word (as [6] do), and ADAM [11] was used to fine-tune all models.",
"Hyperparameters were chosen using a random search over learning rate, batch size, and number of epochs.",
"Code for duplicating all models and experiments is available at https://github.com/swiseman/neighbor-tagging."
],
[
"Main Results",
"The results of our experiments on standard sequence labeling tasks are in Table REF .",
"We first note that all results are quite good, and are competitive with the state of the art.",
"The label-agnostic model tends to underperform the standard fine-tuned BERT model only very slightly, though consistently, and is typically within several tenths of a point in performance.",
"Table: A comparison of fine-tuned BERT and our nearest-neighbor (NN) based approach on standard sequence labeling tasks.",
"From top-to-bottom, NER performance on the CoNLL 2003 data, part-of-speech tagging performance on the Penn Treebank, and universal part-of-speech tagging performance on the Penn Treebank; results use the standard metrics and dataset splits.",
"BERT numbers are from fine-tuning the huggingface BERT implementation, and differ slightly from those in .The results of our zero-shot transfer experiments are in Table REF .",
"We see that in all cases the label-agnostic model outperforms standard fine-tuned BERT, often significantly.",
"In particular, we note that when going from universal POS tags to standard POS tags, the fine-tuned label-agnostic model manages to outperform the standard most-frequent-tag-per-word baseline, which itself obtains slightly less than 92% accuracy.",
"The most dramatic increase in performance, of course, occurs on the Chunking to NER task, where the label-agnostic model is successfully able to use chunking-based training information in copying labels, whereas the parametric fine-tuned BERT model can at best attempt to map NP-chunks to PERSON labels (the most frequent named entity in the dataset).",
"In order to check that the increase in performance is not just due to BERT's pretraining, Table REF also shows the results of the label-agnostic model without fine-tuning (as indicated by “no FT” in the table).",
"In all cases, this leads to a decrease in performance.",
"Table: From top-to-bottom, zero-shot performance of models trained on the CoNLL 2003 data and applied to the fine-grained OntoNotes NER task, on PTB with universal part-of-speech tags and applied to PTB with standard part-of-speech tags, and on the CoNLL 2003 chunking data and applied to the CoNLL 2003 NER task.",
"Above, “no FT” indicates the model was not fine tuned even on the original task."
],
[
"Encouraging Contiguous Copies",
"Although we model token-level label copying, at test time each $\\hat{y}_t$ is predicted by selecting the label type with highest marginal probability, without any attempt to ensure that the resulting sequence $\\hat{y}$ resembles one or a few of the labeled neighbors $y^{\\prime (m)}$ .",
"In this section we therefore consider a decoding approach that allows for controlling the trade-off between prediction confidence and minimizing the number of distinct segments in $\\hat{y}$ that represent direct (segment-level) copies from some neighbor, in the hope that having fewer distinct copied segments in our predictions might make them more interpretable or accurate.",
"We emphasize that the following decoding approach is in fact applicable even to standard sequence labeling models (i.e., non-nearest-neighbor based models), as long as neighbors can be retrieved at test time.",
"Figure: A CoNLL 2003 NER development example, which can be labeled with only two distinct segments.",
"We show the segments used by a model trained on the NER data (top), and by a model trained on the CoNLL chunking data (bottom).To begin with a simple case, suppose we already know the true labels $y$ for a sequence $x$ , and are simply interested in being able to reconstruct $y$ by concatenating as few segments $y^{\\prime }_{i:j}$ that appear in some $y^{\\prime (m)} \\, {\\in } \\, $ as possible.",
"More precisely, define the set $_{}$ to contain all the unique label type sequences appearing as a subsequence of some sequence $y^{\\prime (m)} \\, {\\in } \\, $ .",
"Then, if we're willing to tolerate some errors in reconstructing $y$ , we can use a dynamic program to minimize the number of mislabelings in our now “prediction” $\\hat{y}$ , plus the number of distinct segments used in forming $\\hat{y}$ multiplied by a constant $c$ , as follows: J(t) = 1 k t z : |z| = k J(t-k) + c + j=1k1[yt-k+j zj], where $J(0) \\, {=} \\, 0$ is the base case and $|z|$ is the length of sequence $z$ .",
"Note that greedily selecting sequences that minimize mislabelings may result in using more segments, and thus a higher $J$ .",
"In the case where we do not already know $y$ , but wish to predict it, we might consider a modification of the above, which tries to minimize $c$ times the number of distinct segments used in forming $\\hat{y}$ plus the expected number of mislabelings: J(t) = 1 k t z : |z| = k [J(t-k) + c + j=1k1 - p(yt-k+j = zjx, )], where we have used the linearity of expectation.",
"Note that to use such a dynamic program to predict $\\hat{y}$ we only need an estimate of $p(y_{t{-}k{+}j} \\, {=} \\, z_jx, )$ , which we can obtain as in Section (or from a more conventional model).",
"In Figure REF we plot both the F$_1$ score and the average number of distinct segments used in predicting each $\\hat{y}$ against the $c$ parameter from the dynamic program above, for the CoNLL 2003 NER development data in both the standard and zero-shot settings.",
"First we note that we are able to obtain excellent performance with only about 1.5 distinct segments per prediction, on average; see Figure REF for examples.",
"Interestingly, we also find that using a higher $c$ (leading to fewer distinct segments) can in fact improve performance.",
"Indeed, taking the best values of $c$ from Figure REF (0.4 in the standard setting and 0.5 in the zero-shot setting), we are able to improve our performance on the test set from 89.94 to 90.20 in the standard setting and from 71.74 to 73.61 in the zero shot setting, respectively; see Tables REF and REF .",
"Figure: F 1 _1 performance (top, orange suplots) on the CoNLL 2003 NER development data and the average number of distinct segments per predicted labeling (bottom, blue subplots) as the cc parameter is varied, when the model is trained either (top) on the standard training set or (bottom) on the CoNLL chunking data (i.e., zero-shot performance)."
],
[
"Conclusion",
"We have proposed a simple label-agnostic sequence-labeling model, which performs nearly as well as a standard sequence labeler, but improves on zero-shot transfer tasks.",
"We have also proposed an approach to sequence label prediction in the presence of retrieved neighbors, which allows for discouraging the use of many distinct segments in a labeling.",
"Future work will consider problems where more challenging forms of neighbor manipulation are necessary for prediction."
]
] | 1906.04225 |
[
[
"The metallicity sensitivity of a surface brightness temperature scale"
],
[
"Abstract To obtain the accuracy now sought in the extragalactic distance scale through standard candles and rulers, calibration of stellar photometry must be improved.",
"The sensitivity of the V-K color surface brightness relation is examined here by means of model atmosphere fluxes.",
"It has previously been neglected, but is shown here to be a significant term in the error budget of a recent high precision distance of the Large Magellanic Cloud, an anchor in galaxy distances based on Cepheids."
],
[
"Introduction",
"The goal of 1% accuracy in the extragalactic distance scale places new demands on stellar astrophysics.",
"ESA's $Gaia$ mission has opened up new possibilities in the classes of stars that can be standard candles (e.g.",
"Mould, Clementini, Da Costa 2019).",
"Pietrzynski et al (2019) employ the surface brightness colour relation to attain the required accuracy in the distance of the Large Magellanic Cloud (LMC).",
"Both works require accurate calibration of photometry to temperatures and luminosities.",
"In this paper we explore the metallicity dependence of the surface brightness color relation using the surface fluxes of Kurucz model atmospheres."
],
[
"Surface brightness and effective temperature",
"Eclipsing binaries allow accurate measurements of stellar radii (e.g.",
"Elgueta et al 2018).",
"Physically, we can understand this as combining the definition of effective temperature T$_e$ $L = 4\\pi \\sigma R^2 T_e^4$ where L and R are the stellar luminosity and radius respectively, with a geometric measurement of R. Dividing by stellar surface area, we obtain a flux $F = \\sigma \\theta ^2 T_e^4$ where $\\theta $ is the stellar angular radius and $\\sigma $ is the Stefan-Boltzmann constant.",
"If the flux is measured photometrically, the effective temperature can be estimated from V–K (e.g.",
"di Benedetto et al 1998, 2005).",
"One combines $\\theta $ and R to obtain the distance.",
"In these terms we can express the error propagation: $\\delta \\theta /\\theta ~=~ 2 \\delta T_e/T_e$ If V–K = $f$ (T$_e$ , g, Z), then $\\delta (V-K) = \\frac{\\partial f}{\\partial T_e} \\delta T_e + \\frac{\\partial f}{\\partial g} \\delta g + \\frac{\\partial f}{\\partial Z} \\delta Z$ It is possible to ignore the second term for the time being, supposing the ratio of stellar mass to R$^2$ to be perfectly determined by a spectroscopic eclipsing binary solution and investigate the third term using the predictions of Kurucz model atmospheres.",
"Figure: V–K colour vs temperature.",
"Red symbols are metal rich; blue: metal poor.",
"Colors are calculated as filter integrated flux ratios and normalised to the Sun."
],
[
"Synthetic Color Temperature relation",
"The relation between V–K and T$_e$ can be modeled using Buser & Kurucz (1992) fluxes and Bessell (2005), together with Bessell & Brett (1988), filter responses.",
"The fluxes of these models are not well sampled in the near infrared and were therefore interpolated in the K bandpass.",
"Results are independent of whether linear or parabolic interpolation was used.",
"The dependence of color on metallicity at fixed gravity is illustrated in Figure 1.",
"Similar results are obtained using model atmosphere fluxes with nearly two orders of magnitude more spectral resolution (Allard 2016).",
"These models include TiO in the V bandpass and CO in the K bandpass, and so are superior in their predictions for K giants and M stars.",
"Figure: Modeled V–K color dependence on metallicity for three stellar temperatures and fixed gravity.",
"The metallicity [M/H] is the logarithmic Z value relative to the Sun.",
"Red symbols are T e _e = 4500K.",
"Green: 5000K; blue: 5500K.Spherical models are also available to complement the standard plane parallel atmosphere models (e.g.",
"SATLAS, Lester & Neilson 2008).",
"At one solar mass MARCS modelsmarcs.astro.uu.se by Gustafsson et al (2008) are redder in V–K at 5000K and log g = 3 in the spherical case than the plane parallel case, but $\\partial f/\\partial \\log Z$ is almost identical."
],
[
"Red clump stars",
"Pietrzynski et al (2019) employ eclipsing binaries from the red clump in the LMC.",
"Onozato et al (2019) find a mean value K $\\approx $ 16.82 mag for the red clump in star clusters, and if the LMC is 50 kpc distant, a bolometric correction BC$_K$ = 1.92 mag (Johnson 1966) gives log L/L$_\\odot ~\\approx $ 2.0.",
"For solar mass stars the gravity is log g = 2.2.",
"Figure 2 shows that d(V-K)/dlogZ $\\approx $ 0.2/3 mag/dex.",
"If d(V-K)/dlogT$_e$ = 2/0.176, then dlogT$_e$ / dlogZ = 0.1 $\\times $ 0.176 This means that if an error $\\delta $ logZ = 0.3 is made, then $\\delta $ logT$_e$ =0.434 $\\delta $ T/T = 1.76 $\\times $ 10$^{-3}$ and $\\delta $ T/T = 0.004, corresponding to a 0.8% error in distance."
],
[
"Giant Branch Stars",
"If the stars lie on a giant branch, appropriate gravities for each temperature should be employed.",
"In solar logarithmic units [g] = [M] – [L] + 4[T$_e$ ].",
"Modeling the giant branch as a one solar mass line commencing at the Sun and linear in log L, log T to L$_{tip}$ , T$_{tip}$ , we can write [L] = [L$_{tip}$ ]/[T$_{tip}$ ] [T$_e$ ] If L$_{tip}$ = 2000 L$_\\odot $ and T$_{tip}$ = 3500K, then [g] = 18.9 [T$_e$ ] With these assumptions we derive Figure 3.",
"Figure: Modeled V–K color dependence on metallicity for three stellar temperatures on the schematic red giant branch outlined in §§5.",
"As in the previous figure, red symbols are T e _e = 4500K.",
"Green: 5000K; blue: 5500K."
],
[
"Main sequence stars",
"The surface brightness V–K relation is also employed on the main sequence.",
"We show the modeled gravity dependence of V–K, the second term in the error propagation equation, in Figure 4.",
"Figure: Modeled V–K color dependence on gravity for three stellar temperatures at solar metallicity (above) and approximately one third solar metallicity (below).As in the previous figures, red symbols are T e _e = 4500K.",
"Green: 5000K; blue: 5500K."
],
[
"Conclusion",
"For precision of 1% in distance the metallicities of eclipsing binaries should not be neglected, as the surface brightness color relation is affected by V–K metallicity dependence at this level.",
"In the range they explore, Onozato et al (2019) find that population effects on JHK colors of LMC red clump stars are small.",
"It may be possible to control the metallicity dependence noted here by small correction to the V–K colors in the V bandpass where blanketing by absorption lines is greater than in the infrared.",
"I would like to thank the referee for useful comments."
],
[
"References",
"Allard, F. 2016, SF2A, Proc.",
"Ann.",
"French Soc.",
"Astr & Ap., Reylé, Richard, Cambrésy, Deleuil, Pécontal, Tresse & Vaughlin, eds., p.237 Bessell, M. 2005, ARAA, 43, 293 Bessell, M. & Brett, J.",
"1988, PASP, 100, 1134 Buser, R, & Kurucz, R. 1992, A&A, 264, 557 Di Benedetto, G. 1998, A&A, 339, 858 Di Benedetto, G. 2005, MNRAS, 357, 174 Elgueta, S., Graczyk, D. & Gieren, W. 2018, ASP Conf Series, 514, 119 Gustafsson, B. et al 2008, A&A, 486, 951 Johnson, H. 1966, ARAA, 4, 201 Lester, J.",
"& Neilson, H. 2008, A&A, 491, 633 Mould, J., Clementini, G. & Da Costa, G. 2019, PASA, 36, 1 Onozato, H. et al 2019, MNRAS, 486, 5600 Pietrzynski, G. et al 2019, Nature 567, 200"
]
] | 1906.04320 |
[
[
"Semiparametric estimation for incoherent optical imaging"
],
[
"Abstract The theory of semiparametric estimation offers an elegant way of computing the Cram\\'er-Rao bound for a parameter of interest in the midst of infinitely many nuisance parameters.",
"Here I apply the theory to the problem of moment estimation for incoherent imaging under the effects of diffraction and photon shot noise.",
"Using a Hilbert-space formalism designed for Poisson processes, I derive exact semiparametric Cram\\'er-Rao bounds and efficient estimators for both direct imaging and a quantum-inspired measurement method called spatial-mode demultiplexing (SPADE).",
"The results establish the superiority of SPADE even when little prior information about the object is available."
],
[
"Introduction",
"Two fundamental problems confront incoherent optical imaging: the diffraction limit [1], [2] and the photon shot noise [3], [4].",
"To quantify their effects on the resolution rigorously, the Cramér-Rao bound (CRB) on the error of parameter estimation [5] has been widely used, especially in astronomy and fluorescence microscopy [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].",
"Most previous studies, however, assume that the object has a simple specific shape, such as a point source or two, and only one or few parameters of the object are unknown.",
"Such parametric models may not be justifiable when there is little prior information about the object.",
"Without a parametric model, the CRB seems intractable—infinitely many parameters are needed to specify the object distribution, leading to a Fisher information matrix with infinitely many entries, and then the infinite-dimensional matrix has to be inverted to give the CRB.",
"While there also exist many studies on superresolution that can deal with more general objects [17], [18], [19], [20], they either ignore noise or use noise models that are too simplistic to capture the signal-dependent nature of photon shot noise.",
"To compute the CRB and to evaluate the efficiency of estimators for general objects, here I propose a theory of semiparametric estimation for incoherent optical imaging.",
"Semiparametric estimation refers to the estimation of a parameter of interest in the presence of infinitely many other unknown “nuisance” parameters [21], [22].",
"The method has found many applications in econometrics, biostatistics, and astrostatistics [21].",
"A typical example is the estimation of the mean of a random variable when its probability density is assumed to have finite variance but otherwise arbitrary.",
"Thanks to a beautiful Hilbert-space formalism [21], [22], the semiparametric theory is able to compute the CRB for such problems despite the infinite dimensionality and also evaluate the existence and efficiency of semiparametric estimators.",
"Such problems are exactly the type that bedevil the study of imaging thus far, and here I show how the semiparametric theory can be used to yield similarly elegant results for optical imaging.",
"The optics problem of interest here is the far-field imaging of an object emitting spatially incoherent light [2], [4], with the most important applications being optical astronomy [6], [7], [8], [9] and fluoresence microscopy [10], [11], [12], [13], [14].",
"With a finite numerical aperture, the imaging system introduces a spatial bandwidth limit to the waves, otherwise known as the diffraction limit [1], [2].",
"The standard measurement, called direct imaging, records the intensity of the light on the image plane.",
"Recently, quantum information theory inspired the invention of an alternative measurement called spatial-mode demultiplexing (SPADE) [23], which has been shown theoretically [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51] and experimentally [52], [53], [54], [55], [56], [57], [58] to be superior to direct imaging in resolving two sub-Rayleigh sources and estimating the size and moments of a subdiffraction object.",
"Most of the aforementioned studies, however, assume parametric models for the object.",
"Exceptions include Refs.",
"[55], [24], [25], [26], [27], [50], which consider the estimation of object moments, but the results there are not conclusive—only the CRB for direct imaging was computed exactly [25], while the CRB for SPADE was evaluated only approximately [24], [25], [27].",
"Another problem is the existence and efficiency of unbiased moment estimators; again only approximate results have been obtained so far [24], [25].",
"Building on the established semiparametric theory [21], [22], here I compute the exact semiparametric CRBs and also propose unbiased and efficient moment estimators for both direct imaging and SPADE.",
"These results enable a fair and rigorous comparison of the two measurement methods, which proves the fundamental superiority of SPADE for moment estimation.",
"This paper is organized as follows.",
"Section introduces the Fisher information and the CRB for Poisson processes.",
"Section presents the semiparametric CRB in terms of a Hilbert-space formalism designed for such processes.",
"Section introduces the models of direct imaging and SPADE.",
"Section computes the CRB for moment estimation with direct imaging and proposes an efficient estimator.",
"Section shows how the CRB should be modified for a normalized object distribution.",
"Section computes the CRB for SPADE and also proposes an efficient estimator.",
"Section uses the CRBs to compare the performances of direct imaging and SPADE, demonstrating the superiority of SPADE for subdiffraction objects.",
"Section concludes the paper and points out open issues, while the Appendices detail the technical issues that arise in the main text."
],
[
"Cramér-Rao bound for Poisson processes",
"For optical astronomy [6], [4], [8], [9], fluorescence microscopy [10], [11], [12], [13], [14], and even electron microscopy [15], [16], Poisson noise can be safely assumed.",
"Suppose that each detector in a photodetector array is labeled by $x \\in \\mathcal {X}$ , where $\\mathcal {X}$ denotes the detector space.",
"Assume that the observed process, such as the image recorded by a camera, is a Poisson random measure $n$ on $\\mathcal {X}$ and its $\\sigma $ -algebra $\\Sigma $ , with a mean given by the intensity measure $\\bar{n}$ on the same $(\\mathcal {X},\\Sigma )$ [59].",
"$n(\\mathcal {A})$ for any $\\mathcal {A} \\in \\Sigma $ is then a Poisson variable with mean $\\bar{n}(\\mathcal {A})$ .",
"For example, if $\\mathcal {X} \\subseteq \\mathbb {R}^2$ is a two-dimensional surface, then $\\Sigma $ is the set of all subareas that can be defined on the surface, and $n(\\mathcal {A}) = \\int _{x \\in \\mathcal {A}} dn(x)$ is the detected photon number over the area $\\mathcal {A}$ .",
"For any vectoral function $h:\\mathcal {X} \\rightarrow \\mathbb {R}^{q}$ on the detector space, $\\check{h}(n) &= \\int h(x) dn(x),$ a linear functional of $n$ , is a random variable with statistics $\\mathbb {E}(\\check{h}) &= \\int h(x) d\\bar{n}(x) = \\nu (h),\\\\\\mathbb {V}(\\check{h}) &= \\mathbb {E}(\\check{h}\\check{h}^\\top )- \\mathbb {E}(\\check{h})\\mathbb {E}(\\check{h}^\\top )= \\nu (hh^\\top ),$ where $\\mathbb {E}$ denotes the statistical expectation, $\\mathbb {V}$ denotes the covariance, $\\nu $ denotes the average with respect to the intensity measure $\\bar{n}$ , $\\top $ denotes the matrix transpose, and all vectors in this paper are column vectors.",
"Suppose that $\\bar{n}$ depends on an unknown vectoral parameter $\\theta \\in \\Theta \\subset \\mathbb {R}^{p}$ with $p$ entries and has a density $f(x|\\theta )$ with respect to a dominating measure $\\mu $ such that $f(x|\\theta ) = d\\bar{n}(x|\\theta )/d\\mu (x)$ .",
"The log-likelihood derivatives are given by [60] $\\check{S}_j(n|\\theta ) &=\\int \\frac{\\partial }{\\partial \\theta _j}\\ln f(x|\\theta ) dn(x)-\\frac{\\partial }{\\partial \\theta _j}\\int d\\bar{n}(x|\\theta ).$ As $\\check{S}$ is a linear functional of $n$ , its covariance, called the Fisher information matrix, can be simplified via Eq.",
"() and is given by [60] $J &= \\mathbb {V}(\\check{S}) =\\int S(x) [S(x)]^\\top d\\bar{n}(x) =\\nu (SS^\\top ),$ where $S$ is a vector of detector-space functions given by $S_j(x|\\theta ) &= \\frac{\\partial }{\\partial \\theta _j} \\ln f(x|\\theta ).$ Here $\\mathbb {V}$ , $\\check{S}$ , $\\bar{n}$ , $S$ , and $\\nu $ are all evaluated at the same $\\theta $ , and I assume hereafter that all functions of $\\theta $ are evaluated implicitly at the same $\\theta $ .",
"Each $S_j$ is hereafter called a score function, borrowing the same terminology for $\\check{S}$ in statistics [21], [22].",
"An important distinction is that, whereas $\\mathbb {E}(\\check{S}_j) = 0$ , $\\nu (S_j)$ does have to be zero, since $\\bar{n}$ does not have to be normalized.",
"Let $\\beta (\\theta )$ be a scalar parameter of interest.",
"If $\\beta (\\theta ) = \\theta _k$ for example, then all the other parameters in $\\theta $ are called nuisance parameters.",
"For any unbiased estimator $\\check{\\beta }(n)$ , the CRB on its variance is [5] $\\mathbb {V}(\\check{\\beta }) &\\ge u^\\top J^{-1} u = {\\rm CRB},&u_j &= \\frac{\\partial \\beta }{\\partial \\theta _j}.$ $J^{-1}$ seems intractable if $\\theta $ is infinite-dimensional.",
"The next section introduces a cleverer method."
],
[
"Semiparametric Cramér-Rao bound",
"The key to the semiparametric theory is to treat random variables as elements in a Hilbert space [21], [22].",
"Here I introduce another Hilbert space for detector-space functions on top of the statistical one for the purpose of computing the CRB for Poisson processes.",
"Define an inner product between two scalar functions $h_1,h_2:\\mathcal {X} \\rightarrow \\mathbb {R}$ as $\\left\\langle h_1,h_2\\right\\rangle &= \\nu (h_1h_2) =\\int h_1(x)h_2(x)d\\bar{n}(x),$ and the norm as $||h|| &= \\sqrt{\\left\\langle h,h\\right\\rangle } = \\sqrt{\\nu (h^2)}.$ With the inner product, a Hilbert space $\\mathcal {H}$ can be defined as the set of all square-summable functions, viz., $\\mathcal {H} &= \\left\\lbrace h(x): \\nu (h^2) < \\infty \\right\\rbrace .$ Denote the set of score functions $\\lbrace S_j\\rbrace $ as $S$ in a slight abuse of notation.",
"If the Fisher information $J_{jj} = \\nu (S_j^2) < \\infty $ for all $j$ , $S \\subset \\mathcal {H}$ .",
"Define the tangent space $\\mathcal {T} \\subseteq \\mathcal {H}$ of a parametric model as the linear span of $S$ , or $\\mathcal {T} &= \\left\\lbrace w^\\top S: w \\in \\mathbb {R}^{p}\\right\\rbrace =\\operatorname{span}(S).$ Define also an “influence” function as any $\\tilde{\\beta }\\in \\mathcal {H}$ that satisfies $\\nu (\\tilde{\\beta }S) &= u,$ borrowing the name of a similar concept in statistics [21], [22].",
"The Cauchy-Schwartz inequality $\\nu (\\tilde{\\beta }^2) [w^\\top \\nu (SS^\\top ) w] \\ge (u^\\top w)^2$ with $w = [\\nu (SS^\\top )]^{-1} u$ then yields $\\nu (\\tilde{\\beta }^2) &\\ge u^\\top J^{-1} u,$ the right-hand side of which coincides with the CRB given by Eq.",
"(REF ).",
"Define the efficient influence as the influence function that saturates Eq.",
"(REF ), viz., $\\tilde{\\beta }_{\\rm eff} &= u^\\top J^{-1} S = \\nu (\\tilde{\\beta }S^\\top )\\left[\\nu (SS^\\top )\\right]^{-1} S,\\\\{\\rm CRB} &= \\nu (\\tilde{\\beta }_{\\rm eff}^2).$ Equation (REF ) can be interpreted as the orthogonal projection of any influence function $\\tilde{\\beta }\\in \\mathcal {H}$ that satisfies Eq.",
"(REF ) into $\\mathcal {T}$ , viz., $\\tilde{\\beta }_{\\rm eff} &= \\Pi (\\tilde{\\beta }|\\mathcal {T})= \\operatornamewithlimits{arg\\,min}_{h \\in \\mathcal {T}}||\\tilde{\\beta }- h||.$ Figure REF illustrates this concept.",
"Figure: The efficient influenceβ ˜ eff \\tilde{\\beta }_{\\rm eff} is the orthogonal projection of anyinfluence function β ˜∈ℋ\\tilde{\\beta }\\in \\mathcal {H} that satisfiesEq.",
"() into the tangent space𝒯=span(S)\\mathcal {T} = \\operatorname{span}(S).",
"The norm of β ˜ eff \\tilde{\\beta }_{\\rm eff}gives the CRB.Consider now the semiparametric scenario.",
"For the purpose of this paper, it suffices to assume that the dimension of $\\theta $ is infinite but countable ($p = \\infty $ ).",
"The score functions are still defined in the same way, but now there are infinitely many of them.",
"The tangent space should be modified to be the closed linear span $\\mathcal {T} &= \\operatorname{\\overline{span}}(S),$ so that projection into it is well defined [61], and the semiparametric CRB is still given by Eqs.",
"(REF ), (), and (REF ); see Appendix for a proof.",
"This Hilbert-space approach is tractable when finding a candidate influence function is straightforward and the tangent space is so large that the candidate is already in it or at least very close to it.",
"If the dimension of $\\theta $ is uncountable, the tangent space and the CRB can still be defined via the concept of parametric submodels [21], [22], although it is not needed here.",
"If $\\beta $ can be expressed as a functional $\\beta (f)$ , a useful way of finding an influence function is to consider a functional derivative of $\\beta (f)$ with respect to $h(x)$ defined as $\\dot{\\beta }(f,h)&= \\lim _{\\epsilon \\rightarrow 0}\\frac{\\beta ((1+ \\epsilon h)f)-\\beta (f)}{\\epsilon }\\\\&= \\int \\tilde{\\beta }(x) h(x) f(x) d\\mu (x) = \\nu (\\tilde{\\beta }h),$ which leads to $\\frac{\\partial \\beta }{\\partial \\theta _j}&= \\lim _{\\epsilon \\rightarrow 0} \\frac{\\beta (f + \\epsilon \\partial f/\\partial \\theta _j)-\\beta (f)}{\\epsilon }\\\\&= \\dot{\\beta }(f,S_j)= \\nu (\\tilde{\\beta }S_j)= u_j,$ and the $\\tilde{\\beta }(x)$ function obtained from the functional derivative is an influence function that satisfies Eq.",
"(REF ).",
"The simplest example is the linear functional $\\beta (f) &= \\int \\tilde{\\beta }(x)f(x) d\\mu (x) = \\nu (\\tilde{\\beta }),$ and $\\tilde{\\beta }(x)$ is an influence function.",
"If the tangent space is so large that $\\mathcal {T} = \\mathcal {H}$ , then a square-summable influence function is already in $\\mathcal {H} = \\mathcal {T}$ and therefore efficient.",
"There are often some constraints that make $\\mathcal {T}$ smaller, however, and the CRB is reduced as a result.",
"For example, if the constraint can be expressed as $\\gamma (f) &= 0,$ and its functional derivative is $\\dot{\\gamma }(f,h) &= \\nu (h\\tilde{\\gamma }),$ then $\\frac{\\partial \\gamma (f)}{\\partial \\theta _j} &=\\dot{\\gamma }(f,S_j) = \\nu (\\tilde{\\gamma }S_j) = \\left\\langle \\tilde{\\gamma },S_j\\right\\rangle = 0,$ and it follows that $\\tilde{\\gamma }$ should be placed in the set that spans $\\mathcal {T}^\\perp $ , the orthocomplement of $\\mathcal {T}$ in $\\mathcal {H}$ .",
"In terms of $\\mathcal {T}^\\perp $ , the efficient influence can be evaluated as $\\tilde{\\beta }_{\\rm eff} &= \\tilde{\\beta }-\\Pi (\\tilde{\\beta }|\\mathcal {T}^\\perp ).$ If $\\mathcal {T}^\\perp = \\operatorname{span}(R)$ , then $\\Pi (\\tilde{\\beta }|\\mathcal {T}^\\perp ) &= \\nu (\\tilde{\\beta }R^\\top )\\left[\\nu (RR^\\top )\\right]^{-1} R,$ which is still tractable if $R$ has a low dimension."
],
[
"Incoherent optical imaging",
"Consider a distribution of spatially incoherent sources described by the measure $F$ on the object plane with coordinate $y$ , a far-field paraxial imaging system with point-spread function $\\psi (z-y)$ for the field [2], further processing of the field on the image plane with coordinate $z$ via passive linear optics with Green's function $\\kappa (x,z)$ , and Poisson noise at the output detectors labeled by $x \\in \\mathcal {X}$ , as depicted by Fig.",
"REF .",
"For simplicity, assume one-dimensional imaging such that $y, z \\in \\mathbb {R}$ ; generalization for two-dimensional imaging is possible [24], [25] but not very interesting.",
"The intensity can be described by the mixture model [23], [30], [24], [25], [4] $f(x) &=\\int \\left|\\int \\kappa (x,z)\\psi (z-y)dz\\right|^2dF(y),$ where the image-plane coordinate $z$ is normalized with respect to the magnification factor, both $y$ and $z$ are normalized with respect to the width of the point-spread function such that they are dimensionless, and $\\psi $ is normalized as $\\int |\\psi (x)|^2dx = 1$ .",
"This semiclassical Poisson model can be derived from standard quantum optics [62], [23], [51].",
"Figure: A far-field incoherent imaging system.",
"Seethe main text for definitions.For direct imaging with infinitesimal pixels, $\\kappa (x,z) = \\sqrt{\\tau }\\delta (x-z)$ , where $\\tau $ is a positive conversion factor, $x \\in \\mathcal {X} = \\mathbb {R}$ denotes the position of each pixel, $d\\mu (x) = dx$ , and the image intensity obeys the convolution model $f(x) &= \\int H(x-y)dF(y),&H(x) &= \\tau |\\psi (x)|^2,$ which will be studied in Sec. .",
"The most remarkable physics of the problem lies in the possibility of improving the measurement via optics with a different Green's function $\\kappa $ .",
"Quantum information theory has shown that substantial improvement is possible for subdiffraction objects, and SPADE has been found to be quantum-optimal in many special cases [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51].",
"In one version of SPADE, $\\kappa ^*(q,z)$ is the $q$ th mode function in the point-spread-function-adapted (PAD) basis [36], [25], such that the output intensity is given by $f(q) &= \\int H(q|y) dF(y),\\quad q \\in \\mathcal {X} = \\mathbb {N}_0,\\\\H(q|y) &= \\left|\\int \\kappa (q,z)\\psi (z-y)dz\\right|^2,$ where $\\mu $ is simply the counting measure and $\\kappa $ and $H$ obey special properties, as further discussed in Sec. .",
"For a fair comparison, the quantum efficiencies of direct imaging and SPADE are assumed to be the same, meaning that [25] $\\sum _{q=0}^\\infty H(q|y) &= \\tau ,$ where $\\tau $ is the same factor as that for direct imaging.",
"Then $N &= \\mathbb {E}[n(\\mathcal {X})] = \\bar{n}(\\mathcal {X}) = \\nu (1) = \\tau \\int dF(y),$ the expected photon number received in total, is also the same."
],
[
"Moment estimation\nwith direct imaging",
"Consider the direct-imaging model given by Eq.",
"(REF ).",
"Assume that $H$ can be expanded in a Taylor series as $H(x-y) &= \\sum _{j=0}^\\infty \\frac{(-1)^j}{j!",
"}\\frac{\\partial ^j H(x)}{\\partial x^j} y^j,$ which leads to $f(x) &= \\sum _{j=0}^\\infty \\frac{(-1)^j}{j!",
"}\\frac{\\partial ^j H(x)}{\\partial x^j}\\theta _j,$ where the unknown parameters are the object moments defined by $\\theta _j &= \\int y^j dF(y),\\quad j \\in \\mathbb {N}_0.$ For the CRB to hold, the parameter space should correspond to the condition that $F$ contains an infinite number of point sources with different positions, as discussed in Appendix .",
"Appendix shows a way of reconstructing $F$ from $\\theta $ via an orthogonal-series expansion, following Ref. [51].",
"The score function for each $\\theta _j$ is $S_j(x) &= \\frac{(-1)^j}{j!f(x)}\\frac{\\partial ^j H(x)}{\\partial x^j}.$ It turns out that the tangent space $\\mathcal {T}$ for this problem is equal to the whole Hilbert space $\\mathcal {H}$ under certain technical conditions, as shown in Appendix .",
"Let the parameter of interest be $\\beta &= u^\\top \\theta = \\sum _{j=0}^\\infty u_j\\theta _j,$ where $u$ is independent of $\\theta $ .",
"To find a candidate influence function, a trick [63] is to consider the image moments $\\phi $ given by $\\phi &= \\int \\tilde{\\phi }(x) d\\bar{n}(x) = \\nu (\\tilde{\\phi }),$ where $\\tilde{\\phi }_j(x) &= x^j,\\quad j \\in \\mathbb {N}_0$ are the monomials.",
"Assuming that all the moments of $F$ and $H$ are finite such that all the moments of $f$ are also finite, $\\phi $ can be related to the object moments via $\\phi _j &= \\iint x^j H(x-y)dF(y) dx\\\\&= \\iint (z+y)^j H(z) dF(y) dz\\\\&= \\iint \\sum _{k=0}^j \\begin{pmatrix}j\\\\ k\\end{pmatrix} z^{j-k}y^k H(z) dF(y) dz\\\\&= \\sum _{k=0}^\\infty C_{jk}\\theta _k,$ where $C_{jk} &= 1_{j\\ge k}\\begin{pmatrix}j\\\\ k\\end{pmatrix}\\int H(x) x^{j-k} dx$ and $1_{\\rm proposition} &={\\left\\lbrace \\begin{array}{ll}1 & \\textrm {if proposition is true},\\\\0 & \\textrm {otherwise}.\\end{array}\\right.",
"}$ $C$ is a lower-triangular matrix, and with $C_{jj} =\\int H(x)dx =\\tau > 0$ , $C^{-1}$ is well defined and also lower-triangular even if the dimension of $C$ is infinite, as shown in Appendix .",
"The object moments can then be related to the image moments by $\\theta &= C^{-1}\\phi ,$ and $\\beta $ can be expressed as $\\beta = u^\\top \\theta = u^\\top C^{-1}\\phi = \\nu (u^\\top C^{-1}\\tilde{\\phi }).$ According to Eq.",
"(REF ), an influence function is $\\tilde{\\beta }(x) &=u^\\top C^{-1}\\tilde{\\phi }(x) = u^\\top \\tilde{\\theta }(x),\\\\\\tilde{\\theta }(x) &= C^{-1} \\tilde{\\phi }(x).$ Since $\\mathcal {T} = \\mathcal {H}$ as shown in Appendix , the $\\tilde{\\beta }$ given by Eq.",
"(REF ) belongs to $\\mathcal {H} = \\mathcal {T}$ and is efficient according to Eq.",
"(REF ) as long as it is square-summable.",
"For example, if $u$ contains a finite number of nonzero entries, $\\tilde{\\beta }$ is a polynomial of $x$ and must be square-summable, since all the moments of $f$ are assumed to be finite.",
"The CRB is hence ${\\rm CRB}^{(\\rm direct)}&= \\nu (\\tilde{\\beta }^2) = u^\\top \\nu (\\tilde{\\theta }\\tilde{\\theta }^\\top ) u\\nonumber \\\\&= u^\\top C^{-1}\\nu (\\tilde{\\phi }\\tilde{\\phi }^\\top ) C^{-\\top }u,$ where $C^{-\\top } = (C^{-1})^\\top $ .",
"This result coincides with that derived in Ref.",
"[25] via a more direct but less rigorous method, which is repeated in Appendix for completeness.",
"An unbiased and efficient estimator $\\check{\\beta }(n)$ in terms of the observed process $n$ can be constructed from the efficient influence as $\\check{\\beta }(n) &= \\int \\tilde{\\beta }(x) dn(x) = u^\\top \\check{\\theta }(n),$ where the object moment estimator is $\\check{\\theta }(n) &= C^{-1}\\check{\\phi }(n),&\\check{\\phi }(n) &= \\int \\tilde{\\phi }(x) dn(x).$ $\\check{\\beta }(n)$ is a linear filter of $n$ , so its variance is $\\mathbb {V}(\\check{\\beta }) = \\nu (\\tilde{\\beta }^2)$ , which coincides with the CRB.",
"It is important to note that this estimator does not require any knowledge of the unknown parameters, as $\\check{\\phi }(n)$ is simply the empirical moments of the observed image, and $C^{-1}$ is a lower-triangular matrix that depends on the moments of the point-spread function $H$ .",
"The estimator still works even if the object happens to consist of a finite number of point sources and $\\theta $ is on the boundary of the parameter space, although its efficiency in that case is a more difficult question, as explained in Appendix .",
"Unlike some of the previous studies on superresolution [17], [18], [19], [20], the results here place no restriction on the separations of the point sources and also account for Poisson noise explicitly."
],
[
"Constrained Cramér-Rao bound",
"In imaging, the parameters of interest are often the moments with respect to a normalized object distribution with $\\int dF(y) = 1$ .",
"A simple way of modeling this is to assume that $\\theta _0 = 1$ is known.",
"This constraint also makes the model comparable to those in Refs.",
"[24], [26], [51].",
"Then $N &= \\phi _0 = \\tau \\theta _0$ is known as well, implying the constraint $\\gamma (f) = \\int f(x)dx - N= 0$ .",
"The constraint can be differentiated to yield $\\dot{\\gamma }(f,S_j) = \\nu (S_j) = \\langle S_j,1\\rangle = 0$ , leading to $\\mathcal {T}^\\perp = \\operatorname{span}(1)$ .",
"The new efficient influence, according to Eqs.",
"(REF ) and (REF ), should therefore be $\\tilde{\\beta }_{\\rm eff} &= \\tilde{\\beta }-\\Pi (\\tilde{\\beta }|\\mathcal {T}^\\perp )= \\tilde{\\beta }- \\frac{\\nu (\\tilde{\\beta })}{\\nu (1)} = \\tilde{\\beta }- \\frac{\\beta }{N}.$ The constrained CRB is now ${\\rm CRB}_{\\theta _0 = 1}^{({\\rm direct})} &= \\nu (\\tilde{\\beta }_{\\rm eff}^2)= \\frac{1}{N}\\left[\\nu _0(\\tilde{\\beta }_0^2)-\\beta ^2\\right],\\\\\\tilde{\\beta }_0(x) &= N\\tilde{\\beta }(x) = u^\\top (C/\\tau )^{-1}\\tilde{\\phi }(x),$ where $\\nu _0(h) = \\nu (h)/\\nu (1)$ is the normalized version of $\\nu $ .",
"The CRB is necessarily lowered by the constraint.",
"Other approaches to the constrained CRB yield the same result, as discussed in Appendix .",
"To construct a near-efficient estimator, suppose that $n(\\mathcal {X}) = \\int dn(x) = L > 0$ photons have been detected.",
"Then $dn(x) = \\sum _{l=1}^L 1_{x=X_l}$ , and the photon positions $\\lbrace X_1,X_2,\\dots ,X_L\\rbrace $ are independent and identically distributed according to the probability measure $\\bar{n}/N$ .",
"Consider the estimator $\\check{\\beta }(n) &=\\frac{1}{L} \\int \\tilde{\\beta }_0(x) dn(x) = \\frac{1}{L}\\sum _{l=1}^L \\tilde{\\beta }_0(X_l).$ It is straightforward to show that $\\mathbb {E}(\\check{\\beta }|n(\\mathcal {X})=L) &= \\nu _0(\\tilde{\\beta }_0) = \\beta ,\\\\\\mathbb {V}(\\check{\\beta }|n(\\mathcal {X}) = L)&=\\frac{1}{L}\\left[\\nu _0(\\tilde{\\beta }_0^2)-\\beta ^2\\right],$ which is close to the CRB given by Eq.",
"(REF ) if $L$ is close to its expected value $N$ .",
"The results are then consistent with standard results in semiparametric estimation concerning the moments of a normalized probability measure [21]."
],
[
"Even-moment estimation\nwith SPADE",
"Now consider the SPADE model given by Eqs.",
"(REF ) and () and the Fourier transforms $\\Psi (k) &= \\frac{1}{\\sqrt{2\\pi }}\\int \\psi (z)\\exp (-ikz)dz,\\\\\\Phi _q(k) &= \\frac{1}{\\sqrt{2\\pi }}\\int \\kappa ^*(q,z)\\exp (-ikz) dz.$ Suppose that $\\Phi = \\lbrace \\Phi _q(k)\\rbrace $ is the PAD basis [36], [25] given by $\\Phi _q(k) &= \\sqrt{\\tau } b_q(k)\\Psi (k),\\quad q \\in \\mathbb {N}_0,$ where $b = \\lbrace b_q(k): q \\in \\mathbb {N}_0\\rbrace $ is the set of orthonormal polynomials defined by $\\int \\left|\\Psi (k)\\right|^2 b_q(k) b_r(k) dk = \\delta _{qr}.$ The polynomials exist for all $q \\in \\mathbb {N}_0$ as long as the support of $|\\Psi (k)|^2$ is infinite [64], and the orthonormality of $\\Phi $ ensures that the measurement can be implemented by passive linear optics [23], [30], [36].",
"Equation () becomes $H(q|y) &= \\tau \\left|\\int \\left|\\Psi (k)\\right|^2 b_q(k)\\exp (-iky)dk\\right|^2\\\\&=\\tau \\left|\\int \\left|\\Psi (k)\\right|^2 b_q(k)\\sum _{j=0}^\\infty \\frac{(-iky)^j}{j!}",
"dk\\right|^2.$ As the $b$ polynomials are derived by applying the Gram-Schmidt procedure to the monomials $(1,k,k^2,\\dots )^\\top $ , their basic properties include $\\int |\\Psi (k)|^2 b_q(k) k^r dk= 0$ if $r < q$ , $\\int |\\Psi (k)|^2 b_q(k) k^q dk \\ne 0$ , and $b_q(k) = (-1)^q b_q(-k)$ if $|\\Psi (k)|^2$ is even, as is often the case in optics.",
"These properties lead to $H(q|y) &= \\sum _{j=0}^\\infty C_{qj} y^{2j},$ where $C$ is an upper-triangular matrix ($C_{qj} = 0$ if $j < q$ ) with positive diagonal entries ($C_{qq} > 0$ ).",
"Equation (REF ) becomes $f(q) &= \\sum _{j=0}^\\infty C_{qj}\\theta _{2j},$ which depends on the even moments $\\theta _{2j} &= \\int y^{2j} dF(y),\\quad j \\in \\mathbb {N}_0.$ The score function with respect to each $\\theta _{2j}$ becomes $S_{j}(q) &= \\frac{1}{f(q)}\\frac{\\partial f(q)}{\\partial \\theta _{2j}} = \\frac{C_{qj}}{f(q)}.$ Appendix proves that $\\mathcal {T} = \\operatorname{\\overline{span}}(S) = \\mathcal {H}$ .",
"To find a candidate influence function, suppose that Eq.",
"(REF ) can be inverted to give $\\theta _{2j} &= \\sum _{q=0}^\\infty (C^{-1})_{jq} f(q).$ An influence function for $\\beta = u^\\top \\theta $ according to Eq.",
"(REF ) is therefore $\\tilde{\\beta }(q) &= u^\\top \\tilde{\\theta }(q),&\\tilde{\\theta }_{2j}(q) &= (C^{-1})_{jq}.$ Since $\\mathcal {T} = \\mathcal {H}$ , this $\\tilde{\\beta }$ belongs to $\\mathcal {T}$ and is efficient as long as it is square-summable.",
"The CRB is hence ${\\rm CRB}^{(\\rm SPADE)} &= \\nu (\\tilde{\\beta }^2) =u^\\top \\nu (\\tilde{\\theta }\\tilde{\\theta }^\\top )u\\nonumber \\\\&= u^\\top C^{-1}D C^{-\\top }u,\\\\D_{jk} &= f(j)\\delta _{jk}.$ A more direct but heuristic way of deriving Eq.",
"(REF ) is shown in Appendix .",
"An unbiased and efficient estimator in terms of the observed detector counts $n$ is $\\check{\\beta }(n) &= \\sum _{q=0}^\\infty \\tilde{\\beta }(q) n(q) =u^\\top \\sum _{q=0}^\\infty \\tilde{\\theta }(q) n(q).$ This estimator has a variance $\\mathbb {V}(\\check{\\beta }) = \\nu (\\tilde{\\beta }^2) = {\\rm CRB}^{(\\rm SPADE)}$ , requires no knowledge of any unknown parameter, and still works even if the object happens to consist of a finite number of point sources, with no restriction on their separations.",
"If $\\theta _0 = 1$ , the constrained CRB can be derived in ways similar to Sec.",
"and Appendix .",
"To estimate the odd moments of $F$ via SPADE, variations of the PAD basis are needed [24], [25].",
"The model is much more complicated and a derivation of the CRB and the efficient estimator is too tedious to work out here, but the upshot is the same: it can be shown that the tangent space for the problem encompasses the whole Hilbert space $\\mathcal {H}$ , the efficient influence can be retrieved from the relation $\\beta = \\nu (\\tilde{\\beta })$ , and an unbiased and efficient estimator is $\\check{\\beta }(n) = \\int \\tilde{\\beta }(x) dn(x)$ ."
],
[
"Gaussian point-spread function",
"More explicit results can be obtained and the assumptions can be checked more carefully by assuming the Gaussian point-spread function $\\psi (z) &= \\frac{1}{(2\\pi )^{1/4}}\\exp \\left(-\\frac{z^2}{4}\\right).$ The PAD basis becomes the Hermite-Gaussian basis, and it can be shown that [23], [24], [55] $H(q|y) &= \\tau \\exp \\left(-\\frac{y^2}{4}\\right) \\frac{(y/2)^{2q}}{q!",
"}.$ The $C$ matrix in Eq.",
"(REF ) can be determined by expanding $\\exp (-y^2/4)=\\sum _{j=0}^\\infty (-y^2/4)^j/j!$ , giving $C_{qj} &= 1_{j\\ge q}\\frac{\\tau (-1)^{j-q}}{4^{j}q!",
"(j-q)!",
"}.$ It is not difficult to check that the matrix inverse of $C$ is $\\tilde{\\theta }_{2j}(q) &= (C^{-1})_{jq} =1_{q\\ge j}\\frac{4^{j}q!",
"}{\\tau (q-j)!",
"},$ which is a degree-$j$ polynomial of $q$ .",
"$\\sum _{q=0}^\\infty \\tilde{\\theta }_{2j}(q) f(q)$ is the $j$ th factorial moment of $f$ and indeed equal to $\\theta _{2j}$ , since $H(q|y)$ is Poisson and its factorial moment is $\\sum _{q=0}^\\infty \\tilde{\\theta }_{2j}(q) H(q|y) = y^{2j}$ [65].",
"In general, each degree-$j$ moment of $H(q|y)$ is a degree-$j$ polynomial of $y^2$ , so each degree-$j$ moment of $f(q)$ is a linear combination of the moments of $F$ up to degree $2j$ .",
"All the moments of $f$ are therefore finite as long as all the moments of $F$ are finite.",
"If $u$ has a finite number of nonzero entries, the influence function given by Eqs.",
"(REF ) is a polynomial of $q$ , so $\\nu (\\tilde{\\beta }^2) < \\infty $ , and $\\tilde{\\beta }\\in \\mathcal {H}$ is ensured."
],
[
"Bandlimited point-spread function",
"Another important example is the bandlimited point-spread function given by $\\Psi (k) &= \\frac{1_{|k| < 1}}{\\sqrt{2}}.$ $b$ is then the well known set of Legendre polynomials [66].",
"Appendix shows the detailed calculations; here I list the results only.",
"Equation (REF ) becomes $H(q|y) &= \\tau (2q+1)j_q^2(y),$ where $j_q(y)$ is the spherical Bessel function of the first kind [66].",
"An influence function for estimating $\\theta _{2j}$ with $\\theta _{2j} = \\nu (\\tilde{\\theta }_{2j})$ is $\\tilde{\\theta }_{2j}(q) &=1_{q\\ge j} \\frac{(2j+1)!!(2j-1)!!",
"}{\\tau }\\begin{pmatrix}q+j\\\\ 2j\\end{pmatrix},$ where $!",
"!$ denotes the double factorial [66].",
"$\\tilde{\\theta }_{2j}(q)$ is a degree-$2j$ polynomial of $q$ , so $\\tilde{\\beta }(q)$ is also a polynomial of $q$ if $u$ contains a finite number of nonzero entries.",
"As long as all the moments of $F$ are finite, all the moments of $f$ can also be shown to be finite, and $\\nu (\\tilde{\\beta }^2) < \\infty $ is ensured.",
"Notice that the direct-imaging theory in Sec.",
"breaks down for this bandlimited point-spread function, as the second and higher even moments of $H(x) = \\tau |\\psi (x)|^2 = (\\tau /\\pi )\\operatorname{sinc}^2(x)$ are all infinite.",
"The CRB in that case remains an open problem, although it is possible to apodize the point-spread function optically such that all its moments become finite and the semiparametric estimator given by Eq.",
"(REF ) has a finite variance.",
"For example, if $\\Psi (k) &\\propto 1_{|k|< 1}\\exp \\left(-\\frac{1}{k^2-1}\\right),$ then $\\Psi (k)$ is infinitely differentiable despite the hard bandwidth limit [67] and all the moments of $|\\psi (x)|^2$ are finite [25]."
],
[
"Comparison of imaging methods",
"The advantage of SPADE over direct imaging occurs in the subdiffraction regime, where the width $\\Delta $ of the object distribution $F$ with respect to the origin is much smaller than the width of the point-spread function $\\psi $ [24], [25], [26], [51].",
"As the width of $\\psi $ is normalized as 1, the regime is defined as $\\Delta &\\ll 1,$ and the object moments scale as $\\theta _j &= \\theta _0 O(\\Delta ^j).$ With the attainable CRBs given by Eqs.",
"(REF ) and (REF ) at hand, I can now compare direct imaging and SPADE on the same semiparametric footing.",
"Consider the estimation of a specific moment $\\theta _k$ with $u_j = \\delta _{jk}.$ For direct imaging in the subdiffraction regime, the image becomes close to the point-spread function, viz., $f(x) &\\approx \\theta _0 H(x) = N |\\psi (x)|^2,$ where $N$ , the expected photon number received in total, is given by Eq.",
"(REF ).",
"With $C_{jk} = \\tau O(1)$ and $\\nu (\\tilde{\\phi }\\tilde{\\phi }^\\top ) = N O(1)$ , the CRB becomes ${\\rm CRB}^{(\\rm direct)} &= \\frac{\\theta _0^2}{N}O(1).$ For SPADE on the other hand, notice that the $C$ and $C^{-1}$ matrices are upper-triangular, meaning that $f(q) &= N O(\\Delta ^{2q}),$ and the CRB for estimating $\\theta _{k}$ , where $k$ is even, becomes ${\\rm CRB}^{(\\rm SPADE)}&= \\frac{\\theta _0^2}{N}O(\\Delta ^{k}),$ which is much lower than Eq.",
"(REF ) when $\\Delta \\ll 1$ and $k \\ge 2$ .",
"This is consistent with earlier approximate results [24], [25].",
"An intuitive explanation of the enhancement, as well as a discussion of the limitations of SPADE, can be found in Ref. [51].",
"The constrained CRB with $\\theta _0 = 1$ becomes $[O(\\Delta ^k)-\\theta _k^2]/N = O(\\Delta ^k)/N$ , which is on the same order of magnitude as the fundamental quantum limit [26].",
"More exact and pleasing results can be obtained if $\\psi $ is Gaussian and given by Eq.",
"(REF ).",
"Consider for example the estimation of the second moment $\\theta _2$ .",
"For direct imaging, it can be shown that ${\\rm CRB}^{(\\rm direct)}&=\\frac{1}{\\tau }\\left(2\\theta _0 + 4\\theta _2 + \\theta _4\\right) =\\frac{\\theta _0^2}{N} O(1).$ For SPADE on the other hand, ${\\rm CRB}^{(\\rm SPADE)} &=\\frac{1}{\\tau }\\left(4\\theta _2 + \\theta _4\\right)= \\frac{\\theta _0^2}{N}O(\\Delta ^{2}),$ which not only beats direct imaging by a significant amount in the subdiffraction regime but is in fact superior for all parameter values.",
"To further illustrate the difference between the two methods, suppose that the object happens to be a flat top given by $dF(y) &= \\frac{\\theta _0}{\\Delta } 1_{|y| < \\Delta /2} dy.$ Do note that the semiparametric CRBs do not assume the knowledge of this object shape, which is specified here only for the purpose of plotting examples of the CRBs.",
"With $\\theta _2 = \\theta _0\\Delta ^2/12$ and $\\theta _4 = \\theta _0 \\Delta ^4/80$ , Fig.",
"REF plots Eqs.",
"(REF ) and (REF ) against $\\Delta $ in log-log scale.",
"The gap between the two curves in the $\\Delta \\ll 1$ regime is striking.",
"Figure: The semiparametric CRBs for thesecond moment θ 2 \\theta _2 given by Eqs.",
"() and() versus the object size Δ\\Delta in log-logscale, if the point-spread function is Gaussian and the objecthappens to be a flat top.",
"Both the CRBs and Δ\\Delta are normalizedsuch that they are dimensionless.With the constraint $\\theta _0 = 1$ , the CRBs become ${\\rm CRB}_{\\theta _0=1}^{(\\rm direct)}&=\\frac{1}{N}\\left(2 + 4\\theta _2 + \\theta _4-\\theta _2^2\\right) =\\frac{O(1)}{N},\\\\{\\rm CRB}_{\\theta _0=1}^{(\\rm SPADE)} &=\\frac{1}{N}\\left(4\\theta _2 + \\theta _4-\\theta _2^2\\right)= \\frac{O(\\Delta ^{2})}{N}.$ It is noteworthy that Eq.",
"(REF ) is exactly equal to the quantum limit given by [51], meaning that SPADE is exactly quantum-optimal—at all parameter values—for estimating the second moment.",
"This is consistent with previous results concerning the estimation of two-point separation [23] and object size [24], [28], but note that the previous results assume that the object shape is known, whereas the CRBs and the estimators here assume the opposite."
],
[
"Conclusion",
"The semiparametric theory set forth solves an important and difficult problem in incoherent optical imaging: the evaluation of the CRB and the efficient estimation of object parameters when little prior information about the object is available.",
"The theory gives exact and achievable semiparametric CRBs for both direct imaging and SPADE, establishing the superiority and versatility of SPADE beyond the special parametric scenarios considered by previous studies.",
"Despite the elegant results, the theory has a few shortcomings.",
"On the mathematical side, the conditions for the theory to hold seem difficult to check in the case of direct imaging with a non-Gaussian point-spread function, especially when the point-spread function has infinite moments.",
"It is an open question whether this is merely a technicality or a hint at a whole new regime of statistics.",
"On the practical side, the theory may be accused of assuming ideal conditions for both measurements, such as infinitesimal pixels for direct imaging, the availability of infinitely many modes for SPADE, perfect specification and knowledge of the optical systems, and the absence of excess noise.",
"Reality is necessarily uglier, but the results here remain relevant by serving as fundamental limits (via the data-processing inequality [68], [51]) and offering insights into the essential physics.",
"The theoretical and experimental progress on SPADE and related methods so far [23], [24], [25], [28], [26], [27], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [69], [57], [58], [69], [70], [71] has provided encouragement that the theory has realistic potential, and the general results here should motivate further research into the wider applications of quantum-inspired imaging methods.",
"An interesting future direction is to generalize the semiparametric formalism for quantum estimation [72], [73].",
"By treating the symmetric logarithmic derivatives of the quantum state $\\rho $ as the scores in the $\\mathcal {L}_h^2(\\rho )$ space proposed by Holevo [73] and adopting a geometric picture [74], a quantum generalization of the semiparametric CRB can be envisioned, but whether it can solve any important problem, such as the quantum limit to incoherent imaging [26], [27], remains to be seen."
],
[
"Acknowledgments",
"This work is supported by the Singapore National Research Foundation under Project No.",
"QEP-P7."
],
[
"Proof of the semiparametric CRB\nfor Poisson processes",
"Define the inner product between two random variables $\\check{r}_1$ and $\\check{r}_2$ as $(\\check{r}_1,\\check{r}_2) &= \\mathbb {E}\\left(\\check{r}_1\\check{r}_2\\right),$ and the norm as $|||\\check{r}||| &= \\sqrt{(\\check{r},\\check{r})} = \\sqrt{\\mathbb {E}(\\check{r}^2)}.$ Let the Hilbert space of zero-mean random variables be $\\check{\\mathcal {R}} &= \\left\\lbrace \\check{r}: \\mathbb {E}(\\check{r}) = 0,\\mathbb {E}(\\check{r}^2) < \\infty \\right\\rbrace ,$ and define $\\check{\\mathcal {T}} &= \\operatorname{\\overline{span}}(\\check{S}) \\subseteq \\check{\\mathcal {R}},$ where $\\check{S}$ is defined by Eq.",
"(REF ).",
"Let $\\check{\\delta }\\in \\check{\\mathcal {R}}$ be any random variable that satisfies $\\mathbb {E}(\\check{\\delta }\\check{S}) &= u.$ The semiparametric CRB is [21], [22] $\\mathbb {E}(\\check{\\delta }^2) &\\ge {\\rm CRB} = \\mathbb {E}(\\check{\\delta }_{\\rm eff}^2),\\\\\\check{\\delta }_{\\rm eff} &= \\Pi (\\check{\\delta }|\\check{\\mathcal {T}}) =\\operatornamewithlimits{arg\\,min}_{\\check{h} \\in \\check{\\mathcal {T}}} |||\\check{\\delta }- \\check{h}|||.$ The proof can be done via a Pythagorean theorem [22] without recourse to the Cauchy-Schwartz inequality or the existence of $J^{-1}$ .",
"$\\check{S}_j$ is called a score and $\\check{\\delta }$ an influence in statistics [21], [22]; this paper uses the same terminology for $S$ and $\\tilde{\\beta }$ in light of their resemblance to the statistical quantities.",
"The resemblance can be turned into a precise correspondence for a Poisson random measure by considering the subspace $\\check{\\mathcal {H}} \\subset \\check{\\mathcal {R}}$ of random variables that are linear with respect to $n$ .",
"Any element $\\check{h} \\in \\check{\\mathcal {H}}$ can be expressed as $\\check{h} &= Uh = \\int h(x) \\left[dn(x) - d\\bar{n}(x)\\right],$ where $U$ is a surjective linear map from $\\mathcal {H}$ to $\\check{\\mathcal {H}}$ .",
"Since $(Uh_1,Uh_2) &= \\left\\langle h_1,h_2\\right\\rangle \\quad \\forall h_1,h_2 \\in \\mathcal {H}$ by virtue of Eq.",
"(), $\\check{\\mathcal {H}}$ is isomorphic to $\\mathcal {H}$ and $U$ is unitary [61], and since $\\check{\\mathcal {T}} \\subseteq \\check{\\mathcal {H}}$ and $\\check{S} = US$ , $\\check{\\mathcal {T}}$ is isomorphic to $\\mathcal {T}$ .",
"Picking a $\\check{\\delta }\\in \\check{\\mathcal {H}}$ with $\\check{\\delta }&= U\\tilde{\\beta }=\\int \\tilde{\\beta }(x) \\left[dn(x) - d\\bar{n}(x)\\right]$ leads to $\\mathbb {E}(\\check{\\delta }\\check{S}) &= \\nu (\\tilde{\\beta }S) = u,&\\check{\\delta }_{\\rm eff} &= U \\tilde{\\beta }_{\\rm eff},$ where $\\tilde{\\beta }_{\\rm eff}$ is given by Eq.",
"(REF ) because of Eq.",
"() and the isomorphisms.",
"The CRB becomes ${\\rm CRB} = \\mathbb {E}(\\check{\\delta }_{\\rm eff}^2) = \\nu (\\tilde{\\beta }_{\\rm eff}^2),$ which is Eq.",
"()."
],
[
"The moment parameter space",
"Define an $s\\times s$ Hankel matrix with respect to a real-number sequence $\\theta = (\\theta _0,\\theta _1,\\dots )^\\top $ as $M_{jk}^{(s)}(\\theta ) &= \\theta _{j+k},\\quad j,k \\in \\lbrace 0,1,\\dots ,s-1\\rbrace .$ If $\\theta $ is a moment sequence that arises from a nonnegative measure $F$ , $w^\\top M^{(s)} w = \\int \\left(\\sum _{j=0}^{s-1} w_j y^{j}\\right)^2 dF(y)$ is nonnegative for any real vector $w$ , and all Hankel matrices are positive-semidefinite, viz., $M^{(s)} &\\ge 0\\quad \\forall s \\in \\mathbb {N}_1.$ Conversely, any sequence with Hankel matrices that obey Eq.",
"(REF ) can be expressed in the form of Eq.",
"(REF ) with a nonnegative $F$ by virtue of Hamburger's theorem [75].",
"For the CRB to hold for a $p$ -dimensional $\\theta $ , the parameter space $\\Theta $ should be an open subset of $\\mathbb {R}^p$ [68], [76].",
"Intuitively, the requirement makes sense because all the parameters in $\\theta $ are unknown and $\\theta $ should be allowed to vary in a neighborhood, otherwise the problem would be overparametrized.",
"If $\\Theta $ is not an open subset, the parameter space would be constrained and the CRB may be modified [76].",
"When all the moments are unknown parameters, consider the set $\\Theta &= \\left\\lbrace \\theta : M^{(s)}(\\theta ) > 0\\ \\forall s \\in \\mathbb {N}_1\\right\\rbrace .$ Each $\\theta \\in \\Theta $ corresponds to a measure with infinite support $r = \\infty $ [75].",
"The proof can be done by observing that the polynomial in Eq.",
"(REF ) has at most $s-1$ zeros and the integral is strictly positive for any $w \\ne 0$ if and only if $r \\ge s$ , and therefore the constraint for $\\Theta $ is satisfied if and only if $r = \\infty $ .",
"For $r < \\infty $ , $F$ can be expressed in terms of its support $\\lbrace y_l: 0\\le l \\le r-1, y_l < y_{l+1}\\rbrace $ as $dF(y) &= \\sum _{l=0}^{r-1} F_l 1_{y=y_l},&\\frac{dF(y)}{dy} &= \\sum _{l=0}^{r-1} F_l \\delta (y-y_l).$ In the context of optics, $r$ is the minimum number of point sources that can describe the object distribution.",
"The assumption of Eq.",
"(REF ) as the parameter space is consistent with the infinite-support assumption for semiparametric estimation with mixture models [21], and it also makes intuitive sense, at least as a necessary condition—with $r$ point sources, there are only $2r$ unknown parameters, and the problem would be overparametrized if all the moments are assumed to be unknown.",
"Further inequality constraints on $\\theta $ may be needed to ensure the convergence of the Taylor series in Eqs.",
"(REF ) and (REF ), although they should not affect the CRB as long as $\\theta $ obeys and stays away from them [76].",
"The boundary of $\\Theta $ corresponds to measures with finite support $r < \\infty $ .",
"If $s \\le r$ , then $M^{(s)}> 0$ and $M^{(s)}$ is full-rank (rank $ = s$ ), but if $s > r$ , I can write $M^{(s)} &= V^\\top \\operatorname{diag}(F) V,\\\\V_{jk} &= y_l^k,\\quad \\operatorname{diag}(F)_{jk} =1_{0\\le j\\le r-1}F_j\\delta _{jk}.$ $V$ is the Vandermonde matrix and invertible since $\\lbrace y_l\\rbrace $ are assumed to be distinct [77].",
"With $M^{(s)} \\ge 0$ and $\\operatorname{diag}(F) \\ge 0$ , Sylvester's law of inertia [77] implies that the rank of $M^{(s)}$ is the same as the rank of $\\operatorname{diag}(F)$ , which is $r$ .",
"In other words, the rank of $M^{(s)}$ is $\\min (r,s)$ , and any finite $r$ means that $M^{(s)}$ is rank-deficient and does not satisfy the strict $M^{(s)} > 0$ for all $s > r$ .",
"Whether the CRB still holds for $\\theta $ on the boundary is a difficult question, considering that the parameter space here is infinite-dimensional and it is not obvious how existing finite-dimensional results regarding the CRB on a boundary [76] can be applied."
],
[
"Series expansion of the object\ndistribution",
"Assume that the object measure $F$ can be expressed as the orthogonal series $dF(y) &= \\sum _{j=0}^\\infty \\xi _j g_j(y) dF^{(0)}(y)$ with respect to a reference measure $F^{(0)}$ , where $\\lbrace g_j = \\sum _{k=0}^\\infty G_{jk} y^k: j\\in \\mathbb {N}_0\\rbrace $ are the orthogonal polynomials defined by $g_j(y) &= \\sum _{k=0}^\\infty G_{jk} y^k,&\\int g_j(y) g_k(y) dF^{(0)}(y) &= \\delta _{jk},$ and $G$ is a lower-triangular matrix with nonzero diagonal entries that can be obtained by the Gram-Schmidt procedure.",
"Thus each “Fourier” coefficient $\\xi _j$ can be expressed in terms of the moment parameters as $\\xi _j &= \\int g_j(y) dF(y) = \\sum _{k=0}^\\infty G_{jk}\\theta _k,$ which can be written as $\\xi &= G \\theta , & \\theta &= G^{-1}\\xi .$ Hence each $\\theta $ corresponds to a set of coefficients $\\xi $ that can be used to represent $F$ via Eq.",
"(REF ).",
"It is straightforward to transform the CRBs and the efficient estimators derived in this paper for $\\theta $ to those for $\\xi $ via Eqs.",
"(REF )."
],
[
"Tangent space for the direct-imaging\nmodel",
"Consider the tangent space $\\mathcal {T}$ given by Eq.",
"(REF ) and the score functions given by Eq.",
"(REF ) for direct imaging.",
"First note that $S \\subset \\mathcal {H}$ , as the Fisher information $J_{jj} = \\langle S_j,S_j\\rangle = \\nu (S_j^2)$ is assumed to be finite for all $j$ .",
"Recent calculations in quantum estimation theory suggest that this assumption is reasonable for any measurement [26].",
"To prove $\\mathcal {T} = \\operatorname{\\overline{span}}(S) = \\mathcal {H}$ , the standard method is to show that the only element in $\\mathcal {H}$ orthogonal to $S$ is 0 [61], that is, $\\langle h,S_j\\rangle = 0\\quad \\forall j \\in \\mathbb {N}_0$ implies $h = 0$ (almost everywhere with respect to $\\bar{n}$ ).",
"Here I list a few approaches with various levels of rigor.",
"The first approach is to consider the set of orthogonal polynomials $a &= \\left\\lbrace a_j(x) = A \\tilde{\\phi }(x): j \\in \\mathbb {N}_0,\\left\\langle a_j,a_k\\right\\rangle = \\delta _{jk}\\right\\rbrace ,$ where $A$ is a lower-triangular matrix with nonzero diagonal entries and can be determined by applying the Gram-Schmidt procedure to the monomials $\\tilde{\\phi }(x)$ [64].",
"Under rather general conditions on $f$ , the polynomials form an orthonormal basis of $\\mathcal {H}$ [64], viz., $\\mathcal {H} &= \\operatorname{\\overline{span}}(a),$ and I can write Eq.",
"(REF ) as $\\left\\langle h,S_j\\right\\rangle &= \\sum _{k=0}^\\infty \\left\\langle h,a_k\\right\\rangle \\left\\langle a_k,S_j\\right\\rangle = 0\\quad \\forall j \\in \\mathbb {N}_0,$ or, more compactly, $B^\\top w &= 0, & w_k &= \\langle h,a_k\\rangle , &B_{kj} &= \\langle a_k,S_j\\rangle .$ If the only solution to Eq.",
"(REF ) is $w = 0$ , then the only solution to Eq.",
"(REF ) is $h = 0$ .",
"This is equivalent to the condition that $B^\\top $ is injective.",
"Integration by parts yields $B_{kj} &= \\frac{(-1)^j}{j!",
"}\\int a_k(x) \\frac{\\partial ^jH(x)}{\\partial x^j}dx= \\sum _{l=0}^\\infty A_{kl} C_{lj},$ where $C$ is the same as Eq.",
"(REF ).",
"Since both $A$ and $C$ are lower-triangular with nonzero diagonal entries, $B = AC$ is also lower-triangular with nonzero diagonal entries, and $B^\\top $ has a well defined matrix inverse $(B^\\top )^{-1} = (B^{-1})^\\top = A^{-\\top }C^{-\\top }$ in the sense that $B^\\top (B^\\top )^{-1} = (B^\\top )^{-1} B^\\top = I,$ where $I$ is the identity matrix; see Appendix for details.",
"If the matrices were finite-dimensional, the existence of a matrix inverse would imply $(B^\\top )^{-1}(B^\\top w) = [(B^\\top )^{-1} B^\\top ] w = w,$ and the only solution to Eq.",
"(REF ) would be $w = 0$ .",
"This proof is not entirely satisfactory however, as Eq.",
"(REF ) assumes that the product of the infinite-dimensional matrices is associative.",
"Associativity assumes that the order of the sums involved in the matrix product can be interchanged, but it cannot be guaranteed for infinite-dimensional matrices.",
"In other words, the existence of a matrix inverse for $B^\\top $ may not imply that $B^\\top $ is injective.",
"A more rigorous approach is to define $\\chi _y(x) &=\\sum _{j=0}^\\infty y^j S_j(x),\\quad y \\in \\mathcal {Y} \\subset \\mathbb {R},$ and notice that Eq.",
"(REF ) implies $\\left\\langle h,\\chi _y\\right\\rangle &= \\int h(x)\\sum _{j=0}^\\infty y^j\\frac{(-1)^j}{j!",
"}\\frac{\\partial ^jH(x)}{\\partial x^j} dx\\\\&= \\int h(x) H(x-y)dx = 0 \\quad \\forall y \\in \\mathcal {Y}.$ The unique solution to Eq.",
"(REF ) is $h = 0$ if the family $\\lbrace H(x-y): y \\in \\mathcal {Y}\\rbrace $ satisfies a property called completeness in statistics [5].",
"For example, if $H$ is Gaussian, $\\lbrace H\\rbrace $ is a full-rank exponential family for any open subset $\\mathcal {Y} \\subset \\mathbb {R}$ and therefore complete [5].",
"An even more rigorous formulation of this approach [21] is to treat $\\langle h,\\chi _y\\rangle $ as an operator that maps $h \\in \\mathcal {H}$ to a function of $y$ in another Hilbert space, and then show that the null space of the operator consists of only $h = 0$ .",
"The proof again boils down to the requirement that $\\lbrace H\\rbrace $ should be complete; see Ref.",
"[21]."
],
[
"Inverse of an infinite-dimensional triangular matrix",
"Let $C$ be an infinite-dimensional matrix with entries indexed by $(j,k) \\in \\mathbb {N}_0^2$ .",
"Define its formal matrix inverse $C^{-1}$ as another infinite-dimensional matrix that satisfies $\\sum _{l=0}^\\infty C_{jl}(C^{-1})_{lk} &= \\delta _{jk}.$ If $C$ is lower-triangular with nonzero diagonal entries, viz., $C_{jk} &= 0 \\textrm { if }k > j,&C_{jj} &\\ne 0,$ then $C^{-1}$ can be found by a recursive relation as follows.",
"Define $C^{(s)}$ as the $s\\times s$ upper-left submatrix of $C$ .",
"Write $C^{(s+1)}$ and $(C^{-1})^{(s+1)}$ as the partitions $C^{(s+1)} &= \\begin{pmatrix}C^{(s)} & 0\\\\ c^\\top & C_{ss}\\end{pmatrix},\\\\(C^{-1})^{(s+1)} &= \\begin{pmatrix}(C^{-1})^{(s)} & 0\\\\ d^\\top & (C^{-1})_{ss}\\end{pmatrix}.$ Given $(C^{-1})^{(s)} = (C^{(s)})^{-1}$ , $d^\\top &= - \\frac{c^\\top (C^{(s)})^{-1}}{C_{ss}},&(C^{-1})_{ss} &= \\frac{1}{C_{ss}},$ and the recursion starts from $(C^{-1})^{(1)} = (C^{(1)})^{-1}$ with one element $(C^{-1})_{00} = 1/C_{00}$ .",
"The matrix inverse of an infinite-dimensional upper-triangular matrix can be defined in a similar way.",
"Although the product of infinite-dimensional matrices may not be associative, it can still be proved by induction that $D (Cu) = (DC)u$ for any vector $u$ if $D$ and $C$ are lower-triangular, because $D(Cu) &= \\sum _{k=0}^\\infty D_{jk}\\sum _{l=0}^\\infty C_{kl}u_l= \\sum _{k=0}^j D_{jk}\\sum _{l=0}^k C_{kl}u_l$ involves finite sums only.",
"Thus it is safe to assume that $C^{-1}(Cu) = (C^{-1} C)u = u$ and $C(C^{-1}u) = (C C^{-1}) u = u$ if $C$ is lower-triangular with nonzero diagonal entries."
],
[
"An alternative derivation\nof the Cramér-Rao bound for direct imaging",
"Consider the problem described in Sec. .",
"Since the polynomials given by Eq.",
"(REF ) are an orthonormal basis, the information matrix for the moment parameters can be expressed as $J_{jk} = \\left\\langle S_j,S_k\\right\\rangle = \\sum _{l=0}^\\infty \\langle S_j,a_l\\rangle \\langle a_l,S_k\\rangle ,$ meaning that $J = B^\\top B$ , where $B = AC$ is given by Eq.",
"(REF ).",
"Ignoring the complications of multiplying infinite-dimensional matrices, the CRB becomes $J^{-1} = B^{-1}B^{-\\top } = C^{-1}A^{-1}A^{-\\top }C^{-\\top }.$ To evaluate $A^{-1}A^{-\\top }$ , notice that the orthonormality of $a$ can be written as $\\left\\langle a_j,a_k\\right\\rangle &= \\sum _{l=0}^\\infty \\sum _{m=0}^\\infty A_{jl} \\left\\langle \\tilde{\\phi }_l,\\tilde{\\phi }_m\\right\\rangle A_{km} = \\delta _{jk},$ where $\\tilde{\\phi }$ is the monomials given by Eq.",
"(REF ).",
"In other words, $A \\nu (\\tilde{\\phi }\\tilde{\\phi }^\\top ) A^\\top &= I,&A^{-1}A^{-\\top } &= \\nu (\\tilde{\\phi }\\tilde{\\phi }^\\top ),$ giving $J^{-1} &= C^{-1}\\nu (\\tilde{\\phi }\\tilde{\\phi }^\\top )C^{-\\top }.$ This leads to Eq.",
"(REF ) for the parameter $\\beta = u^\\top \\theta $ ."
],
[
"Alternative approaches\nto the constrained Cramér-Rao bound",
"One way of deriving the constrained CRB if $\\theta _0$ is known is to consider the information matrix $\\tilde{J}$ with respect to the parameters $\\vartheta = (\\theta _1,\\theta _2,\\dots )^\\top $ without $\\theta _0$ .",
"Then $\\theta = (\\theta _0,\\vartheta ^\\top )^\\top $ , and $\\tilde{J}$ can be written as the submatrix of $J$ , or $J &= \\begin{pmatrix}J_{00} & j^\\top \\\\ j & \\tilde{J}\\end{pmatrix},$ where $j$ is a column vector.",
"Ignore the complications of dealing with infinite-dimensional matrices and partition $J^{-1}$ similarly as $J^{-1} &=\\begin{pmatrix}E_{00} & e^\\top \\\\ e & \\tilde{E}\\end{pmatrix}.$ Then it is straightforward to show that $\\tilde{J}^{-1} &= \\tilde{E} - \\frac{e e^\\top }{E_{00}}.$ Let $\\tilde{\\vartheta }= (\\tilde{\\theta }_1,\\tilde{\\theta }_2,\\dots )^\\top $ , and observe that $\\tilde{\\theta }_0 = 1/C_{00}$ from Eqs.",
"(), (REF ), and (REF ).",
"Then Eq.",
"(REF ) implies that $\\tilde{E} &= \\nu (\\tilde{\\vartheta }\\tilde{\\vartheta }^\\top ),\\\\e &= \\nu (\\tilde{\\vartheta }\\tilde{\\theta }_0) =\\frac{\\nu (\\tilde{\\vartheta })}{C_{00}}= \\frac{\\vartheta }{C_{00}},\\\\E_{00} &= \\nu (\\tilde{\\theta }_0 \\tilde{\\theta }_0) = \\frac{\\nu (1)}{C_{00}^2}=\\frac{\\phi _0}{C_{00}^2}.$ Hence $\\tilde{J}^{-1} &= \\nu (\\tilde{\\vartheta }\\tilde{\\vartheta }^\\top ) -\\frac{\\vartheta \\vartheta ^\\top }{\\phi _0},$ which implies Eq.",
"(REF ) if the parameter of interest is defined as $\\beta = u^\\top \\theta $ with $u_0 = 0$ .",
"Yet another way of deriving the constrained CRB can be found in Ref.",
"[76], which can be shown to lead to the same result here."
],
[
"Tangent space for the SPADE model",
"The proof is similar to the first approach in Appendix .",
"Consider $\\mathcal {H} = \\operatorname{\\overline{span}}(a)$ in terms of an obvious orthonormal basis $a &=\\left\\lbrace a_j(q)= \\delta _{jq}/\\sqrt{f(j)}: j \\in \\mathbb {N}_0\\right\\rbrace .$ Any $h\\in \\mathcal {H}$ orthogonal to the $S$ given by Eq.",
"(REF ) obeys $\\left\\langle h,S_{j}\\right\\rangle &= \\sum _{k=0}^\\infty \\left\\langle h,a_k\\right\\rangle \\left\\langle a_k,S_{j}\\right\\rangle = 0\\quad \\forall j \\in \\mathbb {N}_0,$ which can be written as $B^\\top w &= 0, \\quad w_k= \\left\\langle h,a_k\\right\\rangle ,\\\\B_{jk} &= \\left\\langle a_j,S_{k}\\right\\rangle = \\frac{C_{jk}}{\\sqrt{f(j)}}.$ As $C$ is upper-triangular with nonzero diagonal entries and $f > 0$ is assumed, $B^\\top $ is lower-triangular with nonzero diagonal entries, and induction can be used to prove that the only solution to $B^\\top w = 0$ is $w = 0$ , or equivalently $h = 0$ .",
"Hence $\\mathcal {T} = \\operatorname{\\overline{span}}(S) = \\mathcal {H}$ .",
"The proof is easier than the one in Appendix because $B^\\top $ here is lower-triangular rather than upper-triangular.",
"An alternative proof, similar to the second approach in Appendix and Ref.",
"[21] but less fruitful in this case, is to define $\\chi _y(x) &= \\sum _{j=0}^\\infty y^{2j} S_{j}(x),\\quad y \\in \\mathcal {Y} \\subset \\mathbb {R},$ consider $\\left\\langle h,\\chi _y\\right\\rangle &= \\sum _{q=0}^\\infty h(q) H(q|y) = 0,$ and use the completeness of $\\lbrace H(q|y): y \\in \\mathcal {Y}\\rbrace $ to prove the unique solution $h = 0$ .",
"If $H$ is Poisson, for example, then $\\lbrace H\\rbrace $ is a full-rank exponential family and therefore complete [5]."
],
[
"An alternative derivation of the\nCramér-Rao bound for SPADE",
"Consider the problem described in Sec. .",
"With the orthonormal basis given by Eq.",
"(REF ) and the $B$ matrix given by Eq.",
"(REF ), the information matrix with respect to the moment parameters can again be expressed as $J = B^\\top B$ according to Eq.",
"(REF ).",
"With Eq.",
"(REF ), $B^{-1}$ becomes $(B^{-1})_{jq} &= (C^{-1})_{jq}\\sqrt{f(q)}.$ Ignoring the complications of multiplying infinite-dimensional matrices, the CRB $J^{-1} = B^{-1}B^{-\\top }$ is $(J^{-1})_{jk} &=\\sum _{q=0}^\\infty (C^{-1})_{jq}f(q)(C^{-1})_{kq}=C^{-1}DC^{-\\top },$ where $D$ is given by Eq.",
"(), and the CRB for $\\beta = u^\\top \\theta $ coincides with Eq.",
"(REF )."
],
[
" Calculations concerning\nSPADE for a bandlimited point-spread function",
"Consider the point-spread function given by Eq.",
"(REF ).",
"The standard Legendre polynomials are defined in terms of $\\frac{1}{2}\\int _{-1}^1 P_q(k)P_p(k) dk &= \\frac{1}{2q+1}\\delta _{qp},$ such that the orthonormal version is $b_q(k) &= \\sqrt{2q+1} P_q(k).$ The Fourier transform of the polynomial is [66] $\\frac{1}{2}\\int _{-1}^1 b_q(k)\\exp (iky) dk &= \\sqrt{2q+1} j_q(y),$ where $j_q(y)$ is the spherical Bessel function of the first kind [66].",
"Then Eq.",
"(REF ) follows from Eq.",
"(REF ) and (REF ).",
"Let $\\tilde{H}(q|y) = \\frac{H(q|y)}{\\tau } = (2q+1)j_q^2(y).$ From Ref.",
"[66], one can derive the useful formula $\\sum _{q=0}^\\infty \\tilde{H}(q|y) P_q(k) &= \\operatorname{sinc}w={\\left\\lbrace \\begin{array}{ll}(\\sin w)/w, & w \\ne 0,\\\\1, & w = 0,\\end{array}\\right.",
"}\\\\w &= y\\sqrt{2-2k}.$ For example, since $P_q(1) = 1$ , one can check that $\\sum _{q=0}^\\infty \\tilde{H}(q|y) = 1$ in accordance with Eq.",
"(REF ).",
"Using the facts $\\operatorname{sinc}w &=\\frac{1}{2}\\int _{-1}^1 \\exp (iw z) dz=\\sum _{l=0}^\\infty \\frac{(-1)^l w^{2l}}{(2l+1)!",
"},\\\\\\frac{d w}{d k} &= -\\frac{y}{w},\\quad P_q^{(j)}(1) = \\left.\\frac{d ^j P_q(k)}{d k^j}\\right|_{k=1},$ it can also be shown that $\\sum _{q=0}^\\infty \\tilde{H}(q|y) P_q^{(j)}(1) &=\\left.\\frac{d ^j \\operatorname{sinc}w}{d k^j}\\right|_{k=1}= \\frac{y^{2j}}{(2j+1)!!",
"},$ which leads to $\\sum _{q=0}^\\infty f(q)P_q^{(j)}(1)&= \\frac{\\tau \\theta _{2j}}{(2j+1)!!",
"}.$ An influence function for estimating $\\theta _{2j}$ is hence $\\tilde{\\theta }_{2j}(q) &= \\frac{(2j+1)!!",
"}{\\tau } P_q^{(j)}(1),$ which obeys $\\theta _{2j} = \\nu (\\tilde{\\theta }_{2j})$ .",
"To derive an explicit expression for $P_q^{(j)}(1)$ , consider the Rodrigues formula [66] $P_q(k) &= \\frac{1}{2^qq!}",
"\\frac{d ^q}{d k^q}(k^2-1)^q,$ which leads to $P_q(k) &= \\sum _{l=0}^q\\begin{pmatrix}q\\\\ l\\end{pmatrix}\\begin{pmatrix}q+l\\\\ l\\end{pmatrix}\\left(\\frac{k-1}{2}\\right)^l,\\\\P_q^{(j)}(1) &= 1_{q\\ge j}(2j-1)!",
"!\\begin{pmatrix}q+j\\\\ 2j\\end{pmatrix},$ and Eq.",
"(REF ) results.",
"To bound the moments of $\\tilde{H}$ and $f$ , consider a lower bound on the binomial coefficient for $j\\ge 1$ given by [78] $\\begin{pmatrix}q+j\\\\ 2j\\end{pmatrix} &\\ge \\frac{(q+j)^{2j}}{(2j)^{2j}}\\ge \\frac{q^{2j}}{(2j)^{2j}},$ such that each even moment of $\\tilde{H}$ can be bounded as $&\\quad \\sum _{q=0}^\\infty \\tilde{H}(q|y) q^{2j}\\nonumber \\\\&=\\sum _{q=0}^{j-1}\\tilde{H}(q|y) q^{2j} +\\sum _{q=j}^{\\infty }\\tilde{H}(q|y) q^{2j}\\\\&\\le (j-1)^{2j} + \\frac{(2j)^{2j}}{(2j-1)!!",
"}\\sum _{q=0}^\\infty \\tilde{H}(q|y) P_q^{(j)}(1)\\\\&= (j-1)^{2j} + \\frac{(2j)^{2j}y^{2j}}{(2j-1)!!(2j+1)!!",
"}.$ This means that each even moment of $f(q)$ is bounded as $\\nu (q^{2j})& \\le \\tau \\left[(j-1)^{2j}\\theta _0 +\\frac{(2j)^{2j}\\theta _{2j}}{(2j-1)!!(2j+1)!!",
"}\\right].$ With $\\nu (q^0) = \\nu (1) = \\tau \\theta _0$ , $\\nu (q^{2j})<\\infty $ for all $j \\in \\mathbb {N}_0$ as long as $\\theta _0$ and $\\theta _{2j}$ are finite.",
"Odd moments can be bounded via the Cauchy-Schwartz inequality $[\\nu (q^j)]^2 \\le \\nu (1)\\nu (q^{2j})$ .",
"Hence all the moments of $f$ are finite as long as all the moments of $F$ are finite."
]
] | 1906.04578 |
[
[
"Weight Agnostic Neural Networks"
],
[
"Abstract Not all neural network architectures are created equal, some perform much better than others for certain tasks.",
"But how important are the weight parameters of a neural network compared to its architecture?",
"In this work, we question to what extent neural network architectures alone, without learning any weight parameters, can encode solutions for a given task.",
"We propose a search method for neural network architectures that can already perform a task without any explicit weight training.",
"To evaluate these networks, we populate the connections with a single shared weight parameter sampled from a uniform random distribution, and measure the expected performance.",
"We demonstrate that our method can find minimal neural network architectures that can perform several reinforcement learning tasks without weight training.",
"On a supervised learning domain, we find network architectures that achieve much higher than chance accuracy on MNIST using random weights.",
"Interactive version of this paper at https://weightagnostic.github.io/"
],
[
"Introduction",
"In biology, precocial species are those whose young already possess certain abilities from the moment of birth.",
"There is evidence to show that lizard [79] and snake [14], [82] hatchlings already possess behaviors to escape from predators.",
"Shortly after hatching, ducks are able to swim and eat on their own [111], and turkeys can visually recognize predators [29].",
"In contrast, when we train artificial agents to perform a task, we typically choose a neural network architecture we believe to be suitable for encoding a policy for the task, and find the weight parameters of this policy using a learning algorithm.",
"Inspired by precocial behaviors evolved in nature, in this work, we develop neural networks with architectures that are naturally capable of performing a given task even when their weight parameters are randomly sampled.",
"By using such neural network architectures, our agents can already perform well in their environment without the need to learn weight parameters.",
"Figure: Examples of Weight Agnostic Neural Networks: Bipedal Walker (left), Car Racing (right)We search for architectures by deemphasizing weights.",
"In place of training, networks are assigned a single shared weight value at each rollout.",
"Architectures that are optimized for expected performance over a wide range of weight values are still able to perform various tasks without weight training.Decades of neural network research have provided building blocks with strong inductive biases for various task domains.",
"Convolutional networks [61], [25] are especially suited for image processing [17].",
"Recent work [45], [117] demonstrated that even randomly-initialized CNNs can be used effectively for image processing tasks such as superresolution, inpainting and style transfer.",
"Schmidhuber et al.",
"[103] have shown that a randomly-initialized LSTM [49] with a learned linear output layer can predict time series where traditional reservoir-based RNNs [51], [99] fail.",
"More recent developments in self-attention [121] and capsule [100] networks expand the toolkit of building blocks for creating architectures with strong inductive biases for various tasks.",
"Fascinated by the intrinsic capabilities of randomly-initialized CNNs and LSTMs, we aim to search for weight agnostic neural networks, architectures with strong inductive biases that can already perform various tasks with random weights.",
"In order to find neural network architectures with strong inductive biases, we propose to search for architectures by deemphasizing the importance of weights.",
"This is accomplished by (1) assigning a single shared weight parameter to every network connection and (2) evaluating the network on a wide range of this single weight parameter.",
"In place of optimizing weights of a fixed network, we optimize instead for architectures that perform well over a wide range of weights.",
"We demonstrate our approach can produce networks that can be expected to perform various continuous control tasks with a random weight parameter.",
"As a proof of concept, we also apply our search method on a supervised learning domain, and find it can discover networks that, even without explicit weight training, can achieve a much higher than chance test accuracy of $\\sim $ 92% on MNIST.",
"We hope our demonstration of such weight agnostic neural networks will encourage further research exploring novel neural network building blocks that not only possess useful inductive biases, but can also learn using algorithms that are not necessarily limited to gradient-based methods.We released a software toolkit not only to facilitate reproduction, but also to further research in this direction.",
"Refer to the Supplementary Materials for more information about the code repository."
],
[
"Related Work",
"Our work has connections to existing work not only in deep learning, but also to various other fields: Architecture Search Search algorithms for neural network topologies originated from the field of evolutionary computing [116], [43], [18], [26], [34], [58], [9], [76], [126], [77], [2], [64], [87], [92], [124].",
"Our method is based on NEAT [110], an established topology search algorithm notable for its ability to optimize the weights and structure of networks simultaneously.",
"In order to achieve state-of-the-art results, recent methods narrow the search space to architectures composed of basic building blocks with strong domain priors such as CNNs [128], [95], [70], [78], recurrent cells [53], [128], [78] and self-attention [107].",
"It has been shown that random search can already achieve SOTA results if such priors are used [69], [104], [94].",
"The inner loop for training the weights of each candidate architecture before evaluation makes the search costly, although efforts have been made to improve efficiency [91], [10], [71].",
"In our approach, we evaluate architectures without weight training, bypassing the costly inner loop, similar to the random trial approach in [48], [106] that evolved architectures to be more weight tolerant.",
"Bayesian Neural Networks The weight parameters of a BNN [74], [47], [4], [5], [84], [27] are not fixed values, but sampled from a distribution.",
"While the parameters of this distribution can be learned [41], [42], [30], [59], the number of parameters is often greater than the number of weights.",
"Recently, Neklyudov et al.",
"[85] proposed variance networks, which sample each weight from a distribution with a zero mean and a learned variance parameter, and show that ensemble evaluations can improve performance on image recognition tasks.",
"We employ a similar approach, sampling weights from a fixed uniform distribution with zero mean, as well as evaluating performance on network ensembles.",
"Algorithmic Information Theory In AIT [108], the Kolmogorov complexity [56] of a computable object is the minimum length of the program that can compute it.",
"The Minimal Description Length (MDL) [97], [35], [98] is a formalization of Occam's razor, in which a good model is one that is best at compressing its data, including the cost of describing of the model itself.",
"Ideas related to MDL for making neural networks “simple” was proposed in the 1990s, such as simplifying networks by soft-weight sharing [86], reducing the amount of information in weights by making them noisy [47], and simplifying the search space of its weights [102].",
"Recent works offer a modern treatment [7] and application [67], [115] of these principles in the context of larger, deep neural network architectures.",
"While the aforementioned works focus on the information capacity required to represent the weights of a predefined network architecture, in this work we focus on finding minimal architectures that can represent solutions to various tasks.",
"As our networks still require weights, we borrow ideas from AIT and BNN, and take them a bit further.",
"Motivated by MDL, in our approach, we apply weight-sharing to the entire network and treat the weight as a random variable sampled from a fixed distribution.",
"Network Pruning By removing connections with small weight values from a trained neural network, pruning approaches [63], [44], [40], [36], [68], [81], [73], [72], [75] can produce sparse networks that keep only a small fraction of the connections, while maintaining similar performance on image classification tasks compared to the full network.",
"By retaining the original weight initialization values, these sparse networks can even be trained from scratch to achieve a higher test accuracy [23], [65] than the original network.",
"Similar to our work, a concurrent work [127] found pruned networks that can achieve image classification accuracies that are much better than chance even with randomly initialized weights.",
"Network pruning is a complementary approach to ours; it starts with a full, trained network, and takes away connections, while in our approach, we start with no connections, and add complexity as needed.",
"Compared to our approach, pruning requires prior training of the full network to obtain useful information about each weight in advance.",
"In addition, the architectures produced by pruning are limited by the full network, while in our method there is no upper bound on the network's complexity.",
"Neuroscience A connectome [105] is the “wiring diagram” or mapping of all neural connections of the brain.",
"While it is a challenge to map out the human connectome [109], with our 90 billion neurons and 150 trillion synapses, the connectome of simple organisms such as roundworms [122], [120] has been constructed, and recent works [21], [112] mapped out the entire brain of a small fruit fly.",
"A motivation for examining the connectome, even of an insect, is that it will help guide future research on how the brain learns and represents memories in its connections.",
"For humans it is evident, especially during early childhood [50], [114], that we learn skills and form memories by forming new synaptic connections, and our brain rewires itself based on our new experiences [6], [12], [55], [19].",
"The connectome can be viewed as a graph [13], [46], [118], and analyzed using rich tools from graph theory, network science and computer simulation.",
"Our work also aims to learn network graphs that can encode skills and knowledge for an artificial agent in a simulation environment.",
"By deemphasizing learning of weight parameters, we encourage the agent instead to develop ever-growing networks that can encode acquired skills based on its interactions with the environment.",
"Like the connectome of simple organisms, the networks discovered by our approach are small enough to be analyzed."
],
[
"Weight Agnostic Neural Network Search",
"Creating network architectures which encode solutions is a fundamentally different problem than that addressed by neural architecture search (NAS).",
"The goal of NAS techniques is to produce architectures which, once trained, outperform those designed by humans.",
"It is never claimed that the solution is innate to the structure of the network.",
"Networks created by NAS are exceedingly `trainable' – but no one supposes these networks will solve the task without training the weights.",
"The weights are the solution; the found architectures merely a better substrate for the weights to inhabit.",
"To produce architectures that themselves encode solutions, the importance of weights must be minimized.",
"Rather than judging networks by their performance with optimal weight values, we can instead measure their performance when their weight values are drawn from a random distribution.",
"Replacing weight training with weight sampling ensures that performance is a product of the network topology alone.",
"Unfortunately, due to the high dimensionality, reliable sampling of the weight space is infeasible for all but the simplest of networks.",
"Though the curse of dimensionality prevents us from efficiently sampling high dimensional weight spaces, by enforcing weight-sharing on all weights, the number of weight values is reduced to one.",
"Systematically sampling a single weight value is straight-forward and efficient, enabling us to approximate network performance in only a handful of trials.",
"This approximation can then be used to drive the search for ever better architectures.",
"The search for these weight agnostic neural networks (WANNs) can be summarized as follows (See Figure REF for an overview): (1) An initial population of minimal neural network topologies is created, (2) each network is evaluated over multiple rollouts, with a different shared weight value assigned at each rollout, (3) networks are ranked according to their performance and complexity, and (4) a new population is created by varying the highest ranked network topologies, chosen probabilistically through tournament selection [80].",
"The algorithm then repeats from (2), yielding weight agnostic topologies of gradually increasing complexity that perform better over successive generations.",
"Figure: Overview of Weight Agnostic Neural Network SearchWeight Agnostic Neural Network Search avoids weight training while exploring the space of neural network topologies by sampling a single shared weight at each rollout.Networks are evaluated over several rollouts.",
"At each rollout a value for the single shared weight is assigned and the cumulative reward over the trial is recorded.The population of networks is then ranked according to their performance and complexity.The highest ranking networks are then chosen probabilistically and varied randomly to form a new population, and the process repeats.Topology Search The operators used to search for neural network topologies are inspired by the well-established neuroevolution algorithm NEAT [110].",
"While in NEAT the topology and weight values are optimized simultaneously, we ignore the weights and apply only topological search operators.",
"The initial population is composed of sparsely connected networks, networks with no hidden nodes and only a fraction of the possible connections between input and output.",
"New networks are created by modifying existing networks using one of three operators: insert node, add connection, or change activation (Figure REF ).",
"To insert a node, we split an existing connection into two connections that pass through this new hidden node.",
"The activation function of this new node is randomly assigned.",
"New connections are added between previously unconnected nodes, respecting the feed-forward property of the network.",
"When activation functions of hidden nodes are changed, they are assigned at random.",
"Activation functions include both the common (e.g.",
"linear, sigmoid, ReLU) and more exotic (Gaussian, sinusoid, step), encoding a variety of relationships between inputs and outputs.",
"Figure: Operators for Searching the Space of Network TopologiesLeft: A minimal network topology, with input and outputs only partially connected.Middle: Networks are altered in one of three ways.",
"Insert Node: a new node is inserted by splitting an existing connection.",
"Add Connection: a new connection is added by connecting two previously unconnected nodes.",
"Change Activation: the activation function of a hidden node is reassigned.Right: Possible activation functions (linear, step, sin, cosine, Gaussian, tanh, sigmoid, absolute value, invert (i.e.",
"negative linear), ReLU) shown over the range [2,2][2,2].Performance and Complexity Network topologies are evaluated using several shared weight values.",
"At each rollout a new weight value is assigned to all connections, and the network is tested on the task.",
"In these experiments we used a fixed series of weight values ($[-2,-1,-0.5,+0.5,+1,+2]$ ) to decrease the variance between evaluations.Variations on these particular values had little effect, though weight values in the range $[-2,2]$ showed the most variance in performance.",
"Networks whose weight values were set to greater than 3 tended to perform similarly – presumably saturating many of the activation functions.",
"Weight values near 0 were also omitted to reduce computation, as regardless of the topology little to no information is passed to the output.",
"We calculate the mean performance of a network topology by averaging its cumulative reward over all rollouts using these different weight values.",
"Motivated by algorithmic information theory [108], we are not interested in searching merely for any weight agnostic neural networks, but networks that can be described with a minimal description length [97], [35], [98].",
"Given two different networks with similar performance we prefer the simpler network.",
"By formulating the search as a multi-objective optimization problem [57], [83] we take into account the size of the network as well as its performance when ranking it in the population.",
"We apply the connection cost technique from [16] shown to produce networks that are more simple, modular, and evolvable.",
"Networks topologies are judged based on three criteria: mean performance over all weight values, max performance of the single best weight value, and the number of connections in the network.",
"Rather than attempting to balance these criteria with a hand-crafted reward function for each new task, we rank the solutions based on dominance relations [20].",
"Ranking networks in this way requires that any increase in complexity is accompanied by an increase in performance.",
"While encouraging minimal and modular networks, this constraint can make larger structural changes – which may require several additions before paying off – difficult to achieve.",
"To relax this constraint we rank by complexity only probabilistically: in 80% of cases networks are ranked according to mean performance and the number of connections, in the other 20% ranking is done by mean performance and max performance."
],
[
"Experimental Results",
"Continuous Control Weight agnostic neural networks (WANNs) are evaluated on three continuous control tasks.",
"The first, CartPoleSwingUp, is a classic control problem where, given a cart-pole system, a pole must be swung from a resting to upright position and then balanced, without the cart going beyond the bounds of the track.",
"The swingup task is more challenging than the simpler CartPole [11], where the pole starts upright.",
"Unlike the simpler task, it cannot be solved with a linear controller [113], [93].",
"The reward at every timestep is based on the distance of the cart from track edge and the angle of the pole.",
"Our environment is closely based on the one described in [28], [129].",
"The second task, BipedalWalker-v2 [11], is to guide a two-legged agent across randomly generated terrain.",
"Rewards are awarded for distance traveled, with a cost for motor torque to encourage efficient movement.",
"Each leg is controlled by a hip and knee joint in reaction to 24 inputs, including LIDAR sensors which detect the terrain and proprioceptive information such as the agent's joint speeds.",
"Compared to the low dimensional CartPoleSwingUp, BipedalWalker-v2 has a non-trivial number of possible connections, requiring WANNs to be selective about the wiring of inputs to outputs.",
"The third, CarRacing-v0 [11], is a top-down car racing from pixels environment.",
"A car, controlled with three continuous commands (gas, steer, brake) is tasked with visiting as many tiles as possible of a randomly generated track within a time limit.",
"Following the approach described in [39], we delegate the pixel interpretation element of the task to a pre-trained variational autoencoder [54], [96] (VAE) which compresses the pixel representation to 16 latent dimensions.",
"These dimensions are given as input to the network.",
"The use of learned features tests the ability of WANNs to learn abstract associations rather than encoding explicit geometric relationships between inputs.",
"Hand-designed networks found in the literature [38], [39] are compared to the best weight agnostic networks found for each task.",
"We compare the mean performance over 100 trials under 4 conditions: Random weights: individual weights drawn from $\\mathcal {U}(-2,2)$ ; Random shared weight: a single shared weight drawn from $\\mathcal {U}(-2,2)$ ; Tuned shared weight: the highest performing shared weight value in range $(-2,2)$ ; Tuned weights: individual weights tuned using population-based REINFORCE [123].",
"Table: Performance of Randomly Sampled and Trained Weights for Continuous Control TasksWe compare the cumulative reward (average of 100 random trials) of the best weight agnostic network architectures found with standard feed forward network policies commonly used in previous work (i.e.",
", ).The intrinsic bias of a network topology can be observed by measuring its performance using a shared weight sampled from a uniform distribution.By tuning this shared weight parameter we can measure its maximum performance.To facilitate comparison to baseline architectures we also conduct experiments where networks are allowed unique weight parameters and tuned.The results are summarized in Table REF .We conduct several independent search runs to measure variability of results in Supplementary Materials.",
"In contrast to the conventional fixed topology networks used as baselines, which only produce useful behaviors after extensive tuning, WANNs perform even with random shared weights.",
"Though their architectures encode a strong bias toward solutions, WANNs are not completely independent of the weight values – they do fail when individual weight values are assigned randomly.",
"WANNs function by encoding relationships between inputs and outputs, and so while the importance of the magnitude of the weights is not critical, their consistency, especially consistency of sign, is.",
"An added benefit of a single shared weight is that it becomes trivial to tune this single parameter, without requiring the use of gradient-based methods.",
"The best performing shared weight value produces satisfactory if not optimal behaviors: a balanced pole after a few swings, effective if inefficient gaits, wild driving behaviour that cuts corners.",
"These basic behaviors are encoded entirely within the architecture of the network.",
"And while WANNs are able to perform without training, this predisposition does not prevent them from reaching similar state-of-the-art performance when the weights are trained.",
"Figure: Development of Weight Agnostic Neural Network Topologies Over TimeGeneration 8: An early network which performs poorly with nearly all weights.Generation 32: Relationships between the position of the cart and velocity of the pole are established.",
"The tension between these relationships produces both centering and swing-up behavior.Generation 128: Complexity is added to refine the balancing behavior of the elevated pole.As the networks discovered are small enough to interpret, we can derive insights into how they function by looking at network diagrams (See Figure REF ).",
"Examining the development of a WANN which solves CartPoleSwingUp is also illustrative of how relationships are encoded within an architecture.",
"In the earliest generations the space of networks is explored in an essentially random fashion.",
"By generation 32, preliminary structures arise which allow for consistent performance: the three inverters applied to the $x$ position keep the cart from leaving the track.",
"The center of the track is at 0, left is negative, right is positive.",
"By applying positive force when the cart is in a negative position and vice versa a strong attractor towards the center of the track is encoded.",
"The interaction between the regulation of position and the Gaussian activation on $d\\theta $ is responsible for the swing-up behavior, also developed by generation 32.",
"At the start of the trial the pole is stationary: the Gaussian activation of $d\\theta $ is 1 and force is applied.",
"As the pole moves toward the edge the nodes connected to the $x$ input, which keep the cart in the center, begin sending an opposing force signal.",
"The cart's progress toward the edge is slowed and the change in acceleration causes the pole to swing, increasing $d\\theta $ and so decreasing the signal that is pushing the cart toward the edge.",
"This slow down causes further acceleration of the pole, setting in motion a feedback loop that results in the rapid dissipation of signal from $d\\theta $ .",
"The resulting snap back of the cart towards the center causes the pole to swing up.",
"As the pole falls and settles the same swing up behavior is repeated, and the controller is rewarded whenever the pole is upright.",
"As the search process continues, some of these controllers linger in the upright position longer than others, and by generation 128, the lingering duration is long enough for the pole to be kept balanced.",
"Though this more complicated balancing mechanism is less reliable under variable weights than the swing-up and centering behaviors, the more reliable behaviors ensure that the system recovers and tries again until a balanced state is found.",
"Notably, as these networks encode relationships and rely on tension between systems set against each other, their behavior is still consistent even with a wide range of shared weight values.",
"For video demonstrations of the policies learned at various developmental phases of the weight agnostic topologies, please refer to the supplementary website.",
"WANN controllers for BipedalWalker-v2 and CarRacing-v0 (Figure REF , page 1) are likewise remarkable in their simplicity and modularity.",
"The biped controller uses only 17 of the 25 possible inputs, ignoring many LIDAR sensors and knee speeds.",
"The WANN architecture not only solves the task without training the individual weights, but uses only 210 connections, an order of magnitude fewer than commonly used topologies (2804 connections used in the SOTA baseline [38]).",
"The architecture which encodes stable driving behavior in the car racer is also striking in its simplicity (Figure REF , right).",
"Only a sparsely connected two layer network and a single weight value is required to encode competent driving behavior.",
"While the SOTA baseline [39] also gave the hidden states of a pre-trained RNN world model, in addition to the VAE's representation to its controller, our controller operates on the VAE's latent space alone.",
"Nonetheless, it was able to develop a feed-forward controller that achieves a comparable score.",
"Future work will explore removing the feed-forward constraint from the search to allow WANNs to develop recurrent connections with memory states.",
"The networks shows in Figure REF (Page 1) were selected for both performance and readability.",
"In many cases a great deal of complexity is added for only minimal gains in performance, in these cases we preferred to showcase more elegant networks.",
"The final champion networks are shown in Figure REF .",
"Figure: Champion Networks for Continuous Control TasksLeft to Right (Number of Connections): Swing up (52), Biped (210), Car Racing (245)Shown in Figure (Page 1) are high performing, but simpler networks, chosen for clarity.",
"The three network architectures in this figure describe the champion networks whose results are reported.Classification Promising results on reinforcement learning tasks lead us to consider how widely a WANN approach can be applied.",
"WANNs which encode relationships between inputs are well suited to RL tasks: low-dimensional inputs coupled with internal states and environmental interaction allow discovery of reactive and adaptive controllers.",
"Classification, however, is a far less fuzzy and forgiving problem.",
"A problem where, unlike RL, design of architectures has long been a focus.",
"As a proof of concept, we investigate how WANNs perform on the MNIST dataset [60], an image classification task which has been a focus of human-led architecture search for decades [62], [15], [100].",
"Even in this high-dimensional classification task WANNs perform remarkably well (Figure REF , Left).",
"Restricted to a single weight value, WANNs are able to classify MNIST digits as well as a single layer neural network with thousands of weights trained by gradient descent.",
"The architectures created still maintain the flexibility to allow weight training, allowing further improvements in accuracy.",
"Figure: Classification Accuracy on MNIST.Left:WANNs instantiated with multiple weight values acting as an ensemble perform far better than when weights are sampled at random, and as well as a linear classifier with thousands of weights.Right: No single weight value has better accuracy on all digits.",
"That WANNs can be instantiated as several different networks has intriguing possibilities for the creation of ensembles.It is straight forward to sweep over the range of weights to find the value which performs best on the training set, but the structure of WANNs offers another intriguing possibility.",
"At each weight value the prediction of a WANN is different.",
"On MNIST this can be seen in the varied accuracy on each digit (Figure REF , Right).",
"Each weight value of the network can be thought of as a distinct classifier, creating the possibility of using one WANN with multiple weight values as a self-contained ensemble.",
"In the simplest ensemble approach, a collection of networks are created by instantiating a WANN with a range of weight values.",
"Each of these networks is given a single vote, and the ensemble classifies samples according to the category which received the most votes.",
"This approach yields predictions far more accurate than randomly selected weight values, and only slightly worse than the best possible weight.",
"That the result of this naive ensemble is successful is encouraging for experimenting with more sophisticated ensemble techniques when making predictions or searching for architectures.",
"Figure: MNIST classifier network (1849 connections)Not all neurons and connections are used to predict each digit.",
"Starting from the output connection for a particular digit, we can trace the sub-network and also identify which part of the input image is used for classifying each digit.",
"Please refer to the supplementary website for more detailed visualizations."
],
[
"Discussion and Future Work",
"In this work we introduced a method to search for simple neural network architectures with strong inductive biases.",
"Since networks are optimized to perform well using a shared weight over a range of values, this single parameter can easily be tuned to increase performance.",
"Individual weights can be further tuned from a best shared weight.",
"The ability to quickly fine-tune weights is useful in few-shot learning [22] and may find uses in continual learning [89] where agents continually acquire, fine-tune, and transfer skills throughout their lifespan, as in animals [125].",
"Inspired by the Baldwin effect [3], weight tolerant networks have long linked theories of evolution and learning in AI [1], [48], [106].",
"To develop a single WANN capable of encoding many different useful tasks in its environment, one might consider developing a WANN with a strong intrinsic bias for intrinsic motivation [101], [88], [90], and continuously optimize its architecture to perform well at pursuing novelty in an open-ended environment [66].",
"Such a WANN might encode, through a curiosity reward signal, a multitude of skills that can easily be fine-tuned for a particular downstream task in its environment later on.",
"While our approach learns network architectures of increasing complexity by adding connections, network pruning approaches find new architectures by their removal.",
"It is also possible to learn a pruned network capable of performing additional tasks without learning weights [75].",
"A concurrent work [127] to ours learns a supermask where the sub-network pruned using this mask performs well at image recognition even with randomly initialized weights – it is interesting that their approach achieves a similar range of performance on MNIST compared to ours.",
"While our search method is based on evolution, future work may extend the approach by incorporating recent ideas that formulate architecture search in a differentiable manner [71] to make the search more efficient.",
"The success of deep learning is attributed to our ability to train the weights of large neural networks that consist of well-designed building blocks on large datasets, using gradient descent.",
"While much progress has been made, there are also limitations, as we are confined to the space of architectures that gradient descent is able to train.",
"For instance, effectively training models that rely on discrete components [52], [32] or utilize adaptive computation mechanisms [31] with gradient-based methods remain a challenging research area.",
"We hope this work will encourage further research that facilitates the discovery of new architectures that not only possess inductive biases for practical domains, but can also be trained with algorithms that may not require gradient computation.",
"That the networks found in this work do not match the performance of convolutional neural networks is not surprising.",
"It would be an almost embarrassing achievement if they did.",
"For decades CNN architectures have been refined by human scientists and engineers – but it was not the reshuffling of existing structures which originally unlocked the capabilities of CNNs.",
"Convolutional layers were themselves once novel building blocks, building blocks with strong biases toward vision tasks, whose discovery and application have been instrumental in the incredible progress made in deep learning.",
"The computational resources available to the research community have grown significantly since the time convolutional neural networks were discovered.",
"If we are devoting such resources to automated discovery and hope to achieve more than incremental improvements in network architectures, we believe it is also worth trying to discover new building blocks, not just their arrangements.",
"Finally, we see similar ideas circulating in the neuroscience community.",
"A recent neuroscience commentary, “What artificial neural networks can learn from animal brains” [125] provides a critique of how learning (and also meta-learning) is currently implemented in artificial neural networks.",
"Zador [125] highlights the stark contrast with how biological learning happens in animals: “The first lesson from neuroscience is that much of animal behavior is innate, and does not arise from learning.",
"Animal brains are not the blank slates, equipped with a general purpose learning algorithm ready to learn anything, as envisioned by some AI researchers; there is strong selection pressure for animals to restrict their learning to just what is needed for their survival.” [125] This paper is strongly motivated towards these goals of blending innate behavior and learning, and we believe it is a step towards addressing the challenge posed by Zador.",
"We hope this work will help bring neuroscience and machine learning communities closer together to tackle these challenges."
],
[
"Acknowledgments",
"We would like to thank our three reviewers for their helpful comments, and also express gratitude to Douglas Eck, Geoffrey Hinton, Anja Austermann, Jeff Dean, Luke Metz, Ben Poole, Jean-Baptiste Mouret, Michiel Adriaan Unico Bacchiani, Heiga Zen, and Alex Lamb for their thoughtful feedback."
],
[
"Code Release",
"We release a general purpose tool, not only to facilitate reproduction, but also for further research in this direction.",
"Our NumPy [119] implementation of NEAT [110] supports MPI [33] and OpenAI Gym [11] environments.",
"All code used to run these experiments, in addition to the best networks found in each run, is referenced in the interactive article: https://weightagnostic.github.io/"
],
[
"“Have you also thought about trying ... ?”",
"In this section, we highlight things that we have attempted, but did not explore in sufficient depth."
],
[
"Searching for network architecture using a single weight rather than range of weights.",
"We experimented with setting all weights to a single fixed value, e.g.",
"0.7, and saw that the search is faster and the end result better.",
"However, if we then nudge that value by a small amount, to say 0.6, the network fails completely at the task.",
"By training on a wide range of weight parameters, akin to training on uniform samples weight values, networks were able to perform outside of the training values.",
"In fact, the best performing values were outside of this training set."
],
[
"Searching for network architecture using random weights for each connection.",
"This was the first thing we tried, and did not have much luck.",
"We tried quite a few things to get this to work–at one point it seemed like we finally had it, poles were balanced and walkers walking, but it turned out to be a bug in the code!",
"Instead of setting all of the weights to different random values we had set all of the weights to the same random value.",
"It was in the course of trying to understand this result that we began to view and approach the problem through the lens of MDL and AIT."
],
[
"Adding noise to the single weight values.",
"We experimented adding Gaussian noise to the weight values so that each weight would be different, but vary around a set mean at each rollout.",
"We only did limited experiments on swing-up and found no large difference, except with very high levels of noise where it performed poorly.",
"Our intuition is that adding noise would make the final topologies even more robust to changes in weight value, but at the cost of making the evaluation of topologies more noisy (or more rollouts to mitigate the variance between trials).",
"With no clear benefit we chose to keep the approach as conceptually simple as possible–but see this as a logical next step towards making the networks more weight tolerant."
],
[
"Using backpropagation to fine-tune weights of a WANN.",
"We explored the use of autograd packages such as JAX [24] to fine-tune individual weights of WANNs for the MNIST experiment.",
"Performance improved, but ultimately we find that black-box optimization methods such as CMA-ES and population-based REINFORCE can find better solutions for the WANN architectures evolved for MNIST, suggesting that the various activations proposed by the WANN search algorithm may have produced optimization landscapes that are more difficult for gradient-based methods to traverse compared to standard ReLU-based deep network architectures."
],
[
"Why did you choose to use many different activation functions in the same network? Why not just ReLU? Wouldn't too many activations break biological plausibility?",
"Without concrete weight values to lean on, we instead relied on encoding relationships between inputs into the network.",
"This could have been done with ReLUs or sigmoids, but including relationships such as symmetry and repetition allow for more compact networks.",
"We didn't do much experimentation, but our intuition is that the variety of activations is key.",
"That is not to say that all of them are necessary, but we're not confident this could have been accomplished with only linear activations.",
"As for biological corollaries, we're not going to claim that a cosine activation is an accurate model of a how neurons fire–but don't think a feed forward network of instantaneously communicating sigmoidal neurons would be any more biologically plausible."
],
[
"MNIST",
"The MNIST version used in this paper is a downsampled version, reducing the digits from [28x28] to [16x16], and deskewed using the OpenCV library[8].",
"The best MNIST network weight was chosen as the network with the highest accuracy on the training set.",
"To fit into our existing approach MNIST classification is reframed as a reinforcement learning problem.",
"Each sample in MNIST is downsampled to a 16x16 image, deskewed, and pixel intensity normalized between 0 and 1.",
"WANNs are created with input for each of the 256 pixels and one output for each of the 10 digits.",
"At each evaluation networks are fed 1000 samples randomly selected from the training set, and given reward based on the softmax cross entropy.",
"Networks are tested with a variety of shared weight values, maximizing performance over all weights while minimizing the number of connections."
],
[
"Hyperparameters and Setup",
"All experiments but those on Car Racing were performed used 96 core machines on the Google Cloud Platform.",
"As evaluation of the population is embarrassingly parallel, populations were sized as multiples of 96 to make efficient use of all processors.",
"Car Racing was performed on a 64 core machine and the population size used reflects this.",
"The code and setup of the VAE for the Car Racing task is taken from [39], were a VAE with a latent size of 16 was trained following the same procedure as [39].",
"Tournament sizes were scaled in line with the population size.",
"The number of generations were determined after initial experiments to ensure that a majority of runs would converge.",
"Table: NO_CAPTION"
],
[
"Results over multiple independent search runs",
"For each task a WANN search was run 9 times.",
"At regular intervals the network in the population with the best mean performance was compared to that with the previously best found network.",
"If the newer network had a higher mean, the network was evaluated 96 or 64 times (depending on the number of processors on the machine), and if the mean of those evaluations was better than the previous best network, it was kept as the new `best' network.",
"These best networks were kept only for record keeping and did not otherwise interact with the population.",
"These best networks at the end of each run were reevaluated thirty times on each weight in the series $[-2,-1.5,-1,-0.5,0.5,1,1.5,2]$ and the network with the best mean chosen as the champion for more intensive analysis and examination.",
"Shown below are the results of these initial tests, both as individual runs and as distributions of performance over weights.",
"Figure: Swing-up Performance over Multiple Runs.Left: Performance per weight value of best network found in each of 9 runs.Right: Average performance of best networks found at end of each of 9 runs.",
"Performance is shown by top weight, top quartile of weights, top half of weights, and over all weights.Figure: Biped Performance over Multiple Runs.Left: Performance per weight value of best network found in each of 9 runs.Right: Average performance of best networks found at end of each of 9 runs.",
"Performance is shown by top weight, top quartile of weights, top half of weights, and over all weights.Figure: Car Racing Performance over Multiple Runs.Left: Performance per weight value of best network found in each of 9 runs.Right: Average performance of best networks found at end of each of 9 runs.",
"Performance is shown by top weight, top quartile of weights, top half of weights, and over all weights."
],
[
"Optimizing for individual weight parameters",
"In our experiments, we also fine-tuned individual weight parameters for the champion networks found to measure the performance impact of further training.",
"For this, we used population-based REINFORCE, as in Section 6 of [123].",
"Our specific approach is based on the open source estool [37] implementation of population-based REINFORCE.",
"We use a population size of 384, and each agent performs the task 16 times with different initial random seeds for Swing Up Cartpole and Bipedal Walker.",
"The agent’s reward signal used by the policy gradient method is the average reward of the 16 rollouts.",
"For Car Racing, due to the extra computation time required, we instead use a population size of 64 and the average cumulative reward of 4 rollouts to calculate the reward signal.",
"All models trained for 3000 generations.",
"All other parameters are set to the default settings of estool [37].",
"For MNIST, we use the negative of the cross entropy loss as the reward signal, and optimize directly on the training set with population-based REINFORCE."
],
[
"Fixed Topology Baselines",
"For Bipedal Walker, we used the model and architecture available from estool [37] as our baseline.",
"To our knowledge, this baseline currently, at the time of writing, achieves the state-of-the-art average cumulative score (over 100 random rollouts) on Bipedal Walker as reported in [38].",
"In the Swing Up Cartpole task, we trained a baseline controller with 1 hidden layer of 10 units (71 weight parameters), using the same training methodology as the one used to produce SOTA results for the Bipedal Walker task mentioned earlier.",
"We experimented with a larger number of nodes in the hidden layer, and an extra hidden layer, but did not see meaningful differences in performance.",
"For the Car Racing baseline, we used the code and model provided in [39] and treated the 867 parameters of the controller as free weight parameters, while keeping the parameters of the pre-trained VAE and RNN fixed.",
"As of writing, the average cumulative score (over 100 random rollouts) produced by [39] for Car Racing is currently the state-of-the-art.",
"As mentioned in the main text, for simplicity, the WANN controller has access only to the pre-trained VAE, and not to the RNN."
]
] | 1906.04358 |
[
[
"Beyond Folklore: A Scaling Calculus for the Design and Initialization of\n ReLU Networks"
],
[
"Abstract We propose a system for calculating a \"scaling constant\" for layers and weights of neural networks.",
"We relate this scaling constant to two important quantities that relate to the optimizability of neural networks, and argue that a network that is \"preconditioned\" via scaling, in the sense that all weights have the same scaling constant, will be easier to train.",
"This scaling calculus results in a number of consequences, among them the fact that the geometric mean of the fan-in and fan-out, rather than the fan-in, fan-out, or arithmetic mean, should be used for the initialization of the variance of weights in a neural network.",
"Our system allows for the off-line design & engineering of ReLU neural networks, potentially replacing blind experimentation."
],
[
"Introduction",
"The design of neural networks is often considered a black-art, driven by trial and error rather than foundational principles.",
"This is exemplified by the success of recent architecture random-search techniques , , which take the extreme of applying no human guidance at all.",
"Although as a field we are far from fully understanding the nature of learning and generalization in neural networks, this does not mean that we should proceed blindly.",
"In this work, we define a scaling quantity $\\gamma _{l}$ for each layer $l$ that approximates two quantities of interest when considering the optimization of a neural network: The ratio of the gradient to the weights, and the average squared singular value of the corresponding diagonal block of the Hessian for layer $l$ .",
"This quantity is easy to compute from the (non-central) second moments of the forward-propagated values and the (non-central) second moments of the backward-propagated gradients.",
"We argue that networks that have constant $\\gamma _{l}$ are better conditioned than those that do not, and we analyze how common layer types affect this quantity.",
"We call networks that obey this rule preconditioned neural networks.",
"As an example of some of the possible applications of our theory, we: Propose a principled weight initialization scheme that can often provide an improvement over existing schemes; Show which common layer types automatically result in well-conditioned networks; Show how to improve the conditioning of common structures such as bottlenecked residual blocks by the addition of fixed scaling constants to the network."
],
[
"Notation",
"Consider a neural network mapping $x_{0}$ to $x_{L}$ made up of $L$ layers.",
"These layers may be individual operations or blocks of operations.",
"During training, a loss function is computed for each minibatch of data, and the gradient of the loss is back-propagated to each layer $l$ and weight of the network.",
"We prefix each quantity with $\\Delta $ to represent the back-propagated gradient of that quantity.",
"We assume a batch-size of 1 in our calculations, although all conclusions hold using mini-batches as well.",
"Each layer's input activations are represented by a tensor $x_{l}:n_{l}\\times \\rho _{l}\\times \\rho _{l}$ made up of $n_{l}$ channels, and spatial dimensions $\\rho _{l}\\times \\rho _{l}$ , assumed to be square for simplicity (results can be adapted to the rectangular case by using $h_{l}w_{l}$ in place of $\\rho _{l}$ everywhere)."
],
[
"A model of ReLU network dynamics",
"Our scaling calculus requires the use of simple approximations of the dynamics of neural networks, in the same way that simplifications are used in physics to make approximate calculations, such as the assumption of zero-friction or ideal gasses.",
"These assumptions constitute a model of the behavior of neural networks that allows for easy calculation of quantities of interest, while still being representative enough of the real dynamics.",
"To this end, we will focus in this work on the behavior of networks at initialization.",
"Furthermore, we will make strong assumptions on the statistics of forward and backward quantities in the network.",
"These assumptions include: The input to layer $l$ , denoted $x_{l}$ , is a random tensor assumed to contain i.i.d entries.",
"We represent the element-wise uncentered 2nd moment by $E[x_{l}^{2}]$ .",
"The back-propagated gradient of $x_{l}$ is $\\Delta x_{l}$ and is assumed to be uncorrelated with $x_{l}$ and iid.",
"We represent the uncentered 2nd-moment of $\\Delta x_{l}$ by $E[\\Delta x_{l}^{2}]$ .",
"All weights in the network are initialized i.i.d from a centered, symmetric distribution.",
"All bias terms are initialized as zero.",
"Our calculations rely heavily on the uncentered second moments rather than the variance of weights and gradients.",
"This is a consequence of the behavior of the ReLU activation, which zeros out entries.",
"The effect of this zeroing operation is simple when considering uncentered second moments under a symmetric input distribution, as half of the entries will be zeroed, resulting in a halving of the uncentered second moment.",
"In contrast, expressing the same operation in terms of variance is complicated by the fact that the mean after application of the ReLU is distribution-dependent.",
"We will refer to the uncentered second moment just as the “second moment” henceforth."
],
[
"Activation and layer scaling factors",
"Definition 1 The key quantity in our calculus is the activation scaling factor $\\varsigma _{l}$ , of the input activations for a layer $l$ , which we define as: $\\varsigma _{l}=n_{l}\\rho _{l}^{2}E[\\Delta x_{l}^{2}]E[x_{l}^{2}].$ This quantity arises due to its utility in computing other quantities of interest in the network, such as the scaling factors for the weights of convolutional and linear layers.",
"In ReLU networks, many, but not all operations maintain this quantity in the sense that $\\varsigma _{l}=\\varsigma _{l+1}$ for a layer $x_{l+1}=F(x_{l})$ with operation $F$ , under the assumptions of Section .",
"Table REF contains a list of common operations and indicates if they maintain scaling.",
"As an example, consider adding a simple scaling layer of the form $x_{l+1}=\\sqrt{2}x_{l}$ which doubles the second moment during the forward pass and doubles the backward second moment during back-propagation.",
"We can see that: $\\varsigma _{l+1} & =n_{l+1}\\rho _{l+1}^{2}E[\\Delta x_{l+1}^{2}]E[x_{l+1}^{2}]\\\\& =n_{l}\\rho _{l}^{2}\\frac{1}{2}E[\\Delta x_{l}^{2}]\\cdot 2E[x_{l}^{2}]=\\varsigma _{l}$ Our analysis in our work is focused on ReLU networks primarily due to the fact that ReLU non-linearities maintain this scaling factor.",
"Table: Details of LibSVM dataset input/output scaling"
]
] | 1906.04267 |
[
[
"Hypothesis Testing under Subjective Priors and Costs as a Signaling Game"
],
[
"Abstract Many communication, sensor network, and networked control problems involve agents (decision makers) which have either misaligned objective functions or subjective probabilistic models.",
"In the context of such setups, we consider binary signaling problems in which the decision makers (the transmitter and the receiver) have subjective priors and/or misaligned objective functions.",
"Depending on the commitment nature of the transmitter to his policies, we formulate the binary signaling problem as a Bayesian game under either Nash or Stackelberg equilibrium concepts and establish equilibrium solutions and their properties.",
"We show that there can be informative or non-informative equilibria in the binary signaling game under the Stackelberg and Nash assumptions, and derive the conditions under which an informative equilibrium exists for the Stackelberg and Nash setups.",
"For the corresponding team setup, however, an equilibrium typically always exists and is always informative.",
"Furthermore, we investigate the effects of small perturbations in priors and costs on equilibrium values around the team setup (with identical costs and priors), and show that the Stackelberg equilibrium behavior is not robust to small perturbations whereas the Nash equilibrium is."
],
[
"INTRODUCTION",
"In many decentralized and networked control problems, decision makers have either misaligned criteria or have subjective priors, which necessitates solution concepts from game theory.",
"For example, detecting attacks, anomalies, and malicious behavior with regard to security in networked control systems can be analyzed under a game theoretic perspective, see e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].",
"In this paper, we consider signaling games that refer to a class of two-player games of incomplete information in which an informed decision maker (transmitter or encoder) transmits information to another decision maker (receiver or decoder) in the hypothesis testing context.",
"In the following, we first provide the preliminaries and introduce the problems considered in the paper, and present the related literature briefly."
],
[
"Notation",
"We denote random variables with capital letters, e.g., $Y$ , whereas possible realizations are shown by lower-case letters, e.g., $y$ .",
"The absolute value of scalar $y$ is denoted by $|y|$ .",
"The vectors are denoted by bold-faced letters, e.g., $\\mathbf {y}$ .",
"For vector $\\mathbf {y}$ , $\\mathbf {y}^T$ denotes the transpose and $\\Vert \\mathbf {y}\\Vert $ denotes the Euclidean ($L_2$ ) norm.",
"${1}_{\\lbrace D\\rbrace }$ represents the indicator function of an event $D$ , $\\oplus $ stands for the exclusive-or operator, $\\mathcal {Q}$ denotes the standard $\\mathcal {Q}$ -function; i.e., $\\mathcal {Q}(x)={1\\over \\sqrt{2\\pi }}\\int _{x}^{\\infty }\\exp \\lbrace -{t^2\\over 2}\\rbrace {\\rm {d}}t$ , and the sign of $x$ is defined as $ \\text{sgn}(x)={\\left\\lbrace \\begin{array}{ll}-1 & \\text{if }x < 0 \\\\0 & \\text{if }x=0 \\\\1 & \\text{if }x>0\\end{array}\\right.",
"}\\,.$"
],
[
"Preliminaries",
"Consider a binary hypothesis-testing problem: $\\begin{split}\\mathcal {H}_0 : Y = S_0 + N \\;, \\\\\\mathcal {H}_1 : Y = S_1 + N \\;,\\end{split}$ where $Y$ is the observation (measurement) that belongs to the observation set $\\Gamma =\\mathbb {R}$ , $S_0$ and $S_1$ denote the deterministic signals under hypothesis $\\mathcal {H}_0$ and hypothesis $\\mathcal {H}_1$ , respectively, and $N$ represents Gaussian noise; i.e., $N \\sim \\mathcal {N} (0,\\sigma ^2)$ .",
"In the Bayesian setup, it is assumed that the prior probabilities of $\\mathcal {H}_0$ and $\\mathcal {H}_1$ are available, which are denoted by $\\pi _0$ and $\\pi _1$ , respectively, with $\\pi _0+\\pi _1=1$ .",
"In the conventional Bayesian framework, the aim of the receiver is to design the optimal decision rule (detector) based on $Y$ in order to minimize the Bayes risk, which is defined as [14] $r(\\delta ) = \\pi _0 R_0(\\delta ) + \\pi _1 R_1(\\delta ) \\;,$ where $\\delta $ is the decision rule, and $R_i(\\cdot )$ is the conditional risk of the decision rule when hypothesis $\\mathcal {H}_i$ is true for $i\\in \\lbrace 0,1\\rbrace $ .",
"In general, a decision rule corresponds to a partition of the observation set $\\Gamma $ into two subsets $\\Gamma _0$ and $\\Gamma _1$ , and the decision becomes $\\mathcal {H}_i$ if the observation $y$ belongs to $\\Gamma _i$ , where $i\\in \\lbrace 0,1\\rbrace $ .",
"The conditional risks in (REF ) can be calculated as $R_i(\\delta ) = C_{0i}\\mathsf {P}_{0i} + C_{1i} \\mathsf {P}_{1i} \\;,$ for $i\\in \\lbrace 0,1\\rbrace $ , where $C_{ji}\\ge 0$ is the cost of deciding for $\\mathcal {H}_j$ when $\\mathcal {H}_i$ is true, and $\\mathsf {P}_{ji}=\\mathsf {Pr}(y\\in \\Gamma _j|\\mathcal {H}_i)$ represents the conditional probability of deciding for $\\mathcal {H}_j$ given that $\\mathcal {H}_i$ is true, where $i,j\\in \\lbrace 0,1\\rbrace $ [14].",
"It is well-known that the optimal decision rule $\\delta $ which minimizes the Bayes risk is the following test, known as the likelihood ratio test (LRT): $\\delta : \\Bigg \\lbrace \\pi _1 (C_{01}-C_{11}) p_1(y) \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\pi _0 (C_{10}-C_{00})p_0(y) \\;,$ where $p_i(y)$ represents the probability density function (PDF) of $Y$ under $\\mathcal {H}_i$ for $i\\in \\lbrace 0,1\\rbrace $ [14].",
"If the transmitter and the receiver have the same objective function specified by (REF ) and (REF ), then the signals can be designed to minimize the Bayes risk corresponding to the decision rule in (REF ).",
"This leads to a conventional formulation which has been studied intensely in the literature [14], [15].",
"On the other hand, it may be the case that the transmitter and the receiver can have non-aligned Bayes risks.",
"In particular, the transmitter and the receiver may have different objective functions or priors: Let $C^t_{ji}$ and $C^r_{ji}$ represent the costs from the perspective of the transmitter and the receiver, respectively, where $i,j\\in \\lbrace 0,1\\rbrace $ .",
"Also let $\\pi _i^t$ and $\\pi _i^r$ for $i\\in \\lbrace 0,1\\rbrace $ denote the priors from the perspective of the transmitter and the receiver, respectively, with $\\pi _0^j+\\pi _1^j=1$ , where $j\\in \\lbrace t,r\\rbrace $ .",
"Here, from transmitter's and receiver's perspectives, the priors are assumed to be mutually absolutely continuous with respect to each other; i.e., $\\pi _i^t=0\\Rightarrow \\pi _i^r=0$ and $\\pi _i^r=0\\Rightarrow \\pi _i^t=0$ for $i\\in \\lbrace 0,1\\rbrace $ .",
"This condition assures that the impossibility of any hypothesis holds for both the transmitter and the receiver simultaneously.",
"The aim of the transmitter is to perform the optimal design of signals $\\mathcal {S}=\\lbrace S_0,S_1\\rbrace $ to minimize his Bayes risk; whereas, the aim of the receiver is to determine the optimal decision rule $\\delta $ over all possible decision rules $\\Delta $ to minimize his Bayes risk.",
"The Bayes risks are defined as follows for the transmitter and the receiver: $r^j(\\mathcal {S},\\delta ) = \\pi _0^j (C^j_{00} \\mathsf {P}_{00} + C^j_{10} \\mathsf {P}_{10}) + \\pi _1^j (C^j_{01} \\mathsf {P}_{01} + C^j_{11} \\mathsf {P}_{11})\\;,$ for $j\\in \\lbrace t,r\\rbrace $ .",
"Here, the transmitter performs the optimal signal design problem under the power constraint below: $\\mathbb {S}\\triangleq \\lbrace \\mathcal {S}=\\lbrace S_0,S_1\\rbrace :| S_0 | ^2 \\le P_0 \\,,\\; | S_1 | ^2 \\le P_1\\rbrace \\;,$ where $P_0$ and $P_1$ denote the power limits [14].",
"Although the transmitter and the receiver act sequentially in the game as described above, how and when the decisions are made and the nature of the commitments to the announced policies significantly affect the analysis of the equilibrium structure.",
"Here, two different types of equilibria are investigated: Nash equilibrium: the transmitter and the receiver make simultaneous decisions.",
"Stackelberg equilibrium : the transmitter and the receiver make sequential decisions where the transmitter is the leader and the receiver is the follower.",
"In this paper, the terms Nash game and the simultaneous-move game will be used interchangeably, and similarly, the Stackelberg game and the leader-follower game will be used interchangeably.",
"In the simultaneous-move game, the transmitter and the receiver announce their policies at the same time, and a pair of policies $(\\mathcal {S}^*, \\delta ^*)$ is said to be a Nash equilibrium [16] if $\\begin{split}r^t(\\mathcal {S}^*, \\delta ^*) &\\le r^t(\\mathcal {S}, \\delta ^*) \\quad \\forall \\,\\mathcal {S} \\in \\mathbb {S}\\;, \\\\r^r(\\mathcal {S}^*, \\delta ^*) &\\le r^r(\\mathcal {S}^*, \\delta ) \\quad \\forall \\,\\delta \\in \\Delta \\;.\\end{split}$ As noted from the definition in (REF ), under the Nash equilibrium, each individual player chooses an optimal strategy given the strategies chosen by the other player.",
"However, in the leader-follower game, the leader (transmitter) commits to and announces his optimal policy before the follower (receiver) does, the follower observes what the leader is committed to before choosing and announcing his optimal policy, and a pair of policies $(\\mathcal {S}^*, \\delta ^*_{\\mathcal {S}^*})$ is said to be a Stackelberg equilibrium [16] if $\\begin{split}&r^t(\\mathcal {S}^*, \\delta ^*_{\\mathcal {S}^*}) \\le r^t(\\mathcal {S}, \\delta ^*_\\mathcal {S}) \\quad \\forall \\,\\mathcal {S} \\in \\mathbb {S}\\;, \\\\&\\text{where } \\delta ^*_\\mathcal {S} \\text{ satisfies} \\\\&r^r(\\mathcal {S}, \\delta ^*_\\mathcal {S}) \\le r^r(\\mathcal {S}, \\delta _\\mathcal {S}) \\quad \\forall \\,\\delta _\\mathcal {S} \\in \\Delta \\,.\\end{split}$ As observed from the definition in (REF ), the receiver takes his optimal action $\\delta ^*_\\mathcal {S}$ after observing the policy of the transmitter $\\mathcal {S}$ .",
"Further, in the Stackelberg game (also often called Bayesian persuasion games in the economics literature, see [17] for a detailed review), the leader cannot backtrack on his commitment, but he has a leadership role since he can manipulate the follower by anticipating the actions of the follower.",
"If an equilibrium is achieved when $\\mathcal {S}^*$ is non-informative (e.g., $S_0^*=S_1^*$ ) and $\\delta ^*$ uses only the priors (since the received message is useless), then we call such an equilibrium a non-informative (babbling) equilibrium [18]."
],
[
"Two Motivating Setups",
"We present two different scenarios that fit into the binary signaling context discussed here and revisit these setups throughout the paperBesides the setups discussed here (and the throughout the paper), the deception game can also be modeled as follows.",
"In the deception game, the transmitter aims to fool the receiver by sending deceiving messages, and this goal can be realized by adjusting the transmitter costs as $C^t_{00}>C^t_{10}$ and $C^t_{11}>C^t_{01}$ ; i.e, the transmitter is penalized if the receiver correctly decodes the original hypothesis.",
"Similar to the standard communication setups, the goal of the receiver is to truly identify the hypothesis; i.e., $C^r_{00}<C^r_{10}$ and $C^r_{11}<C^r_{01}$ .."
],
[
"Subjective Priors",
"In almost all practical applications, there is some mismatch between the true and an assumed probabilistic system/data model, which results in performance degradation.",
"This performance loss due to the presence of mismatch has been studied extensively in various setups (see e.g.,[19], [20], [21] and references therein).",
"In this paper, we have a further salient aspect due to decentralization, where the transmitter and the receiver have a mismatch.",
"We note that in decentralized decision making, there have been a number of studies on the presence of a mismatch in the priors of decision makers [22], [23], [24].",
"In such setups, even when the objective functions to be optimized are identical, the presence of subjective priors alters the formulation from a team problem to a game problem (see [25] for a comprehensive literature review on subjective priors also from a statistical decision making perspective).",
"With this motivation, we will consider a setup where the transmitter and the receiver have different priors on the hypotheses $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , and the costs of the transmitter and the receiver are identical.",
"In particular, from transmitter's perspective, the priors are $\\pi _0^t$ and $\\pi _1^t$ , whereas the priors are $\\pi _0^r$ and $\\pi _1^r$ from receiver's perspective, and $C_{ji}=C^t_{ji}=C^r_{ji}$ for $i,j\\in \\lbrace 0,1\\rbrace $ .",
"We will investigate equilibrium solutions for this setup throughout the paper."
],
[
"Biased Transmitter Cost",
"A further application will be for a setup where the transmitter and the receiver have misaligned objective functions.",
"Consider a binary signaling game in which the transmitter encodes a random binary signal $x=i$ as $\\mathcal {H}_i$ by choosing the corresponding signal level $S_i$ for $i\\in \\lbrace 0,1\\rbrace $ , and the receiver decodes the received signal $y$ as $u=\\delta (y)$ .",
"Let the priors from the perspectives of the transmitter and the receiver be the same; i.e., $\\pi _i=\\pi _i^t=\\pi _i^r$ for $i\\in \\lbrace 0,1\\rbrace $ , and the Bayes risks of the transmitter and the receiver be defined as $r^t(\\mathcal {S},\\delta )=\\mathbb {E}[{1}_{\\lbrace 1=(x\\oplus u\\oplus b)\\rbrace }]$ and $r^r(\\mathcal {S},\\delta )=\\mathbb {E}[{1}_{\\lbrace 1=(x\\oplus u)\\rbrace }]$ , respectively, where $b$ is a random variable with a Bernoulli distribution; i.e., $\\alpha \\triangleq \\mathsf {Pr}(b=0)=1-\\mathsf {Pr}(b=1)$ , and $\\alpha $ can be translated as the probability that the Bayes risks (objective functions) of the transmitter and the receiver are aligned.",
"Then, the following relations can be observed: $r^t(\\mathcal {S},\\delta )&=\\mathbb {E}[{1}_{\\lbrace 1=(x\\oplus u\\oplus b)\\rbrace }]=\\alpha (\\pi _0\\mathsf {P}_{10}+\\pi _1\\mathsf {P}_{01})+(1-\\alpha )(\\pi _0\\mathsf {P}_{00}+\\pi _1\\mathsf {P}_{11}) \\\\&\\Rightarrow \\quad C^t_{01}=C^t_{10}=\\alpha \\text{ and } C^t_{00}=C^t_{11}=1-\\alpha \\;, \\\\r^r(\\mathcal {S},\\delta )&=\\mathbb {E}[{1}_{\\lbrace 1=(x\\oplus u)\\rbrace }]=\\pi _0\\mathsf {P}_{10}+\\pi _1\\mathsf {P}_{01} \\\\&\\Rightarrow \\quad C^r_{01}=C^r_{10}=1 \\text{ and } C^r_{00}=C^r_{11}=0 \\;.$ Note that, in the formulation above, the misalignment between the Bayes risks of the transmitter and the receiver is due to the presence of the bias term $b$ (i.e., the discrepancy between the Bayes risks of the transmitter and the receiver) in the Bayes risk of the transmitter.",
"This can be viewed as an analogous setup to what was studied in a seminal work due to Crawford and Sobel [18], who obtained the striking result that such a bias term in the objective function of the transmitter may have a drastic effect on the equilibrium characteristics; in particular, under regularity conditions, all equilibrium policies under a Nash formulation involve information hiding; for some extensions under quadratic criteria please see [26] and [27]."
],
[
"Related Literature",
"In game theory, Nash and Stackelberg equilibria are drastically different concepts.",
"Both equilibrium concepts find applications depending on the assumptions on the leader, that is, the transmitter, in view of the commitment conditions.",
"Stackelberg games are commonly used to model attacker-defender scenarios in security domains [28].",
"In many frameworks, the defender (leader) acts first by committing to a strategy, and the attacker (follower) chooses how and where to attack after observing defender's choice.",
"However, in some situations, security measures may not be observable for the attacker; therefore, a simultaneous-move game is preferred to model such situations; i.e., the Nash equilibrium analysis is needed [29].",
"These two concepts may have equilibria that are quite distinct: As discussed in [26], [17], in the Nash equilibrium case, building on [18], equilibrium properties possess different characteristics as compared to team problems; whereas for the Stackelberg case, the leader agent is restricted to be committed to his announced policy, which leads to similarities with team problem setups [30], [27], [31].",
"However, in the context of binary signaling, we will see that the distinction is not as sharp as it is in the case of quadratic signaling games [26], [17].",
"Standard binary hypothesis testing has been extensively studied over several decades under different setups [14], [15], which can also be viewed as a decentralized control/team problem involving a transmitter and a receiver who wish to minimize a common objective function.",
"However, there exist many scenarios in which the analysis falls within the scope of game theory; either because the goals of the decision makers are misaligned, or because the probabilistic model of the system is not common knowledge among the decision makers.",
"A game theoretic perspective can be utilized for hypothesis testing problem for a variety of setups.",
"For example, detecting attacks, anomalies, and malicious behavior in network security can be analyzed under the game theoretic perspective [2], [3], [4], [5], [6].",
"In this direction, the hypothesis testing and the game theory approaches can be utilized together to investigate attacker-defender type applications [7], [8], [9], [11], [12], [13], [10], multimedia source identification problems [32], inspection games [33], [34], [35], and deception games[36].",
"In [8], a Nash equilibrium of a zero-sum game between Byzantine (compromised) nodes and the fusion center (FC) is investigated.",
"The strategy of the FC is to set the local sensor thresholds that are utilized in the likelihood-ratio tests, whereas the strategy of Byzantines is to choose their flipping probability of the bit to be transmitted.",
"In [9], a zero-sum game of a binary hypothesis testing problem is considered over finite alphabets.",
"The attacker has control over the channel, and the randomized decision strategy is assumed for the defender.",
"The dominant strategies in Neyman-Pearson and Bayesian setups are investigated under the Nash assumption.",
"The authors of [34], [35] investigate both Nash and Stackelberg equilibria of a zero-sum inspection game where an inspector (environmental agency) verifies, with the help of randomly sampled measurements, whether the amount of pollutant released by the inspectee (management of an industrial plant) is higher than the permitted ones.",
"The inspector chooses a false alarm probability $\\alpha $ , and determines his optimal strategy over the set of all statistical tests with false alarm probability $\\alpha $ to minimize the non-detection probability.",
"On the other side, the inspectee chooses the signal levels (violation strategies) to maximize the non-detection probability.",
"[10] considers a complete-information zero-sum game between a centralized detection network and a jammer equipped with multiple antennas and investigates pure strategy Nash equilibria for this game.",
"The fusion center (FC) chooses the optimal threshold of a single-threshold rule in order to minimize his error probability based on the observations coming from multiple sensors, whereas the jammer disrupts the channel in order to maximize FC's error probability under instantaneous power constraints.",
"However, unlike the setups described above, in this work, we assume an additive Gaussian noise channel, and in the game setup, a Bayesian hypothesis testing setup is considered in which the transmitter chooses signal levels to be transmitted and the receiver determines the optimal decision rule.",
"Both players aim to minimize their individual Bayes risks, which leads to a nonzero-sum game.",
"[36] investigates the perfect Bayesian Nash equilibrium (PBNE) solution of a cyber-deception game in which the strategically deceptive interaction between the deceivee (privately-informed player, sender) and the deceiver (uninformed player, receiver) are modeled by a signaling game framework.",
"It is shown that the hypothesis testing game admits no separating (pure, fully informative) equilibria, there exist only pooling and partially-separating-pooling equilibria; i.e., non-informative equilibria.",
"Note that, in [36], the received message is designed by the deceiver (transmitter), whereas we assume a Gaussian channel between the players.",
"Further, the belief of the receiver (deceivee) about the priors is affected by the design choices of the transmitter (deceiver), unlike this setup, in which constant beliefs are assumed.",
"Within the scope of the discussions above, the binary signaling problem investigated here can be motivated under different application contexts: subjective priors and the presence of a bias in the objective function of the transmitter compared to that of the receiver.",
"In the former setup, players have a common goal but subjective prior information, which necessarily alters the setup from a team problem to a game problem.",
"The latter one is the adaptation of the biased objective function of the transmitter in [18] to the binary signaling problem considered here.",
"We discuss these further in the following."
],
[
"Contributions",
"The main contributions of this paper can be summarized as follows: (i) A game theoretic formulation of the binary signaling problem is established under subjective priors and/or subjective costs.",
"(ii) The corresponding Stackelberg and Nash equilibrium policies are obtained, and their properties (such as uniqueness and informativeness) are investigated.",
"It is proved that an equilibrium is almost always informative for a team setup, whereas in the case of subjective priors and/or costs, it may cease to be informative.",
"(iii) Furthermore, robustness of equilibrium solutions to small perturbations in the priors or costs are established.",
"It is shown that, the game equilibrium behavior around the team setup is robust under the Nash assumption, whereas it is not robust under the Stackelberg assumption.",
"(iv) For each of the results, applications to two motivating setups (involving subjective priors and the presence of a bias in the objective function of the transmitter) are presented.",
"In the conference version of this study [1], some of the results (in particular, the Nash and Stackelberg equilibrium solutions and their robustness properties) appear without proofs.",
"Here we provide the full proofs of the main theorems and also include the continuity analysis of the equilibrium.",
"Furthermore, the setup and analysis presented in [1] are extended to the multi-dimensional case and partially to the case with an average power constraint.",
"The remainder of the paper is organized as follows.",
"The team setup, the Stackelberg setup, and the Nash setup of the binary signaling game are investigated in Sections II, Section III, and Section IV, respectively.",
"In Section V, the multi-dimensional setup is studied, and in Section VI, the setup under an average power constraint is investigated.",
"The paper ends with Section VII, where some conclusions are drawn and directions for future research highlighted."
],
[
"TEAM THEORETIC ANALYSIS: CLASSICAL SETUP with IDENTICAL COSTS and PRIORS",
"Consider the team setup where the costs and the priors are assumed to be the same and available for both the transmitter and the receiver; i.e., $C_{ji}=C^t_{ji}=C^r_{ji}$ and $\\pi _i=\\pi _i^t=\\pi _i^r$ for $i,j\\in \\lbrace 0,1\\rbrace $ .",
"Thus the common Bayes risk becomes $r^t(\\mathcal {S},\\delta )=r^r(\\mathcal {S},\\delta )=\\pi _0 (C_{00} \\mathsf {P}_{00} + C_{10} \\mathsf {P}_{10}) + \\pi _1 (C_{01} \\mathsf {P}_{01} + C_{11} \\mathsf {P}_{11})$ .",
"The arguments for the proof of the following result follow from the standard analysis in the detection and estimation literature [14], [15].",
"However, for completeness, and for the relevance of the analysis in the following sections, a proof is included.",
"Theorem 2.1 Let $\\tau \\triangleq {\\pi _0 (C_{10}-C_{00}) \\over \\pi _1 (C_{01}-C_{11})}$ .",
"If $\\tau \\le 0$ or $\\tau =\\infty $ , the team solution of the binary signaling setup is non-informative.",
"Otherwise; i.e., if $0<\\tau <\\infty $ , the team solution is always informative.",
"The players adjust $S_0$ , $S_1$ , and $\\delta $ so that $r^t(\\mathcal {S},\\delta )=r^r(\\mathcal {S},\\delta )$ is minimized.",
"The Bayes risk of the transmitter and the receiver in (REF ) can be written as followsNote that we are still keeping the parameters of the transmitter and the receiver as distinct in order to be able to utilize the expressions for the game formulations.",
": $\\begin{split}r^j(\\mathcal {S},\\delta ) &= \\pi _0^j C^j_{00} + \\pi _1^j C^j_{11} + \\pi _0^j (C^j_{10}-C^j_{00})\\mathsf {P}_{10} + \\pi _1^j (C^j_{01}-C^j_{11})\\mathsf {P}_{01} \\;, \\end{split}$ for $j\\in \\lbrace t,r\\rbrace $ .",
"Here, first the receiver chooses the optimal decision rule $\\delta ^*_{S_0,S_1}$ for any given signal levels $S_0$ and $S_1$ , and then the transmitter chooses the optimal signal levels $S_0^*$ and $S_1^*$ depending on the optimal receiver policy $\\delta ^*_{S_0,S_1}$ .",
"Assuming non-zero priors $\\pi _0^t, \\pi _0^r, \\pi _1^t$ , and $\\pi _1^r$ , the different cases for the optimal receiver decision rule can be investigated by utilizing (REF ) as follows: If $C^r_{01}>C^r_{11}$ , if $C^r_{10}>C^r_{00}$ , the LRT in (REF ) must be applied to determine the optimal decision.",
"if $C^r_{10}\\le C^r_{00}$ , the left-hand side (LHS) of the inequality in (REF ) is always greater than the right-hand side (RHS); thus, the receiver always chooses $\\mathcal {H}_1$ .",
"If $C^r_{01}=C^r_{11}$ , if $C^r_{10}>C^r_{00}$ , the LHS of the inequality in (REF ) is always less than the RHS; thus, the receiver always chooses $\\mathcal {H}_0$ .",
"if $C^r_{10}=C^r_{00}$ , the LHS and RHS of the inequality in (REF ) are equal; hence, the receiver is indifferent of deciding $\\mathcal {H}_0$ or $\\mathcal {H}_1$ .",
"if $C^r_{10}<C^r_{00}$ , the LHS of the inequality in (REF ) is always greater than the RHS; thus, the receiver always chooses $\\mathcal {H}_1$ .",
"If $C^r_{01}<C^r_{11}$ , if $C^r_{10}\\ge C^r_{00}$ , the LHS of the inequality in (REF ) is always less than the RHS; thus, the receiver always chooses $\\mathcal {H}_0$ .",
"if $C^r_{10}<C^r_{00}$ , the LRT in (REF ) must be applied to determine the optimal decision.",
"The analysis above is summarized in Table REF : Table: Optimal decision rule analysis for the receiver.As it can be observed from Table REF , the LRT is needed only when $\\tau \\triangleq {\\pi _0^r (C^r_{10}-C^r_{00}) \\over \\pi _1^r (C^r_{01}-C^r_{11})}$ takes a finite positive value; i.e., $0<\\tau <\\infty $ .",
"Otherwise; i.e., $\\tau \\le 0$ or $\\tau =\\infty $ , since the receiver does not consider any message sent by the transmitter, the equilibrium is non-informative.",
"For $0<\\tau <\\infty $ , let $\\zeta \\triangleq \\text{sgn}(C^r_{01}-C^r_{11})$ $($ notice that $\\zeta =\\text{sgn}(C^r_{01}-C^r_{11})=\\text{sgn}(C^r_{10}-C^r_{00})$ and $\\zeta \\in \\lbrace -1,1\\rbrace )$ .",
"Then, the optimal decision rule for the receiver in (REF ) becomes $\\delta : \\Bigg \\lbrace \\zeta { p_1(y)\\over p_0(y)} \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta {\\pi _0^r (C^r_{10}-C^r_{00}) \\over \\pi _1^r (C^r_{01}-C^r_{11})} = \\zeta \\tau \\;.$ Let the transmitter choose optimal signals $\\mathcal {S}=\\lbrace S_0,S_1\\rbrace $ .",
"Then the measurements in (REF ) become $\\mathcal {H}_i : Y \\sim \\mathcal {N} (S_i,\\sigma ^2)$ for $i\\in \\lbrace 0,1\\rbrace $ , as $N\\sim \\mathcal {N} (0,\\sigma ^2)$ , and the optimal decision rule for the receiver is obtained by utilizing (REF ) as $\\delta ^*_{S_0,S_1} &: \\Bigg \\lbrace \\zeta y(S_1-S_0) \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\left(\\sigma ^2\\ln (\\tau )+{S_1^2-S_0^2\\over 2}\\right)\\;.$ Since $\\zeta Y(S_1-S_0)$ is distributed as $\\mathcal {N} \\Big (\\zeta (S_1-S_0)S_i,(S_1-S_0)^2\\sigma ^2\\Big )$ under $\\mathcal {H}_i$ for $i\\in \\lbrace 0,1\\rbrace $ , the conditional probabilities can be written based on (REF ) as follows: $\\mathsf {P}_{10}&=\\mathsf {Pr}(y\\in \\Gamma _1|\\mathcal {H}_0)=\\mathsf {Pr}(\\delta (y)=1|\\mathcal {H}_0)=1-\\mathsf {Pr}(\\delta (y)=0|\\mathcal {H}_0)=1-\\mathsf {P}_{00}\\nonumber \\\\&=\\mathcal {Q}\\left(\\zeta \\left({\\sigma \\ln (\\tau )\\over |S_1-S_0|}+{|S_1-S_0|\\over 2\\sigma }\\right)\\right)\\;,$ and similarly, $\\mathsf {P}_{01}$ can be derived as $\\mathsf {P}_{01}=\\mathcal {Q}\\left(\\zeta \\left(-{\\sigma \\ln (\\tau )\\over |S_1-S_0|}+{|S_1-S_0|\\over 2\\sigma }\\right)\\right)$ .",
"By defining $d\\triangleq {|S_1-S_0|\\over \\sigma }$ , $\\mathsf {P}_{10}=\\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau )\\over d}+{d\\over 2}\\right)\\right)$ and $\\mathsf {P}_{01}=\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d}+{d\\over 2}\\right)\\right)$ can be obtained.",
"Then, the optimum behavior of the transmitter can be found by analyzing the derivative of the Bayes risk of the transmitter in (REF ) with respect to $d$ : $\\begin{split}{\\mathrm {d}\\,r^t(\\mathcal {S},\\delta )\\over \\mathrm {d}\\,d}&= -{1\\over \\sqrt{2\\pi }}\\exp \\left\\lbrace -{(\\ln \\tau )^2\\over 2d^2}\\right\\rbrace \\exp \\left\\lbrace -{d^2\\over 8}\\right\\rbrace \\\\&\\quad \\;\\times \\Bigg (\\pi _0^t\\zeta (C^t_{10}-C^t_{00})\\tau ^{-{1\\over 2}}\\left(-{\\ln \\tau \\over d^2}+{1\\over 2}\\right)+\\pi _1^t\\zeta (C^t_{01}-C^t_{11})\\tau ^{1\\over 2}\\left({\\ln \\tau \\over d^2}+{1\\over 2}\\right)\\Bigg )\\;.\\end{split}$ In (REF ), if we utilize $C_{ji}=C^t_{ji}=C^r_{ji}$ , $\\pi _i=\\pi _i^t=\\pi _i^r$ and $\\tau ={\\pi _0 (C_{10}-C_{00}) \\over \\pi _1 (C_{01}-C_{11})}$ , we obtain the following: ${\\mathrm {d}\\,r^t(\\mathcal {S},\\delta )\\over \\mathrm {d}\\,d}&=-{1\\over \\sqrt{2\\pi }}\\exp \\left\\lbrace -{(\\ln \\tau )^2\\over 2d^2}\\right\\rbrace \\exp \\left\\lbrace -{d^2\\over 8}\\right\\rbrace \\sqrt{\\pi _0\\pi _1(C_{10}-C_{00})(C_{01}-C_{11})}<0 \\;.$ Thus, in order to minimize the Bayes risk, the transmitter always prefers the maximum $d$ , i.e., $d^*={\\sqrt{P_0}+\\sqrt{P_1}\\over \\sigma }$ , and the equilibrium is informative.",
"Remark 2.1 (i) Note that there are two informative equilibrium points which satisfy $d^*={\\sqrt{P_0}+\\sqrt{P_1}\\over \\sigma }$ : $(S_0^*,S_1^*)=\\left(-\\sqrt{P_0},\\sqrt{P_1}\\right)$ and $(S_0^*,S_1^*)=\\left(\\sqrt{P_0},-\\sqrt{P_1}\\right)$ , and the decision rule of the receiver is chosen based on the rule in (REF ) accordingly.",
"Actually, these equilibrium points are essentially unique; i.e., they result in the same Bayes risks for the transmitter and the receiver.",
"(ii) In the non-informative equilibrium, the receiver chooses either $\\mathcal {H}_0$ or $\\mathcal {H}_1$ as depicted in Table REF .",
"Since the message sent by the transmitter has no effect on the equilibrium, there are infinitely many ways of signal selection, which implies infinitely many equilibrium points.",
"However, all these points are essentially unique; i.e., they result in the same Bayes risks for the transmitter and the receiver.",
"Actually, if the receiver always chooses $\\mathcal {H}_i$ , the Bayes risks of the players are $r^j(\\mathcal {S},\\delta )=\\pi ^j_0C^j_{i0}+\\pi _1^jC_{i1}^j$ for $i\\in \\lbrace 0,1\\rbrace $ and $j\\in \\lbrace t,r\\rbrace $ ."
],
[
"STACKELBERG GAME ANALYSIS",
"Under the Stackelberg assumption, first the transmitter (the leader agent) announces and commits to a particular policy, and then the receiver (the follower agent) acts accordingly.",
"In this direction, first the transmitter chooses optimal signals $\\mathcal {S}=\\lbrace S_0,S_1\\rbrace $ to minimize his Bayes risk $r^t(\\mathcal {S},\\delta )$ , then the receiver chooses an optimal decision rule $\\delta $ accordingly to minimize his Bayes risk $r^r(\\mathcal {S},\\delta )$ .",
"Due to the sequential structure of the Stackelberg game, besides his own priors and costs, the transmitter also knows the priors and the costs of the receiver so that he can adjust his optimal policy accordingly.",
"On the other hand, besides his own priors and costs, the receiver knows only the policy and the action (signals $\\mathcal {S}=\\lbrace S_0,S_1\\rbrace $ ) of the transmitter as he announces during the game-play; i.e., the costs and priors of the transmitter are not available to the receiver."
],
[
"Equilibrium Solutions",
"Under the Stackelberg assumption, the equilibrium structure of the binary signaling game can be characterized as follows: Theorem 3.1 If $\\tau \\triangleq {\\pi _0^r (C^r_{10}-C^r_{00}) \\over \\pi _1^r (C^r_{01}-C^r_{11})}\\le 0$ or $\\tau =\\infty $ , the Stackelberg equilibrium of the binary signaling game is non-informative.",
"Otherwise; i.e., if $0<\\tau <\\infty $ , let $d\\triangleq {|S_1-S_0|\\over \\sigma }$ , $d_{\\max }\\triangleq {\\sqrt{P_0}+\\sqrt{P_1}\\over \\sigma }$ , $\\zeta \\triangleq \\text{sgn}(C^r_{01}-C^r_{11})$ , $k_0\\triangleq \\pi _0^t\\zeta (C^t_{10}-C^t_{00})\\tau ^{-{1\\over 2}}$ , and $k_1\\triangleq \\pi _1^t\\zeta (C^t_{01}-C^t_{11})\\tau ^{1\\over 2}$ .",
"Then, the Stackelberg equilibrium structure can be characterized as in Table REF , where $d^*=0$ stands for a non-informative equilibrium, and a nonzero $d^*$ corresponds to an informative equilibrium.",
"Table: Stackelberg equilibrium analysis for 0<τ<∞0<\\tau <\\infty .Before proving Theorem REF , we make the following remark: Remark 3.1 As we observed in Theorem REF , for a team setup, an equilibrium is almost always informative (practically, $0<\\tau <\\infty $ ), whereas in the case of subjective priors and/or costs, it may cease to be informative.",
"By applying the same case analysis as in the proof of Theorem REF , it can be deduced that the equilibrium is non-informative if $\\tau \\le 0$ or $\\tau =\\infty $ (see Table REF ).",
"Thus, $0<\\tau <\\infty $ can be assumed.",
"Then, from (REF ), $r^t(\\mathcal {S},\\delta )$ is a monotone decreasing (increasing) function of $d$ if $k_0\\left(-{\\ln \\tau \\over d^2}+{1\\over 2}\\right)+k_1\\left({\\ln \\tau \\over d^2}+{1\\over 2}\\right)$ , or equivalently $d^2(k_0+k_1)-2\\ln \\tau \\,(k_0-k_1)$ is positive (negative) $\\forall d$ , where $k_0$ and $k_1$ are as defined in the theorem statement.",
"Therefore, one of the following cases is applicable: if $\\ln \\tau \\;(k_0-k_1)<0$ and $k_0+k_1\\ge 0$ , then $d^2 (k_0+k_1)>2\\ln \\tau (k_0-k_1)$ is satisfied $\\forall d$ , which means that $r^t(\\mathcal {S},\\delta )$ is a monotone decreasing function of $d$ .",
"Therefore, the transmitter tries to maximize $d$ ; i.e., chooses the maximum of $|S_1-S_0|$ under the constraints $| S_0 | ^2 \\le P_0$ and $| S_1 | ^2 \\le P_1$ , hence $d^*=\\max {|S_1-S_0|\\over \\sigma }={\\sqrt{P_0}+\\sqrt{P_1}\\over \\sigma }=d_{\\max }$ , which entails an informative equilibrium.",
"if $\\ln \\tau \\;(k_0-k_1)<0$ , $k_0+k_1<0$ , and $d_{\\max }^2<\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ , then $r^t(\\mathcal {S},\\delta )$ is a monotone decreasing function of $d$ .",
"Therefore, the transmitter maximizes $d$ as in the previous case.",
"if $\\ln \\tau \\;(k_0-k_1)<0$ , $k_0+k_1<0$ , and $d_{\\max }^2\\ge \\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ , since $d^2(k_0+k_1)-2\\ln \\tau \\,(k_0-k_1)$ is initially positive then negative, $r^t(\\mathcal {S},\\delta )$ is first decreasing and then increasing with respect to $d$ .",
"Therefore, the transmitter chooses the optimal $d^*$ such that $(d^*)^2=\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ which results in a minimal Bayes risk $r^t(\\mathcal {S},\\delta )$ for the transmitter.",
"This is depicted in Figure REF .",
"if $\\ln \\tau \\;(k_0-k_1)\\ge 0$ and $k_0+k_1<0$ , then $d^2 (k_0+k_1)<2\\ln \\tau (k_0-k_1)$ is satisfied $\\forall d$ , which means that $r^t(\\mathcal {S},\\delta )$ is a monotone increasing function of $d$ .",
"Therefore, the transmitter tries to minimize $d$ ; i.e., chooses $S_0=S_1$ so that $d^*=0$ .",
"In this case, the transmitter does not provide any information to the receiver and the decision rule of the receiver in (REF ) becomes $\\delta : \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau $ ; i.e., the receiver uses only the prior information, thus the equilibrium is non-informative.",
"if $\\ln \\tau \\;(k_0-k_1)\\ge 0$ , $k_0+k_1\\ge 0$ , and $d_{\\max }^2<\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ , then $r^t(\\mathcal {S},\\delta )$ is a monotone increasing function of $d$ .",
"Therefore, the transmitter chooses $S_0=S_1$ so that $d^*=0$ .",
"Similar to the previous case, the equilibrium is non-informative.",
"if $\\ln \\tau \\;(k_0-k_1)\\ge 0$ , $k_0+k_1\\ge 0$ , and $d_{\\max }^2\\ge \\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ , $r^t(\\mathcal {S},\\delta )$ is first an increasing then a decreasing function of $d$ , which makes the transmitter choose either the minimum $d$ or the maximum $d$ ; i.e., he chooses the one that results in a lower Bayes risk $r^t(\\mathcal {S},\\delta )$ for the transmitter.",
"If the minimum Bayes risk is achieved when $d^*=0$ , then the equilibrium is non-informative; otherwise (i.e., when the minimum Bayes risk is achieved when $d^*=d_{\\max }$ ), the equilibrium is an informative one.",
"There are three possible cases: $\\zeta (1-\\tau )>0$ : If $d^*=0$ , since $\\delta : \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau $ , the receiver always chooses $\\mathcal {H}_1$ , thus $\\mathsf {P}_{10}=\\mathsf {P}_{11}=1$ and $\\mathsf {P}_{00}=\\mathsf {P}_{01}=0$ .",
"Then, from (REF ), $r^t(\\mathcal {S},\\delta )=\\pi _0^t C^t_{00} + \\pi _1^t C^t_{11} + \\pi _0^t (C^t_{10}-C^t_{00})$ .",
"If $d^*=d_{\\max }$ , by utilizing (REF ) and (REF ), $r^t(\\mathcal {S},\\delta )=\\pi _0^t C^t_{00} + \\pi _1^t C^t_{11} + \\pi _0^t (C^t_{10}-C^t_{00})\\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right) + \\pi _1^t (C^t_{01}-C^t_{11})\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right)$ .",
"Then the decision of the transmitter is determined by the following: $&\\pi _0^t (C^t_{10}-C^t_{00})\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}\\nonumber \\\\&\\qquad \\qquad \\pi _0^t (C^t_{10}-C^t_{00})\\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right)+ \\pi _1^t (C^t_{01}-C^t_{11})\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right) \\nonumber \\\\&\\pi _0^t (C^t_{10}-C^t_{00})\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}\\pi _1^t (C^t_{01}-C^t_{11})\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right) \\nonumber \\\\\\begin{split}&\\zeta k_0\\tau \\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }} \\zeta k_1\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right) \\,.\\end{split}$ For (REF ), there are two possible cases: $\\zeta =1$ and $0<\\tau <1$ : Since $\\ln \\tau (k_0-k_1)\\ge 0\\Rightarrow k_0-k_1\\le 0 $ and $k_0+k_1\\ge 0$ , $k_1\\ge 0$ always.",
"Then, (REF ) becomes $&{k_0\\tau \\over k_1}\\mathcal {Q}\\left(-{\\ln (\\tau )\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)-\\mathcal {Q}\\left(-{\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}0 \\,.$ $\\zeta =-1$ and $\\tau >1$ : Since $\\ln \\tau (k_0-k_1)\\ge 0\\Rightarrow k_0-k_1\\ge 0 $ and $k_0+k_1\\ge 0$ , $k_0\\ge 0$ always.",
"Then, (REF ) becomes $&{k_1\\over k_0\\tau }\\mathcal {Q}\\left({\\ln (\\tau )\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)-\\mathcal {Q}\\left({\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}0 \\,.$ $\\zeta (1-\\tau )=0 \\Leftrightarrow \\tau =1$ : Since $k_0+k_1\\ge 0$ and $d^2(k_0+k_1)-2\\ln \\tau \\,(k_0-k_1)\\ge 0$ , $r^t(\\mathcal {S},\\delta )$ is a monotone decreasing function of $d$ , which implies $d^*=d_{\\max }$ and informative equilibrium.",
"$\\zeta (1-\\tau )<0$ : If $d^*=0$ , since $\\delta : \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau $ , the receiver always chooses $\\mathcal {H}_0$ , thus $\\mathsf {P}_{00}=\\mathsf {P}_{01}=1$ and $\\mathsf {P}_{10}=\\mathsf {P}_{11}=0$ .",
"Then, from (REF ), $r^t(\\mathcal {S},\\delta )=\\pi _0^t C^t_{00} + \\pi _1^t C^t_{11} + \\pi _1^t (C^t_{01}-C^t_{11})$ .",
"If $d^*=d_{\\max }$ , by utilizing (REF ) and (REF ), $r^t(\\mathcal {S},\\delta )=\\pi _0^t C^t_{00} + \\pi _1^t C^t_{11} + \\pi _0^t (C^t_{10}-C^t_{00})\\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right) + \\pi _1^t (C^t_{01}-C^t_{11})\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right)$ .",
"Then, similar to the analysis in case-a), the decision of the transmitter is determined by the following: $\\begin{split}&\\zeta k_1\\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau )\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}\\zeta k_0\\tau \\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\right) \\,.\\end{split}$ For (REF ), there are two possible cases: $\\zeta =-1$ and $0<\\tau <1$ : Since $\\ln \\tau (k_0-k_1)\\ge 0\\Rightarrow k_0-k_1\\le 0 $ and $k_0+k_1\\ge 0$ , $k_1\\ge 0$ always.",
"Then, (REF ) becomes $&{k_0\\tau \\over k_1}\\mathcal {Q}\\left(-{\\ln (\\tau )\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)-\\mathcal {Q}\\left(-{\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}0 \\,.$ $\\zeta =1$ and $\\tau >1$ : Since $\\ln \\tau (k_0-k_1)\\ge 0\\Rightarrow k_0-k_1\\ge 0 $ and $k_0+k_1\\ge 0$ , $k_0\\ge 0$ always.",
"Then, (REF ) becomes $&{k_1\\over k_0\\tau }\\mathcal {Q}\\left({\\ln (\\tau )\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)-\\mathcal {Q}\\left({\\ln (\\tau )\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}0 \\,.$ Thus, by combining all the cases, the comparison of the transmitter Bayes risks for $d^*=0$ and $d^*=d_{\\max }$ reduces to the following rule: $\\begin{split}&\\left({k_1\\over k_0\\tau }\\right)^{\\text{sgn}(\\ln (\\tau ))}\\mathcal {Q}\\left({|\\ln (\\tau )|\\over d_{\\max }}-{d_{\\max }\\over 2}\\right)-\\mathcal {Q}\\left({|\\ln (\\tau )|\\over d_{\\max }}+{d_{\\max }\\over 2}\\right)\\overset{d^*=d_{\\max }}{\\underset{d^*=0}{\\gtreqless }}0\\,.\\end{split}$ The most interesting case is Case-3 in which $\\ln \\tau \\;(k_0-k_1)<0, k_0+k_1<0,$ and $d_{\\max }^2\\ge \\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ , since in all other cases, the transmitter chooses either the minimum or the maximum distance between the signal levels.",
"Further, for classical hypothesis-testing in the team setup, the optimal distance corresponds to the maximum separation [14].",
"However, in Case-3, there is an optimal distance $d^*=\\sqrt{\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |}<d_{\\max }$ that makes the Bayes risk of the transmitter minimum as it can be seen in Figure REF .",
"Figure: The Bayes risk of the transmitter versus dd when C 00 t =0.6,C 10 t =0.4,C 01 t =0.4,C 11 t =0.6,C 00 r =0,C 10 r =0.9,C 01 r =0.4,C 11 r =0,π 0 t =0.25,π 0 r =0.25,P 0 =1,P 1 =1 C^t_{00}=0.6, C^t_{10}=0.4, C^t_{01}=0.4, C^t_{11}=0.6, C^r_{00}=0, C^r_{10}=0.9, C^r_{01}=0.4, C^r_{11}=0, \\pi _0^t=0.25, \\pi _0^r=0.25, P_0=1, P_1=1, and σ=0.1\\sigma = 0.1.",
"The optimal d * =|2lnτ(k 0 -k 1 ) (k 0 +k 1 )|=0.4704<d max =20d^*=\\sqrt{\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |}=0.4704<d_{max}=20 and its corresponding Bayes risk r t =0.5379r^t=0.5379 are indicated by the star.Remark 3.2 Similar to the team setup analysis, for every possible case in Table REF , there are more than one equilibrium points, and they are essentially unique since the Bayes risks of the transmitter and the receiver depend on $d$ .",
"In particular, (i) for $d^*=d_{\\max }$ , the equilibrium is informative, $(S_0^*,S_1^*)=\\left(-\\sqrt{P_0},\\sqrt{P_1}\\right)$ and $(S_0^*,S_1^*)=\\left(\\sqrt{P_0},-\\sqrt{P_1}\\right)$ are the only possible choices for the transmitter, which are essentially unique, and the decision rule of the receiver is chosen based on the rule in (REF ).",
"(ii) for $d^*=\\sqrt{\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |}$ , the equilibrium is informative, there are infinitely many choices for the transmitter and the receiver, and all of them are essentially unique; i.e., they result in the same Bayes risks for the transmitter and the receiver.",
"(iii) for $d^*=0$ or $\\tau \\notin (0,\\infty )$ , the equilibrium is non-informative and there are infinitely many equilibrium points which are essentially unique; see Remark REF -(ii)."
],
[
"Continuity and Robustness to Perturbations around the Team Setup",
"We now investigate the effects of small perturbations in priors and costs on equilibrium values.",
"In particular, we consider the perturbations around the team setup; i.e., at the point of identical priors and costs.",
"Define the perturbation around the team setup as $\\epsilon =\\lbrace \\epsilon _{\\pi 0},\\epsilon _{\\pi 1},\\epsilon _{00},\\epsilon _{01},\\epsilon _{10},\\epsilon _{11}\\rbrace \\in \\mathbb {R}^6$ such that $\\pi _i^t=\\pi _i^r+\\epsilon _{\\pi i}$ and $C_{ji}^t=C_{ji}^r+\\epsilon _{ji}$ for $i,j\\in \\lbrace 0,1\\rbrace $ (note that the transmitter parameters are perturbed around the receiver parameters which are assumed to be fixed).",
"Then, for $0<\\tau <\\infty $ , at the point of identical priors and costs, small perturbations in both priors and costs imply $k_0=(\\pi _0^r+\\epsilon _{\\pi 0})\\zeta (C^r_{10}-C^r_{00}+\\epsilon _{10}-\\epsilon _{00})\\tau ^{-{1\\over 2}}$ and $k_1=(\\pi _1^r+\\epsilon _{\\pi 1})\\zeta (C^r_{01}-C^r_{11}+\\epsilon _{01}-\\epsilon _{11})\\tau ^{1\\over 2}$ .",
"Since, for $0<\\tau <\\infty $ , $k_0=k_1=\\sqrt{\\pi ^r_0\\pi ^r_1}\\sqrt{(C^r_{10}-C^r_{00})(C^r_{01}-C^r_{11})}>0$ at the point of identical priors and costs, it is possible to obtain both positive and negative $(k_0-k_1)$ by choosing the appropriate perturbation $\\epsilon $ around the team setup.",
"Then, as it can be observed from Table REF , even the equilibrium may alter from an informative one to a non-informative one; hence, under the Stackelberg equilibrium, the policies are not continuous with respect to small perturbations around the point of identical priors and costs, and the equilibrium behavior is not robust to small perturbations in both priors and costs."
],
[
"Subjective Priors",
"Referring to Section REF , for $0<\\tau <\\infty $ , the related parameters can be found as follows (note that the equilibrium is non-informative if $\\tau \\le 0$ or $\\tau =\\infty $ ): $\\tau &={\\pi _0^r (C_{10}-C_{00}) \\over \\pi _1^r (C_{01}-C_{11})} \\,,\\\\k_0&=\\pi _0^t\\sqrt{\\pi _1^r\\over \\pi _0^r}\\sqrt{(C_{10}-C_{00})(C_{01}-C_{11})}\\,,\\\\k_1&=\\pi _1^t\\sqrt{\\pi _0^r\\over \\pi _1^r}\\sqrt{(C_{10}-C_{00})(C_{01}-C_{11})}\\,.$ Since $k_0+k_1>0$ , depending on the values of $\\ln \\tau \\;(k_0-k_1)$ , $d_{\\max }^2$ , and $\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ , Case-1, Case-5 or Case-6 of Theorem REF may hold as depicted in Table REF .",
"Here, the decision rule in Case-6 is the same as (REF ).",
"Table: Stackelberg equilibrium analysis of subjective priors case for 0<τ<∞0<\\tau <\\infty ."
],
[
"Biased Transmitter Cost",
"Based on the arguments in Section REF , the related parameters can be found as follows: $\\tau ={\\pi _0\\over \\pi _1} \\,,\\; k_0=\\sqrt{\\pi _0\\pi _1}(2\\alpha -1)\\,,\\;k_1=\\sqrt{\\pi _0\\pi _1}(2\\alpha -1)\\,.$ Then, $\\ln \\tau \\;(k_0-k_1)=0$ and $k_0+k_1=2\\sqrt{\\pi _0\\pi _1}(2\\alpha -1)$ ; hence, either Case-4 or Case-6 of Theorem REF applies.",
"Namely, if $\\alpha <1/2$ (Case-4 of Theorem REF applies), the transmitter chooses $S_0=S_1$ to minimize $d$ and the equilibrium is non-informative; i.e., he does not send any meaningful information to the transmitter and the receiver considers only the priors.",
"If $\\alpha =1/2$ , the transmitter has no control on his Bayes risk, hence the equilibrium is non-informative.",
"Otherwise; i.e., if $\\alpha >1/2$ (Case-6 of Theorem REF applies), the equilibrium is always informative.",
"In other words, if $\\alpha >1/2$ , the players act like a team.",
"As it can be seen, the informativeness of the equilibrium depends on $\\alpha =\\mathsf {Pr}(b=0)$ , the probability that the Bayes risks of the transmitter and the receiver are aligned."
],
[
"NASH GAME ANALYSIS",
"Under the Nash assumption, the transmitter chooses optimal signals $\\mathcal {S}=\\lbrace S_0,S_1\\rbrace $ to minimize $r^t(\\mathcal {S},\\delta )$ , and the receiver chooses optimal decision rule $\\delta $ to minimize $r^r(\\mathcal {S},\\delta )$ simultaneously.",
"In this Nash setup, the transmitter and the receiver do not need to know the priors and the costs of each other; they need to know only their own priors and costs while calculating the best response to a given action of other player.",
"Further, there is no commitment between the transmitter and the receiver.",
"Due to this difference, the equilibrium structure and robustness properties of the Nash equilibrium show significant differences from the ones in the Stackelberg equilibrium, as stated in the following.",
"In the analysis, we assume deterministic policies for the transmitter and receiver, and we restrict the receiver to use only the single-threshold rules.",
"Although a single-threshold rule is sub-optimal for the receiver in general, it is always optimal for Gaussian densities, and always optimal for uni-modal densities under the maximum likelihood decision rule [14], [37]."
],
[
"Equilibrium Solutions",
"Under the Nash assumption, the equilibrium structure of the binary signaling game can be characterized as follows: Theorem 4.1 Let $\\tau \\triangleq {\\pi _0^r (C^r_{10}-C^r_{00}) \\over \\pi _1^r (C^r_{01}-C^r_{11})}$ and $\\zeta \\triangleq \\text{sgn}(C^r_{01}-C^r_{11})$ , $\\xi _0 \\triangleq {C^t_{10}-C^t_{00}\\over C^r_{10}-C^r_{00}}$ , and $\\xi _1 \\triangleq {C^t_{01}-C^t_{11}\\over C^r_{01}-C^r_{11}}$ .",
"If $\\tau \\le 0$ or $\\tau =\\infty $ , then the Nash equilibrium of the binary signaling game is non-informative.",
"Otherwise; i.e., if $0<\\tau <\\infty $ , the Nash equilibrium structure is as depicted in Table REF .",
"Table: Nash equilibrium analysis for 0<τ<∞0<\\tau <\\infty .Let the transmitter choose any signals $\\mathcal {S}=\\lbrace S_0,S_1\\rbrace $ .",
"Assuming nonzero priors $\\pi _0^t, \\pi _0^r, \\pi _1^t$ and $\\pi _1^r$ , the optimal decision for the receiver is given by (REF ).",
"By applying the same extreme case analysis as in the proof of Theorem REF , the equilibrium is non-informative if $\\tau \\le 0$ or $\\tau =\\infty $ (see Table REF ); thus, $0<\\tau <\\infty $ can be assumed.",
"Now assume that the receiver applies a single-threshold rule; i.e., $\\delta : \\Bigg \\lbrace a y \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\eta $ where $a\\in \\mathbb {R}$ and $\\eta \\in \\mathbb {R}$ .",
"Remark 4.1 Note that for $a=0$ , the receiver chooses either always $\\mathcal {H}_0$ or always $\\mathcal {H}_1$ without considering the value of $y$ , which implies a non-informative equilibrium.",
"Therefore, $S_0^*=S_1^*$ , $a^*=0$ , and $\\eta ^*=\\zeta (\\tau -1)$ (i.e., the decision rule of the receiver is $\\delta ^* : \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau $ ) constitute a non-informative equilibrium regardless of the values of the priors and costs of the players.",
"Thus, due to the remark above, it can be assumed that $a\\ne 0$ holds.",
"Since $aY \\sim \\mathcal {N} \\Big (a S_i,a^2\\sigma ^2\\Big )$ under $\\mathcal {H}_i$ for $i\\in \\lbrace 0,1\\rbrace $ , the conditional probabilities are $\\mathsf {P}_{10}=\\mathcal {Q}\\left(\\eta -aS_0\\over |a|\\sigma \\right)$ and $\\mathsf {P}_{01}=\\mathcal {Q}\\left(-{\\eta -aS_1\\over |a|\\sigma }\\right)$ .",
"Then, the Bayes risk of the transmitter becomes $r^t(\\mathcal {S},\\delta ) = \\pi _0^t C^t_{00} &+ \\pi _1^t C^t_{11} + \\pi _0^t (C^t_{10}-C^t_{00})\\mathcal {Q}\\left(\\eta -aS_0\\over |a|\\sigma \\right)+ \\pi _1^t (C^t_{01}-C^t_{11})\\mathcal {Q}\\left(-{\\eta -aS_1\\over |a|\\sigma }\\right) \\,.$ Since the power constraints are $|S_0|^2 \\le P_0$ and $|S_1|^2 \\le P_1$ , the signals $S_0$ and $S_1$ can be regarded as independent, and the optimum signals $\\mathcal {S}=\\lbrace S_0,S_1\\rbrace $ can be found by analyzing the derivative of the Bayes risk of the transmitter with respect to the signals: ${\\partial \\,r^t(\\mathcal {S},\\delta )\\over \\partial \\,S_i}&= {\\text{sgn}(a)\\over \\sqrt{2\\pi }\\sigma }\\pi _i^t (C^t_{1i}-C^t_{0i})\\exp \\left\\lbrace -{1\\over 2}\\left(\\eta -aS_i\\over |a|\\sigma \\right)^2\\right\\rbrace \\,.$ Then, for $i\\in \\lbrace 0,1\\rbrace $ , the following cases hold: $C^t_{1i}=C^t_{0i}$ $\\Rightarrow $ $S_i$ has no effect on the Bayes risk of the transmitter.",
"$C^t_{1i}\\ne C^t_{0i}$ $\\Rightarrow $ $r^t(\\mathcal {S},\\delta )$ is a decreasing (increasing) function of $S_i$ if $a(C^t_{1i}-C^t_{0i})$ is negative (positive); thus the transmitter chooses the optimal signal levels as $S_0=-\\text{sgn}(a)\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}$ and $S_1=\\text{sgn}(a)\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}$ .",
"By using the expressions above, the cases can be listed as follows: $\\tau \\le 0$ or $\\tau =\\infty $ $\\Rightarrow $ The equilibrium is non-informative.",
"$C^t_{10}=C^t_{00}$ (and/or $C^t_{01}=C^t_{11}$ ) $\\Rightarrow $ $S_0$ (and/or $S_1$ ) has no effect on the Bayes risk of the transmitter; thus it can arbitrarily be chosen by the transmitter.",
"In this case, if the transmitter chooses $S_0=S_1$ ; i.e., he does not send anything useful to the receiver, and the receiver applies the decision rule $\\delta : \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau $ ; i.e., he only considers the prior information (totally discards the information sent by the transmitter).",
"Therefore, there exists a non-informative equilibrium.",
"Notice that, since $0<\\tau <\\infty $ is assumed, $\\zeta =\\text{sgn}(C^r_{01}-C^r_{11})=\\text{sgn}(C^r_{10}-C^r_{00})$ is obtained.",
"Now, assume that the decision rule of the receiver is $\\delta : \\Bigg \\lbrace a y \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\eta $ .",
"Then, the transmitter selects $S_0=-\\text{sgn}(a)\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}$ and $S_1=\\text{sgn}(a)\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}$ as optimal signals, and the decision rule becomes (REF ).",
"By combining the best responses of the transmitter and the receiver, $a&=\\zeta (S_1-S_0)=\\zeta \\text{sgn}(a)\\left(\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}+\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}\\right) \\nonumber \\\\\\Rightarrow & \\text{sgn}(a) = \\zeta \\text{sgn}(a) \\text{sgn}\\left(\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}+\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}\\right) \\nonumber \\\\\\Rightarrow & \\underbrace{\\text{sgn}(C^t_{01}-C^t_{11})\\over \\text{sgn}(C^r_{01}-C^r_{11})}_{=\\text{sgn}(\\xi _1)}\\sqrt{P_1}+\\underbrace{\\text{sgn}(C^t_{10}-C^t_{00})\\over \\text{sgn}(C^r_{10}-C^r_{00})}_{=\\text{sgn}(\\xi _0)}\\sqrt{P_0} > 0$ is obtained.",
"Here, unless (REF ) is satisfied, the best responses of the transmitter and the receiver cannot match each other.",
"Then, there are four possible cases: $\\xi _0<0$ and $\\xi _1<0$ $\\Rightarrow $ (REF ) cannot be satisfied; thus, the best responses of the transmitter and the receiver do not match each other, which results in the absence of a Nash equilibrium for $a\\ne 0$ .",
"However, as discussed in Remark REF , $S_0^*=S_1^*$ , $a^*=0$ , and $\\eta ^*=\\zeta (\\tau -1)$ always constitute a non-informative equilibrium.",
"$\\xi _0<0$ and $\\xi _1>0$ $\\Rightarrow $ (REF ) is satisfied only when $\\sqrt{P_1}>\\sqrt{P_0}$ .",
"If $\\sqrt{P_1}<\\sqrt{P_0}$ , (REF ) cannot be satisfied and the best responses of the transmitter and the receiver do not match each other, which results in the absence of a Nash equilibrium for $a\\ne 0$ .",
"However, due to Remark REF , for $a=0$ , there always exist non-informative equilibria.",
"If $\\sqrt{P_1}=\\sqrt{P_0}$ (which implies $S_0=S_1$ ), then the receiver applies $\\delta : \\Bigg \\lbrace \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\zeta \\tau $ as in Case-2, and the receiver chooses either always $\\mathcal {H}_0$ or always $\\mathcal {H}_1$ .",
"Hence, there exists a non-informative equilibrium; i.e., the transmitter sends dummy signals, and the receiver makes a decision without considering the transmitted signals.",
"$\\xi _0>0$ and $\\xi _1<0$ $\\Rightarrow $ (REF ) is satisfied only when $\\sqrt{P_0}>\\sqrt{P_1}$ .",
"If $\\sqrt{P_0}<\\sqrt{P_1}$ , (REF ) cannot be satisfied and the best responses of the transmitter and the receiver do not match each other, which results in the absence of a Nash equilibrium for $a\\ne 0$ .",
"However, due to Remark REF , for $a=0$ , there always exist non-informative equilibria.",
"If $\\sqrt{P_0}=\\sqrt{P_1}$ (which implies $S_0=S_1$ ), then the receiver applies $\\delta : \\Bigg \\lbrace \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\zeta \\tau $ as in Case-2, and the equilibrium is non-informative.",
"$\\xi _0>0$ and $\\xi _1>0$ $\\Rightarrow $ (REF ) is always satisfied; thus, the consistency is established, and there exists an informative equilibrium.",
"As it can be deduced from Table REF , as the costs related to both hypotheses are aligned$\\xi _i$ is the indicator that the transmitter and the receiver have similar preferences about hypothesis $\\mathcal {H}_i$ ; i.e., if $\\xi _i>0$ , then both the transmitter and the receiver aim to transmit and decode the hypothesis $\\mathcal {H}_i$ correctly (or incorrectly).",
"If $\\xi _i<0$ , then the transmitter and the receiver have conflicting goals over hypothesis $\\mathcal {H}_i$ ; i.e., one of them tries to achieve the correct transmission and decoding, whereas the goal of the other player is the opposite.",
"for the transmitter and the receiver, the Nash equilibrium is informative.",
"If the power limit corresponding to the hypothesis that has aligned costs for the transmitter and receiver is greater than the power limit of the other hypothesis, again, there exists an informative equilibrium.",
"For the other cases, there may exist non-informative equilibrium.",
"Remark 4.2 (i) We emphasize that, under the Nash formulation, while calculating the best responses, the transmitter and the receiver do not need to know the priors and the costs of each other.",
"In particular, for a given decision rule of the receiver $\\delta : \\Bigg \\lbrace a y \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\eta $ , the best response of the transmitter is $S_0^{\\mathrm {BR}}=-\\text{sgn}(a)\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}$ and $S_1^{\\mathrm {BR}}=\\text{sgn}(a)\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}$ .",
"similarly, for a given signal design $S_0$ and $S_1$ of the transmitter, the best response of the receiver is $a^{\\mathrm {BR}}=\\zeta (S_1-S_0)$ and $\\eta ^{\\mathrm {BR}}=\\zeta \\left(\\sigma ^2\\ln (\\tau )+{(S_1)^2-(S_0)^2\\over 2}\\right)$ .",
"(ii) As shown in Theorem REF , at the informative Nash equilibrium, the transmitter selects $S_0^*=-\\text{sgn}(a^*)\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}$ and $S_1^*=\\text{sgn}(a^*)\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}$ , and the decision rule of the receiver is $\\delta ^* : \\Bigg \\lbrace a^* y \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\eta ^*$ , where $a^*=\\zeta (S_1^*-S_0^*)$ and $\\eta ^*=\\zeta \\left(\\sigma ^2\\ln (\\tau )+{(S_1^*)^2-(S_0^*)^2\\over 2}\\right)$ .",
"Similar to the team and Stackelberg setup analyses, the informative equilibrium is essentially unique in the Nash case, too; i.e., if $(S_0^*,S_1^*,a^*,\\eta ^*)$ is an equilibrium point, then $(-S_0^*,-S_1^*,-a^*,\\eta ^*)$ is another equilibrium point, and they both result in the same Bayes risks for the transmitter and the receiver.",
"(iii) For the non-informative equilibrium, as discussed in Remark REF , the optimal strategies of the transmitter and the receiver are determined by $S_0^*=S_1^*$ , $a^*=0$ , and $\\eta ^*=\\zeta (\\tau -1)$ ; which results in essentially unique equilibria (see Remark REF -(ii)).",
"Even though the transmitter and the receiver do not know the private parameters of each other, they can achieve (converge) to an equilibrium.",
"Note that, due to Remark REF -(i), for any arbitrary receiver strategy $(a,\\eta )$ , the best response of the transmitter $(S_0^{\\mathrm {BR}},S_1^{\\mathrm {BR}})$ is one of the four possibilities: $(\\sqrt{P_0},\\sqrt{P_1})$ , $(-\\sqrt{P_0},\\sqrt{P_1})$ , $(\\sqrt{P_0},-\\sqrt{P_1})$ , or $(-\\sqrt{P_0},-\\sqrt{P_1})$ .",
"Then, the corresponding best responses of the receiver are characterized by $(a^{\\mathrm {BR}}_1,\\eta ^{\\mathrm {BR}})$ , $(a^{\\mathrm {BR}}_2,\\eta ^{\\mathrm {BR}})$ , $(-a^{\\mathrm {BR}}_2,\\eta ^{\\mathrm {BR}})$ , or $(-a^{\\mathrm {BR}}_1,\\eta ^{\\mathrm {BR}})$ , respectively, where $a^{\\mathrm {BR}}_1\\triangleq \\zeta (\\sqrt{P_1}-\\sqrt{P_0})$ , $a^{\\mathrm {BR}}_2\\triangleq \\zeta (\\sqrt{P_1}+\\sqrt{P_0})$ , and $\\eta ^{\\mathrm {BR}}=\\zeta \\left(\\sigma ^2\\ln (\\tau )+{P_1-P_0\\over 2}\\right)$ .",
"By continuing these iterations, the best responses of the transmitter and the receiver can be combined and (REF ) is obtained.",
"If their private parameters (priors and costs) satisfy the condition of the unique informative equilibrium in Table IV, their best responses match each other, so the best-response dynamics converges to an equilibrium (e.g., $(a,\\eta )\\rightarrow (\\sqrt{P_0},\\sqrt{P_1})\\rightarrow (a^{\\mathrm {BR}}_1,\\eta ^{\\mathrm {BR}})\\rightarrow (\\sqrt{P_0},\\sqrt{P_1})\\rightarrow \\cdots $ ).",
"Otherwise, the optimal strategies (best responses) of the transmitter and the receiver oscillate between two best responses; e.g., $(a,\\eta )\\rightarrow (\\sqrt{P_0},\\sqrt{P_1})\\rightarrow (a^{\\mathrm {BR}}_1,\\eta ^{\\mathrm {BR}})\\rightarrow (-\\sqrt{P_0},-\\sqrt{P_1})\\rightarrow (-a^{\\mathrm {BR}}_1,\\eta ^{\\mathrm {BR}})\\rightarrow (\\sqrt{P_0},\\sqrt{P_1})\\rightarrow \\cdots $ .",
"Then, they deduce that there exist only non-informative equilibria, in which $S_0^*=S_1^*$ , $a^*=0$ , and $\\eta ^*=\\zeta (\\tau -1)$ (see Remark REF -(iii)).",
"Note that, when $a\\ne 0$ , the misalignment between the costs can even induce a scenario, in which there exists no equilibrium.",
"For $a\\ne 0$ , the main reason for the absence of a non-informative (babbling) equilibrium under the Nash assumption is that in the binary signaling game setup, the receiver is forced to make a decision.",
"Using only the prior information, the receiver always chooses one of the hypothesis.",
"By knowing this, the transmitter can manipulate his signaling strategy for his own benefit.",
"However, after this manipulation, the receiver no longer keeps his decision rule the same; namely, the best response of the receiver alters based on the signaling strategy of the transmitter, which entails another change of the best response of the transmitter.",
"Due to such an infinite recursion, the optimal policies of the transmitter and the receiver keep changing, and thus, there does not exist a pure Nash equilibrium unless $a=0$ ; i.e., due to Remark REF , there always exist non-informative equilibria with $S_0^*=S_1^*$ , $a^*=0$ , and $\\eta ^*=\\zeta (\\tau -1)$ ."
],
[
"Continuity and Robustness to Perturbations around the Team Setup",
"Similar to that in Section REF for the Stackelberg setup, the effects of small perturbations in priors and costs on equilibrium values around the team setup are investigated for the Nash setup as follows: Define the perturbation around the team setup as $\\epsilon =\\lbrace \\epsilon _{\\pi 0},\\epsilon _{\\pi 1},\\epsilon _{00},\\epsilon _{01},\\epsilon _{10},\\epsilon _{11}\\rbrace \\in \\mathbb {R}^6$ such that $\\pi _i^t=\\pi _i^r+\\epsilon _{\\pi i}$ and $C_{ji}^t=C_{ji}^r+\\epsilon _{ji}$ for $i,j\\in \\lbrace 0,1\\rbrace $ (note that the transmitter parameters are perturbed around the receiver parameters which are assumed to be fixed).",
"Then, for $0<\\tau <\\infty $ , at the point of identical priors and costs, small perturbations in priors and costs imply $\\xi _0 = {C^r_{10}-C^r_{00}+\\epsilon _{10}-\\epsilon _{00}\\over C^r_{10}-C^r_{00}}$ and $\\xi _1 = {C^r_{01}-C^r_{11}+\\epsilon _{01}-\\epsilon _{11}\\over C^r_{01}-C^r_{11}}$ .",
"As it can be seen, the Nash equilibrium is not affected by small perturbations in priors.",
"Further, since $\\xi _0=\\xi _1=1$ at the point of identical priors and costs for $0<\\tau <\\infty $ , as long as the perturbation $\\epsilon $ is chosen such that $\\Big \\vert {\\epsilon _{10}-\\epsilon _{00}\\over C^r_{10}-C^r_{00}}\\Big \\vert <1$ and $\\Big \\vert {\\epsilon _{01}-\\epsilon _{11}\\over C^r_{01}-C^r_{11}}\\Big \\vert <1$ , we always obtain positive $\\xi _0$ and $\\xi _1$ in Table REF .",
"Thus, under the Nash assumption, the equilibrium behavior is robust to small perturbations in both priors and costs.",
"For the continuity analysis, first consider a non-informative equilibrium; i.e., the policies are $S_0^*=S_1^*$ , $a^*=0$ , and $\\eta ^*=\\zeta (\\tau -1)$ , which are independent of the values of the priors and costs of the players.",
"Thus, consider when $a\\ne 0$ ; i.e., an informative equilibrium: if the priors and costs are perturbed around the team setup, $S_0=-\\text{sgn}(a)\\text{sgn}(C^r_{10}-C^r_{00}+\\epsilon _{10}-\\epsilon _{00})\\sqrt{P_0}$ and $S_1=\\text{sgn}(a)\\text{sgn}(C^r_{01}-C^r_{11}+\\epsilon _{01}-\\epsilon _{11})\\sqrt{P_1}$ are obtained.",
"As long as the perturbation $\\epsilon $ is chosen such that $\\Big \\vert {\\epsilon _{10}-\\epsilon _{00}\\over C^r_{10}-C^r_{00}}\\Big \\vert <1$ and $\\Big \\vert {\\epsilon _{01}-\\epsilon _{11}\\over C^r_{01}-C^r_{11}}\\Big \\vert <1$ , the changes in $\\eta $ , $S_0$ and $S_1$ are continuous with respect to perturbations; actually, the values of the equilibrium parameters remain constant; i.e., either $(S_0^*,S_1^*,a^*,\\eta ^*)=\\left(-\\zeta \\sqrt{P_0},\\zeta \\sqrt{P_1}, (\\sqrt{P_0}+\\sqrt{P_1}),\\zeta \\left(\\sigma ^2\\ln (\\tau )+{S_1^2-S_0^2\\over 2}\\right)\\right)$ or the essentially equivalent one $(S_0^*,S_1^*,a^*,\\eta ^*)=\\left(\\zeta \\sqrt{P_0},-\\zeta \\sqrt{P_1}, -(\\sqrt{P_0}+\\sqrt{P_1}),\\zeta \\left(\\sigma ^2\\ln (\\tau )+{S_1^2-S_0^2\\over 2}\\right)\\right)$ holds.",
"Thus, the policies are continuous with respect to small perturbations around the point of identical priors and costs."
],
[
"Subjective Priors",
"The related parameters are $\\tau ={\\pi _0^r (C_{10}-C_{00}) \\over \\pi _1^r (C_{01}-C_{11})}$ , $\\xi _0 =1$ , and $\\xi _1=1$ .",
"Thus, if $\\tau <0$ or $\\tau =\\infty $ , the equilibrium is non-informative; otherwise, there always exists a unique informative equilibrium."
],
[
"Biased Transmitter Cost",
"Based on the arguments in Section REF , the related parameters can be found as follows: $C^t_{01}&=C^t_{10}=\\alpha \\text{ and } C^t_{00}=C^t_{11}=1-\\alpha \\,,\\\\C^r_{01}&=C^r_{10}=1 \\text{ and } C^r_{00}=C^r_{11}=0 \\,,\\\\\\tau &={\\pi _0 (C^r_{10}-C^r_{00}) \\over \\pi _1 (C^r_{01}-C^r_{11})}={\\pi _0\\over \\pi _1} \\,,\\\\\\xi _0 &= {C^t_{10}-C^t_{00}\\over C^r_{10}-C^r_{00}} =2\\alpha -1 \\,,\\\\\\xi _1 &= {C^t_{01}-C^t_{11}\\over C^r_{01}-C^r_{11}}=2\\alpha -1\\,.$ If $\\alpha >1/2$ (Case-3-d of Theorem REF applies), the players act like a team and the equilibrium is informative.",
"If $\\alpha =1/2$ (Case-2 of Theorem REF applies), the equilibrium is non-informative.",
"Otherwise; i.e., if $\\alpha <1/2$ (Case-3-a of Theorem REF applies), there exist non-informative equilibria.",
"As it can be seen, the existence of the equilibrium depends on $\\alpha =\\mathsf {Pr}(b=0)$ , the probability that the Bayes risks of the transmitter and the receiver are aligned."
],
[
"EXTENSION to the MULTI-DIMENSIONAL CASE",
"When the transmitter sends a multi-dimensional signal over a multi-dimensional channel, or the receiver takes multiple samples from the observed waveform, the scalar analysis considered heretofore is not applicable anymore; thus, the vector case can be investigated.",
"In this direction, the binary hypothesis-testing problem aforementioned can be modified as $\\mathcal {H}_0 : \\mathbf {Y}= \\mathbf {S}_0 + \\mathbf {N}\\;,\\nonumber \\\\\\mathcal {H}_1 : \\mathbf {Y}= \\mathbf {S}_1 + \\mathbf {N}\\;,$ where $\\mathbf {Y}$ is the observation (measurement) vector that belongs to the observation set $\\Gamma =\\mathbb {R}^n$ , $\\mathbf {S}_0$ and $\\mathbf {S}_1$ denote the deterministic signals under hypothesis $\\mathcal {H}_0$ and hypothesis $\\mathcal {H}_1$ , such that $\\mathbb {S}\\triangleq \\lbrace \\mathcal {S}:\\Vert \\mathbf {S}_0 \\Vert ^2 \\le P_0 \\,,\\; \\Vert \\mathbf {S}_1 \\Vert ^2 \\le P_1\\rbrace $ , respectively, and $\\mathbf {N}$ represents a zero-mean Gaussian noise vector with the positive definite covariance matrix $\\Sigma $ ; i.e., $\\mathbf {N}\\sim \\mathcal {N} (\\mathbf {0},\\Sigma )$ .",
"All the other parameters ($\\pi _i^k$ and $C^k_{ji}$ for $i,j\\in \\lbrace 0,1\\rbrace $ and $k\\in \\lbrace t,r\\rbrace $ ) and their definitions remain unchanged."
],
[
"Team Setup Analysis",
"Theorem 5.1 Theorem REF also holds for the vector case: if $0<\\tau <\\infty $ , the team solution is always informative; otherwise, there exist only non-informative equilibria.",
"Let the transmitter choose optimal signals $\\mathcal {S}=\\lbrace \\mathbf {S}_0,\\mathbf {S}_1\\rbrace $ .",
"Then the measurements become $\\mathcal {H}_i : \\mathbf {Y}\\sim \\mathcal {N} (\\mathbf {S}_i,\\Sigma )$ for $i\\in \\lbrace 0,1\\rbrace $ .",
"As in the scalar case in Theorem REF , the equilibrium is non-informative for $\\tau \\le 0$ or $\\tau =\\infty $ ; hence, $0<\\tau <\\infty $ can be assumed.",
"Similar to (REF ), the optimal decision rule for the receiver is obtained by utilizing (REF ) as $\\delta ^*_{\\mathbf {S}_0,\\mathbf {S}_1} &: \\Bigg \\lbrace \\zeta { p_1(\\mathbf {y})\\over p_0(\\mathbf {y})} \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta {\\pi _0^r (C^r_{10}-C^r_{00}) \\over \\pi _1^r (C^r_{01}-C^r_{11})} \\triangleq \\zeta \\tau \\nonumber \\\\&:\\Bigg \\lbrace \\zeta { {1\\over \\sqrt{(2\\pi )^n|\\Sigma |}}\\exp \\left\\lbrace -{1\\over 2}(\\mathbf {y}-\\mathbf {S}_1)^T\\Sigma ^{-1}(\\mathbf {y}-\\mathbf {S}_1)\\right\\rbrace \\over {1\\over \\sqrt{(2\\pi )^n|\\Sigma |}}\\exp \\left\\lbrace -{1\\over 2}(\\mathbf {y}-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {y}-\\mathbf {S}_0)\\right\\rbrace } \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau \\nonumber \\\\\\begin{split}&:\\Bigg \\lbrace \\zeta (\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}\\mathbf {y}\\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\left(\\ln (\\tau )+{1\\over 2}(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1+\\mathbf {S}_0)\\right)\\;.\\end{split}$ Since, under hypothesis $\\mathcal {H}_i$ , $\\zeta (\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}\\mathbf {Y}\\sim \\mathcal {N} \\left(\\zeta (\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}\\mathbf {S}_i,(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1-\\mathbf {S}_0)\\right)$ for $i\\in \\lbrace 0,1\\rbrace $ , by defining $d^2\\triangleq (\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1-\\mathbf {S}_0)$ , the conditional probabilities can be written as follows: $\\mathsf {P}_{10}&=\\mathcal {Q}\\left(\\zeta {\\ln (\\tau )+{1\\over 2}(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1+\\mathbf {S}_0-2\\mathbf {S}_0)\\over \\sqrt{(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1-\\mathbf {S}_0)}}\\right)=\\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau )\\over d}+{d\\over 2}\\right)\\right)\\,, \\nonumber \\\\\\mathsf {P}_{01}&=1-\\mathcal {Q}\\left(\\zeta {\\ln (\\tau )+{1\\over 2}(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1+\\mathbf {S}_0-2\\mathbf {S}_1)\\over \\sqrt{(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1-\\mathbf {S}_0)}}\\right)=1-\\mathcal {Q}\\left(\\zeta \\left({\\ln (\\tau ) \\over d} -{d\\over 2}\\right)\\right)\\nonumber \\\\&=\\mathcal {Q}\\left(\\zeta \\left(-{\\ln (\\tau )\\over d}+{d\\over 2}\\right)\\right)\\;.$ Notice that the conditional probabilities are the same in (REF ) and (REF ); therefore, in the vector case, the equilibrium is always informative, and the transmitter always prefers the maximum distance similar to the scalar case.",
"However, selecting optimal vector signals is not as trivial as in the scalar case; see [14] for details.",
"Since the eigenvector with the largest (smallest) eigenvalue of $\\Sigma $ corresponds to the direction, along which the noise is most (least) powerful, signaling in the least noisy direction results in the highest signal-to-noise power ratio for the system.",
"Accordingly, the optimum signals are $\\mathbf {S}_0 = \\pm \\sqrt{P_0} {\\nu _{\\min }\\over \\Vert \\nu _{\\min }\\Vert }$ and $\\mathbf {S}_1 = \\mp \\sqrt{P_1} {\\nu _{\\min }\\over \\Vert \\nu _{\\min }\\Vert }$ , which corresponds to $d_{\\max }^2={(\\sqrt{P_0}+\\sqrt{P_1})^2\\over \\lambda _{\\min }}$ , where $\\lambda _{\\min }$ is the minimum eigenvalue of $\\Sigma $ and $\\nu _{\\min }$ is the eigenvector corresponding to $\\lambda _{\\min }$ [14]."
],
[
"Stackelberg Game Analysis",
"Theorem 5.2 Let $d\\triangleq \\sqrt{(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1-\\mathbf {S}_0)}$ and $d_{\\max }^2\\triangleq {(\\sqrt{P_0}+\\sqrt{P_1})^2\\over \\lambda _{\\min }}$ , where $\\lambda _{\\min }$ is the minimum eigenvalue of $\\Sigma $ .",
"Then Theorem REF also holds for the vector case.",
"The proof of Theorem REF can be applied by modifying the definitions of $d$ and $d_{\\max }$ as in the statement.",
"For $d^*=d_{\\max }$ , the method described in the proof of Theorem REF can be applied for the optimal signal selection, whereas, for $d^*=0$ , by choosing $\\mathbf {S}_0=\\mathbf {S}_1$ , the non-informative equilibrium can be achieved.",
"Further, for Case-3 of Theorem REF , in order to achieve $(d^*)^2=\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |<d_{\\max }^2$ , the signals can be chosen in the direction of $\\nu _{\\min }$ , that is, the eigenvector corresponding to $\\lambda _{\\min }$ .",
"Accordingly, $\\mathbf {S}_0=(-\\sqrt{P_0}+t) {\\nu _{\\min }\\over \\Vert \\nu _{\\min }\\Vert }$ and $\\mathbf {S}_1=(-\\sqrt{P_0}+d^*+t) {\\nu _{\\min }\\over \\Vert \\nu _{\\min }\\Vert }$ for $t\\in [0,\\sqrt{P_1}+\\sqrt{P_0}-d^*]$ are possible optimal signal pairs.",
"Similarly, $\\mathbf {S}_0=(\\sqrt{P_0}-t) {\\nu _{\\min }\\over \\Vert \\nu _{\\min }\\Vert }$ and $\\mathbf {S}_1=(\\sqrt{P_0}-d^*-t) {\\nu _{\\min }\\over \\Vert \\nu _{\\min }\\Vert }$ for $t\\in [0,\\sqrt{P_1}+\\sqrt{P_0}-d^*]$ consist of another set of possible optimal signal pairs.",
"Note that it may be possible to find optimal signal pairs $\\lbrace \\mathbf {S}_0,\\mathbf {S}_1\\rbrace \\in \\mathbb {S}$ that satisfy $(\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}(\\mathbf {S}_1-\\mathbf {S}_0)=\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |$ in any other direction rather than the direction of $\\nu _{\\min }$ ; however, finding a single pair that corresponds to an equilibrium would be sufficient."
],
[
"Nash Game Analysis",
"Theorem 5.3 Theorem REF also holds for the vector case.",
"Let the transmitter choose any signals $\\mathcal {S}=\\lbrace \\mathbf {S}_0,\\mathbf {S}_1\\rbrace $ .",
"Assuming nonzero priors $\\pi _0^t, \\pi _0^r, \\pi _1^t$ and $\\pi _1^r$ , the optimal decision rule for the receiver is given by (REF ).",
"Similar to the team case analysis in Section REF , the equilibrium is non-informative if $\\tau \\le 0$ or $\\tau =\\infty $ ; thus, $0<\\tau <\\infty $ can be assumed.",
"Now assume that the receiver applies a single-threshold rule; i.e., $\\delta : \\Bigg \\lbrace \\mathbf {a}^T \\mathbf {y}\\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\eta $ where $\\mathbf {a}\\in \\mathbb {R}^n$ and $\\eta \\in \\mathbb {R}$ .",
"Remark 5.1 Note that for $\\mathbf {a}=\\mathbf {0}$ , the receiver chooses either always $\\mathcal {H}_0$ or always $\\mathcal {H}_1$ without considering the value of $\\mathbf {y}$ , which implies a non-informative equilibrium.",
"Therefore, $\\mathbf {S}_0^*=\\mathbf {S}_1^*$ , $\\mathbf {a}^*=\\mathbf {0}$ , and $\\eta ^*=\\zeta (\\tau -1)$ (i.e., the decision rule of the receiver is $\\delta ^* : \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau $ ) constitute a non-informative equilibrium regardless of the values of the priors and costs of the players.",
"Thus, due to the remark above, it can be assumed that $\\mathbf {a}\\ne \\mathbf {0}$ holds.",
"Since $\\mathbf {a}^T \\mathbf {Y}\\sim \\mathcal {N} \\Big (\\mathbf {a}^T \\mathbf {S}_i,\\mathbf {a}^T\\Sigma \\mathbf {a}\\Big )$ under $\\mathcal {H}_i$ for $i\\in \\lbrace 0,1\\rbrace $ , the conditional probabilities are $\\mathsf {P}_{10}=\\mathcal {Q}\\left(\\eta -\\mathbf {a}^T \\mathbf {S}_0\\over \\sqrt{\\mathbf {a}^T\\Sigma \\mathbf {a}}\\right)$ and $\\mathsf {P}_{01}=\\mathcal {Q}\\left(-{\\eta -\\mathbf {a}^T \\mathbf {S}_1\\over \\sqrt{\\mathbf {a}^T\\Sigma \\mathbf {a}}}\\right)$ .",
"Then, the Bayes risk of the transmitter becomes $r^t(\\mathcal {S},\\delta ) = \\pi _0^t C^t_{00} + \\pi _1^t C^t_{11} &+ \\pi _0^t (C^t_{10}-C^t_{00})\\mathcal {Q}\\left(\\eta -\\mathbf {a}^T \\mathbf {S}_0\\over \\sqrt{\\mathbf {a}^T\\Sigma \\mathbf {a}}\\right) + \\pi _1^t (C^t_{01}-C^t_{11})\\mathcal {Q}\\left(-{\\eta -\\mathbf {a}^T \\mathbf {S}_1\\over \\sqrt{\\mathbf {a}^T\\Sigma \\mathbf {a}}}\\right) \\,.$ Since the power constraints are $\\Vert \\mathbf {S}_0 \\Vert ^2 \\le P_0$ and $\\Vert \\mathbf {S}_1 \\Vert ^2 \\le P_1$ , the signals $\\mathbf {S}_0$ and $\\mathbf {S}_1$ can be regarded as independent.",
"Since $\\mathcal {Q}$ function is a monotone decreasing, the following cases hold for $i\\in \\lbrace 0,1\\rbrace $ : $C^t_{1i}<C^t_{0i}$ $\\Rightarrow $ Then, $r^t(\\mathcal {S},\\delta )$ is a decreasing function of $\\mathbf {a}^T \\mathbf {S}_i$ , thus the transmitter always chooses $\\mathbf {a}^T \\mathbf {S}_i$ as maximum subject to $\\Vert \\mathbf {S}_i \\Vert ^2 \\le P_i$ ; i.e., $\\mathbf {S}_i=\\sqrt{P_i}{\\mathbf {a}\\over \\Vert \\mathbf {a}\\Vert }$ .",
"$C^t_{1i}=C^t_{0i}$ $\\Rightarrow $ Then $\\mathbf {S}_i$ has no effect on the Bayes risk of the transmitter.",
"$C^t_{1i}>C^t_{0i}$ $\\Rightarrow $ Then, $r^t(\\mathcal {S},\\delta )$ is an increasing function of $\\mathbf {a}^T \\mathbf {S}_i$ , thus the transmitter always chooses $\\mathbf {a}^T \\mathbf {S}_i$ as minimum subject to $\\Vert \\mathbf {S}_i \\Vert ^2 \\le P_i$ ; i.e., $\\mathbf {S}_i=-\\sqrt{P_i}{\\mathbf {a}\\over \\Vert \\mathbf {a}\\Vert }$ .",
"Thus, the the optimal signals can be characterized as $S_0=-\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}{\\mathbf {a}\\over \\Vert \\mathbf {a}\\Vert }$ and $S_1=\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}{\\mathbf {a}\\over \\Vert \\mathbf {a}\\Vert }$ .",
"By using the expressions above, the cases can be listed as follows: $\\tau \\le 0$ or $\\tau =\\infty $ $\\Rightarrow $ The equilibrium is non-informative.",
"$C^t_{10}=C^t_{00}$ (and/or $C^t_{01}=C^t_{11}$ ) $\\Rightarrow $ $\\mathbf {S}_0$ (and/or $\\mathbf {S}_1$ ) has no effect on the Bayes risk of the transmitter, thus it can arbitrarily be chosen by the transmitter.",
"In this case, if the transmitter chooses $\\mathbf {S}_0=\\mathbf {S}_1$ ; i.e., he does not send anything useful to the receiver, and the receiver applies the decision rule $\\delta : \\zeta \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }} \\zeta \\tau $ ; i.e., he only considers the prior information (totally discards the information sent by the transmitter).",
"Then there exists a non-informative equilibrium.",
"Notice that, since $0<\\tau <\\infty $ is assumed, $\\zeta =\\text{sgn}(C^r_{01}-C^r_{11})=\\text{sgn}(C^r_{10}-C^r_{00})$ is obtained.",
"Now, assume that the decision rule of the receiver is $\\delta : \\Bigg \\lbrace \\mathbf {a}^T \\mathbf {y}\\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\eta $ .",
"Then, the transmitter selects $S_0=-\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}{\\mathbf {a}\\over \\Vert \\mathbf {a}\\Vert }$ and $S_1=\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}{\\mathbf {a}\\over \\Vert \\mathbf {a}\\Vert }$ as optimal signals, and the decision rule becomes (REF ).",
"By combining the best responses of the transmitter and the receiver, $\\mathbf {a}^T&=\\zeta (\\mathbf {S}_1-\\mathbf {S}_0)^T\\Sigma ^{-1}=\\zeta {\\mathbf {a}^T\\over \\Vert \\mathbf {a}\\Vert }\\left(\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}+\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}\\right)\\Sigma ^{-1} \\nonumber \\\\\\Rightarrow & \\mathbf {a}^T\\mathbf {a}= {\\mathbf {a}^T\\Sigma ^{-1}\\mathbf {a}\\over \\Vert \\mathbf {a}\\Vert } \\zeta \\left(\\text{sgn}(C^t_{01}-C^t_{11})\\sqrt{P_1}+\\text{sgn}(C^t_{10}-C^t_{00})\\sqrt{P_0}\\right) \\nonumber \\\\\\Rightarrow & \\underbrace{\\text{sgn}(C^t_{01}-C^t_{11})\\over \\text{sgn}(C^r_{01}-C^r_{11})}_{=\\text{sgn}(\\xi _1)}\\sqrt{P_1}+\\underbrace{\\text{sgn}(C^t_{10}-C^t_{00})\\over \\text{sgn}(C^r_{10}-C^r_{00})}_{=\\text{sgn}(\\xi _0)}\\sqrt{P_0} > 0 \\,.$ Notice that the expressions in (REF ) and (REF ) of Theorem REF are the same, and Remark REF and Remark REF are equivalent; hence, the Nash equilibrium solution of Theorem REF also holds for the vector case."
],
[
"EXTENSION TO a SCENARIO with an AVERAGE POWER CONSTRAINT",
"Besides the peak power constraint considered in the previous sections, the average power constraint can be assumed at the transmitter side.",
"Before presenting the technical results, we provide the following lemma which will be utilized in the equilibrium analyses of the team and Stackelberg setups.",
"Lemma 6.1 The optimal solutions to the optimization problem $\\underset{{S_0,S_1}}{\\text{sup}} ~ (S_1-S_0)^2 ~~~\\text{s.t.}",
"~~ \\beta _0S_0^2+\\beta _1S_1^2 \\le P, ~~ \\beta _0,\\beta _1\\in \\mathbb {R}_{>0}$ are $(S_0^*,S_1^*)=\\left(-\\sqrt{{\\beta _1\\over \\beta _0(\\beta _0+\\beta _1)}P},\\sqrt{{\\beta _0\\over \\beta _1(\\beta _0+\\beta _1)}P}\\right)$ and $(S_0^*,S_1^*)=\\left(\\sqrt{{\\beta _1\\over \\beta _0(\\beta _0+\\beta _1)}P},-\\sqrt{{\\beta _0\\over \\beta _1(\\beta _0+\\beta _1)}P}\\right)$ .",
"Observe the following inequalities: $\\beta _0\\beta _1(S_1-S_0)^2&=\\beta _0\\beta _1\\left(S_1^2-2S_1S_0+S_0^2\\right)\\overset{(a)}{\\le } \\beta _0\\beta _1\\left(S_1^2+2|S_1||S_0|+S_0^2\\right) \\nonumber \\\\&= \\beta _0\\beta _1\\left(S_1^2+S_0^2\\right)+2|\\beta _0S_0||\\beta _1S_1|\\overset{(b)}{\\le } \\beta _0\\beta _1\\left(S_1^2+S_0^2\\right)+\\beta _0^2S_0^2+\\beta _1^2S_1^2 \\nonumber \\\\&= \\beta _0\\left(\\beta _0S_0^2+\\beta _1S_1^2\\right) + \\beta _1\\left(\\beta _0S_0^2+\\beta _1S_1^2\\right)\\overset{(c)}{\\le } (\\beta _0+\\beta _1)P \\,.$ Here, (b) follows from the inequality for the arithmetic and geometric mean, and the equality holds iff $\\beta _1^2S_1^2=\\beta _0^2S_0^2$ .",
"For (a), the equality holds iff $S_1S_0\\le 0$ ; and for (c), the equality holds iff $\\beta _0S_0^2+\\beta _1S_1^2=P$ .",
"Thus, the upper bound of $(S_1-S_0)^2$ can be achieved with optimal solutions $(S_0^*,S_1^*)=\\left(-\\sqrt{{\\beta _1\\over \\beta _0(\\beta _0+\\beta _1)}P},\\sqrt{{\\beta _0\\over \\beta _1(\\beta _0+\\beta _1)}P}\\right)$ or $(S_0^*,S_1^*)=\\left(\\sqrt{{\\beta _1\\over \\beta _0(\\beta _0+\\beta _1)}P},-\\sqrt{{\\beta _0\\over \\beta _1(\\beta _0+\\beta _1)}P}\\right)$ so that $(S_1^*-S_0^*)^2={\\beta _0+\\beta _1\\over \\beta _0\\beta _1}P$ .",
"Consider a transmitter with an average power constraint; i.e., the transmitter performs the optimal signal design problem under the power constraint below: $\\mathbb {S}\\triangleq \\lbrace \\mathcal {S}=\\lbrace S_0,S_1\\rbrace :\\pi _0^t| S_0 | ^2 + \\pi _1^t| S_1 | ^2\\le P_{\\mathrm {avg}}\\rbrace \\;,$ where $P_{\\mathrm {avg}}$ denotes the average power limit."
],
[
"Team Theoretic Analysis",
"In order to minimize the Bayes risk, the transmitter always prefers the maximum $d={|S_1-S_0|\\over \\sigma }$ .",
"Thus, by Lemma REF , the optimal signal levels are chosen as either $(S_0^*,S_1^*)=\\left(-\\sqrt{{\\pi _1^t\\over \\pi _0^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}},\\sqrt{{\\pi _0^t\\over \\pi _1^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}}\\right)$ or $(S_0^*,S_1^*)=\\left(\\sqrt{{\\pi _1^t\\over \\pi _0^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}},-\\sqrt{{\\pi _0^t\\over \\pi _1^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}}\\right)$ .",
"The corresponding optimal decision rule of the receiver is chosen based on the rule in (REF ) accordingly.",
"Actually, the equilibrium points are essentially unique; i.e., they result in the same Bayes risks for the transmitter and the receiver."
],
[
"Stackelberg Game Analysis",
"Similar to the team setup analysis, for every possible case in Table REF , there are more than one equilibrium points, and they are essentially unique since the Bayes risks of the transmitter and the receiver depend on $d$ .",
"For example, for $d^*=d_{\\max }\\triangleq {\\sqrt{{\\pi _0^t+\\pi _1^t\\over \\pi _0^t\\pi _1^t}P_{\\mathrm {avg}}}\\over \\sigma }$ , $(S_0^*,S_1^*)=\\left(-\\sqrt{{\\pi _1^t\\over \\pi _0^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}},\\sqrt{{\\pi _0^t\\over \\pi _1^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}}\\right)$ and $(S_0^*,S_1^*)=\\left(\\sqrt{{\\pi _1^t\\over \\pi _0^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}},-\\sqrt{{\\pi _0^t\\over \\pi _1^t(\\pi _0^t+\\pi _1^t)}P_{\\mathrm {avg}}}\\right)$ are the only possible choices for the transmitter, and the decision rule of the receiver is chosen based on the rule in (REF ).",
"However, for $d^*=0$ , there are infinitely many choices for the transmitter and the receiver, and all of them are essentially unique; i.e., they result in the same Bayes risks for the transmitter and the receiver.",
"A similar argument holds for $d^*=\\sqrt{\\Big |{2\\ln \\tau (k_0-k_1)\\over (k_0+k_1)}\\Big |}$ ; i.e., there are infinitely many choices for the transmitter and the receiver, and all of them are essentially unique."
],
[
"Nash Game Analysis",
"For $0<\\tau <\\infty $ , if the receiver applies a single-threshold ruleDue to Remark REF , $S_0^*=S_1^*$ , $a^*=0$ , and $\\eta ^*=\\zeta (\\tau -1)$ always constitute a non-informative equilibrium regardless of the values of the priors and costs of the players; i.e., $\\delta : \\Bigg \\lbrace a y \\overset{\\mathcal {H}_1}{\\underset{\\mathcal {H}_0}{\\gtreqless }}\\eta $ where $a\\in \\mathbb {R}-\\lbrace 0\\rbrace $ , and $\\eta \\in \\mathbb {R}$ , after analyzing the derivative of the Bayes risk of the transmitter in (REF ) with respect to the signals, the following can be obtained: $C^t_{1i}=C^t_{0i}$ $\\Rightarrow $ $S_i$ has no effect on the Bayes risk of the transmitter.",
"$C^t_{1i}<C^t_{0i}$ or $C^t_{1i}>C^t_{0i}$ $\\Rightarrow $ $r^t(\\mathcal {S},\\delta )$ is a decreasing (increasing) function of $S_i$ if $a(C^t_{1i}-C^t_{0i})$ is negative (positive); thus the transmitter chooses the optimal signal level $S_i$ as large as possible in absolute value.",
"Therefore, the transmitter prefers to utilize the maximum possible total power; i.e., the power constraint can be considered as $\\pi _1^tS_1^2+\\pi _0^tS_0^2=P_{\\mathrm {avg}}$ rather than $\\pi _1^tS_1^2+\\pi _0^tS_0^2\\le P_{\\mathrm {avg}}$ .",
"By using the analysis above, the cases can be listed as follows: $C^t_{1i}=C^t_{0i}$ $\\Rightarrow $ If $C^t_{1j}=C^t_{0j}$ also holds for $j\\ne i$ , then neither $S_0$ nor $S_1$ changes the Bayes risk of the transmitter; thus, there exists a non-informative equilibrium.",
"Otherwise; i.e., $C^t_{1j}\\ne C^t_{0j}$ for $j\\ne i$ , the transmitter chooses the optimal signal levels as $S_i=0$ and $S_j=-\\text{sgn}\\left(a(C^t_{1j}-C^t_{0j})\\right)\\sqrt{P_{\\mathrm {avg}}\\over \\pi _j^t}$ , and the equilibrium is informative.",
"$C^t_{10}\\ne C^t_{00}$ and $C^t_{11}\\ne C^t_{01}$ $\\Rightarrow $ Since the transmitter adjust the signal levels such that $\\pi _1^tS_1^2+\\pi _0^tS_0^2=P$ , the optimal signals must be in the form of $S_0=-\\text{sgn}\\big (a(C^t_{10}-C^t_{00})\\big )x$ and $S_1=\\text{sgn}\\big (a(C^t_{01}-C^t_{11})\\big )\\sqrt{P_{\\mathrm {avg}}-\\pi _0^tx^2\\over \\pi _1^t}$ for $x\\in \\left[0,\\sqrt{P_{\\mathrm {avg}}\\over \\pi _0^t}\\right]$ .",
"Then, the Bayes risk of the transmitter in (REF ) can be expressed as $\\begin{split}r^t(\\mathcal {S},\\delta ) &= \\pi _0^t C^t_{00} + \\pi _1^t C^t_{11} + \\pi _0^t (C^t_{10}-C^t_{00})\\mathcal {Q}\\left(\\eta +|a|\\text{sgn}\\left(C^t_{10}-C^t_{00}\\right)x\\over |a|\\sigma \\right)\\\\&\\qquad + \\pi _1^t (C^t_{01}-C^t_{11})\\mathcal {Q}\\left(-{\\eta -|a|\\text{sgn}\\left(C^t_{01}-C^t_{11}\\right)\\sqrt{P_{\\mathrm {avg}}-\\pi _0^tx^2\\over \\pi _1^t}\\over |a|\\sigma }\\right) \\,.\\end{split}$ Note that the convexity of $r^t(\\mathcal {S},\\delta )$ in (REF ) with respect to $x$ changes depending on the other parameters (i.e., priors, costs and the receiver policy); hence, the optimal $x$ cannot be expressed in a closed form.",
"Let $x^*$ be an optimal solution to (REF ); i.e., $x^*=\\arg \\min _{x\\in \\left[0,\\sqrt{P_{\\mathrm {avg}}\\over \\pi _0^t}\\right]} r^t(\\mathcal {S},\\delta )$ , which implies that the optimal signal levels are $S_0=-\\text{sgn}\\big (a(C^t_{10}-C^t_{00})\\big )x^*$ and $S_1=\\text{sgn}\\big (a(C^t_{01}-C^t_{11})\\big )\\sqrt{P_{\\mathrm {avg}}-\\pi _0^t(x^*)^2\\over \\pi _1^t}$ .",
"Then, similar to (REF ), the following condition on the existence of an equilibrium can be obtained: $\\underbrace{\\text{sgn}(C^t_{01}-C^t_{11})\\over \\text{sgn}(C^r_{01}-C^r_{11})}_{=\\text{sgn}(\\xi _1)}\\sqrt{P_{\\mathrm {avg}}-\\pi _0^t(x^*)^2\\over \\pi _1^t}+\\underbrace{\\text{sgn}(C^t_{10}-C^t_{00})\\over \\text{sgn}(C^r_{10}-C^r_{00})}_{=\\text{sgn}(\\xi _0)}x^* > 0 \\,.$ Here, similar to the analysis under the individual power constraint in Theorem REF , unless (REF ) is satisfied, the best responses of the transmitter and the receiver cannot match each other.",
"In particular, $\\xi _0<0$ and $\\xi _1<0$ $\\Rightarrow $ There does not exist a Nash equilibrium for $a\\ne 0$ ; however, due to Remark REF , for $a=0$ , there always exist non-informative equilibria.",
"$\\xi _0<0$ and $\\xi _1>0$ $\\Rightarrow $ If $\\sqrt{P_{\\mathrm {avg}}-\\pi _0^t(x^*)^2\\over \\pi _1^t}>x^* \\Rightarrow x^* < \\sqrt{P_{\\mathrm {avg}}}$ , then the Nash equilibrium is informative.",
"If $x^*=\\sqrt{P_{\\mathrm {avg}}}$ , there exists a non-informative equilibrium.",
"Otherwise; i.e., if $x^*>\\sqrt{P_{\\mathrm {avg}}}$ , there does not exist a Nash equilibrium for $a\\ne 0$ ; however, due to Remark REF , for $a=0$ , there always exist non-informative equilibria.",
"$\\xi _0>0$ and $\\xi _1<0$ $\\Rightarrow $ If $x^* > \\sqrt{P_{\\mathrm {avg}}}$ , then the Nash equilibrium is informative.",
"If $x^*=\\sqrt{P_{\\mathrm {avg}}}$ , there exists a non-informative equilibrium.",
"Otherwise; i.e., if $x^*<\\sqrt{P_{\\mathrm {avg}}}$ , there does not exist a Nash equilibrium for $a\\ne 0$ ; however, due to Remark REF , for $a=0$ , there always exist non-informative equilibria.",
"$\\xi _0>0$ and $\\xi _1>0$ $\\Rightarrow $ There exists an informative Nash equilibrium."
],
[
"CONCLUDING REMARKS",
"In this paper, we considered binary signaling problems in which the decision makers (the transmitter and the receiver) have subjective priors and/or misaligned objective functions.",
"Depending on the commitment nature of the transmitter to his policies, we formulated the binary signaling problem as a Bayesian game under either Nash or Stackelberg equilibrium concepts and established equilibrium solutions and their properties.",
"We showed that there can be informative or non-informative equilibria in the binary signaling game under the Stackelberg and Nash assumptions, and derived the conditions under which an informative equilibrium exists.",
"We also studied the effects of small perturbations around the team setup (with identical priors and costs) and showed that the game equilibrium behavior around the team setup is robust under the Nash assumption, whereas it is not robust under the Stackelberg assumption.",
"The binary setup considered here can be extended to the $M$ -ary hypothesis testing setup, and the corresponding signaling game structure can be formed in order to model a game between players with a multiple-bit communication channel.",
"The extension to more general noise distributions is possible: the Nash equilibrium analysis holds identically when the noise distribution leads to a single-threshold test.",
"Finally, in addition to the Bayesian approach considered here, different cost structures and parameters can be introduced by investigating the game under Neyman-Pearson and mini-max criteria."
]
] | 1906.04577 |
[
[
"Probabilistic Forecasting with Temporal Convolutional Neural Network"
],
[
"Abstract We present a probabilistic forecasting framework based on convolutional neural network for multiple related time series forecasting.",
"The framework can be applied to estimate probability density under both parametric and non-parametric settings.",
"More specifically, stacked residual blocks based on dilated causal convolutional nets are constructed to capture the temporal dependencies of the series.",
"Combined with representation learning, our approach is able to learn complex patterns such as seasonality, holiday effects within and across series, and to leverage those patterns for more accurate forecasts, especially when historical data is sparse or unavailable.",
"Extensive empirical studies are performed on several real-world datasets, including datasets from JD.com, China's largest online retailer.",
"The results show that our framework outperforms other state-of-the-art methods in both accuracy and efficiency."
],
[
"Introduction",
"Time series forecasting plays a key role in many business decision-making scenarios, such as managing limited resources, optimizing operational processes, among others.",
"Most existing forecasting methods focus on point forecasting, i.e., forecasting the conditional mean or median of future observations.",
"However, probabilistic forecasting becomes increasingly important as it is able to extract richer information from historical data and better capture the uncertainty of the future.",
"In retail business, probabilistic forecasting of product supply and demand is fundamental for successful procurement process and optimal inventory planning.",
"Also, probabilistic shipment forecasting, i.e., generating probability distributions of the delivery volumes of packages, is the key component of the consequent logistics operations, such as labor resource planning and delivery vehicle deployment.",
"In such circumstances, instead of predicting individual or a small number of time series, one needs to predict thousands or millions of related series.",
"Moreover, there are many more challenges in real-world applications.",
"For instance, new products emerge weekly on retail platforms and one often needs to forecast the demand of products without historical shopping festival data (e.g., Black Friday in North America, “11.11” shopping festival in China).",
"Furthermore, forecasting often requires the consideration of exogenous variables that have significant influence on future demand (e.g., promotion plans provided by operations teams, accurate weather forecasts for brick and mortar retailers).",
"Such forecasting problems can be extended to a variety of domains.",
"Examples include forecasting the web traffic for internet companies , the energy consumption for individual households, the load for servers in a data center and traffic flows in transportation domain .",
"Classical forecasting methods, such as ARIMA and exponential smoothing , are widely employed for univariate base-level forecasting.",
"To incorporate exogenous covariates, several extensions of these methods have been proposed, such as ARIMAX (AutoRegressive Integrated Moving Average with Explanatory Variable) and dynamic regression models .",
"These models are well-suited for applications in which the structure of the data is well understood and there is sufficient historical data.",
"However, working with thousands or millions of series requires prohibitive labor and computing resources for parameter estimation.",
"Moreover, they are not applicable in situations where historical data is sparse or unavailable.",
"Models based on Recurrent neural network (RNN) and the sequence to sequence (Seq2Seq) framework , have achieved great success in many different sequential tasks such as machine translation , language modeling and recently time series forecasting , , , , , .",
"For example, in the forecasting competition community, the Seq2Seq model based on a gated recurrent unit (GRU) won the Kaggle web traffic forecasting competition .",
"A hybrid model that combines exponential smoothing method and RNN won the M4 forecasting competition, which consists of 100,000 series with different seasonal patterns .",
"However, training with back propagation through time (BPTT) algorithm often hampers efficient computation.",
"In addition, training RNN can be remarkably difficult , .",
"Dilated causal convolutional architectures, e.g., Wavenet , offers an alternative for modeling sequential data.",
"By stacking layers of dilated causal convolutional nets, receptive fields can be increased, and the long-term correlations can be captured without violating the temporal orders.",
"In addition, in dilated causal convolutional architectures, the training process can be performed in parallel, which guarantees computation efficiency.",
"Most Seq2Seq frameworks or Wavenet are autoregressive generative models that factorize the joint distribution as a product of the conditionals.",
"In this setting, a one-step-ahead prediction approach is adopted, i.e., first a prediction is generated by using the past observations, and the generated result is then fed back as the ground truth to make further forecasts.",
"More recent research shows that non-autoregressive approaches or direct prediction strategy, predicting observations of all time steps directly, can achieve better performances , , .",
"In particular, non-autoregressive models are more robust to mis-specification by avoiding error accumulation and thus yield better prediction accuracy.",
"Moreover, training over all the prediction horizons can be parallelized.",
"Having reviewing all these challenges and developments, in this paper, we propose the Deep Temporal Convolutional Network (DeepTCN), a non-autoregressive probabilistic forecasting framework for large collections of related time series.",
"The main contributions of the paper are as follows: We propose a CNN-based forecasting framework that provides both parametric and non-parametric approaches for probability density estimation.",
"The framework, being able to learn latent correlation among series and handle complex real-world forecasting situations such as data sparsity and cold starts, shows high scalability and extensibility.",
"The model is very flexible and can include exogenous covariates such as additional promotion plans or weather forecasts.",
"Extensive empirical studies show our framework compares favorably to state-of-the-art methods in both point forecasting and probabilistic forecasting tasks.",
"The rest of this paper is organized as follows.",
"Section provides a brief review of related work on time series forecasting and deep learning methods for forecasting.",
"In Section , we describe the proposed forecasting method, including the neural network architectures, the probabilistic forecasting framework, and the input features.",
"We demonstrate the superiority of the proposed approach via extensive experiments in Section and conclude the paper in Section ."
],
[
"Related Work",
"Earlier studies on time series forecasting are mostly based on statistical models, which are mainly generative models based on state space framework such as exponential smoothing, ARIMA models and several other extensions.",
"For these methods, and provide a comprehensive overview in the context of univariate forecasting.",
"In recent years, large number of related series are emerging in the routine functioning of many companies.",
"Not sharing information from other time series, traditional univariate forecasting methods fit a model for each individual time series, and thus cannot learn across similar time series.",
"Moreover, numerous researchers have shown that pure machine learning methods could not outperform statistical approaches in forecasting individual time series, the reasons for which can be attributed to overfitting and non-stationarity , .",
"Therefore, methods that can provide forecasting on multiple series jointly have received increasing attention in the last few years .",
"Both RNNs and CNNs have been shown to be able to model complex nonlinear feature interactions and yield substantial forecasting performances, especially when many related time series are available , , , , .",
"For example, Long Short-Term Memory (LSTM), one type of RNN architecture, won the CIF2016 forecasting competition for monthly time series .",
"compare a variety of RNNs in their performances in the Short Term Load Forecasting problem.",
"investigate the application of CNNs to financial time series forecasting.",
"To better understand the uncertainty of the future, probabilistic forecasting with deep learning models has attracted increasing attention.",
"DeepAR , which trains an auto-regressive RNN model on a rich collection of similar time series, produces more accurate probabilistic forecasts on several real-world data sets.",
"The deep state space models (DeepState), presented by , combine state space models with deep learning and can retain data efficiency and interpretability while learning the complex patterns from raw data.",
"Under a similar scheme, propose the combination of deep neural networks and Gaussian Process.",
"More recently, propose SQF-RNN, a probabilistic framework to model conditional quantile functions with isotonic splines, which allows more flexible output distributions.",
"Most of these probabilistic forecasting frameworks are autoregressive models, which use recursive strategy to generate multi-step forecasts.",
"In neural machine translation, non-autoregressive translation (NAT) models have achieved significant speedup at the cost of slightly inferior accuracy compared to autoregressive translation models .",
"For example, propose a non-autoregressive framework based on dilated causal convolution and the empirical study on multiple datasets shows that the framework outperforms generic recurrent architectures such as LSTMs and GRUs.",
"In forecasting applications, non-autoregressive approaches have also been shown to be less biased and more robust.",
"Recently, present a multi-horizon quantile recurrent forecaster to combine sequential neural nets and quantile regression .",
"By training on all time points at the same time, their framework can significantly improve the training stability and the forecasting performances of recurrent nets.",
"Our method differs from the aforementioned approaches in the following ways.",
"First, stacked dilated causal convolutional nets are constructed to represent the encoder and model the stochastic process of historical observations of series.",
"Instead of applying gating mechanism (e.g., in Wavenet ), residual blocks are used for the dilated causal convolutional nets to extract information of historical observations and help achieve superior forecasting accuracy.",
"Second, inspired by the dynamic regression models such as ARIMAX, in the decoder part, a novel variant of the residual neural network is proposed to incorporate information from both past observations and exogenous covariates.",
"Finally, our model enjoys the flexibility to embrace a variety of probability density estimation approaches."
],
[
"Method",
"A general probabilistic forecasting problem for multiple related time series can be described as follows: Given a set of time series ${\\mathbf {y}}_{1:t} = \\lbrace y_{1:t}^{(i)}\\rbrace _{i=1}^N$ , we denote the future time series as ${\\mathbf {y}}_{(t+1):(t+\\Omega )} = \\lbrace y_{(t+1):(t+\\Omega )}^{(i)}\\rbrace _{i=1}^N$ , where $N$ is the number of series, $t$ is the length of the historical observations and $\\Omega $ is the length of the forecasting horizon.",
"Our goal is to model the conditional distribution of the future time series $P\\left({\\mathbf {y}}_{(t+1):(t+\\Omega )}|{{\\mathbf {y}}_{1:t}}\\right)$ .",
"Classical generative models are often used to model time series data, which factorize the joint probability of future observations given the past information as the product of conditional probabilities: $P\\left({\\mathbf {y}}_{(t+1):(t+\\Omega )}|{\\mathbf {y}}_{1:t}\\right) = \\prod \\limits _{\\omega =1}^{\\Omega } p({\\mathbf {y}}_{t+\\omega }|{\\mathbf {y}}_{1:t+\\omega -1}),$ where each future observation is conditioned on the observations at all previous timestamps.",
"In practice, the generative models may face some challenges when applied to real-world forecasting scenarios such as demand forecasting for online retailers.",
"In addition to the efficiency issue in both training and forecasting stages, there is also an error accumulation problem as each prediction is fed back as the ground-truth to forecast longer horizons, in which process errors may accumulate.",
"Instead of applying the classical generative approach, our framework forecasts the joint distribution of future observations directly: $P\\left({\\mathbf {y}}_{(t+1):(t+\\Omega )}|{\\mathbf {y}}_{1:t}\\right) =\\prod \\limits _{\\omega =1}^{\\Omega } p({\\mathbf {y}}_{t+\\omega }|{\\mathbf {y}}_{1:t}).$ While time series data usually have systematic patterns such as trend and seasonality, it is also crucial that a forecasting framework allows covariates $X_{t+\\omega }^{(i)}~(\\mathrm {where}~\\omega = 1,...,\\Omega ~\\mathrm {and}~i = 1, ..., N)$ that include additional information to the direct forecasting strategy in Equation REF .",
"The joint distribution of the future incorporating the covariates becomes: $P\\left({\\mathbf {y}}_{(t+1):(t+\\Omega )}|{\\mathbf {y}}_{1:t}\\right) =\\prod \\limits _{\\omega =1}^{\\Omega } p({\\mathbf {y}}_{t+\\omega }|{\\mathbf {y}}_{1:t}, X_{t+\\omega }^{(i)}, i=1,...,N).$ Under the above settings, the challenge becomes to design a neural network framework that incorporates the historical observations ${\\mathbf {y}}_{1:t}$ and the covariates $X_{t+\\omega }^{(i)}$ .",
"In the following sections, we describe how we extend the idea of the dynamic regression model (e.g., the ARIMAX model) to build a direct forecasting framework for multiple time series by applying dilated causal convolutions and residual neural networks.",
"We will then describe the probabilistic forecasting framework in detail and some practical considerations of the input features."
],
[
"Neural network architecture",
"Dynamic regression models (e.g., ARIMAX) extend the classical time series model to include both information from past observations and exogenous variables ().",
"A way to represent dynamic regression models is as follows: $y_{t}^{(i)} = \\nu _B(X_{t}^{(i)})+n_t^{(i)}.$ where $\\nu _B(\\cdot )$ is a transfer function that describes how the changes in exogenous variables $X_{t}^{(i)}$ are transferred to $y_t^{(i)}$ , and $n_t^{(i)}$ is a stochastic time series process, e.g., the ARIMA process, which captures a forecast of $y_t^{(i)}$ using historical information.",
"To extend the dynamic regression model to multiple time series forecasting scenario, we propose a variant of residual neural network , .",
"Its main difference from the original resnet is that the new block allows for two inputs – one input for the historical observations and the other for exogenous variables.",
"for convenience, we refer it as resnet-v in the rest of the paper.",
"Section REF provides more details of the module resnet-v.",
"In this paper, we propose the Deep Temporal Convolutional Network (DeepTCN).",
"The entire architecture of DeepTCN is presented in Figure REF .",
"The high-level architecture is similar to the classical Seq2Seq framework.",
"In the encoder part, stacked dilated causal convolutions are constructed to model the stochastic process of historical observations and output $h_t^{(i)}$ .",
"Then, the module resnet-v in the decoder part incorporates the latent output $h_t^{(i)}$ and future exogenous variables $X_{t+\\omega }^{(i)}$ , and outputs another latent output.",
"Finally, a dense layer is applied to map the output of resnet-v and to produce the probabilistic forecasts of future observations.",
"In the following sections, we provide further details for each module.",
"Figure: (a) Architecture of DeepTCN.",
"Encoder part: stacked dilated causal convolutional nets are constructed to capture the long-term temporal dependencies.Decoder part: the decoder includes a variant of residual block (referred as resnet-v, shown as ⊕\\oplus ) and an output dense layer.",
"The module resnet-v is designed to integrate output of stochastic process of historical observations and future covariates.",
"Then the output dense layer is adopted to map the output of resnet-v into our final forecasts.",
"(b) Encoder module.",
"Residual blocks are taken as the ingredient.",
"Each residual block consists of two layers of dilated causal convolutions, the first of which is followed by a batch normalization and ReLU and the second of which is follow by another batch normalization.",
"The output is taken as the input of the residual block, followed by another ReLU.",
"(c) Decoder module.",
"h t (i) h_{t}^{(i)} is the output of the encoder, X t+ω (i) X_{t+\\omega }^{(i)} are the future covariates, and R(·)R(\\cdot ) is the nonlinear function applied on X t+ω (i) X_{t+\\omega }^{(i)}.",
"For the residual function R(·)R(\\cdot ), we first apply a dense layer and a batch normalization to project the future covariates.",
"Then a ReLU activation is applied followed by another dense layer and batch normalization.Causal convolutions are convolutions where the output at time $t$ can only be obtained from the inputs that are no later than $t$ .",
"Dilation causal convolutions allow the filter to be applied over an area larger than its length by skipping the input values with a certain step .",
"In the case of univariate series, given a single-dimensional input sequence $x$ , the output (feature map) $s$ at location $t$ of a dilated convolution with kernel $w$ can be expressed as: $s(t) = (x*_{d}w)(t) = \\sum \\limits _{k=0}^{K-1} w(k)x(t-d\\cdot k),$ where $d$ is the dilation factor, and $K$ is the size of the kernel.",
"Stacking multiple dilated convolutions enable networks to have very large receptive fields and to capture long-range temporal dependencies with a smaller number of layers.",
"The left part of Figure REF is an example of dilated causal convolutions with dilation factors $d=\\lbrace 1,2,4,8\\rbrace $ , where the filter size $K=2$ and a receptive field of size 16 is reached by staking four layers.",
"Figure REF shows the basic module for each layer of the encoder, where both of two dilated convolutions inside the module have the same kernel size $K$ and dilation factor $d$ .",
"Instead of implementing the classical gating mechanism in Wavenet , in which a dilated convolution is followed by a gating activation, residual blocks are taken as the ingredient.",
"As shown in Figure REF , each residual block consists of two layers of dilated causal convolutions, the first of which is followed by a batch normalization and rectified nonlinear unit (ReLU) while the second of which is followed by another batch normalization .",
"The output after the second batch normalization layer is taken as the input of the residual block, followed by a second ReLU.",
"Residual blocks have been proven to help efficiently train and stabilize the network, especially when the input sequence is very long.",
"More importantly, non-linearity gained by the rectified linear unit (ReLU) achieves better prediction accuracy in most of the empirical studies.",
"Various Natural Language Processing (NLP) tasks also support the above conclusion ."
],
[
"Decoder: Residual neural network",
"The decoder includes two parts.",
"The first part is the variant of residual neural network, the module resnet-v.",
"The second part is a dense layer that maps the output of the resnet-v to the probabilistic forecasts.",
"As mentioned before, the module resnet-v allows for two inputs (one for the historical information and the other for exogenous variables), and is designed to capture the information of these two inputs.",
"It can be written as: $\\delta _{t+\\omega }^{(i)} = R(X_{t+\\omega }^{(i)}) + {h_t^{(i)}},$ where $h_t^{(i)}$ is the latent output by the encoder, $X_{t+\\omega }^{(i)}$ are the future covariates and $\\delta _{t+\\omega }^{(i)}$ is the latent output of resnet-v. $R(\\cdot )$ is the residual function applied on $X_{t+\\omega }^{(i)}$ .",
"Hence the nonlinear function $R(\\cdot )$ plays the role of transfer function in dynamic regression model and explains the residuals between ground truth and predictions solely determined by the encoder part (e.g, the promotion effects on online retailer platforms or weather forecast for brick and mortar retailers).",
"Figure REF shows the structure of the resnet-v. For the residual function $R(\\cdot )$ , we first apply a dense layer and a batch normalization to project the future covariates.",
"Then a ReLU activation is applied followed by another dense layer and batch normalization.",
"Finally, an output dense layer maps the latent variable $\\delta _{t+\\omega }^{(i)}$ to produce the final output $Z$ that corresponds to the probabilistic estimation of interest.",
"In the next section, we describe how we construct the probabilistic forecasting framework via neural networks in the output dense layer.",
"Neural networks enjoy the flexibility to produce multiple outputs.",
"In the DeepTCN framework, for each future observation, the output dense layer in the decoder can produce $m$ outputs: $Z=(z^1,..., z^m)$ , which represent the parameter set of the hypothetical distribution of interest.",
"Take Gaussian distribution as an example, for the $\\omega $ -th future observation of the $i$ -th series, $y_{t+\\omega }^{(i)}$ , the output layer produces two outputs (the mean and the standard deviation), which gives $Z_{t+\\omega }^{(i)}=(\\mu _{t+\\omega }^{(i)}, \\sigma _{t+\\omega }^{(i)})$ , where $\\mu _{t+\\omega }^{(i)}$ is the expectation of $y_{t+\\omega }^{(i)}$ and $\\sigma _{t+\\omega }^{(i)}$ is the standard deviation.",
"Therefore, the probabilistic forecasts can be described as: $P\\left(y_{t+\\omega }^{(i)} \\right)\\sim G(\\mu _{t+\\omega }^{(i)}, \\sigma _{t+\\omega }^{(i)}).$ More specifically, we consider two probabilistic forecasting frameworks in this paper.",
"The first one is the parametric framework, in which probabilistic forecasts of future observations can be achieved by directly predicting the parameters of the hypothetical distribution (e.g., the mean and the standard deviation for Gaussian distribution) based on maximum likelihood estimation.",
"The second one is non-parametric, which produces a set of forecasts corresponding to quantile points of interest with $Z$ representing the quantile forecasts.",
"In practice, whether to choose the parametric approach or the non-parametric approach depends on the application context.",
"The parametric approach requires the assumption of a specific probability distribution while the non-parametric approach is distribution-free and thus is usually more robust.",
"However, a decision-making scenario may rely on the sum of probabilistic forecasts for a certain period.",
"For example, an inventory replenishment decision may depend on the distribution of the sum of demand for the next few days.",
"In such cases, the non-parametric approach will not work since the output (e.g., the quantiles) is not additive over time and the parametric approach has its advantage of being flexible in obtaining such information by sampling from the estimated distributions."
],
[
"Non-parametric approach",
"In the non-parametric framework, forecasts can be obtained by quantile regression.",
"In quantile regression , denoting the observation and the prediction for a specific quantile level $q$ as $y$ and $\\hat{y}^{q}$ respectively, models are trained to minimized the quantile loss, which is defined as $L_q(y, \\hat{y}^{q} ) = q(y - \\hat{y}^{q})^{+}+(1-q)(\\hat{y}^{q} - y)^{+},$ where $(y)^{+} = \\max (0,y)$ and $q \\in (0,1)$ .",
"Given a set of quantile levels $Q=(q_1,...,q_m)$ , the $m$ corresponding forecasts can be obtained by minimizing the total quantile loss defined as $L_{Q} = \\sum \\limits _{j=1}^m L_{q_j} \\left(y, \\hat{y}^{q_j} \\right).$"
],
[
"Parametric approach",
"For the parametric approach, given the predetermined distribution (e.g., Gaussian distribution), the maximum likelihood estimation is applied to estimate the corresponding parameters.",
"Take Gaussian distribution as an example, for each target value $y$ , the network outputs the parameters of the distribution, namely the mean and the standard deviation, denoted by $\\mu $ and $\\sigma $ , respectively.",
"The negative log-likelihood function is then constructed as the loss function: $L_{G} &=& -\\log \\ell (\\mu ,\\sigma \\,| y) \\nonumber \\\\& = & -\\log \\left( (2\\pi \\sigma ^2)^{-1/2}\\exp \\left[-(y-\\mu )^2/(2\\sigma ^2) \\right] \\right)\\nonumber \\\\&=&\\frac{1}{2}\\log (2 \\pi )+\\log (\\sigma ) + \\frac{(y-\\mu )^2}{2\\sigma ^2}.",
"\\nonumber $ We can extend this approach to a variety of probability distribution families.",
"For example, we can choose negative-binomial distribution for long-tail products, which is traditionally used for modeling over-dispersed count data and has been shown to perform well in empirical studies , , , .",
"It is worth mentioning that some parameters of a certain distribution (e.g., the standard deviation in Gaussian distribution) must satisfy the condition of positivity.",
"To accomplish this, we apply “Soft ReLU” activation, namely the transformation $ \\hat{z} = \\log (1+\\exp (z))$ , to ensure positivity ."
],
[
"Input features",
"There are typically two kinds of input features: time-dependent features (e.g., product price and day-of-the-week) and time-independent features (e.g., product_id, product brand and category).",
"Time-independent covariates such as product_id contain series-specific information.",
"Including these covariates helps capture the scale level and seasonality for each specific series.",
"To capture seasonality, we use hour-of-the-day, day-of-the-week, day-of-the-month for hourly data, day-of-the-year for daily data and month-of-year for monthly data.",
"Besides, we use hand-crafted holiday indicators for shopping festival such as “11.11”, which enables the model to learn spikes due to scheduled events.",
"Dummy variables such as product_id and day-of-the-week are mapped to dense numeric vectors via embedding , .",
"We find that the model is able to learn more similar patterns across series by representation learning and thus improves the forecasting accuracy for related time series, which is especially useful for series with little or no historical data.",
"In the case of new products or new warehouses without sufficient historical data, we perform zero padding to ensure the desired length of the input sequence."
],
[
"Datasets",
"We evaluate the performance of DeepTCN on five datasets.",
"More specifically, within the DeepTCN framework, two models – the non-parametric model that predicts the quantiles and the parametric Gaussian likelihood model – are applied for the forecasting performance evaluation.",
"We refer to them as DeepTCN-Quantile and DeepTCN-Gaussian, respectively, for the rest of the paper.",
"Table REF shows the details of the five datasets.",
"JD-demand and JD-shipment are from JD.com, which correspond to two forecasting tasks for online retailers, namely demand forecasting of regional product sales and shipment forecasting of the daily delivery volume of packages in retailers' warehouses.",
"Since it is inevitable for new products or warehouses to emerge, the training periods for these two datasets can range from zero to several years and the corresponding forecasting tasks involve situations such as cold-starts and data sparsity.",
"We also use three public datasets that have been widely used in various time series forecasting studies for accuracy comparison, namely electricity, traffic and parts.",
"The electricity dataset contains hourly time series of the electricity consumption of 370 customers.",
"The traffic dataset is a collection of the occupancy rates (between 0 and 1) of 963 car lanes from San Francisco bay area freeways.",
"The parts dataset is comprised of 1,046 time series representing monthly demand of spare parts in a US car company.",
"A more detailed description of these datasets can be found in Appendix ."
],
[
"Accuracy comparison",
"Current baseline models for JD.com's datasets include seasonal ARIMA (SARIMA) and lightGBM, a gradient boosting tree method that has been empirically proven to be a highly effective approach in predictive modeling.",
"These two online models are deployed and continuously improved to provide more accurate forecasts and to better serve the consequent business operations (e.g., inventory replenishment).",
"A more detailed description including the features and parameters used in these two models can be found in Appendix .",
"For the public datasets, we compare DeepTCN with DeepAR as implemented using the student-$t$ distribution , SQF-RNN and DeepState ."
],
[
"Evaluation metrics",
"To evaluate probabilistic forecasting, given $N$ time series $\\lbrace y^{(i)}\\rbrace _{i=1}^N$ and the prediction range $\\lbrace t+1, t+2,...,t+\\Omega \\rbrace $ , we use the normalized sum of quantile losses , which is denoted as $QL_q = \\frac{\\sum _{i,t} L_q(y_t^{(i)}, \\hat{y}_t^{(i)})}{\\sum _{i,t}|y_{t}^{(i)}|},$ where $L_q(\\cdot )$ is defined in Equation REF .",
"We refer to $QL_q$ as the $q$ -quantile loss.",
"The evaluation metrics used in our experiments for point forecasting include the Symmetric Mean Absolute Percent Error (SMAPE), the Normalized Root Mean Square Error (NRMSE) and the Mean Absolute Scaled Error (MASE), which are defined as follows.",
"Note that for the MASE, the value $m$ is the seasonal frequency.",
"$SMAPE &=& \\frac{1}{N(\\Omega -t)}\\sum \\limits _{i, t} \\left|\\frac{2\\left(y_{t}^{(i)}-\\hat{y}_{t}^{(i)}\\right)}{y_{t}^{(i)}+\\hat{y}_{t}^{(i)}}\\right|, \\nonumber \\\\NRMSE &=& \\frac{\\sqrt{\\frac{1}{N(\\Omega -t)}\\sum \\limits _{i,t} \\left(y_{t}^{(i)}-\\hat{y}_{t}^{(i)}\\right)^2}}{\\frac{1}{N(\\Omega -t)}\\sum \\limits _{i,t} \\left|y_{t}^{(i)}\\right|}, \\nonumber \\nonumber \\\\MASE &=& \\frac{1}{N(\\Omega -t)}\\sum \\limits _{i}\\frac{\\sum \\limits _{t} \\left|y_{t}^{(i)}-\\hat{y}_{t}^{(i)}\\right|}{\\frac{1}{T-m}\\sum \\limits _{t=m+1}^T \\left|y_{t}^{(i)}- y_{t-m}^{(i)}\\right|}, \\nonumber $ where $y_{t}^{(i)}$ is the true value of series $i$ at time step $t$ , and $\\hat{y}_{t}^{(i)}$ is the corresponding prediction value."
],
[
"Results on JD.com's datasets",
"We start with comparing the probabilistic forecasting results of DeepTCN against the online SARIMA and lightGBM models on JD.com datasets over two testing periods: Oct 2018 and Nov 2018.",
"In particular, China's largest shopping festival “11.11” lasts from Nov 1 to Nov 12, during which Nov 11 is the biggest promotion day.",
"We use the 0.5-quantile loss and 0.9-quantile loss as the evaluation metrics, which are referred to as QL50 and QL90, respectively.",
"The model DeepTCN-Quantile is trained to predict $q$ -quantiles with $q \\in \\lbrace 0.5, 0.9\\rbrace $ .",
"For the model DeepTCN-Gaussian, the quantile predictions are obtained by calculating the percent point function of Gaussian distribution (the inverse of cumulative density function) at $0.5$ and $0.9$ quantiles.",
"The comparison results of JD-demand and JD-shipment are illustrated in Table REF .",
"As we can see, both DeepTCN-Quantile and DeepTCN-Gaussian achieve better results than the two online models.",
"In particular, DeepTCN-Quantile performs the best.",
"One possible reason is that DeepTCN-Gaussian is constructed based on the Gaussian likelihood, but these datasets do not necessarily follow the assumption of normal distribution.",
"On the contrary, the model DeepTCN-Quantile, in light of the distribution-free nature, generates better forecasts by minimizing the quantile loss directly.",
"Table: Comparison of probabilistic forecasts on JD-demand and JD-shipment datasets.",
"The quantile losses QL50/QL90 are evaluated against online models over two testing periods – Oct 2018 and Nov 2018.We then present in Table REF an accuracy comparison of point forecasting between our model and the other two baseline models including SARIMA and lightGBM.",
"The point forecasting results of DeepTCN-Quantile are achieved with the non-parametric approach that predicts the $0.5$ quantiles.",
"In Table REF , All-Data consists of all series in the dataset; Long-series includes series with historical data longer than two years; Short-series are those starting after 2018, i.e., without historical shopping festival data.",
"We can see that DeepTCN-Quantile achieves consistently the best accuracy with regard to all the metrics across all data groups.",
"In particular, when historical shopping festival data is not available, the performance of SARIMA and lightGBM become much worse (the result in Short-series), while DeepTCN-Quantile maintains the same performance level.",
"Table: Point forecasting accuracy comparison on NRMSE, SMAPE and MASE of different subgroups of JD-shipment in Nov 2018.",
"All-Data represents all series with the length of training periods ranging from zero to four years; Long-series includes the warehouses with historical data of more than two years; Short-series indicates series starting after 2018, namely those with no historical shopping festival data.To gain a better understanding of the performance improvement exhibited by the proposed DeepTCN framework, we show in Figure REF three cases of probabilistic forecasts generated by SARIMA and DeepTCN-Quantile.",
"Case A and Case B are two demand forecasting examples of Oct 2018 and Nov 2018, respectively, while Case C is an example of shipment forecasting of Nov 2018.",
"It is shown that for both tasks, DeepTCN-Quantile generates more accurate uncertainty estimation.",
"Moreover, SARIMA postulates increasing uncertainty over time while the uncertainty estimation of DeepTCN-Quantile is well learned from the data.",
"For example, the uncertainty of SARIMA during the shopping festival period is huge due to both promotion activities and intense market competition.",
"Figure: Probabilistic forecasts of SARIMA and DeepTCN-Quantile for three cases (randomly chosen for illustration purposes).",
"Case A and Case B show the forecasting results of two fast-moving products; Case C shows the forecasting results of the daily delivery volume of packages from one warehouse.",
"The ground truth, and the [10%, 90%] prediction intervals of SARIMA and DeepTCN-Quantile are also shown in different colors.",
"(For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.",
")Finally, we perform a qualitative analysis on JD-shipment dataset over the testing period of Nov 2018.",
"We choose this dataset because 1) it consists of series whose magnitudes of volume are high and stable, and 2) the testing period involves China's biggest shopping festival “11.11”.",
"As mentioned before, the occurrence of this festival results in a spike for the shipment volume, in which the forecasting tasks become more challenging.",
"In Figure REF , we illustrate cases of point forecasting under three different scenarios.",
"“11.11” is the major promotion day and we can observe a spike in the true volume.",
"In Cases A-1 and A-2 , where historical data of more than two years is available, all models can learn a similar volume pattern, including the spike on “11.11”.",
"However, SARIMA and lightGBM in Cases B-1 and B-2 fail to capture the spike on “11.11” due to lack of sufficient training data for historical festivals.",
"Cases C-1 and C-2 are selected to demonstrate how these models handle cold-start forecasting.",
"It turns out that DeepTCN-Quantile stands out for this situation as it is able to capture both scale and shape patterns of the new warehouses by learning data from other warehouses with similar store-specific features.",
"Figure: Point forecasts of DeepTCN-Quantile, SARIMA and lightGBM for six cases (randomly chosen from JD-shipment for illustration purposes).",
"Cases A-1 and A-2 are examples with historical data of more than two years;cases B-1 and B-2 show instances without previous shopping festival data; cases C-1 and C-2 illustrate cold-start forecasting namely the forecasting of time series with little historical data, e.g., less than three days.Note that Nov 11 is one of China's biggest promotion days.",
"(For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)"
],
[
"Results on the public datasets",
"Our first experiment on public datasets is to evaluate the performance of DeepTCN regarding probabilistic forecasting.",
"For electricity and traffic dataset, we implement a 24-hour ahead forecasting task for last seven days based on a rolling-window approach as described in .",
"For parts dataset, we evaluate the performance for last 12 months.",
"It is worth noting that we use the same model trained on the data before the first prediction window rather than retraining the model after updating the forecasts.",
"In all forecasting experiments, we train the DeepTCN-Quantile models to predict $q$ -quantiles with $q \\in \\lbrace 0.5, 0.9\\rbrace $ , and for the model DeepTCN-Gaussian, the quantile predictions are obtained by calculating the percent point function of Gaussian distribution at 0.5 and 0.9 quantiles, which is the same approach as applied in JD.com's datasets.",
"Table: Accuracy comparison of probabilistic forecasting on public datasets.",
"The numbers in the table are the QL50/QL90 results.Table REF illustrates the probabilistic forecasting results obtained by these models.",
"It can be seen that the probabilistic forecasting results of both DeepTCN-Quantile and DeepTCN-Gaussian outperform other state-of-the-art models on traffic and parts datasets.",
"One possible reason is that there exists high correlation within these two datasets, and DeepTCN takes an advantage by learning the non-linear correlation among series, while traditional models such as ETS and auto.arima could not learn the shared patterns across the time series.",
"Table: Accuracy comparison of point forecasting on public datasets.To evaluate the performance of point forecasting, we comparing DeepTCN (DeepTCN that predicts the 0.5 quantiles) against ETS, DeepAR-$t$ and SQF-RNN and calculate the metrics NRMSE, SMAPE and MASE over the electricity and traffic datasets.",
"As can be seen from Table REF , for traffic dataset with highly correlated series, DeepTCN-Quantile achieves the best forecasting accuracy on NRMSE and MASE, while it performs the best on MASE for electricity.",
"Figure: ℒ 1 \\mathcal {L}_1 loss over 200 epochs of three different architectures with 5, 6 and 7 encoder layers.Given that the stochastic process of historical observations is modeled by the stacked dilated causal convolutions in the encoder part of our DeepTCN framework, we now perform sensitivity analysis taking the traffic dataset as an example, to explore the effect of the number of encoder layers on the model performance.",
"In this experiment, we set the filter size of the dilated causal convolutions as $k=2$ and implement three model architectures: (1) a 5-layer architecture with dilation factors $d= \\lbrace 1,2,4,8,16\\rbrace $ , (2) a 6-layer architecture with dilation factors $d = \\lbrace 1,2,4,8,16,32\\rbrace $ and (3) a 7-layer architecture with dilation factors $d = \\lbrace 1,2,4,8,16,20,32\\rbrace $ .",
"Notice that in our experiment of traffic dataset, the length of input sequence is $7 \\times 24=168$ , which is the hourly data of the previous week.",
"And for each layer of the dilated causal convolutions, the kernel size times the dilation factor cannot exceed the length of the input sequence, thus we set the dilation factor of the third model as $d = \\lbrace 1,2,4,8,16,20,32\\rbrace $ to ensure the input length of each layer is sufficient.",
"Figure REF shows the $\\mathcal {L}_1$ loss of these three models over 200 epochs.",
"It can be seen that both the 6-layer and 7-layer architectures perform better than the 5-layer architecture.",
"One reason is that the 5-layer architecture is relatively shallow to fully model the information from the historical observations.",
"However, as one can see, the difference between using 6-layer and 7-layers is small, meaning that as long as one uses enough number of layers, the difference in result is quite small.",
"Such phenomenon is quite consistent across other test cases.",
"Thus our method is quite robust with respect to the model parameters."
],
[
"Conclusion",
"We present a convolutional-based probabilistic forecasting framework for multiple related time series and show both non-parametric and parametric approaches to model the probabilistic distribution based on neural networks.",
"Our solution can help in the design of practical large-scale forecasting applications, which involves situations such as cold-starts and data sparsity.",
"Results from both industrial datasets and public datasets show that the framework yields superior performance compared to other state-of-the-art methods in both point and probabilistic forecasting."
],
[
"Acknowledgements",
"Yanfei Kang's research were supported by the National Natural Science Foundation of China (No.",
"11701022)."
],
[
"Dataset",
" JD-demand.",
"The JD-demand dataset is a collection of 50,000 time series of regional demand which involves around 6,000 products of 3C (short for communication, computer and consumer electronics) category from seven regions of China.",
"The dataset is gathered from 201411-1 to 2018121-1.",
"The features set for JD-demand includes historical demand and the product-specific information (e.g., region_id, product categories, brand, the corresponding product price and promotions).",
"JD-shipment.",
"The JD-shipment dataset includes about 1450 time series from 2014101-1 to 2018121-1, including new series (warehouses) that emerge with the development of the companies' business.",
"The covariates consist of historical demand, the warehouse specific info including geographic and metropolitan informations (e.g., geo_region, city) and warehouse categories (e.g.",
"food, fashion, appliances).",
"Electricity.",
"The electricity dataset describes the series of the electricity consumption of 370 customers, which can be accessed at https://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20112014, The electricity usage values are recorded per 15 minutes from 2011 to 2014.",
"We select the data of the last three years.",
"By aggregating the records of the same hour, we use the hourly consumption data of size $N \\times T = 370 \\times 26304$ , where $N$ is the number of time series and $T$ is the length .",
"Traffic.",
"The traffic dataset describes the occupancy rates (between 0 and 1) of 963 car lanes from San Francisco bay area freeways, which can be accessed at https://archive.ics.uci.edu/ml/datasets/PEMS-SF The measurements are carried out over the period from 200811-1 to 20090330-1 and are sampled every 10 minutes.",
"The original dataset was split into training and test parts, and the daily order was shuffled.",
"The total datasets were merged and rearranged to make sure it followed the calendar order.",
"Hourly aggregation was applied to obtain hourly traffic data .",
"Finally, we get the dataset of size $N \\times T = 963 \\times 10560$ , with the occupancy rates at each station described by a time series of length $10,560$ .",
"Parts.",
"The parts dataset includes 2,674 time series supplied by a US car company, which represents the monthly sales for slow-moving parts and covers a period of 51 months.",
"The data can be accessed at http://www.exponentialsmoothing.net/supplements#data.",
"After applying two filtering rules as follows: Removing series possessing fewer than ten positive monthly demands.",
"Removing series having no positive demand in the first 15 and final 15 months.",
"There are finally 1,046 time series left and a more detailed description can be find in ."
],
[
"Baselines",
"Forecasting in industrial applications often relies on a combination of univariate forecasting models and machine-learning based methods.",
"SARIMA: Seasonal ARIMA (SARIMA) is a widely used time series forecasting model which extends the ARIMA model by including additional seasonal term and is capable of modeling seasonal behaviors from the data .",
"Currently, SARIMA is applied to JD-shipment dataset and fast-moving products with historical data of length more than 14 in JD-demand dataset.",
"The model is implemented with Python's package pmdarima and the best parameters are automatically select based on the criterion of minimizing the AICs .",
"The predictions at confidence level {10%, 90%} are taken as the probabilistic forecasts in our experiments.",
"lightGBM: Gradient boosting tree method has been empirically proven to be a highly effective approach in predictive modeling.",
"As one of efficient implementation of the gradient boosting tree algorithm, lightGBM has gained popularity of being the winning algorithm in numerous machine learning competitions, like Kaggle Competition .",
"lightGBM is applied to both JD-demand dataset and JD-shipment dataset.",
"The features for forecasting on JD-shipment are presented in Table REF .",
"A grid-search is used to find the best values of parameters like learning rate, the depth-of-tree based on the offline evaluation on data from both last month and the same month of last year.",
"Table: lightGBM feature listsTable: Dataset details and deepTCN parameters"
],
[
"Experiment details",
"The current model is implemented with Mxnet and its new high-level interface Gluon.",
"We trained our model on a GPU server with one Tesla P40 and 16 CPU (3.4 GHz).",
"Multiple-GPU can be applied to speed up and achieve better training efficiency in real industrial application.",
"The code for public datasets is available from https://github.com/oneday88/deepTCN.",
"For the JD.com's datasets, the training range and prediction horizon are both 31 days.",
"We implement two models for both JD-demand and JD-shipment datasets.",
"One model is trained on the data before Oct 2018 and produces forecasting on Oct 2018; the other one is trained on the data before Nov 2018 and produces forecasting on Nov 2018.",
"For the parts dataset, we use the first 39 months as training data and the last 12 months for evaluation.",
"A rolling window approach with window size =4 is adopted.",
"The training and prediction range are both 12 months and a rolling window approach with window size 4 is adopted.",
"For both electricity and traffic datasets, the training range and prediction range are selected as 168 hours and 24 hours respectively.",
"For electricity dataset, we use only samples taken in December of 2011, 2012 and 2013 as training data, as we assume that this small data set is sufficient for the task of forecasting electricity consumption during the last seven days of December 2014.",
"For traffic dataset, we train models on all the data before last seven days.",
"For each dataset, we fit the model on the training data and evaluate the corresponding metrics on the testing data after every epoch.",
"When the training process is complete, we pick the model that gains the best evaluation results on the test set.",
"Convolution-related hyper-parameters, such as kernel size, number of channels and dilation length, are selected according to different tasks and datasets.",
"The most important principle for choosing kernel size and dilation length is to make sure that the encoder (stacked residual blocks) has sufficiently large receptive field, namely long effective history of the time series.",
"The number of channels at each convolution layer is determined by the number of input features and is kept fixed for all residual blocks.",
"We manually tune for each dataset training-related hyper-parameters, including batch size and learning rate, in order to achieve the best performance on both evaluation metrics and running time.",
"A more detailed description of parameters is presented in Table REF ."
]
] | 1906.04397 |
[
[
"How much entanglement can be created in a closed system?"
],
[
"Abstract In a closed system, the total number of particles is fixed.",
"We ask how much does this conservation law restrict the amount of entanglement that can be created.",
"We derive a tight upper bound on the bipartite entanglement entropy in closed systems, and find what a maximally entangled state looks like in such a system.",
"Finally, we illustrate numerically on an isolated system of one-dimensional fermionic gas, that the upper bound can be reached during its unitary evolution, when starting in a pure state that emulates a thermal state with high enough temperature.",
"These results are in accordance with current experiments measuring R\\'enyi-2 entanglement entropy, all of which employ a particle-conserving Hamiltonian, where our bound acts as a loose bound, and will become especially important for bounding the amount of entanglement that can be spontaneously created, once a direct measurement of entanglement entropy becomes feasible."
],
[
"=1 How much entanglement can be created in a closed system?",
"Dana Faiez [email protected] Department of Physics, University of California, Santa Cruz, California 95064, USA Dominik Šafránek [email protected] SCIPP and Department of Physics, University of California, Santa Cruz, California 95064, USA In a closed system, the total number of particles is fixed.",
"We ask how much does this conservation law restrict the amount of entanglement that can be created.",
"We derive a tight upper bound on the bipartite entanglement entropy in closed systems, and find what a maximally entangled state looks like in such a system.",
"Finally, we illustrate numerically on an isolated system of one-dimensional fermionic gas, that the upper bound can be reached during its unitary evolution, when starting in a pure state that emulates a thermal state with high enough temperature.",
"These results are in accordance with current experiments measuring Rényi-2 entanglement entropy, all of which employ a particle-conserving Hamiltonian, where our bound acts as a loose bound, and will become especially important for bounding the amount of entanglement that can be spontaneously created, once a direct measurement of entanglement entropy becomes feasible.",
"Entanglement is one of the most intriguing characteristics of quantum systems.",
"It evolved from its perception as a mathematical artifact, as a result of EPR paradox [1], to becoming closely related and applicable to the fields of condensed matter [2], [3], [4], [5], [6], [7], quantum information [8], [9], [10], [11], [12], [13], [14], quantum metrology [15], [16], [17], [18], [19], [20], and quantum gravity [21], [22], [23], [24], [25].",
"In the field of quantum information, entangled states are the backbone of quantum information protocols as they are considered a resource for tasks such as quantum teleportation [9], [26], cryptography [8], and dense coding [27].",
"In these quantum information protocols, more entanglement usually leads to a better performance.",
"Therefore, it is important to set precise upper bounds on how much entanglement is in principle available in performing these tasks [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39].",
"As different tasks require different types of entangled states, numerous measures of entanglement have been introduced [40], [41], [42], [43].",
"The most prominent measure of entanglement is entanglement entropy [44], [33], [34].",
"It is defined as the von Neumann entropy of the reduced density matrix ${\\hat{\\rho }}_A=\\mathrm {tr}_B[|\\psi \\rangle \\langle \\psi |]$ , where $|\\psi \\rangle $ denotes the state of the composite system, $S_{\\mathrm {ent}}\\equiv S({\\hat{\\rho }}_A).$ This is a valuable measure as it draws a direct connection between density matrix and the amount of non-local correlations present in a given system.",
"Entanglement entropy also gained significant attention in the past few decades due to the discovery of its geometric scaling in thermal state as well as ground states (famously known as the volume law [45] and the area law [46], [47], [48] respectively), and its use for characterizing quantum phase transition [2], [49], [50], [51].",
"Despite its importance, this quantity has proven extremely difficult to probe experimentally, and related Rényi-2 entanglement entropy has been measured instead [52], [53], [54].",
"However, an experimental proposal for measuring the entanglement entropy has been put forward recently [55], opening new exciting possibilities.",
"There exists a general bound on entanglement entropy.",
"For a pure state of a bipartite system, it is straight forward to show that $S_{\\mathrm {ent}}\\equiv S({\\hat{\\rho }}_A)=S({\\hat{\\rho }}_B)$ .",
"This leads to [25], $S_{\\mathrm {ent}}\\le \\ln \\min \\Big \\lbrace \\dim \\mathcal {H}_A,\\dim \\mathcal {H}_B\\Big \\rbrace .$ However, one could wonder whether Eq.",
"(REF ) is stringent enough for systems with additional conservation laws, that effectively restrict some degrees of freedom.",
"For example, consider a system of 2 fermionic particles contained on a lattice comprising of 6 sites, partitioned into two sublattices of 3 sites.",
"Since there can be any number 0, 1, and 2 particles in each sublattice, the upper bound on entanglement entropy given by Eq.",
"(REF ) is $S_{\\mathrm {ent}}\\le \\ln \\big (\\binom{3}{0}+\\binom{3}{1}+\\binom{3}{2}\\big )=\\ln 7$ , yet because of the conservation law, this could be considerably larger than the actually achievable entropy.",
"This is important because, among the aforementioned quantum tasks, those that incorporate massive particles — such as the constituents of condensed matter systems — often exhibit constrains such as the conservation of the total number of particles or charge [56], [52], [53], [57], [58], [59], [54], [60].",
"Such restrictions are described by superselection rules [61], [62].",
"It has been suggested that these restrictions can in fact be used as a resource and can enhance the security of quantum communication [63], [64], [61], [65], [66] and measurement accuracy [67], [68], [69], [70].",
"However, among the vast literature on quantum information protocols, specific bounds on entanglement entropy in the presence of superselection rules are not sufficient.",
"Given the commonality of these conservation laws and recent efforts in probing entanglement entropy experimentally, it is an incentive to provide precise bounds for this quantity.",
"In this Rapid Communication, we derive a general tight upper bound on entanglement entropy for closed systems (in thermodynamic sense), which are defined as those where the total number of particles stays constant.",
"This can be applied to quantum systems evolved with a time-independent or a time-dependent Hamiltonian, as long as this evolution conserves the total number of particles.",
"Figure: A 2-dimensional lattice of size L=24L=24 sites and n=8n=8 particles is shown.",
"The subsystems AA and BB are also depicted as red and blue regions respectively.",
"The smaller subsystem AA has M=6M=6 sites and n A =2n_A=2 particles in this example.Bound on entanglement entropy.— For a bipartite system of $n$ spinless particles moving on a system of $L$ number of sites, which is partitioned into two subsystems $A$ and $B$ ($\\mathcal {H}=\\mathcal {H}_A\\otimes \\mathcal {H}_B$ ) with $M$ and $L-M$ number of sites (see Fig.",
"REF ), assuming that the state of the composite system is pure and that $n\\le M\\le L-M$ , the entanglement entropy is bounded by $S_{\\mathrm {ent}}\\le \\ln \\sum _{{n_{\\!A}}=0}^n \\min \\Big \\lbrace \\dim \\mathcal {H}_A^{({n_{\\!A}})},\\dim \\mathcal {H}_B^{(n-{n_{\\!A}})}\\Big \\rbrace ,$ where $\\mathcal {H}_A^{({n_{\\!A}})}$ denotes the Hilbert spaces of exactly ${n_{\\!A}}$ particles contained in the subsystem $A$ , and $\\mathcal {H}_B^{(n-{n_{\\!A}})}$ denotes the Hilbert space of exactly $n-{n_{\\!A}}$ particles contained in subsystem $B$ respectively.",
"This is a tight bound, meaning that it can be saturated with a specific wave function of $n$ particles.",
"An application of this result is shown in Fig.",
"REF .",
"Figure: A maximally entangled state is such where one has the maximal uncertainty about the state of the full system, but determining the state of subsystem AA also determines the state of subsystem BB with certainty.",
"This means that when constructing such a state, none of the orthogonal states spanning subsystems AA and BB can be used twice.",
"But since the conservation law prohibits matching states whose particle numbers do not add up to the total number of particles, the maximal entanglement entropy is lower than initially expected.",
"In this figure, one of the maximally entangled states |ψ〉=1 5(|000101〉+|001001〉+|010100〉+|100010〉+|110000〉)|\\psi \\rangle =\\frac{1}{\\sqrt{5}}(|000101\\rangle +|001001\\rangle +|010100\\rangle +|100010\\rangle +|110000\\rangle ) for the example mentioned in the introduction is shown, leading to S ent (max) =ln3 0 + 3 1 + 3 0=ln5S_{\\mathrm {ent}}^{(\\max )}=\\ln \\big (\\binom{3}{0}+\\binom{3}{1}+\\binom{3}{0}\\big )=\\ln 5.The above formula can be generalized to include cases $n>M$ , but the fermionic and bosonic cases must be treated separately.",
"For fermionic systems (or systems of hard-core bosons) $\\dim \\mathcal {H}_A^{({n_{\\!A}})}=\\binom{M}{{n_{\\!A}}}$ which leads to $S_{\\mathrm {ent}}\\le \\ln \\sum _{{n_{\\!A}}=\\max \\lbrace 0,n-L+M\\rbrace }^{\\min \\lbrace n,M\\rbrace } \\min \\bigg \\lbrace \\binom{M}{{n_{\\!A}}},\\binom{L-M}{n-{n_{\\!A}}}\\bigg \\rbrace ,$ while for bosonic systems $\\dim \\mathcal {H}_A^{({n_{\\!A}})}=\\binom{M+{n_{\\!A}}-1}{{n_{\\!A}}}$ which leads to $S_{\\mathrm {ent}}\\le \\ln \\sum _{{n_{\\!A}}=0}^n \\min \\bigg \\lbrace \\binom{M+{n_{\\!A}}-1}{{n_{\\!A}}},\\binom{L-M+n-{n_{\\!A}}-1}{n-{n_{\\!A}}}\\bigg \\rbrace .$ Assuming that $n\\le M$ , the Hilbert space of $n$ particles contained on lattice of $L$ sites can be decomposed as $\\mathcal {H}=\\bigoplus _{{n_{\\!A}}=0}^n\\mathcal {H}_A^{({n_{\\!A}})}\\otimes \\mathcal {H}_B^{(n-{n_{\\!A}})}.$ This means that any wavefunction $|\\psi \\rangle \\in \\mathcal {H}$ can be written as $|\\psi \\rangle =\\sum _{{n_{\\!A}}=0}^n a_{{n_{\\!A}}}|\\psi _{n_{\\!A}}\\rangle ,$ where $|\\psi _{n_{\\!A}}\\rangle \\in \\mathcal {H}_A^{({n_{\\!A}})}\\otimes \\mathcal {H}_B^{(n-{n_{\\!A}})}$ .",
"Applying the Schmidt decomposition, we can write each of these vectors as $|\\psi _{n_{\\!A}}\\rangle =\\sum _{i=1}^{d_{{n_{\\!A}}}} b_i^{({n_{\\!A}})}|\\chi _i^{({n_{\\!A}})}\\rangle \\otimes |\\phi _i^{(n-{n_{\\!A}})}\\rangle $ where $d_{n_{\\!A}}=\\min \\big \\lbrace \\dim \\mathcal {H}_A^{({n_{\\!A}})}, \\dim \\mathcal {H}_B^{(n-{n_{\\!A}})}\\big \\rbrace $ , and $\\lbrace |\\chi _i^{({n_{\\!A}})}\\rangle \\rbrace _{i=1}^{d_{{n_{\\!A}}}}$ and $\\lbrace |\\phi _i^{(n-{n_{\\!A}})}\\rangle \\rbrace _{i=1}^{d_{{n_{\\!A}}}}$ form orthogonal sets, and $\\lbrace b_i^{({n_{\\!A}})}\\rbrace _{i=1}^{d_{{n_{\\!A}}}}$ are real, non-negative scalars.",
"Also any two vectors $|\\chi _i^{({{n_{\\!A}}})}\\rangle $ and $|\\chi _j^{(\\tilde{n}_{\\!A})}\\rangle $ , ${n_{\\!A}}\\ne \\tilde{n}_{\\!A}$ , are orthogonal to each other, because they belong into subspaces associated with different eigenvalues ${n_{\\!A}}$ of a Hermitian operator $\\hat{N}_A$ (measuring the number of particles in sublattice $A$ ).",
"The same argument can be made for vectors $|\\phi _i^{(n-{n_{\\!A}})}\\rangle $ using $\\hat{N}_B$ .",
"This allows us to compute the reduced density matrix, ${\\hat{\\rho }}_A=\\mathrm {tr}_B[|\\psi \\rangle \\langle \\psi |]=\\sum _{{n_{\\!A}}=0}^n\\sum _{i=1}^{d_{{n_{\\!A}}}}|a_{{n_{\\!A}}}|^2|b_i^{({n_{\\!A}})}|^2|\\chi _i^{({n_{\\!A}})}\\rangle \\langle \\chi _i^{({n_{\\!A}})}|,$ and since vectors $|\\chi _i^{({n_{\\!A}})}\\rangle $ are orthogonal to each other, we can also compute the entanglement entropy as $S_{\\mathrm {ent}}\\equiv S({\\hat{\\rho }}_A)=-\\sum _{{n_{\\!A}}=0}^n\\sum _{i=1}^{d_{{n_{\\!A}}}}|a_{{n_{\\!A}}}|^2|b_i^{({n_{\\!A}})}|^2 \\ln |a_{{n_{\\!A}}}|^2|b_i^{({n_{\\!A}})}|^2.$ Using Jensen's theorem on the strictly concave function $f(x)=\\ln x$ , which is a standard procedure for bounding the Shannon entropy, we derive $S_{\\mathrm {ent}}\\le \\ln \\sum _{{n_{\\!A}}=0}^n d_{n_{\\!A}},$ which proves the theorem for $n\\le M$ .",
"The inequality is saturated if and only if all the probabilities are equal, i.e., $|a_{{n_{\\!A}}}|^2|b_i^{({n_{\\!A}})}|^2=\\bigg (\\!\\sum _{{n_{\\!A}}=0}^n d_{n_{\\!A}}\\bigg )^{-1}$ for all ${n_{\\!A}}$ and $i$ .",
"Considering decomposition (REF ), this equation is the sufficient and necessary condition for the state to be maximally entangled in a closed system.",
"Now let us include cases of $n\\ge M$ .",
"For a fermionic system, the three cases to consider are: $n\\le M\\le L-M$ , $M\\le n \\le L-M$ , and $M\\le L-M < n$ .",
"Combined, for any $n\\le L$ the Hilbert space can be decomposed as $\\mathcal {H}=\\bigoplus _{{n_{\\!A}}=\\max \\lbrace 0,n-L+M\\rbrace }^{\\min \\lbrace n,M\\rbrace } \\mathcal {H}_A^{({n_{\\!A}})}\\otimes \\mathcal {H}_B^{(n-{n_{\\!A}})}.$ The rest of the analysis proceeds analogously and leads to $S_{\\mathrm {ent}}\\le \\ln \\!\\!\\!",
"\\sum _{{n_{\\!A}}=\\max \\lbrace 0,n-L+M\\rbrace }^{\\min \\lbrace n,M\\rbrace }\\!\\!\\!",
"\\min \\Big \\lbrace \\dim \\mathcal {H}_A^{({n_{\\!A}})},\\dim \\mathcal {H}_B^{(n-{n_{\\!A}})}\\Big \\rbrace ,$ with equality if and only if $|a_{{n_{\\!A}}}|^2|b_i^{({n_{\\!A}})}|^2=\\big (\\!\\sum _{{n_{\\!A}}=\\max \\lbrace 0,n-L+M\\rbrace }^{\\min \\lbrace n,M\\rbrace } d_{n_{\\!A}}\\big )^{-1}$ for all ${n_{\\!A}}$ and $i$ .",
"Considering that $\\dim \\mathcal {H}_A^{({n_{\\!A}})}=\\binom{M}{{n_{\\!A}}}$ (combination: the number of ways we can distribute $n_A$ particles in a sublattice of $M$ sites, where no repetition is possible due to the Pauli exclusion principle or hard-core condition) and $\\dim \\mathcal {H}_B^{(n-{n_{\\!A}})}=\\binom{L-M}{n-{n_{\\!A}}}$ , we obtain Eq.",
"(REF ).",
"For a bosonic system, the decomposition of Hilbert space is identical to Eq.",
"(REF ) for any $n$ .",
"The formula therefore remains the same, and considering that for a bosonic system we have $\\dim \\mathcal {H}_A^{({n_{\\!A}})}=\\binom{M+{n_{\\!A}}-1}{{n_{\\!A}}}$ (combination with repetition: the number of ways we can distribute $n_A$ particles in a sublattice of $M$ sites, where multiple particles can be in a single site) and $\\dim \\mathcal {H}_B^{(n-{n_{\\!A}})}=\\binom{L-M+n-{n_{\\!A}}-1}{n-{n_{\\!A}}}$ , we obtain Eq.",
"(REF ).",
"The condition for the maximally entangled state, Eq.",
"(REF ), has an interesting implication.",
"It gives prediction for the number of particles in each of the subsystems: if the state is maximally entangled, then the probability of measuring ${n_{\\!A}}$ particles in sublattice $A$ (which must be the same as the probability of measuring $n-{n_{\\!A}}$ particles in sublattice $B$ ) is equal to $p_{{n_{\\!A}}}=|a_{{n_{\\!A}}}|^2=\\frac{d_{n_{\\!A}}}{\\sum _{{n_{\\!A}}=\\max \\lbrace 0,n-L+M\\rbrace }^{\\min \\lbrace n,M\\rbrace } d_{n_{\\!A}}},$ $d_{n_{\\!A}}=\\min \\big \\lbrace \\binom{M}{{n_{\\!A}}},\\binom{L-M}{n-{n_{\\!A}}}\\big \\rbrace $ , for the fermionic gas, and $p_{{n_{\\!A}}}=|a_{{n_{\\!A}}}|^2=\\frac{d_{n_{\\!A}}}{\\sum _{{n_{\\!A}}=0}^n d_{n_{\\!A}}},$ $d_{n_{\\!A}}=\\min \\big \\lbrace \\binom{M+{n_{\\!A}}-1}{{n_{\\!A}}},\\binom{L-M+n-{n_{\\!A}}-1}{n-{n_{\\!A}}}\\big \\rbrace $ , for the bosonic gas.",
"The mean number of particles in sublattice $A$ is $\\overline{{n_{\\!A}}}=\\sum _{{n_{\\!A}}=\\max \\lbrace 0,n-L+M\\rbrace }^{\\min \\lbrace n,M\\rbrace } p_{{n_{\\!A}}}{n_{\\!A}}$ and $\\overline{{n_{\\!A}}}=\\sum _{{n_{\\!A}}=0}^n p_{{n_{\\!A}}}{n_{\\!A}}$ (while $\\overline{n_{\\!B}}=n-\\overline{{n_{\\!A}}}$ ) for the fermionic and the bosonic gas respectively.",
"Therefore, if a state of a closed system does not satisfy these properties, it cannot be maximally entangled This is necessary, but not a sufficient condition for the state to be maximally entangled.",
"This means that even if the Eqs.",
"(REF ) or (REF ) are satisfied, the state does not have to be maximally entangled.",
"The sufficient and necessary condition is given by Eq.",
"(REF ) for $n\\le M$ and its generalizations for cases $n\\ge M$.",
"One can also notice that the derived bound stops depending on the total system size $L$ if it is large enough.",
"Specifically, for fermionic systems and $L\\ge \\max \\bigg \\lbrace \\max _{{n_{\\!A}}\\in \\lbrace 0,\\dots ,\\min \\lbrace n,M\\rbrace \\rbrace }\\bigg \\lbrace \\binom{M}{{n_{\\!A}}}\\bigg \\rbrace ,n\\bigg \\rbrace +M,$ the bound becomes $S_{\\mathrm {ent}}\\le \\ln \\bigg (1+ \\sum _{{n_{\\!A}}=0}^{\\min \\lbrace n,M\\rbrace -1} \\binom{M}{{n_{\\!A}}}\\bigg ),$ which no longer depends on $L$ .",
"If in addition $n\\ge M$ , then $S_{\\mathrm {ent}}\\le \\ln \\sum _{{n_{\\!A}}=0}^{M} \\binom{M}{{n_{\\!A}}}=\\ln 2^M.$ which is equal to the maximal entropy of subsystem $\\mathcal {H}_A$ .",
"This is the same result that could be recovered from the original bound, Eq.",
"(REF ).",
"Therefore, for fermionic systems with large enough baths (subsystems $B$ ), and a large number of particles, these bounds are the same.",
"The same does not hold for bosonic systems however, for which $S_{\\mathrm {ent}}\\le \\ln \\Big (1+ \\sum _{{n_{\\!A}}=0}^{n-1} \\binom{M+{n_{\\!A}}-1}{{n_{\\!A}}}\\Big )<\\ln \\sum _{{n_{\\!A}}=0}^{n} \\binom{M+{n_{\\!A}}-1}{{n_{\\!A}}}=\\ln \\dim \\mathcal {H}_A$ , irrespective of $n$ , for large $L$ and $M>1$ .",
"Thus for closed bosonic systems, our bound is always better.",
"It also turns out that Eq.",
"(REF ) is the value of the bound in the thermodynamic limit, where both the number of particles $n$ and size of the system $L$ grow to infinity, but the particle density $c=n/L$ remains constant, while keeping $M$ constant.",
"This can be shown by dividing condition (REF ) by $n$ and taking the limit, which gives $c\\le 1$ , which must be by definition satisfied for any spinless fermionic system.",
"Figure: A 1-dimensional lattice of size L=5L=5 sites and n=3n=3 particles is shown.",
"The right hand side of the figure illustrates the hopping terms tt or t ' t^{\\prime } i.e., particles move to the nearest-neighbor (NN) and next-nearest-neighbor (NNN) sites respectively.",
"The left hand side of the figure shows the interactions of strengths VV and V ' V^{\\prime } between NN and NNN respectively.Achievability of the bound in 1D fermionic lattice.— Here, we illustrate the derived upper bound (REF ) in a simulation.",
"We specifically focus on the case where $n<M\\le L-M$ .",
"The other cases turned out to be very similar, and we shall not show them here.",
"We consider a system of $n$ spin-less fermions in a 1-dimensional lattice of size $L$ , with Hamiltonian $\\begin{split}\\hat{H} = \\sum _{i=1}^{L} [-t({f}_{i}^{\\dagger }{f}_{i+1}+h.c.",
")+V{n}_{i}^{f}{n}_{i+1}^{f}\\\\-t^{\\prime }({f}_{i}^{\\dagger }{f}_{i+2}+h.c.",
")+V^{\\prime }{n}_{i}^{f}{n}_{i+2}^{f}],\\end{split}$ where $f_{i}$ and $f_{i}^{\\dagger }$ are fermionic annihilation and creation operators for site $i$ and ${n}_{i}^{f} = {f}_{i}^{\\dagger }{f}_{i}$ is the local density operator.",
"The nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping terms are respectively $t$ and $t^{\\prime }$ and the interaction strengths are $V$ and $V^{\\prime }$ as illustrated in Fig.",
"REF .",
"We choose this Hamiltonian since it has been extensively studied in the literature [72], [73], [74], [75], [76], [77], and because it is an archetypal example of both non-integrable (generic; $t^{\\prime },V^{\\prime }\\ne 0$ ) and integrable ($t^{\\prime }=V^{\\prime }=0$ ) quantum systems.",
"In the simulation depicted in Fig.",
"REF , we take $t=t^{\\prime }=1.9$ , $V=V^{\\prime }=0.5$ It does not matter much which particular values we choose, as long as $t,t^{\\prime },V,V^{\\prime }\\ne 0$ , the evolution is qualitatively the same., and cases with NN hopping only, and with interaction only.",
"The total number of particles is $n=3$ , and we take subsystem $A$ to be the $M=4$ sites on the left side of the chain, while the full system size $L$ , and inverse temperature $\\beta =1/T$ are both varied.",
"We take the initial state to be the complex random pure thermal state (RPTS) (also known as the thermal pure quantum or canonical thermal pure quantum state [79], [80], [45]), which we define as $|\\psi \\rangle =\\frac{1}{\\sqrt{Z}}\\sum _E c_E e^{-\\beta E/2}|E\\rangle ,$ where $|E\\rangle $ 's are the eigenstates of the total Hamiltonian, computed using exact diagonalization.",
"The coefficients $\\lbrace c_E\\rbrace $ are random complex numbers, $c_E\\equiv (x_E+iy_E)/\\sqrt{2}$ , with $x_E$ and $y_E$ obeying the standard normal distribution $\\mathcal {N}(0,1)$ , and $Z=\\sum _E |c_E|^2 e^{-\\beta E}$ is the normalization constant.",
"This state emulates a thermal state, while being pure.",
"This initial state is entangled but not maximally entangled.",
"For instance, in the case of $\\beta =0.01$ and a given initial RPTS, the initial entanglement entropies associated with system sizes $L=[8,9,10,11,12,13]$ are $S_{\\mathrm {ent}}=[2.031,2.091,2.118,2.122,2.080,2.020]$ .",
"The initial state is then evolved as $|\\psi _{\\tau }\\rangle =e^{-i \\hat{H}\\tau }|\\psi \\rangle $ .",
"Figure: S ent (max) =max τ S ent (|ψ τ 〉)S_{\\mathrm {ent}}^{(\\max )}=\\max _{\\tau }S_{\\mathrm {ent}}(|\\psi _{\\tau }\\rangle ) for different values of LL and β\\beta , n=3n=3 particles, for a subsystem AA being fixed as the left M=4M=4 sites of the chain.",
"In the low temperature limit, β=2\\beta =2, the initial state is close to being a ground state.",
"In the high temperature limit, β=0.01\\beta =0.01, the initial state becomes a random pure state , , , ,in which all the energy coefficients are equal on average.",
"We show cases of (1) non-integrable system (t=t ' =1.9t=t^{\\prime }=1.9, V=V ' =0.5V=V^{\\prime }=0.5) and varying temperatures, and cases of high temperature with (2) NN hopping only (t=1.9t=1.9, t ' =V=V ' =0t^{\\prime }=V=V^{\\prime }=0), and with (3) interaction only (V=V ' =0.5V=V^{\\prime }=0.5, t=t ' =0t=t^{\\prime }=0).",
"In cases (1) and (2) and high temperature, the maximum value of S ent S_{\\mathrm {ent}} reaches exactly the theoretical bound (dashed line) for all LL, but not in case (3).",
"This shows that the non-zero hopping term is the most important for reaching the maximum.To find the maximum value that $S_{{\\mathrm {ent}}}$ can achieve during its time evolution, we use the simplex search algorithm.",
"For a given $L$ and $\\beta $ , we initialize the state in 6 different complex RPTS, and find the maxima for each initial state by maximizing over phases $\\phi _E=E \\tau $ that appear in the time evolution of the wavefunction, $\\hat{U}_{\\tau } = e^{-i\\hat{H}\\tau }$ .",
"As long as the differences of $E$ 's are irrational (or close to being to), this method must give the same result as maximizing over all times $\\tau $ .",
"We do this because maximizing over the time by simply evolving the system would take an extremely long time.",
"We then plot the mean value of these six maxima as well as the standard deviation (depicted as error bars) in Fig.",
"REF .",
"The theoretical bound, Eq.",
"(REF ) for the case where $n<M\\le L-M$ , is plotted in the same figure (dashed line) for various values of $L$ .",
"As Fig.",
"REF illustrates, in the non-integrable case for $\\beta =0.01$ the theoretical bound (REF ) is saturated exactly.",
"For large $L$ the bound flattens, as expected from Eq.",
"(REF ).",
"As the temperature drops, the system can no longer achieve this bound, but the maximum entanglement entropy still stays approximately constant for large $L$ .",
"Interestingly, we found that the upper bound is reached during the unitary evolution also for integrable systems with NN interaction and hopping ($t^{\\prime }=V^{\\prime }=0$ ), or even for systems with just the NN hopping term ($t^{\\prime }=V=V^{\\prime }=0$ ), but not when there is no hopping, which we can summarize as “As long as there is some hopping, in both cases of integrable or non-integrable systems, the bound is achieved during the unitary evolution, if the temperature is high enough.” Regarding the average number of particles in the subsystem, states for which $S_{\\mathrm {ent}}$ has reached its maximum were measured to have $\\overline{{n_{\\!A}}}= [1.50,1.55,1.58,1.58,1.58,1.58]$ for $L=[8,9,10,11,12,13]$ , which corresponds exactly to the prediction of Eq.",
"(REF ).",
"Conclusion and applications.— We derived a tight upper bound on entanglement entropy for bipartite systems with a conserved number of spinless particles.",
"We showed numerically that at high temperature, the maximum entanglement entropy of a fermionic lattice in fact reaches this upper bound during its time evolution.",
"Furthermore, by studying the maximally entangled states, we found that measuring the particle number in one of the subsystems can serve as a simple test of whether the state can be maximally entangled.",
"In contrast to Ref.",
"[34], which derived bounds for the average entanglement entropy of all eigenstates of quadratic fermionic Hamiltonians, our bound holds for any state in any spinless fermionic and bosonic system with a conserved number of particles, irrespective of the Hamiltonian being used.",
"Our results can be also directly transferred to lattices of identical spin-1/2 particles with the total spin conserved, where spin-up and spin-down take the place of a particle and a hole respectively, or to lattices of qubits consisting of different energy states (such as cold atoms [4], [52], [53], [84], trapped ions [85], [86], [54], or superconducting qubits [87], [88], [89], [90]), when the total energy, and therefore also the total number of excited states is conserved, while neglecting the interaction energy.",
"Using that for a pure state ${\\hat{\\rho }}_{AB}$ , $S({\\hat{\\rho }}_{AB})=0$ and $S({\\hat{\\rho }}_A)=S({\\hat{\\rho }}_B)$ , a noteworthy consequence of the bound, Eqs.",
"(REF ) and (REF ), is that it sets a lower bound on conditional entropy $S(A|B)_{{\\hat{\\rho }}} = S({\\hat{\\rho }}_{AB}) - S({\\hat{\\rho }}_B)$ [91], [92], [93] which gives a sufficient and necessary condition for teleportation [94], and an upper bound on the mutual information $I(A;B)=S({\\hat{\\rho }}_A)+S({\\hat{\\rho }}_B)-S({\\hat{\\rho }}_{AB})$ [95], [92], [96], which determines the largest possible rate of communication [97], [98], [99], and has applications in quantum machine learning [100], [101], [102].",
"Another implication of this result is with regards to Rényi entropies of higher order.",
"For a general density matrix, entanglement entropy (Rényi entropy of order $\\alpha =1$ ) is related to Rényi entropies of higher order, $S_{\\alpha >1}$ , by inequality $S_{{\\mathrm {ent}}}({\\hat{\\rho }}) \\ge S_{\\alpha >1}({\\hat{\\rho }})$ [103].",
"This means that the upper bound on entanglement entropy found in this Rapid Communication could be taken as a loose upper bound on $S_{\\alpha >1}({\\hat{\\rho }})$ .",
"This is important due to the existence of experiments involving measurements on Rényi-2 entanglement entropy $S_{\\alpha =2}$ [52], [53], [54], which allows us to compare our bound with experimental data.",
"Ref.",
"[52] used a system of ultracold bosonic atoms trapped in an optical lattice, evolving by the Bose–-Hubbard Hamiltonian.",
"The maximum Rényi entropy of a ground state for a system of $L=4$ sites and $n=4$ particles, and various sizes of subsystems $M=[1,2,4]$ was obtained from Fig.",
"4 in [52] (including an offset of about $0.5$ ) as $S_{\\alpha =2}=[0.6,0.9,0]$ , which is below the bound $S_{{\\mathrm {ent}}}^{(\\mathrm {bound})}=[1.6,2.2,0]$ calculated from Eq.",
"(REF ).",
"The maximal achieved entropy obtained from Fig.",
"6 in [52] for $L=n=2$ and $M=1$ is $S_{\\alpha =2}=0.8$ which is much closer to the bound $S_{{\\mathrm {ent}}}^{(\\mathrm {bound})}=1.1$ .",
"Ref.",
"[53] focused on measuring the Rényi entropy of an evolving system using the same model.",
"The maximum values of the Rényi entropy read out from Fig.",
"3 in [53] for $L=n=6$ and $M=[1,2,3,6]$ are $S_{\\alpha =2}=[0.8,1.9,2.6,0]$ , while the bound gives $S_{{\\mathrm {ent}}}^{(\\mathrm {bound})}=[1.9,3.0,3.4,0]$ .",
"Finally, Ref.",
"[54] used a system of trapped ions, each carrying a spin, evolved by an XY Hamiltonian which conserves the total spin.",
"This model is therefore mathematically identical to a lattice of spinless fermions.",
"$L=10$ atoms were prepared in the Néel state ($n=5$ ), and after $5\\ \\mathrm {ms}$ the Rényi entropy was read out for $M=[1,2,3,4,5,6,7,8,9]$ (Fig.",
"2 in [54]) at values scattered around $S_{\\alpha =2}\\approx [0.6,1.3,1.7,2.1,2.4,2.3,1.9,1.5,0.8]$ (recalculated by changing the base of logarithm $\\log _2\\rightarrow \\ln $ ).",
"These values are comparable but two of them are slightly higher than the bound $S_{{\\mathrm {ent}}}^{(\\mathrm {bound})}=[0.7,1.4,2.1,2.8,3.5,2.8,2.1,1.4,0.7]$ calculated from Eq.",
"(REF ), due to inadvertently introduced decoherence (the total Rényi entropy was $0.5$ at the time of measurement).",
"We conclude that our results are in accordance with current experiments, and will become especially useful for bounding the amount of entanglement spontaneously created in closed systems, once a direct measurement of entanglement entropy becomes feasible.",
"We are grateful to Joshua M. Deutsch and Anthony Aguirre for fruitful discussions, and thank Joshua M. Deutsch for providing his code for simulating 1-dimensional fermionic lattice.",
"DŠ acknowledges support from Foundational Questions Institute (FQXi.org) and from the Faggin Presidential Chair Fund."
]
] | 1906.04234 |
[
[
"Photoproduction Reactions and Non-Strange Baryon Spectroscopy"
],
[
"Abstract We review the last two decades of using photon beams to measure the production of mesons, and in particular the information that can be obtained on the spectrum of light, non-strange baryons.",
"This is a compendium of experimental results, which should be used as a complement to theoretical reviews of the subject.",
"Lists of data sets are given, together with a comprehensive set of references.",
"An indication of the impact of the data is presented with a summary of the results."
],
[
"Introduction",
"Measurements of pion photoproduction on both proton and quasi-free neutron targets have a very long history, starting about 70 years ago with the discovery of the pion by the University of Bristol group [1].",
"Two years later, at the 1949 Spring Meeting of the National Academy of Sciences, a preliminary account was given of some observations of mesons produced by the 335-MeV photon beam from the Berkeley synchrotron [2].",
"Starting with the use of bremsstrahlung facilities, pioneering results for $\\gamma p\\rightarrow \\pi ^0p$ [3], [4], [5], [6], [7], $\\gamma p\\rightarrow \\pi ^+ n$ [8], [9], [10], [11], and $\\gamma n\\rightarrow \\pi ^-p$ [12] were obtained.",
"Despite all the shortcomings of the first measurements (such as large normalization uncertainties, wide energy and angular binning, limited angular coverage and so on), these data were crucial for the discovery of the first excited nucleon state, the $\\Delta (1232)3/2^+$ , [13].",
"Whilst the ability of photoproduction measurements to deliver information on baryon resonances had been shown from an early stage, most of the light baryon spectrum states and their properties were subsequently obtained by pion-nucleon scattering.",
"Until the end of the 1970s, meson photoproduction was essentially only able to confirm pion scattering data, without adding a substantial amount of additional information.",
"Indeed, the evolution of particle physics towards energies beyond the regime in which hadronic states are the relevant degree of freedom suggested to some that the study of the light baryon spectrum had come to an end, if not a conclusion.",
"This was summarized in a 1983 review article “Baryon Spectroscopy\" by Hey and Kelly [14] who stated in their introduction: “Baryon spectroscopy is now thirty years old and perhaps approaching a mid-life crisis.",
"For it is inevitable in such a fast-moving field as high energy particle physics, that experiments have moved on beyond the resonance region to higher energies and different priorities.",
"Thus it is probably no exaggeration to say that we now have essentially all the experimental data relevant to the low-energy baryon spectrum, that we are ever likely to obtain.\"",
"Figure: Stacked histogram of full database for single meson photoproduction γN→mB\\gamma N \\rightarrow m B. m = (π\\pi , η\\eta , η ' \\eta ^{\\prime }, KK, ω)\\omega ), B = (nn, pp, Λ\\Lambda , Σ\\Sigma ).",
"Light shaded – cross sections, dark shaded – polarization data.",
"Experimental data from the SAID database .Armed with the benefit of hindsight, we beg to differ!",
"The 1980s saw several advances in accelerator technology that enabled the production of photon beams of the order of a GeV in energy, whose energy could be accurately enough determined through the tagging of degraded electrons in bremsstrahlung, or via laser backscattering from electron beams.",
"These facilities initially concentrated on photonuclear research, but as soon as the threshold for pion production was reached, it became clear that photon beams for hadron physics research was a reality.",
"Nevertheless, it took a while for this potential to be realized, which is clearly demonstrated in Figure REF .",
"This plot shows the increase in the worldwide dataset for photoproduction reactions as a function of year.",
"One can readily see that by the time of the Hey and Kelly review [14] (1983), the amount of new data being obtained was indeed tailing off, so their pessimism about more data was at the time well-founded.",
"It took until the turn of the 21st century before a substantial increase was seen.",
"The beginning of the exponential rise in the number of data points around 1996 therefore serves as a starting point for this current review.",
"The plot also does not indicate the relative improvement in the accuracy of the data, which can only be appreciated by delving into the relevant literature.",
"Where initial measurements showed rough energy and angular dependencies, more recent results have been obtained that allow energy scans and fits to angular distributions that allow sophisticated partial wave analyses (PWA) that were previously only possible with pion scattering data.",
"The scope of this review may seem to be somewhat narrow (a particular set of reactions and only the lightest sector of the baryon spectrum).",
"However, we have limited ourselves to this scope not only to avoid an enormous task of covering all of baryon spectroscopy, but to point out that our knowledge of the light baryon spectrum is not yet complete and that there is a vigorous amount of activity devoted to extracting as much information as possible from the most recent, precise and statistically accurate measurements.",
"In addition, measurements of photoproduction reactions, and in particular those on pseudoscalar meson photoproduction including polarization have now been carried out.",
"It is therefore timely to review this work.",
"In this review, we concentrate on the measurements of physical quantities, and the information that can be extracted from them.",
"We are less concerned with theoretical interpretations other than the identification of new resonances, and leave a discussion of different models to other excellent reviews (e.g., Ref. [16]).",
"In this sense, we are taking a phenomenological point of view, but our aim is to tie together the many different experimental results over the last couple of decades, and present this unified overview as a starting point for further serious assaults on the understanding of the light baryon spectrum from first principles.",
"We start with an overview of formalism for dealing with measured data in Section , followed in Section by a description of how information can be extracted from the data.",
"In Section , we review various experimental facilities that have been used to obtain the data sets, which are described and sorted by final state in Section .",
"Some concluding remarks are given in Section ."
],
[
"Formalism for Photoproduction Reactions",
"Experiments only ever measure counts.",
"For a specific beam intensity, hitting a target with a specific density of scattering centers in a specific state of polarization reacting to give a specific final state, whose particles have specific spin orientations, all that an experiment will do is to register counts.",
"The registered counts are subject to the efficiency of the detection apparatus, both in sensitivity and in correctly identifying the desired combination of particles.",
"Advances in experimental technologies are aimed at improving this efficiency so that more complicated measurements can be performed.",
"In the last couple of decades there have been many such advances that have been relevant to photoproduction reactions, including: control and polarization of photon beams, development of polarized gas and solid targets, construction of large solid angle detectors, development of higher rate data acquisition systems and of data analysis and statistical techniques.",
"What is recorded by an experiment is most likely a distribution of counts in the space of independent kinematic variables, which includes the effect of potentially complicated resolution effects due to the detection apparatus.",
"The data analysis process tries to minimize the resolution effects and to quantify the associated uncertainties (systematic uncertainties).",
"The processed data are then used to estimate physically meaningful quantities, either by binning the counts in one or more dimensions, or by treating the data event-by-event.",
"In any case, there is always uncertainty associated with a finite number of counts (statistical uncertainties).",
"What are commonly referred to as observables are usually theoretical constructs of physically meaningful quantities, and are derived from a consideration of the contributing quantum mechanical amplitudes.",
"Being able to extract information at the amplitude level is therefore seen as a goal of these campaigns, since no more information is available to us, even in principle.",
"Since amplitudes are complex functions, there is always an unknown phase.",
"A number of amplitude schemes are commonly employed, and the concept of combining observables to realize a complete experiment has arisen over the years, which would allow the extraction of all relevant amplitudes up to an unknowable phase.",
"However, given that the observables themselves are related to distributions of measured counts, it is worth stressing that the concept of a complete experiment is only mathematically meaningful.",
"In practical terms, one does not “observe” observables.",
"One measures counts, either as total intensities or as asymmetries for experimental configurations that can be constructed with combinations of polarized beam, target and recoils.",
"The extensive work done to study the theoretically complete experiment [17], [18], [19] can perhaps best be utilized by combining it with an approach to quantify the information content of polarization measurements [20], as a guide to developing the most informative measurements.",
"In this section we describe both the formalism and how to extract estimates of observables from measurements.",
"We then indicate how this information can be utilized to gain insight into the light baryon resonance spectrum.",
"We concentrate on single pseudoscalar meson photoproduction, since it is the most straightforward reaction in terms of measurement and formalism, to give a flavour of the relevant issues.",
"Double pseudoscalar meson and vector meson photoproduction require more complicated formalisms, and we will refer the reader to the relevant literature in the interests of saving space."
],
[
"Single Pseudoscalar Meson Photoproduction",
"Single pseudoscalar meson photoproduction involves the interaction of a photon with a free proton, a bound neutron or a whole nucleus.",
"For studies of the baryon spectrum, we are normally interested in the first two of these.",
"So a spin-1 particle (the photon, two helicity states) and a spin-$\\frac{1}{2}$ particle (the nucleon) react to give a spin-0 particle (the pseudoscalar meson) and a spin-$\\frac{1}{2}$ particle (the recoiling baryon).",
"This gives eight spin combinations, of which four are possible within the parity-conserving strong interaction that has taken place.",
"The four combinations are represented as amplitudes, the exact form of which is a matter of choice.",
"Common options are CGLN [21], helicity amplitudes [22] and transversity amplitudes [17].",
"Within any of these bases, there are 16 possible bilinear combinations that are referred to as the “observables”.",
"To illustrate this in detail, a completely general expression for the cross section of these reactions following Ref.",
"[23], with the explicit dependence on the observables, is given below: L5cm C1cm p9.5cm >(1a) C1cm ${red}{\\mathbf {d\\sigma ^{B,T,R}}}(\\vec{P^\\gamma },\\vec{P^T},\\vec{P^R},\\phi )$ $=$ $\\frac{1}{2} \\left\\lbrace {red}{\\mathbf {d\\sigma _0}} \\left[ 1 - P_L^{\\gamma }P^T_y P^R_{y^\\prime } \\cos 2(\\alpha - \\phi ) \\right] \\right.$ Single spin observables $ \\quad + {red}{\\mathbf {\\Sigma }} \\left[- P_L^{\\gamma }\\cos 2(\\alpha - \\phi ) + P^T_y P^R_{y^\\prime } \\right]$ $ \\quad + {red}{\\mathbf {T}} \\left[ P^T_y - P_L^{\\gamma }P^R_{y^\\prime } \\cos 2(\\alpha - \\phi ) \\right]$ $ \\quad + {red}{\\mathbf {P}} \\left[ P^R_{y^\\prime } - P_L^{\\gamma }P^T_y \\cos 2(\\alpha - \\phi ) \\right]$ Beam-Target observables $ \\quad + {red}{\\mathbf {E}} \\left[ -P_{\\odot }^{\\gamma }P^T_z + P_L^{\\gamma }P^T_x P^R_{y^\\prime }\\sin 2(\\alpha - \\phi ) \\right]$ $ \\quad + {red}{\\mathbf {G}} \\left[ P_L^{\\gamma }P^T_z \\sin 2(\\alpha - \\phi ) + P_{\\odot }^{\\gamma }P^T_x P^R_{y^\\prime } \\right]$ $ \\quad + {red}{\\mathbf {F}} \\left[ P_{\\odot }^{\\gamma }P^T_x + P_L^{\\gamma }P^T_z P^R_{y^\\prime } \\sin 2(\\alpha - \\phi ) \\right]$ $ \\quad + {red}{\\mathbf {H}} \\left[ P_L^{\\gamma }P^T_x \\sin 2(\\alpha - \\phi ) - P_{\\odot }^{\\gamma }P^T_x P^R_{y^\\prime } \\right]$ Beam-Recoil observables $ \\quad + {red}{\\mathbf {C_{x^\\prime }}} \\left[ P_{\\odot }^{\\gamma }P^R_{x^\\prime } - P_L^{\\gamma }P^T_y P^R_{z^\\prime } \\sin 2(\\alpha - \\phi ) \\right]$ $ \\quad + {red}{\\mathbf {C_{z^\\prime }}} \\left[ P_{\\odot }^{\\gamma }P^R_{z^\\prime } - P_L^{\\gamma }P^T_y P^R_{x^\\prime } \\sin 2(\\alpha - \\phi ) \\right]$ $ \\quad + {red}{\\mathbf {O_{x^\\prime }}} \\left[ P_L^{\\gamma }P^R_{x^\\prime } \\sin 2(\\alpha - \\phi ) + P_{\\odot }^{\\gamma }P^T_y P^R_{z^\\prime } \\right]$ $ \\quad + {red}{\\mathbf {O_{z^\\prime }}} \\left[ P_L^{\\gamma }P^R_{z^\\prime } \\sin 2(\\alpha - \\phi ) - P_{\\odot }^{\\gamma }P^T_y P^R_{x^\\prime } \\right]$ Target-Recoil observables $ \\quad + {red}{\\mathbf {L_{x^\\prime }}} \\left[ P^T_z P^R_{x^\\prime } + P_L^{\\gamma }P^T_x P^R_{z^\\prime }\\cos 2(\\alpha - \\phi ) \\right]$ $ \\quad + {red}{\\mathbf {L_{z^\\prime }}} \\left[ P^T_z P^R_{z^\\prime } - P_L^{\\gamma }P^T_x P^R_{x^\\prime }\\cos 2(\\alpha - \\phi ) \\right]$ $ \\quad + {red}{\\mathbf {T_{x^\\prime }}} \\left[ P^T_x P^R_{x^\\prime } + P_L^{\\gamma }P^T_z P^R_{z^\\prime }\\cos 2(\\alpha - \\phi ) \\right]$ $ \\left.",
"\\quad + {red}{\\mathbf {T_{z^\\prime }}} \\left[ P^T_x P^R_{z^\\prime } - P_L^{\\gamma }P^T_z P^R_{x^\\prime }\\cos 2(\\alpha - \\phi ) \\right] \\right\\rbrace $ Figure: The definitions of laboratory and event axes, as wellas azimuthal angles.",
"The common laboratory, center-of-mass andevent zz-axis is directed out of the page.",
"The lab xx- and yy-axesare in the horizontal and vertical directions, and the event yy-axisis normal to the reaction plane.In these equations, $\\sigma _0$ denotes unpolarized cross section, $P^\\gamma _L$ denotes degree of linear photon polarization, $P^\\gamma _{\\odot }$ denotes degree of circular photon polarization, $P^T_{x,y,z}$ and $P^R_{x^\\prime ,y^\\prime ,z^\\prime }$ describe target and recoil baryon polarization components.",
"The angle $\\phi $ is the azimuthal angle of the reaction plane, which is defined in the diagram in Figure REF .",
"${red}{\\mathbf {\\Sigma }}$ , ${red}{\\mathbf {T}}$ , and ${red}{\\mathbf {P}}$ are the single beam, target and recoil spin asymmetries.",
"${red}{\\mathbf {E}}, {red}{\\mathbf {G}}, {red}{\\mathbf {H}}$ and ${red}{\\mathbf {F}}$ are the beam-target double spin asymmetries; ${red}{\\mathbf {C_{x^\\prime }}}, {red}{\\mathbf {C_{z^\\prime }}}, {red}{\\mathbf {O_{x^\\prime }}}$ and ${red}{\\mathbf {O_{z^\\prime }}}$ are the beam-recoil double spin asymmetries; ${red}{\\mathbf {T_{x^\\prime }}}, {red}{\\mathbf {T_{z^\\prime }}}, {red}{\\mathbf {L_{x^\\prime }}}$ , and ${red}{\\mathbf {L_{z^\\prime }}}$ are the target-recoil double spin asymmetries.",
"The primes refer to a coordinate system in which $\\hat{z^{\\prime }}$ is parallel to the pseudoscalar meson momentum, $\\hat{y^{\\prime }}$ is normal to the scattering plane and $\\hat{x^{\\prime }} = \\hat{y^{\\prime }}\\times \\hat{z^{\\prime }}$ .",
"The unprimed coordinate system has $\\hat{z}$ parallel the photon momentum, $\\hat{y}$ is normal to the scattering plane and $\\hat{x} = \\hat{y}\\times \\hat{z}$ .",
"In Equations ( REFREF ) to ( REFREF ), it can be seen that each observable enters twice.",
"This means that there are always experimental configurations that can be used to extract the values, some of which require triple polarization measurements.",
"Whilst not strictly required, the extraction of observables from two experimental configurations is desirable in order to reduce systematic uncertainties."
],
[
"Two Pseudoscalar Meson Photoproduction",
"Reactions such as $\\gamma p\\rightarrow p\\pi ^+\\pi ^-$ , as well as any other two pseudoscalar mesons reactions, have three body final states and therefore have more independent kinematic variables and observables.",
"Figure REF illustrates the momenta involved.",
"Formalism relating amplitudes and observables for these reactions is discussed detail in Ref. [24].",
"Writing a full cross section formula in component form is not practical, so we illustrate a more special case with a vector notation.",
"In case of polarized photons and polarized target, using notation consistent with the previous section, the cross section can be written as : ${red}{\\mathbf {d\\sigma ^{B,T}}}(\\vec{P^\\gamma },\\vec{P^T},x_i)= {red}{\\mathbf {d\\sigma _0}} \\lbrace (1 &+ \\vec{P^T}\\cdot {red}{\\mathbf {\\vec{P}}}) \\nonumber \\\\&+ P_{\\odot }^{\\gamma }({red}{\\mathbf {I^{\\odot }}} + \\vec{P^T}\\cdot {red}{\\mathbf {\\vec{P}^{\\odot }}}) \\nonumber \\\\&+ P_L^{\\gamma }[\\sin 2(\\alpha - \\phi ) ({red}{\\mathbf {I^s}} + \\vec{P^T}\\cdot {red}{\\mathbf {\\vec{P}^s}}) \\nonumber \\\\&+ \\cos 2(\\alpha - \\phi ) ({red}{\\mathbf {I^c}} + \\vec{P^T}\\cdot {red}{\\mathbf {\\vec{P}^c}})], \\rbrace $ where: ${red}{\\mathbf {d\\sigma _0}}$ is the unpolarized cross section; $\\alpha - \\phi $ is the angle between photon polarization and reaction plane; $x_i$ represents all the kinematic variables; $P_{\\odot }^{\\gamma }, P_L^{\\gamma }$ are the degrees of circular or linear photon polarization; $\\vec{P^T}$ is the target nucleon polarization $(P^T_x, P^T_y, P^T_z)$ ; Track changes is on The observables in this case are: ${red}{\\mathbf {I^{\\odot , s, c}}}$ single spin beam asymmetries associated with polarized photons; ${red}{\\mathbf {\\vec{P}}}$ target asymmetry ${red}{\\mathbf {(P_x, P_y, P_z)}}$ ; ${red}{\\mathbf {\\vec{P}^{\\odot ,s,c}}}$ double spin asymmetries ${red}{\\mathbf {(P^{\\odot }_x, P^{\\odot }_y, P^{\\odot }_z)}}$ , ${red}{\\mathbf {(P^s_x, P^s_y, P^s_z)}}$ ,${red}{\\mathbf {(P^c_x, P^c_y, P^c_z)}}$ .",
"In these reactions there are a total of 64 possible observables.",
"In practice, however, it would be extremely challenging to extract all of these with reasonable accuracy, so published experiments tend to concentrate on a few of them."
],
[
"Vector Meson Photoproduction",
"With a spin-1 vector meson in the final state, the number of underlying helicity amplitudes is 12, which would require a total of 23 independent observables at each energy and angle to extract.",
"As with the full suite of two-pion spin observables, it may never be practical to extract all of them.",
"The decay angular distributions of the vector mesons can be examined to extract some of the spin density matrix elements (SDMEs).",
"A comprehensive guide to this formalism is given in Ref. [25].",
"The SDMEs are defined in the rest frame of the vector meson, however the effects of resonances and other mechanisms relevant to the low energy baryon spectrum require that spin observables be measured in the $\\gamma $ N center of mass frame [26].",
"In this review, we restrict ourselves to $\\omega $ photoproduction; $\\rho $ photoproduction is predominantly analysed in the context of two pion photoproduction, and other light vector mesons such as the $\\phi (1020)$ and $K^{\\ast }(892)$ have hitherto had limited impact on studies of the light baryon spectrum.",
"The number of counts $N$ registered in a detector of efficiency $\\varepsilon $ , subtending solid angle $d\\Omega $ and in a measurement of luminosity ${\\cal L}$ for a total time $T$ , is given by $N = \\varepsilon ^{-1} \\int _0^T {\\cal L}dt \\int \\dfrac{d\\sigma \\left( \\theta , \\phi \\right)}{d\\Omega } d\\Omega ,$ where the efficiency is the ratio of the number of particles of interest identified by the detector to the number of the particles passing through the solid angle, the luminosity is a (possibly time-dependent) product of beam flux and density of scattering centers.",
"The process also depends on beam energy.",
"To simplify notation we write that for a specific experimental configuration $i$ , $N_i = \\varepsilon _i^{-1} {\\cal L}_i \\sigma _i,$ where it is implicit that ${\\cal L}_i$ is an integrated luminosity for the configuration, and that $\\sigma _i$ is the differential cross section, which could depend on energy and scattering anglesWe will simply refer to these quantities as “luminosity” and “cross section” hereafter..",
"The efficiency and the luminosity are experiment-dependent, whereas the cross section contains all the physics information and is a link to theoretical models of the reaction.",
"The main observable for any reaction of interest is the cross section, and its determination as a function of energy and angle requires careful setup and handling of the beam, target and detector systems, in order to obtain an accurate value for the luminosity and efficiency of the experiment.",
"If the experiment is setup so that the spin configuration of beam, target or recoils is not fixed then the cross section represents a sum over initial spins and an average over final spins.",
"If the experiment does contain an element of polarization, then the distribution of cross section will contain additional dependence on the kinematics of the reaction and the degrees of polarization.",
"Since theoretical models of cross sections are calculated from coherent sums of amplitudes that are dependent on the individual spin combinations of beam, target and recoiling products, it is desirable to evaluate these as well.",
"Table summarizes the distributions for the various experimental configurations, where we again limit the discussion to single pseudoscalar meson photoproduction.",
"The main point of this table is to illustrate that as more elements of the experimental configuration are polarized, the more complicated is the dependence of the intensity distribution on the number of observables.",
"C2cm C2cm C1.5cm p9.5cm >(5a) C1cm width=.75 Expressions for cross sections for different experiments.",
"3cConfiguration 2[8]*Cross section formula, $\\sigma /\\sigma _0$ (lr)1-3 Beam Target Recoil Beam Target Recoil Cross section formula, $\\sigma /\\sigma _0$ 5rContinued on next page 6*Unpolarized 2*Unpolarized N 1 (r)4-5 Y $1 + {red}{\\mathbf {P}} P^R_{y^\\prime }$ [0.7](lr)3-5 2*Longitudinal N 1 (r)4-5 Y $1 + {red}{\\mathbf {P}} P^R_{y^\\prime } +\\left({red}{\\mathbf {L_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {L_{z^\\prime }}} P^R_{z^\\prime }\\right) P^T_z$ [0.7](lr)3-5 2*Transverse N $1 + {red}{\\mathbf {T}} P^T_T \\sin (\\beta -\\phi ) $ (r)4-5 [0.5em] Y $\\begin{aligned}& 1 + {red}{\\mathbf {P}} P^R_{y^\\prime }+\\left({red}{\\mathbf {\\Sigma }} P^R_{y^\\prime } + {red}{\\mathbf {T}}\\right) P^T_T \\sin (\\beta -\\phi ) \\\\& \\hspace{5.0pt}+\\left({red}{\\mathbf {T_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {T_{z^\\prime }}} P^R_{z^\\prime }\\right) P^T_T \\cos (\\beta - \\phi )\\end{aligned}$ 6*[-4.0em]Circular 2*Unpolarized N 1 (r)4-5 Y $1 + {red}{\\mathbf {P}} P^R_{y^\\prime } +\\left({red}{\\mathbf {C_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {C_{z^\\prime }}} P^R_{z^\\prime }\\right) P_{\\odot }^{\\gamma }$ [0.7](lr)3-5 2*Longitudinal N $1 - {red}{\\mathbf {E}} P_{\\odot }^{\\gamma }P^T_y$ (r)4-5 [0.5em] Y $\\begin{aligned}& 1 + {red}{\\mathbf {P}} P^R_{y^\\prime } + \\left({red}{\\mathbf {L_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {L_{z^\\prime }}} P^R_{z^\\prime }\\right) P^T_z \\\\& \\hspace{5.0pt}+ \\left\\lbrace {red}{\\mathbf {C_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {C_{z^\\prime }}} P^R_{z^\\prime }- \\left({red}{\\mathbf {E}} + {red}{\\mathbf {H}} P^R_{y^\\prime }\\right) P^T_z\\right\\rbrace P_{\\odot }^{\\gamma }\\end{aligned}$ [0.5em] [0.7](lr)3-5 2*Transverse N $1 + {red}{\\mathbf {T}} P^T_T \\sin (\\beta -\\phi ) +{red}{\\mathbf {F}} P_{\\odot }^{\\gamma }P^T_T \\cos (\\beta -\\phi )$ (r)4-5 [0.5em] Y $\\begin{aligned}& 1 + {red}{\\mathbf {P}} P^R_{y^\\prime }+ \\left({red}{\\mathbf {\\Sigma }} P^R_{y^\\prime } + {red}{\\mathbf {T}}\\right) P^T_T \\sin (\\beta -\\phi ) \\\\& \\hspace{5.0pt}+ \\left({red}{\\mathbf {T_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {T_{z^\\prime }}} P^R_{z^\\prime }\\right) P^T_T \\cos (\\beta -\\phi ) \\\\& \\hspace{5.0pt}+\\left\\lbrace {red}{\\mathbf {C_{x^\\prime }}} P^R_{x^\\prime }+ {red}{\\mathbf {C_{z^\\prime }}} P^R_{z^\\prime }+ \\left({red}{\\mathbf {F}} + {red}{\\mathbf {G}}P^R_{y^\\prime } \\right) P^T_T \\cos (\\beta -\\phi ) \\right.\\\\& \\left.",
"\\qquad + \\left({red}{\\mathbf {O_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {O_{z^\\prime }}} P^R_{x^\\prime }\\right) P^T_T \\sin (\\beta -\\phi )\\right\\rbrace P_{\\odot }^{\\gamma }\\end{aligned}$ [0.5em] 6*[-5.0em]Linear 2*Unpolarized N $1 - {red}{\\mathbf {\\Sigma }} P_L^{\\gamma }\\cos 2(\\alpha -\\phi )$ (r)4-5 [0.5em] Y $\\begin{aligned}& 1 + {red}{\\mathbf {P}} P^R_{y^\\prime } -\\left\\lbrace {red}{\\mathbf {\\Sigma }} + {red}{\\mathbf {T}} P^R_{y^\\prime }\\right\\rbrace P_L^{\\gamma }\\cos 2(\\alpha -\\phi ) \\\\& \\hspace{5.0pt}+\\left\\lbrace {red}{\\mathbf {O_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {O_{z^\\prime }}} P^R_{z^\\prime }\\right\\rbrace P_L^{\\gamma }\\sin 2(\\alpha -\\phi ) \\end{aligned}$ [0.5em] [0.7](lr)3-5 2*Longitudinal N $1 - {red}{\\mathbf {\\Sigma }} P_L^{\\gamma }\\cos 2(\\alpha -\\phi ) +{red}{\\mathbf {G}} P^T_y P_L^{\\gamma }\\sin 2(\\alpha -\\phi )$ (r)4-5 [0.5em] Y $ \\begin{aligned}&1 + {red}{\\mathbf {P}} P^R_{y^\\prime } +\\left({red}{\\mathbf {L_{x^\\prime }}} P^R_{x^\\prime } +{red}{\\mathbf {L_{z^\\prime }}} P^R_{z^\\prime }\\right) P^T_z \\\\& \\; -\\left\\lbrace {red}{\\mathbf {\\Sigma }} + {red}{\\mathbf {T}} P^R_{y^\\prime } +\\left({red}{\\mathbf {T_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {T_{z^\\prime }}} P^R_{x^\\prime }\\right) P^T_z\\right\\rbrace P_L^{\\gamma }\\cos 2(\\alpha -\\phi ) \\\\& \\; +\\left\\lbrace \\left({red}{\\mathbf {F}} P^R_{y^\\prime } + {red}{\\mathbf {G}}\\right) P^T_z +{red}{\\mathbf {O_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {O_{z^\\prime }}} P^R_{z^\\prime }\\right\\rbrace P_L^{\\gamma }\\sin 2(\\alpha -\\phi )\\end{aligned}$ [0.5em] [0.7](lr)3-5 [0.5em] 2*[-2.0em]Transverse N $\\begin{aligned}& 1 + {red}{\\mathbf {T}} P^T_T \\sin (\\beta -\\phi ) \\\\& \\hspace{5.0pt}-\\left\\lbrace {red}{\\mathbf {\\Sigma }}+{red}{\\mathbf {P}} P^T_T \\sin (\\beta -\\phi )\\right\\rbrace P_L^{\\gamma }\\cos 2(\\alpha -\\phi ) \\\\& \\hspace{5.0pt}+{red}{\\mathbf {H}} P^T_T \\cos (\\beta -\\phi ) P_L^{\\gamma }\\sin 2(\\alpha -\\phi )\\end{aligned}$ [0.5em] (r)4-5 [0.5em] Y $\\begin{aligned}& 1 - P_L^{\\gamma }P^R_{y^\\prime } P^T_T \\sin (\\beta -\\phi )\\cos 2(\\alpha -\\phi ) + {red}{\\mathbf {P}} P^R_{y^\\prime } \\\\& \\hspace{5.0pt}+\\left({red}{\\mathbf {T_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {T_{z^\\prime }}} P^R_{z^\\prime }\\right) P^T_T \\cos (\\beta -\\phi ) \\\\& \\hspace{5.0pt}+\\left({red}{\\mathbf {\\Sigma }} P^R_{y^\\prime } + {red}{\\mathbf {T}}\\right) P^T_T \\sin (\\beta -\\phi ) \\\\& \\hspace{5.0pt}-\\left\\lbrace {red}{\\mathbf {\\Sigma }}+ {red}{\\mathbf {T}} P^R_{y^\\prime }+ {red}{\\mathbf {P}} P^T_T \\sin (\\beta -\\phi ) \\right.",
"\\\\& \\left.",
"\\qquad - \\left({red}{\\mathbf {L_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {L_{z^\\prime }}} P^R_{x^\\prime }\\right) P^T_T \\cos (\\beta -\\phi )\\right\\rbrace P_L^{\\gamma }\\cos 2(\\alpha -\\phi ) \\\\& \\hspace{5.0pt}+\\left\\lbrace {red}{\\mathbf {O_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {O_{z^\\prime }}} P^R_{z^\\prime }+ \\left({red}{\\mathbf {E}} P^R_{y^\\prime } + {red}{\\mathbf {H}}\\right) P^T_T \\cos (\\beta -\\phi ) \\right.",
"\\\\& \\left.",
"\\qquad - \\left({red}{\\mathbf {C_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {C_{z^\\prime }}} P^R_{x^\\prime }\\right) P^T_T \\sin (\\beta -\\phi )\\right\\rbrace P_L^{\\gamma }\\sin 2(\\alpha -\\phi )\\end{aligned}$ [0.5em] Rather than measuring cross-sections for specific polarization configurations, a common technique is to access them by measuring asymmetries.",
"Defining in general the notation for asymmetry in the number of counts between two experimental configurations $i$ and $j$ $A_N = \\dfrac{N_i - N_j}{N_i + N_j}= \\dfrac{\\varepsilon _i^{-1} {\\cal L}_i \\sigma _i - \\varepsilon _j^{-1} {\\cal L}_j \\sigma _j}{\\varepsilon _i^{-1} {\\cal L}_i \\sigma _i + \\varepsilon _j^{-1} {\\cal L}_j \\sigma _j},$ and introducing the further notation $A_{\\cal L}=\\frac{{\\cal L}_i-{\\cal L}_j}{{\\cal L}_i+{\\cal L}_j};\\qquad A_\\varepsilon =\\frac{\\varepsilon _i-\\varepsilon _j}{\\varepsilon _i+\\varepsilon _j};\\qquad A_\\sigma =\\frac{\\sigma _i-\\sigma _j}{\\sigma _i+\\sigma _j};$ we find $A_N = \\dfrac{A_\\sigma + A_{\\cal L}- A_\\varepsilon - A_\\sigma A_{\\cal L}A_\\varepsilon }{1 - A_{\\cal L}A_\\varepsilon - A_\\sigma A_\\varepsilon + A_\\sigma A_{\\cal L}}.$ In most cases, the difference in efficiency between two settings will be close to, if not identically, zero, and the expression simplifies to $A_N = \\dfrac{A_\\sigma + A_{\\cal L}}{1 + A_\\sigma A_{\\cal L}},$ which shows that if $A_{\\cal L}$ can be made small (i.e., the luminosity in the two settings is roughly equal), the main driver in the asymmetry of counts will be in $A_\\sigma $ , which contains the physics quantities of interest.",
"For a given setting ${\\cal S}$ of a configuration of beam and target polarization, the cross section formula can be written in a simple form $\\sigma = u + {\\cal S}v,$ where $u$ is a function of everything that does not depend on the setting ${\\cal S}$ and $v$ is a function of everything that does depend on it.",
"If we have two settings, ${\\cal S}_i$ and ${\\cal S}_j$ then $A_\\sigma = \\frac{\\left( {\\cal S}_i - {\\cal S}_j \\right) v}{2u + \\left( {\\cal S}_i + {\\cal S}_j \\right) v},$ so that if we can arrange ${\\cal S}_j = - {\\cal S}_i$ this would maximally isolate the function $v$ in the asymmetry.",
"This may not be possible to achieve in practice, so if the best we can do is ${\\cal S}_j = 2\\delta - {\\cal S}_i$ , where $\\delta $ represents half the difference in degree of polarization between the two settings, then $A_\\sigma = \\frac{\\left( s_i + \\delta \\right) v}{u + \\delta v},$ where $s_i \\in [0,1]$ is the degree of polarization in setting ${\\cal S}_i$ .",
"To make this less abstract, we give in Table some examples of $A_\\sigma $ s for a range of beam and target polarization settings.",
"For clarity we take $\\delta = 0$ , so that $A_\\sigma = s_i v / u$ but note the straightforward extension to Eq.",
"(REF ) if the degree of polarization is different between settings.",
"We include the terms related to recoil polarization measurement, which can be removed if recoil polarization is not determined (i.e., set $P^R_{x^\\prime }=P^R_{y^\\prime }=P^R_{z^\\prime }=0$ ).",
"Note that in some cases, such as the identification of $\\Lambda $ s from the decay to $\\pi p$ by detecting the pion or proton, there will be sensitivity to recoil polarization, so those terms cannot be removed.",
"C1.5cm C1.5cm C3.5cm p8.5cm >(13a) C1cm width=.75 Expressions for asymmetries for different experiments.",
"The definitions of angles are shown in figure REF .",
"Configurations are labeled U, C and L for unpolarized, circular and linear polarized photon beams; U, L and T for unpolarized, longitudinal and transverse target polarization.",
"Where degrees of polarization are labelled +ve or -ve, this refers to their direction with respect to an axis: lab $x$ , $y$ or $z$ for target polarization; photon beam direction for circular photon polarization.",
"3cConfiguration 2[8]*Asymmetry formula, $A_\\sigma $ (lr)1-3 Beam Target Settings Beam Target Settings Asymmetry formula, $A_\\sigma $ 5rContinued on next page [0.5em] 2*[-1.5em]U L $P^T_z\\;+$ ve; $P^T_z\\;-$ ve $\\dfrac{P^T_z \\left({red}{\\mathbf {L_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {L_{z^\\prime }}} P^R_{z^\\prime } \\right)}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}$ [1.0em] T $\\beta = 0;\\;\\beta =\\pi $ $\\begin{aligned}\\dfrac{ P^T_T}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}& \\left\\lbrace \\left( {red}{\\mathbf {T_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {T_{z^\\prime }}} P^R_{z^\\prime } \\right) \\cos \\phi \\right.",
"\\\\& \\left.",
"\\hspace{5.0pt}-\\left( {red}{\\mathbf {\\Sigma }} P^R_{y^\\prime } + {red}{\\mathbf {T}}\\right) \\sin \\phi \\right\\rbrace \\end{aligned}$ [0.5em] [0.5em] 3*[-3.45em]C U $P_{\\odot }^{\\gamma }\\;+$ ve; $P_{\\odot }^{\\gamma }\\;-$ ve $\\dfrac{P_{\\odot }^{\\gamma }\\left({red}{\\mathbf {C_{x^\\prime }}} P^R_{x^\\prime } + {red}{\\mathbf {C_{z^\\prime }}} P^R_{z^\\prime } \\right)}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}$ [1.0em] L ($P_{\\odot }^{\\gamma }\\;+$ ve, $P^T_z\\;+$ ve | $P_{\\odot }^{\\gamma }\\;-$ ve, $P^T_z\\;-$ ve); ($P_{\\odot }^{\\gamma }\\;+$ ve, $P^T_z\\;-$ ve | $P_{\\odot }^{\\gamma }\\;-$ ve, $P^T_z\\;+$ ve) $\\dfrac{P_{\\odot }^{\\gamma }P^T_z \\left({red}{\\mathbf {E}} + {red}{\\mathbf {H}} P^R_{y^\\prime } \\right)}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}$ [1.0em] T ($P_{\\odot }^{\\gamma }\\;+$ ve, $\\beta =0$ | $P_{\\odot }^{\\gamma }\\;-$ ve, $\\beta =\\pi $ ); ($P_{\\odot }^{\\gamma }\\;+$ ve, $\\beta =0$ | $P_{\\odot }^{\\gamma }\\;-$ ve, $\\beta =\\pi $ ) $\\begin{aligned}\\dfrac{ P_{\\odot }^{\\gamma }P^T_T}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}& \\left\\lbrace \\left( {red}{\\mathbf {F}} + {red}{\\mathbf {G}} P^R_{y^\\prime } \\right)\\cos \\phi \\right.",
"\\\\& \\left.",
"\\hspace{5.0pt}-\\left( {red}{\\mathbf {O_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {O_{z^\\prime }}} P^R_{x^\\prime }\\right)\\sin \\phi \\right\\rbrace \\end{aligned}$ [0.5em] [0.5em] [0.5em] 3*[-4.2em]L U $\\alpha =0$ ; $\\alpha =\\frac{\\pi }{2}$ $\\begin{aligned}\\dfrac{ - P_L^{\\gamma }}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}& \\left\\lbrace \\left( {red}{\\mathbf {\\Sigma }} + {red}{\\mathbf {T}} P^R_{y^\\prime } \\right)\\cos 2 \\phi \\right.",
"\\\\& \\left.",
"\\hspace{5.0pt}+\\left( {red}{\\mathbf {O_{x^\\prime }}} P^R_{x^\\prime } -{red}{\\mathbf {O_{z^\\prime }}} P^R_{z^\\prime }\\right)\\sin 2 \\phi \\right\\rbrace \\end{aligned}$ [1.0em] L ($\\alpha =0$ , $P^T_z\\;+$ ve | $\\alpha =\\frac{\\pi }{2}$ , $P^T_z\\;-$ ve); ($\\alpha =0$ , $P^T_z\\;-$ ve | $\\alpha =\\frac{\\pi }{2}$ , $P^T_z\\;+$ ve) $\\begin{aligned}\\dfrac{ - P_L^{\\gamma }P^T_z}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}& \\left\\lbrace \\left( {red}{\\mathbf {T_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {T_{z^\\prime }}} P^R_{x^\\prime } \\right)\\cos 2 \\phi \\right.",
"\\\\& \\left.",
"\\hspace{5.0pt}+\\left( {red}{\\mathbf {F}} P^R_{y^\\prime } + {red}{\\mathbf {G}}\\right)\\sin 2 \\phi \\right\\rbrace \\end{aligned}$ [1.0em] T ($\\alpha =0$ , $\\beta =0$ | $\\alpha =\\frac{\\pi }{2}$ , $\\beta =\\pi $ ); ($\\alpha =0$ , $\\beta =0$ | $\\alpha =\\frac{\\pi }{2}$ , $\\beta =\\pi $ ) $\\begin{aligned}\\dfrac{ P_L^{\\gamma }P^T_T}{1 + {red}{\\mathbf {P}} P^R_{y^\\prime }}& \\left\\lbrace \\left( P^R_{y^\\prime } + {red}{\\mathbf {P}} \\right)\\sin \\phi \\cos 2 \\phi \\right.",
"\\\\& +\\left( {red}{\\mathbf {L_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {L_{z^\\prime }}} P^R_{x^\\prime }\\right)\\cos \\phi \\cos 2 \\phi \\\\& +\\left( {red}{\\mathbf {C_{x^\\prime }}} P^R_{z^\\prime } - {red}{\\mathbf {C_{z^\\prime }}} P^R_{x^\\prime } \\right)\\sin \\phi \\sin 2 \\phi \\\\& \\left.",
"+\\left( {red}{\\mathbf {E}} P^R_{y^\\prime } + {red}{\\mathbf {H}} \\right)\\cos \\phi \\sin 2 \\phi \\right\\rbrace \\end{aligned}$ [0.5em] Tables and show that in practice observables are always measured in combinations.",
"The final, but most technically challenging measurement, given by Eq.",
"(5) in Table is perhaps the nearest one could claim to being a “complete experiment\" as it is sensitive to a “complete set\" of observables, but note that it is additionally sensitive to several more observables.",
"The more important challenge is to perform measurements with sufficient accuracy.",
"A rule of thumb is that pseudoscalar photoproduction observables need to be measured to better than $\\pm \\sim 0.5$ to provide any information."
],
[
"How to Extract Parameters of Nucleon Resonances from the Photoproduction Data",
"Very simply put, one constructs a data model whose parameters are explicitly or implicitly related to physical parameters such as masses, branching ratios and coupling constants.",
"The data model can be constructed from a physics model of the reaction.",
"Physics models can vary from simply describing a single reaction channel at the tree level, to complicated coupled-channel models that require the analysis of any reaction that can kinematically contribute to a final state.",
"The advantage of a single-channel reaction model is that it is relatively straightforward to calculate and to obtain a rough idea of the main contributions from resonances.",
"The disadvantage of this is that the extracted parameters are more difficult to interpret when comparing results for different channels.",
"A coupled-channels approach on the other hand allows one to extract coupling constants and other parameters in such a way as to be consistent between channels, at the expense of having to estimate sometimes hundreds of parameters, which requires heavy computational resource.",
"In doing this there are a number of complications.",
"For instance, how does one choose which resonant states to include?",
"This is a model comparison problem, since adding more resonances will mean the addition of more parameters, thereby making a fit to the data easier.",
"On the other hand, an Occam's razor approach to keep the model as simple as possible should act to reduce the number of resonances that require to be invoked.",
"Alternatively one may want to extract information in a “model independent” way.",
"By analyzing distributions in energy and angle, a partial-wave analysis (PWA) can be carried out in which the intensity and phase of each partial wave can be examined to determine the contributions of different resonances.",
"Again, there is a model comparison issue with the question of how many partial waves to include in fits.",
"Originally, PWA arose as the technology to determine the amplitudes of a reaction through fitting scattering data.",
"This is a non-trivial mathematical problem – looking for a solution of an ill-posed problem, as described in Hadamard [27] and Tikhonov [28].",
"Resonances appeared as a by-product (bound states objects with definite quantum numbers, mass, lifetime and so on).",
"Standard PWA reveals resonances that are not too wide ($\\Gamma <$ 500 MeV) and possess a large enough elastic branching ratio (BR $>$ 4%).",
"It is possible, however, to miss narrow resonances with $\\Gamma <$ 30 MeV [29].",
"Whether one wants to extract physics from the data by fitting model parameters or projecting out partial waves, there is a choice as to how to use the data.",
"If the phenomenology group is well enough connected with the experiments, it can be possible to construct likelihood functions on an event-by-event basis.",
"This approach does require high numbers of events for the results to be robust, but means that quantities are not averaged over regions of phase space.",
"A more common interface between experiment and theory is for the experimenters to report the values of observables, which have been binned in energy and angles.",
"At the current levels of accuracy, both approaches are yielding similar results.",
"The main objectives of PWA schemes, apart from establishing the existence of resonances, are to derive estimates of resonance properties such as mass, width, branching ratios, couplings, etc.",
"Calling an object a resonance implies that there is a resonant frequency and an associated width that characterizes the state.",
"By analogy with mechanical resonances Breit-Wigner (BW) parameters, mass and width, can be used to describe each resonance, but their exact values depend on the model-dependent method of extraction.",
"The preferred approach, as described in the Review of Particle Physics [30], is for an analysis to estimate the position of poles in the complex energy plane."
],
[
"Reactions on Neutron Targets",
"Only with good data on both proton and neutron targets, can one hope to disentangle the isoscalar and isovector electromagnetic couplings of various $N^\\ast $ and $\\Delta ^\\ast $ resonances, as well as the isospin properties of non-resonant background amplitudes [31], [32].",
"Unfortunately, there is no free neutron target.",
"The radiative decay width of neutral baryons may be extracted from $\\pi ^-$ and $\\pi ^0$ photoproduction from neutrons, but in practice one can only use a target containing a bound neutron.",
"To extract relevant information one requires the use of model-dependent final-state interaction (FSI) corrections [33], [31].",
"There is no way to isolate FSI experimentally [34], [35].",
"At lower energies (E < 700 MeV), there are data for the inverse $\\pi ^-$ photoproduction reaction, $\\pi ^-p\\rightarrow \\gamma n$ .",
"This process is free from complications associated with a deuteron target.",
"However, there is a major disadvantage of using $\\pi ^-p\\rightarrow \\gamma n$ : there is a large background from $\\pi ^-p\\rightarrow \\pi ^0n\\rightarrow \\gamma \\gamma n$ reactions, whose cross section is 5 to 500 times larger than $\\pi ^-p\\rightarrow \\gamma n$ .",
"Studies of the $\\gamma n\\rightarrow \\pi ^-p$ and $\\gamma n\\rightarrow \\pi ^0n$ reactions can be carried out in quasi-free kinematics with deuteron targets.",
"The reactions $\\gamma d\\rightarrow \\pi ^-p(p)$ and $\\gamma d\\rightarrow \\pi ^0n(p)$ in these kinematics have a fast, knocked-out nucleon and a slow proton spectator, and the slow proton is assumed not to be involved in the pion production process.",
"In this quasi-free region, the reaction mechanism corresponds to the “dominant\" impulse approximation (IA) diagram in Figure REF (a) with the slow proton emerging from the deuteron vertex.",
"Here, the differential cross section on the deuteron can be related to that on the neutron target in a well understood way [34], [35].",
"Figure REF illustrates this dominant IA diagram, as well as the leading terms of FSI corrections.",
"An energy and angle dependent FSI correction factor, $R(E,\\theta )$ , can be defined as the ratio between the sum of three dominant diagrams in Figure REF and IA (the first of the diagrams).",
"This can then be applied to the experimental $\\gamma d$ data to get a two-body cross section for $\\gamma n\\rightarrow \\pi ^-p$ and $\\gamma n\\rightarrow \\pi ^0n$ .",
"The GWU SAID database contains phenomenological amplitudes for the reactions $\\pi N\\rightarrow \\pi N$ [36], $NN\\rightarrow NN$ [37], and $\\gamma N\\rightarrow \\pi N$ [38].",
"The GW-ITEP group, for example, used these amplitudes as inputs to calculate the dominant diagrams of the GWU-ITEP FSI approach.",
"The full Bonn potential [39] was then used for the deuteron description, which includes the Fermi motion of nucleons.",
"The GWU-ITEP FSI calculations [34] are available over a broad energy range (threshold to E = 2.7 GeV), and for the full CM angular range ($\\theta = 0^\\circ $ to $180^\\circ $ ).",
"Overall, the FSI correction factor $R < 1.00$ , while its value varies from 0.70 to 0.90 depending on the kinematics.",
"The behavior of $R$ is very smooth vs pion production angle.",
"There is a sizable FSI effect from the S-wave part of pp-FSI at small angles.",
"$R(E,\\theta )$ is used as the FSI correction factor for the CLAS quasi-free $\\gamma d\\rightarrow \\pi ^-pp$ cross section averaged over the laboratory photon energy bin width citeCH12,MA17.",
"Note that the FSI correction grows rapidly to the forward direction ($\\theta < 30^\\circ $ ).",
"There are currently few measurements in this regime, so the uncertainty due to FSI for this reaction at forward angles does not cause too much concern.",
"The contribution of uncertainty in FSI calculations to the overall systematic normalization uncertainty is estimated to be about 2-3% (the sensitivity to the deuteron wave-function is 1% and to the number of steps in the integration of the five-fold integrals is 2%).",
"For the CLAS measurements, no sensitivity was found to the value of proton momentum used to determine whether or not it is a spectator.",
"The $\\gamma n\\rightarrow \\pi ^0n$ measurement is much more complicated than the case of $\\gamma n\\rightarrow \\pi ^-p$ because the $\\pi ^0$ can come from both neutron and proton initial states.",
"The GW-ITEP studies have shown that photoproduction cross sections from protons and neutrons are generally not equal [35].",
"For $\\pi ^0$ photoproduction on proton and neutron targets we have $A(\\gamma p\\rightarrow \\pi ^0p) = A_v + A_s~~~~{\\mathrm {a}nd}~~~~A(\\gamma n\\rightarrow \\pi ^0n) = A_v - A_s,$ where $A_v$ and $A_s$ are the isovector and isoscalar amplitudes, respectively.",
"Therefore, if $A_s\\ne 0$ the $\\gamma p$ and $\\gamma n$ amplitudes are not equal.",
"Figure REF shows that proton and neutron cross sections are very close to each other in the $\\Delta (1232)3/2^+$ region ($A_s = 0$ ).",
"At higher energies, however, the contributions from $N(1440)1/2^+$ and $N(1535)1/2^-$ become important, the isoscalar amplitude does not equal zero, and the difference between proton and neutron differential cross sections becomes more clearly visible.",
"That means in general that one cannot simply use the ratio between free and bound proton data to be indicative of the ratio between free and bound neutron data.",
"Measurements using bound neutrons will thus always carry significant model-dependent uncertainty.",
"Figure: The IA (M a1 M_{a1}, M a2 M_{a2}), NN-FSI (M b M_b), and πN\\pi N (M c1 M_{c1}, M c2 M_{c2}) diagrams for the reaction γd→πN\\gamma d\\rightarrow \\pi N. Wavy, solid, dashed and double lines correspond to thephoton, nucleons, pion, and deuteron, respectively.Unfortunately, there are currently no FSI calculations for polarized measurements on neutron targets.",
"In the absence of these calculations, for PWA one can only assume that the effects of FSI on polarization observables are small.",
"There is some indirect proof that this assumption is reasonable, since several PWAs can successfully fit the polarized measurements in the world database (see, for instance, [40]).",
"Figure: The differential cross sections of the γp→π 0 p\\gamma p\\rightarrow \\pi ^0p (red solid curves) and γn→π 0 n\\gamma n\\rightarrow \\pi ^0n (blue dashedcurves) reaction reactions at several photon energies (a) E = 340 MeV,(b) E = 630 MeV, and (c) E = 787 MeV, which correspond toΔ(1232)3/2 + \\Delta (1232)3/2^+, N(1440)1/2 + N(1440)1/2^+, and N(1535)1/2 - N(1535)1/2^- regions,respectively (Ref.",
")."
],
[
"Experimental Facilities ",
"In this section, we provide a brief description and references to experimental facilities that were the main contributors of the photoproduction data over last two decades.",
"Some of them used bremsstrahlung to generate real photons, others used laser Compton backscattering.",
"Some detectors were optimized for charged particles, others for neutrals.",
"In that respect they are complimentary to each other."
],
[
"CEBAF",
"The Thomas Jefferson National Accelerator Facility (TJNAF) commonly known as Jefferson Lab or JLab is the home of the Continuous Electron Beam Accelerator Facility, CEBAF.",
"This is a race track shaped machine that consists of two linear accelerators joined together with a pair of arc sections.",
"For the results reported here, the electron beam made up to five passes through the machine and gained energy up to 6 GeV.",
"The extracted beam was delivered to end stations known as Hall A, Hall B and Hall C. The electron beam can be highly polarized.",
"The majority of photoproduction data at CEBAF was obtained in Hall B with CLAS detector.",
"Recently CEBAF was upgraded and its energy doubled.",
"Now it can accelerate electrons up to 12 GeV.",
"One more experimental hall, Hall D, was added.",
"The CEBAF Large Acceptance Spectrometer (CLAS) was a magnetic spectrometer with a toroidal magnetic field [41].",
"It has since been upgraded to CLAS12 to cope with the increased electron beam energy in Hall B.",
"The new detector has a slightly different configuration to the older CLAS, although some of the original detector subsystems have been refurbished and retained.",
"The toroidal field bends particles of different charge either towards or away from the beam direction, which results in some asymmetry of the acceptance for opposite charges.",
"The magnetic field is produced by six superconducting coils positioned around the beam.",
"Essentially it may be considered as six independent spectrometers.",
"The gaps between each pair of the coils are filled with detector packages.",
"Each package has six multilayer drift chambers for charged particle tracking.",
"The momentum resolution for charge particles from tracking depends on the angle and magnetic field setting and on average was $\\Delta p/p \\sim 0.5-1\\%$ .",
"Polar angle resolution is about 1 mrad or better.",
"Azimuthal angle resolution is about 4 mrad.",
"They drift chambers followed by gas Čerenkov counters for electron pion separation covering forward angles up to $45^\\circ $ .",
"Further out there is an array of TOF scintillation counters that were used for charged particles identification.",
"TOF counters cover polar angle rage from $8^\\circ $ to $142^\\circ $ and full range of azimuthal angles.",
"The solid angle for charged particles was about 60% of $4\\pi $ .",
"The last detector in a package is an electromagnetic calorimeter.",
"It is a sampling calorimeter made of alternating layers of lead and plastic scintillators.",
"The total thickness is 16 radiation lengths.",
"The sampling fraction is approximately 0.3 for electrons of 3 GeV and greater, and for smaller energies, there is a monotonic decrease to about 0.25 for electrons of 0.5 GeV.",
"The energy resolution was $\\sigma /E = 10.3\\% / \\sqrt{E(GeV)}$ .",
"In order to get coordinates of the shower the scinitillator strips were arranged to provide three views crossing each other at $60^\\circ $ .",
"The calorimeters covered angles from $8^\\circ $ to $45^\\circ $ .",
"The design of CLAS was optimized for charged particles.",
"The unpolarized or circularly photons were produced via bremsstrahlung on a thin gold foil.",
"Coherent bremsstrahlung on a diamond radiator was used to produce linearly polarized photons.",
"Tagging of bremsstrahlung photons was done by the Hall B tagging spectrometer [42] with a tagging range from 20% to 95% of the electron beam energy.",
"The focal plane was instrumented with a two-layer scintillation hodoscope.",
"The first layer consisted of 384 overlapping counters providing the energy of the post-bremsstrahlung electron with an accuracy of $\\sim 0.001$ of the electron beam energy.",
"The second layer of 61 counters provided timing information.",
"The target was placed in the center of the detector and was surrounded by a scintillation start counter.",
"CLAS could operate with various types of targets: unpolarized gas, liquid and solid targets.",
"Two different frozen spin polarized targets were used in photoproduction experiments.",
"One, FROST [43], with butanol as a target material was used for experiments with polarized protons.",
"It allowed for longitudinal and transverse polarization of protons.",
"The second target, HDIce [44], was used for experiments with longitudinally polarized protons and deuterons."
],
[
"ESRF",
"The European Synchrotron Radiation Facility (ESRF) is the most intense source of synchrotron-generated light.",
"After the ESRF pre-injector LINAC a 200 MeV electrons injected into the booster synchrotron which accelerate them to 6 GeV.",
"They then injected in a 6 GeV storage ring where they can be used for physics."
],
[
"GRAAL",
"One of the Collaborative research beam lines at ESRF hosted GRenoble Anneau Accélérateur Laser (GRAAL) facility [45].",
"Photons were produced by Compton backscattering of laser light from the electron beam.",
"The tagged photon energy spectrum at GRAAL extended from 600 MeV to 1500 MeV.",
"The core of the facility was a large solid angle detector (La$\\gamma $ range).",
"The central part of La$\\gamma $ range was a BGO calorimeter which covered polar angles $25^\\circ - 155^\\circ $ and full range of azimuthal angle.",
"In the center of the calorimeter there was a plastic scintillator barrel and internal tracker made of two cylindrical multiwire proportional chambers (MWPC).",
"The forward polar angles below $25^\\circ $ were covered by two pairs of planar MWPC and double wall of plastic scintillators followed by shower wall consisting of four layers of lead and plastic scintillators.",
"The calorimeter had excellent energy resolution for photons and electrons, 3% at 1 GeV.",
"It also had good response for protons below 300 MeV.",
"Charged particles could be tracked by MWPCs.",
"Neutrons could be detected either in BGO calorimeter or forward wall.",
"The entire apparatus was optimized for the detection of mesons decaying to photons but could also detect charged particles.",
"GRAAL is no longer in operation.",
"The BGO calorimeter has been moved to Bonn and became a part of the new BGO-OD setup [46]."
],
[
"MAMI",
"The Mainz Microtron, MAMI, is an accelerator for electron beams run by the Institute for Nuclear Physics of the University of Mainz, and is used extensively for hadron physics experiments.",
"It is a continuous wave accelerator system.",
"Over the years it went through a chain of upgrades.",
"The latest incarnation is MAMI-C, which can accelerate electrons up to 1508 MeV.",
"Experimental area A2 is dedicated to experiments with tagged bremsstrahlung photons.",
"Linearly polarized photons are produced via coherent bremsstrahlung on a diamond radiator.",
"The tagging is done by the Glasgow tagger [47].",
"It was originally built for MAMI-B with maximum energy of 833 MeV.",
"To improve energy resolution it was later complimented by a microscope [48] with increase energy resolution over a smaller range of electron energies.",
"After MAMI-C went into operation the tagger was upgraded for use with beam energy of 1500 MeV [49].",
"The tagging range is 5 – 93% of the electron beam energy.",
"The energy resolution without the microscope is 4 MeV for a 1500 MeV incident beam.",
"The microscope improves energy the resolution by a factor of 6 in the 60 MeV energy range."
],
[
"DAPHNE",
"DAPHNE (Detecteur à grande Acceptance pour la PHysique photoNucleaire Experimentale) is a large acceptance tracking detector for intermediate-energy hadrons comprising a vertex detector surrounded by a segmented calorimeter [50].",
"The detector consists of three principal parts, arranged as a set of coaxially .",
"In the center there is a vertex detector which is surrounded by a charged-particle detector consisting of several layers of scintillator which is itself surrounded by a lead-aluminium-scintillator sandwich designed to detect neutral particles.",
"It covers polar angles from $21^\\circ $ to $159^\\circ $ and has full azimuthal angle coverage.",
"Now DAPHNE is no longer in operation."
],
[
"TAPS",
"TAPS (Two Arm Photon Spectrometer) [51] is a detector array of 384 individual modules of hexagonal shaped detectors.",
"Each detector module is a telescope consisting of a BaF$_2$ crystal and a separate plastic scintillator in front of it.",
"It can be used for charged/neutral separation and charged particle identification.",
"The energy resolution of TAPS is $\\sigma /E = 0 .59\\% /\\sqrt{E_\\gamma } +1.9\\%$ where $E_\\gamma $ is given in GeV.",
"The position resolution is about 2 cm.",
"TAPS was originally designed to detect two photon decays of $\\pi ^0$ and $\\eta $ mesons.",
"Recently TAPS was split in two pieces which were used separately with other detectors, the Crystal Ball and the Crystal Barrel."
],
[
"Crystal Ball/TAPS",
"The latest experimental setup in A2 is a combination of Crystal Ball and half of TAPS.",
"The details of the most recent configuration of the setup can be found in Ref. [52].",
"The Crystal Ball (CB) was originally built by Stanford Linear Accelerator Center (SLAC) [53].",
"It consists of 672 optically isolated NaI(Tl) crystals with a thickness of 15.7 radiation lengths .",
"The crystals are arranged to form a sphere covering 93% of the full solid angle.",
"The energy resolution for electromagnetic showers is described as $\\Delta E/E = 0.02/(E/{\\rm GeV})^{0.36}$ .",
"The accuracy of the shower direction reconstruction is about $\\sigma _\\theta \\sim 2-3^\\circ $ in polar angle and $\\sigma _\\varphi \\sim 2^\\circ / \\sin \\theta $ .",
"In the center of the CB there is a barrel of 24 scintillation counters surrounding the target.",
"It measures energy losses of the charged particle and can be used in $\\Delta E/E$ analysis for charged particles identification and also to separate charged particles from neutrals.",
"The forward angles $\\theta = 1 - 20^\\circ $ are covered by a half of the TAPS which is placed 1.5 m downstream of CB center.",
"The combined solid angle of CB and TAPS is 97% of 4$\\pi $ .",
"This setup can be used with both polarized and unpolarized targets.",
"This facility is operational and continues data taking."
],
[
"ELSA",
"The electron accelerator ELektronen-Stretcher-Anlage (ELSA) [54] is operated by the university of Bonn.",
"It has three stages: injector LINACs, booster synchrotron and stretcher ring.",
"It can deliver beams of polarized or unpolarized electrons with energies up to 3.5 GeV.",
"Real photon beam is produced via bremsstrahlung.",
"The linearly polarized beam is produced via coherent bremsstrahlung.",
"The bremsstrahlung photons are tagged with tagging hodoscope.",
"The accuracy of the photon energy is 0.4% of electron beam energy."
],
[
"SAPHIR",
"SAPHIR (Spectrometer Arrangement for PHoton Induced Reactions) [55] was a large solid angle detector at the Bonn accelerator ELSA.",
"SAPHIR was a magnetic spectrometer with a dipole magnet.",
"The photon beam entered through a hole in the magnet yoke.",
"The space between the magnet poles was occupied by the Central Drift Chamber (CDC) for charged particle tracking.",
"The target was placed in the center of the CDC.",
"For better tracking and momentum resolution there were also three planar drift chambers, two on the sides and one in the forward direction.",
"The momentum resolution of about 6.5% was achieved at 1.0 GeV/$c$ particle momentum.",
"The use of the forward drift chamber improved the momentum resolution considerably up to 2% at 1.8 GeV/$c$ .",
"There were three planes of scintillation counter hodoscopes, two on the sides and one in the forward direction.",
"The hodoscopes in coincidence with tagging system produced the trigger and were used for particle identification by measuring time-of-flight (TOF).",
"Downstream of the forward TOF there was an array of electromagnetic shower counters (EMC).",
"The energy resolution of the EMC was found to be $13\\%/\\sqrt{E}$ , where E in GeV.",
"Now SAPHIR is no longer in operation."
],
[
"CBELSA",
"The central part of the setup is the Crystal Barrel [56], the calorimeter that was used at the Low Energy Antiproton Ring (LEAR) at CERN.",
"In its original configuration it consisted of 1380 CsI(Tl) crystals.",
"The length of each crystal is 16.1 radiation lengths.",
"The crystals are grouped in 26 rings ($\\Delta \\theta = 6^\\circ $ ), where the larger rings consist of 60 crystals ($\\Delta \\varphi = 6^\\circ $ ), the six smallest rings contain 30 crystals ($\\Delta \\varphi = 12^\\circ $ ).",
"It covers angles from $12^\\circ $ to $168^\\circ $ with respect to the beam direction resulting in 97.8% coverage of the solid angle.",
"During the first configuration change the three forward rings were taken out and part of TAPS, MiniTAPS, was installed to extend coverage to smaller angles down to $1^\\circ $ .",
"During the second configuration change the forward crystals ($\\theta < 27^\\circ $ ) were covered by plastic scintillators in front of each crystal for charged particle identification.",
"Inside the calorimeter, a three-layer inner detector with 513 scintillating fibers was installed.",
"More details about the most recent version of the setup can be found in Ref. [57].",
"This setup is optimized for detection of multiphoton events.",
"CBELSA is active and continues data taking."
],
[
"BGO-OD",
"The BGO-OD [46] is a new experiment at ELSA.",
"It consists of a central detector enclosing the target in the angular range $10 - 155^\\circ $ .",
"This is complemented by a large aperture forward magnetic spectrometer covering the angular range from approximately $2^\\circ $ to $12^\\circ $ .",
"The main component of the central detector is BGO calorimeter formerly used at GRAAL.",
"A segmented plastic scintillator barrel and a double layer cylindrical MWPC placed inside the calorimeter enable tracking and identification of charged particles.",
"The forward spectrometer consists of a large aperture dipole magnet sandwiched between tracking detectors.",
"Front tracking upstream of the magnet is performed with two sets of scintillating fibre detectors.",
"Eight double layers of drift chambers serve for rear tracking downstream of the magnet.",
"Several new components are to be added.",
"The BGO-OD was commissioned in 2016."
],
[
"Spring-8",
"SPring-8 is a large synchrotron radiation facility located in Harima Science Park City, Japan.",
"The name “SPring-8\" is derived from “Super Photon ring-8 GeV\".",
"As the name implies it is an 8 GeV electron storage ring.",
"Among many other applications it is used for hadronic physics and photoproduction in particular."
],
[
"LEPS",
"Backward Compton scattering of laser light from a high energy electron beam is used to produce high energy photons.",
"This type of beam line was constructed at Spring-8 and is called “Laser-electron-photon\" (LEP).",
"If the laser light is polarized then the produced high energy photons also polarized.",
"The photons were tagged by detecting scattered electron.",
"The initial version of this facility could provide photons with energies up to 2.4 GeV.",
"The first detector, LEPS [58], was designed to study $\\phi $ -meson photoproduction in forward angles.",
"It is a magnetic spectrometer with a dipole magnet.",
"The vertex detector is located upstream of the magnet and consists silicon strip detectors and drift chambers.",
"Downstream of the magnet there were two sets of drift chambers, one on each side of the beam.",
"Particle identification is done using TOF.",
"The LEP beam line has been upgraded to increase the intensity of the photon beam and extend the energy range up to 2.9 GeV [59]."
],
[
"LEPS2",
"This approach was used to construct the second LEP beam line, LEPS2 [60].",
"LEPS1 had acceptance limitation to forward angles only.",
"To overcome this limitation a new detectors needed to be constructed for LEPS2.",
"One of the detectors aimed to study $\\eta ^\\prime $ mesic nuclei is BGOegg [61].",
"The detector is optimized for detection of photons.",
"It is an egg-shaped electromagnetic calorimeter.",
"It consists of 1320 BGO crystals of 20 radiation lengths.",
"It has a polar angle coverage from $24^\\circ $ to $144^\\circ $ and complete azimuthal coverage.",
"The energy resolution is 1.3% at 1 GeV and position resolution is 3.1 mm.",
"To detect charged particles the scintillation hodoscopes and cylindrical drift chambers are installed in the center of the calorimeter.",
"The second detector for LEPS2 is a solenoid spectrometer [60].",
"It is designed to detect both charged particles and photons.",
"It is a solenoid magnet with a 0.9 T field.",
"Tracking of charged particles is done by the Time projection chamber and forward drift chambers.",
"The tracking detectors are surrounded by a barrel of resistive plate chambers (RPC).",
"RPCs have very good timing resolution and are used for particle identification by measuring TOF.",
"For particle momenta above 1 GeV in addition to TOF the aerogel Čerenkov counters are used.",
"The outer most detector is barrel electromagnetic calorimeter, Barrel $\\gamma $ .",
"It is a sampling lead/plastic scintillator calorimeter with a thickness of 14.3 radiation lengths and covers polar angles $30 - 110^\\circ $ ."
],
[
" Available Experimental Data on Meson Photoproduction",
"In this section, we give an overview of available experimental data of meson photoproduction.",
"The source of the data is SAID database [15] which is to date the most comprehensive.",
"The data are organized by the final state.",
"The number of data points accumulated thus far makes it pointless to try to plot each of them in this review.",
"Instead we plot for each channel, in the style of Figure REF , the number of data points as a function of hadronic mass $W$ , and as a function of year.",
"The data are split into unpolarized and polarized stacked histograms and are mean to convey the relative amount available from each channel, as well as an indication of the progress in measurements over time.",
"For convenience we list the thresholds for the relevant photoproduction reactions.",
"They can be found in Table .",
"Since most of the photoproduction data were obtained within last two decades, we concentrate on this period.",
"We also limit discussion to the center of mass energies $W \\le 2.55$ GeV (E$_\\gamma \\le 3$ GeV).",
"Figures REF through REF show energy distribution for 1996 through 2018 (left) and time distributions (right).",
"Tables REF through REF provide references to all relevant experiments from 1996 through 2018.",
"They are organized by reaction and include observable, energy and angular range, number of the experimental data and a reference to original publication.",
"We have not included total cross sections because they were not directly measured but obtained by integration of differential ones and depend on the angular range of differential quantities measurements and extrapolation procedure.",
"For the reaction channels with limited amount of measurements we show only tables.",
"For double meson production we don't provide tables but rather just list experiments, their energy range and extracted observables.",
"The reason for this is following.",
"Since these are not binary reaction there are many possibles choices of the kinematic variables The same data can be binned differently depending on what is the goal of the analysis.",
"In many cases the event by event likelihood analysis was used without any binning.",
"-2pt L3cm C2.5cm C3cm 3rContinued on next page width=.75 Threshold energies.",
"Reaction W (MeV) E$_\\gamma $ (MeV) $\\gamma p\\rightarrow \\pi ^0p$ 1073.2 144.7 $\\gamma n\\rightarrow \\pi ^0n$ 1074.5 144.7 $\\gamma n\\rightarrow \\pi ^-p$ 1077.8 148.4 $\\gamma p\\rightarrow \\pi ^+n$ 1079.1 151.4 $\\gamma p\\rightarrow \\eta p$ 1487.4 707.6 $\\gamma n\\rightarrow \\eta n$ 1486.1 707.8 $\\gamma p\\rightarrow K^+\\Lambda $ 1609.4 911.1 $\\gamma n\\rightarrow K^0\\Lambda $ 1613.3 915.3 $\\gamma p\\rightarrow K^+\\Sigma ^0$ 1686.3 1046.2 $\\gamma p\\rightarrow K^0\\Sigma ^+$ 1687.0 1047.4 $\\gamma n\\rightarrow K^0\\Sigma ^0$ 1690.2 1050.6 $\\gamma n\\rightarrow K^+\\Sigma ^-$ 1691.1 1052.1 $\\gamma n\\rightarrow \\omega n$ 1722.2 1108.6 $\\gamma p\\rightarrow \\omega p$ 1720.9 1109.1 $\\gamma p\\rightarrow \\eta ^\\prime p$ 1896.0 1446.6 $\\gamma p\\rightarrow \\pi ^0\\pi ^0p$ 1208.2 308.8 $\\gamma p\\rightarrow \\pi ^+\\pi ^-p$ 1217.4 320.7 $\\gamma p\\rightarrow \\pi ^0\\eta p$ 1621.1 931.3"
],
[
"Single Pion Photoproduction ",
"The first experimental study of single pion photoproduction have started just two years after discovery of pion.",
"It has the lowest threshold and at low energies it is dominated by $\\Delta $ .",
"The amount of data vs. energy essentially follows the cross section For pion photoproduction, there is a dis-balance between $\\pi ^0$ p and $\\pi ^+$ n measurements, $\\pi ^+$ n/$\\pi ^0$ p = 20%.",
"While pion photoproduction on the neutron much less known vs on the proton, n/p = 31% [15].",
"Figure: Database for γp→π 0 p\\gamma p\\rightarrow \\pi ^0p.",
"Left: Experimental data from the SAID database selected for 1996 through 2018.",
"Right: Amount of data as a function of time.",
"Full SAID database.",
"The data shown as stacked histogram.",
"Light shaded – cross sections, dark shaded – polarization data.-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow \\pi ^0p$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"$\\Delta _{13} = (d\\sigma /d\\Omega )_{1/2}- (d\\sigma /d\\Omega )_{3/2}$ .",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 40%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 14*$d\\sigma /d\\Omega $ 1074-109110-170171MAMI [62] 1075-113618-162600MAMI [63] 1122-15373-1781129MAMI [64] 1131-122770-13073BNL [65] 1136-195715-1657978MAMI [66] 1209-137655-12067MAMI [67] 1217-243932-1481089ELSA [68] 1277-127770-17824MAMI [69] 1386-194245-168861GRAAL [45] 1390-153145-11997MAMI [70] 1455-153826-154799MAMI [71] 1465-250541-148620CEBAF [38] 1810-254234-80580CEBAF [72] 1934-2300129-167112Spring-8 [73] 11*$\\Sigma $ 1075-112625-155220MAMI [63] 1086-108630-1507MAMI [62] 1131-130660-15084BNL [65] 1154-130611-170353MAMI [64] 1216-144831-1581403MAMI [74] 1349-170285-125158Yerevan [75] 1384-191045-171441GRAAL [45] 1523-186937-156135ELSA [76] 1621-1998 5-165249ELSA [77] 1717-209132-148700CEBAF [78] 1946-2280129-16748Spring-8 [73] 3*P 1471-161351-163152ELSA [79] 1527-234959-13529CEBAF [80] 2084-246896-1433CEBAF [81] 4*T 1073-12915-1754343MAMI [82] 1179-139853-12752ELSA [83] 1306-188830-162397MAMI [52] 1471-247929-163601ELSA [79] 2*G 1232-123270-1103MAMI [84] 1438-182219-161318ELSA [85] H 1472-161351-163154ELSA [79] F 1306-188830-162397MAMI [52] E 1426-225922-158456ELSA [86] 2*$\\Delta _{13}$ 1209-137659-12262MAMI [67] 1390-153144-12378MAMI [70] 3*$C_{x^{\\prime }}$ 1322-184175-14045MAMI [87] 1527-234959-13528CEBAF [80] 2084-246896-1433CEBAF [81] 2*$C_{z^{\\prime }}$ 1527-234959-13525CEBAF [80] 2084-246896-1433CEBAF [81] Figure: Database for γp→π + n\\gamma p\\rightarrow \\pi ^+n.The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow \\pi ^+n$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"$\\Delta _{13} = (d\\sigma /d\\Omega )_{1/2}-(d\\sigma /d\\Omega )_{3/2}$ .",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 51%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 9*$d\\sigma /d\\Omega $ 1080-108146-13445TRIUMF/SAL [88] 1104-131331-157205MAMI [67] 1162-127772-14339MAMI [89] 1178-129245-135160MAMI [90] 1193-2201112-1791267ELSA [91] 1323-153345-155203MAMI [92] 1497-250532-148618CEBAF [93] 1714-235450-9010CEBAF [94] 1934-252411-49174Spring-8 [95] 6*$\\Sigma $ 1178-129220-17085BNL [65] 1178-129245-135160MAMI [90] 1416-168848-15492GRAAL [96] 1543-190147-160237GRAAL [97] 1722-209132-148386CEBAF [78] 1946-249611-4984Spring-8 [95] G 1232-123230-1306MAMI [84] E 1250-223020-148900CEBAF [98] 2*$\\Delta _{13}$ 1104-131335-153129MAMI [67] 1323-152450-150102MAMI [92] -2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma n\\rightarrow \\pi ^-p$ below W = 2.55 GeV (E$_\\gamma $ = 3.1 GeV).",
"Experimental data from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 4%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 5*$d\\sigma /d\\Omega $ 1191-152641-148300BNL [99] 1203-131858-133104MAMI [100] 1311-236626-1358428CEBAF [101] 1690-255133-157699CEBAF [102], [103] 1720-235650-901CEBAF [94] $\\Sigma $ 1516-189433-16399GRAAL [104] E 1500-230026-154266CEBAF [40] Figure: Database for γn→π 0 n\\gamma n\\rightarrow \\pi ^0n.The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma n\\rightarrow \\pi ^0n$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 24%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref $d\\sigma /d\\Omega $ 1300-190032-162969MAMI [105] $\\Sigma $ 1484-191253-164216GRAAL [106] E 1312-188846-154151MAMI [105]"
],
[
"$\\eta $ and {{formula:a46e6614-93d5-498e-8431-1b54d51ae2a0}} photoproduction",
"Since $\\eta $ and $\\eta ^\\prime $ are iso-singlets their photoproduction may not be directly coupled to $\\Delta $ resonances but only to the excitation of $N^\\ast $ Figure: Database for γp→ηp\\gamma p\\rightarrow \\eta p.The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow \\eta p$ below W = 2.55 GeV (E$_\\gamma $ = 3.1 GeV).",
"$\\Delta _{13} = (d\\sigma /d\\Omega )_{1/2}-(d\\sigma /d\\Omega )_{3/2}$ .",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 6%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 10*$d\\sigma /d\\Omega $ 1488-187018-1622400MAMI [107] 1488-195717-1635880MAMI [108] 1490-191132-162487GRAAL [109] 1492-173926-154180LNS [110] 1528-212046-134190CEBAF [111] 1528-212033-1481012CEBAF [112] 1533-251018-139631ELSA [113] 1533-153770-702MAMI [114] 1685-237018-162680ELSA [115] 1994-2300130-16232Spring-8 [116] 3*$\\Sigma $ 1496-190933-161150GRAAL [109] 1569-184551-14834ELSA [117] 1700-208046-134201CEBAF [118] 2*T 1492-171933-14550ELSA [83] 1497-184824-156144MAMI [119] F 1497-184824-156144MAMI [119] E 1525-212546-15469CEBAF [120] $\\Delta _{13}$ 1533-153770-70129MAMI [114] -2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma n\\rightarrow \\eta n$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 15%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 3*$d\\sigma /d\\Omega $ 1483-232226-154200ELSA [121] 1487-207051-151279ELSA [122] 1492-187518-162880MAMI [123] $\\Sigma $ 1506-189432-16599GRAAL [124] E 1505-188237-143135MAMI [125] Figure: Database for γp→η ' p\\gamma p\\rightarrow \\eta ^\\prime p. The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow \\eta ^\\prime p$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 7%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 5*$d\\sigma /d\\Omega $ 1898-195626-154120MAMI [108] 1917-233637-14334ELSA [126] 1925-238032-146524CEBAF [112] 1934-235026-154200ELSA [115] 1935-224946-134111CEBAF [127] 2*$\\Sigma $ 1903-191220-15914GRAAL [128] 1904-208046-13460CEBAF [118]"
],
[
"Kaon photoproduction",
"Whilst the cross section for kaon photoproduction is a couple of orders of magnitude smaller than pion photoproduction, these channels have been seen as the “golden” channels in recent times for a number of reasons.",
"A different coupling of kaons to light baryon resonances had been hypothesized as a means of discovering more resonances [129].",
"More importantly, especially with the $K\\Lambda $ final state, the self-analyzing property of the $\\Lambda $ through its weak decay means that information on the recoil polarization is readily obtainable in the final state.",
"Together with the advances in photon beam and target polarization, this has meant that a large number of polarization observables have been extracted across the resonance region.",
"Such data have been shown to be extremely useful in fitting model parameters and establishing the existence of resonances.",
"The plot in Figure REF indicates that very few kaon photoproduction data were available before the start of the century.",
"Initial measurements by SAPHIR [130], [131], SPring-8 [132], [133], [134] and GRAAL [135], [136] have been added to by a comprehensive campaign of measurements by CLAS [137], [138], [139], [140], [141].",
"Figure: Database for γp→K + Λ\\gamma p\\rightarrow K^+\\Lambda .",
"The notation is the same as in Figure .It should be noted that, at the time of writing, a recently published paper by the BES Collaboration [142], and a study of kaon photoproduction at CLAS [143] have cast doubt on the previously quoted value of the weak decay parameter $\\alpha _-$ of the $\\Lambda $ .",
"The value obtained by both analyses is significantly higher than the number quoted in the current PDG [30].",
"As such, this means that the polarization observables that depend on $\\alpha _-$ (beam asymmetry, beam-recoil observables) could be systematically too high, and analyses that depend on a fit to them should be examined to establish whether this change would make a difference to the final results.",
"-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow K^+\\Lambda $ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 40%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 7*$d\\sigma /d\\Omega $ 1610-239018-162701ELSA [131] 1612-189666-1431306MAMI [144] 1617-229032-148920CEBAF [137] 1617-210826-15490ELSA [130] 1625-239527-1541674CEBAF [139] 1628-253326-1431377CEBAF [145] 1934-231013-4178Spring-8 [132] 5*$\\Sigma $ 1649-190631-14466GRAAL [135] 1721-218037-134314CEBAF [141] 1946-230013-4945Spring-8 [133] 1946-228013-4930Spring-8 [132] 2041-223818-324Spring-8 [134] 6*P 1617-229026-154233CEBAF [137] 1625-254526-1431497CEBAF [139] 1649-190631-14466GRAAL [135] 1660-201741-13912ELSA [130] 1660-228034-14630ELSA [131] 1721-218037-134314CEBAF [141] 2*T 1649-190631-14466GRAAL [136] 1721-218037-134314CEBAF [141] $C_{x^{\\prime }}$ 1678-245432-139144CEBAF [138] $C_{z^{\\prime }}$ 1678-245432-139146CEBAF [138] 2*$O_{x}$ 1649-190631-14466GRAAL [136] 1721-218037-134314CEBAF [141] 2*$O_{z}$ 1649-190631-14466GRAAL [136] 1721-218037-134314CEBAF [141] Figure: Database for γp→K + Σ 0 \\gamma p\\rightarrow K^+\\Sigma ^0.",
"The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow K^+\\Sigma ^0$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 42%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 9*$d\\sigma /d\\Omega $ 1695-254526-1801576CEBAF [140] 1695-239018-162656ELSA [131] 1702-229032-139778CEBAF [137] 1703-189666-1431130MAMI [144] 1713-253326-1431279CEBAF [145] 1716-237026-154120ELSA [146] 1716-209726-154920ELSA [130] 1934-231013-4178Spring-8 [132] 1934-231018-49144Spring-8 [147] 6*$\\Sigma $ 1737-217037-124127CEBAF [141] 1755-190618-13842GRAAL [135] 1822-218537-14310ELSA [146] 1946-230013-4945Spring-8 [133] 1946-228013-4930Spring-8 [132] 1946-230013-4972Spring-8 [147] 6*P 1728-255027-163355CEBAF [140] 1737-217037-124127CEBAF [141] 1743-202941-13912ELSA [130] 1743-228041-13916ELSA [131] 1756-229026-13497CEBAF [137] 1762-185139-130 8GRAAL [135] T 1737-217037-124127CEBAF[141] $C_{x}$ 1787-245437-13471CEBAF [138] $C_{z}$ 1787-245437-13472CEBAF [138] $O_{x}$ 1737-217037-124127CEBAF [141] $O_{z}$ 1737-217037-124127CEBAF [141] Figure: Database for γp→K 0 Σ + \\gamma p\\rightarrow K^0\\Sigma ^+.",
"The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow K^0\\Sigma ^+$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 21%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 3*$d\\sigma /d\\Omega $ 1730-188529-15150MAMI [148] 1743-189820-15618ELSA [149] 2062-226346-13448ELSA [146] 3*P 1730-188529-15149MAMI [148] 1822-182230-1504ELSA [149] 2073-207330-1504ELSA [146] Figure: Database for γn→K + Σ - \\gamma n\\rightarrow K^+\\Sigma ^-.",
"The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma n\\rightarrow K^+\\Sigma ^-$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 9%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 2*$d\\sigma /d\\Omega $ 1745-253534-151285CEBAF [150] 1934-231018-49 144Spring-8 [147] $\\Sigma $ 1946-230013-49 36Spring-8 [147] -2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma n\\rightarrow K^0 \\Lambda $ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"There are no unpolarized measurements.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref $d\\sigma /d\\Omega $ 1645-251641-130360CEBAF [151] E 1700-202053-1276CEBAF [152] -2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma n\\rightarrow K^0\\Sigma ^0$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"There are no unpolarized measurements.",
"Observable W (MeV)$\\theta $ (deg) Data Lab Ref E 1700-202053-1276CEBAF [152]"
],
[
"$\\omega $ photoproduction",
"There was no $\\omega $ photoproduction data before 2003.",
"A substantial amount of data was accumulated since the.",
"All major facilities (CLAS, CBELSA, Crysta Ball at MAMI, GRAAL) made their contributions.",
"Based on these data it was found that excitation of nucleon resonance plays important role in $\\omega $ photoproduction.",
"The quality of the data near threshold gives access to a variety of interesting physics aspects.",
"As an example, an estimation of the $\\omega $ N scattering length $\\alpha _{\\omega p}$ is provided [153].",
"Figure: Database for γp→ωp\\gamma p\\rightarrow \\omega p. The notation is the same as in Figure .-2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma p\\rightarrow \\omega p$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"SDME is spin-density matrix element.",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"Polarized data contribution is 72%.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref 5*$d\\sigma /d\\Omega $ 1723-238013-159307ELSA [154] 1725-254524-1471148CEBAF [155] 1725-187221-159300MAMI [153] 1736-213118-139121ELSA [156] 1756-235011-162648ELSA [157] 4*$\\Sigma $ 1720-201719-15131ELSA [158] 1743-217415-14581CEBAF [159] 1744-209832-148492CEBAF [160] 1750-190313-16728GRAAL [161] P 1150-205053-18050CEBAF [162] T 1796-245837-180143CEBAF [159] G 1778-177837-1415ELSA [163] 2*E 1743-230029-151104CEBAF [164] 1749-225628-15195ELSA [163] F 1250-275037-180160CEBAF [162] H 1150-205053-18050CEBAF [162] 2*SDME 1725-254523-1474592CEBAF [112] 1756-235018-151891ELSA [157] -2pt C3cm L2.5cm C2cm r L2.5cm C1cm 6rContinued on next page width=.75 Data for $\\gamma n\\rightarrow \\omega n$ below W = 2.55 GeV (E$_\\gamma $ = 3 GeV).",
"Experimental data are from the SAID database [15] selected for 1996 through 2018.",
"There are no polarized measurements.",
"Observable W (MeV) $\\theta $ (deg) Data Lab Ref $d\\sigma /d\\Omega $ 1762-213618-13991ELSA [156]"
],
[
"Photoproduction of two pseudoscalar mesons",
"As photon energy increases all the single meson production cross sections decline, but the two pion cross section increases followed by $\\eta \\pi $ etc.",
"Once we get above 1.6 GeV two pion production becomes dominant.",
"The two meson final state provides a link to the final states $N\\rho $ , $N\\sigma $ , and more complex states such as $N^\\ast \\pi $ , $\\Delta \\pi $ , $\\Delta \\eta $ etc.",
"The latter final states may result from the excitation of a higher mass resonance, with a sequential decay chain to an intermediate lighter resonance and one meson, followed by the decay to the ground state nucleon and a second meson.",
"The first total cross section measurements of $\\pi ^+\\pi ^-$ photoproduction were carried out in the late 1960s with untagged photon beams of energies up to 1 GeV incident on bubble chambers [165], [166].",
"The critical requirement for double meson production experiments is large solid angle coverage, the capability of detecting multiparticle events and high energy beams of tagged photons.",
"This only became available in mid 90s.",
"The first \"new era\" electronic experiment measuring two pion photoproduction was performed with DAPHNE at MAMI [167].",
"This experiment extracted total cross sections for three double pion channels: $\\sigma _{tot}(p\\pi ^+\\pi ^-)$ , $\\sigma _{tot}(n\\pi ^+\\pi ^0)$ , and $\\sigma _{tot}(p\\pi ^0\\pi ^0)$ .",
"The measurements were done for photon energies from 400 to 800 MeV.",
"SAPHIR extended the photon energy range for $\\pi ^+\\pi ^-$ up to 2.6 GeV.",
"In this experiment they were able to extract differential cross sections and use Dalitz-plot analysis to isolate different contributions [168].",
"The first polarization measurements for this reaction were done by CLAS [169].",
"That experiment used circularly polarized photon beam and extracted the helicity asymmetry $I^c$ for photon energies from 1.35 to 2.30 GeV.",
"The latest measurements of this channel were done by CLAS [170].",
"This experiment covered the range of the center of mass energies from 1.6 to 2.0 GeV.",
"High statistics allowed for the first time the extraction of nine 1-fold differential cross section and the determination of photocouplings of some known resonances.",
"For the $\\pi ^0\\pi ^0$ channel a series of experiments were performed at MAMI-B with TAPS on a proton target [171], [172] and a deuteron target [173] from threshold to 820 MeV photon energies.",
"Then measurements were continued with the Crystal Ball/TAPS [174] combination.",
"The addition of the Crystal Ball allowed the access of $\\pi ^0\\pi ^+$ channel as well.",
"With an extended energy reach of MAMI-C, the measurement with Crystal Ball/TAPS was performed up to 1.4 GeV [175], [176].",
"GRAAL extended measurements up to 1.5 GeV photon energies and in addition to the cross section they also took advantage of the linearly polarized photon beam and extracted $\\Sigma $ beam asymmetry for this reaction [177].",
"Meanwhile the CBELSA collaboration did not stand aside and joined the effort [178], [179], [180], [181], further extending the energy reach up to 2.5 GeV.",
"They also contributed polarization measurements of $I^s$ and $I^c$ [181].",
"The natural next step after $\\pi ^0\\pi ^0$ from the experimental point of view was to study $\\pi ^0\\eta $ , which has similar topology.",
"The first measurement of this reaction channel was reported by GRAAL.",
"As usual for the GRAAL photon energy range up to 1.5 GeV, they presented total and differential cross section together with beam asymmetry $\\Sigma $ [182].",
"This was followed up by CBELSA in a series of measurements covering photon energies up to 2.5 GeV [183], [184], [185], [186].",
"This experiment produced total and differential cross sections together with polarization observables $\\Sigma $ , $I^s$ and $I^c$ .",
"Crystal Ball/TAPS at MAMI-C measured total and differential cross sections [187], which was followed by beam-target polarization measurements [188]."
],
[
" What Have We Learned from these Data so far",
"In the previous sections of this review, we have presented all the experimental photoproduction data obtained in the last two decades.",
"We conclude by summarizing how this plethora of data has expanded our knowledge of nucleon excited states.",
"Tables and compare the non-strange baryon summary tables from the PDG for the 1996 [189] and 2018 [30] editions.",
"Figures REF and REF complement the tables by showing the spectra of states graphically, where masses and widths are represented by solid lines and boxes, respectively, and the star rating is represented by the shading.",
"The first thing one notices while looking at these tables and figures is that none of the listed states has been left untouched, with one exception alone: the nucleon ground state.",
"The tables show only the \"star status\" of the resonances.",
"Quite often the knowledge of the resonance parameters improves while \"star status\" remains unchanged.",
"The latest edition of PDG lists nine new states.",
"Three states which have not received confirmation have been removed.",
"The most of the changes are in $N^\\ast $ table, and not so much in the table of $\\Delta ^\\ast $ 's.",
"Most new information on nucleon resonances over the last two decades has come from photoproduction experiments, while in the past it was mostly from $\\pi N$ scattering.",
"Figure: Comparison of the N * N^\\ast spectrum from the PDG of 1996 with 2018 editions.Nature gives us an additional powerful tool: an isospin filter.",
"Photoproduction of the final states with isospin $I = 0$ mesons ($\\eta $ , $\\eta ^\\prime $ , $\\omega $ ), or $I = 0$ baryons, $\\Lambda $ 's, cannot be directly coupled to $\\Delta $ 's.",
"As can be seen from the Table , most of the changes come exactly for these final states.",
"New columns for $N\\omega $ and $N\\eta ^\\prime $ have been added.",
"Couplings to these states were not known previously.",
"Double meson production established couplings of several resonances to the $\\sigma N$ decay channel, which again was not known previously.",
"Double meson production data also allowed the identification of sequential decays and established couplings of some of the higher mass $\\Delta ^\\ast $ -resonances to $\\Delta \\eta $ , which were not known before.",
"These advances did not occur easily.",
"It took time and effort for the information in the newly accumulated data sets to be translated into new knowledge of the baryon spectrum.",
"As we described earlier, the renaissance of photoproduction started around mid 1990's.",
"The first major overhaul of the non-strange baryon table happened in 2012 [190].",
"This represented the point at which the amount of new data needed to make an impact reached a critical mass.",
"One remarkable example is new evidence for the $\\Delta (2200){7/2}^-$ .",
"This was a poorly known “1-star\" state with only visible couplings to $N\\pi $ .",
"New high accuracy polarization data from pion photoproduction were then added to the database.",
"A coupled channel analysis revealed this resonance coupling to many channels: $\\pi ^+n$ , $\\pi ^0p$ , $K\\Sigma $ , $\\pi ^0\\pi ^0p$ , $\\pi ^0\\eta p$ [191].",
"In the latest edition of PDG, its status was upgraded to “3-star\".",
"This example also demonstrates the strength of of the coupled channel approach to the data.",
"To conclude, it would no be exaggeration to say that non-strange baryon spectroscopy is quite healthy today.",
"Several “missing\" resonances have been found.",
"New photoproduction data keep coming and there are no signs of a decline any time soon.",
"5pt l l| l| l| l l l l l l l l l l $*$$*$$*$$*$ 13lExistence is certain $*$$*$$*$ 13lExistence is very likely $*$$*$ 13lEvidence of existence is fair $*$ 13lEvidence of existence is poor 13rContinued on next page $*$$*$$*$$*$ 13lExistence is certain $*$$*$$*$ 13lExistence is very likely $*$$*$ 13lEvidence of existence is fair $*$ 13lEvidence of existence is poor Comparison of $N^\\ast $ summary tables from PDG for the years 1996 and 2018.",
"“ — ” means the cell is not present for that year.",
"2*Particle 2*$J^P$ 2*YearOverall9cStatus as seen in status $N\\gamma $ $N\\pi $ $\\Delta \\pi $ $N\\sigma $ $N\\eta $ $\\Lambda K$ $\\Sigma K$ $N\\rho $ $N\\omega $ $N\\eta ^\\prime $ 2*$N$ 2*$1/2^+$ 1996 $*$$*$$*$$*$ 2018 $*$$*$$*$$*$ 2*$N(1440)$ 2*$1/2^+$ 1996 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$ $*$$*$$*$ — $*$ $*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ 2*$N(1520)$ 2*$3/2^-$ 1996 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ — $*$ $*$$*$$*$$*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$ $*$$*$$*$$*$ 2*$N(1535)$ 2*$1/2^-$ 1996 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$ — $*$$*$$*$$*$ $*$$*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ $*$ $*$$*$$*$$*$ 2*$N(1650)$ 2*$1/2^-$ 1996 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ — $*$ $*$$*$$*$ $*$$*$ $*$$*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ $*$ $*$$*$$*$$*$ $*$ 2*$N(1675)$ 2*$5/2^-$ 1996 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ — $*$ $*$ $*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$ $*$ 2*$N(1680)$ 2*$5/2^+$ 1996 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ — $*$$*$$*$$*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$ $*$ 2*$N(1700)$ 2*$3/2^-$ 1996 $*$$*$$*$ $*$$*$ $*$$*$$*$ $*$$*$ — $*$ $*$$*$ $*$ $*$ — — 2018 $*$$*$$*$ $*$$*$ $*$$*$$*$ $*$$*$$*$ $*$ $*$ $*$ 2*$N(1710)$ 2*$1/2^+$ 1996 $*$$*$$*$ $*$$*$$*$ $*$$*$$*$ $*$$*$ — $*$$*$ $*$$*$ $*$ $*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$ $*$$*$$*$ $*$$*$ $*$ $*$ $*$ 2*$N(1720)$ 2*$3/2^+$ 1996 $*$$*$$*$$*$ $*$$*$ $*$$*$$*$$*$ $*$ — $*$ $*$$*$ $*$ $*$$*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ $*$ $*$ $*$$*$$*$$*$ $*$ $*$ $*$ 2*$N(1860)$ 2*$5/2^+$ 1996 — — — — — — — — — — — 2018 $*$$*$ $*$ $*$$*$ $*$ $*$ 2*$N(1875)$ 2*$3/2^-$ 1996 — — — — — — — — — — — 2018 $*$$*$$*$ $*$$*$ $*$$*$ $*$ $*$$*$ $*$ $*$ $*$ $*$ $*$ 2*$N(1880)$ 2*$1/2^+$ 1996 — — — — — — — — — — — 2018 $*$$*$$*$ $*$$*$ $*$ $*$$*$ $*$$*$ $*$ $*$$*$ $*$$*$ $*$$*$ 2*$N(1895)$ 2*$1/2^-$ 1996 — — — — — — — — — — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$ $*$ $*$ $*$$*$$*$$*$ $*$$*$ $*$$*$ $*$ $*$$*$ $*$$*$$*$$*$ 2*$N(1900)$ 2*$3/2^+$ 1996 $*$$*$ $*$$*$ — $*$$*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$ $*$$*$ $*$ $*$ $*$$*$ $*$$*$ $*$ $*$$*$ 2*$N(1900)$ 2*$7/2^+$ 1996 $*$$*$ $*$ $*$$*$ — $*$ $*$ $*$ — — 2018 $*$$*$ $*$$*$ $*$$*$ $*$ $*$ $*$ 2*$N(2000)$ 2*$5/2^+$ 1996 $*$$*$ $*$$*$ $*$ — $*$ $*$ $*$ $*$$*$ — — 2018 $*$$*$ $*$$*$ $*$ $*$$*$ $*$ $*$ $*$ 2*$N(2040)$ 2*$3/2^+$ 1996 — — — — — — — — — — — 2018 $*$ $*$ 2*$N(2060)$ 2*$5/2^+$ 1996 — — — — — — — — — — — 2018 $*$$*$$*$ $*$$*$$*$ $*$$*$ $*$ $*$ $*$ $*$$*$ $*$ $*$ 2*$N(2080)$ 2*$3/2^-$ 1996 $*$$*$ $*$ $*$$*$ — $*$ $*$ — — 2018 — — — — — — — — — — — 2*$N(2090)$ 2*$1/2^-$ 1996 $*$ $*$ — — — 2018 — — — — — — — — — — — 2*$N(2100)$ 2*$1/2^+$ 1996 $*$$*$ $*$$*$ $*$ — $*$ $*$ $*$ $*$$*$ — — 2018 $*$$*$ $*$$*$ $*$ $*$$*$ $*$ $*$ $*$ 2*$N(2120)$ 2*$3/2^-$ 1996 — — — — — — — — — — — 2018 $*$$*$$*$ $*$$*$$*$ $*$$*$ $*$$*$ $*$$*$ $*$$*$$*$ $*$ $*$ 2*$N(2190)$ 2*$7/2^-$ 1996 $*$$*$$*$$*$ $*$ $*$$*$$*$$*$ — $*$ $*$ $*$ $*$$*$ — — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$ $*$ $*$$*$ $*$ $*$ $*$ 2*$N(2200)$ 2*$5/2^-$ 1996 $*$$*$ $*$$*$ — $*$ $*$ — — 2018 — — — — — — — — — — — 2*$N(2220)$ 2*$9/2^+$ 1996 $*$$*$$*$$*$ $*$$*$$*$$*$ — $*$ — — 2018 $*$$*$$*$$*$ $*$$*$ $*$$*$$*$$*$ $*$ $*$ $*$ 2*$N(2250)$ 2*$9/2^-$ 1996 $*$$*$$*$$*$ $*$$*$$*$$*$ — $*$ — — 2018 $*$$*$$*$$*$ $*$$*$ $*$$*$$*$$*$ $*$ $*$ $*$ 2*$N(2300)$ 2*$1/2^+$ 1996 — — — — — — — — — — — 2018 $*$$*$ $*$$*$ 2*$N(2570)$ 2*$5/2^-$ 1996 — — — — — — — — — — — 2018 $*$$*$ $*$$*$ 2*$N(2600)$ 2*$11/2^-$ 1996 $*$$*$$*$ $*$$*$$*$ — — — 2018 $*$$*$$*$ $*$$*$$*$ — — — 2*$N(2700)$ 2*$13/2^+$ 1996 $*$$*$ $*$$*$ — — — 2018 $*$$*$ $*$$*$ — — — Figure: Comparison of the Δ * \\Delta ^\\ast spectrum from the PDG of 1996with 2018 editions.5pt l l| l| l| l l l l l l l l l l $*$$*$$*$$*$ 10lExistence is certain $*$$*$$*$ 10lExistence is very likely $*$$*$ 10lEvidence of existence is fair $*$ 10lEvidence of existence is poor 10rContinued on next page $*$$*$$*$$*$ 10lExistence is certain $*$$*$$*$ 10lExistence is very likely $*$$*$ 10lEvidence of existence is fair $*$ 10lEvidence of existence is poor Comparison of $\\Delta ^\\ast $ summary tables from PDG for the years 1996 and 2018.",
"“ — ” means the cell is not present for that year.",
"2*Particle 2*$J^P$ 2*YearOverall6cStatus as seen in status $N\\gamma $ $N\\pi $ $\\Delta \\pi $ $\\Sigma K$ $N\\rho $ $\\Delta \\eta $ 2*$\\Delta (1232)$ 2*$3/2^+$ 1996 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ 2*$\\Delta (1600)$ 2*$3/2^+$ 1996 $*$$*$$*$ $*$$*$ $*$$*$$*$ $*$$*$$*$ $*$ — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ 2*$\\Delta (1620)$ 2*$1/2^-$ 1996 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ 2*$\\Delta (1700)$ 2*$3/2^-$ 1996 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$ $*$ $*$$*$ — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$ $*$ 2*$\\Delta (1750)$ 2*$1/2^+$ 1996 $*$ $*$ — 2018 $*$ $*$ $*$ $*$ 2*$\\Delta (1900)$ 2*$1/2^-$ 1996 $*$$*$$*$ $*$ $*$$*$$*$ $*$ $*$ $*$$*$ — 2018 $*$$*$$*$ $*$$*$$*$ $*$$*$$*$ $*$ $*$$*$ $*$ 2*$\\Delta (1905)$ 2*$5/2^+$ 1996 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$$*$ $*$ $*$$*$ — 2018 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$ $*$ $*$ $*$$*$ 2*$\\Delta (1910)$ 2*$1/2^+$ 1996 $*$$*$$*$$*$ $*$ $*$$*$$*$$*$ $*$ $*$ $*$ — 2018 $*$$*$$*$$*$ $*$$*$$*$ $*$$*$$*$$*$ $*$$*$ $*$$*$ $*$ 2*$\\Delta (1920)$ 2*$3/2^+$ 1996 $*$$*$$*$ $*$ $*$$*$$*$ $*$$*$ $*$ — 2018 $*$$*$$*$ $*$$*$$*$ $*$$*$$*$ $*$$*$$*$ $*$$*$ $*$$*$ 2*$\\Delta (1930)$ 2*$5/2^-$ 1996 $*$$*$$*$ $*$$*$ $*$$*$$*$ $*$ — 2018 $*$$*$$*$ $*$$*$$*$ $*$$*$$*$ $*$ $*$ 2*$\\Delta (1940)$ 2*$3/2^-$ 1996 $*$ $*$ — 2018 $*$$*$ $*$ $*$$*$ $*$ $*$ 2*$\\Delta (1950)$ 2*$7/2^+$ 1996 $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$ $*$ — 2018 $*$$*$ $*$$*$$*$$*$ $*$$*$$*$$*$ $*$$*$ $*$$*$$*$ 2*$\\Delta (2000)$ 2*$5/2^+$ 1996 $*$$*$ $*$$*$ — 2018 $*$$*$ $*$ $*$$*$ $*$ $*$ 2*$\\Delta (2150)$ 2*$1/2^-$ 1996 $*$ $*$ — 2018 $*$ $*$ 2*$\\Delta (2200)$ 2*$7/2^-$ 1996 $*$ $*$ — 2018 $*$$*$$*$ $*$$*$$*$ $*$$*$ $*$$*$$*$ $*$$*$ 2*$\\Delta (2300)$ 2*$9/2^+$ 1996 $*$$*$ $*$$*$ — 2018 $*$$*$ $*$$*$ 2*$\\Delta (2350)$ 2*$5/2^-$ 1996 $*$ $*$ — 2018 $*$ $*$ 2*$\\Delta (2390)$ 2*$7/2^+$ 1996 $*$ $*$ — 2018 $*$ $*$ 2*$\\Delta (2400)$ 2*$9/2^-$ 1996 $*$$*$ $*$$*$ — 2018 $*$$*$ $*$$*$ $*$$*$ 2*$\\Delta (2420)$ 2*$11/2^+$ 1996 $*$$*$$*$$*$ $*$$*$$*$$*$ — 2018 $*$$*$$*$$*$ $*$ $*$$*$$*$$*$ 2*$\\Delta (2750)$ 2*$13/2^-$ 1996 $*$$*$ $*$$*$ — 2018 $*$$*$ $*$$*$ 2*$\\Delta (2950)$ 2*$15/2^+$ 1996 $*$$*$ $*$$*$ — 2018 $*$$*$ $*$$*$"
],
[
"Acknowledgements",
"The work of D.I.",
"was supported by the United Kingdom's Science and Technology Facilities Council (STFC) from grant number ST/P004458/1.",
"The work of I.S.",
"was supported in part by the US Department of Energy Grant DE–SC0016583.",
"This material in part is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract No.",
"DE–AC05–06OR23177.",
"Notice: Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No.",
"DE–AC05–06OR23177.",
"The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S. Government purposes."
]
] | 1906.04228 |
[
[
"Design and integration of a parallel, soft robotic end-effector for\n extracorporeal ultrasound"
],
[
"Abstract Objective: In this work we address limitations in state-of-the-art ultrasound robots by designing and integrating a novel soft robotic system for ultrasound imaging.",
"It employs the inherent qualities of soft fluidic actuators to establish safe, adaptable interaction between ultrasound probe and patient.",
"Methods: We acquire clinical data to determine the movement ranges and force levels required in prenatal foetal ultrasound imaging and design the soft robotic end-effector accordingly.",
"We verify its mechanical characteristics, derive and validate a kinetostatic model and demonstrate controllability and imaging capabilities on an ultrasound phantom.",
"Results: The soft robot exhibits the desired stiffness characteristics and is able to reach 100% of the required workspace when no external force is present, and 95% of the workspace when considering its compliance.",
"The model can accurately predict the end-effector pose with a mean error of 1.18+/-0.29mm in position and 0.92+/-0.47deg in orientation.",
"The derived controller is, with an average position error of 0.39mm, able to track a target pose efficiently without and with externally applied loads.",
"Ultrasound images acquired with the system are of equally good quality compared to a manual sonographer scan.",
"Conclusion: The system is able to withstand loads commonly applied during foetal ultrasound scans and remains controllable with a motion range similar to manual scanning.",
"Significance: The proposed soft robot presents a safe, cost-effective solution to offloading sonographers in day-to-day scanning routines.",
"The design and modelling paradigms are greatly generalizable and particularly suitable for designing soft robots for physical interaction tasks."
],
[
"Introduction",
"It is commonly accepted that sonographers are exposed to an increased risk in repetitive strain injury [1], [2], [3].",
"A representative study amongst diagnostic medical sonographers and vascular technologists indicates that a significant majority of sonographers experience pain while performing ultrasound scans [4].",
"This suggests a high demand to improve ergonomics and offload sonographers during clinical scan procedures.",
"Recent investigations show that besides diagnostic sonography, there is an increased demand for intraoperative transthoracic [5], [6] and transoesophegal [7] ultrasound imaging, particularly for cardiac and lung procedures.",
"Sonographers performing intraoperative ultrasound in for example cardiac catheterization procedures have therefore presumably an increased risk of radiation exposure [8].",
"Figure: Soft robotic end-effector (SEE) performing ultrasound scan on abdominal prenatal phantomAutomating diagnostic and intraoperative ultrasound procedures through robot-guidance or -assistance can help address the aforementioned problems and lay the groundwork for more intelligent image acquisition.",
"Robotic ultrasound guidance has found particular application in procedures involving steering orthopaedic [9] or minimally-invasive surgical tools [10] and biopsy needles [11].",
"Various robotic hardware solutions have been proposed.",
"Researchers have adopted robotic platforms originally aimed at collaborative scenarios in industrial settings, such as Universal Robot’s UR-series [12], [13] or the KUKA LWR [9] and LBR iiwa [14], [15].",
"A commercial robotic manipulator has been released (LBR Med, KUKA AG, Augsburg, Germany) which is suitable for use in clinical environments due to its conformity with medical device safety (ISO 60601) and medical software regulations (ISO 62304).",
"Current research suggests that such robots can be applied in diagnostics to autonomously perform aorta measurements [16], in combination with previously acquired MRI scans to autonomously find standard view-planes [17] and in intraoperative procedures to autonomously track surgical tools [18], amongst others.",
"Whilst such robotic platforms allow for great flexibility through a large workspace and high manipulability, the use of large-scale robotic manipulators can pose various disadvantages for clinical integration.",
"Diagnostic ultrasound scans are divided into their respective body area of interest.",
"For an individual procedure such as a lower abdominal ultrasound scan, a robotic system is therefore only required to achieve a workspace to cover a fraction of the human body.",
"This yields that common robotic manipulators could be oversized for such applications, which unnecessarily poses risks to patient safety.",
"Despite high degrees of electrical safety, a mechanical system with a high mass can potentially be more dangerous [19].",
"To address this issue, researchers developed customized solutions which are tailored to the application-specific requirements of diagnostic and interventional sonography.",
"Researchers [20], [21], [22] have proposed a mechanism which achieves a high degree of probe manipulability and safety.",
"The robot actuation has been moved to the base of the system, thus minimizing its size and weight.",
"Other systems have been developed which separate the probe positioning into two stages: approximate probe placement and finer view-plane adjustments.",
"The first can be achieved by a passive positioning mechanism, which is operated by a clinician, while the latter is obtained with an active end-effector.",
"A system based on cables which are driven by McKibben actuators has been proposed [23].",
"The antagonistic configuration of the cables is employed to position the ultrasound probe on a patient.",
"The system is tele-operated by a sonographer.",
"Researchers from Waseda University first proposed this concept and corresponding design in [24], in which the end-effector is driven through a parallel mechanism.",
"Similarly, a consortium of researchers have developed a system with active end-effector with the aim of remote tele-diagnosis [25], [26], [27].",
"The system has since been trialled for remote scans [28] and translated to a commercial product (MELODY, AdEchoTech, Naveil, France).",
"Despite the scanning being performed remotely, the design of the system suggests, however, that the assisting operator is still required to apply the necessary force to maintain a stable contact.",
"Maintaining stable mechanical coupling between ultrasound probe and patient tissue is of paramount importance for ensuring a high-quality image.",
"Approaches to achieve this involve controlling the contact force directly or establishing an elastic contact between the position-controlled device and the patient.",
"While the first has been researched extensively [29], [30] and can be commonly found in various forms of industrial applications, the latter has found more attention in recent years due to an increased demand in cost effective force control and -limiting solutions for human robot collaboration tasks [31], [32].",
"Series-elastic actuators have been developed to provide passive compliance in actuated robotic joints [33].",
"While providing a degree of compliance, this has the disadvantage that a collision or undesired contact in a direction other than the joint axis cannot be compensated for.",
"We have trialled safety clutches for the use in ultrasound robots which exhibit compliant behaviour once disengaged through an excess force [34], [35], [36].",
"This, however, renders the system uncontrollable and requires reengaging the clutch mechanism for further operation.",
"In this work, we make use of an elastic soft robotic system, which is aimed at overcoming aforementioned limitations.",
"Soft robotics technologies have opened up new design paradigms for robotic systems through the use of elastic and deformable materials and structures [37], [38].",
"Soft robotics systems are commonly designed to interact with or conform to environmental contacts.",
"This allows soft robotic manipulators to exhibit highly dexterous manoeuvrability in for example surgical [39], [40], [41] or search and rescue operations [42].",
"In these scenarios, however, soft robots are not applied to tasks which require significant loadbearing capabilities, predominantly due to their low stiffness.",
"To bridge the trade-off between manoeuvrability and stiffness, research has been driven towards systems with variable stiffness capabilities.",
"A comprehensive overview of stiffening technologies is given in [43].",
"For applications in which softness is desired, high loadings are demanded and stiffening mechanisms are not suitable, soft robotic systems tend to be combined with external constraints to ensure structural integrity.",
"This is commonly found in exoskeleton research and rehabilitation robotics.",
"Examples include full body, soft exosuits [44], lower limb exoskeletons [45] and hand exoskeletons for post-stroke rehabilitation [46], [47].",
"In our previous work, we identified the advantages of soft robotics technology in ultrasound interaction tasks compared to rigid state-of-the-art robots and showed an initial proof-of-concept of a parallel soft robotic end-effector with the right characteristics for medical ultrasound tasks [48].",
"We now derive a novel soft robotic end-effector which is capable of safely acquiring standard views in extracorporeal diagnostic foetal ultrasound (US).",
"We select foetal US as an initial application due to its high demands to robot safety.",
"We evaluate the performance of our system with respect to derived specifications and show that the proposed system is capable of acquiring a set of standard view-plane required for the assessment of the foetus.",
"The robot utilizes linear soft fluidic actuators (SFAs) which are arranged in parallel around the ultrasound probe to provide high axial loadbearing capabilities and high lateral compliance, thus enabling adaptability and safety in the patient interaction.",
"The individual contributions of this study are: Figure: Proposed design of the soft robotic end-effector (a) and workflow (b) for obtaining a desired view through manual placement in the approximate region of interest (i) and active steering of the probe towards the desired view-plane (ii).",
"Clinical investigation to determine workspace and force requirements for view-plane adjustments in foetal diagnostic ultrasound imaging.",
"Design and verification of a soft robotic end-effector which satisfies the derived clinical requirements in workspace and force.",
"It employs robust linear soft fluidic actuators, for which a novel injection-based fabrication is derived, and undesired twist is prevented through a mesh constraint.",
"Definition and validation of a lumped stiffness model to describe the motion of the soft robotic end-effector in the absence and presence of external loading.",
"The controllability and imaging capabilities of the integrated system are validated in position control and US phantom experiments respectively.",
"The paper is structured in the following way.",
"In Section REF the system requirements are determined, and the robot design is introduced.",
"Based on the design of the system, Section REF derives a kinetostatic model.",
"Methodologies for the actuation and control of the system are presented in Section REF .",
"In Section the mechanical properties of the system and its workspace are evaluated.",
"Results are presented in section .",
"The proposed model is validated and the position controller performance, as well as the imaging capabilities of the system, are assessed."
],
[
"Methods",
"Prenatal foetal ultrasound is a routine diagnostic procedure for pregnant women to determine birth defects and abnormalities in the foetus.",
"Common checks include measuring the foetus’ biparietal diameter (BPD), its head and abdominal circumferences (HC and AC) as well as its femur length (FL) [49].",
"In this work we focus on obtaining HC, AC and FL standard view-planes.",
"We establish the clinical requirements to the contact force and movement range of the ultrasound probe for bespoke application and derive a suitable design for a soft robotic end-effector (SEE).",
"Figure: Braided nylon mesh uncrimped (a) and crimped (b)."
],
[
"Clinical data acquisition and processing",
"Pregnant women between 18 to 24 weeks of gestation underwent research ultrasound scans at St Thomas’ Hospital (Study title: Intelligent Fetal Imaging and Diagnosis (iFIND)-2: Further Ultrasound and MR Imaging, Study reference: 14/LO/1806).",
"Trained sonographers performed the foetal ultrasound scan using a standard ultrasound probe (X6-1, Philips, Amsterdam, Netherlands) which is connected to an ultrasound scanner (EPIQ7, Philips, Amsterdam, Netherlands).",
"The probe was placed in a holder as detailed in [50].",
"This holder incorporated an electromagnetic (EM) tracking sensor (Aurora, NDI, Ontario, Canada) and six axis force-torque sensor (Nano 17, ATI, Apex, USA), which allowed measurements of the position and orientation of the probe, and the force applied at the probe face to be measured throughout the scan.",
"The recorded tracking and force data of six patients were analysed by extracting time ranges during which standard fetal anomaly views were imaged.",
"These included HC, AC and FL views.",
"Each time range consisted of the few seconds when the sonographer had placed the probe in the correct anatomical region and was adjusting the probe to find the ideal view.",
"For each view the tracking data were analysed to find the range of positions and orientations in the three axes separately.",
"The X and Y axes show movement in the horizontal plane of the scanning bed (left to right on the patient, and foot to head, respectively), and the Z axis shows vertical movement.",
"Orientation ranges are given in probe coordinates, with yaw showing axial rotation, pitch showing elevational tilting out of the image plane, and roll showing in-plane lateral tilting.",
"Forces were analysed by dividing the measured force vector into normal and tangential components applied to the surface.",
"The local surface angle was determined at each measurement by fitting an ellipsoidal shape to the tracking data of the scan.",
"The 95th percentile of the forces measured within a time range gives an indication of the maximum force that must be applied by the probe."
],
[
"Mechanism requirements and synthesis",
"Following the results of the clinical data analysis, it is found that the soft robotic end-effector must at least satisfy the following requirements Be able to withstand mean axial and transversal contact forces of 8.01N and 4.42N without significant deterioration of the imaging view.",
"Achieve an axial extension along Z of 5.22mm and transversal translations in X and Y of 7.75mm.",
"Achieve rotations of 5.08$$ around X and Y.",
"Figure: Free body diagram of SEE model definitionTo maintain a high degree of safety when interacting with the device, the SEE should furthermore comprise of a low transversal stiffness.",
"This allows both the operating clinician and patient to manually displace the probe in case of discomfort.",
"As the investigated system is compliant, its deflection has to be considered when determining if a position is achievable.",
"Taking into account normal and tangential forces applied during the scanning, the system must satisfy the following conditions $\\delta _{SEE} \\ge \\delta _{req} +\\delta _{f} \\quad \\text{with} \\quad \\delta _{f} = K^{-1}_{min}f_{req}$ Where $\\delta _{f}$ is a deformation induced by external forces, $f_{req}$ is a vector of the required forces and $K_{min}$ is the minimum system stiffness throughout the workspace.",
"$\\delta _{req}$ and $\\delta _{SEE}$ are vectors of the required and achievable translations respectively.",
"As only tip forces are considered in this work, tilting effects induced by external moments at the SEE tip are ignored and forces are assumed to only affect the tip position.",
"A soft robotic design based on soft fluidic actuators (SFAs), which have previously been presented in [48], is proposed.",
"It is comprised of two rigid platforms which serve as base and transducer holder respectively.",
"The platforms are connected through a set of three soft fluidic actuators which are arranged in a parallel fashion at 120 intervals.",
"To allow for sufficient space for the ultrasound transducer cable, the actuators are tilted at an angle of 15$$ .",
"An overview of the design is shown in Fig.",
"REF a).",
"Whilst a rigid mechanism of such configuration would be over-constrained and thus unable to move, the elasticity of the SFAs allows the SEE to perform bending (coupled translation and rotation) and axial extension motions.",
"As the SFAs are tilted, axial extension causes the SFAs to bend into an S-shaped configuration.",
"This allows for the SEE to be axially compliant whilst exhibiting a high degree of load-bearing capabilities, which is further investigated in Section REF .",
"Furthermore, curving into an S-shaped configuration eliminates the possibility of unstable buckling to occur in the SFAs, as shown in Section REF .",
"A common problem in such a proposed soft robotic system is the low stiffness along its twist axis.",
"To improve the stability of the system against twist deformations, a nylon fibre mesh is attached to base and transducer platforms, which acts as a mechanical constraint between the two.",
"To reduce unwanted buckling behaviour, crimps can be added to the mesh by deforming and heat-treating it.",
"Examples of uncrimped and crimped meshes are shown in Fig.",
"REF .",
"Thus, axial rotation of the ultrasound transducer is not considered in this study, as it could be added by simply applying a rotating mechanism to the base of the SEE, which would function as a stiff rotational axis in conjunction with the mesh constraint.",
"The workflow of imaging using the SEE is shown in Fig.",
"REF b).",
"Once the SEE is manually placed in the approximate area of the target view using a passive positioning arm, it is fixed on the patient.",
"The ultrasound probe is then actively steered either in a tele-operated manner by a sonographer or in an autonomous fashion using pose or image feedback.",
"As the loadbearing is achieved by the SEE, contact forces the sonographer is required to apply are minimized, which presumably has an impact on the ergonomics of the sonographer.",
"To determine the ultrasound probe pose under internal fluid volume variation and external loading a kinetostatic model is derived according to [51].",
"A free body diagram of the model is shown in Fig.",
"REF .",
"In the following, a vector denoted as $w_f$ represents a 6 degree of freedom wrench in an arbitrary frame $f$ such that $w_f=[F_x^f,F_y^f,F_z^f,M_x^f,M_y^f,M_z^f ]^T$ with forces $F$ and moments $M$ .",
"Similarly, $\\tau _f$ denotes a reaction wrench in the local SFA frame, which is of the same form as $w_f$ .",
"Vectors noted as $\\delta x_f$ indicate infinitesimally small displacements in frame f of the form $\\delta x_f=[u_x^f,u_y^f,u_z^f,v_x^f,v_y^f,v_z^f ]^T$ with translations $u$ and rotations $v$ .",
"Let $w_{ext}$ be a vector of forces and moments applied to the tip of the ultrasound transducer.",
"Under static equilibrium conditions, the following holds for a single actuator $w_{ext}=w_\\theta + w_V$ Where $w_\\theta $ is the wrench caused by the elastic deformation of the SFA and $w_V$ is the reaction wrench caused by the constrained hydraulic chamber.",
"Both are expressed in the tip frame of the system.",
"The tip wrenches $w_\\theta $ and $w_V$ can be expressed relative to their local frames by $\\begin{split}w_\\theta & =J_\\theta (x) \\tau _\\theta \\\\ w_V & =J_V(x)\\tau _V\\end{split}$ Where $\\tau _\\theta $ is a vector of local reaction forces and moments caused by the SFA deformation and $\\tau _V$ is the uniaxial reaction force of the volumetric constraint in the actuator.",
"The matrices $J_\\theta (x)$ and $J_V(x)$ are defined by $\\begin{aligned}J_\\theta (x) &=\\begin{bmatrix}R(x) & 0\\\\0 & R(x)\\end{bmatrix}Ad \\\\J_V(x) &=\\begin{bmatrix}R(x) & 0\\\\0 & R(x)\\end{bmatrix}Ad_z=\\begin{bmatrix}R(x) & 0\\\\0 & R(x)\\end{bmatrix}\\hat{H}\\end{aligned}$ $R(x)$ is the rotation matrix of the current tip deflection.",
"Matrix $Ad$ is the wrench transformation matrix relating the local SFA frame to the tip frame by $Ad =\\begin{bmatrix}R_0 & 0 \\\\D_0R_0 & R_0\\end{bmatrix}$ Where $R_0$ is the spatial rotation of the respective frame and $D_0$ is the cross-product matrix with the translation vector $d_0 = [d_x, d_y, d_z]$ .",
"$\\hat{H}$ is for a single SFA a 6x1 vector containing the third column of $Ad$ .",
"Considering the elastic behaviour of the SFA, its reaction force $\\tau _\\theta $ caused by an infinitesimally small, local displacement $\\delta x_\\theta $ can be written as $\\tau _\\theta =K_\\theta \\delta x_\\theta $ Where the SFA stiffness $K_\\theta $ is defined as a Timoshenko beam element with $K_\\theta =\\begin{bmatrix}\\frac{12EI}{(1+\\Phi )L^3} & 0 & 0 & 0 & \\frac{6EI}{(1+\\Phi )L^2} & 0\\\\0 & \\frac{12EI}{(1+\\Phi )L^3} & 0 & \\frac{-6EI}{(1+\\Phi )L^2} & 0 & 0 \\\\0 & 0 & \\frac{EA}{L} & 0 & 0 & 0\\\\0 & \\frac{-6EI}{(1+\\Phi )L^2} & 0 & \\frac{(4+\\Phi )EI}{(1+\\Phi )L} & 0 & 0\\\\\\frac{6EI}{(1+\\Phi )L^2} & 0 & 0 & 0 & \\frac{(4+\\Phi )EI}{(1+\\Phi )L} & 0\\\\0 & 0 & 0 & 0 & 0 & \\frac{GJ}{L}\\end{bmatrix}$ $L$ describes the length of the SFA, $A$ it’s cross-sectional area, $E$ its Young’s modulus, $I$ the area moment of inertia, $G$ its shear modulus and $J$ the torsion constant.",
"The Timoshenko coefficient $\\Phi $ is defined as $\\Phi =\\frac{12EI}{\\frac{A}{\\alpha }GL^3}$ with the Timoshenko coefficient $\\alpha $ .",
"An overview of the SFA constants is given in Table REF .",
"Table: SFA model parametersWhilst parameters $L$ and $A$ are obtained from the SFA geometry, the torsion constant of a beam with circular cross-section can be expressed as $J = 0.5\\pi r^4$ and its Timoshenko coefficient is defined as $5/6$ [52].",
"The shear modulus $G$ is approximated as half the Young's Modulus.",
"For a given SFA volume, the kinematic relationship between an infinitesimal small volume change $\\delta V$ of the SFA and the displacement of the ultrasound tip frame is given by $\\delta V/a=J_V^T \\delta x_{tip}$ Where $a$ is the cross-sectional area of the fluid actuation channel.",
"The kinematic motion of the tip frame caused by the SFA deflection can be defined as $\\delta x_{\\theta }=J_\\theta ^T \\delta x_{tip}$ Substituting Equation REF into REF yields $\\tau _\\theta =K_\\theta J_\\theta ^T \\delta x_{tip}$ Applying Equations REF and REF , the static equilibrium condition in Equation REF can be written as $w_{ext}=J_\\theta K_\\theta J^T_\\theta \\delta x_{tip} + J_V\\tau _V$ Equation REF can be combined with the imposed kinematic constraint defined by Equation REF to a linear equation system of the form $\\begin{bmatrix}w_{ext}\\\\\\delta V/a\\end{bmatrix} =\\begin{bmatrix}J_\\theta K_\\theta J_\\theta ^T & J_V\\\\J_V^T & 0\\end{bmatrix}\\begin{bmatrix}\\delta x_{tip}\\\\\\tau _V\\end{bmatrix}$ The deflection of the ultrasound transducer tip and internal reaction of the system can consequently be found through matrix inversion $\\begin{bmatrix}\\delta x_{tip}\\\\\\tau _V\\end{bmatrix} =\\begin{bmatrix}J_\\theta K_\\theta J_\\theta ^T & J_V\\\\J_V^T & 0\\end{bmatrix}^{-1}\\begin{bmatrix}w_{ext}\\\\\\delta V/a\\end{bmatrix}$ The formulation can be expanded to a number of $n$ SFAs by considering a lumped stiffness $K$ in the probe tip frame.",
"As the actuators are aligned in a parallel configuration, it can be defined by $K = \\sum _{i=1}^{n}J_\\theta ^i K_\\theta ^i {J_\\theta ^i}^T$ The matrix $J_V$ is adopted by appending the respective columns of the wrench transformation matrix of actuator $i$ $Ad_z^i$ to $\\hat{H}$ such that $^nJ_V = \\begin{bmatrix}R(x) & 0\\\\0 & R(x)\\end{bmatrix}[Ad_z^1, Ad_z^2, ..., Ad_z^n]$ The kinematic constraint relationship then becomes $\\delta V/a = ^n\\!J_V^T x_{tip}$ Where $\\delta V$ is an $n \\times 1$ vector of SFA volume changes.",
"Consequently, $\\tau _V$ is expanded to an $n \\times 1$ vector containing $n$ local reactions in the form $\\tau _V=[\\tau _{V,1},\\tau _{V,2}...,\\tau _{V,n}]^T$ .",
"To account for changes in matrices $J_\\theta $ and $J_V$ for a given motion, the model is solved numerically by dividing the applied external wrench and induced volume vectors into small increments $[\\Delta w_{ext}, \\Delta V]^T$ .",
"After each iteration, $R(x)$ is updated according to the previous tip pose.",
"Figure: Actuation unit (a) with syringe pumps (b) and controller systemFor the given number of three SFAs, the update rule for the numerical solution is defined by $\\begin{bmatrix}x^{k}_{tip}\\\\\\tau _V^{k}\\end{bmatrix} =\\begin{bmatrix}x^{k+1}_{tip}\\\\\\tau _V^{k+1}\\end{bmatrix} +\\begin{bmatrix}K & ^3J^k_V\\\\^3{J^k_V}^T & 0\\end{bmatrix}^{-1}\\begin{bmatrix}\\Delta w_{ext}\\\\\\Delta V/a\\end{bmatrix}$ For iteration step $k$ ."
],
[
"Actuation and control",
"The SEE is actuated by inflating respective SFAs with a working fluid.",
"As shown in our previous work [53], we utilize custom hydraulic syringe pumps (Fig.",
"REF b)) which are driven by stepper motors (Nema 17, Pololu Corporation, Las Vegas, USA) to induce volume changes in the SFAs.",
"The pumps are controlled with a microcontroller (Teensy 3.5, PJRC, Sherwood, USA) which communicates via a serial interface with a PC running ROS (Intel Core I7-7700HQ, XPS15 9560, Dell, Texas, USA).",
"The PC generates demand velocities or positions for the microcontroller and solves the previously-defined kinetostatic model to determine the system Jacobian for a given pose.",
"Furthermore, the laptop handles interfaces with peripherals such as a joystick for teleoperation (Spacemouse Compact, 3dconnexion, Monaco) and an electromagnetic tracking (EM) system for closed-loop position control (Aurora, NDI, Ontario, Canada).",
"The linear soft fluidic actuators which are utilized to drive the system have first been conceptualized in our previous work [48].",
"They are comprised of a silicone rubber body (Dragonskin 10-NV, SmoothOn Inc, Pennsylvania, USA) and stiffer silicone rubber endcaps (SmoothSil 945, SmoothOn Inc, Pennsylvania, USA).",
"A helical constraint is inserted into the silicone to counteract radial expansion of the actuator upon inflation.",
"This, in combination with the stiff endcaps, allows for the actuators to maintain its form and only expand in the direction of actuation.",
"The moulding process of creating SFAs has been significantly improved from our previous work.",
"For the radial constraint an extension spring (Fig.",
"REF (v)) is used.",
"The liquid silicone rubber is injected through an inlet (Fig.",
"REF (ii)) using a syringe instead of being poured into the mould.",
"This has the significant advantage for the user to be able to pre-assemble the mould without having to manually wind the constraint helix, as it has been commonly done in soft fluidic actuators [54].",
"In combination with the injection of the silicone this could reduce variations in the fabrication process.",
"A drawing of a finished actuator is shown in Fig.",
"REF (vii).",
"The combination of radial constraint and stiff endcaps allows for the actuators to be driven efficiently with a volumetric input without exhibiting a nonlinear relationship between input volume and output length change due to bulging, which is investigated in Section REF .",
"Figure: Overview of mould components (i)-(vi) and drawing of final SFA (vii)In this work, two methods for controlling the ultrasound probe pose are investigated.",
"A joystick-based teleoperated open-loop controller is implemented to allow a sonographer to steer the probe according to the acquired ultrasound image stream.",
"For this purpose, the aforementioned joystick is used.",
"The axial motion of the joystick is linked to a translation of the SEE in Z-direction while the two tilt axes of the joystick are mapped to the X- and Y-rotation axes of the SEE.",
"The high-level controller generates syringe pump velocities according to $\\dot{V}_d = J_V^T v_{cart}$ Where $\\dot{V}_d$ is the desired SFA velocity, $v_{cart}$ the target velocity in Cartesian space and $J_V^T$ the actuation matrix of the system which has been derived in Section REF .",
"black A closed-loop controller is integrated to drive the ultrasound probe tip position according to EM tracker feedback.",
"For this purpose, a high-level trajectory generator continuously updates the demand position for the position controller, which generates in return demand volumes for the three syringe pumps according to the control law $\\Delta {V}_d = J_V^T U$ Where $\\Delta V_d$ is the desired change in volume and $U$ the control signal.",
"A linear PI controller of the form $U = K_PX_e + K_I\\int X_e dt$ is employed, where $X_e = X_d - X_c$ .",
"$X_d$ and $X_c$ are demanded and measured probe tip position respectively.",
"blackThe gain matrices $K_P=diag(k_P,k_P,k_P)$ and $K_I=diag(k_I,k_I,k_I)$ contain the gain constants $k_P$ and $k_I$ , which have been verified experimentally and are defined as $0.3 \\frac{\\text{ml}}{\\text{mm}}$ and $0.03 \\frac{\\text{ml}\\cdot s}{\\text{mm}}$ respectively.",
"The target points are generated at 2Hz while both the position controller and the kinetostatic model are updated at 30Hz.",
"The low-level step generation for driving the syringe pumps is achieved with an update rate of 6kHz."
],
[
"SFA characterization",
"Using the blackthree SFAs to control the SEE pose in an open-loop configuration requires the volume-extension relation to be predictable for any given point in time.",
"blackFrom the radial mechanical constraint incorporated in the SFA design it is assumed that the relationship between induced volume and SFA length change is linear.",
"To verify this, the extension behaviour of a single SFA is investigated for different working fluid changes using a linear rail setup.",
"The position of the tip of the actuator is equipped with a slider and tracked using a linear potentiometer.",
"Contact friction between the linear bearings and rails is minimized using lubrication and friction forces are therefore neglected in the evaluation of the results.",
"Volume and extension data are tracked and synchronized using ROS."
],
[
"Stiffness characterization",
"blackAs the SEE is highly compliant, knowledge of its deformability under external loads is required to determine its efficacy to the given task.",
"To verify the structural behaviour of the SEE under contact forces required for the clinical application, the stiffness of the system is characterized with the setup shown in Fig.",
"REF .",
"The SEE is mounted to a base plate and its tip is connected through a force-torque sensor (Gamma, ATI, Apex, USA) to a robot manipulator (UR3, Universal Robots, Odense, Denmark).",
"To determine the stiffness of the SEE in a given direction, the manipulator moves the SEE in said direction and the resulting reaction force is measuredblack.",
"The robot allows for an accurate, repeatable displacement of the SEE in a defined direction, thus isolating the desired DOFs.",
"The payload of the system is with 3kg sufficiently high to withstand the induced reaction forces caused by the elastic deformation of the SEE.",
"The motions are repeated 10 times for each configuration.",
"The linearized relationship between reaction force and manipulator displacement corresponds to the stiffness of the SEE.",
"The mesh reinforcement’s effect on the axial twist stiffness is determined by twisting the SEE repeatedly by 10 and measuring the z-axis moment.",
"This is done for a configuration without mesh reinforcement, mesh reinforcement without crimps and mesh reinforcement with crimps.",
"Figure: Experimental setup for stiffness characterizationFigure: SEE moving in contact with soft rubber patch.",
"blackThe tip pose change with respect to the SEEs origin is highlighted with an arrow.The directional lateral stiffness is obtained by displacing the SEE tip radially in a defined direction over a distance of 10mm.",
"This is repeated for four inflation levels (25%, 50%, 75% and 100% of the maximum SFA volume) and for directions between 0$$ and 345 in 15 increments around the z-axis.",
"The axial stiffness which corresponds to each extension is determined by displacing the SEE tip in negative z-direction by 1.5mm for 25% and 50% inflation, and by 2.5mm for 75% and 100% extension."
],
[
"Workspace and repeatability",
"blackTo verify whether the attainable motions of the SEE satisfy the imposed clinical requirements for the ultrasound probe motion, the workspace of the SEE is mapped for achievable volumetric inputs.",
"The blackSEE pose is measured using an electromagnetic tracker (6DOF Reference, Aurora, NDI, Ontario, Canada) which is attached to the side of the SEE tip.",
"The pose of the ultrasound probe tip is calculated with the known homogeneous transformation between tracker and tip.",
"The SFA volumes are varied between 0% and 100% in 10% increments and the resulting static tip pose is determined with respect to its deflated state.",
"The repeatability in positioning the tip of the SEE is determined by repeatedly approaching defined SFA volume states and measuring the tip pose.",
"A set of 6 states is defined and the resultant trajectory is executed 50 times."
],
[
"Model validation",
"The derived model is validated by comparing the workspace and corresponding SFA volumes to the calculated tip pose of the SEE.",
"blackFor this purpose tip poses are calculated for each configuration achieved in Section REF and the error between model and measurement is determined."
],
[
"Indentation behaviour",
"blackWhilst the abdomen exhibits an increased stiffness with the duration of the pregnancy and thus counteracts indentation of the ultrasound probe, deep tissue indentation in the early weeks can affect the positioning behaviour of the SEE.",
"To verify the effect a soft tissue-like contact has on the SEE, a soft mechanical phantom is created.",
"The cylindrical phantom is moulded from a layer of Ecoflex Gel and a structural layer of Ecoflex 00-30 (SmoothOn Inc, Pennsylvania, USA).",
"The tip of the SEE is controlled to perform a line trajectory from its negative to positive x-axis limits at 60% inflation.",
"The tip pose is monitored with a magnetic tracker and contact forces between SEE and phantom are measured using aforementioned force sensor at the base of the phantom.",
"The manipulator is used to test for different indentation depths from 0mm to 15mm in 5mm increments.",
"Figure: Sonographer performing SEE-assisted ultrasound scanning of a prenatal abdominal phantom (iii).",
"The SEE (i) is attached to a passive arm (ii) and manually placed on the phantom.",
"A joystick (iv) is used to manipulate the ultrasound probe under visual guidance of the acquired image (v)."
],
[
"Controllability",
"blackTo achieve a desired view-plane in the ultrasound image, the probe attached to the SEE needs to be steerable accurately across the patient's body.",
"The controllability of the SEE is verified with the closed-loop position control system described in Section REF .",
"Target trajectories are defined as isosceles triangles with a base of 12.33mm and height of 10mm.",
"For the tilted trajectory, the triangle is titled about one of its sides by 19$$ .",
"The trajectory is tested in a planar and tilted configuration and tracked 3 times each.",
"To determine the controllability under an external load, a stiff silicone rubber patch is created as shown in Fig.",
"blackREF .",
"The patch is lubricated and positioned with its center at the tip of the SEE.",
"To ensure contact with the patch, an initial axial force of 5N is generated by displacing the patch and running the position controller.",
"This is repeated for planar and tilted configurations, where each trajectory is tracked 3 times."
],
[
"Sonographer-guided teleoperation",
"The imaging capabilities of an ultrasound transducer guided by the SEE are verified using a prenatal abdominal phantom (SPACE FAN-ST, Kyoto Kagaku, Japan).",
"The SEE is equipped with an abdominal ultrasound probe (X6-1, Philips, Amsterdam, Netherlands) which is connected to an ultrasound scanner (EPIQ7, Philips, Amsterdam, Netherlands).",
"A passive positioning arm (Field Generator Mounting Arm, NDI, Ontario, Canada) is used to manually position the SEE in the region of interest on the phantom.",
"The sonographer uses the provided ultrasound image feedback to steer the SEE with the connected joystick towards a desired view-plane.",
"The target view-planes are manually acquired using a handheld ultrasound probe.",
"An overview of the experimental setup is shown in Fig.",
"REF ."
],
[
"Clinical data",
"blackThe results of the clinical data acquisition are presented in Table REF .",
"For each subject the maximum observed motion range in translation and rotation of the ultrasound probe is presented for the HC, AC and FL standard views.",
"The presented forces correspond to the 95th percentile of the occurring normal and tangential force magnitudes.",
"A time series of the probe pose and force data obtained for subject 5 is shown in Fig.",
"REF .",
"For subject 2 only HC and AC views were obtained.",
"Translations and rotations are shown with respect to the patient bed.",
"The normal force is assumed to be acting only in negative probe direction and the tangential force shows the vector magnitude of the tangential forces in X and Y. blackTo obtain workspace requirements which are compatible with the obtained forces, it is divided into transversal and axial movements and transversal rotations.",
"In this study, axial rotations of the probe are ignored.",
"Workspace requirements for the SEE are consequently obtained by selecting the larger translation between X and Y for the transversal $\\delta _{req}^{tr}$ and the translation in Z for the axial motion $\\delta _{req}^{ax}$ , thus resulting in a required cylindrical workspace of radius $\\delta _{req}^{tr}$ and height $\\delta _{req}^{ax}$ .",
"For the orientation, the required rotation is defined by $\\theta _{req}^{tr}$ .",
"The mean required workspace from the clinical data is therefore $\\begin{split}\\delta _{req} = &[\\delta _{req}^{ax}, \\delta _{req}^{tr}]^T = [5.22\\text{mm}, 7.75\\text{mm}]^T\\\\\\theta _{req} = &5.08\\end{split}$ Corresponding maximum tilts of pitch and roll are in ranges of $\\pm 9.8$ and $\\pm 12.9$ .",
"The maximum occurring normal and tangential forces are 20.77N and $\\pm $ 10.67N respectively.",
"Table: Range of motion and contact force required to obtain a desired view in foetal ultrasound.",
"Values used to generate the required SEE workspace are marked in blue.Figure: SFA pressure (a) and extension (b) under increasing working fluid volume blackfor different inflation levels.Figure: Change in transversal stiffness with the direction of the applied force for different extensions (a).",
"Change in stiffness with extension for axial (i) and transversal stiffness (ii) (b).",
"Change in stiffness with bending for axial (i) and transversal stiffness (ii) (c).Figure: Measured compression force of the SEE f meas f_{meas} at 0% (a), 50% (b) and 100% axial extension with it's corresponding linear interpolation f lin f_{lin}.",
"For each configuration the compressed SEE is depicted and the SFA centerlines are highlighted."
],
[
"SFA characterization",
"The results of the SFA characterization are shown in Fig.",
"REF .",
"The hydraulic pressure under SFA inflation and the resulting extension are shown in Fig.",
"REF a) and REF b) respectively.",
"blackHysteresis is mainly observable in the fluid pressure.",
"The mean deviation from the centerline between loading and unloading is 3.82$\\pm $ 1.63kPa for the pressure and 0.14$\\pm $ 0.05mm for the extension across the different inflation cycles.",
"A maximum deviation due to hysteresis is observable in the pressure when inflated to 100% with 9.28kPa and when inflated to 50% at 0.44mm.",
"The volume-extension curve of the SFA can be separated into two regions, a nonlinear (0ml to $\\approx $ 1.25ml) and a linear region ($\\approx $ 1.25ml to 5ml).",
"In the linear region, the relationship can be can be approximated with a first order polynomial as $\\Delta L(\\Delta V)=6.61\\text{mm}/\\text{ml} - 5.52\\text{mm}$ .",
"blackThe interpolation is used to determine the relationship between the SFA length change and the input volume change of the form $a = {\\Delta V}/{\\Delta L} = \\pi \\cdot 6.9^2 \\text{mm}^2$ .",
"As the proportion of the nonlinear region compared to the overall extension of the SFA is small, it is ignored for the following investigations.",
"SFAs are therefore assumed to be pre-extended with a volume of 1.25ml.",
"Figure: Workspace of the SEE in position (a-b) and orientation (c-d).",
"blackThe required workspace δ req \\delta _{req} without and with consideration of the deflected tip pose δ f \\delta _f are indicated.",
"A cross-section view along the dotted lines shows the coupling between position and orientation in the performed bending motions (e), in which the dashed lines indicate iso-volume lines for V 2 =V 3 V_2=V_3."
],
[
"Stiffness",
"The results of the twist stiffness characterization for each mesh configuration are shown in Table REF , where $\\mu $ and $\\sigma $ are the mean and standard deviation of the twist stiffness $K_{tw}$ respectively.",
"The application of a nylon mesh helps to significantly stiffen the torsional axis of the system by 184%.",
"A crimped mesh can further improve the torsional stiffness to 299% of its original value.",
"Table: Twist stiffnessThe results of the lateral stiffness characterization under inflation of the SEE are shown in Fig.",
"REF a) in polar coordinates.",
"The radius indicates the magnitude of the stiffness in the given direction.",
"The axial and averaged lateral stiffness of the SEE under axial extension are presented in Fig.",
"REF b).",
"The data are presented alongside their corresponding spline interpolations.",
"Both decrease monotonically with the axial stiffness starting from a maximum of 34.83N/mm and reaching a minimum of 14.41N/mm at 100% extension.",
"The transversal stiffness decreases at a comparable rate from 3.21N/mm at 25% down to 1.51N/mm at 100% extension.",
"The stiffness variation under bending of the SEE is shown in Fig.",
"REF c) with the visualized trends interpolated by splines.",
"Whilst the the transversal stiffness decreases monotonically from 3.15N/mm to 1.77N/mm, the axial stiffness decreases from 21.15N/mm at 0.3$$ tilt to a minimum of 9.99N/mm at 10$$ followed by an increase in stiffness to 18.75N/mm at 13.75$$ .",
"The presented data is employed to determine the minimum stiffness of the system throughout the workspace to infer possiblly occurring tip pose deviations from external forces.",
"It can be seen that the system reaches a minimum axial stiffness of 14.41N/mm and transversal stiffness of 1.51N/mm, both in a straight and fully extended configuration.",
"Despite high loads along the axial direction of the SEE no discontinuous buckling behaviour of the SFAs is observable.",
"This is demonstrated in Fig.",
"REF .",
"The force-displacement relationships and their corresponding linear interpolations are shown for 0%, 50% and 100% extension and depictions of the SEE at the corresponding maximum loads are presented.",
"Whilst a slight increase in the nonlinearity between force and displacement is observable for 100% extension (the corresponding mean absolute errors between data and linear interpolation are 0.84N, 0.62N and 1.16N for 0%, 50% and 100% extension) no discontinuities are identifiable.",
"The depictions of the deformed SEE show how the forced S-shape bending of the SFAs helps to prevent buckling.",
"An increase in axial force only causes the curvature of the S-bend to increase."
],
[
"Workspace",
"The workspace of the SEE in position and orientation is shown in Fig.",
"REF .",
"The figures show the tip pose acquired by the EM tracker for any given SFA configuration.",
"The required workspace in position and orientation, $\\delta _{req}$ and $\\theta _{req}$ , obtained in Section REF from clinical data is projected into the center of the SEE workspace.",
"The deflected workspace $\\delta _f$ is calculated from the results obtained in Section REF .",
"It can be seen that the SEE exhibits an minimum transversal stiffness of $1.51\\text{N}/\\text{mm}$ and a minimum axial stiffness of $14.41\\text{N}/\\text{mm}$ at 100% extension.",
"Taking into account the mean external load applied to the tip, a possible additional deflection of $\\delta _f =\\begin{bmatrix}14.41& 0\\\\0& 1.51\\end{bmatrix}^{-1}\\begin{bmatrix}8.01\\\\4.42\\end{bmatrix}=\\begin{bmatrix}0.56\\\\2.95\\end{bmatrix}$ Thus, the workspace the SEE is required to achieve extends correspondingly to $\\hat{\\delta }_{req} = \\delta _{req} + \\delta _f =\\begin{bmatrix}5.78\\\\10.68\\end{bmatrix}$ Whilst in some instances larger motions have to be achieved, the derived values represent a baseline motion range desirable from the SEE.",
"blackTo quantify whether the SEE is able to reach the desired workspace, the intersections between requirement and SEE workspace volumes are computed.",
"It can be seen that for the unloaded requirements in translation and rotation, $\\delta _{req}$ and $\\theta _{req}$ , the SEE can accomplish 100% of the workspace.",
"For the workspace adapted to account for an external force $\\hat{\\delta }_{req}$ , the robot achieves 95.18% of the required workspace.",
"It is shown that a maximum combined lateral deflection of 19.01mm can be reached along the principal plane of $SFA_3$ , which is about 4.5% lower than the maximum transversal motion blackobserved in manual scanning.",
"The maximum extension of the SEE of 22.13mm is reached for a full inflation of all SFAs blackand exceeds the demanded axial translation of 10.22mm as well as the transversal translation of 19.91mm determined from the clinical data.",
"The maximum tilt of the SEE is reached along the principal plane of $SFA_1$ with 14.02$$ black, which is $\\approx 9\\%$ greater than the maximum demanded tilt of 12.9$$ .",
"A maximum axial torsion of 1.03$$ occurs.",
"Compared to the tilt ranges in X and Y the twist is significantly lower and will therefore be ignored in the following investigations.",
"blackThe coupling between translation and rotation, the bending, of the SEE upon actuator inflation is shown in Fig.",
"REF e) for a cross-section of the workspace along the central x-z-plane in translation and the corresponding y-axis of the rotational workspace.",
"It can be seen that with the amount of transversal translation, the rotation of the tip increases, whilst axial extension has no effect on the rotation.",
"Table: RepeatabilityThe results of the positioning repeatability evaluation are presented in Table REF .",
"The table indicates the mean Euclidean errors in position and orientation with their respective standard deviations from the given pose for the 50 repetitions blackwith respect to the mean pose for the given configuration, $\\mu (x(C_j))$ .",
"blackFor a configuration $C_j$ , for instance, the Euclidean error $||\\delta _e||$ is computed as $||\\delta _e|| = \\sum _{i=1}^{n=50}\\frac{||x_i-\\mu (x(C_j))||}{n}$ The pose $x_i$ for a given configuration $C_j$ is obtained by averaging the measured static tip pose over a period 4 seconds.",
"The orientation error $||\\theta _e||$ and both corresponding standard deviations are calculated in the same manner.",
"Whilst it can be seen that the measured accuracy of the SEE is with $\\approx 0.1$ mm in position and $0.05$ orientation slightly below the rated accuracy of the EM tracking system (0.48mm and 0.30$$ RMS [55]), it can be seen that averaging the pose data over 4 seconds reduces noise-related variance in the data.",
"The samples are normally distributed across the workspace and thus the time-averaged mean is assumed to represent the tip pose sufficiently.",
"Figure: Workspace generated with model in position (a) and orientation (b).",
"The colour indicates the normalized Euclidean error in the given state with respect to the maximum deviation from the modelblack, which is 2.37mm in position and 2.46 in orientation."
],
[
"Model validation",
"The results of the model validation are shown in Fig.",
"REF and summarized in Table REF black, where $\\mu $ refers to the mean error, $\\sigma $ to the standard deviation and $max$ to the maximum error.",
"The estimated workspace of the SEE generated with the kinetostatic model is shown in Fig.",
"REF .",
"The colour of each marker indicates the Euclidean distance between the calculated point and the corresponding measured pose normalized to the maximum error in position and orientation respectively, namely 2.37mm and 2.46$$ .",
"blackThe Young's modulus of the SFA material $E$ and its area moment of intertia $I$ have been manually tuned to minimize the Euclidean error in position and orientation.",
"The obtained values are shown in Table REF .",
"Overall, the model validation shows with a mean Euclidean error of $1.18\\pm 0.29$ mm in position and $0.92\\pm 0.47$ in orientation good results in predicting the tip pose under SFA extension.",
"Table: Model validationFigure: Effect of axial loading on transversal motionFigure: Example of tracked trajectory under external loading.",
"The normal force is represented with scale of 0.2mm/N"
],
[
"Contact experiment",
"The motion constraint induced by an indentation contact is investigated.",
"Fig.",
"REF shows the constraint of the mean X-displacement and Y-tilt for a given motion over 10 repetitions normalized to blacktheir respective maximum valueblacks of 12.92mm in position and 8.67$$ in orientation when no external force is present, as well as their corresponding linear interpolations.",
"The transversal force applied by the SEE is measured with the force torque sensor.",
"For both, the displacement and the tilt, the magnitude declines monotonically.",
"Whereas the displacement reaches a minimum at 27.84%, the tilt remains less affected by the lateral force with a minimum of 55.35%.",
"Linearizing the trends yield a decrease of 14.09$\\%/\\text{N}$ for the displacement and only 8.56$\\%/\\text{N}$ for the tilt.",
"Table: Position control results"
],
[
"Position control",
"An example of a tracked trajectory with external loading is shown in Fig.",
"REF .",
"The position controller tracks the desired position accurately with marginally larger tracking error around the corners of the triangular path.",
"The quantitative results of the controller evaluation for the three executions are presented in Table REF for both the unloaded and loaded trajectories, blackwhere, as in Section REF , $\\mu $ refers to the mean error, $\\sigma $ to the standard deviation and $max$ to the maximum error in the respective direction.",
"The results indicate a higher mean error for the z-direction regardless of the configuration, which is also observable in the visualization above.",
"Figure: Ultrasound images acquired by sonographer (a-c) and SEE (d-f) for HC (a,d), AC (b,e) and FL (c,f) measurements"
],
[
"Teleoperation and image-acquisition",
"The images obtained through manual ultrasound probe placement and steering with the SEE are presented in Fig.",
"REF .",
"Anatomical structures of the foetus phantom are clearly visible throughout all images with minor shadowing on the left side of the FL standard view-plane, outside of the region of interest.",
"In both cases, the regions of interest are centered in the image.",
"Moreover, the contrast in the robot-acquired images is similar to the one in the manually-obtained images."
],
[
"Discussion",
"In this work we developed a soft robotic ultrasound imaging system blackto offload sonographers in day-to-day scanning routines.",
"The system addresses the issue of providing a stable contact between the ultrasound probe and patient, which could help improve sonographers’ ergonomics particularly with respect to work-related musculoskeletal disorders which arise from stresses induced by repeated manual probe handling.",
"The robot allows for tele-operated scanning and provides a platform for advanced imaging approaches.",
"It is designed in form of an end-effector which is manually positioned in the area of interest and actively steered towards the desired view-plane.",
"Due to its inherent compliance, the SEE is able to maintain contact while exhibiting sufficient axial stiffness to ensure mechanical coupling for the ultrasound image acquisition blackwhich is verified by acquiring standard views on a foetal ultrasound phantom.",
"The system shows with its high axial and low lateral stiffness good applicability to foetal ultrasound scanning.",
"Despite the quick decline of stiffness with axial extension, the SEE is with $14.41\\text{N}/\\text{mm}$ axial stiffness at full extension still capable to apply sufficiently high forces to the patient without significant deformation, blackwhich is approximately 1.44mm at maximum axial load of 20.77N.",
"The lower lateral stiffness allows for the system to be adaptable to the contact surface and to be moved away in case of discomfort in the patient blackwhilst being sufficiently high to counteract transversal loads occurring during the intervention.",
"It can be seen that for the fully extended SEE the transversal displacement at a maximum occurring load of 10.67N reaches 7.1mm.",
"blackThe compliance of the system allows for deformation upon external motion when clamped onto a patient.",
"Thus, the resulting contact force is significantly lower compared to a rigid system.",
"It furthermore exhibits a low mass which could be beneficial in the dynamic safety of the system [19].",
"blackIf the stiffness in the axial direction of the probe needs to be adjusted or the tip force controlled, the system can be equipped either with a force sensor at the base to estimate tip forces or serve as a sensor itself [56].",
"While in the first case the tip pose change during the operation needs to be accounted for to accurately determine the external force, either by an accurate model or pose feedback, the second case can make use of the deformable structure of the robot paired with the kinematic constraints induced by the actuation channels to infer the external force.",
"We have shown that the integration of a braided nylon mesh, which has previously only been used to avoid ballooning in SFAs, can significantly improve the twist stiffness of the SEE to up to three times in comparison to the mesh-free system.",
"blackThe use braided meshes is a highly versatile design approach and shows the potential to become a de facto standard in reinforcing not only soft robotics system but also continuum robots against unwanted external twists induced by contact wrenches, thus enabling such robots for a wider range of applications.",
"blackThe workspace achieved by the SEE covers without external loading the blackaverage translation and rotation motion ranges required to achieve a desired view, as shown from clinical data.",
"Loading the probe with the contact forces measured in clinical scans and assuming the lowest possible stiffness of the system reduces the achieved workspace to about 95.18% of the blackmean required range.",
"Whilstblack, for example, the maximum translation of the SEE is at 19.01mm significantly higher than the required deflected motion of 10.68mm, the non-homogeneous shape of the SEE workspace dictates the limitations in covering the required translation range.",
"This limitation could be addressed by adding a linear stage to the base of the SEE to allow for axial translation without sacrificing the softness of the system.",
"blackMoreover, an axial rotation stage could be added to allow for more complex probe motions.",
"blackA high variability in the monitored ultrasound probe motion ranges can be observed across the obtained views and subjects.",
"Whilst, on average, relatively small maximum deflections are observed, in some instances significantly larger motions occur.",
"This is indicated by the high standard deviations in the motion ranges of the respective axes.",
"Further research needs to be conducted into the exact metrics of the ultrasound probe motions and whether the designed system can satisfy those metrics.",
"Additional considerations such as the coupling between different motion axes then need to be accounted for.",
"Another factor in the feasibility of a desired view is the accuracy of the manual placement of the passive positioning arm.",
"If the accuracy is low and the view is out of reach of the end-effector, the passive arm could either be repositioned manually or additional DOFs could be added to the system.",
"More accurate methods should be employed in evaluating the manual probe motions.",
"The use of a percentile is difficult for the given data due to the high variability in the times required to obtain desired views, as seen in the presented time series for the motions of subject 5 in Fig.",
"10 for example.",
"Thus, a larger scale and more streamlined data acquisition needs to be conducted.",
"blackWe showed that the combination of SFAs and hydraulic actuation exhibits good properties for the SEE to be driven in an open-loop configuration.",
"The relationship between SFA length and input volume is highly linear and only shows 0.14$\\pm $ 0.05mm deviation due to hysteresis, thus allowing for an accurate prediction of the kinematic constraints imposed on the SEE.",
"This compliments the derived kinetostatic model, which is able to accurately predict the SEE tip motion with an accuracy of 1.18mm in position and 0.92$$ in orientation as a function of the induced working fluid volume.",
"The model deviates more along the boundaries of the workspace, which could be caused by the larger deflection of the SFAs and resultant nonlinearieties caused by the bending of the actuators.",
"This could be addressed by extending the model to a nonlinear approach, as we have for example demonstrated in [56] for a soft continuum robot.",
"blackThe repeatability lies with 0.1mm in position and 0.05$$ in orientation slightly below the rated accuracy of the measurement system.",
"As the obtained measurements are expressed relative to a mean pose, averaged over time and normally distributed, it is assumed that these values still represent the true pose well.",
"The high repeatability and should allow for accurate positioning of the SEE in view-plane finding applications.",
"blackThe system maintains stability and controllability well when in contact with a tissue-like soft silicone rubber patch.",
"We showed that the implemented closed-loop position controller is able to track target trajectories accurately with a mean position error of 0.35mm with only marginally increased errors in the tracking accuracy of 0.44mm when a contact force applied.",
"In scenarios where EM tracking is not available, the ultrasound image could be used to provide pose feedback.",
"This could then employed as a substitute for the position feedback in the closed-loop controller.",
"The coupling between position and orientation is an obvious limitation in the usability of the design.",
"It can be seen, however, that the mechanical properties of the surface contact greatly affect the coupling behaviour.",
"We have shown that an indenting contact reduces the lateral motion of the ultrasound probe significantly more than the tilt.",
"It can easily be seen that a very stiff coupling in combination with the minimal contact friction caused by the application of ultrasound gel greatly reduces the tilt capabilities of the system while allowing for lateral sliding.",
"It can therefore be assumed that in practice the coupling can be reduced by varying the axial pressure applied to the patient.",
"This is supported by the findings of the tele-operated image acquisition in Section REF and will be investigated further in future research."
],
[
"Conclusion",
"The SEE design proposed in this work blackshows a novel approach to applying soft robotics technologies in medical ultrasound imaging.",
"We have shown that under certain conditions the SEE satisfies the requirements imposed by the clinical application.",
"The derived kinetostatic model mimics adequately the behaviour of the physical robot and the integrated system is capable of tracking target trajectories accurately and obtaining high-quality ultrasound images of a prenatal ultrasound phantom.",
"In our future work, we will make use of the hydraulic actuation to integrate a force-controlled system through intrinsic force sensing, as shown in our previous work [56].",
"[Error behaviour in repeatability validation] For each achieved configuration in the repeatability validation experiment, the pose data is averaged over a period of 4 seconds.",
"The resulting data for the displacement in Z-direction upon reaching configuration $C_4$ is presented in Fig.",
"REF a).",
"The corresponding distribution of measurements is shown in Fig.",
"REF b).",
"It can be seen that the readings follow a normal distribution around a mean of 5.36mm with a standard deviation of 0.03mm.",
"A $\\chi ^2$ goodness-of-fit is performed to determine the suitability of describing the individual readings for a given pose as a normal distribution.",
"Across all configurations, the mean $p$ value associated with the fit is 0.34$\\pm $ 0.28.",
"It is therefore concluded that this hypothesis holds across the workspace and thus, the time-averaged pose is a suitable indicator for the true pose of the SEE.",
"Figure: Sampling of EM tracker data for defined pose over 4sec (a) and corresponding distribution of measurements (b)."
],
[
"Safety considerations",
"The use of a soft robotic system can help to greatly reduce the contact contact forces upon undesired patient or motions of the robot itself.",
"The build-up of contact force with a clamping contact between robot and the patient constrained by the patient bed can lead to discomfort and potentially injury [57].",
"To determine an approximate occurring force for a patient motion of 1cm against a rigid robot, we can calculate the following.",
"The Young’s modulus for visceral contents can be approximated by $E_{vis} = 8.42$ kPa [58].",
"Assuming a circular contact of 10mm radius ($r$ ) with a tissue thickness ($d$ ) of 10mm, the stiffness of the visceral contents can be determined as $K_{vis}={E \\pi r^2}/d=39.37 \\text{N}/\\text{mm}$ If the patient moves against a stationary rigid robot over the distance $\\Delta x=10$ mm, the contact force experienced by patient and robot is $f_{rigid}= K_{vis}\\cdot \\Delta x=39.37\\text{N}/\\text{mm}\\cdot 10\\text{mm}=393.7\\text{N}$ For the soft robot, the system stiffness is combined in form of two serially-connected springs.",
"In case of the lowest transversal stiffness of the soft robot ($K_{SEE}=1.51 $ N/mm), one can compute for the combined stiffness $K_{comb}=(1/K_{vis} +1/K_{SEE} )^{-1}=1.45 \\text{N}/\\text{mm}$ The resulting force build-up upon contact is then only 21.69N.",
"Considering the reduction in contact force when exposed to an involuntary patient or clinician motion, it can be assumed that the use of soft robots instead of rigid ones could greatly reduce contact forces when a patient is exposed to a clamping contact.",
"This work was supported by the Wellcome Trust IEH Award [102431], the iFIND project, and by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/R013977/1."
]
] | 1906.04526 |
[
[
"Study of Compressed Randomized UTV Decompositions for Low-Rank Matrix\n Approximations in Data Science"
],
[
"Abstract In this work, a novel rank-revealing matrix decomposition algorithm termed Compressed Randomized UTV (CoR-UTV) decomposition along with a CoR-UTV variant aided by the power method technique is proposed.",
"CoR-UTV computes an approximation to a low-rank input matrix by making use of random sampling schemes.",
"Given a large and dense matrix of size $m\\times n$ with numerical rank $k$, where $k \\ll \\text{min} \\{m,n\\}$, CoR-UTV requires a few passes over the data, and runs in $O(mnk)$ floating-point operations.",
"Furthermore, CoR-UTV can exploit modern computational platforms and can be optimized for maximum efficiency.",
"CoR-UTV is also applied for solving robust principal component analysis problems.",
"Simulations show that CoR-UTV outperform existing approaches."
],
[
"Introduction",
"Low-rank matrix approximations play an increasingly important role in signal processing and its applications.",
"Such compact representations which retain the key features of a high-dimensional matrix provide a significant reduction in memory requirements, and more importantly, computational costs when the latter scales, e.g., according to a high-degree polynomial, with the dimensionality.",
"Matrices with low-rank structures have found many applications in background subtraction , , system identification , IP network anomaly detection , , latent variable graphical modeling , subspace clustering , and sensor and multichannel signal processing , , , , , , , , , , , , , , , , , , .",
".",
"Singular value decomposition (SVD) and the rank-revealing QR (RRQR) decomposition , are among the most commonly used algorithms for computing a low-rank approximation of a matrix.",
"On the other hand, a UTV decomposition is a compromise between the SVD and the RRQR decomposition with the virtues of both: UTV (i) is more efficient than the SVD, and (ii) provides information on the numerical null space of the matrix .",
"Given a matrix $\\bf A$ , the UTV algorithm computes a decomposition ${\\bf A =UTV}^T$ , where ${\\bf U}$ and ${\\bf V}$ have orthonormal columns, and ${\\bf T}$ is triangular (either lower or upper triangular).",
"These deterministic algorithms, however, are computationally expensive for large data sets.",
"Furthermore, standard techniques for their computation are challenging to parallelize in order to utilize advanced computer architectures , .",
"Recently developed algorithms for low-rank approximations based on random sampling schemes, however, have been shown to be remarkably efficient, highly accurate and robust, and are known to outperform existing algorithms in many practical situations , , , .",
"The power of randomized algorithms lies in that (i) they are computationally efficient, and (ii) their main operations can be optimized for maximum efficiency on modern architectures.",
"This work presents a novel randomized rank-revealing method termed compressed randomized UTV (CoR-UTV) decomposition.",
"Given a large and dense rank-$k$ matrix ${\\bf A} \\in \\mathbb {R}^{m \\times n}$ , CoR-UTV computes a low-rank approximation $\\hat{\\bf A}_\\text{CoR}$ of $\\bf A$ such that $\\hat{\\bf A}_\\text{CoR}={\\bf UTV}^T,$ where ${\\bf U}$ and ${\\bf V}$ have orthonormal columns, and ${\\bf T}$ is triangular (either lower or upper, whichever is preferred).",
"CoR-UTV only requires a few passes through data, for a matrix stored externally, and runs in $O(mnk)$ floating-point operations (flops).",
"The operations of CoR-UTV involve matrix-matrix multiplication, the QR and RRQR decompositions.",
"Due to recently developed Communication-Avoiding QR algorithms , , , which can perform the computations with optimal/minimum communication costs, CoR-UTV can be optimized for peak machine performance on modern architectures.",
"We illustrate, through numerical examples, that CoR-UTV is rank-revealer and provides a highly accurate low-rank approximation to a given matrix.",
"Furthermore, we apply CoR-UTV to solve the robust principal component analysis (robust PCA) problem , , i.e., to decompose a given matrix with grossly corrupted entries into a low-rank matrix plus a sparse matrix of outliers.",
"The rest of this paper is structured as follows.",
"In Section , we introduce the mathematical model of the data and discuss related works.",
"In Section , we describe the proposed CoR-UTV method in detail.",
"In Section , we develop an algorithm for robust PCA using CoR-UTV.",
"In Section , we present and discuss simulation results and conclusions are given in Section ."
],
[
"Mathematical Model and Related Works",
"Given a matrix ${\\bf A} \\in \\mathbb {R}^{m \\times n}$ , where $m \\ge n$ , with numerical rank $k$ , its SVD is defined as: $\\begin{aligned}{\\bf A} = & {\\bf U}_\\text{A}{\\bf \\Sigma }_\\text{A}{\\bf V}_\\text{A}^T= & \\underbrace{\\begin{bmatrix} {{\\bf U}_k \\quad {\\bf U}_0}\\end{bmatrix}}_{{\\bf U}_\\text{A} \\in \\mathbb {R}^{m \\times n}}\\underbrace{\\begin{bmatrix}{\\bf \\Sigma }_k & 0 \\\\0 & {\\bf \\Sigma }_0\\end{bmatrix}}_{{\\bf \\Sigma }_\\text{A} \\in \\mathbb {R}^{n \\times n}}\\underbrace{\\begin{bmatrix}{{\\bf V}_k \\quad {\\bf V}_0} \\end{bmatrix}^T}_{{\\bf V}_\\text{A}^T \\in \\mathbb {R}^{n \\times n}},\\end{aligned}$ where ${\\bf U}_k \\in \\mathbb {R}^{m \\times k}$ , ${\\bf U}_0 \\in \\mathbb {R}^{m \\times n-k}$ , ${\\bf V}_k \\in \\mathbb {R}^{n \\times k}$ and ${\\bf V}_0 \\in \\mathbb {R}^{n \\times n-k}$ have orthonormal columns, ${\\bf \\Sigma }_k \\in \\mathbb {R}^{k \\times k}$ and ${\\bf \\Sigma }_0 \\in \\mathbb {R}^{n-k \\times n-k}$ are diagonal matrices containing the singular values, i.e., ${\\bf \\Sigma }_k=\\text{diag}(\\sigma _1, ...,\\sigma _k)$ and ${\\bf \\Sigma }_0 =\\text{diag}(\\sigma _{k+1}, ...,\\sigma _n)$ .",
"$\\bf A$ can be written as ${\\bf A} = {\\bf A}_k+{\\bf A}_0$ , where ${\\bf A}_k = {\\bf U}_k{\\bf \\Sigma }_k{\\bf V}_k^T$ , and ${\\bf A}_0 = {\\bf U}_0{\\bf \\Sigma }_0{\\bf V}_0^T$ .",
"The SVD constructs the optimal rank-$k$ approximation ${\\bf A}_k$ to ${\\bf A}$ , i.e., $\\begin{aligned}&\\ \\Vert {\\bf A} - {\\bf A}_k\\Vert _2 = \\sigma _{k+1}, \\\\&\\ {\\Vert {\\bf A} - {\\bf A}_k\\Vert _F} = \\sqrt{\\sigma _{k+1}^2 +...+ \\sigma _{n}^2},\\end{aligned}$ where ${\\Vert {\\cdot }\\Vert _2}$ and ${\\Vert {\\cdot }\\Vert _F}$ denote the spectral norm and the Frobenius norm, respectively.",
"In this paper we focus on the matrix $\\bf A$ defined.",
"The SVD is highly accurate for computing singular subspaces and singular values.",
"However, its computation is costly for large data sets.",
"Moreover, standard techniques for its computation are challenging to parallelize in order to take advantage of modern processors , .",
"An economical version of the SVD is the partial SVD based on Krylov subspace methods, such as the Lanczos and Arnoldi algorithms, which constructs an approximate SVD of an input matrix, for instance $\\bf A$ , at a cost $O(mnk)$ .",
"However, the partial SVD suffers from two drawbacks: (i) it is numerically unstable , , and (ii) it does not lend itself to parallel implementations , , which makes it unsuitable for modern architectures.",
"Other approaches for low-rank matrix approximations include the RRQR and the UTV decompositions .",
"Even though the QR with column pivoting (QRCP) and UTV decompositions provide highly accurate approximations to $\\bf A$ , they suffer from two drawbacks: (i) they are costly, i.e., $O(mn^2)$ , and (ii) methods for their computation are challenging to parallelize and hence they cannot exploit modern computational platforms , .",
"Recently developed algorithms for low-rank approximations based on randomization , , , , have attracted significant attention.",
"The randomized algorithms project a large input matrix onto a lower dimensional space using a random matrix, and apply deterministic methods on the smaller matrix to give an approximation of the matrix.",
"Hence (i) they are computationally efficient, and (ii) lend themselves to parallel implementation.",
"Halko et al.",
"proposed randomized SVD (R-SVD) in which a smaller matrix is formed by linear combinations of columns of the given matrix.",
"The low-rank approximation is then given through the SVD of a reduced-size matrix.",
"Gu applied a slightly modified version of the R-SVD algorithm to improve subspace iteration methods, and presents a new error analysis.",
"Another algorithm proposed in , which we call two-sided randomized SVD (TSR-SVD), is a single-pass method, i.e., it required only one pass through data.",
"It captures most attributes of the data by means of forming the smaller matrix through linear combinations of both rows and columns of the given matrix, and then applies the SVD for further computations.",
"The work in proposed a randomized algorithm termed subspace-orbit randomized SVD (SOR-SVD).",
"SOR-SVD alternately projects the matrix onto its column and row space.",
"The matrix is then transformed into a lower dimensional space, and a truncated SVD follows in order to construct an approximation.",
"TSR-SVD gives poor approximation compared to the optimal SVD due to the single-pass strategy.",
"SOR-SVD has shown better performance than TSR-SVD, however both methods apply the SVD on the reduced-size matrix.",
"This computation may be burdensome in terms of communication cost for large matrices.",
"In this work, we develop a randomized algorithm for low-rank approximation that with comparable flops (i) outperforms the TSR-SVD in terms of accuracy, and (ii) can utilize advanced computer architectures better than TSR-SVD as well as SOR-SVD."
],
[
"Compressed Randomized UTV Decompositions",
"In this section, we present a randomized rank-revealing decomposition algorithm termed compressed randomized UTV (CoR-UTV) decomposition , which computes a low-rank approximation of a given matrix.",
"We focus on the matrix $\\bf A$ with $m \\ge n$ , where CoR-UTV, in the form of (REF ), produces an upper triangular middle matrix $\\bf T$ .",
"The modifications required for a CoR-UTV for the case $m < n$ that produces a lower triangular middle matrix $\\bf T$ is straightforward."
],
[
"Proposed CoR-UTV Decompositions",
"Given the matrix ${\\bf A}$ and an integer $k\\le \\ell <\\text{min}\\lbrace m,n\\rbrace $ , the basic version of CoR-UTV is computed as follows: using a random number generator, we form a matrix ${\\bf \\Psi } \\in \\mathbb {R}^{n \\times \\ell }$ with entries drawn independent, identically distributed (i.i.d.)",
"from the standard Gaussian distribution.",
"We then compute the matrix product: ${\\bf C}_1 = {\\bf A}{\\bf \\Psi },$ where ${\\bf C}_1 \\in \\mathbb {R}^{m \\times \\ell }$ is, in fact, a projection onto the subspace spanned by columns of ${\\bf A}$ .",
"Having ${\\bf C}_1$ , we form ${\\bf C}_2 \\in \\mathbb {R}^{n \\times \\ell }$ : ${\\bf C}_2 = {\\bf A}^T{\\bf C}_1,$ where ${\\bf C}_2$ is, in fact, a projection onto the subspace spanned by rows of ${\\bf A}$ .",
"Using a QR decomposition, we factor ${\\bf C}_1$ and ${\\bf C}_2$ such that: ${\\bf C}_1 = {\\bf Q}_1{\\bf R}_1, \\quad \\text{and} \\quad {\\bf C}_2 ={\\bf Q}_2{\\bf R}_2,$ where ${\\bf Q}_1$ and ${\\bf Q}_2$ are approximate bases for $\\mathcal {R}({\\bf A})$ and $\\mathcal {R}({\\bf A}^T)$ , respectively.",
"We now compress $\\bf A$ by left and right multiplications by the orthonormal bases obtained, forming the matrix ${\\bf D} \\in \\mathbb {R}^{\\ell \\times \\ell }$ : ${\\bf D}={\\bf Q}_1^T{\\bf A}{\\bf Q}_2,$ We then compute a QRCP of ${\\bf D}$ : ${\\bf D} = \\widetilde{\\bf Q}\\widetilde{\\bf R}\\widetilde{\\bf P}^T.$ The CoR-UTV-based low-rank approximation of $\\bf A$ is given by $\\hat{\\bf A}_\\text{CoR}= {\\bf UTV}^T,$ where ${\\bf U}={\\bf Q}_1 \\widetilde{\\bf Q} \\in \\mathbb {R}^{m \\times \\ell }$ and ${\\bf V}={\\bf Q}_2 \\widetilde{\\bf P} \\in \\mathbb {R}^{n\\times \\ell }$ construct approximations to the $\\ell $ leading left and right singular vectors of $\\bf A$ , respectively, and ${\\bf T}=\\widetilde{\\bf R}\\in \\mathbb {R}^{\\ell \\times \\ell }$ is upper triangular with diagonals approximating the first $\\ell $ singular values of $\\bf A$ .",
"CoR-UTV requires three passes through data, for a matrix stored externally, but it can be altered to revisit the data only once.",
"To this end, the compressed matrix $\\bf D$ (REF ) can be approximated as follows: both sides of the currently unknown ${\\bf D}={\\bf Q}_1^T{\\bf A}{\\bf Q}_2$ are postmultiplied by ${\\bf Q}_2^T{\\bf \\Psi }$ .",
"Having defined ${\\bf A}\\approx {\\bf A}{\\bf Q}_2{\\bf Q}_2^T$ and ${\\bf C}_1 = {\\bf A}{\\bf \\Psi }$ , then $ {\\bf D}_\\text{approx} = {\\bf Q}_1^T{\\bf C}_1({\\bf Q}_2^T{\\bf \\Psi })^\\dagger $ .",
"CoR-UTV is accurate for matrices whose singular values display some decay, however in applications where the data matrix has a slowly decaying singular spectrum, it may produce a poor approximation compared to the SVD.",
"Thus, we incorporate $q$ steps of a power iteration , to improve the accuracy of the algorithm in these circumstances.",
"Given the matrix ${\\bf A}$ , and integers $k\\le \\ell < n$ and $q$ , the resulting algorithm is described in Alg.",
"REF .",
"CoR-UTV with Power Method [1] Matrix $\\ {\\bf A} \\in \\mathbb {R}^{m \\times n}$ , integers $k$ , $\\ell $ and $q$ .",
"A rank-$\\ell $ approximation.",
"Draw a standard Gaussian matrix ${\\bf C}_2 \\in \\mathbb {R}^{n \\times \\ell }$ ; $i=$ 1: $q+1$ Compute ${\\bf C}_1 = {\\bf A}{\\bf C}_2$ ; Compute ${\\bf C}_2 = {\\bf A}^T{\\bf C}_1$ ; Compute QR decompositions ${\\bf C}_1 = {\\bf Q}_1{\\bf R}_1$ , ${\\bf C}_2 = {\\bf Q}_2{\\bf R}_2$ ; Compute ${\\bf D}={\\bf Q}_1^T{\\bf A}{\\bf Q}_2$ or ${\\bf D}_\\text{approx} = {\\bf Q}_1^T{\\bf C}_1({\\bf Q}_2^T{\\bf C}_2)^\\dagger $ ; Compute a QRCP ${\\bf D} = \\widetilde{\\bf Q}\\widetilde{\\bf R}\\widetilde{\\bf P}^T$ or ${\\bf D}_\\text{approx} = \\widetilde{\\bf Q}\\widetilde{\\bf R}\\widetilde{\\bf P}^T$ ; Form the CoR-UTV-based low-rank approximation of $\\bf A$ : $\\hat{\\bf A}_\\text{CoR}= {\\bf UTV}^T$ ; ${\\bf U}={\\bf Q}_1 \\widetilde{\\bf Q},{\\bf T}=\\widetilde{\\bf R}$ ,${\\bf V}={\\bf Q}_2\\widetilde{\\bf P}^T$ ."
],
[
"Computational Complexity",
"The cost of an algorithm involves both arithmetic, i.e., the number of flops, and communication, i.e., data movement either between different levels of a memory hierarchy or between processors .",
"On multicore and accelerator-based computers, for a data matrix stored externally, the communication cost becomes substantially more expensive compared to the arithmetic , .",
"The randomized algorithms operate on a compressed version of the data matrix rather than a matrix itself and therefore can be organized to exploit modern computational environments better than their classical counterparts.",
"To decompose $\\bf A$ , the simple version of CoR-UTV incurs the following costs: Step 1 costs $n\\ell $ , Step 2 costs $2mn\\ell $ , Step 3 costs $2mn\\ell $ , Step 4 costs $2m\\ell ^2 + 2n\\ell ^2$ , Step 5 costs $m\\ell ^2+2mn\\ell $ (if $\\bf D$ is approximated by ${\\bf D}_\\text{approx}$ , this step costs $2m\\ell ^2 + 2n\\ell ^2 +3\\ell ^3$ ), Step 6 costs $8/3\\ell ^3$ , Step 7 costs $2m\\ell ^2 + 2n\\ell $ .",
"The dominant cost of Steps 1-7 occurs when multiplying $\\bf A$ and ${\\bf A}^T$ with the corresponding matrices.",
"Thus $C_\\text{CoR-UTV} = O(mn\\ell ).$ The sample size parameter $\\ell $ is typically close to the minimal rank $k$ .",
"The simple form of CoR-UTV requires either three or two passes (when $\\bf D$ is approximated by ${\\bf D}_\\text{approx}$ ) through data to factor $\\bf A$ .",
"When the power method is incorporated, CoR-UTV requires either $(2q+3)$ or $(2q+2)$ passes (when $\\bf D$ is approximated by ${\\bf D}_\\text{approx}$ ) over the data with arithmetic costs of $(2q+3)C_\\text{CoR-UTV}$ or $(2q+2)C_\\text{CoR-UTV}$ , respectively.",
"In addition to matrix-matrix multiplications and QR decompositions, CoR-UTV performs one QRCP on an $\\ell \\times \\ell $ matrix, however TSR-SVD and SOR-SVD perform an SVD on the $\\ell \\times \\ell $ matrix.",
"The SVD is more expensive than QRCP and, furthermore, recently developed QRCP algorithms based on randomization can perform the factorization with minimum communication costs , , , while standard techniques to compute an SVD are challenging for parallelization , .",
"Hence for large matrices to be factored on high performance computing architectures, where the compressed $\\ell \\times \\ell $ matrix does not fit into fast memory, the execution time to compute CoR-UTV can be substantially less than those of TSR-SVD and SOR-SVD.",
"This is an advantage of CoR-UTV over TSR-SVD and SOR-SVD."
],
[
"Robust PCA with CoR-UTV",
"This section describes how to solve the robust PCA problem using the proposed CoR-UTV method.",
"Robust PCA , represents an input low-rank matrix ${\\bf M} \\in \\mathbb {R}^{m \\times n}$ whose fraction of entries being corrupted, as a linear superposition of a low-rank matrix ${\\bf L}$ and a sparse matrix of outliers ${\\bf S}$ such as ${\\bf M=L+S}$ , by solving the following convex program: $\\begin{aligned}&{\\text{minimize}_{\\bf (L, S)}} \\ {\\Vert {\\bf L}\\Vert _* + \\lambda \\Vert {\\bf S}\\Vert _1} \\\\&{\\text{subject to}} \\ {\\bf M} = {\\bf L} + {\\bf S},\\end{aligned}$ where ${\\Vert \\mbox{\\bf B}\\Vert _*} \\triangleq \\sum _i\\sigma _i (\\mbox{\\bf B}) $ is the nuclear norm of any matrix $\\mbox{\\bf B}$ , ${\\Vert \\mbox{\\bf B}\\Vert _1} \\triangleq \\sum _{ij} |\\mbox{\\bf B}_{ij}|$ is the $\\ell _{1}$ -norm of $\\mbox{\\bf B}$ , and $\\lambda >0$ is a tuning parameter.",
"One efficient method to solve (REF ) is the method of augmented Lagrange multipliers (ALM) .",
"The ALM method yields the optimal solution, however its bottleneck is computing the costly SVD at each iteration to approximate the low-rank component $\\bf L$ of $\\bf M$ , .",
"To address this issue and to speed up the convergence of the ALM method, the work in proposes a few techniques including predicting the principal singular space dimension, a continuation technique , and a truncated SVD by using PROPACK package .",
"The modified algorithm substantially improves the convergence speed, however the truncated SVD employed uses the Lanczos algorithm that (i) is unstable, and (ii) due to the limited data reuse in its operations, has very poor performance on modern architectures , , , .",
"To address this issue, by considering the original objective function proposed in , , , we apply CoR-UTV as a surrogate to the truncated SVD to solve the robust PCA problem.",
"We adopt the continuation technique , , which increases $\\mu $ in each iteration.",
"The proposed ALM-CoRUTV method is given in Alg.",
".",
"Robust PCA solved by ALM-CoRUTV [1] Matrix ${\\bf M}, \\lambda , \\mu , {\\bf Y}_0 = {\\bf S}_0 = 0, j=0$ .",
"Low-rank plus sparse matrix.",
"the algorithm does not converge Compute ${\\bf L}_{j+1} = \\mathcal {C}_{\\mu _j^{-1}}({\\bf M} - {\\bf S}_j +\\mu _j^{-1} {\\bf Y}_j)$ ; Compute ${\\bf S}_{j+1} = \\mathcal {S}_{\\lambda \\mu _j^{-1}}({\\bf M} - {\\bf L}_{j+1} +\\mu _j^{-1} {\\bf Y})$ ; Compute ${\\bf Y}_{j+1} = {\\bf Y}_j +\\mu _j({\\bf M} - {\\bf L}_{j+1}- {\\bf S}_{j+1})$ ; Update $\\mu _{j+1} = \\text{max}(\\rho \\mu _j, {\\bar{\\mu }})$ ; $\\bf L^*$ and $\\bf S^*$ .",
"In Alg.",
", for any matrix $\\bf B$ with a CoR-UTV decomposition described in Section , $\\mathcal {C}_\\delta ({\\bf B})$ refers to a CoR-UTV thresholding operator defined as: $\\mathcal {C}_\\delta ({\\bf B})={\\bf U}(:,1:r){\\bf T}(1:r,:){\\bf V}^T,$ where $r$ is the number of diagonals of $\\bf T$ greater than $\\delta $ , $\\mathcal {D}_\\delta ({\\bf B})$ refers to a singular value thresholding operator defined as $\\mathcal {D}_\\delta ({\\bf B}) ={\\bf U}_\\text{B}\\mathcal {S}_\\delta ({\\bf \\Sigma }_\\text{B}){\\bf V}_\\text{B}^T$ , where $\\mathcal {S}_\\delta (x) ={\\text{sgn}(x)\\text{max}}(|x| - \\delta , 0)$ is a shrinkage operator , $\\lambda $ , $\\mu _0$ , ${\\bar{\\mu }}$ , $\\rho $ , ${\\bf Y}_0$ , and ${\\bf S}_0$ are initial values.",
"The main operation of ALM-CoRUTV is computing CoR-UTV in each iteration, which is efficient in terms of flops, $O(mnk)$ , and can be computed with minimum communication costs."
],
[
"Numerical Experiments",
"In this section, we present simulations that evaluate the performance of CoR-UTV for approximating a low-rank input matrix.",
"We show that CoR-UTV provides highly accurate singular values and low-rank approximations, and compare CoR-UTV against competing algorithms from the literature.",
"We also employ CoR-UTV for solving the robust PCA problem."
],
[
"Rank-Revealing Property & Singular Values Estimation",
"We first show that CoR-UTV (i) is rank revealer, i.e., the gap in the singular value spectrum of the matrix is revealed, and (ii) provides highly accurate singular values.",
"For the randomized algorithms considered, namely CoR-UTV, TSR-SVD, and SOR-SVD, the results presented are averaged over 20 trials.",
"Each trial was run with the same input matrix with an independent draw of the test matrix.",
"Due to space constraints, we only consider one class of low-rank matrices, and for simplicity we focus on a square matrix.",
"We construct a noisy rank-$k$ matrix ${\\bf A}$ of order $10^3$ generated as ${\\bf A} ={\\bf U\\Sigma V}^T + 0.1\\sigma _k{\\bf E}$ , where ${\\bf U}$ and ${\\bf V}$ are random orthonormal matrices, ${\\bf \\Sigma }$ is diagonal containing the singular values $\\sigma _i$ s that decrease linearly from 1 to $10^{-9}$ , $\\sigma _{k+1}=...=\\sigma _{10^3}=0$ , and ${\\bf E}$ is a normalized Gaussian matrix.",
"We set $k=20$ .",
"We compare the singular values of the matrix computed by CoR-UTV against those of competing methods such as the SVD , QRCP , UTV and TSR-SVD .",
"For CoR-UTV and TSR-SVD, we arbitrarily set the sample size parameter to $\\ell =2k$ .",
"Both algorithms require the same number of passes over $\\bf A$ , either two or $2q+2$ when the power method is used, to perform a factorization.",
"Figure: Comparison of singular values.",
"q=0q=0 (left), and q=2q=2 (right).The results are shown in Fig.",
"REF .",
"It is observed that (i) CoR-UTV strongly reveals the numerical rank $k$ , (ii) with no power iterations ($q=0$ ), CoR-UTV provides very good approximations to singular values and outperforms TSR-SVD in approximating both leading and trailing singular values, (iii) with $q=2$ , CoR-UTV delivers singular values as accurate as the optimal SVD, (iv) QRCP only suggests the gap in the singular spectrum, and gives a fuzzy approximation to singular values of the matrix."
],
[
"Low-Rank Approximation",
"We now compare the low-rank approximation constructed by our method against those of the SVD, QRCP, TSR-SVD, and SOR-SVD .",
"We construct a rank-$k$ approximation ${\\hat{\\bf A}}_\\text{out}$ to ${\\bf A}$ by varying the sample size parameter $\\ell $ with the rank fixed, and calculate the error: $e_k = \\Vert {\\bf A} - \\hat{\\bf A}_{\\text{out}}\\Vert _F.$ The results are shown in Fig.",
"REF .",
"It is observed that (i) when $q=0$ , CoR-UTV and SOR-SVD show similar performances, while TSR-SVD shows the worst performance, (ii) when $q=2$ , the errors resulting from CoR-UTV show no loss of accuracy compared to the optimal SVD.",
"In this case, QRCP has the poorest performance.",
"Figure: Comparison of low-rank approximationerrors.",
"q=0q=0 (left), and q=2q=2 (right)."
],
[
"Robust Principal Component Analysis",
"Here, we examine the efficiency and efficacy of ALM-CoRUTV in Alg.",
"for recovering the low-rank and sparse components of data.",
"We compare the results obtained with those of the efficient inexact ALM method by , called InexactALM hereafter.",
"Robust PCA represents an input low-rank matrix ${\\bf M} \\in \\mathbb {R}^{m \\times n}$ , whose a fraction of entries being corrupted, as a linear superposition of a low-rank matrix ${\\bf L}$ and a sparse matrix of outliers ${\\bf S}$ such as ${\\bf M=L+S}$ , by solving the following convex program: $\\begin{aligned}&{\\text{minimize}_{\\bf (L, S)}} \\ {\\Vert {\\bf L}\\Vert _* + \\lambda \\Vert {\\bf S}\\Vert _1} \\\\&{\\text{subject to}} \\ {\\bf M} = {\\bf L} + {\\bf S},\\end{aligned}$ where ${\\Vert \\mbox{\\bf B}\\Vert _*} \\triangleq \\sum _i\\sigma _i (\\mbox{\\bf B}) $ is the nuclear norm of any matrix $\\mbox{\\bf B}$ , ${\\Vert \\mbox{\\bf B}\\Vert _1} \\triangleq \\sum _{ij} |\\mbox{\\bf B}_{ij}|$ is the $\\ell _{1}$ -norm of $\\mbox{\\bf B}$ , and $\\lambda >0$ is a tuning parameter , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .",
"The matrix ${\\bf L}$ is generated as ${\\bf L}={\\bf U}{\\bf V}^T$ , where ${\\bf U}$ , ${\\bf V} \\in \\mathbb {R}^{n \\times k}$ have standard Gaussian distributed entries.",
"The error matrix ${\\bf S}$ has $s$ non-zero entries independently drawn from the set $\\lbrace $ -80, 80$\\rbrace $ .",
"We apply the ALM-CoRUTV and InexactALM algorithms to $\\bf M$ to recover ${\\bf L}$ and ${\\bf S}$ .",
"The numerical results are summarized in Table REF , where the rank of $\\bf L$ $r({\\bf L})=0.05\\times n$ and $s = \\Vert {\\bf S}\\Vert _0=0.05\\times n^2$ .",
"In our experiments, we adopt the initial values suggested in .",
"The algorithms are terminated when $ {\\Vert {\\bf M}-{\\bf L}^{out}-{\\bf S}^{out}\\Vert _F}< 10^{-5}{\\Vert {\\bf M}\\Vert _F}$ is satisfied, where $({\\bf L}^{out}, {\\bf S}^{out})$ is the pair of output of either algorithm.",
"In the Table, $Time(s)$ refers to the runtime in seconds, $Iter.$ refers to the number of iterations, and $\\zeta ={\\Vert {\\bf M}-{\\bf L}^{out}-{\\bf S}^{out}\\Vert _F}/{{\\Vert {\\bf M}\\Vert _F}}$ refers to the relative error.",
"Table: Conclusions"
]
] | 1906.04572 |
[
[
"Self-Supervised Learning for Contextualized Extractive Summarization"
],
[
"Abstract Existing models for extractive summarization are usually trained from scratch with a cross-entropy loss, which does not explicitly capture the global context at the document level.",
"In this paper, we aim to improve this task by introducing three auxiliary pre-training tasks that learn to capture the document-level context in a self-supervised fashion.",
"Experiments on the widely-used CNN/DM dataset validate the effectiveness of the proposed auxiliary tasks.",
"Furthermore, we show that after pre-training, a clean model with simple building blocks is able to outperform previous state-of-the-art that are carefully designed."
],
[
"Introduction",
"Extractive summarization aims at shortening the original article while retaining the key information through the way of selection sentences from the original articles.",
"This paradigm has been proven effective by many previous systems [4], [16], [15], [3].",
"In order to decide whether to choose a particular sentence, the system should have a global view of the document context, e.g., the subject and structure of the document.",
"However, previous works [17], [2], [31], [30] usually directly build an end-to-end training system to learn to choose sentences without explicitly modeling the document context, counting on that the system can automatically learn the document-level context.",
"Figure: An example for the Mask pre-training task.",
"A sentence is masked in the original paragraph, and the model is required to predicted the missing sentence from the candidate sentences.We argue that it is hard for these end-to-end systems to learn to leverage the document context from scratch due to the challenges of this task, and a well pre-trained embedding model that incorporates document context should help on this task.",
"In recent years, extensive works [22], [19], [13], [23], [7], [26], [5], [14], [21] have been done in learning the word or sentence representations, but most of them only use a sentence or a few sentences when learning the representation, and the document context can hardly be included in the representation.",
"Hence, we introduce new pre-training methods that take the whole document into consideration to learn the contextualized sentence representation with self-supervision.",
"Self-supervised learning [24], [8], [1], [28] is a newly emerged paradigm, which aims to learn from the intrinsic structure of the raw data.",
"The general framework is to construct training signals directly from the structured raw data, and use it to train the model.",
"The structure information learned through the process can then be easily transformed and benefit other tasks.",
"Thus self-supervised learning has been widely applied in structured data like text [20], [6], [23], [7], [29] and images [8], [1], [28], [11].",
"Since documents are well organized and structured, it is intuitive to employ the power of self-supervised learning to learn the intrinsic structure of the document and model the document-level context for the summarization task.",
"In this paper, we propose three self-supervised tasks (Mask, Replace and Switch), where the model is required to learn the document-level structure and context.",
"The knowledge learned about the document during the pre-training process will be transferred and benefit on the summarization task.",
"Particularly, The Mask task randomly masks some sentences and predicts the missing sentence from a candidate pool; The Replace task randomly replaces some sentences with sentences from other documents and predicts if a sentence is replaced.",
"The Switch task switches some sentences within the same document and predicts if a sentence is switched.",
"An illustrating example is shown in Figure REF , where the model is required to take into account the document context in order to predict the missing sentence.",
"To verify the effectiveness of the proposed methods, we conduct experiments on the CNN/DM dataset [9], [18] based on a hierarchical model.",
"We demonstrate that all of the three pre-training tasks perform better and converge faster than the basic model, one of which even outperforms the state-of-the-art extractive method NeuSum [31].",
"The contributions of this work include: $\\bullet $ To the best of our knowledge, we are the first to consider using the whole document to learn contextualized sentence representations with self-supervision and without any human annotations.",
"$\\bullet $ We introduce and experiment with various self-supervised approaches for extractive summarization, one of which achieves the new state-of-the-art results with a basic hierarchical model.",
"$\\bullet $ Benefiting from the self-supervised pre-training, the summarization model is more sample efficient and converges much faster than those trained from scratch.",
"Figure: The structure of the Basic Model.",
"We use LSTM and self-attention module to encode the sentence and document respectively.",
"X i X_i represent the word embedding for sentence ii.",
"S i S_i and D i D_i represent the independent and document involved sentence embedding for sentence ii respectively."
],
[
"Basic Model",
"As shown in Figure REF , our basic model for extractive summarization is mainly composed of two parts: a sentence encoder and a document-level self-attention module.",
"The sentence encoder is a bidirectional LSTM [10], which encodes each individual sentence $X_i$ (a sequence of words) and whose output vector at the last step is viewed as the sentence representation $S_i$ .",
"Given the representations of all the sentences, a self-attention module [27] is employed to incorporate document-level context and learn the contextualized sentence representation $D_i$ for each sentence.We leave the combination of different architectures such as replacing the self-attention module with LSTM for future work.",
"Finally, a linear layer is applied to predict whether to choose the sentence to form the summary."
],
[
"Self-supervised Pre-training Methods",
"In this section, we will describe three self-supervised pre-training approaches.",
"Through solving each pre-training task, the model is expected to learn the document-level contextualized sentence embedding model from the raw documents, which will then be used to solve the downstream summarization task.",
"Note that we are only pretraining the sentence encoder and document-level self-attention module of the basic model for extractive summarization."
],
[
"Mask",
"Similar to the task of predicting missing word, the Mask task is to predict the masked sentence from a candidate pool.",
"Specifically, we first mask some sentences within a document with the probability $P_m$ and put these masked sentences ($\\bf {x^m_1}, \\bf {x^m_2}, \\cdots , \\bf {x^m_t}$ ) into a candidate pool $T^m$ .",
"The model is required to predict the correct sentence from the pool for each masked position $i$ .",
"We replace the sentence in the masked position $i$ with a special token $\\langle $ unk$\\rangle $ and compute its document contextualized sentence embedding $D_i$ .",
"We use the same sentence encoder in the basic model to obtain the sentence embedding $S^m$ for these candidate sentences in $T^m$ .",
"We score each candidate sentence $j$ in $T^m$ by using the cosine similarity: $\\Theta (i, j) = \\cos (D_{i}, S^m_{j})$ To train the model, we adopt a ranking loss to maximize the margin between the gold sentence and other sentences: $\\ell _{m} = \\text{max}\\lbrace 0, \\gamma - \\Theta (i, j) + \\Theta (i, k)\\rbrace $ where $\\gamma $ is a tuned hyper-parameter, $j$ points to the gold sentence in $T^m$ for the masked position $i$ , and $k$ points to another non-target sentence in $T^m$ ."
],
[
"Replace",
"The Replace task is to randomly replace some sentences (with probability $P_r$ ) in the document with sentences from other documents, and then predict if a sentence is replaced.",
"Particularly, we use sentences from $10,000$ randomly chosen documents to form a candidate pool $T^r$ .",
"Each sentence in the document will be replaced with probability $P_r$ by a random sentence in $T^r$ .",
"Let $C_r$ be the set of positions where sentences are replaced.",
"We use a linear layer $f_r$ to predict if the sentence is replaced based on the document embedding $D$ , and minimize the MSE loss: $\\ell _r = \\text{MSE}(f_r(D_i),~y^r_i)$ where $y^r_i=1$ if $i\\in C_r$ (i.e., the sentence in position $i$ has been replaced), otherwise $y^r_i=0$ ."
],
[
"Switch",
"The Switch task is similar to the Replace task.",
"Instead of filling these selected sentences with sentences out of the document, this task chooses to use sentences within the same document by switching these selected sentences, i.e., each selected sentence will be put in another position within the same document.",
"Let $C_s$ be the set of positions where the sentences are switched.",
"Similarly, we use a linear layer $f_s$ to predict if a sentence is switched and minimize the MSE loss: $\\ell _s = \\text{MSE}(f_s(D_i),~y^s_i)$ where $y^s_i=1$ if $i\\in C_s$ , otherwise $y^s_i=0$ ."
],
[
"Experiment",
"To show the effectiveness of the pre-training method (Mask, Replace and Switch), we conduct experiments on the commonly used dataset CNN/DM [9], [18], and compare them with a popular baseline Lead3 [25], which selects first three sentences as the summary, and the state-of-the-art extractive summarization method NeuSum [31], which jointly scores and selects sentences using pointer network.",
"Figure: This figure shows the Rouge-2 score for each pre-training method and the basic model on the development set during the training process.",
"We put the result for Rouge-1 and Rouge-L score in Appendix"
],
[
"Model and training details",
"We use the rule-based system from [31] to label sentences in a document, e.g., sentences to be extracted will be labeled as 1.",
"Rouge scoreWe use PyRouge https://pypi.org/project/pyrouge/ to compute the Rouge score.",
"[12] is used to evaluate the performance of the model, and we report Rouge-1, Rouge-2, and Rouge-L as in prior work.",
"We use the pre-trained glove embedding [22] with 100 dimensions to initialize the word embedding.",
"A one-layer bidirectional LSTM [10] is used as the sentence encoder, and the size of hidden state is 200.",
"A 5-layer Transformer encoder [27] with 4 heads is used as the document-level self-attention module.",
"A linear classification layer is used to predict whether to choose the sentence.",
"The training process consists of two phrases.",
"First, we use the pre-training task to pre-train the basic model using the raw article from the CNN/DM dataset without labels.",
"Second, we fine-tune the pre-trained model for the extractive summarization task using the sentence labels.",
"The learning rate is set as $0.0001$ in the pre-training phase and $0.00001$ in the fine-tune phase.",
"We train each pre-training task until it is converged or the number of training epochs reaches the upper bound 30.",
"We set the probability to mask, replace or switch sentences as $0.25$ .",
"Table: The Rouge scores for the basic model, baselines, pre-training methods, and analytic experiments.",
"All of our Rouge scores have a 95%95\\% confidence interval of at most ±0.25\\pm 0.25 as reported by the official ROUGE script.",
"The best result is marked in bold, and those that are not significantly worse than the best are marked with * ^*."
],
[
"Results",
"We show the Rouge score on the development set during the training process in Figure REF , and present the best Rouge score for each method in Table REF .",
"All pre-training methods improve the performance compared with the Basic model.",
"Especially, Switch method achieves the best result on all the three evaluations compared with other pre-training methods, and is even better than the state-of-the-art extractive model NeuSumWe use code from https://github.com/magic282/NeuSum to train the model, and evaluate it using our evaluation script.",
"Results using their script (only include Rouge-1 and Rouge-2) is put in Appendix REF ..",
"In the terms of convergence, the Mask, Replace and Switch task takes $21, 24, 17$ epochs in the training phase respectively, and $18, 13, 9$ epochs to achieve the best performance in the fine-tune phase.",
"The basic model takes 24 epochs to obtain the best result.",
"From Figure REF , we can see that the Switch task converges much faster than the basic model.",
"Even adding on the epochs taken in the pre-training phase, Switch method (26 epochs) takes roughly the same time as the Basic model (24 epochs) to achieve the best performance."
],
[
"Reuse only the sentence encoder",
"Our basic model has mainly two components: a sentence encoder and a document-level self-attention module.",
"The sentence encoder focuses on each sentence, while document-level self-attention module incorporates more document information.",
"To investigate the role of the document-level self-attention module, we only reuse the sentence encoder of the pre-train model, and randomly initialize the document-level self-attention module.",
"The results is shown in Table REF as SentEnc.",
"We can see that using the whole pre-training model (Switch $0.25$ ) can achieve better performance, which indicates the model learn some useful document-level information from the pre-training task.",
"We notice that only using the sentence encoder also get some improvement over the basic model, which means that the pre-training task may also help to learn the independent sentence representation."
],
[
"On the sensitivity of hyper-parameter",
"In this part, we investigate the sensitivity of the model to the important hyper-parameter $P_w$ , i.e., the probability to switch sentences.",
"In the previous experiment, we switch sentences with probability $0.25$ .",
"We further try the probability of $0.15$ and $0.35$ , and show the results in Table REF as Switch $0.15$ and Switch $0.35$ .",
"We can see Switch $0.15$ achieve basically the same result as Switch $0.25$ , and Switch $0.35$ is slightly worse.",
"So the model is not so sensitive to the hyper-parameter of the probability to switch sentences, and probability between $0.15$ and $0.25$ should be able to work well."
],
[
"Conclusion",
"In this paper, we propose three self-supervised tasks to force the model to learn about the document context, which will benefit the summarization task.",
"Experiments on the CNN/DM verify that through the way of pre-training on our proposed tasks, the model can perform better and converge faster when learning on the summarization task.",
"Especially, through the Switch pre-training task, the model even outperforms the state-of-the-art method NeuSum [31].",
"Further analytic experiments show that the document context learned by the document-level self-attention module will benefit the model in summarization task, and the model is not so sensitive to the hyper-parameter of the probability to switch sentences."
],
[
"Rouge-1 and Rouge-L results",
"The Rouge-1 and Rouge-L results are shown in Figure REF , from which we can see that the Switch method achieves the best performance."
]
] | 1906.04466 |
[
[
"Near horizon symmetries, emergence of Goldstone modes and thermality"
],
[
"Abstract For a long time it is believed that black hole horizon are thermal and quantum mechanical in nature.",
"The microscopic origin of this thermality is the main question behind our present investigation, which reveals possible importance of near horizon symmetry.",
"It is this symmetry which is assumed to be spontaneously broken by the background spacetime, generates the associated Goldstone modes.",
"In this paper we construct a suitable classical action for those Goldstone modes, and show that all the momentum modes experience nearly the same inverted harmonic potential, leading to an instability.",
"Thanks to the recent conjectures on the chaos and thermal quantum system, particularly in the context of an inverted harmonic oscillator system.",
"Going into the quantum regime, the system of large number of Goldstone modes with the aforementioned instability is shown to be inherently thermal.",
"Interestingly the temperature of the system also turns out to be proportional to that of the well known horizon temperature.",
"Therefore, we hope our present study can illuminate an intimate connection between the horizon symmetries and the associated Goldstone modes as a possible mechanism of the microscopic origin of the horizon thermality."
],
[
"Introduction: BMS symmetry and Goldstone modes",
"Black hole is a fascinating object in Einstein's theory of gravity.",
"Even though it exists for a long time, we still do not understand it fully.",
"One of the important properties of it is its very nature as a thermal object.",
"Over the last few years various efforts being made to understand this thermality, transpired to the fact that symmetries, their breaking and associated Goldstone modes may play fundamental role in understanding this subject.",
"In this paper we try to establish this connection between symmetry and thermality of the black holes with some encouraging results.",
"Symmetry breaking phenomena is ubiquitous in nature.",
"Across a large span of physical problems in particle physics, cosmology and condensed matter physics, it is not only the symmetry, but it's spontaneous breaking also plays very crucial role in understanding the low energy properties.",
"Symmetries in nature are broadly classified into two categories.",
"A symmetry which acts globally on the physical fields are called global symmetry.",
"Most importantly for each global continuous symmetry there exists associated conserved charge which encodes important properties of the system under consideration.",
"Another class of symmetry which acts locally on the fields, generally known as gauge symmetry, makes the description of the system redundant.",
"Unlike global continuous symmetries, gauge symmetry does not have associated non-trivial conserved charge [1][2].",
"However, as compared to the global symmetry the most striking property of a global continuous symmetry lies in its spontaneous breaking phenomena which plays very important role in understanding the low energy behaviour of the system under consideration.",
"If a global continuous symmetry of a system breaks spontaneously, associated Goldstone boson mode emerges, whose dynamics will characterise the underlying states and their properties of the system [3].",
"On the other hand breaking of gauge symmetry is inherently inconsistent with the theory under consideration.",
"In the present paper, we will be trying to understand the dynamics of the Goldstone boson modes associated with a special class of global symmetry arising at the boundary of a spacetime with nontrivial gravitational background.",
"The generic underlying symmetry of a gravitational theory is spacetime diffeomophism which is a set of local general coordinate transformation.",
"Therefore, diffeomrophism can be thought of as a gauge symmetry of the gravitational theory.",
"However, it is well known that a gauge symmetry in the bulk acts as a non-trivial global symmetry at the boundary.",
"Therefore, even if gravitational theory can be formulated as a gauge theory, theory of Goldstone modes can still be applicable and information about the microscopic gravitational states may be extracted from the boundary global symmetry.",
"Symmetries near the boundary of a spacetime has been the subject of interest for a long time [4]-[14].",
"One of the popular and important examples of such a bulk-boundary correspondence is the well known global Bondi-Metzner-Sachs (BMS) group [4], [5], [6] of transformation.",
"BMS group is an infinite dimensional global symmetry transformation which acts non-trivially on the asymptotic null boundary of an asymptotically flat spacetime.",
"The original study [5] was done on the asymptotic null boundary of an asymptotically flat spacetime.",
"Subsequently the analysis on another null boundary namely the event horizon of a black hole spacetime has been studied [15]-[17].",
"The basic idea is to find out the generators which preserve the boundary structure of a spacetime of our interest under diffeomorphism.",
"Usually, one encounters two types of generators, one is super-translation associated with time reparametrization and other one is super-rotation associated with angular rotation.",
"Over the years it is observed that these generators can play a crucial role in understanding the horizon entropy of a black hole [18]-[28].",
"Since then there is a constant effort to understand these symmetries and its role to uncover the microscopic structure of the horizon thermodynamics.",
"Although there is no significant progress till now, but the motivation is still there which led to some of the recent attempts [20]-[34].",
"Moreover in the series of remarkable papers [35]-[41] a deep connection between ward identities associated with the aforementioned BMS supertranslation symmetries and Weinberg's soft graviton theorem has been unraveled.",
"It is argued that the soft photons are the Goldstone boson modes arising due to spontaneous breaking of the asymptotic symmetries.",
"Hence an equivalence has been established between Wienberg's soft photon theorem and BMS symmetries [40].",
"More interestingly the same BMS transformation is shown to be closely related with the gravitational memory effect [37][42].",
"Subsequently same effect has been shown to arise near the black hole horizon as well [43].",
"In the present paper our focus will be on the Killing horizon specifically in Rindler and Schwarzschild background.",
"Those horizons behave like another null boundary where bulk diffeomorphism acts non-trivially in terms of BMS-like global symmetry [44] [45].",
"Associated with those global symmetry on the horizon, black hole microstates have been conjectured to be played by the soft hairs which are essentially the Goldstone boson modes associated with the symmetry broken by the macroscopic black hole state [46]-[53].",
"Although the appearance of Goldstone modes in the context of BMS symmetry exits, its dynamical behavior has not been studied in a concrete way.",
"It is believed that the dynamics of those modes should play crucial role in understanding the microscopic nature of the black holes.",
"Having set this motivation, in the present paper we will study the dynamics of those Goldstone modes following the standard procedure.",
"In order to clarify and better understand the methodology of our calculation, let us consider the emergence of Goldstone boson mode for a well known U(1) invariant complex scalar field theory with the following Lagrangiain, ${\\cal L} = 1/2 (\\partial _{\\mu } \\phi \\partial ^{\\mu } \\phi ^{\\dagger }) - V(\\phi \\phi ^{\\dagger })$ .",
"The background solution such as $\\phi _0 = c$ naturally breaks U(1) symmetry which transforms the vacuum as $\\phi _0^{\\prime } \\rightarrow e^{i \\pi (x)} \\phi _0 = c + i c \\pi (x) .$ Now we can identify the $\\pi (x)$ as Goldstone boson field, and calculate the Lagrangian as follows ${\\cal L}_{\\pi } &=& \\frac{1}{2} (\\partial _{\\mu } \\phi _0^{\\prime } \\partial ^{\\mu } \\phi _0^{\\prime \\dagger }) - V(\\phi _0^{\\prime }\\phi _0^{\\prime \\dagger }) \\nonumber \\\\&=& \\partial _{\\mu } (c + i c \\pi (x)) \\partial ^{\\mu } (c-i c \\pi (x)) - V(c + i c \\pi (x)) \\nonumber \\\\&=& \\frac{c^2}{2} \\partial _{\\mu } \\pi (x)) \\partial ^{\\mu } \\pi (x) + \\cdots .$ The last expression should be the leading order Goldstone boson Lagrangian associated with the broken U(1) symmetry (more detail can be found in [54]).",
"Throughout our following discussions, we will use this analogy to understand the dynamics of the Goldstone mode in the gravity sector.",
"In the first half of our paper we consider Rindler space time with flat spatial section.",
"In the later half we consider the asymptotically flat Schwarzchild black hole.",
"Once we have a gravitational background, we first identify the global symmetry associated with the null boundary surface [15]-[17] [55] imposing the appropriate boundary conditions.",
"Boundary conditions are such that the near horizon form of the metric remains invariant after the symmetry transformation.",
"However, macroscopic quantities such as mass, charge and angular momentum characterizing the physical states of a black hole under consideration will change under those symmetry transformation.",
"Such phenomena can be understood as a spontaneous breaking of the aforementioned boundary global symmetry by the black hole background.",
"We, therefore, expects the associated dynamical Goldstone boson modes.",
"As mentioned earlier in this paper we will study the dynamics of those Goldstone boson modes which may shed some light on the possible microscopic states of the black holes."
],
[
"Rindler Background",
"In this section we will consider the simplest background and try to understand the symmetry breaking phenomena as described in the introduction.",
"The Rindler metric, in the Gaussian null coordinate is expressed as $ds^2 = -2 r \\alpha dv^2 + 2 dv dr + \\delta _{AB} dx^A dx^B~.$ The Rindler horizon is located at $r=0$ .",
"$\\alpha $ is the acceleration parameter which characterizes the macroscopic state of the background spacetime.",
"Symmetry properties of the horizon can be extracted from the following fall off and gauge conditions, $£_\\zeta g_{rr}= 0, \\ \\ \\ £_\\zeta g_{vr}=0, \\ \\ \\ £_\\zeta g_{Ar}=0~;\\\\£_\\zeta g_{vv} \\approx \\mathcal {O}(r); \\ \\ \\ £_\\zeta g_{vA} \\approx \\mathcal {O}(r); \\ \\ \\ £_\\zeta g_{AB} \\approx \\mathcal {O}(r)~.$ Here, $£_\\zeta $ corresponds to the Lie variation for the diffeomorphism $x^a\\rightarrow x^a+\\zeta ^a$ .",
"The above conditions are satisfied for the following form of the diffeomorphism vector, $\\zeta ^a \\partial _a=&& F(v,y,z) \\partial _v -r \\partial _v F(v,y,z) \\partial _r\\nonumber \\\\&&-r \\partial ^{A} F(v,y,z) \\partial _A~.$ Note that in this case we have only one diffemorphism parameter $F$ which characterizes the symmetry of the Rindler horizon.",
"Since for constant $F$ , it essentially gives the time translation, the general form of this time diffemorphsim which acts non-trivially on the $r=0$ hypersurface, is called supertranslation.",
"For details of this analysis, we refer to [55], [16], [17].",
"We shall see below that under the diffeomorphism (REF ) some of the metric coefficients will transform.",
"This can be thought as similar to the transformation (REF ) which breaks the $U(1)$ symmetry.",
"The corrections in the metric coefficients are determined by the supertranslation parameter $F$ .",
"Consequently the macroscopic parameters of the original metric will be modified and therefore with the analogy of $U(1)$ symmetry breaking, this can be regarded as the breaking of horizon boundary symmetry.",
"Hence one can promote the parameter $F$ as Goldstone mode.",
"Analogous to the U(1) Goldstone mode, here also we shall propose the underlying theory of F to be determined by the Einstein-Hilbert action.",
"We shall find the leading order correction to this action due this aforesaid diffeomorphism which will, as for $U(1)$ symmetry breaking Goldstone, ultimately determine the dynamics of the “Goldstone modes” ($F$ ) in our present context.",
"In order to study the dynamics, let us first find the modified metric which are consistent with the aforementioned gauge (REF ) and fall-off () conditions.",
"Important point to remember that the Lie variation of the metric component in our analysis is defined up to the linear order in $\\zeta ^a$ and hence we express the form of $\\zeta ^a$ (REF ) valid up to linear order in $F$ .",
"Under this diffeomorphism vector (REF ), the modified metric takes the following form: $g^{\\prime }_{ab} &=& \\Big [ g^{(0)}_{ab} + £_\\zeta g^0_{ab} \\Big ] dx^a dx^b \\nonumber \\\\&=& -2 r \\alpha dv^2 + 2 dv dr + \\delta _{AB} dx^A dx^B \\nonumber \\\\&+& \\Big [-2r\\Big (\\alpha \\partial _{v} F + \\partial ^2_v F\\Big )\\Big ] dv^2 \\nonumber \\\\&+& \\Big [-2r\\Big (\\alpha \\ \\partial _{A}F + \\partial _{A} \\partial _v F\\Big )\\Big ] dv dx^A\\nonumber \\\\&+&\\big [- 2 r \\partial _A \\partial _B F \\Big ] dx^A dx^B~.$ In the above, $g^{(0)}_{ab}$ is the original unperturbed metric (REF ), whereas all linear in $F$ terms are incorporated in $h_{ab}$ .",
"Under the following supertranslation symmetry transformation, $v^{\\prime }= v + F(v,x^A)~,~x^{\\prime A} = x^A - r\\partial ^A F(v,x^A) ,$ we can clearly see the macroscopic state parameter $\\alpha $ of the original Rindler background transforms into $\\alpha \\rightarrow \\alpha +\\Big (\\alpha \\partial _{v} F + \\partial ^2_v F\\Big ) .$ Therefore, this change of macroscopic state by the symmetry transformation can be understood as a breaking of the boundary symmetry of the Rindler spacetime [51].",
"As $F$ is the parameter associated with the broken symmetry generator, following the standard procedure of Goldstone mode analysis, we promote $F$ as a Goldstone boson field.",
"However, all the measure will be done with respect to the usual unprimed coordinate, and dynamics of the mode is defined on the $r=0$ hyper-surface.",
"Since $\\alpha $ appears as a Lagrange multiplier in the Hamiltonian formulation, one usually chooses the gauge where variation of alpha is zero everywhere [51], [52].",
"However strictly speaking this is not a generic choice.",
"It is sufficient to set the variation of $\\alpha $ to be zero only at the boundary for consistency, $\\delta \\alpha (-\\infty ,x^A) = \\lim _{v\\rightarrow -\\infty } \\Big (\\alpha \\partial _{v} F + \\partial ^2_v F\\Big ) =0, $ where horizon is located at $v\\rightarrow -\\infty $ .",
"One of the obvious choices to satisfy the above condition is to set the total variation $\\delta \\alpha $ to be zero everywhere [56].",
"This naturally set the boundary condition at the horizon and furthermore makes the field $F$ non-dynamical.",
"Therefore, we believe this restrictive condition does not capture the full potential of the Goldstone modes.",
"Our goal of this paper is to go beyond and understand the dynamics of this Goldstone modes, which could be the potential candidate for the underlying degrees of freedom of the black hole.",
"Therefore, we first construct an appropriate Lagrangian of this mode and finally at the solution level we set the boundary condition such that Eq.",
"(REF ) is automatically satisfied at the horizon.",
"Important to note that if one allows the fluctuation of $\\alpha $ even at the boundary, one needs to take care of the appropriate boundary terms (e.g.",
"see [57][58]), which will be discussed in a separate paper."
],
[
"Dynamical equation for $F$",
"As we have already pointed out, in order to study the dynamics for $F$ we propose the Lagrangian $\\mathcal {L}_{F}$ associated with the newly perturbed metric (REF ) near the $r=0$ surface: $\\mathcal {L}_{F} = \\sqrt{-g^{\\prime }} R^{\\prime } ~.$ Here $R^{\\prime }$ is the Ricci scalar calculated for the newly constructed metric $g^{\\prime }_{ab}$ (REF ) and $g^{\\prime }$ is the corresponding determinant.",
"To study the dynamics of the Goldstone mode associated with the horizon symmetry, first we will compute the above Lagrangian eq.",
"(REF ) at arbitrary $r$ value in the bulk spacetime and then we take the limit $r\\rightarrow 0$ .",
"This procedure is similar to the stretched horizon approached in black hole thermodynamics (for example see the discussion in section 4 of [21]).",
"In this approach, if we are interested to find any quantity on a particular surface (say $r=0$ ), one usually first calculate the same just away from this surface (say $r=\\epsilon $ , with $\\epsilon $ is very small).",
"After that the obtained value is derived by taking the limit $\\epsilon \\rightarrow 0$ .",
"Now we are in a position to expand our Lagrangian (REF ) in terms of the transformed metric (REF ).",
"If the background metric components are $g_{ab}=g_{ab}^{(0)}+h_{ab}$ , with $h_{ab}$ as a small fluctuation, in general the Taylor series expansion of the Lagrangian around background metric $g_{ab}=g_{ab}^{(0)}$ can be written as $\\mathcal {L_{F}} = \\mathcal {L_{F}}(g_{ab}^{(0)}) + h_{ab}\\Big (\\frac{\\delta \\mathcal {L_{F}}}{\\delta g_{ab}}\\Big )^{(0)} + h_{ab}h_{cd}\\Big (\\frac{\\delta ^2\\mathcal {L_{F}}}{\\delta g_{ab} \\delta g_{cd}}\\Big )^{(0)}+\\dots $ The first term of the above equation obviously does not contribute to the dynamics.",
"Given the background metric to be a solution of equation of motion, the second term vanishes as it is essentially proportional to the Einstein's equation of motion.",
"Third term introduces the quadratic form for the Goldstone field $F$ .",
"For our purpose of the present paper, we will restrict only to Lagrangian for the Goldstone mode which is at the quadratic order.",
"All the higher order in $F$ -terms we left for our future discussions.",
"The final form of the Lagrangian (REF ) after taking the near Horizon limit comes out as: ${\\mathcal {L}_{F}} &=& \\lim _{r \\rightarrow 0} \\Big (\\sqrt{-g^{\\prime }} R^{\\prime }\\Big )\\nonumber \\\\&=& \\Big [-6 \\alpha ^2 \\partial _y F \\partial _y F -6 \\alpha ^2 \\partial _z F \\partial _z F+4 \\alpha \\partial _v F \\partial ^2_y F \\nonumber \\\\ & -& 12 \\alpha \\partial _z F \\partial _v \\partial _z F - 6(\\partial _v \\partial _z F)^2 - 12 \\alpha \\partial _y F \\partial _v \\partial _y F \\nonumber \\\\ &-& 6(\\partial _v \\partial _y F)^2 + 4 \\partial ^2_y F \\partial ^2_v F + 4 \\partial ^2_z F (\\alpha \\partial _v F + \\partial ^2_v F) \\Big ]~.\\nonumber \\\\$ Since ${\\mathcal {L}_{F}}$ is calculated on a $r=$ constant surface, the action can be defined as the integration of the above Lagrangian on $v,y$ and $z$ .",
"Th induced horizon geometry has flat spacial section, we, therefore, consider the following generic form of $F$ : $F_{mn} = f_{mn}(v) \\frac{1}{\\alpha } \\exp \\Big [i (my+nz)\\Big ]~.$ Hence the general solution for Goldstone mode would be, $F(v,y,z) = \\sum _{m,n} C_{mn} F_{mn}~.",
"$ Here we need to find the form of $f_{mn}(v)$ from the solution of the equation of motion obtained from (REF ).",
"It is quite obvious that substitution of the above ansatz and then integrating over transverse coordinates one can get an one dimensional action which determine the evaluation of $f_{mn}(v)$ with respect to $v$ .",
"Since the total derivative terms in action keep the dynamics unchanged, it may be verified that under the integration of transverse coordinates third, fourth, sixth and ninth terms are total derivative terms with respect to $v$ .",
"So, those terms can be neglected.",
"Ignoring total derivative terms the final form of the Lagrangian (REF ) is given as; ${\\mathcal {L}_{F}} &=& \\Big [-6 \\alpha ^2 \\partial _y F \\partial _y F -6 \\alpha ^2 \\partial _z F \\partial _z F- 6(\\partial _v \\partial _y F)^2 \\nonumber \\\\ &-& 6(\\partial _v \\partial _z F)^2+ 4 \\partial ^2_y F \\partial ^2_v F + 4 \\partial ^2_z F \\partial ^2_v F \\Big ]~.",
"$ Before proceeding further, here we want to mention an important point of our proposed form of the Goldstone boson Lagrangian.",
"Since the modified metric (REF ) has been constructed by taking into account a particular type of diffeomorphism, one always concludes that the Lagrangian must be invariant upto a total derivative.",
"Contribution of the total derivative term vanishes over the closed boundaries which encloses the bulk of the manifold.",
"For instance, in the case of $\\sqrt{-g}R$ , the variation of it under diffeomorphism $x^a\\rightarrow x^a+\\xi ^a$ leads to $\\sqrt{-g}\\nabla _a(R\\xi ^a)=\\partial _a(\\sqrt{-g}R\\xi ^a)$ , which is a total derivative term.",
"In this analysis we are interested to build a theory on the horizon (i.e.",
"$r=0$ ) and horizon is a part of the closed boundary of the bulk manifold.",
"Therefore it is expected that the total derivative term will give non-vanishing contribution on a part of the closed surfaces such as horizon.",
"In the case of $\\sqrt{-g}R$ , the boundary term on $r=$ constant surface in the variation of action is given by $\\int d^3x\\hat{n}_a\\xi ^a \\sqrt{-g}R$ , where $\\hat{n}_a$ is the normal to the surface with components $(0,1,0,0)$ .",
"Therefore, on this surface our proposal for the Lagrangian density (loosely call it as Lagrangian) is $\\sqrt{-g}R$ .",
"This is precisely considered here.",
"The Lagrangian (REF ) is not the one which is defined for the whole spacetime, rather it is calculated on the $r=$ constant surface, and hence coming out to be non-trivial.",
"In this sense our proposed Lagrangian does not carry any ambiguity and correctly describe the dynamics of the Goldstone mode $F$ associated with the super-transnational symmetry near the horizon.",
"Next we concentrate on Gibbons-Hawking-York (GHY) boundary term $\\mathcal {S}_2 =- \\frac{1}{8\\pi G}\\int d^3x \\sqrt{h}K~,$ which is usually added to the action in order to define a proper variation of the action.",
"The trace of the extrinsic curvature of the boundary surface ($r\\rightarrow 0$ ) is given by $K = -\\nabla _a N^a$ , where $N^a$ is considered as the unit normal to the $r=constant$ hyper-surface.",
"For metric (REF ), its lower component is given by $ N_a =(0, 1/\\sqrt{2r(\\alpha +\\alpha \\partial _v F +\\partial ^2_{v} F)},0,0) $ .",
"Therefore in the near horizon limit ($r \\rightarrow 0$ ), one gets the following form of the action coming from the GHY term: $&&\\mathcal {S}_2= -\\frac{1}{8 \\pi G}\\int d^3 x\\Big [\\alpha +\\Big (\\alpha \\partial _v F + \\frac{1}{2}\\partial ^2_v F + \\frac{1}{2 \\alpha } \\partial ^3_v F\\Big )\\nonumber \\\\&&+ \\frac{1}{2 \\alpha ^2}\\Big (\\alpha ^2 \\partial _v F \\partial ^2_v F + \\alpha (\\partial ^2_v F)^2 + \\alpha \\partial _v F \\partial ^3_v F\\nonumber \\\\&&+ \\partial ^2_v F \\partial ^3_v F \\Big )\\Big ]~.$ However, we observed that this terms does not contribute to our required equation of motion.",
"In fact, the above boundary term in the action is turned out to be related to the horizon entropy which is discussed in appendix ().",
"Note that the aforesaid Lagrangian (REF ) contains higher derivative terms of $F$ .",
"Therefore, the theory of Goldstone boson modes emerging on the boundary of a gravitational theory turns out to be higher derivative in nature.",
"However, if we want to trace back the origin of this higher derivative action, it is from the diffeomorphically transformed metric components which already contains the derivative term.",
"However, we will see those higher derivative terms will be crucial for our subsequent discussions on the horizon properties.",
"This connection could be an interesting topic to investigate further.",
"However, one of the important point here is that at the background label system is not Lorentz invariant.",
"The generalized Euler-Lagrangian equation, defined for higher derivative theory, is: $\\frac{\\partial L}{\\partial F} - \\partial _{\\mu }(\\frac{\\partial L}{\\partial (\\partial _{\\mu } F)}) + \\partial _{\\mu } \\partial _{\\nu } (\\frac{\\partial L}{\\partial (\\partial _{\\mu } \\partial _{\\nu } F)}) =0 .$ With this the equation of motion is found to be $3 \\alpha ^2 \\partial ^2_y F +3 \\alpha ^2 \\partial ^2_z F -4 \\partial ^2_y \\partial ^2_v F -4 \\partial ^2_z \\partial ^2_v F =0~.$ Important to note again, the contribution on the equation of motion comes only from from (REF ).",
"GHY (REF ) term does not contribute.",
"Substitution of (REF ) in (REF ) yields $ (m^2 +n^2) [\\partial ^2_{v}f_{mn}(v) -\\frac{3 \\alpha ^2}{4} f_{mn}(v)]= 0~.$ Important point to note that every individual mode ($m,n$ ), will follow the same equation of motion of a simple oscillator in an inverted harmonic potential.",
"The solution will be, $f_{mn}(v) && = A \\exp \\Big [(\\sqrt{3/4}) \\alpha v\\Big ] + B \\exp \\Big [-(\\sqrt{3/4}) \\alpha v \\Big ] \\nonumber \\\\ &+ & f_1 (y,z) \\delta _{m,0} \\delta _{n,0}~,$ for all $m,n$ .",
"In the above, $A$ and $B$ are arbitrary constants to be determined.",
"So far we talked about the classical dynamics of the Goldstone mode.",
"It is apparent that the system is unstable because of the inverted harmonic potential at least at the tree level Lagrangian.This is also apparent from the solution (REF ).",
"As we are interested to the near horizon region where $v\\rightarrow -\\infty $ , the above solution grows rapidly and makes the mode unstable.",
"Therefore, the appropriate boundary condition one can set is $B=0$ , leading to $F_{mn} (v,y,z) &&= [A \\exp [(\\sqrt{3}/2) \\alpha v] \\nonumber \\\\ &+& f_1 (y,z) \\delta _{m,0} \\delta _{n,0} ] \\ \\frac{1}{\\alpha }\\exp [i (my+nz)]~.\\nonumber \\\\ $ Interesting this is precisely the boundary condition which satisfies the condition of vanishing fluctuation of surface gravity $\\delta \\alpha =0$ at the horizon defined by the Eq.",
"(REF ).",
"We already know that the horizon is a special place in the entire spacetime region, as any two hypothetical observers spatially separated by the horizon can never communicated to each other.",
"Therefore, it would have been unusual, had there been just simple stable free field like Lagrangian for the Goldstone modes.",
"The connection between these special nature of the horizon and emergence of instability is the subject of study for a long time.",
"Our goal of this paper would be to shed some light on this issue.",
"Does the emergence of the inverted harmonic potential has anything to do with the thermal nature of the black hole horizon?",
"Of course in order to understand this, we need to go to beyond the classical regime.",
"In the next section we will try to make this connection considering a recent proposal [59][60]."
],
[
"Thermal behaviour of the field solution",
"In this section we consider the quantum mechanical treatment of the Goldstone boson mode discussed so far.",
"It has recently been conjectured that Lyapunov exponent $\\lambda $ of a thermal quantum system, in presence of quantum chaos, is bounded by the temperature $T$ of the system as $\\lambda \\le 2\\pi T/\\hbar $ [61].",
"Based on this result further conjecture has been made in the reference [60][62] which says that a chaotic system with a definite Lyapunov exponent could be fundamentally thermal by reversing the above inequality.",
"To justify the argument, one of the interesting example the author has studied is the semi-classical dynamics of a particle in an inverted harmonic potentialThe choice of the inverted harmonic oscillator stems from the fact that the particle motion is unstable under this potential and hence, at the classical level, any small perturbation can lead induction of chaos in the motion (for example, see [64])., and showed that the quantum correction induces an energy emission by the particle under study obeying thermal probability distribution.",
"Therefore, the connection between the semi-classical chaotic system and the thermal nature is emerged.",
"Interestingly, for our present system, each individual Goldstone boson mode behaves like an inverted harmonic oscillator.",
"Hence, the aforesaid connection between the thermal emission and the semi-classical chaotic dynamics could be a potential reason for the thermal nature of the black hole horizon.",
"Even more interestingly, every individual Goldstone boson mode parametrized by $(m,n)$ see the same inverted potential, which may also indicate the universality of the thermal nature of the horizon.",
"Our present claim is ambitious and exciting which needs detailed future exploration.",
"Before we resort to our discussion of thermal nature of the black hole, let us briefly describe the connection between the thermality and the inverted harmonic oscillator, following from the reference [59][60].",
"These are connected with the finite quantum mechanical transition probability through a potential barrier.",
"The equation of motion of the particle moving in a harmonic potential is given by, $\\mu \\ddot{x} -\\omega x =0 $ Here potential $V= -\\frac{\\omega x^2}{2}$ , and $\\mu $ is the mass of the particle.",
"Important case would be, if one considers the energy of the particle $E < 0$ , for which the potential energy of the particle is greater than its kinetic energy.",
"With this energy if the particle travels toward the potential from the left ($x<0$ ), classically it cannot pass through the potential towards right $(x >0$ ).",
"However, quantum mechanically the particle will have finite tunneling probability to go across the potential barrier.",
"Therefore, the particle will have finite probability of transmission through the barrier even for $E<0$ .",
"In the similar manner for $E>0$ , the particle will have finite quantum mechanical probability of reflection off the barrier, which otherwise was not possible classically.",
"Therefore, to describe the above quantum mechanical phenomena the appropriate Hamiltonian for the wave function $\\Phi (x)$ associated with the particle is expressed as $H = -\\frac{{\\hbar }^2}{2}\\frac{d^2}{d x^2} - \\frac{{\\omega } x^2}{2}$ with the Schrödinger equation, $-\\frac{{\\hbar }^2}{2}\\frac{d^2 \\Phi }{d x^2} - \\frac{{\\omega } x^2}{2} \\Phi =E \\Phi ~.$ $E$ is the energy of the paticle.",
"The well known expression for the probability of transmission($P_T$ ) and the reflection ($P_R$ ) using WKB approximation (detail in [63]) are given as, $P_{T/R} = \\frac{1}{e^{\\frac{2 \\pi }{\\hbar } \\sqrt{\\frac{\\mu }{ \\omega }} |E|} +1} = \\frac{1}{ e^{\\beta |E|} +1}~.$ An interesting interpretation of this expression is that for large absolute value of the energy $E$ , probability amplitude from classical path to quantum transmission or reflection will be $\\exp [-\\beta |E|]$ .",
"Therefore, the quantum harmonic oscillator system can be mapped to a two level system with temperature $T$ , whose ground state is represented as the classical trajectories and excited state is quantum one.",
"And the temperature of the system can be easily identified as $T = \\frac{\\hbar }{2 \\pi } \\sqrt{\\frac{\\omega }{\\mu }} ~.$ For further detail of this interesting interpretation the reader can look into the reference [59] [60]).",
"In this context it is worth to mention that recently one of the authors of this paper also showed by an independent and completely different way that the inverted harmonic oscillator gives rise to temperature at the quantum level [65].",
"In our present analysis we have obtained the dynamical equation of motion for individual mode as given in (REF ).",
"Comparing this with Eq.",
"(REF ) one can easily conclude that the dynamics of the mode along $v$ is governed by inverted harmonic oscillator potential.",
"To clarify our analogy, each mode $f_{mn}(v)$ can be thought of as the position $x(t)$ of a particle of mass unity with $v$ playing the role of time coordinate as $t$ .",
"Therefore, we have following equivalence table: $f_{mn} \\equiv x; \\,\\,\\,\\,\\ v \\equiv t;$ accompanied by the identifications $\\mu \\equiv 1;\\,\\,\\,\\,\\ \\omega \\equiv \\frac{3 \\alpha ^2}{4}~.$ Hence by the earlier argument, we can conclude that each mode, at the quantum level, is thermal.",
"The temperature is evaluated as (REF ) with the following substitutions: $\\mu =1$ and $\\omega =(3\\alpha ^2/4)$ .",
"Therefore in our case it is given by $T = \\frac{\\hbar }{2 \\pi } \\frac{\\sqrt{3}\\alpha }{2} ~.$ Even more interestingly what is emerged from our present calculation that all the modes with quantum number $(p,q)$ are degenerate with respect to $E$ .",
"This observation seems to suggest that the horizon under study can carry entropy because of those degenerate quantum states.",
"However, in order to have finite entropy, we need to have an upper limit on the value of $(p,q)$ , which must be proportional to the only scale available in the theory namely Planck scale.",
"Our naive analysis based on [59], shows that semi-classical Goldstone boson dynamics can capture the well known thermal behaviour of the horizon.",
"Moreover, the temperature turned out to be proportional to the acceleration of the Rindler frame.",
"This is an important observation as we know that the Rindler horizon is thermal with respect to its own frame.",
"In this case also the temperature is proportional to $\\alpha $ , known as Unruh temperature [66].",
"However, the proportionality constant appeared to be different.",
"Another important outcome of our analysis is the emergence of infinite number of degenerate states which can be associated with the entropy on this horizon.",
"We will take up this issue in our future publication.",
"The microscopic origin of the horizon thermodynamics is a subject of intensive research for a long time.",
"Our present analysis hints towards an important fact that the BMS like symmetry near the horizon could play important role in understanding the thermal nature and possible origin of the underlying microscopic states of a black hole.",
"Motivated by our analysis, in the subsequent section we will discuss about the Schwarzschild black hole."
],
[
"Schwarzschild black hole",
"So far we have discussed about the dynamics of Goldstone boson mode in Rindler background.",
"To this end we perform similar analysis considering Schwarzschild black hole background.",
"The near horizon geometry of the Schwarzchild black is again Rindler, however, with two dimensional sphere at each point.",
"Therefore, we expect similar behavior of the Goldstone mode for this case as well.",
"As we go along we also notice the main differences with flat Rindler case.",
"The Schwarzschild metric in Eddington-Finkelstein coordinate ($v,r,\\theta ,\\phi $ ) is expressed as, $ds^2 = -(1- 2M/r) dv^2 + 2 dv dr + r^2 \\gamma _{AB} dx^A dx^B~.$ The event horizon is located at $r=2 M$ .",
"$M$ is the mass of the black hole which characterizes the macroscopic state of the background spacetime.",
"Asymptotic symmetry properties of the horizon can be extracted from similar fall off and gauge conditions for the metric components, $£_\\zeta g_{rr}= 0, \\ \\ \\ £_\\zeta g_{vr}=0, \\ \\ \\ £_\\zeta g_{Ar}=0~;\\\\£_\\zeta g_{vv} \\approx \\mathcal {O}(r); \\ \\ \\ £_\\zeta g_{vA} \\approx \\mathcal {O}(r); \\ \\ \\ £_\\zeta g_{AB} \\approx \\mathcal {O}(r)~.$ Here, $£_\\zeta $ corresponds to the Lie variation for the diffeomorphism $x^a\\rightarrow x^a+\\zeta ^a$ .",
"The primary motivation to consider the aforementioned conditions is essentially to preserve the form of the metric under the diffeomorphism.",
"As has already been observed in our previous case, those differmorphsim in turn renormalizes the state of the black hole parameter such as mass $M$ of the Schwarzschild black hole.",
"Similar to our previous analysis after solving the above gauge fixing conditions with the imposed fall-off conditions, the diffeomorphism vectors turned out to be, $\\zeta ^a \\partial _a= F(v,x^A) \\partial _v -(r-2M) \\partial _v F \\partial _r \\nonumber \\\\ + (1/r -1/2M) \\gamma ^{AB} \\partial _{B} F \\ \\partial _A .$ Again we have one unknown function $F$ which is identified as supertranslation generator.",
"Under this transformation the background metric takes of following form [48], $g^{\\prime }_{ab} &=& \\Big [g^{(0)}_{ab} + £_\\zeta g^0_{ab} \\Big ] dx^a dx^b\\nonumber \\\\&=& -(1- 2M/r) dv^2 + 2 dv dr + r^2 \\gamma _{AB} dx^A dx^B \\nonumber \\\\&+& \\Big [2M/r(1- 2M/r) \\partial _v F - 2 (1- 2M/r)\\partial _v F \\nonumber \\\\&& -2 (r-2M) \\partial ^2_v F \\Big ] dv^2 + \\Big [-(1-2M/r) \\partial _{A} F \\nonumber \\\\ &&-(r-2M) \\partial _{A}\\partial _v F + r^2 \\partial _{A}\\partial _v F (1/r - 1/2M) \\Big ] dv dx^A \\nonumber \\\\ &+& \\Big [-2 (2M-r)r \\gamma _{AB} \\partial _v F\\nonumber \\\\ &-& (1/r -1/2M)(\\partial _E F \\gamma ^{DE} \\partial _D \\gamma _{AB}\\nonumber \\\\ &+& \\gamma _{AD} \\ \\partial _B (\\partial _E F \\gamma ^{DE})) \\Big ] dx^A dx^B~.$ As has already been discussed for the Rindler metric with flat spatial section, for the present case the modification $h_{ab}$ due to following super-translation, $v^{\\prime }= v + F~;~x^{\\prime A} = x^A + (1/r -1/2M) \\gamma ^{AB} \\partial _{B} F ,$ the macroscopic black hole state parameter $M$ renormalizes to, $\\frac{1}{M} \\rightarrow \\frac{1}{M} +\\frac{1}{M}\\Big (\\partial _{v} F + 4 M \\partial ^2_v F\\Big ) .$ Therefore, this change of macroscopic state by the symmetry transformation can similarly be understood as a breaking of the boundary super-translation symmetry with $F$ as the broken symmetry generator.",
"Following the same procedure as for the Rindler case, the Lagrangian $\\mathcal {L}_{F}$ of the Goldstone mode on the horizon surface takes the following form, $\\mathcal {L_{F}}&=& \\Big [ \\frac{-3}{2 (2M)^2} \\csc \\theta \\ \\partial _{\\phi } F \\partial _{\\phi } F - \\frac{3}{2 (2M)^2} \\sin \\theta \\ \\partial _{\\theta } F \\partial _{\\theta } F \\nonumber \\\\ &+& 4 \\sin \\theta \\ \\partial _{v} F \\partial _{v} F - \\frac{3}{M} \\csc \\theta \\ \\partial _{\\phi } F \\partial _{v} \\partial _{\\phi } F \\nonumber \\\\ &+& \\frac{1}{M} \\cos \\theta \\ \\partial _{\\theta } F \\partial _{v} F -\\frac{3}{M} \\sin \\theta \\ \\partial _{\\theta } F \\partial _{\\theta } \\partial _{v} F \\nonumber \\\\ &+& 4 \\cos \\theta \\ \\partial _{\\theta } F \\partial ^2_{v} F + \\frac{1}{M} \\csc \\theta \\ \\partial ^2_{\\phi } F \\partial _v F\\nonumber \\\\ & +& 4 \\csc \\theta \\ \\partial ^2_{\\phi } F \\partial ^2_{v} F + \\frac{1}{M} \\sin \\theta \\ \\partial ^2_{\\theta } F \\partial _v F\\nonumber \\\\ & +& 4 \\sin \\theta \\ \\partial ^2_{\\theta } F \\partial ^2_{v} F - 6 \\csc \\theta \\ (\\partial _v \\partial _{\\phi } F)^2 \\nonumber \\\\ &-& 6 \\sin \\theta \\ (\\partial _v \\partial _{\\theta } F)^2 + 8 \\sin \\theta \\ \\partial _v F \\partial ^2_v F \\Big ]~.$ Neglecting total derivative terms, we can write final Lagrangian as, $\\mathcal {L_{F}}&=& \\Big [ \\frac{-3}{2 (2M)^2} \\csc \\theta \\ \\partial _{\\phi } F \\partial _{\\phi } F - \\frac{3}{2 (2M)^2} \\sin \\theta \\ \\partial _{\\theta } F \\partial _{\\theta } F \\nonumber \\\\ &+& 4 \\sin \\theta \\ \\partial _{v} F \\partial _{v} F + 4 \\cos \\theta \\ \\partial _{\\theta } F \\partial ^2_{v} F \\nonumber \\\\ &+& 4 \\csc \\theta \\ \\partial ^2_{\\phi } F \\partial ^2_{v} F + 4 \\sin \\theta \\ \\partial ^2_{\\theta } F \\partial ^2_{v} F\\nonumber \\\\ &-& 6 \\csc \\theta \\ (\\partial _v \\partial _{\\phi } F)^2 - 6 \\sin \\theta \\ (\\partial _v \\partial _{\\theta } F)^2 \\Big ]~.$ Here the non-vanishing lower components of $N^a$ is given by $ N_r = \\frac{1}{\\sqrt{f(r)- (2M/r)f(r) \\partial _v F +2 f(r)\\partial _v F +2rf(r) \\partial ^2_v F}}~,$ where $f(r)=1-2M/r$ .",
"Hence for GHY boundary term the action can be expressed as, $\\mathcal {S}_2 &=& - \\frac{M}{8 \\pi G}\\int d^3 x \\sin \\theta \\Big [1 + (\\partial _v F + 2M \\partial ^2_v F)\\nonumber \\\\ &+& (2M \\partial _v F \\partial ^2_v F + 8 M^2 (\\partial ^2_v F)^2 + 8 M^2 \\partial _v F \\partial ^3_v F\\nonumber \\\\ &+& 32 M^3 \\partial ^2_v F \\partial ^3_v F) \\Big ]~,$ which is again does not contribute to the equation of motion as was the case for Rindler space.",
"The dynamics of the Goldstone mode will be governed by the action corresponding to $\\mathcal {L_{F}}$ , and the equation of motion is given by, $&& -8 \\sin \\theta \\partial ^2_v F + \\frac{3}{(2M)^2} \\cos \\theta \\partial _{\\theta } F + \\frac{3}{(2M)^2} \\sin \\theta \\partial ^2_{\\theta } F \\nonumber \\\\ &+& \\frac{3}{(2M)^2} \\csc \\theta \\partial ^2_{\\phi } F -16 \\sin \\theta \\partial ^2_v \\partial ^2_{\\theta } F-16 \\cos \\theta \\partial ^2_v \\partial _{\\theta } F\\nonumber \\\\ &-& 16 \\csc \\theta \\partial ^2_v \\partial ^2_{\\phi } F = 0~.$ In this analysis full metric has been considered.",
"Since we are interested in the near horizon symmetries, the near horizon metric could be enough to obtain the same result.",
"For completeness, we explicitly demonstrated this in Appendix .",
"Since the action has the rotational symmetry, we can take the following solution ansatz for Goldstone boson mode in terms of spherical harmonic, $F(v,\\theta ,\\phi )=\\frac{1}{k} \\sum _{lm} {c_{lm}}f_{lm}(v) Y_{lm}(\\theta ,\\phi ),$ with $c_{lm}$ are constant coefficients and $f_{lm}$ are the time dependent mode function.",
"This is consistent with the spherically symmetric Schwarzschild geometry.",
"The factor ${1}/{k} = {4M}$ is introduced for dimensional reason.",
"Substituting the form of $F$ (REF ) in (REF ) we get following equation of motion for $f_{lm}(v)$ $ && [2l(l+1) -1] \\partial ^2_{v} f_{lm} - \\frac{3}{32 M^2} l(l+1) f_{lm} = 0 .$ Since the near horizon geometry of the Scwarzchild black hole is Rindler with sphere as spatial section, one notices some significant differences in the mode dynamics governed by eq.",
"(REF ) and that of the previous case in eq.",
"(REF ).",
"Most importantly, for spatial spherical geometry the effective potential perceived by every individual mode parametrized by $(l,m)$ is no longer universal but dependent upon the angular momentum $l$ .",
"Before we discuss the implications of this dependence, let us take a look at the behaviour of individual modes.",
"For $l=0$ mode, the equation reduces to, $\\partial ^2_{v} f_{00}(v) =0~.$ The solution of the above equation is $f_{00} = c_1(x^A) v + c_2(x^A)$ .",
"By choosing $c_1=0$ , the final solution will be $f_{00}(v) = c_2(x^A)$ .",
"For all remaining modes $l\\ge 1$ , we get the inverted harmonic oscillator potential similar to our previous case.",
"One important difference is the angular momentum dependence of the inverted harmonic potential.",
"Therefore, the universality of all the modes with respect to their time dynamics is lost as opposed to our previous study in Rindler metric with spatial section.",
"However, it can be checked that numerically the inverted potential depends very weakly on the value of $l$ , which we will discuss in terms of temperature in the next subsection.",
"Nonetheless, the mode equation looks likes, $\\partial ^2_{v} f_{lm} -k^2 \\Omega ^2 f_{lm}(v) = 0~,$ where, $\\Omega = \\sqrt{\\frac{3 l (l+1)}{ 2(2 l(l+1)-1)}}~.$ We get the inverted harmonic oscillator potential similar to our previous case.",
"The complete solution for all modes can therefore be, for $l=0$ ; $F(x^A) = \\sum _{lm} \\frac{1}{k} c_2(x^A) Y_{lm}(x^A);$ for $l \\ge 1$ , $F(v,x^A) = \\sum _{lm} \\frac{A}{k} e^{ k v} Y_{lm}(x^A)~.$ Hereafter we can proceed along the same line as discussed before.",
"Important difference would be the state dependent inverted harmonic potential $V_{harmonic} = - \\frac{1}{2} \\Omega (l)^2 {k}^2 f_{lm}^2~.$ Therefore, strictly speaking for the present case degenerate states will be only for $m$ within $(-l,l)$ .",
"However, let us point out that if we consider numerical values into consideration, the value of $\\Omega $ is confined within a very narrow region $\\sqrt{\\frac{3}{4}}\\le \\Omega (l)\\le 1~.$ Hence, one can approximately consider all the quantum states of the Goldstone boson parametrized by $(l,m)$ with $l\\ge 1$ , are quasi-degenerate.",
"Unlike the previous case for the Rindler spacetime with flat spatial section, the emission probability for the present case would be identified with Boltzmann distribution with temperature, $T_l = \\frac{\\hbar }{8 \\pi M} \\Omega (l) ,$ which will weakly depend upon the value of angular momentum quantum number $l$ .",
"Interestingly for $l=1$ mode the above expression came out exactly same as usual black hole temperature $T_{BH}$ , given by the Hawking expression [67].",
"However considering other modes we can define an average temperature $T_{avg} = \\frac{\\hbar }{8 \\pi M} \\left(\\frac{\\sum _l \\Omega (l)}{\\sum _l 1}\\right) = \\frac{\\hbar }{8 \\pi M} \\left(\\sqrt{\\frac{3}{4}}\\right) = \\frac{\\sqrt{3}}{2} T_{BH}~,$ Here again we observed that the Goldstone modes are inherently thermal in nature.",
"The obtained temperature is proportional to the Hawking expression for that of the Schwarzschild horizon.",
"From the analysis so far what we can infer is that since the origin of the Goldstone modes are associated with the breaking of symmetries of the horizon, those modes can be a potential candidate for the microscopic states of a black hole.",
"Quantum mechanically all those states turned out to be thermal with a specific temperature.",
"However, origin of different expressions for the temperature compared with that of the usual Hawking temperature needs to be explored in detail.",
"Furthermore, nature of degeneracy of those Goldstone states appears to be dependent upon the spacetime background.",
"Such as for Rindler spacetime with plane symmetric horizon, all the modes emerged as degenerate and, therefore, each mode fills the same temperature.",
"On the other hand for Schwarzschild black hole this is not the case as the degeneracy of states has been lifted by the less symmetric spherical horizon.",
"Nevertheless, we hope that this thermal nature of the Goldstone modes at the quantum level can be inferred for all types of horizon.",
"We keep this for our future project."
],
[
"Summary and conclusions",
"Microscopic origin of the thermodynamic nature of the black hole is one of fundamental questions in the theory of gravity.",
"It is obvious that within the framework of Einsteinian gravity this question can not be answered.",
"However, the recent understanding of infrared behavior of gravity opens up a new avenue towards understanding this question.",
"In the gravitational theory, one of the interesting infrared properties is the emergence of infinite dimensional symmetry at null infinity which leads to soft graviton theorem.",
"Over the years it has been observed that analogous symmetry exists near the null horizon which can play important role in explaining the microscopic origin of horizon thermodynamics.",
"Here we particularly concentrated on the BMS-like symmetry in the near horizon region.",
"Under the diffeomorphism symmetry, appropriate boundary conditions are imposed in such a way that the near horizon form of the metric remains unchanged.",
"It is observed that in this process the macroscopic parameters, like mass (surface gravity), get modified.",
"This change in macroscopic parameters is argued to be the phenomena of symmetry breaking on the horizon and corresponding parameter can be viewed as the Goldstone mode.",
"In the present paper our main effort was to explore the dynamics of these Goldstone modes.",
"For the purpose of our present study, we consider two simple gravitational background.",
"One is simple Rindler spacetime with flat Killing horizon and the other one is Schwarzschild black hole.",
"Our preliminary investigation at tree level reveals that the horizon is indeed a special place where the dynamics of the Goldstone mode in momentum space is governed by inverse harmonic potential.",
"As mentioned earlier, in the framework of classical Einsteinian gravity it is difficult to understand this situation as those modes are simply unstable.",
"Interestingly, at the quantum level this instability [60] can have a nice interpretation in terms of inherent thermality in connection with its chaotic behaviour, which may provide us a first glimpse of microscopic view of the horizon thermodynamics.",
"Interestingly, for both the gravitational backgrounds, as expected the temperature turned out to be proportional to the surface gravity which is similar to the expression (except a numerical factor) given by Unruh [66] and Hawking [67].",
"This led us to think that these Goldstone modes might be candidates for the microscopic description of the horizon thermality.",
"Even more interestingly, we found out large number of degenerate states for Rindler and qusi-degenerate states for Schwarzschild black holes which may be responsible for the horizon entropy.",
"We will take up these issues in more detail in future publication.",
"So far we have considered the black hole spacetime which are static and hence generating only one Goldstone field.",
"However for a gravitational background having intrinsic rotation such as Kerr spacetime, corresponding analysis of the Goldstone mode dynamics will be more effective.",
"This is because in this case there will be more than one symmetry generator.",
"This topic is now under investigation.",
"Finally, we want to mention that since the Goldstone modes are thermal in nature, it might be interesting to look at BMS symmetry in this way hoping that such an analysis will be able to shed some light towards the microscopic description of horizon thermodynamics."
],
[
"Near horizon analysis of Schwarzschild black hole",
"As mentioned in the main text, in this section we will argue that same dynamical equations and solution for the Goldstone modes can be obtained starting from the near horizon metric of the Schwarzschild black hole.",
"Now we can easily check that the action constructed from the near horizon metric will contains three types of terms.",
"The ones which are independent and linear in $F$ , can be traced back from their origin which can be transmitted to the fact that the near horizon geometry of the Schwarzschild black hole is Rindler times a sphere, and it does not satisfy the background Einstein's equation.",
"We, therefore, ignore those terms as they can also be made total derivative.",
"Non-trivial dynamics of the Goldstone modes are attributed to second order term in $F$ in the action, and it can be easily checked that those terms are exactly the same as in (REF ) up to a total derivative.",
"As a result with a proper prescription, full spacetime geometry as well as near horizon geometry of the Schwarzschild background are giving rise to the same Goldstone mode dynamics."
],
[
" Surface Hamiltonian and heat content",
"In the main text we have constructed the GHY boundary term in the action formulation which did not contribute in the dynamics of the Goldstone mode.",
"However an important analysis is left to be discussed there.",
"It is well known that the boundary term of the Einstein-Hilbert action in gravitational theroy leads to surface Hamiltonian which is directly related to the heat content of the Horizon (detail discussion is given in ([68] [69])).",
"The expression of the surface Hamiltonian comes out to be the product of temperature and entropy of the horizon.",
"Keeping this in mind we can write surface Hamiltonian corresponding to the GHY boundary term (REF ): $H_{sur} = -\\frac{\\partial S_2}{\\partial v}~.",
"$ Now substituting the solution (REF ) in the expression (REF ) and integrating the boundary term , the Hamiltonian (REF ) comes out: $H_{sur}=\\frac{\\bar{A}}{8 \\pi G} \\Big [ \\alpha + f_3(\\alpha )e^{(f_2(\\alpha ) v)}\\Big ]~,$ $ \\bar{A}$ denotes the transverse area of the Rindler horizon.",
"In the second part function $f_3(\\alpha )$ comes from the first and higher order time derivative of $F$ in the boundary term of the action.",
"Whereas $f_2 (\\alpha )$ denotes all the square, cubic and higher order terms in the expression.",
"Now near horizon (where $v \\rightarrow -\\infty $ ) the second terms in the above expression vanishes.",
"Hence the result comes out as: $H_{sur} = \\frac{1}{8 \\pi G} \\bar{A} \\alpha = TS~, $ where $S = \\bar{A} /4 G $ and $T = \\alpha /2\\pi $ are the Horizon entropy and temperature respectively.The result clearly indicates that irrespective of Goldstone modes $F_{m,n}$ , the GHY boundary term in the action is related to the heat content of the horizon.",
"Similar conclusion can be drawn for Schwarzschild black hole horizon also."
]
] | 1906.04489 |
[
[
"Network-based Fake News Detection: A Pattern-driven Approach"
],
[
"Abstract Fake news gains has gained significant momentum, strongly motivating the need for fake news research.",
"Many fake news detection approaches have thus been proposed, where most of them heavily rely on news content.",
"However, network-based clues revealed when analyzing news propagation on social networks is an information that has hardly been comprehensively explored or used for fake news detection.",
"We bridge this gap by proposing a network-based pattern-driven fake news detection approach.",
"We aim to study the patterns of fake news in social networks, which refer to the news being spread, spreaders of the news and relationships among the spreaders.",
"Empirical evidence and interpretations on the existence of such patterns are provided based on social psychological theories.",
"These patterns are then represented at various network levels (i.e., node-level, ego-level, triad-level, community-level and the overall network) for being further utilized to detect fake news.",
"The proposed approach enhances the explainability in fake news feature engineering.",
"Experiments conducted on real-world data demonstrate that the proposed approach can outperform the state of the arts."
],
[
"Introduction",
"With “post-truth” named as the Word of the Year in 2016 by the Oxford Dictionary, discussion around fake news has sparked, especially in the period around the 2016 U.S. presidential election and the U.K. Brexit referendum .",
"The rise of social media and its popularity play an indispensable role in this surge of interest.",
"Social media breaks the physical distance barrier among individuals, and provides rich platforms for users to participate and discuss online news, where the most popular story during the critical months of the 2016 U.S. presidential election campaign (“Pope Francis Shocks World, Endorses Donald Trump for President, Releases Statement”, which was fake news) can generate 960,000 shares, reactions, and comments on Facebook .",
"The situation becomes worse with the existence of an echo chamber effect on social media, where the biased information can be amplified and reinforced .",
"Meanwhile, studies have shown that humans can be irrational and vulnerable differentiating between truth and falsehood when overloaded with deceptive information; studies in social psychology and communications have demonstrated that human ability to detect deception is only slightly better than chance - with a mean accuracy of 54% over 1,000 participants in over 100 experiments .",
"Various manual fact-checking websites and platforms (e.g., PolitiFactand Snopes) have emerged to serve the public on this matter.",
"Nevertheless, manual fact-checking does not scale well with the volume of newly created information, especially on social media, hence motivating the need for automatic fake news detection.",
"[1]https://www.politifact.com/ [2]https://www.snopes.com/ Current research on automatic fake news detection heavily relies on news content .",
"These studies have significantly contributed to fake news detection (see “Related Work” in Section ) while often face multiple challenges.",
"First, the traditional approach to detect fake news is to use a knowledge-based fact-checking system , .",
"The system compares relational knowledge extracted from to-be-verified news content with that stored in a knowledge graph, often a ground truth dataset collected from the Web , .",
"However, the most serious issue by using such system is that it can only detect false news instead of fake news (i.e., intentionally false news) .",
"Second, another common approach is to use a style-based fake news detection system by assuming that fake news exhibits a distinguishable writing style from that of the truth , where malicious entities can disguise the writing style to bypass these linguistic models.",
"Recently, neural networks and deep learning techniques have been well developed to detect fake news by incorporating multi-modal or social-network data, e.g., images within news content , and users (news spreaders) , , ; nevertheless, these models often face the problems with computational efficiency or interpretability .",
"Present Work: Considering that social-network data related to news propagation and spreaders has hardly been comprehensively explored (across network levels) and used in an explainable way for fake news detection, we propose a network-based pattern-driven fake news detection model, robust against manipulations by malicious entities on news content.",
"To that end, our work aims to utilize patterns in fake news dissemination on social networks, which reveal that compared to the truth, fake news can (i) spread farther and (ii) attract more spreaders, where these spreaders are often (iii) more strongly engaged with the news and (iv) more densely connected within the network.",
"Machine learning features representing these patterns are designed at different levels of a network (i.e., node-, ego-, triad-, community-, and network-level), which will be further used within a supervised learning framework to detect fake news.",
"Overall, the specific contributions of this paper are as follows: A network-based pattern-driven approach is proposed, which can detect fake news in an explainable way.",
"Experiments conducted on real-world data demonstrate that the proposed approach can perform comparatively well compared to the state of the art.",
"Fake news patterns in social networks are investigated and summarized, which relate to the news being spread, spreaders of the news, and relationships among the news spreaders.",
"Empirical studies and social psychological theories are provided to validate and interpret the existence of these patterns; Fake news patterns are represented and quantified across multiple network levels, i.e., node, ego, triad, community, and the overall network level.",
"Experimental results indicate that the proposed approach can perform stably with limited available network information, which makes it suitable for fake news early detection.",
"The rest of this paper is organized as follows.",
"Section reviews current fake news detection research.",
"Fake news patterns in social networks are summarized and represented in Section .",
"Experiments are conducted and presented in Section .",
"Section concludes the paper."
],
[
"Related Work",
"As an emerging topic, the development of fake news detection is in its early stages, where the existing research can be generally grouped into content-based and network-based fake news detection.",
"Content-based Fake News Detection.",
"Content-based fake news detection investigates news content.",
"One traditional way of detection is based on knowledge, often represented as a set of (Subject, Predicate, Object) triples , .",
"Knowledge-based approaches aim to assess news authenticity by comparing the knowledge extracted from to-be-verified news content with true knowledge (i.e., ground truth) , .",
"Such ground truth is generally provided in a knowledge graph such as Knowledge Vault , which contains massive manually processed relational knowledge from the open Web.",
"However, the timeliness and completeness of knowledge graphs are still open issues, and importantly, such approaches developed can only detect false news rather than fake news (intentionally false news) .",
"Another common way is based on writing style, a set of self-defined [non-latent] features well representing news writing style.",
"Style features can be those capturing content structure at various language levels such as discourse level by employing rhetorical structure theory , ; or those capturing specific attributes in the content such as sentiment and readability , , , which can be supported by forensic psychological theories such as Undeutsch hypothesis .",
"Such fundamental theories are a double-edged sword for content-based fake news detection: features inspired can help achieve explainable fake news detection, while some linguistic cues that they reveal might not be applicable for news articles (e.g, non-immediacy) .",
"In addition to non-latent features, fake news detection based on latent representation of news content has been well developed recently, where neural networks such as Convolutional Neural Network (CNN) have been utilized to automatically select content features.",
"Nevertheless, these features are often difficult to be comprehended.",
"While content-based approaches can detect fake news by analyzing news content from various perspectives, auxiliary information revealed in news propagation, e.g., news spreaders, is not considered.",
"In addition, approaches can be sensitive to news content when heavily relying on it, where malicious entities might manipulate the results of detection by disguising their writing styles.",
"Hence, network-based fake news detection has been emerged recently.",
"Network-based Fake News Detection.",
"Network-based fake news detection utilizes social context information revealed in news propagation.",
"In general, it investigates two types of networks: homogeneous and heterogeneous networks.",
"Homogeneous networks contain single type of nodes and edges.",
"A typical example is the stance network, which represents the stance (e.g., for or against) similarity among news or posts of news.",
"Based on such network, Jin et al.",
"evaluate news credibility by mining the stance correlations within a graph optimization framework .",
"Another typical example of homogeneous networks is the propagation graph (tree), which presents post-repost relationships for each news article on social media, e.g., tweets and retweets on Twitter , .",
"Using propagation trees, for instance, Vosoughi et al.",
"discover that fake news spreads faster, farther and more broadly than the truth .",
"Heterogeneous networks have multiple types of nodes or edges.",
"By exploring relationships among entities such as news articles, publishers, users (spreaders) and user posts, PageRank-like algorithm , matrix/tensor factorization , , and Recurrent Neural Networks (RNN) , have been developed for fake news detection.",
"In general, our work is a complement of the current [network-based] studies.",
"Compared to current studies, our work investigates a homogeneous network, the friendship network.",
"To our best knowledge, studying fake news with respect to the friendship network is yet to be explored, which allows one to better understand news spreaders and their social relationships on various network levels.",
"Additionally, we aim to detect fake news in an explainable way - by utilizing its propagation characteristics on social networks, which will be detailed in the next section.",
"Table: Key Notations"
],
[
"Fake News Patterns and Representation in Networks",
"Fake news dissemination in networks exhibits distinguishable patterns from the diffusion of true news.",
"In this section, we summarize these patterns and discuss social psychological theories that can explain the existence of these patterns.",
"In terms of fake news patterns, we demonstrate ways to represent news articles as a set of features across network levels (i.e., node-, triad-, community- and network-level), which can be further utilized to detect fake news within a supervised machine learning framework.",
"Broadly speaking, fake news patterns involved in this study relate to (1) the news being spread (Section REF and Section REF ), (2) spreaders of the news (Section REF ), and (3) relationships among the news spreaders (Section REF ).",
"Before further elaboration, we first define Fake News Network (FNN) in Definition REF .",
"Definition 1 (Fake News Network, FNN) Fake News Network (FNN) is a subgraph $\\mathrm {G}_{\\mathcal {F}}=(\\mathrm {V}_{\\mathcal {F}}, \\mathrm {E}_{\\mathcal {F}})$ of the social network $\\mathrm {G}=(\\mathrm {V},\\mathrm {E})$ , where $\\mathrm {V}_{\\mathcal {F}} \\in \\mathrm {V}$ are the users that have engaged with fake news $\\mathcal {F}$ , and $\\mathrm {E}_{\\mathcal {F}} \\in \\mathrm {E}$ represents the relationships among these users.",
"True News Network (TNN) is similarly defined, which is denoted as $\\mathrm {G}_{\\mathcal {T}}=(\\mathrm {V}_{\\mathcal {T}}, \\mathrm {E}_{\\mathcal {T}})$ for a true news event $\\mathcal {T}$ .",
"The key notations in this section are presented in Table REF ."
],
[
"More-Spreader Pattern",
"Evidence has been provided that fake news is in general more “popular” than true news within the same population of users.",
"For instance, during the critical months of the 2016 U.S. presidential election campaign, top twenty frequently-discussed fake election stories generated 8,711,000 shares, reactions, and comments on Facebook, ironically, greater than the total of 7,367,000 for the top twenty most-discussed election stories posted by nineteen major news medium .",
"Fake news popularity can be attributed to two reasons.",
"First, as stated by information gap theory , rather than telling the truth, fake news creators make great efforts to produce an information gap between the news content and individuals' knowledge.",
"Such information gap produces the feeling of deprivation labeled curiosity, which motivates individuals to obtain the missing information to reduce such feeling.",
"Secondly, to greatly influence online users, those who can benefit from fake news often create or recruit malicious accounts (e.g., bots ) to spread or discuss the fake content.",
"For example, millions of malicious accounts have participated in 2016 U.S. presidential election online discussions.https://comprop.oii.ox.ac.uk/research/public-scholarship/resource-for-understanding-political-bots/ News popularity can be characterized in terms of the number of users that spread such news, where Vosoughi et al.",
"have empirically validated that: Pattern 1 (More-Spreader Pattern) More users spread fake news than true news.",
"To capture the number of news spreaders, we investigate the number and proportion of (I) general (i.e., non-attributed) spreaders and (II) specific (i.e., attributed) spreaders in news propagation.",
"I.",
"General (Non-Attributed) Spreaders.",
"In general, the More-Spreaders Pattern can be quantified by the number of users involved in spreading each fake or true news story.",
"This number is basically the number of nodes within each FNN and TNN, which we use as a feature.",
"II.",
"Specific (Attributed) Spreaders.",
"Principles like homophily and social validation theory suggest that in a social network, users with similar characteristics tend to become connected or form groups and exhibit similar behavior.",
"These observations imply that spreaders of fake (true) news stories may also share some similar attributes; hence, allowing one to distinguish fake news from true news by studying specific users (i.e., with specific attributes) participated in news dissemination.",
"Here we consider (a) user susceptibility [to fake news] and (b) user influence, both of which are attributes that can be computed with information provided by FNNs and TNNs.",
"a.",
"User Susceptibility.",
"We investigate user susceptibility to fake news based on (i) the number of involvements in the propagation of different fake news and (ii) the frequency of such involvements.",
"Number of Involvements.",
"Susceptibility in terms of involvements is defined as the proportion of fake news among all news that user $v_i$ has participated in spreading, which is denoted as $\\mathbf {S}(v_i)$ : ${{\\mathbf {S}(v_i) = \\frac{\\sum _j \\mathbf {B} (v_i \\in \\mathrm {V}_{\\mathcal {F}_j})}{\\sum _k \\mathbf {B} (v_i \\in \\mathrm {V}_{\\mathcal {T}_k})+\\sum _j \\mathbf {B} (v_i \\in \\mathrm {V}_{\\mathcal {F}_j})}}},$ where $\\mathbf {B} (v_i \\in \\mathrm {V}_{X}) = 1$ if $v_i \\in \\mathrm {V}_{X}$ , otherwise $\\mathbf {B} (v_i \\in \\mathrm {V}_{X}) = 0$ .",
"$\\mathbf {S}(v_i)= 1$ ($\\mathbf {S}(v_i)= 0$ ) indicates that all news stories spread through $v_i$ are fake (true), i.e., $v_i$ is completely susceptible (immune) to fake news.",
"Frequency of Involvements.",
"Consider the special case where a user spreads a true news story once and a fake news story multiple times, this user may need to be considered more susceptible than a user who posts each story once.",
"Hence, as an alternative way, we define user susceptibility as the ratio between the spreading frequency of fake news stories and that of all news stories a user has spread.",
"Mathematically, $\\hspace*{-11.38109pt}\\mathbf {S}(v_i) =\\frac{\\sum _j \\mathbf {B} (v_i \\in \\mathrm {V}_{\\mathcal {F}_j}) \\mathbf {T}(v_i,\\mathcal {F}_j)}{\\sum _k \\mathbf {B} (v_i \\in \\mathrm {V}_{\\mathcal {T}_k}) \\mathbf {T}(v_i,\\mathcal {T}_k)+\\sum _j \\mathbf {B} (v_i \\in \\mathrm {V}_{\\mathcal {F}_j}) \\mathbf {T}(v_i,\\mathcal {F}_j)},$ where $\\mathbf {T}(v_i,X)$ is the number of times that $v_i$ has spread news story $X$ .",
"Being assigned with a susceptibility score $\\mathbf {S}(v_i)$ , users can be further labeled as susceptible ($\\mathbf {S}(v_i)>\\theta $ ) or normal ($\\mathbf {S}(v_i)<\\theta $ ) based on a fixed threshold value $\\theta \\in [0,1]$ .",
"Such labeling allows us to represent More-Spreaders Pattern by recording the (i) number and (ii) proportion of susceptible spreaders (nodes) in each FNN or TNN, as well as the (iii) number and (iv) proportion of normal spreaders within each FNN and TNN.",
"We include (i-iv) as features representing the pattern.",
"Without such labeling one can represent spreaders involved in each FNN or TNN in terms of their mean and median of susceptibility scores, which are also considered into our feature set.",
"b.",
"User Influence.",
"An approximation of a node (user) influence can be computed via a centrality score within the network.",
"One can consider the following well-established criteria for computing centrality: (i) [in-, out-] degrees, (ii) [in-, out-] closeness, (iii) betweenness, (iv) PageRank score, (v) hub and authority score, all of which use the connections among nodes to identify their positions within the network.",
"We avoid grouping users into influential and non-influential users as many parameters will be introduced (each centrality measure requires a threshold value), which in turn can affect the performance of fake news detection.",
"Therefore, based on each centrality measure, we directly calculate the mean and median user influence within each FNN and TNN, and include both as features."
],
[
"Farther-Distance Pattern",
"In addition to the number of users that spread news articles, news popularity can be also characterized by how far the news can spread, which leads to the corresponding pattern: Pattern 2 (Farther-Distance Pattern) Fake news spreads farther than true news.",
"This pattern has been observed and validated by Vosoughi et al.",
"; they found that the propagation trees of fake news are generally deeper than that of truth, i.e., an original post referring to a fake news event is often more iteratively forwarded than a true news event.",
"On the other hand, given a news story, how far it spreads can be approximated by computing the shortest “distance” between the two most distant spreaders (nodes) within the corresponding FNN or TNN (i.e., network diameter).",
"To represent Farther-Distance Pattern and calculate such “distance”, we investigate (I) shortest (geodesic) distance which refers to the paths existing between two nodes, and (II) effective distance which considers the information flow between two nodes .",
"I. Geodesic Distance.",
"Based on geodesic distance, the diameter of each FNN and TNN is equivalent to the shortest path length between the two most distant spreaders within the network.",
"II.",
"Effective Distance.",
"Besides conventional shortest distance, we introduce effective distance to help assess the network diameter, which was initially proposed by Brockmann and Helbing .",
"The initial binary (unweighted) FNNs and TNNs is hence transformed into weighted networks, where the weights are determined by the volume of information flow among nodes.",
"Given a network, the effective distance among nodes is defined as follows.",
"Definition 2 (Effective Distance) Given a network $\\mathrm {G}$ , we assume $\\mathrm {F}$ denotes the flow matrix whose entities $\\mathrm {F}_{ij}$ represent the volume of information flow from node $i$ to node $j$ .",
"Based on the flow matrix, the effective distance $d_{\\mathsf {Eff}}(i,j)$ from node $i$ to node $j$ is defined as $d_{\\mathsf {Eff}}(i,j) = 1-\\log \\frac{\\mathrm {F}_{ij}}{\\sum _l \\mathrm {F}_{lj}},$ where $d_{\\mathsf {Eff}}(i,j)$ satisfies $d_{\\mathsf {Eff}(i,j)}\\ge 1$ .",
"Information flow has been defined differently in various networks.",
"For instance, it can be the passenger flux in global mobility networks or the transport flow in transportation networks .",
"In FNNs and TNNs it is the news flow among nodes (users) in the network which can be defined as (i) the total number of news stories both users have spread, i.e., $\\mathrm {F}_{ij}=\\sum _X \\mathbf {B}(e_{ij} \\in \\mathrm {E}_X)$ , or (ii) the overall number of times both users have at least spread the same news stories, i.e., $\\mathrm {F}_{ij}=\\sum _X \\mathbf {B}(e_{ij} \\in \\mathrm {E}_X) \\times \\min \\lbrace \\mathbf {T}(u_i, X), \\mathbf {T}(u_j, X) \\rbrace $ .",
"The diameter of each FNN and TNN based on effective distance is then equivalent to the minimum [sum of] effective distance between the two most distant spreaders within the network.",
"We include diameters computed using geodesic and effective distances as features representing Farther-Distance Pattern."
],
[
"Stronger-Engagement Pattern",
"The statistics in have revealed that fake news stories can engage users more compared to true news stories.",
"Note that a user may decide to engage with a fake news story (e.g., post it) more than one time, such “more engagements” can be attributed to the number of users engaging with fake news, which has been summarized as More-Spreader Pattern investigated in Section REF , and/or to the number of times each user engages with a fake news story, leading to the following pattern: Pattern 3 (Stronger-Engagement Pattern) Spreaders engage more strongly with fake news than with true news.",
"To quantify the “engagements” of users for each news story, one can concentrate on (I) group level engagements, i.e., the engagements of all spreaders, and (II) individual level engagements, i.e., the engagements of a single spreader.",
"I.",
"Group Engagements.",
"On a group level, quantifying spreader engagements for a certain news story can be equivalent to counting the total number of times that the news story has been spread.",
"With specific user attributes (susceptible or normal), such engagements can be further quantified as the (i) number or (ii) proportion of times that the news story has been spread by susceptible users, as well as (iii) number or (iv) proportion of times that the news story has been spread by normal users.",
"II.",
"Individual Engagements.",
"Individual engagements of a news story can be evaluated by the average spreading frequencies of (susceptible, normal, all) users who have participated in the news propagation.",
"In this case, the impact of the number of such news spreaders (i.e., More-Spreaders Pattern) is divided and removed.",
"All above ways of representing fake news patterns are on the level of nodes, e.g., individual engagement, and the whole network, e.g., network diameter.",
"Next we will specify how to represent Denser-Networks Pattern for fake news detection, which will be represented at different network levels: ego, triad and community."
],
[
"Denser-Network Pattern",
"Research in social psychology such as homophily and social validation theory suggests that connected users in social networks share similar attributes, interests and behaviors, e.g., sharing the same news article.",
"On the other hand, malicious users often form cohesive groups, taking collective action that are more purposeful than normal users , .",
"These fundamental theories suggest the possibility to assume that fake and true news articles can be distinguished by the relationships among their corresponding spreaders, which can be summarized as: Pattern 4 (Denser-Network Pattern) Fake news spreaders form denser networks compared to truth spreaders.",
"To capture the “density” of connections among news spreaders, we analyze news networks at different levels: (I) ego, (II) triad and (III) community levels.",
"I. Ego Level.",
"At the ego level, to compute density of networks formed by users that have engaged with a certain news story, we look at the numbers and proportions of connections that these users (spreaders) have (i) generally formed with other spreaders, and (ii) specifically with other normal or susceptible spreaders.",
"i.",
"General Ego Relations.",
"We include as a feature the total number of ego relationships among spreaders for each news story, i.e., the number of edges within each FNN and TNN ($|\\mathrm {E}_X|$ ).",
"To eliminate the impact of the number of news spreaders (i.e., More-Spreaders Pattern), for each FNN or TNN $\\mathrm {G}_X$ we also record $|\\mathrm {E}_X|/|\\mathrm {V}_X|$ and $|\\mathrm {E}_X|/\\binom{|\\mathrm {V}_X|}{2}$ , which calculate the average number of ego relationships per spreader and network density, respectively.",
"Here, ${|\\mathrm {V}_X|}$ is the number of spreaders (nodes) in $\\mathrm {G}_X$ and $\\binom{|\\mathrm {V}_X|}{2}$ is the number of edges within a fully connected version of $\\mathrm {G}_X$ .",
"ii.",
"Specific Ego Relations.",
"Labeling users as susceptible or normal allows one to group all directed ego relationships into four subsets: (1) $\\mathrm {E}_{NN}$ containing relationships from a normal user to a normal user, (2) $\\mathrm {E}_{NS}$ containing relationships from a normal user to a susceptible one, (3) $\\mathrm {E}_{SN}$ containing relationships from a susceptible user to a normal one, (4) $\\mathrm {E}_{SS}$ containing relationships from a susceptible user to a susceptible one.",
"We include the number and proportion of each type of edges within a FNN or TNN as features being used for fake news detection.",
"In addition, each edge $e_{ij}$ can be also classified into one of the following set: (1) $\\mathrm {E}_{\\triangle >0}$ if $\\triangle = \\mathbf {S}(v_i)-\\mathbf {S}(v_j)>0$ , (2) $\\mathrm {E}_{\\triangle =0}$ if $\\mathbf {S}(v_i)-\\mathbf {S}(v_j)=0$ , (3) $\\mathrm {E}_{\\triangle <0}$ if $\\mathbf {S}(v_i)-\\mathbf {S}(v_j)<0$ which does not require partitioning users into susceptible or normal ones.",
"We also include as features the number and proportion of each above type of edges within a FNN or TNN.",
"II.",
"Triad Level.",
"Triads (a set of three connected users) are the most basic subgraphs of networks.",
"Similar to our study at the ego level, we investigate (i) general triads and (ii) specific triads formed between [susceptible and normal] users within networks.",
"i.",
"General Triads.",
"One simple way to represent the Denser-Network pattern is to directly count the total number of triads $|\\mathrm {Tr}_X|$ within a $\\mathrm {G}_X$ .",
"Similarly, to control for More-Spreaders Pattern, we also include as features the value of $|{\\mathrm {Tr}_X}|/|\\mathrm {V}_X|$ and $|\\mathrm {Tr}_X|/\\binom{|\\mathrm {V}_X|}{3}$ where $\\binom{|\\mathrm {V}_X|}{3}$ is the number of triads within a fully connected version of $\\mathrm {G}_X$ .",
"Figure: Specific Triads.",
"NN indicates normal users and SS indicates susceptible users.",
"A→BA\\rightarrow B denotes AA follows BB.ii.",
"Specific Triads.",
"Regarding each user as either a susceptible or normal user, we can have twelve different triads to be further explored (shown in Figure REF ).",
"We include as features the number and proportion of every type of triads within each FNN and TNN.",
"III.",
"Community Level.",
"In networks, a community structure refers to the occurrence of groups of nodes in a network that are more densely connected internally than with the rest of the network.",
"Therefore, the number and proportion of communities within each FNN and TNN can be used to represent Denser-Networks Pattern and, broadly speaking, should be negatively correlated to the network density.",
"As features, we include the number of communities $|\\mathrm {M}_X|$ within each FNN and TNN, and the proportion of communities (assuming in the worst case each node is its own community) which removes the impact of the number of news spreaders, i.e., the value of $|\\mathrm {M}_X| / |\\mathrm {V}_X|$ .",
"Note that $|\\mathrm {M}_X|$ can be obtained either from (i) global or (ii) local perspective.",
"From a global perspective, communities that nodes (spreaders) belong to within a FNN or TNN, as a subgraph of the social network, are based on the structure of the overall social network.",
"From a local perspective, communities can be detected within a FNN or TNN.",
"We include counts and proportion features for both types of communities.",
"Table: Network-based Pattern-driven Feature Set for Fake News DetectionIntegrated Representation of Patterns.",
"To represent each fake news pattern, we have used network information such as network diameters, the number of news spreaders (size), and the number of relationships among the spreaders (density).",
"Networks with various diameters, sizes and densities exhibit various overall structures.",
"Hence, the overall network structure can be regarded as the integrated representation of all related patterns.",
"On the other hand, including such “structure” features to detect fake news helps to evaluate if the fake news patterns and their representations defined in this section have well captured the difference of dissemination between fake news and the truth.",
"To quantify such “structure”, one can compare the similarities among FNNs and TNNs, where graph kernel and graph embedding , methods can be useful.",
"Here, we consider FNNs and TNNs as labeled graphs for further comparison, where node labels can be either (i) user identities or (ii) user attributes (susceptible or normal).",
"Overall, Table REF presents all features defined and involved in our work to detect fake news, and their corresponding formulations for reproducibility.",
"Fake news patterns in networks have been specified as well as how they can be represented as a set of quantifiable and meaningful features.",
"In this section, various experiments are conducted to verify the effectiveness of the proposed approach in detecting fake news.",
"We first present the experimental setup in Section REF , followed by the evaluations of experimental results in Section REF .",
"Table: Data Statistics"
],
[
"Experimental Setup",
"We detail data used in experiments in Section REF , followed by how data is prepared for experiments in Section REF , and the baselines which the proposed approach is compared with in Section REF .",
"Our experiments are conducted on two public benchmark datasets of fake news detection .",
"News articles in these datasets are collected from PolitiFact and BuzzFeed, respectively.",
"Ground truth labels (true or fake) of news articles in both datasets are provided by fact-checking experts.",
"In addition to (i) news content and labels, both datasets also provide information on (ii) social network of Twitter which contains Twitter users and their following relationships, i.e., user-user relationships, and (iii) how the news has propagated (tweeted/re-tweeted) by users, i.e., news-user relationships.",
"Based on the original datasets, we further identify triads and communities in the social network.",
"Communities are detected using Louvain algorithm, a fast and widely-accepted modularity-based community detection algorithm .",
"Statistics of two datasets are shown in Table REF ."
],
[
"Data Preparation",
"Following dataset collection, feature values are computed for both datasets, which will be utilized in a supervised learning framework for fake news detection.",
"However, an extra step is necessary to take when computing user susceptibility scores.",
"In Section , two ways are defined for determining user susceptibility [to fake news] (see Equation (REF ) and (REF ), respectively).",
"Both ways rely on the historical information of users on how they previously engaged in fake news dissemination, where the news labels (true or fake) are necessary in the calculation.",
"To avoid information leakage (i.e., features having an unfair prior knowledge of labels), when dividing a dataset into the training and testing one, we dynamically calculate user susceptibility by using the historical information provided in training dataset, rather than the whole dataset.",
"For users with no historical information in training dataset, we treat their susceptibility as the threshold value, indicating that their susceptibility to fake news is unknown."
],
[
"Baselines",
"The performance of the proposed method is compared with several benchmark fake news detection methods on the same datasets.",
"These methods include (1) content-based (linguistic) models, which rely on non-latent (, ) or latent representation (, ) of news content, (2) network-based models (), which investigate information revealed in news propagation, and hybrid models (), which utilize both content and network information to detect fake news.",
"I. Pérez-Rosas et al.",
"propose a comprehensive linguistic model for fake news detection, involving the following features: (i) $n$ -grams (i.e., uni-grams and bi-grams) and (ii) CFGs based on TF-IDF encoding; (iii) word and phrase proportions referring to all categories provided by LIWC; and (iv) readability.",
"Features are computed and used to predict fake news within a supervised machine learning framework.",
"II.",
"Zhou et al. .",
"In our previous study, forensic psychological theories are studied and used to detect fake news in a supervised learning framework, which provide the evidence of distinguishing fake news in content style from the truth.",
"Such content style is captured by the frequency of (i) lexicons relying on Bag-Of-Words (BOW) model, (ii) Part-Of-Speech (POS) tags and Context Free Grammers (CFGs) at syntax-level, (iii) Rhetorical Relationships (RRs) at discourse-level, and by assessing a set of theory-driven (iv) DisInformation-related Attributes (DIAs) and (v) ClickBait-related Attributes (CBAs) at semantic-level.",
"III.",
"Castillo et al.",
"design features that exploit information from user profiles, tweets and propagation trees to evaluate news credibility within a supervised learning framework.",
"Specifically, these features are based on (i) quantity, sentiment, hash-tag and URL information from user tweets, (ii) user profiles such as registration age, (iii) news topics through mining tweets of users, and (iv) propagation trees (e.g., the number of propagation trees for each news topic).",
"IV.",
"Shu et al.",
"detect fake news by exploring and embedding the relationships among news articles, publishers and spreaders on social media.",
"Such embedding involves (i) news content by using non-negative matrix factorization, (ii) users on social media, (iii) news-user relationships (i.e., user engagements in spreading news articles), and (iv) news-publisher relationships (i.e., publisher engagements in publishing news articles).",
"Fake news detection is then conducted within a semi-supervised machine learning framework.",
"Additionally, fake news detection based on latent representation of news articles is also investigated in comparative studies, where we consider as baselines supervised classifiers with features being (V) Word2Vec and (VI) Doc2Vec embedding of news articles.",
"Figure: General Performance of Fake News Detection by Using Different Classifiers, where random forests perform best among all on both datasets.Table: General Performance of Fake News Detection Methods.",
"The proposed network-based approach can perform relatively well compared to the content-based (, , , ) and network-based approaches () among baselines.",
"Compared to the hybrid one (), the proposed approach can be comparable with it and can outperform it when introducing the linguistic features in the proposed approach (“Our Approach + ”).Table: Pattern Performance in Fake News Detection.More-Spreader Pattern and Stronger-Engagement Pattern perform best compared to the others when being separately utilized to detect fake news.When combining different patterns, their performance is in general better than when separately using them, and than when using network similarity, as a mix of patterns from a higher view.Table: Conclusion"
]
] | 1906.04210 |
[
[
"Direct Characterization of Spectral Stability of Small Amplitude\n Periodic Waves in Scalar Hamiltonian Problems Via Dispersion Relation"
],
[
"Abstract Various approaches to studying the stability of solutions of nonlinear PDEs lead to explicit formulae determining the stability or instability of the wave for a wide range of classes of equations.",
"However, these are typically specialized to a particular equation and checking the stability conditions may not be not straightforward.",
"We present results for a large class of problems that reduce the determination of spectral stability of a wave to a simple task of locating zeros of explicitly constructed polynomials.",
"We study spectral stability of small-amplitude periodic waves in scalar Hamiltonian problems as a perturbation of the zero-amplitude case.",
"A necessary condition for stability of the wave is that the unperturbed spectrum is restricted to the imaginary axis.",
"Instability can come about through a Hamiltonian-Hopf bifurcation, i.e., of a collision of purely imaginary eigenvalues of the Floquet spectrum of opposite Krein signature.",
"In recent work on the stability of small-amplitude waves the dispersion relation of the unperturbed problem was shown to play a central role.",
"We demonstrate that the dispersion relation provides even more explicit information about wave stability: we construct a polynomial of half the degree of the dispersion relation, and its roots directly characterize not only collisions of eigenvalues at zero-amplitude but also an agreement or a disagreement of their Krein signatures.",
"Based on this explicit information it is possible to detect instabilities of non-zero amplitude waves.",
"In our analysis we stay away from the possible instabilities at the origin of the spectral plane corresponding to modulation or Benjamin-Fair instability.",
"Generalized KdV and its higher-order analogues are used as illustrating examples."
],
[
"Introduction",
"We study the spectral stability of small-amplitude periodic traveling waves in scalar Hamiltonian partial differential equations: $u_t = \\partial _x \\frac{\\delta H}{\\delta u}\\, .$ Here $u = u(x,t) = u(x+L,t), \\quad x \\in [0,L], \\quad t > 0, \\qquad \\mbox{and} \\quad H = \\int _0^L \\mathcal {H} (u, u_x, \\dots ) \\, dx$ is the Hamiltonian with density $\\mathcal {H}$ .",
"Without loss of generality, we let $L = 2\\pi $ .",
"This class of equations includes the Korteweg–de Vries (KdV), the generalized and modified KdV equations, the Kawahara equation, and other equations that arise in the study of dispersive problems in water waves, plasma physics etc.",
"[1], [13].",
"We assume that (REF ) has a trivial solution, i.e., ${\\delta H}/{\\delta u} = 0$ for $u = 0$ , and $H$ has an expansion $H = H^0 + H^1$ , where $H^0$ is the quadratic part of $H$ and $H^1$ contains the higher order terms: $H^0 = -\\frac{1}{2} \\int _0^{2\\pi } \\sum _{j=0}^{N} \\alpha _j (\\partial _x^{j} u)^2\\, .$ As a consequence, all linear terms in (REF ) are of odd degree, as even degree terms would introduce dissipation.",
"We assume that $N$ is a finite positive integer, and $\\alpha _j \\in \\mathbb {R}$ .",
"These assumptions exclude problems like the Whitham equation [7] ($N = \\infty $ ) which remains a topic of investigation.",
"The now-standard approach to examine the stability of waves in Hamiltonian problems with symmetries is the theory developed by Vakhitov and Kolokolov [25] and Grillakis, Shatah, and Strauss [9], [10], which allows for the determination of spectral stability of waves of arbitrary amplitude.",
"In that setup, spectral stability implies orbital (nonlinear) stability under certain conditions, emphasizing the importance of the spectral information of the linearized problem.",
"Extensions of these results are found in [16], [18], [22], [23].",
"Periodic problems within the same framework were considered in [15], [11].",
"The use of any of these results relies on index theory requiring additional information about the PDE.",
"That information is typically provided, for instance, by assuming something about the dimension of the kernel of the linearized problem.",
"For small-amplitude waves extra information is often obtained through a perturbation of the zero-amplitude problem.",
"We avoid index theory and study directly the collision of eigenvalues.",
"The parallel work [24] illustrates how small-amplitude information is used to characterize the (in)stability of the waves.",
"Here, we reduce the spectral stability problem for small-amplitude waves to the investigation of zeros of certain recurrently-defined polynomials, which appear in the theory of proper polynomial mappings [3] and in orthogonal polynomial theory [21].",
"To our knowledge, the connection between stability theory and these polynomials is new to the literature.",
"Our approach allows us to rigorously analyze the stability of KdV-type equations, including the generalized KdV equation (gKdV), its higher order analogues, and also the two-term balanced KdV equation.",
"The results agree with the existing literature of spectral stability of periodic waves for gKdV and in the case of balanced high-order KdV equations they confirm and extend the analytical and numerical predictions in [24].",
"Our method is closely related to the results in [7], where the spectrum of small-amplitude periodic solutions of Hamiltonian PDEs is determined directly from the dispersion relation of the PDE linearized about the zero solution.",
"Our theory adds to the results in [7], and provides a simple and, importantly, a natural framework for studying the spectral stability of waves by perturbative methods.",
"We refer the reader to [7] and [24] for a number of numerical illustrations of the results presented here for KdV-type equations.",
"The spectral stability of small-amplitude waves bifurcating from the trivial solution $u=0$ at a critical velocity $c=c_0$ can be examined using regular perturbation theory of the spectrum of (REF ) linearized about $u=0$ at $c = c_0$ .",
"Our assumptions guarantee that $u=0$ is spectrally stable, i.e., the spectrum of the linearized problem is restricted to the imaginary axis, since (REF ) is Hamiltonian.",
"In the periodic setting the whole spectrum of the zero-amplitude problem is needed.",
"However, Floquet theory [17] allows to decompose the continuous spectrum to an infinite union of sets of discrete eigenvalues of eigenvalue problems parametrized by the Floquet multiplier $\\mu $ .",
"An important scenario for instability of small-amplitude waves on the bifurcation branch comes about through Hamiltonian-Hopf bifurcations [20], [26] producing symmetric pairs of eigenvalues off the imaginary axis, i.e., exponentially growing and therefore unstable modes.",
"Such bifurcations require non-simple eigenvalues of the linearized problem at zero amplitude, i.e., “collided eigenvalues”.",
"Furthermore, such colliding eigenvalues can split off from the imaginary axis only if they have opposite Krein signatures [20], [19].",
"Note that we stay away from the origin of the spectral plane and thus we do not consider modulation or Benjamin-Feir instability.",
"Both the location of the eigenvalues and their Krein signatures are characterized by the dispersion relation of the linearized problem [7].",
"We show that even the collision of eigenvalues and the agreement of their signatures is directly characterized by the dispersion relation.",
"This characterization is through the roots of a polynomial, which is a reduction of the dispersion relation to a polynomial approximately half its degree.",
"This is a surprising fact as it is by no means clear why such a characterization is possible, as the collisions of eigenvalues and their types are not itself objects that can be identified directly algebraically, particularly with a simpler algebraic relation than the eigenvalues themselves."
],
[
"General Setting",
"We follow the steps outlined in Section III of [7].",
"We use a coordinate transformation $x \\rightarrow x-ct$ to a frame moving with the wave, $\\partial _t u = \\partial _x \\frac{\\delta H}{\\delta u} + c\\partial _ x u = \\partial _x \\left( \\frac{\\delta H}{\\delta u} + c u\\right) =\\partial _x \\frac{\\delta H_c}{\\delta u},$ where $H_c$ is the modified Hamiltonian.",
"The quadratic part of $H_c$ is $H_c^0 = \\frac{c}{2} \\int _0^{2\\pi } u^2 \\, dx - \\frac{1}{2} \\int _0^{2\\pi } \\sum _{j=0}^{N} \\alpha _j (\\partial _x^{j} u)^2 \\, dx\\, .$ Traveling wave solutions of (REF ) are stationary solutions $U(x)$ of (REF ) and stationary points of $H_c$ ."
],
[
"Perturbation from the trivial state. Dispersion relation",
"For all $c\\in \\mathbb {R}$ , eq.",
"(REF ) has the trivial solution $u(x,t) = 0$ .",
"We linearize (REF ) about the zero solution to obtain an equation for the perturbation $v = v(x,t)$ from the trivial state $\\partial _t v = c\\partial _x v - \\sum _{j=0}^{N} (-1)^j \\alpha _j \\partial _x^{2j+1} v \\, .$ We decompose $v$ into a Fourier series in $x$ , $v = \\sum _{k=-\\infty }^{\\infty } \\exp (ikx) \\hat{v}_k$ , to obtain decoupled evolution equations for each of the Fourier coefficients $\\hat{v}_k = \\hat{v}_k(t)$ : $\\partial _t \\hat{v}_k=-i \\Omega (k)\\hat{v}_k\\qquad k \\in \\mathbb {Z},$ where $\\Omega (k)$ is given by $\\Omega (k) = \\omega (k) - ck = \\sum _{j=0}^N \\eta _j k^{2j+1}\\, , \\qquad \\omega (k) = \\sum _{j=0}^N \\alpha _j k^{2j+1}\\,, \\qquad \\eta _j = \\alpha _j - c\\delta _{j1}\\, ,$ is the dispersion relation of (REF ), obtained by letting $v(x,t) = \\exp (ikx - i\\Omega t)$ in (REF ).",
"Here $\\omega = \\omega (k)$ is the dispersion relation in the original frame of reference corresponding to (REF )–(REF ).",
"Note that $\\omega (k)$ is an odd function."
],
[
"Non-zero amplitude branches",
"Next, we discuss non-zero amplitude periodic solution branches of (REF ) bifurcating from the trivial state.",
"A requirement for this is that a non-trivial stationary solution of (REF ) exists, i.e., $\\Omega (k) = 0$ , for $k \\in \\mathbb {N}$ , since we have imposed that the solutions are $2\\pi $ periodic.",
"Thus $c = c_k = \\frac{\\omega (k)}{k}, \\qquad k \\in \\mathbb {N}.$ For simplicity, we assume that a unique bifurcating branch emanates from $c=c_k$ .",
"The solutions with $k > 1$ are $2\\pi / k$ periodic.",
"We focus on $k = 1$ , i.e., $c = \\omega (1)$ .",
"The cases with $k >1$ may be treated analogously (see Section REF for a discussion of the $k\\ge 2$ in the context of gKdV equation)."
],
[
"Floquet theory at zero amplitude",
"Using Floquet theory [4], [17] the spectral stability of the non-trivial solution $U = U(x)$ of (REF ) on the bifurcation branch starting at $c$ is determined by the growth rates of perturbations of the form $v(x,t) = e^{\\lambda t}V(x),~~V(x) = e^{i\\tilde{\\mu } x}\\sum _{n = -\\infty }^{\\infty } a_n e^{i nx}\\, .$ Here $\\tilde{\\mu } \\in (-1/2, 1/2]$ is the Floquet exponent.",
"Using (REF ) for the zero-amplitude case, $\\lambda = \\lambda _n^{(\\tilde{\\mu })}= -i \\Omega (n+\\tilde{\\mu }) = - i \\omega (n+\\tilde{\\mu }) + i (n+\\tilde{\\mu }) c, \\qquad n \\in \\mathbb {Z}\\, .$ The expression (REF ) is an explicit expression for the spectrum of the linearized stability problem for solutions of zero amplitude.",
"Next, we examine how the spectrum of the linearization changes as the solution bifurcates away from zero amplitude."
],
[
"Collisions of eigenvalues, Hamiltonian-Hopf bifurcations",
"After Floquet decomposition (REF ), the elements of the spectrum become eigenvalues of the $\\tilde{\\mu }$ -parameterized operator obtained by replacing $\\partial _x\\rightarrow \\partial _x+i \\tilde{\\mu }$ in the linear stability problem.",
"The eigenfunctions associated with these eigenvalues are (quasi)periodic and are bounded on the whole real line, see [17], [6] for details.",
"For zero amplitude, the spectrum (REF ) is on the imaginary axis.",
"Instabilities for small amplitude come about through collisions of purely imaginary eigenvalues at zero amplitude for a fixed value of $\\tilde{\\mu }$ .",
"Away from the origin, eigenvalues generically split off from the axis through the Hamiltonian-Hopf bifurcations [20], [26] as the solution amplitude increases.",
"Each such Hamiltonian-Hopf bifurcation produces a pair of eigenvalues off the imaginary axis that is symmetric with respect to the imaginary axis, thus yielding an exponentially growing eigenmode.",
"From (REF ), it is easy to detect eigenvalue collisions away from the origin.",
"They correspond to solutions of $\\lambda _{n_1}^{(\\tilde{\\mu })} = \\lambda _{n_2}^{(\\tilde{\\mu })} \\ne 0$ , $n_1,n_2 \\in \\mathbb {Z}$ , $n_1 \\ne n_2$ , $\\tilde{\\mu } \\in (-1/2, 1/2]$ , i.e., $-i\\Omega (n_1+\\tilde{\\mu }) = -i\\omega (n_1+\\tilde{\\mu }) + i (n_1+\\tilde{\\mu }) c = -i\\omega (n_2+\\tilde{\\mu }) + i (n_2+\\tilde{\\mu }) c =- i\\Omega (n_2 +\\tilde{\\mu })\\, ,$ where $c=c_1$ is given by (REF ) with $k=1$ .",
"Solving this equation results in values of $\\tilde{\\mu }$ and $n_1$ for which $\\lambda ^{(\\tilde{\\mu })}_{n_1}$ is an eigenvalue colliding with another one.",
"Typically this is done by solving (REF ) for $\\tilde{\\mu }$ for different fixed $n_1$ ."
],
[
"Krein signature",
"A necessary condition for two eigenvalues colliding on the imaginary axis to cause a Hamiltonian-Hopf bifurcation is that the eigenvalues have opposite Krein signatures.",
"The Krein signature is the sign of the energy of the eigenmode associated with the eigenvalue.",
"For a collision of eigenvalues to produce an instability this energy needs to be indefinite: a definite sign would entail bounded level sets of the energy, leading to perturbations remaining bounded.",
"For Hamiltonian systems with quadratic part given by (REF ) the eigenmode of the form $v(x,t) = a_n \\exp \\left[i(n+\\tilde{\\mu }) x + \\lambda _n^{(\\tilde{\\mu })}t\\right] + \\mbox{c.c.",
"}$ , where c.c.",
"stands for complex conjugate of the preceding term, contributes to $H_c^0$ the relative energy (see [7]) $H_c^0|_{(n, \\tilde{\\mu })} \\sim - |a_p|^2\\, \\frac{\\Omega (n+\\tilde{\\mu })}{n+\\tilde{\\mu }}.$ Thus the Krein signature of $\\lambda _n^{(\\tilde{\\mu })}$ is given by $\\kappa (\\lambda _n^{(\\tilde{\\mu })}) = -\\mathop {\\rm sign}\\nolimits \\left( \\frac{\\Omega (n+\\tilde{\\mu })}{n+\\tilde{\\mu }} \\right)\\, .$ A simple characterization of agreement of the signatures of two colliding eigenvalues $\\lambda _{n_1}^{(\\tilde{\\mu })}$ and $\\lambda _{n_2}^{(\\tilde{\\mu })}$ immediately follows.",
"Proposition 1 Let two eigenvalues $\\lambda _{n_1}^{(\\tilde{\\mu })} = \\lambda _{n_2}^{(\\tilde{\\mu })} = \\lambda \\ne 0$ , $n_1 \\ne n_2$ , of the Bloch wave decomposition (REF ) of (REF ) coincide, i.e., (REF ) holds.",
"Then the product of Krein signatures of the eigenvalues is characterized by the sign of the quantity $q = q_{n_1,n_2}^{(\\tilde{\\mu })} = \\frac{\\Omega (n_1+\\tilde{\\mu })}{n_1+\\tilde{\\mu }} \\cdot \\frac{\\Omega (n_2+\\tilde{\\mu })}{n_2+\\tilde{\\mu }}= \\frac{|\\lambda |^2}{(n_1+\\tilde{\\mu })(n_2+\\tilde{\\mu })}\\,$ Let $Z = Z_{n_1,n_2}^{(\\tilde{\\mu })} = (n_1+\\tilde{\\mu }) (n_2+\\tilde{\\mu })$ .",
"Since $\\lambda \\ne 0$ the sign of $Z$ characterizes an agreement of Krein signatures of the coinciding eigenvalues: $\\kappa (\\lambda _{n_1}^{(\\tilde{\\mu })}) \\kappa (\\lambda _{n_2}^{(\\tilde{\\mu })}) = \\mathop {\\rm sign}\\nolimits (q) = \\mathop {\\rm sign}\\nolimits \\left[(n_1+\\tilde{\\mu })(n_2+\\tilde{\\mu })\\right] = \\mathop {\\rm sign}\\nolimits (Z)\\, .$ We denote $\\mu := n_2 + \\tilde{\\mu }, \\qquad \\mbox{and} \\qquad \\triangle n:= n_1 - n_2\\, .$ Here $\\triangle n> 0$ .",
"Then $Z = \\mu (\\triangle n+\\mu )$ and the collision condition (REF ) reduces to $\\Omega (\\triangle n+ \\mu ) = \\Omega (\\mu )\\, .$"
],
[
"Recurrent Sequences of Polynomials",
"Before we revisit (REF ) in the next section, we need to define some particular recurrent sequences of polynomials.",
"Lemma 1 Let $a, b \\in \\mathbb {C}$ , $m \\in \\mathbb {N}_0$ , and $t_m = a^m + (-b)^m\\, .$ Then $t_{m+1} = (a-b) t_m + (ab)t_{m-1}\\, ,\\qquad m \\ge 1.$ $t_{m+1} = (a-b)(a^m + (-1)^m b^m) + ab (a^{m-1} + (-1)^{m-1} b^{m-1}) =(a-b)t_m + (ab)t_{m-1}\\, .$ Since $t_0 = 2$ and $t_1 = a-b$ , by induction all $t_m$ can be written as polynomials in the two variables $a-b$ and $ab$ , $t_m = t_m (a-b,ab)$ .",
"Further, $t_m$ is a homogeneous polynomial in $a$ and $b$ of degree $m$ .",
"We introduce $s_m$ by $t_m = (a-b)^m s_m (\\gamma )$ , i.e., $s_m = s_m(\\gamma ) := \\frac{t_m (a-b,ab)}{(a-b)^m}\\, , \\qquad \\mbox{with $\\gamma := \\displaystyle \\frac{ab}{(a-b)^2}$.",
"}$ The sequence $s_m$ is characterized recursively by $s_{m+1} & = & s_m + \\gamma s_{m-1}\\, , \\quad m \\ge 1, \\qquad s_0= 2, \\quad s_1 = 1, $ which shows that $s_m$ is a polynomial in $\\gamma $ of degree $m/2$ ($m$ even) or $(m-1)/2$ ($m$ odd).",
"One can easily see that $s_2(\\gamma ) = 1+ 2\\gamma , \\quad s_3(\\gamma ) = 1 + 3\\gamma , \\quad s_4(\\gamma ) = 1 + 4\\gamma + 2\\gamma ^2, \\quad s_5(\\gamma ) = 1 + 5\\gamma + 5\\gamma ^2\\, , \\\\s_6(\\gamma ) = 1+ 6\\gamma + 9 \\gamma ^2 + 3\\gamma ^3, \\qquad s_7(\\gamma ) = 1 + 7\\gamma + 14\\gamma ^2 + 7 \\gamma ^3\\, .$ Solving the recurrence relationship, $s_m (\\gamma ) = \\psi _+^m + \\psi _-^m\\, , \\quad m \\ge 0, \\qquad \\psi _{\\pm } := \\frac{1}{2}\\left(1 \\pm \\sqrt{1 + 4\\gamma }\\right)\\, .$ That implies $s_m(0)=1,~~s_m(-1/4)=2^{1-m}.$ Note that $s_m(\\gamma )$ is increasing on $(-1/4,0)$ as $s_m^{\\prime }(\\gamma ) = \\frac{m}{\\sqrt{1+4\\gamma }} (\\psi _+^{m-1} - \\psi _-^{m-1}) > 0\\, .$ A few lemmas characterizing the behavior of $s_m(\\gamma )$ are proved in the Appendix."
],
[
"Reduction of the Equation for Signatures of Colliding Eigenvalues",
"We prove that for scalar Hamiltonian problems (REF )–(REF ) of order $2N+1$ , the polynomial equation (REF ) characterizing the collision of eigenvalues with indices $n+\\mu $ and $\\mu $ at zero-amplitude resulting in Hamiltonian-Hopf bifurcations, and thus instability of non-zero amplitude periodic waves, can be expressed as a polynomial of degree $N$ in a real variable $\\gamma $ with coefficients independent of $\\mu $ , where $\\gamma $ is defined as $\\gamma :=\\frac{ \\mu (\\triangle n + \\mu )}{(\\triangle n)^2}\\, .",
"$ Theorem 1 Let $\\Omega := \\Omega (k)=\\sum _{j=0}^N \\eta _j k^{2j+1}\\, ,$ be an odd polynomial of degree $2N+1$ , $\\eta _j \\in \\mathbb {C}$ for $j=0,\\dots , N$ .",
"Then $\\Omega (\\triangle n+\\mu ) - \\Omega (\\mu ) = \\sum _{j=0}^{N} \\eta _j ({\\triangle n})^{2j+1}s_{2j+1}\\left( \\gamma \\right)\\, ,$ where the polynomial $s_{2j+1} = s_{2j+1}(\\gamma )$ of degree $j$ is defined recurrently by (REF ).",
"The claim follows immediately by (REF ) and Lemma REF by setting $a := \\triangle n+ \\mu $ and $b:=\\mu $ : $\\Omega (a) - \\Omega (b) = \\sum _{j=0}^{N} \\eta _j (a^{2j+1} - b^{2j+1})= \\sum _{j=0}^{N} \\eta _j t_{2j+1}(a-b,ab)=\\sum _{j=0}^{N} \\eta _j ({\\triangle n})^{2j+1}s_{2j+1}(\\gamma )\\,.$ As before, the collision condition (REF ) expressed using (REF ) is solved for $\\gamma $ for different fixed values of $\\triangle n$ .",
"After solving for $\\gamma $ , it is necessary to check that $\\gamma $ gives rise to a real value of $\\mu $ by solving the quadratic equation with the unknown $\\mu $ : $\\mu (\\mu + \\triangle n) = \\gamma (\\triangle n)^2 \\, .$ Thus $\\mu _{1,2} = \\frac{-\\triangle n\\pm \\sqrt{(\\triangle n)^2 + 4\\gamma (\\triangle n)^2 }}{2} = \\frac{\\triangle n}{2}\\left(-1 \\pm \\sqrt{1 + 4\\gamma } \\right)\\, .$ By Proposition (REF ) we are interested in negative values of $\\gamma $ that characterize a possible coincidence of two eigenvalues of opposite signature, as $\\gamma $ has by (REF ) the same sign as $Z$ in (REF ).",
"Then any root $\\gamma \\in [-1/4,0)$ corresponds to a collision of two eigenvalues of opposite signature.",
"If $\\gamma < -1/4$ , $\\gamma $ does not correspond to a collision of two purely imaginary eigenvalues as $\\mu $ is not real.",
"If $\\gamma >0$ then there is a collision of two eigenvalues of the same signature.",
"If $\\gamma = 0$ the collision is located at the origin of the spectral plane, i.e., it does not correspond to the Hamiltonian-Hopf bifurcation.",
"We have proved the following main theorem characterizing the spectral stability of small-amplitude traveling waves of (REF ).",
"Theorem 2 Consider a scalar $2\\pi $ -periodic Hamiltonian partial differential equation of the form (REF ) and assume that $u = 0$ is a spectrally stable solution.",
"Let (REF ) be the dispersion relation of the equation linearized about $u = 0$ in a reference frame moving with the velocity $c$ .",
"Then a branch of traveling wave solutions of (REF ) with velocity $c$ bifurcates from the trivial solution at $c = \\omega (1)$ , see (REF ).",
"A necessary condition for a Hamiltonian-Hopf bifurcation at zero-amplitude characterizing a loss of spectral stability of small-amplitude solutions on the bifurcating branch is that $\\sum _{j=0}^{N} \\eta _j ({\\triangle n})^{2j+1}s_{2j+1}(\\gamma ) = 0\\,$ has a root $\\gamma $ , $\\gamma \\in [-1/4, 0)$ ."
],
[
"Generalized KdV Equations",
"As a simple example illustrating an application of Theorem REF to study spectral stability of small-amplitude periodic traveling waves, we consider the generalized KdV equation (gKdV) $\\partial _t v + \\alpha \\partial _x^3 v + \\partial _x f(v) = 0\\, ,$ and the generalized higher-order KdV equation ($p \\ge 2$ ) $\\partial _t v + \\alpha \\partial _x^{2p+1} v + \\partial _x f(v) = 0\\, .$ Here we assume $f(0) = 0$ and periodic boundary conditions, $x\\in [0,2\\pi ]$ .",
"Within this work we study high-frequency instabilities, staying away from the origin in the spectral plane, i.e., we do not discuss the modulational or Benjamin-Feir instability.",
"For simplicity we consider (REF ) first and then discuss the case of (REF ) as the reduction process and the results are completely analogous.",
"We will pay particular attention to the case of KdV equation with $f(v) = v^2$ in (REF ).",
"For a detailed history of stability results of periodic traveling waves for KdV, mKdV (equation (REF ) with $f(u) = u^3$ ), and gKdV we refer the reader to [2], [15], see also [11], [14], [5], see also [7], Section 3.1, for numerical results illustrating the theory developed here.",
"The results in the literature can be shortly summarized as: periodic traveling waves are spectrally stable away from the origin of the spectral plane (with the exception of cn solutions to mKdV), and also nonlinearly orbitally stable with respect to certain classes of perturbations.",
"The techniques used to prove the results for KdV are based on its integrability.",
"The dispersion relation of the linearization of (REF ) in the traveling frame is $\\Omega = \\Omega (k) = ck + \\alpha k^3\\, .$ Branches of small-amplitude waves are bifurcating from the trivial solution for the critical values of $c$ for which $\\Omega (k) = 0$ for a nonzero integer value of $k$ : $c_k = -\\alpha k^2\\, .$ Let us now fix $k\\in \\mathbb {Z}/\\lbrace 0\\rbrace $ and set $c = c_k$ .",
"The condition for a collision of eigenvalues (REF ) has the form $c\\triangle n+ \\alpha \\left[(\\triangle n)^3 + 3\\triangle n\\mu (\\triangle n+\\mu )\\right] = 0\\, .$ According to Theorem REF equation (REF ) can be rewritten in the form (REF ), i.e., $c(\\triangle n) + \\alpha (\\triangle n)^3 (1+3\\gamma ) = 0\\, .$ The root $\\gamma $ of (REF ) that characterizes the nature of collisions of eigenvalues at zero amplitude is given by $\\gamma =- \\frac{c}{3\\alpha (\\triangle n)^2} - \\frac{1}{3} = \\frac{1}{3}\\left( \\frac{k^2}{(\\triangle n)^2} - 1\\right).$ The condition $-1/4 \\le \\gamma < 0$ can be expressed as $-\\frac{3}{4} \\le \\frac{k^2}{(\\triangle n)^2} - 1 < 0, \\qquad \\mbox{i.e.,} \\qquad \\frac{1}{4}(\\triangle n)^2 \\le k^2 < (\\triangle n)^2\\, ,$ or equivalently $k^2 < (\\triangle n)^2 \\le 4k^2\\, , \\qquad \\mbox{and thus} \\qquad |k| < |\\triangle n| \\le 2|k|\\, .$ It is easy to see that in this special case the equality in the upper bound in (REF ) corresponds to a collision of eigenvalues $\\lambda $ with indices $n_1+\\tilde{\\mu } = 1$ and $n_2+\\tilde{\\mu } = -1$ in (REF ).",
"But $\\Omega (1) = \\Omega (-1) = 0$ for (REF –REF ).",
"Thus the collision of opposite signature eigenvalues corresponding to the root $\\gamma = -1/4$ in this particular case is located at the origin of the spectral plane and thus it is not associated with the Hamiltonian-Hopf bifurcation.",
"Thus the instability condition is $|k| < |\\triangle n| < 2|k|\\, .$ Since the stability results are independent of $\\alpha $ without loss of generality we assume $\\alpha = 1$ in the rest of this section."
],
[
"gKdV Equation. Solutions with base period ${2\\pi }$",
"First, we consider KdV, i.e., $f(x) = u^2$ , as the linear analysis is identical for all $f(x)$ satisfying $f(0) = 0$ and the characterization of the collision condition in Theorem REF does not dependent on the form of nonlinearity.",
"In that case, the solution branch indexed by $k=1$ bifurcating at $c_1 = -1$ from the trivial solution corresponds to the cnoidal waves with base-period $2\\pi $ , see [7], Section 3.1, for the solution formula, numerical results, and analysis.",
"The condition (REF ) implies that collisions of eigenvalues with opposite Krein signature at zero-amplitude happen only for two eigenmodes of the form (REF ) with Fourier indices $n_1, n_2$ , $\\triangle n = n_1 - n_2$ , where $1 < \\triangle n < 2$ .",
"As no such $\\triangle n$ exists the small-amplitude cnoidal waves of base-period $2\\pi $ are spectrally stable (away from the origin of the spectral plane).",
"This is in agreement with the results obtained in [2] and [7], Section 3.1, step 5.",
"The same result is true for any nonlinearity $f(x)$ , including the case of mKdV, and thus, not accounting for a possible modulational instability, small-amplitude periodic traveling waves with base period $2\\pi $ are spectrally stable for gKdV (REF )."
],
[
"KdV Equation. Solutions with base period $2 \\pi /k$",
"We discuss the case $k\\ge 2$ .",
"Solutions on the branch bifurcating from the trivial solution at $c_k = -k^2$ also correspond in the case of KdV to the cnoidal wave solutions, as the cnoidal waves comprise all periodic traveling waves to KdV.",
"However, these solutions are subharmonic compared to the solutions on the branch with index $k=1$ , i.e., their base-period is $2\\pi /k$ .",
"One way to see this is to consider (REF ) with $f(v) = v^2$ in the frame traveling with velocity $c$ : $v_t +\\alpha v_{xxx} + (v^2)_x + cv_x = 0\\, .$ We set $y = \\frac{x}{k}, \\qquad \\tau = \\frac{t}{k^3}, \\qquad u = k^2 v, \\qquad \\tilde{c} = k^2c.$ Then (REF ) transforms to $u_{\\tau } + \\alpha u_{yyy} + (u^2)_y + \\tilde{c}u = 0\\, .$ Thus any solution $v(x,t)$ of (REF ) with the base period $2\\pi $ traveling with velocity $c$ corresponds 1-to-1 to a solution $u(y,\\tau )$ of (REF ) with the base period $2\\pi / k $ traveling with velocity $\\tilde{c} = c k^2$ .",
"The $k$ -repetition of $2\\pi /k$ -periodic solution of (REF ) is also a $2\\pi $ -periodic solution of (REF ) that is equivalent to (REF ) with $c = c_k$ .",
"This relation allows to identify through (REF ) the branch of $2\\pi $ periodic solutions of (REF ) bifurcating at $c=c_k$ with the branch of solutions of the same equation bifurcating at $c = c_1$ , i.e., the branch of solutions of (REF ) bifurcating at $c= c_k$ consists of properly rescaled multicopies of the solutions of the same equation located on the branch bifurcating at $c=c_1$ .",
"Therefore perturbations that are subharmonic for $k=1$ are co-periodic for $k\\ge 2$ , etc.",
"This leads to more eigenvalue collisions for $k\\ge 2$ than for $k=1$ since the co-periodic spectrum, e.g.",
"the spectrum for $k\\ge 2$ for the Floquet multiplier $\\mu =0$ includes (after a proper rescaling) the union of the spectrum for $k=1$ and $\\mu =0$ , $\\mu = 1/k$ , $\\mu = 2/k, \\dots $ .",
"Figure: Illustration of the relation () of the spectrum σ (2) \\sigma ^{(2)} (left) and σ (1) \\sigma ^{(1)} (right) for KdV equation.",
"Individual curves correspond to different values of nn with the index nn indicated.",
"The spectrum partitions σ μ \\sigma _{\\mu } correspond to all λ\\lambda for a given μ\\mu .Displayed are λ=λ n (μ) \\lambda = \\lambda _n^{(\\mu )} values for μ=-0.4\\mu = -0.4 (k=2k=2, left) and μ=-0.2\\mu = -0.2 and μ=0.3\\mu = 0.3 (k=1k=1, right).For better visibility we have removed the branches with indices nn, -2≤n≤3-2\\le n\\le 3 (k=2k=2) and -1≤n≤1-1\\le n \\le 1 (k=1k=1), all undisplayed branches lie close to the horizontal axis.",
"Note the scaling factor 8 on the λ\\lambda axis (left) for σ (2) \\sigma ^{(2)} compared to σ (1) \\sigma ^{(1)} (right).As an illustration consider the case $k=2$ .",
"The spectrum of the linearized problem is given by $\\sigma ^{(2)} = \\displaystyle \\bigcup _{\\mu \\in (-1/2,1/2]} \\sigma ^{(k=2)}_{\\mu } =\\left\\lbrace \\lambda _n^{(\\mu )}; \\ \\lambda _n^{(\\mu )} = - i \\left[4(n+\\mu ) - (n+\\mu )^3\\right], n \\in \\mathbb {Z}\\right\\rbrace \\,.$ On the other hand, the spectrum for $k=1$ is given by $\\sigma ^{(1)} = \\displaystyle \\bigcup _{\\mu \\in (-1/2,1/2]} \\sigma ^{(k=1)}_{\\mu } =\\left\\lbrace \\lambda _n^{(\\mu )}; \\ \\lambda _n^{(\\mu )} = - i \\left[(n+\\mu ) - (n+\\mu )^3\\right], n \\in \\mathbb {Z}\\right\\rbrace \\,.$ It is easy to see (see Fig.",
"REF for a visualization) that for all $\\mu \\in (-1/2,1/2]$ $\\frac{1}{8}\\sigma _{\\mu }^{(k=2)} = \\sigma _{\\mu /2}^{(k=1)} \\cup \\sigma _{\\mu /2 + 1/2}^{(k=1)}\\, .$ Here multiplication of the set by a scalar means multiplication of each of its elements by the scalar and we use the periodicity $\\sigma _{\\mu } = \\sigma _{\\mu +1}$ for all $\\mu \\in \\mathbb {R}$ to properly define the second term $\\sigma _{\\mu /2 + 1/2}^{(k=1)}$ .",
"The condition (REF ) indicates that there are collisions of the eigenvalues of opposite signature at zero amplitude for modes of the form (REF ) for Fourier indices $n_1, n_2$ , with $\\triangle n = n_1 - n_2$ satisfying $\\triangle n \\in \\lbrace k+1, \\dots , 2k-1\\rbrace $ and that is for $k\\ge 2$ a non-empty set.",
"Generically, this would imply spectral instability of the waves.",
"However, none of these collisions unfold for non-zero amplitude to a Hamiltonian-Hopf bifurcation.",
"Such bifurcations are not possible as according to [2] all periodic traveling wave solutions to KdV are spectrally stable.",
"As a collision of eigenvalues of opposite Krein signature is only a necessary condition for a Hamiltonian-Hopf bifurcation, the analysis presented here does not allow to see this phenomenon directly.",
"Some indication can be found in the fact that these new collisions at $c = c_k$ correspond to collisions of opposite signature eigenvalues arising from different components (as opposed to from the same component) of the union on the right hand side of (REF ).",
"The different spectrum partitions and associated eigenspaces do not interact with each other, see [8] and [Kollár & Miller, preprint 2018] for a throughout discussion of avoided Hamiltonian-Hopf bifurcations.",
"It is possible to see within the analysis presented here that the collisions of the opposite Krein signature eigenvalues of the $2\\pi / k$ periodic solutions are just an artifact of the $2\\pi $ periodic setting, i.e., when one considers the stability of the $2\\pi / k$ periodic solutions as the stability of its $k$ -repetition in the $2\\pi $ periodic frame in (REF ).",
"Due to the periodic character of the solution the stability of such a $k$ -repetition is equivalent to the stability of a single $2\\pi / k$ periodic repetition in (REF ).",
"But we have proved above that the waves with period $L$ considered on the interval $[0,L]$ are spectrally stable (this corresponds to $k = 1$ for (REF ) where we have set without loss of generality $L = 2\\pi $ ).",
"Therefore the $2\\pi / k$ periodic waves are spectrally stable and all collisions at zero amplitude of (REF ) at $c = c_k$ are only due to multi-coverage of the spectrum $\\sigma ^{(k)}$ as in (REF ).",
"The same argument can be used for gKdV with the nonlinearity $f(v) =v^n$ , $n \\ge 2$ .",
"However, in regard to the spectral stability of small-amplitude waves lying on branches bifurcating at $c = c_k$ for $k \\ge 2$ for a general $f(v)$ , $f(0) = 0$ , we can only conclude that there are collisions of the opposite signature eigenvalues at zero amplitude.",
"A lack of a transformation analogous to (REF ), that requires existence of a positive $r$ such that $f(au) = a^r f(u)$ for all $a \\in \\mathbb {R}$ , does not allow to rule out the potential Hamiltonian-Hopf bifurcations."
],
[
"Higher-order gKdV Equation",
"A similar analysis can be performed for the higher-order gKdV equation (REF ).",
"In that case $\\Omega (k) = -ck + (-1)^{p+1} \\alpha k^p$ and $c_k = (-1)^p \\alpha k^{p-1}$ .",
"The relation $\\Omega (n+\\mu ) = \\Omega (\\mu )$ reduces to a polynomial equation of degree $p$ for $\\gamma $ .",
"Similarly as for $p=1$ it is possible for $p=2$ to explicitly show that all the waves on the branch $k=1$ are spectrally stable, as none of the roots of $\\Omega (\\triangle n+\\mu ) = \\Omega (\\mu )$ in terms of $\\gamma $ are located in the interval $(-1/4,0)$ .",
"To see this one needs to determine for which integer values of $\\triangle n$ the roots of $-k^4 + (\\triangle n)^4\\left( 1 + 5\\gamma + 5\\gamma ^2\\right) = 0$ lie in the interval $\\gamma \\in (-1/4,0)$ .",
"A short calculation reveals that the condition reduces to $|k| < |\\triangle n| < 2|k|$ , i.e.",
"the same condition as for $p=1$ analyzed above leading to stability for $k=1$ .",
"The same statement can be proved for any $p\\ge 1$ for which the equation for $\\gamma $ has the form $-k^{2p} + (\\triangle n)^{2p} s_{2p+1}(\\gamma ) = 0\\, .$ There $s_{2p+1}(-1/4) = 2^{-2p}$ and $s_{2p+1}(0) = 1$ by (REF ), and also $s_{2p+1}(\\gamma )$ is continuous on $[-1/4,0]$ and increasing on $(-1/4,0)$ by (REF ).",
"Therefore the roots of (REF ) lie in the interval $\\gamma \\in (-1/4,0)$ if and only if $|k| < |\\triangle n| < 2|k|$ .",
"Hence the small-amplitude periodic traveling wave solutions to (REF ) with the base period $2\\pi $ ($k=1$ ) are spectrally stable, except perhaps with respect to modulational perturbations.",
"The question of spectral stability of small-amplitude wave solutions to (REF ) with the base period $2\\pi /k$ , $k\\ge 2$ is not addressed here."
],
[
"Balanced Higher Order KdV equations",
"We demonstrate the full power of Theorem REF on a more complicated example.",
"Here we explicitly characterize stability regions for small-amplitude periodic traveling wave solutions of KdV-type equations with two balanced linear terms of odd order: $u_t = \\partial _x f(u)+ A\\, \\partial ^{2q+1}_x u + B \\, \\partial _x^{2p+1} u,$ subject to periodic boundary conditions.",
"Here $p > q$ are positive integers, $A, B \\in \\mathbb {R}$ are non-zero coefficients, and $f(u)$ is a smooth function of $u$ and its spatial derivatives with $f(0) = 0$ , containing no linear terms.",
"The literature on this topic is limited.",
"Most relevant is [12], where $f(u)\\sim u^2$ (the Kawahara equation), and the period of the solutions is not fixed.",
"It is concluded there that for solutions for which the amplitude scales as the 1.25-th power of the speed, solutions are spectrally stable.",
"No conclusion is obtained for other solutions.",
"Our investigation does not require this scaling, nor does it restrict the type of nonlinearity.",
"Also relevant is [15], where the typical stability approach of [11] is extended to systems with singular Poisson operator like (REF ), but the theory is not applied to (REF ).",
"A mostly numerical investigation of equations like (REF ) is undertaken in [24].",
"As stated, our theory builds almost exclusively on [7] and our rigorous results agree with numerical results in [24] where the special case $p = 2$ , $q = 1$ , and $A, B > 0$ was considered.",
"Traveling wave solutions $u=U(x-ct)$ with wave velocity $c$ satisfy $-c U^{\\prime } =\\partial _x f(U)+ A U^{(2q+1)} + B U^{(2p+1)}.$ The spectral stability of small-amplitude waves that bifurcate at zero amplitude from the trivial solution $U=0$ is characterized by the growth of the solutions of the linear equation $v_t = c v_x+A v_{(2q+1)x} + B v_{(2p+1)x},$ with dispersion relation $\\Omega = \\Omega _{p,q}(k) = -ck - A (-1)^q k^{2q+1} - B (-1)^p k^{2p+1}=-ck-\\alpha k^{2q+1}+\\beta k^{2p+1},$ where we have introduced $\\alpha = A (-1)^q, \\qquad \\qquad \\beta = - B(-1)^p.$ Without loss of generality, we assume that $\\alpha > 0$ .",
"If not, the transformation $x \\rightarrow - x$ (i.e., $k\\rightarrow -k$ ), and $c \\rightarrow -c$ can be used to switch the sign of $\\alpha $ .",
"The scaling symmetry of the equation allows us to equate $\\alpha =1$ hereafter.",
"The choice of opposite signs in front of $\\alpha $ and $\\beta $ in (REF ) is intuitive: if $\\alpha $ and $\\beta $ have opposite sign the Hamiltonian energy (REF ) is definite and all eigenvalues have the same signature.",
"This rules out Hamiltonian-Hopf bifurcations and the spectral instabilities following from them.",
"In other words, the interesting case for our considerations is that both $\\alpha $ and $\\beta $ are positive.",
"Lastly, since we study bifurcations from the first Fourier mode $k = 1$ , $c = \\beta -\\alpha =\\beta -1$ .",
"According to Theorem REF , eigenvalue collisions at zero-amplitude are characterized by the roots $\\gamma $ of $\\triangle nR(\\gamma ) := -c\\triangle n- (\\triangle n)^{2q+1} s_{2q+1}(\\gamma ) + \\beta (\\triangle n)^{2p+1} s_{2p+1}(\\gamma ) = 0.$ This is rewritten as $\\beta \\left[(\\triangle n)^{2p} s_{2p+1}(\\gamma ) - 1 \\right]- \\left[ (\\triangle n)^{2q} s_{2q+1}(\\gamma ) - 1\\right]= 0.$ Our goal is to find the parameter range $(\\beta , \\triangle n)$ for which the root $\\gamma $ of (REF ) satisfies $\\gamma \\in [-1/4,0)$ .",
"The results obtained in the next section are graphically summarized in Fig.",
"REF .",
"Figure: Spectral stability regimes of the small-amplitude 2π2\\pi periodic traveling waves for the Kawahara equation (), p=2p=2, q=1q=1, α=1\\alpha = 1, k=1k=1.",
"Unstable pairs (▵n,β)(\\triangle n, \\beta ) are indicated by the dashed line segments, stable pairs(▵n,β)(\\triangle n, \\beta ) are above the curve β=β -1/4 (▵n)\\beta = \\beta _{-1/4}(\\triangle n) and below the curve β=β 0 (▵n)\\beta = \\beta _0(\\triangle n) given by ()–() for ▵n≥3\\triangle n\\ge 3, by () for ▵n=2\\triangle n= 2, and by ()(\\ref {betan1}) for ▵n=1\\triangle n= 1.An important role is played by the interval end points $\\gamma = 0$ and $\\gamma = -1/4$ .",
"By (REF ) for $\\gamma = 0$ we have $\\beta ((\\triangle n)^{2p} - 1) - ((\\triangle n)^{2q} - 1) = 0$ and therefore we set $\\beta _0 = \\beta _0(\\triangle n) = \\frac{(\\triangle n)^{2q} -1}{(\\triangle n)^{2p} - 1}.$ On the other hand (REF ) reduces for $\\gamma = -1/4$ by (REF ) to $\\beta _{-1/4} = \\beta _{-1/4}(\\triangle n) = \\left[\\left(\\displaystyle \\frac{\\triangle n}{2}\\right)^{2q} - 1\\right]/ \\left[\\left(\\displaystyle \\frac{\\triangle n}{2}\\right)^{2p} - 1\\right] \\, .$ It follows immediately from Lemma REF that for $\\triangle n\\ge 3$ , $\\beta _0(\\triangle n)<\\beta _{-1/4}(\\triangle n)$ , since this inequality may be rewritten as $f_{2p,2q}(\\triangle n)<f_{2p,2q}(2)$ (in the notation of the Lemma)."
],
[
"Collisions of eigenvalues of opposite signature",
"Since the thresholds $\\gamma = 0$ and $\\gamma = -1/4$ correspond, respectively, to $\\beta = \\beta _0(\\triangle n)$ and $\\beta = \\beta _{-1/4}(\\triangle n)$ , where $\\beta _0(\\triangle n) < \\beta _{-1/4}(\\triangle n)$ , one may conjecture (for $\\triangle n\\ge 3$ , since for $\\triangle n=1, 2$ either $\\beta _0$ or $\\beta _{-1/4}$ is not defined) that collisions of eigenvalues of opposite Krein signature happen for $\\beta \\in (\\beta _0(\\triangle n), \\beta _{-1/4}(\\triangle n)]$ .Such a result would follow from monotonicity properties of the location of roots $\\gamma $ with respect to $\\beta $ .",
"Alternatively, we use an argument that proves that $\\beta _0$ and $\\beta _{-1/4}$ are the bounds of the stability region.",
"For $\\beta < \\beta _0(\\triangle n)$ one expects collisions of eigenvalues of the same signature and finally for $\\beta > \\beta _{-1/4}(\\triangle n)$ one expects no collisions as the roots $\\mu $ of (REF ) are not real (see Fig.",
"REF ).",
"As we prove next, this is true.",
"The cases $\\triangle n= 1$ and $\\triangle n= 2$ are treated separately.",
"See [24] for detailed numerical results (wave profiles and Fourier coefficients, spectrum diagrams) in the case $p=2$ , $q=1$ and $f(u) = u^2$ (Kawahara equation), particularly numerical simulations at non-zero amplitude confirming presence of Hamiltonian-Hopf bifurcations (and thus spectral instability) that completely agree with the collisions of opposite Krein signature eigenvalues at zero-amplitude described here.",
"In the numerical experiments all such collisions studied actually yielded the bifurcation.",
"Figure: Parameter regimes for β\\beta , β≤β 0 (▵n)\\beta \\le \\beta _0(\\triangle n), β∈(β 0 (▵n),β -1/4 (▵n)]\\beta \\in (\\beta _0(\\triangle n), \\beta _{-1/4}(\\triangle n)], and β>β -1/4 (▵n)\\beta > \\beta _{-1/4}(\\triangle n).Theorem 3 Case $\\mathbf {\\triangle n\\ge 3}$ .",
"Let $p, q$ , $p > q$ , be positive integers and let $\\triangle n$ is an integer, $\\triangle n\\ge 3$ .",
"The presence and character of collisions of eigenvalues of the linearized problem (REF ) at zero amplitude at $c = c_1=\\beta -\\alpha $ depends on the difference of the indices of the Fourier modes $\\triangle n$ of the perturbation in the following way: (i) If $\\triangle n$ is such that $\\beta < \\beta _0(\\triangle n)$ , then there is a collision of eigenvalues of the same signature, i.e., there is a root of (REF ) with $\\gamma > 0$ and there is no root with $\\gamma \\in [-1/4,0)$ ; (ii) If $\\triangle n$ is such that $\\beta _0(\\triangle n) < \\beta \\le \\beta _{-1/4}(\\triangle n)$ , then there is a collision of eigenvalues of opposite signature, i.e., there is a root $\\gamma $ of (REF ) such that $\\gamma \\in [-1/4, 0)$ ; (iii) If $\\triangle n$ is such that $\\beta _{-1/4}(\\triangle n) < \\beta $ , then there is no collision of eigenvalues, i.e., all roots $\\gamma $ of (REF ) satisfy $\\gamma < -1/4$ .",
"Part (ii).",
"We show that for all $\\triangle n\\ge 3$ and $\\beta _0(\\triangle n) < \\beta \\le \\beta _{-1/4}(\\triangle n)$ there exists $\\gamma \\in [-1/4, 0)$ satisfying $R(\\gamma ) = 0$ .",
"Therefore by (REF ), in such a parameter regime there is a collision of eigenvalues of opposite Krein signature.",
"It is easy to see that $R(0) = \\beta [(\\triangle n)^{2p}-1] - [(\\triangle n)^{2q}-1] > \\beta _0 [(\\triangle n)^{2p}-1] - [(\\triangle n)^{2q}-1] = 0,$ and, $R( -1/4)& =&\\beta \\left( \\frac{(\\triangle n)^{2p}}{2^{2p}} - 1\\right) - \\left(\\frac{(\\triangle n)^{2q}}{2^{2q}} - 1\\right) \\\\& \\le &\\beta _{-1/4} \\left( \\frac{(\\triangle n)^{2p}}{2^{2p}} - 1\\right) - \\left(\\frac{(\\triangle n)^{2q}}{2^{2q}} - 1\\right) = 0.$ Thus $R(0) > 0 \\ge R\\left( -1/4 \\right)$ and the polynomial $R(\\gamma )$ has a real root $\\gamma \\in [-1/4, 0)$ .",
"Part (i).",
"Since $\\beta < \\beta _0(\\triangle n) < \\beta _{-1/4}(\\triangle n)$ the same argument as in Part (ii) yields $R(-1/4) < 0$ .",
"Also, $R(0) = \\beta [(\\triangle n)^{2p}-1] - [(\\triangle n)^{2q}-1] < \\beta _0 [(\\triangle n)^{2p}-1] - [(\\triangle n)^{2q}-1] = 0\\, .$ We prove that $R(\\gamma ) = \\beta [(\\triangle n)^{2p} s_{2p+1}(\\gamma ) -1] - [(\\triangle n)^{2q} s_{2q+1}(\\gamma ) - 1] < 0$ for all $\\gamma \\in [-1/4, 0]$ .",
"By Lemma REF for $\\triangle n\\ge 3$ and $p \\ge 1$ , $(\\triangle n)^{2p}s_{2p+1}(\\gamma ) \\ge \\frac{3^{2p}}{2^{2p+1}} > 1 \\, .$ Thus for all $\\gamma \\in [-1/4,0)$ and $\\beta < \\beta _0$ , $R(\\gamma ) & =& \\beta [(\\triangle n)^{2p}s_{2p+1}(\\gamma ) - 1] - [\\triangle n)^{2q}s_{2q+1}(\\gamma ) - 1] \\nonumber \\\\& <& \\beta _0(\\triangle n) [(\\triangle n)^{2p}s_{2p+1}(\\gamma ) - 1] - [(\\triangle n)^{2q}s_{2q+1}(\\gamma ) - 1] \\, .$ We prove that the right-hand side of (REF ) is non-positive.",
"This is equivalent to $\\beta _0(\\triangle n) = \\frac{(\\triangle n)^{2q}-1}{(\\triangle n)^{2p}-1} \\le \\frac{(\\triangle n)^{2q}s_{2q+1}(\\gamma ) - 1}{(\\triangle n)^{2p}s_{2p+1}(\\gamma ) - 1}\\, ,$ or to $s_{2q+1} \\ge s_{2p+1}[1-\\theta (\\triangle n)] + \\theta (\\triangle n)\\, , \\qquad \\mbox{where} \\quad \\theta (n) : = \\frac{(n)^{2p}-(n)^{2q}}{(n)^{2p+2q}-(n)^{2q}}.$ Clearly $0 < \\theta (n) < 1$ .",
"Since $s_{2p+1} < 1$ it suffices to prove (REF ) for $\\triangle n$ that maximizes $\\theta (\\triangle n)$ , $\\triangle n\\ge 2$ .",
"However, by Lemma REF for $p > q \\ge 1$ , $\\max _{n \\ge 2} \\theta (n) = \\theta (2)$ and it suffices to prove $s_{2q+1} \\ge s_{2p+1} (1-\\theta (2)) + \\theta (2)$ , i.e., $s_{2q+1} 2^{2q} (2^{2p} - 1) \\ge s_{2p+1} 2^{2p} (2^{2q} -1) + 2^{2p} - 2^{2q}.$ Therefore (REF ) follows directly from Lemma REF as it is equivalent for $p > q \\ge 1$ to $\\frac{2^{2q} s_{2q+1} - 1}{2^{2q} - 1} \\ge \\frac{2^{2p} s_{2p+1} - 1}{2^{2p} - 1} \\, .$ Hence we proved $R(\\gamma ) < 0$ for all $\\gamma \\in [-1/4,0]$ .",
"On the other hand $R(\\gamma )$ is an even order polynomial with a positive leading coefficient, i.e., $R(\\gamma ) \\rightarrow \\infty $ as $\\gamma \\rightarrow \\infty $ .",
"Therefore there exists $\\gamma _0 > 0$ such that $R(\\gamma _0) = 0$ .",
"Such a root corresponds by (REF ) to a real value of $\\mu $ .",
"Therefore in this regime there is a collision of two eigenvalues of the same signature.",
"Part (iii).",
"Note that $R(0) >0$ .",
"We show that $R(\\gamma ) >0$ for $\\gamma \\ge -1/4$ .",
"First, $R( -1/4) =\\beta \\left( \\frac{n^{2p}}{2^{2p}} - 1\\right) - \\left(\\frac{n^{2q}}{2^{2q}} - 1\\right) >\\beta _{-1/4} \\left( \\frac{n^{2p}}{2^{2p}} - 1\\right) - \\left(\\frac{n^{2q}}{2^{2q}} - 1\\right) = 0\\, .$ For $\\gamma \\ge -1/4$ , $R(\\gamma ) &= &\\beta \\left[ (\\triangle n)^{2p}s_{2p+1}(\\gamma ) - 1\\right] - \\left[ (\\triangle n)^{2q} s_{2q+1}(\\gamma ) - 1\\right] \\nonumber \\\\& > &\\beta _{-1/4} \\left[ (\\triangle n)^{2p}s_{2p+1}(\\gamma ) - 1\\right] - \\left[ (\\triangle n)^{2q} s_{2q+1}(\\gamma ) - 1\\right]\\, ,$ since, by Lemma REF , $(\\triangle n)^{2p}s_{2p+1}(\\gamma ) \\ge 1$ .",
"We prove that $\\frac{(\\triangle n/2)^q-1}{(\\triangle n/2)^p-1}\\ge \\frac{(\\triangle n)^{q} s_{q+1}(\\gamma ) - 1}{ (\\triangle n)^{p}s_{p+1}(\\gamma ) - 1}\\, ,$ for any $p > q$ .",
"From (REF ), with $p \\rightarrow 2p$ and $q \\rightarrow 2q$ , we obtain $R(\\gamma ) > 0$ for $\\gamma \\ge -1/4$ .",
"Denote $m = \\triangle n/2 \\ge 1$ and $u_j = 2^j s_{j+1}$ for $j \\ge 0$ to rewrite (REF ) as $u_q \\le u_p (1-\\omega (m)) + \\omega (m)\\, , \\qquad \\mbox{where} \\quad \\omega (m) = \\frac{m^p-m^q}{m^{p+q}-m^q}.$ By Lemma REF , the sequence $\\omega (m)\\in (0,1)$ , is non-increasing for $m \\ge 1$ .",
"Also, by Lemma REF , $u_p = 2^p s_{p+1} \\ge 1$ , and (REF ) follows from $u_q \\le u_p [1-\\omega (1)] + \\omega (1)$ , where $\\omega (1) = (p-q)/p$ .",
"Equation (REF ) reduces to $ (u_q - 1)/q \\le (u_p-1)/p$ , for $p > q \\ge 1$ .",
"In terms of $s_q(\\gamma )$ this is equivalent to $\\frac{2^q s_{q+1}(\\gamma ) - 1}{q} \\le \\frac{2^p s_{p+1}(\\gamma ) -1}{p}, \\qquad \\mbox{for $p > q \\ge 1$},$ which follows for $\\gamma \\ge -1/4$ from Lemma REF , since monotonicity of the positive sequence $\\displaystyle \\frac{2^ms_{m +1}-1}{m(m+1)}$ directly implies monotonicity of the sequence $\\displaystyle \\frac{2^ms_{m +1}-1}{m}$ .",
"Thus $R(\\gamma ) > 0$ for all $\\gamma \\ge -1/4$ and $R(\\gamma )$ has no roots in $[-1/4, \\infty )$ and there are no collisions of eigenvalues in this regime.",
"For $\\triangle n=1$ , we use a similar argument.",
"For $\\triangle n= 1$ and $\\gamma = 0$ , $R(0) = 0$ .",
"Hence $\\gamma = 0$ is always a root of $R(\\gamma ) = 0$ , corresponding to the relationThese eigenvalues are present due to symmetries; they do not leave the imaginary axis.",
"$\\Omega (1) = 0 = \\Omega (0)$ .",
"For $p > q > 0$ , denote $\\beta _0^{(\\triangle n=1)} = \\frac{2q+1}{2p+1}\\, ,\\qquad \\mbox{and} \\qquad \\beta _{-1/4}^{(\\triangle n=1)} = \\frac{1-2^{-2q}}{1 - 2^{-2p}}.$ Theorem 4 Case $\\mathbf {\\triangle n=1}$ .",
"Let $p, q$ be positive integers with $p>q$ .",
"For the linearized problem (REF ) at zero amplitude with $c = c_1$ , the presence and the character of eigenvalue collisions depend on the difference $\\triangle n$ of the indices of the Fourier modes of the perturbation as follows: (i) for $\\beta < \\beta ^{(\\triangle n=1)}_0$ , eigenvalues of the same signature collide, i.e., there is a root of (REF ) with $\\gamma > 0$ and there is no root with $\\gamma \\in [-1/4,0)$ ; (ii) for $\\beta ^{(\\triangle n=1)}_0 < \\beta < \\beta ^{(\\triangle n=1)}_{-1/4}$ , eigenvalues of opposite signature collide, i.e., there is a root $\\gamma $ of (REF ) so that $\\gamma \\in [-1/4, 0)$ ; (iii) for $\\beta _{-1/4}^{(\\triangle n=1)} < \\beta $ , eigenvalues do not collide, i.e., $\\gamma < -1/4$ , for all roots $\\gamma $ of (REF ).",
"First, we show that $\\beta _0^{(\\triangle n=1)}< \\beta _{-1/4}^{(\\triangle n=1)}$ , which follows from the function $f(y)=(1-2^{-y})/(1+y)$ being decreasing for $y>2$ .",
"Its derivative has the numerator $(1+y)2^{-y}\\ln 2+2^{-y}-1$ , which is negative at $y=2$ , and itself has a derivative that is negative for $y>2$ .",
"Next, for $\\beta \\le \\beta _{-1/4}^{(\\triangle n=1)}$ , $R(-1/4)& = & \\beta \\left(s_{2p+1}(-1/4)-1\\right) -\\left( s_{2q+1}(-1/4)-1\\right)=\\beta (2^{-2p}-1) - (2^{-2q} - 1) \\nonumber \\\\& \\ge & \\beta _{-1/4}^{(n=1)} (2^{-2p}-1) - (2^{-2q} - 1) = 0\\, ,$ where equality holds only for $\\beta = \\beta _{-1/4}^{(\\triangle n=1)}$ .",
"On the other hand, if $\\beta > \\beta _{-1/4}^{(\\triangle n=1)}$ then $R(-1/4) < 0$ .",
"Further, for $\\gamma = 0$ and all values of $\\beta $ , $R(0) = 0$ .",
"Finally, for $\\gamma \\in [-1/4,0)$ $R^{\\prime }(0) = \\beta (2p+1) - (2q+1).$ Therefore, for $\\beta < \\beta _0^{(\\triangle n=1)}$ , $R(0) = 0, \\qquad R^{\\prime }(0) < 0,$ and, for $\\beta > \\beta _0^{(\\triangle n=1)}$ , $R(0) = 0, \\qquad R^{\\prime }(0) > 0.$ Note that $R(0) = R^{\\prime }(0) = 0$ for $\\beta = \\beta ^{(\\triangle n=1)}_0$ .",
"Part (i).",
"By (REF ) one has $R(-1/4) >0$ , and by (REF ) $R(0) = 0$ and $R^{\\prime }(0) < 0$ .",
"We prove that $R(\\gamma ) >0$ for all $\\gamma \\in [-1/4,0)$ .",
"Thus $R = R(\\gamma )$ does not have any roots in $(-1/4,0)$ .",
"Moreover, $R(\\gamma )$ is an odd-degree polynomial with a positive leading coefficient, $R(\\gamma ) \\rightarrow \\infty $ as $\\gamma \\rightarrow \\infty $ and $R(0) =0$ and $R^{\\prime }(0) < 0$ .",
"Therefore $R$ has a positive root.",
"Assume $\\gamma \\in [-1/4,0)$ and $\\beta < \\beta ^{(\\triangle n=1)}_0$ .",
"Then, using (), $R(\\gamma ) = \\beta (s_{2p+1}(\\gamma ) -1) - (s_{2q+1}(\\gamma )-1) > \\beta ^{(\\triangle n=1)}_0 (s_{2p+1}(\\gamma ) -1) - (s_{2q+1}(\\gamma )-1)\\, .$ To establish $R(\\gamma ) > 0$ it is enough to prove $\\beta ^{(\\triangle n=1)}_{0} \\le \\frac{s_{2q+1}(\\gamma )-1}{s _{2p+1}(\\gamma ) - 1}, \\qquad \\mbox{for $\\gamma \\in [-1/4,0)$.",
"}$ By Lemma REF one has $s_m(\\gamma ) < 1$ for $m\\ge 2$ , $\\gamma \\in [-1/4,0)$ .",
"Hence (REF ) can be rewritten as $\\frac{s _{2p+1}(\\gamma ) - 1}{2p+1} \\ge \\frac{s_{2q+1}(\\gamma )-1}{2q+1},$ which follows for $p > q > 0$ and $\\gamma \\in [-1/4, 0)$ from Lemma REF .",
"Therefore $R(\\gamma ) > 0$ for $\\gamma \\in [-1/4,0)$ .",
"Part (ii).",
"By (REF ) one has $R(-1/4) >0$ , and by (REF ) $R(0) = 0$ , $R^{\\prime }(0) > 0$ .",
"Therefore there exist a $\\gamma \\in (-1/4,0)$ such that $R(\\gamma ) = 0$ .",
"Part (iii).",
"In this case $R(-1/4) < 0$ , and by (REF ) $R(0) = 0$ and $R^{\\prime }(0) > 0$ .",
"We prove that $R(\\gamma ) < 0$ for $\\gamma \\in [-1/4,0)$ and $R(\\gamma ) > 0$ for $\\gamma >0$ .",
"Therefore $R(\\gamma )$ does not have a non-zero root for $\\gamma \\ge -1/4$ .",
"First assume that $\\gamma \\in [-1/4,0)$ .",
"Then $\\beta > \\beta _{-1/4}^{(\\triangle n=1)}$ implies, using (), $R(\\gamma ) =\\beta (s_{2p+1}(\\gamma ) - 1) - (s_{2q+1}(\\gamma ) - 1) < \\beta _{-1/4}^{(\\triangle n=1)} (s_{2p+1}(\\gamma ) - 1) - (s_{2q+1}(\\gamma ) - 1)\\, .$ It suffices to prove $\\beta ^{(\\triangle n=1)}_{-1/4} \\ge \\frac{s_{2q+1}(\\gamma )-1}{s _{2p+1}(\\gamma ) - 1}, \\qquad \\mbox{for $\\gamma \\in [-1/4,0)$,}$ to establish $R(\\gamma ) < 0$ .",
"The inequality (REF ) is rewritten as $\\frac{s _{2p+1}(\\gamma ) - 1}{2^{-2p}-1} \\ge \\frac{s_{2q+1}(\\gamma )-1}{2^{-2q}-1},$ which follows from Lemma REF .",
"Thus $R(\\gamma ) < 0$ for $\\gamma \\in [-1/4,0)$ .",
"Next, we assume $\\gamma >0$ .",
"With $\\beta > \\beta _{-1/4}^{(\\triangle n=1)}$ and using (), $R(\\gamma ) =\\beta (s_{2p+1}(\\gamma ) - 1) - (s_{2q+1}(\\gamma ) - 1) > \\beta _{-1/4}^{(n=1)} (s_{2p+1}(\\gamma ) - 1) - (s_{2q+1}(\\gamma ) - 1)\\, .$ It suffices to prove $\\frac{s _{2p+1}(\\gamma ) - 1}{2^{-2p}-1} \\le \\frac{s_{2q+1}(\\gamma )-1}{2^{-2q}-1},$ which follows from Lemma REF .",
"Thus $R(\\gamma ) > 0$ for $\\gamma > 0$ .",
"It is easy to see that for $\\triangle n= 2$ , $R(-1/4)= 0$ .",
"Thus $\\gamma = -1/4$ is a root of $R(\\gamma ) = 0$ for all $\\beta $ .",
"It corresponds to the fact that $\\Omega (-1) = 0 = \\Omega (1)$ , i.e., there is a collision of two eigenvalues of opposite Krein signature at the origin for all $\\beta $ .",
"This collision is due to the symmetries of the problem and these eigenvalues do not leave the imaginary axis in the weakly nonlinear regime.",
"Thus this collision does not affect stability.",
"We focus on the remaining roots of $R(\\gamma ) = 0$ .",
"We denote $\\beta _{0}^{(\\triangle n=2)} = \\frac{2^{2q}-1}{2^{2p}-1}\\, ,\\qquad \\mbox{and} \\qquad \\beta _{-1/4}^{(\\triangle n=2)} = \\frac{(2q+1)2q}{(2p+1)2p}\\, .$ The inequality $\\beta _0^{(\\triangle n=2)}<\\beta _{-1/4}^{(\\triangle n=2)}$ follows similarly to $\\beta _0^{(\\triangle n=1)}<\\beta _{-1/4}^{(\\triangle n=1)}$ , in the proof of the previous theorem.",
"Theorem 5 Case $\\mathbf {\\triangle n=2}$ .",
"Let $p, q$ , $p > q$ , be positive integers.",
"For the linearized problem (REF ) at zero amplitude at $c = c_1$ the presence and the character of collisions of eigenvalues depends on the Fourier-mode parameter $n$ of the perturbation in the following way: (i) for $\\beta < \\beta ^{(\\triangle n=2)}_0$ , eigenvalues of the same signature collide, i.e.",
"there is a root of (REF ) with $\\gamma > 0$ and there is no root with $\\gamma \\in (-1/4,0)$ ; (ii) for $\\beta ^{(\\triangle n=2)}_0 < \\beta < \\beta ^{(\\triangle n=2)}_{-1/4}$ , eigenvalues of the opposite signature collide, i.e.",
"there is a root $\\gamma $ of (REF ) such that $\\gamma \\in (-1/4, 0)$ ; (iii) for $\\beta _{-1/4}^{(\\triangle n=2)} < \\beta $ , eigenvalues do not collide, i.e.",
"all roots $\\gamma $ of (REF ) satisfy $\\gamma \\le -1/4$ .",
"Part (i).",
"We prove that $R(\\gamma ) < 0$ , for $\\gamma \\in (-1/4,0)$ .",
"First, $R(\\gamma )$ is an odd-degree polynomial and $R(\\gamma ) \\rightarrow \\infty $ as $\\gamma \\rightarrow \\infty $ and $R(0) =0$ and $R^{\\prime }(0) < 0$ .",
"Thus $R$ has a root $\\gamma > 0$ .",
"Assume $\\gamma \\in [-1/4,0)$ and $\\beta < \\beta ^{(\\triangle n=2)}_0$ .",
"Then $R(\\gamma ) &= & \\beta (2^{2p}s_{2p+1}(\\gamma ) -1) - (2^{2q}s_{2q+1}(\\gamma )-1)\\\\& <& \\beta ^{(\\triangle n=2)}_0 (2^{2p}s_{2p+1}(\\gamma ) -1) - (2^{2q}s_{2q+1}(\\gamma )-1)\\, .$ To establish $R(\\gamma ) < 0$ it suffices to prove $\\beta ^{(\\triangle n=2)}_{0} \\le \\frac{2^{2q}s_{2q+1}(\\gamma )-1}{2^{2p} s _{2p+1}(\\gamma ) - 1}, \\qquad \\mbox{for $\\gamma \\in (-1/4,0]$.",
"}$ This inequality is rewritten as $\\frac{2^{2p}s _{2p+1}(\\gamma ) - 1}{2^{2p}-1} \\le \\frac{2^{2q}s_{2q+1}(\\gamma )-1}{2^{2q}-1},$ which follows from Lemma REF .",
"Therefore $R(\\gamma ) < 0$ for $\\gamma \\in (-1/4,0]$ .",
"Part (ii).",
"First, $R(0) &= & \\beta (2^{2p} s_{2p+1}(0) - 1) - (2^{2q}s_{2q+1} - 1)= \\beta (2^{2p}-1) - (2^{2q}-1) \\\\& >& \\beta _0^{(\\triangle n=2)} (2^{2p} - 1) - (2^{2q} - 1) = 0.$ Next we show that $\\lim _{\\gamma \\rightarrow -1/4^+} R^{\\prime }(\\gamma ) < 0$ .",
"Indeed, for $\\gamma > -1/4$ , we have $R^{\\prime }(\\gamma ) &= & \\beta \\frac{2p+1}{\\sqrt{1+4\\gamma }} 2^{2p} (\\psi _+^{2p} - \\psi _-^{2p}) - 2^{2q} (\\psi _+^{2q} - \\psi _-^{2q})\\\\& <& \\beta _{-1/4}^{(n=2)} \\frac{2p+1}{\\sqrt{1+4\\gamma }} \\, 2^{2p} (\\psi _+^{2p} - \\psi _-^{2p}) - \\frac{2q+1}{\\sqrt{1+4\\gamma }} 2^{2q} (\\psi _+^{2q} - \\psi _-^{2q})$ as $\\psi _+^2 > \\psi _-^2 \\ge 0$ .",
"The result follows from l'Hopital's rule, since $\\lim _{\\gamma \\rightarrow -1/4^+}&&\\!\\!\\!\\!\\frac{(2q+1) 2^{2q} (\\psi _+^{2q}(\\gamma ) - \\psi _-^{2q}(\\gamma ))}{(2p+1)2^{2p} (\\psi _+^{2p}(\\gamma ) - \\psi _-^{2p}(\\gamma ))}\\\\&&~~~~=\\lim _{\\gamma \\rightarrow -1/4^+}\\frac{2q(2q+1) 2^{2q}\\frac{1}{\\sqrt{1+4\\gamma }} (\\psi _+^{2q-1}(\\gamma ) + \\psi _-^{2q-1}(\\gamma ))}{2p(2p+1) 2^{2p} \\frac{1}{\\sqrt{1+4\\gamma }} (\\psi _+^{2p-1}(\\gamma ) + \\psi _-^{2p-1}(\\gamma ))} \\\\&&~~~~= \\lim _{\\gamma \\rightarrow -1/4^+}\\frac{2q(2q+1) 2^{2q} s_{2q-1}(\\gamma )}{2p(2p+1) 2^{2p} s_{2p-1}(\\gamma )} \\\\&&~~~~= \\frac{2q(2q+1) 2^{2q} 2^{-(2q-2)}}{2p(2p+1) 2^{2p} 2^{-(2p-2)}} = \\frac{2q(2q+1)}{2p(2p+1)}=\\beta _{-1/4}^{(\\triangle n=2)}.$ Thus $R(\\gamma )< 0$ for $\\gamma \\in (-1/4,-1/4+\\varepsilon )$ , $\\varepsilon > 0$ , small.",
"Since $R(0) > 0$ , there exists $\\gamma \\in (-1/4,0)$ so that $R(\\gamma ) = 0$ .",
"Part (iii).",
"We show that $R(\\gamma ) > 0$ for $\\gamma > -1/4$ .",
"One has $R(\\gamma ) &= & \\beta (2^{2p} s_{2p+1}(\\gamma ) - 1) - (2^{2q}s_{2q+1} (\\gamma ) - 1)\\\\&>& \\beta _{-1/4}^{(\\triangle n=2)} (2^{2p} s_{2p+1}(\\gamma ) - 1) - (2^{2q}s_{2q+1} (\\gamma ) - 1)\\, .$ We show that $\\beta _{-1/4}^{(\\triangle n=2)} = \\frac{2q(2q+1)}{2p(2p+1)} \\ge \\frac{2^{2q}s_{2q+1} (\\gamma ) - 1}{2^{2p} s_{2p+1}(\\gamma ) - 1},$ which is equivalent to $\\frac{2^{2p}s_{2p+1}(\\gamma ) - 1}{2p(2p+1)} \\ge \\frac{2^{2q} s_{2q+1} (\\gamma ) - 1}{2q(2q+1)}.$ This inequality follows from Lemma REF .",
"Therefore $R(\\gamma ) = 0$ has no roots $\\gamma > -1/4$ for $\\beta > \\beta _{-1/4}^{(\\triangle n=2)}$ ."
],
[
"Appendix",
"Lemma 2 Let $\\alpha > 0$ .",
"The function $g(x) = \\frac{x\\alpha ^x}{\\alpha ^x - 1}$ is increasing on $(0,\\infty )$ .",
"The condition $g^{\\prime }(x) > 0$ is equivalent to $\\alpha ^x = e^{x \\ln \\alpha } > 1 + x \\ln \\alpha $ .",
"This follows directly from the Taylor expansion of $e^x$ at $x = 0$ with equality reached for $x = 0$ .",
"Lemma 3 Let $a > b > 0$ .",
"Define $f (n) = f_{a,b}(n) = \\frac{n^{a-b} - 1}{n^a-1}.$ We define $f(1) = \\lim _{n \\rightarrow 1} f(n) = (a-b)/a$ .",
"Then $f(n)$ is a decreasing function on $[1,\\infty )$ .",
"The inequality $f^{\\prime }(n) < 0$ is equivalent to $a(n^b - 1) < b (n^a- 1)$ , i.e., $\\frac{a}{b} < \\frac{n^a-1}{n^b-1}\\, .$ The estimate (REF ) for $n > 1$ follows from the fact that the function $h(n) = \\frac{n^a - 1}{n^b-1}, \\qquad a > b > 0,$ is increasing on $[1,\\infty )$ , where $h(1) = \\lim _{n\\rightarrow 1} h(n) = a/b$ .",
"The inequality $h^{\\prime }(n) > 0$ reduces to $\\frac{an^a}{n^a-1} > \\frac{b n^b}{n^b - 1},$ which holds for $a > b > 0$ and $n > 1$ by Lemma REF .",
"Lemma REF follows by continuity of $h(n)$ at $n=1$ .",
"Lemma 4 Let $s_m(\\gamma )$ be as above.",
"Then $s_m(\\gamma ) & \\ge & 2^{-(m-1)}, \\qquad \\mbox{for all $\\gamma \\ge -1/4$ and $m \\ge 0$,} \\\\s_m(\\gamma ) & < & 1, \\qquad \\quad \\quad \\ \\ \\, \\mbox{for all $\\gamma \\in [-1/4,0)$ and $m \\ge 2$,} \\\\s_m (\\gamma ) & > & 1, \\qquad \\quad \\quad \\ \\ \\, \\mbox{for all $\\gamma >0$ and $m \\ge 2$.",
"}$ First, for $\\gamma \\ge -1/4$ , $s_m(\\gamma )$ is an increasing function of $\\gamma $ since $s_m^{\\prime }(\\gamma ) = (m/\\sqrt{1+4\\gamma }) \\left( \\psi _+^{m-1}(\\gamma ) - \\psi _-^{m-1} (\\gamma )\\right) > 0$ .",
"The inequality (REF ) follows from this and $s_m(-1/4)=2^{1-m}$ .",
"Equation () follows from the fact that $\\psi _{\\pm } \\in (0,1)$ for $\\gamma \\in [-1/4, 0)$ .",
"Hence $s_{m+1}(\\gamma ) < s_m(\\gamma )$ for all $m \\ge 0$ .",
"Then $s_1(\\gamma ) = 1$ yields the claim.",
"Finally, we prove ().",
"For $m=2$ and $m=3$ , $s_2(\\gamma ) = 1 + 2\\gamma > 1$ , and $s_3(\\gamma ) = 1 + 3 \\gamma > 1$ for $\\gamma >0$ .",
"Then () follows directly from (REF ).",
"Lemma 5 For all $m \\ge 0$ and $\\gamma \\ge -1/4$ , $s_{m+2}(\\gamma ) & \\ge & -\\gamma s_m(\\gamma ), \\\\s_{m+1}(\\gamma ) &\\ge & s_{m}(\\gamma )/2, \\\\s_{m+1}(\\gamma ) &\\le & \\left[1+m(1+4\\gamma )\\right]s_m (\\gamma )/2.$ The inequality (REF ) is equivalent to $s_{m+2} - s_{m+1} + (s_{m+1} + \\gamma s_m) \\ge 0$ .",
"Using the recurrence relation (REF ), it reduces to $2s_{m+2} - s_{m+1}\\ge 0$ , i.e., $2s_{m+2} \\ge s_{m+1}$ , $m \\ge 0$ .",
"Thus (REF ) and () are equivalent except for () with $m = 0$ , which is trivially satisfied ($2s_1 = 2 = s_0$ ).",
"Also note that $s_m(\\gamma ) \\ge 0$ for $m \\ge 0$ and $\\gamma \\ge 0$ and (REF ) is satisfied for $\\gamma \\ge 0$ .",
"In the rest of the proof of (REF ), we assume that $m \\ge 1$ and $\\gamma \\in [-1/4,0)$ .",
"We shift $m \\rightarrow m+1$ in (), $m \\ge 0$ , which becomes $\\left(\\psi _+ - \\frac{1}{2}\\right) \\psi _+^{m+1} +\\left(\\psi _- - \\frac{1}{2}\\right)\\psi _-^{m+1} \\ge 0 \\, .$ Since $\\psi _- = 1-\\psi _+$ for $\\gamma \\in [-1/4,0)$ , (REF ) is equivalent to $\\left(\\psi _+ - \\frac{1}{2}\\right) \\left[ \\psi _+^{m+1} - \\psi _-^{m+1} \\right] \\ge 0 \\, ,$ which is satisfied for $\\gamma \\in [-1/4,0)$ since $\\psi _+\\ge 1/2$ and $\\psi _+>\\psi _-$ .",
"This proves () and (REF ).",
"We turn to ().",
"Note that () holds for $m=0$ .",
"For $m \\ge 1$ , first we consider $\\gamma \\ge 0$ .",
"Using (REF ), $2(s_m + \\gamma s_{m-1}) \\le \\left[m(1+4\\gamma ) + 1\\right] s_m,$ i.e., $2\\gamma s_{m-1} \\le \\left[m(1+4\\gamma ) - 1\\right] s_m = (m-1) s_m + 4m\\gamma s_m.$ But $m\\ge 1$ and $s_m \\ge 0$ .",
"Therefore $(m-1)s_m \\ge 0$ and (REF ) follows from $2\\gamma s_{m-1} \\le 4m \\gamma s_m$ , i.e., $s_{m} \\ge s_{m-1}/2m$ , which holds, according to ().",
"Next, consider $\\gamma \\in [-1/4,0)$ .",
"We write () as $2s_{m+1} - s_m \\le m (1+4\\gamma ) s_m$ , and use (REF ) to obtain $\\psi _+^m \\left(\\psi _+ - \\frac{1}{2}\\right) + \\psi _-^m \\left( \\psi _--\\frac{1}{2}\\right) \\le \\frac{m (1+4\\gamma )}{2} (\\psi _+^m + \\psi _-^m)\\, .$ Using $\\psi _+ + \\psi _- = 1$ , $\\left(\\psi _+ - \\frac{1}{2}\\right) (\\psi _+^m - \\psi _-^m) \\le \\frac{m (1+4\\gamma )}{2} (\\psi _+^m + \\psi _-^m)\\, .$ Since $\\psi _+ - \\frac{1}{2} = \\frac{\\sqrt{1+4\\gamma }}{2},$ Equation () is equivalent to $(\\psi _+^m - \\psi _-^m) \\le m \\sqrt{1+4\\gamma }(\\psi _+^m + \\psi _-^m),$ or $\\psi _+^m \\left( 1 - m \\sqrt{1+4\\gamma }\\right) \\le \\psi _-^m \\left( 1 + m \\sqrt{1+4\\gamma }\\right).$ Both $\\psi _+$ and $1 + m\\sqrt{1+4\\gamma }$ are positive, and $\\frac{\\psi _-}{\\psi _+} = \\frac{1-\\sqrt{1+4\\gamma }}{1+\\sqrt{1+4\\gamma }} = \\frac{1+2\\gamma - \\sqrt{1+4\\gamma }}{-2\\gamma }.$ It follows that proving (REF ) is equivalent to proving $\\frac{1-m\\sqrt{1+4\\gamma }}{1 +m\\sqrt{1+4\\gamma }} \\le \\left( \\frac{1+2\\gamma - \\sqrt{1+4\\gamma }}{-2\\gamma }\\right)^m \\, .$ We prove (REF ) by induction for $m \\ge 0$ .",
"For $m = 0$ , (REF ) is trivially satisfied.",
"Assume that (REF ) holds for $m$ .",
"Using this, we have to show that (REF ) holds for $m+1$ .",
"This amounts to showing that $\\frac{1-m\\sqrt{1+4\\gamma }}{1 +m\\sqrt{1+4\\gamma }} \\, \\frac{1+2\\gamma - \\sqrt{1+4\\gamma }}{-2\\gamma }\\ge \\frac{1-(m+1)\\sqrt{1+4\\gamma }}{1 +(m+1)\\sqrt{1+4\\gamma }}.$ Multiplying (REF ) by all (positive) denominators simplifies to an inequality which holds for all $\\gamma \\in [-1/4,0)$ : $m(m+1)(1+4\\gamma )^{3/2}\\left( 1 - \\sqrt{1+4\\gamma }\\right) \\ge 0.$ Lemma 6 For all $m\\ge 2$ , $-\\gamma (2^m - 1)s_{m-1}(\\gamma ) + s_{m+1} (\\gamma ) &\\ge & 1\\, ,\\ \\mbox{for $\\gamma \\in [-1/4,0]$.}",
"\\\\-\\gamma (2^m - 1)s_{m-1}(\\gamma ) + s_{m+1} (\\gamma ) &\\le & 1\\, ,\\ \\mbox{for $\\gamma \\ge 0$.",
"}$ We prove(REF ) using induction.",
"For $m=2$ and $m=3$ $-\\gamma (2^2 - 1) s_1(\\gamma ) + s_3(\\gamma ) = -3\\gamma + 1 + 3\\gamma & =& 1, \\\\-\\gamma (2^3 - 1)s_2(\\gamma ) + s_4(\\gamma ) = 1 - 3\\gamma (1+4\\gamma ) & \\ge & 1.$ Assume (REF ) holds for some $m\\ge 3$ , i.e., $-\\gamma (2^m -1)s_{m-1} + s_{m+1} \\ge 1.$ By Lemma REF , $s_{m} + \\gamma s_{m-2} \\ge 0$ .",
"Using (REF ) this becomes $s_{m-1} + 2 \\gamma s_{m-2} \\ge 0$ .",
"After multiplication by $2^m - 1 >0$ , we obtain the equivalent form $(2^m - 1) s_{m-1} + 2 \\gamma (2^{m}-1) s_{m-2} =(2^m - 1) s_{m-1} + \\gamma (2^{m+1}-2) s_{m-2} \\ge 0,$ which, using (REF ), is rewritten as $2^{m} s_{m-1} + \\gamma (2^{m+1} - 1) s_{m-2} - s_{m} \\ge 0.$ Multiplying (REF ) by $-\\gamma \\ge 0$ and adding (REF ) gives $-\\gamma (2^{m+1} -1) (s_{m-1} + \\gamma s_{m-2}) + ( s_{m+1} + \\gamma s_m) \\ge 1,$ which is rewritten as $-\\gamma (2^{m+1} -1) s_m +s_{m+2}\\ge 1\\, .$ This concludes the proof of the second induction step.",
"Next we prove ().",
"The statement is true for $m = 2$ and $m=3$ : $-\\gamma (2^2 - 1) s_1(\\gamma ) + s_3(\\gamma ) = 1\\, , \\qquad \\quad -\\gamma (2^3 - 1)s_2(\\gamma ) + s_4(\\gamma ) = 1 - 3\\gamma - 12 \\gamma ^2 \\le 1.$ Assume () holds for some $m\\ge 3$ , i.e., $-\\gamma (2^m -1)s_{m-1} + s_{m+1} \\le 1 .$ By Lemma REF , $s_{m} + \\gamma s_{m-2} \\ge 0$ or equivalently $s_{m-1} + 2 \\gamma s_{m-2} \\ge 0$ , so that $(2^m - 1) s_{m-1} + 2 \\gamma (2^{m}-1) s_{m-2} =(2^m - 1) s_{m-1} + \\gamma (2^{m+1}-2) s_{m-2} \\ge 0.$ This is rewritten as $2^{m} s_{m-1} + \\gamma (2^{m+1} - 1) s_{m-2} - s_{m} \\ge 0.$ We reverse this inequality by multiplying it by $-\\gamma \\le 0$ , and add (REF ) to it to obtain $-\\gamma (2^m -1)s_{m-1} + s_{m+1}-\\gamma 2^m s_{m-1} - \\gamma (2^{m+1}-1) \\gamma s_{m-2} + \\gamma s_{m} \\le 1,$ which reduces to $-\\gamma (2^{m+1} -1) s_m +s_{m+2}\\le 1\\, .$ This concludes the proof of the second induction step.",
"Lemma 7 The sequence $\\frac{2^m s_{m+1} (\\gamma ) - 1}{2^m -1}, \\qquad \\qquad m \\ge 1,$ is non-increasing in $m$ for $\\gamma \\in [-1/4,0]$ .",
"We prove that for $m \\ge 1$ , $\\frac{2^m s_{m+1} - 1}{2^m - 1} \\ge \\frac{2^{m+1} s_{m+2} - 1}{2^{m+1} - 1} \\, .$ Using the recurrence relation (REF ), this is equivalent to $s_{m+1} \\ge \\gamma (2^{m+1} - 2) s_{m} + 1 ~~\\iff ~~s_{m+2} - \\gamma (2^{m+1}-1) s_{m} \\ge 1,$ which follows directly from Lemma REF .",
"Lemma 8 The sequence $\\frac{2^m s_{m+1} (\\gamma ) - 1}{m(m+1)}, \\qquad \\qquad m \\ge 1,$ is nondecreasing in $m$ for $\\gamma \\ge -1/4$ .",
"We use induction to show that for $m \\ge 1$ $\\frac{2^m s_{m+1} - 1}{m(m+1)} \\le \\frac{2^{m+1} s_{m+2} - 1}{(m+1)(m+2)},$ or equivalently, for $m\\ge 1$ , $(m+2)2^{m} s_{m+1} \\le m 2^{m+1} s_{m+2} + 2\\, .$ The inequality (REF ) holds for $m= 1$ as $6s_2 = 6(1+2\\gamma ) = 4 (1+3\\gamma ) + 2 = 4s_3+2$ .",
"Using (REF ) to expand $s_{m+2}$ in (REF ) we obtain $(m+2)2^{m} s_{m+1} \\le m 2^{m+1} (s_{m+1} + \\gamma s_{m}) + 2,$ and (REF ) is equivalent to $2^m s_{m+1} - \\gamma m 2^{m+1} s_m \\le (m-1) 2^m s_{m+1} + 2\\, .$ It suffices to prove that $2^m s_{m+1} - \\gamma m 2^{m+1} s_m \\le (m+1)2^{m-1} s_m \\, ,$ since the induction assumption (REF ) for $m \\rightarrow m-1$ implies $(m+1)2^{m-1} s_m \\le (m-1) 2^m s_{m+1} + 2$ .",
"But (REF ) follows directly from () of Lemma REF as it is equivalent to $2s_{m+1} \\le \\left[1 + m(1+4\\gamma )\\right] s_m$ .",
"Finally, we prove two lemmas that provide bounds for growth of the sequence $\\left\\lbrace s_{m}(\\gamma )-1\\right\\rbrace $ .",
"Lemma 9 The sequence $(s_{m} (\\gamma ) - 1)/m, \\qquad \\qquad m \\ge 3,$ is non-decreasing in $m$ for $\\gamma \\in [-1/4,0)$ .",
"The statement is equivalent to $(m+1) s_{m} \\le m s_{m+1} +1$ , which we prove by induction.",
"First, for $m = 3$ we have $4s_3 < 3s_4 + 1$ , i.e., $4(1+3\\gamma ) < 3 (1+4\\gamma + 2 \\gamma ^2)+1$ which holds for $\\gamma \\ne 0$ .",
"Assume that the statement holds for $m \\rightarrow m-1$ , i.e., $ms_{m-1} \\le (m-1) s_{m} +1$ , which is equivalent to $s_m \\le m(s_m - s_{m-1})+1$ .",
"Thus $s_m \\le m\\gamma s_{m-2} + 1$ .",
"However, for $\\gamma \\in [-1/4,0)$ and $m \\ge 2$ one has $0 < s_{m-1} < s_{m-2}$ and thus $s_m \\le m \\gamma s_{m-1} + 1$ .",
"The claim follows by an application of (REF ) to $s_{m-1}$ .",
"Lemma 10 The sequence $(s_{m+1} (\\gamma ) - 1)/(2^{-m}-1), \\qquad \\qquad m \\ge 1,$ is (i) non-decreasing in $m$ , for $\\gamma \\in [-1/4,0)$ ; (ii) non-increasing in $m$ , for $\\gamma > 0$ .",
"First, we prove (i), which is equivalent to $(2^{m+1}-2)s_{m+2} +1 \\le (2^{m+1} - 1)s_{m+1}$ .",
"Using (REF ) in the form $s_{m+2} = s_{m+1} + \\gamma s_m$ , this reduces to $s_{m+1} - 2\\gamma (2^m - 1)s_m \\ge 1$ .",
"This follows directly from a combination of $-\\gamma (2^m -1) s_{m-1} + s_{m+1} \\ge 1$ , which holds for all $m \\ge 2$ , and $\\gamma \\in [-1/4,0)$ by Lemma REF and $s_{m-1} \\le 2 s_m$ (see ()).",
"Next we prove (ii) by an analogous argument.",
"We have to show that $(2^{m+1}-2)s_{m+2} +1 \\ge (2^{m+1} - 1)s_{m+1}$ , which reduces (by (REF ) in the form $s_{m+2} = s_{m+1} + \\gamma s_m$ ) to $s_{m+1} - 2\\gamma (2^m - 1)s_m \\le 1$ .",
"This follows from $-\\gamma (2^m -1) s_{m-1} + s_{m+1} \\le 1$ (by Lemma REF ) and $s_{m-1} \\le 2 s_m$ (by ()) for all $m \\ge 2$ .",
"This work was supported by the Slovak Research and Development Agency under the contract No.",
"APVV-14-0378, by the Scientific Grant Agency of the Slovak Republic under the grant 1/0755/19 (RK) and by the National Science Foundation under grant number NSF-DMS-1522677 (BD).",
"Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.",
"The authors wish to thank Casa Mathemática Oaxaca and Erwin Schrődinger Institute for their hospitality during the development of the ideas for this work.",
"To appear in SIAM Journal on Mathematical Analysis.",
"Published on arXiv with permission of The Society for Industrial and Apllied Mathematics (SIAM)."
]
] | 1906.04453 |
[
[
"Mean estimation and regression under heavy-tailed distributions--a\n survey"
],
[
"Abstract We survey some of the recent advances in mean estimation and regression function estimation.",
"In particular, we describe sub-Gaussian mean estimators for possibly heavy-tailed data both in the univariate and multivariate settings.",
"We focus on estimators based on median-of-means techniques but other methods such as the trimmed mean and Catoni's estimator are also reviewed.",
"We give detailed proofs for the cornerstone results.",
"We dedicate a section on statistical learning problems--in particular, regression function estimation--in the presence of possibly heavy-tailed data."
],
[
"Introduction",
"Arguably the most fundamental problem of statistics is that of estimating the expected value $\\mu $ of a random variable $X$ based on a sample of $n$ independent, identically distributed draws from the distribution of $X$ .",
"The obvious choice of an estimator is, of course, the empirical mean.",
"Its properties are well understood by classical results of probability theory.",
"However, from the early days on, statisticians have been concerned about the quality of the empirical mean, especially when the distribution may be heavy-tailed or outliers may be present in the data.",
"This concern gave rise to the area of robust statistics that addresses the problem of mean estimation (and other statistical problems) for such data.",
"Classical references include Huber [38], Huber and Ronchetti [39], Hampel, Ronchetti, Rousseeuw, and Stahel [30], Tukey [77].",
"Motivated by applications in machine learning and data science, in recent years there has been increased interest in constructing mean and regression function estimates with the requirement that the estimators should achieve high accuracy with a large confidence.",
"The best achievable accuracy/confidence tradeoff is much better understood today and the aim of this paper is to survey some of the recent advances.",
"We primarily focus on the mean estimation problem, both in the univariate and multivariate settings.",
"We offer detailed discussion of what the best performance one may expect is, describe a variety of estimators, and analyze their performance.",
"We pay special attention to a simple but powerful methodology based on median-of-means techniques.",
"We also address one of the basic problems of statistical learning theory, namely regression function estimation.",
"We show how the technology introduced for mean estimation may be used to construct powerful learning algorithms that achieve essentially optimal performance under mild assumptions.",
"The paper is organized as follows.",
"In Section we address the simplest, univariate mean estimation problem.",
"We focus on sub-Gaussian estimators and explore their possibilities and limitations.",
"Section is dedicated to the significantly more challenging multivariate problem.",
"We extend the notion of sub-Gaussian estimators to the multivariate setting and analyze various estimators.",
"In Section we study the problem of estimating the mean of an entire class of random variables with the requirement that all estimators have a high accuracy simultaneously over the entire class.",
"We show how such estimators may be constructed and use these ideas in a general framework of mean estimation.",
"Finally, Section is dedicated to applying these techniques to regression function estimation."
],
[
"Estimating the mean of a real random variable",
"In this section we examine the classical problem of estimating the mean of a random variable.",
"Let $X_1,\\ldots ,X_n$ be independent, identically distributed real random variables with mean $\\mu =\\mathbb {E}X_1$ .",
"Upon observing these random variables, one would like to estimate $\\mu $ .",
"An estimator $\\widehat{\\mu }_n=\\widehat{\\mu }_n(X_1,\\ldots ,X_n)$ is simply a measurable function of $X_1,\\ldots ,X_n$ .",
"The quality of an estimator may be measured in various ways.",
"While most of the early statistical work focused on expected risk measures such as the mean-squared error $\\mathbb {E}\\left[ \\left( \\widehat{\\mu }_n - \\mu \\right)^2 \\right]~,$ such risk measures may be misleading.",
"Indeed, if the difference $|\\widehat{\\mu }_n - \\mu |$ is not sufficiently concentrated, the expected value does not necessarily reflect the “typical” behavior of the error.",
"For such reasons, we prefer estimators $\\widehat{\\mu }_n$ that are close to $\\mu $ with high probability.",
"Thus, our aim is to understand, for any given sample size $n$ and confidence parameter $\\delta \\in (0,1)$ , the smallest possible value $\\epsilon =\\epsilon (n,\\delta )$ such that $ \\mathbb {P}\\left\\lbrace \\left| \\widehat{\\mu }_n - \\mu \\right| > \\epsilon \\right\\rbrace \\le \\delta ~.$ It it important to stress that (REF ) is a non-asymptotic criterion: one would like to obtain quantitative estimates on the way the accuracy $\\epsilon $ scales with the confidence parameter $\\delta $ and the sample size $n$ .",
"This type of estimate is reminiscent to the pac (Probably Approximately Correct) framework usually adopted in statistical learning theory, see Valiant [79], Vapnik and Chervonenkis [82], Blumer, Ehrenfeucht, Haussler, and Warmuth [9].",
"The most natural choice of a mean estimator is the standard empirical mean $\\overline{\\mu }_n=\\frac{1}{n}\\sum _{i=1}^nX_i~.$ The behavior of the empirical mean is well understood.",
"For example, if the $X_i$ have a finite second moment and $\\sigma ^2$ denotes their variance, then the mean-squared error of $\\overline{\\mu }_n$ equals $\\sigma ^2/n$ .",
"On the other hand, the central limit theorem guarantees that this estimator has Gaussian tails, asymptotically, when $n\\rightarrow \\infty $ .",
"Indeed, $\\mathbb {P}\\left\\lbrace \\left|\\overline{\\mu }_n - \\mu \\right|>\\frac{\\sigma \\Phi ^{-1}(1-\\delta /2)}{\\sqrt{n}}\\right\\rbrace \\rightarrow \\delta ~,$ where $\\Phi (x)=\\mathbb {P}\\lbrace G\\le x\\rbrace $ is the cumulative distribution function of a standard normal random variable $G$ .",
"One may easily see (e.g., using the fact that for $t \\ge 1$ , $\\exp (-t^2/2) \\le t\\exp (-t^2/2)$ ), that for all $x\\ge 0$ , $1-\\Phi (x) \\le e^{-x^2/2}~.$ This implies that $\\Phi ^{-1}(1-\\delta /2)\\le \\sqrt{2\\log (2/\\delta )}$ , and the central limit theorem asserts that $\\lim _{n\\rightarrow \\infty }\\mathbb {P}\\left\\lbrace \\left|\\overline{\\mu }_n - \\mu \\right|>\\frac{\\sigma \\sqrt{2\\log (2/\\delta )}}{\\sqrt{n}}\\right\\rbrace \\le \\delta ~.$ However, this is an asymptotic estimate and not the quantitative one we were hoping for.",
"Still, our goal is to obtain non-asymptotic performance bounds of the same form.",
"In particular, we say that a mean estimator $\\widehat{\\mu }_n$ is $L$ -sub-Gaussian if there is a constant $L>0$ , such that for all sample sizes $n$ and with probability at least $1-\\delta $ , $\\left|\\widehat{\\mu }_n - \\mu \\right| \\le \\frac{L\\sigma \\sqrt{\\log (2/\\delta )}}{\\sqrt{n}}~.$ It is worth noting here the well-known fact that if all one knows is that the unknown distribution is Gaussian, then the sample mean is optimal for all sample sizes and confidence levels $\\delta $ .",
"(See Catoni [14] for a precise statement.)",
"Moreover, the following observation, established by Devroye, Lerasle, Lugosi, and Oliveira [22], shows that (REF ) is essentially the best that one can hope for in general, even if one is interested in a fixed confidence level: Theorem 1 Let $n>5$ be a positive integer.",
"Let $\\mu \\in \\mathbb {R}$ , $\\sigma >0$ and $\\delta \\in (2e^{-n/4},1/2)$ .",
"Then for any mean estimator $\\widehat{\\mu }_n$ , there exists a distribution with mean $\\mu $ and variance $\\sigma ^2$ such that $\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n - \\mu \\right|> \\sigma \\sqrt{\\frac{\\log (1/\\delta )}{n}} \\right\\rbrace \\ge \\delta ~.$ Proof.",
"To derive the “minimax” lower bound, it suffices to consider two distributions, $P_+,P_-$ , both concentrated on two points, defined by $P_+(\\lbrace 0\\rbrace ) = P_-(\\lbrace 0\\rbrace ) = 1-p~, \\qquad P_+(\\lbrace c\\rbrace ) = P_-(\\lbrace -c\\rbrace ) = p~,$ where $p\\in [0,1]$ and $c>0$ .",
"Note that the means of the two distributions are $\\mu _{P_+}= pc$ and $\\mu _{P_-}= -pc$ and both have variance $\\sigma ^2=c^2p(1-p)$ .",
"For $i=1,\\ldots ,n$ , let $(X_i,Y_i)$ be independent pairs of real-valued random variables such that $\\mathbb {P}\\lbrace X_i=Y_i=0\\rbrace = 1-p\\quad \\text{and} \\quad \\mathbb {P}\\lbrace X_i=c, Y_i=-c\\rbrace = p~.$ Note that $X_i$ is distributed as $P_+$ and $Y_i$ is distributed as $P_-$ .",
"Let $\\delta \\in (0,1/2)$ .",
"If $\\delta \\ge 2e^{-n/4}$ and $p =(1/(2n))\\log (2/\\delta )$ , then (using $1-p\\ge \\exp (-p/(1-p))$ ), $\\mathbb {P}\\lbrace X_1^n = Y_1^n\\rbrace =(1-p)^n \\ge 2\\delta ~.$ Let $\\widehat{\\mu }_n$ be any mean estimator, possibly depending on $\\delta $ .",
"Then ${\\max \\left(\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n (X_1^n) - \\mu _{P_+}\\right| > cp\\right\\rbrace ,\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n(Y_1^n) - \\mu _{P_-}\\right| > cp \\right\\rbrace \\right)}\\\\& & \\ge \\frac{1}{2}\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n (X_1,\\ldots ,X_n) - \\mu _{P_+}\\right| > cp\\quad \\text{or} \\quad \\left|\\widehat{\\mu }_n(Y_1,\\ldots ,Y_n) - \\mu _{P_-}\\right| > cp \\right\\rbrace \\\\& &\\ge \\frac{1}{2}\\mathbb {P}\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) = \\widehat{\\mu }_n(Y_1,\\ldots ,Y_n)\\rbrace \\\\& &\\ge \\frac{1}{2} \\mathbb {P}\\lbrace X_1,\\ldots ,X_n = Y_1,\\ldots ,Y_n\\rbrace \\ge \\delta ~.$ From $\\sigma ^2=c^2p(1-p)$ and $p\\le 1/2$ we have that $cp\\ge \\sigma \\sqrt{p/2}$ , and therefore $\\max \\left(\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n (X_1,\\ldots ,X_n) - \\mu _{P_+}\\right| >\\sigma \\sqrt{\\frac{\\log \\frac{2}{\\delta }}{n}}\\right\\rbrace ~,\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n(Y_1,\\ldots ,Y_n) - \\mu _{P_-}\\right| > \\sigma \\sqrt{\\frac{\\log \\frac{2}{\\delta }}{n}}\\right\\rbrace \\right) \\ge \\delta ~.$ Theorem REF follows.",
"With Theorem REF in mind, our aim is to consider both univariate and multivariate situations and design estimators that perform with sub-Gaussian error rate.",
"The meaning of sub-Gaussian error rate in the multivariate case is explained in Section .",
"Naturally, the first order of business is to check whether the obvious choice of a mean estimator—the empirical mean—is $L$ -sub-Gaussian for some $L$ .",
"On the one hand, it is easy to see that under certain conditions on the distribution of the $X_i$ , it does exhibit a sub-Gaussian performance.",
"Indeed, if the $X_i$ are such that there exists $L>0$ such that for all $\\lambda >0$ $\\mathbb {E}e^{\\lambda (X_i-\\mu )} \\le e^{\\sigma ^2\\lambda ^2/L^2}~,$ then the empirical mean $\\widehat{\\mu }_n$ is $L$ -sub-Gaussian for all $\\delta \\in (0,1)$ , as it is easily seen by the Chernoff bound.",
"On the other hand, assumptions of this type are quite restrictive and impose strong conditions on the decay of the tail probabilities of the $X_i$ .",
"Specifically, it is equivalent to the fact that for every $p\\ge 2$ , $\\left(\\mathbb {E}|X_i-\\mu |^p\\right)^{1/p} \\le L^\\prime \\sqrt{p} \\left(\\mathbb {E}|X_i-\\mu |^2\\right)^{1/2}$ , where $c_1 L \\le L^\\prime \\le c_2 L $ for suitable absolute constants $c_1$ and $c_2$ (see, e.g., [10]).",
"When the $X_i$ 's do not exhibit such a tail decay, the empirical mean need not be sub-Gaussian.",
"For example, if one only assumes that $\\sigma $ exists (i.e., the variance of the $X_i$ is finite) then the bound implied by Chebyshev's inequality, that is, that with probability at least $1-\\delta $ , $ \\left| \\overline{\\mu }_n - \\mu \\right| \\le \\sigma \\sqrt{\\frac{1}{n\\delta }}~,$ is essentially the best that one can hope for.",
"Although the bound from (REF ) decays with the sample size at the optimal rate of $O(n^{-1/2})$ , the dependence on the confidence parameter $\\delta $ is exponentially worse than in (REF ).",
"We refer to Catoni [14] for a precise formulation and a simple example that (almost) saturates Chebyshev's inequality.",
"This leads to an inevitable conclusion: if one is looking for a mean estimator that is sub-Gaussian for any random variable that has a well-defined mean and variance, then one must find alternatives to the sample mean.",
"As it happens, and perhaps surprisingly, there exist mean estimators that achieve a sub-Gaussian performance for all distributions with a finite variance.",
"Two quite different estimators are presented and analyzed in the next two sections."
],
[
"The median-of-means estimator",
"The median-of-means estimator presented next has been proposed in different forms in various papers, see Nemirovsky and Yudin [69], Hsu [35], Jerrum, Valiant, and Vazirani [40], Alon, Matias, and Szegedy [1].",
"The definition of the median-of-means estimator calls for partitioning the data into $k$ groups of roughly equal size, computing the empirical mean in each group, and taking the median of the obtained values.",
"Formally, recall that the median of $k$ real numbers $x_1,\\ldots ,x_k\\in \\mathbb {R}$ is defined as $M(x_1,\\ldots ,x_k) = x_i$ where $x_i$ is such that $\\left| \\lbrace j\\in [k]\\,:\\,x_j\\le x_i\\rbrace \\right| \\ge \\frac{k}{2} \\quad \\text{and} \\quad \\left| \\lbrace j\\in [k]\\,:\\,x_j\\ge x_i\\rbrace \\right| \\ge \\frac{k}{2}~.$ (If several indices $i$ fit the above description, we take the smallest one.)",
"Now let $1\\le k\\le n$ and partition $[n]=\\lbrace 1,\\dots ,n\\rbrace $ into $k$ blocks $B_1,\\ldots ,B_k$ , each of size $|B_i|\\ge \\lfloor n/k\\rfloor \\ge 2$ .",
"Given $X_1,\\ldots ,X_n$ , compute the sample mean in each block $Z_j=\\frac{1}{|B_j|}\\sum _{i\\in B_j}X_i$ and define the median-of-means estimator by $\\widehat{\\mu }_n= M(Z_1,\\ldots ,Z_k).$ To grasp intuitively why this estimator works, note that for each block, the empirical mean is an unbiased estimator of the mean, with controlled standard deviation $\\sigma /\\sqrt{n/k}$ .",
"Hence, the median of the distribution of the blockwise empirical mean lies within $\\sigma /\\sqrt{n/k}$ from the expectation.",
"Now the empirical median is a highly concetrated estimator of this median.",
"A performance-bound of the estimator is established next.",
"For simplicity, assume that $n$ is divisible by $k$ so that each block has $m=n/k$ elements.",
"Theorem 2 Let $X_1,\\ldots ,X_n$ be independent, identically distributed random variables with mean $\\mu $ and variance $\\sigma ^2$ .",
"Let $m,k$ be positive integers assume that $n=mk$ .",
"Then the median-of-means estimator $\\widehat{\\mu }_n$ with $k$ blocks satisfies $\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n-\\mu \\right| > \\sigma \\sqrt{4/m} \\right\\rbrace \\le e^{-k/8}~.$ In particular, for any $\\delta \\in (0,1)$ , if $k= \\left\\lceil 8 \\log (1/\\delta ) \\right\\rceil $ , then, with probability at least $1-\\delta $ , $\\left|\\widehat{\\mu }_n-\\mu \\right| \\le \\sigma \\sqrt{\\frac{32\\log (1/\\delta )}{n}}~.$ Proof.",
"By Chebyshev's inequality, for each $j=1,\\ldots ,k$ , with probability at least $3/4$ , $\\left| Z_j - \\mu \\right| \\le \\sigma \\sqrt{\\frac{4}{m}}~.$ Thus, $\\left|\\widehat{\\mu }_n-\\mu \\right| > \\sigma \\sqrt{4/m}$ implies that at least $k/2$ of the means $Z_j$ are such that $\\left| Z_j - \\mu \\right| > \\sigma \\sqrt{4/m}$ .",
"Hence, $\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n-\\mu \\right| > \\sigma \\sqrt{4/m} \\right\\rbrace & \\le & \\mathbb {P}\\left\\lbrace {\\mathrm {Bin}}(k,1/4)\\ge \\frac{k}{2} \\right\\rbrace \\\\& & \\text{(where ${\\mathrm {Bin}}(k,1/4)$ is a binomial $(k,1/4)$ random variable)} \\\\& = & \\mathbb {P}\\left\\lbrace {\\mathrm {Bin}}(k,1/4) - \\mathbb {E}{\\mathrm {Bin}}(k,1/4) \\ge \\frac{k}{4} \\right\\rbrace \\\\& \\le & e^{-k/8}\\quad \\text{(by Hoeffding's inequality \\cite {Hoe63}).",
"}$ Theorem REF shows that the median-of-means estimator has a sub-Gaussian performance with $L=8$ for all distributions with a finite variance.",
"However, it is important to point out that the estimator $\\widehat{\\mu }_n$ depends on the confidence level $\\delta $ as the number of blocks $k$ is chosen as a function of $\\delta $ .",
"This is not a desirable property, since for different values of the confidence parameter $\\delta $ , one obtains a different point estimator.",
"However, as it is shown in Section REF below, there do not exist sub-Gaussian estimators that are independent of the confidence level, unless one is willing to assume more than just the finiteness of the second moment of the underlying distribution.",
"The results of Bubeck, Cesa-Bianchi, and Lugosi [12] and Devroye, Lerasle, Lugosi, and Oliveira [22] show that the median-of-means estimator may also be used even if the distribution of the $X_i$ has an infinite variance but has a finite moment of order $1+\\alpha $ for some $\\alpha \\in (0,1)$ .",
"Theorem 3 Let $\\alpha \\in (0,1]$ and let $X_1,\\ldots ,X_n$ be independent, identically distributed random variables with mean $\\mu $ and $(1+\\alpha )$ -th central moment $M=\\mathbb {E}\\left[|X_i-\\mu |^{1+\\alpha }\\right]$ .",
"Let $m,k$ be positive integers and assume that $n=mk$ .",
"Then the median-of-means estimator with $k =\\left\\lceil 8 \\log (2/\\delta ) \\right\\rceil $ blocks satisfies $\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n-\\mu \\right| > 8 \\left(\\frac{12M^{1/\\alpha }\\log (1/\\delta )}{n}\\right)^{\\alpha /(1+\\alpha )} \\right\\rbrace \\le \\delta ~.$ Moreover, for any mean estimator $\\widehat{\\mu }_n$ , there exists a distribution with mean $\\mu $ and $(1+\\alpha )$ -th central moment $M$ such that $\\mathbb {P}\\left\\lbrace \\left|\\widehat{\\mu }_n-\\mu \\right| > \\left(\\frac{M^{1/\\alpha }\\log (2/\\delta )}{n}\\right)^{\\alpha /(1+\\alpha )} \\right\\rbrace \\ge \\delta ~.$ The proof of the first part follows by showing that if $c(\\alpha )$ is an appropriate constant that depends only on $\\alpha $ and $\\eta \\ge c(\\alpha ) \\left(\\mathbb {E}|X_i-\\mu |^{1+\\alpha }\\right)^{1/(1+\\alpha )} \\left(\\frac{1}{m}\\right)^{\\alpha /(1+\\alpha )},$ then $\\mathbb {P}\\left( \\left|\\frac{1}{m}\\sum _{i=1}^m X_i - \\mu \\right| \\ge \\eta \\right) \\le 0.2~.$ The proof of the second statement goes along the lines of Theorem REF .",
"We finish this section by showing that if the distribution of $X$ has a finite moment of order $2+\\alpha $ for some $\\alpha >0$ , then the median-of-means estimator has a sub-Gaussian performance under a much wider range of choices for the parameter $k$ that counts the number of blocks.",
"The following bound is due to Minsker and Strawn [68].",
"For simplicity of the exposition, we only consider the case $\\alpha =1$ .",
"Theorem 4 Let $X_1,\\ldots ,X_n$ be independent, identically distributed random variables with mean $\\mu $ , variance $\\sigma ^2$ , and third central moment $\\rho =\\mathbb {E}|X-\\mu |^3$ .",
"Let $m,k$ be positive integers and assume that $n=mk$ .",
"Assume that $\\sqrt{\\frac{\\log (2/\\delta )}{2k}} + \\frac{\\rho }{2\\sigma ^3\\sqrt{m}} \\le 1/4~.$ Then the median-of-means estimator $\\widehat{\\mu }_n$ with $k$ blocks satisfies that, with probability at least $1-\\delta $ , $\\left|\\widehat{\\mu }_n-\\mu \\right| \\le \\frac{1}{c} \\left( \\sigma \\sqrt{\\frac{\\log (2/\\delta )}{2n}} +\\frac{\\rho k}{2\\sigma ^2 n} \\right)~,$ where $c=\\phi (\\Phi ^{-1}(3/4))$ is a constant.",
"Here $\\phi $ and $\\Phi $ denote the standard normal density and distribution functions.",
"Observe that the first term on the right-hand side of the bound is of the sub-Gaussian form.",
"The second term is smaller than the first whenever the number $k$ of blocks satisfies $k\\le \\frac{2\\sigma ^3}{\\rho }\\sqrt{n\\log (2/\\delta )}~.$ In particular, $k\\le \\frac{2\\sigma ^3}{\\rho }\\sqrt{n}$ suffices to get a sub-Gaussian performance.",
"This is nice since with such a choice the estimator does not depend on the value of the confidence parameter $\\delta $ and the estimator is sub-Gaussian simultaneously for the entire range of values of $\\delta $ permitted by the condition (REF ).",
"Also, note that the number of blocks may be chosen to be much larger than the choice suggested by Theorem REF .",
"In particular, $k$ can be as large as a constant multiple of $\\sqrt{n}$ .",
"In that case the median-of-means estimator is sub-Gaussian simultaneously for all $\\delta \\ge e^{-c_0\\sqrt{n}}$ for an appropriate constant $c_0$ .",
"The price to pay is the extra assumption of the existence of the third moment.",
"Minsker and Strawn [68] also prove that, when $k=o(\\sqrt{n})$ , then, under the assumptions of Theorem REF , $\\sqrt{n}\\left(\\widehat{\\mu }_n-\\mu \\right)$ is asymptotically normal with mean zero and variance $\\sigma ^2\\pi /2$ .",
"Proof.",
"Note that $\\widehat{\\mu }_n \\in [\\mu -a,\\mu +a]$ if $a>0$ is such that $\\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{Z_j-\\mu \\le a} \\ge \\frac{1}{2}\\quad \\text{and} \\quad \\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{Z_j-\\mu \\ge -a} \\ge \\frac{1}{2}~.$ We show that, with probability at least $1-\\delta $ , one may take $a=\\frac{1}{c} \\left( \\sigma \\sqrt{\\frac{\\log (2/\\delta )}{2n}} +\\frac{\\rho k}{2\\sigma ^2 n} \\right)~.$ To this end, note that $\\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{Z_j-\\mu \\le a}& =&\\frac{1}{k} \\sum _{j=1}^k \\left( \\mathbb {1}_{Z_j-\\mu \\le a} -\\mathbb {P}\\left\\lbrace Z_j-\\mu \\le a \\right\\rbrace \\right) \\\\& & + \\left( \\mathbb {P}\\left\\lbrace Z_1-\\mu \\le a \\right\\rbrace - \\mathbb {P}\\left\\lbrace G \\frac{\\sigma }{\\sqrt{m}} \\le a \\right\\rbrace \\right) \\\\& & + \\mathbb {P}\\left\\lbrace G \\frac{\\sigma }{\\sqrt{m}} \\le a \\right\\rbrace \\\\& & \\text{(where $G$ is a standard normal random variable).",
"}$ First note that, by Hoeffding's inequality, with probability at least $1-\\delta /2$ , $\\frac{1}{k} \\sum _{j=1}^k \\left( \\mathbb {1}_{Z_j-\\mu \\le a} -\\mathbb {P}\\left\\lbrace Z_j-\\mu \\le a \\right\\rbrace \\right) \\ge - \\sqrt{\\frac{\\log (2/\\delta )}{2k}}~.$ For the second term on the right-hand side, we may use the Berry-Esseen theorem (see Shevtsova [72]) that implies that $\\mathbb {P}\\left\\lbrace Z_1-\\mu \\le a \\right\\rbrace - \\mathbb {P}\\left\\lbrace G \\frac{\\sigma }{\\sqrt{m}} \\le a \\right\\rbrace \\ge - \\frac{\\rho }{2\\sigma ^3\\sqrt{m}}~.$ Hence, we have that, with probability at least $1-\\delta /2$ , $\\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{Z_j-\\mu \\le a} \\ge \\mathbb {P}\\left\\lbrace G\\frac{\\sigma }{\\sqrt{m}} \\le a \\right\\rbrace -\\sqrt{\\frac{\\log (2/\\delta )}{2k}}- \\frac{\\rho }{2\\sigma ^3\\sqrt{m}}~.$ Thus, $(1/k)\\sum _{j=1}^k \\mathbb {1}_{Z_j-\\mu \\le a} \\ge \\frac{1}{2}$ with probability at least $1-\\delta /2$ , whenever $a$ is such that $\\mathbb {P}\\left\\lbrace G\\le a \\frac{\\sqrt{m}}{\\sigma } \\right\\rbrace \\ge \\frac{1}{2} +\\sqrt{\\frac{\\log (2/\\delta )}{2k}} + \\frac{\\rho }{2\\sigma ^3\\sqrt{m}}~.$ If $\\sqrt{\\frac{\\log (2/\\delta )}{2k}} + \\frac{\\rho }{2\\sigma ^3\\sqrt{m}}\\le 1/4$ then it suffices to consider values of $a$ with $a\\sqrt{m}/\\sigma \\le \\Phi ^{-1}(3/4)$ .",
"Then $\\mathbb {P}\\left\\lbrace G\\le a \\frac{\\sqrt{m}}{\\sigma } \\right\\rbrace \\ge \\frac{1}{2} +c \\frac{a\\sqrt{m}}{\\sigma }$ with $c=\\phi (\\Phi ^{-1}(3/4))$ .",
"Hence, we may take $a= \\frac{\\sigma }{c\\sqrt{m}} \\left(\\sqrt{\\frac{\\log (2/\\delta )}{2k}} +\\frac{\\rho }{2\\sigma ^3\\sqrt{m}} \\right)= \\frac{1}{c} \\left( \\sigma \\sqrt{\\frac{\\log (2/\\delta )}{2n}} +\\frac{\\rho k}{2\\sigma ^2 n} \\right)~.$ The same argument shows that, with probability at least $1-\\delta /2$ , $\\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{Z_j-\\mu \\ge -a} \\ge \\frac{1}{2}$ for the choice of $a$ above."
],
[
"Catoni's estimator",
"Next we present a completely different approach for constructing a mean estimator, introduced and analyzed by Catoni [14].",
"To introduce Catoni's idea, note first that the empirical mean $\\overline{\\mu }_n$ is just the solution $y\\in \\mathbb {R}$ of the equation $\\sum _{i=1}^n \\left( X_i-y\\right) = 0~.$ Catoni proposed to replace the left-hand side of the equation above by another strictly decreasing function of $y$ of the form $R_{n,\\alpha }(y) = \\sum _{i=1}^n \\psi \\left(\\alpha ( X_i-y)\\right)~,$ where $\\psi :\\mathbb {R}\\rightarrow \\mathbb {R}$ is an antisymmetric increasing function and $\\alpha \\in \\mathbb {R}$ is a parameter.",
"The idea is that if $\\psi (x)$ increases much slower than $x$ , then the effect of “outliers” present due to heavy tails is diminished.",
"Catoni offers a whole range of “influence” functions $\\psi $ .",
"For the ease of exposition, we single out one specific choice, namely $\\psi (x) = \\left\\lbrace \\begin{array}{ll}\\log (1+x+x^2/2)& \\text{if} \\ x\\ge 0 \\\\-\\log (1-x+x^2/2) & \\text{if} \\ x < 0~.\\end{array} \\right.$ We define Catoni's mean estimator $\\widehat{\\mu }_{\\alpha ,n}$ as the unique value $y$ such that $R_{n,\\alpha }(y)=0$ with this choice of $\\psi $ .",
"Since $\\psi (x)\\le \\log (1+x+x^2/2)$ for all $x\\in \\mathbb {R}$ , we have, for all $y\\in \\mathbb {R}$ , $\\mathbb {E}\\left[ e^{R_{n,\\alpha }(y)} \\right] & \\le & \\left( \\mathbb {E}\\left[ 1+ \\alpha (X-y) + \\frac{\\alpha ^2 (X-y)^2}{2} \\right]\\right)^n\\\\& = & \\left( 1 + \\alpha (\\mu -y) + \\frac{\\alpha ^2 \\left(\\sigma ^2+ (\\mu -y)^2\\right)}{2} \\right)^n \\\\& \\le & \\exp \\left( n\\alpha (\\mu -y) + \\frac{n\\alpha ^2 \\left(\\sigma ^2+ (\\mu -y)^2\\right)}{2}\\right)~,$ whenever the $X_i$ have a finite variance $\\sigma ^2$ .",
"Thus, by Markov's inequality, we have that, for any fixed $y\\in \\mathbb {R}$ and $\\delta \\in (0,1)$ , $\\mathbb {P}\\left\\lbrace R_{n,\\alpha }(y) \\ge n\\alpha (\\mu -y) + \\frac{n\\alpha ^2 \\left(\\sigma ^2+ (\\mu -y)^2\\right)}{2} + \\log (1/\\delta )\\right\\rbrace \\le \\delta ~.$ Suppose that the parameter $\\alpha $ is such that $\\alpha ^2\\sigma ^2 + 2\\log (1/\\delta )/n \\le 1$ .",
"Then the quadratic polynomial of $y$ $n\\alpha (\\mu -y) + \\frac{n\\alpha ^2 \\left(\\sigma ^2+ (\\mu -y)^2\\right)}{2}+ \\log (1/\\delta )$ has at least one root.",
"In particular, taking the smaller root $y_+ = \\mu + \\frac{\\frac{\\alpha \\sigma ^2}{2} + \\frac{\\log (1/\\delta )}{n\\alpha }}{\\frac{1}{2}+ \\frac{1}{2}\\sqrt{1-\\alpha ^2\\sigma ^2 - \\frac{2\\log (1/\\delta )}{n}}}~,$ we have that $R_{n,\\alpha } (y_+) < 0$ with probability at least $1-\\delta $ .",
"Since $R_{n,\\alpha }(y)$ is strictly decreasing, this implies that $\\widehat{\\mu }_{\\alpha ,n} < y_+$ with probability at least $1-\\delta $ .",
"A symmetric argument shows that $\\widehat{\\mu }_{\\alpha ,n} > y_-$ with probability at least $1-\\delta $ , where $y_- = \\mu - \\frac{\\frac{\\alpha \\sigma ^2}{2} + \\frac{\\log (1/\\delta )}{n\\alpha }}{\\frac{1}{2}+ \\frac{1}{2}\\sqrt{1-\\alpha ^2\\sigma ^2 - \\frac{2\\log (1/\\delta )}{n}}}~.$ Now by straightforward bounding, and choosing the parameter $\\alpha $ to optimize the bounds, we obtain the following performance estimate.",
"Theorem 5 Let $X_1,\\ldots ,X_n$ be independent, identically distributed random variables with mean $\\mu $ and variance $\\sigma ^2$ .",
"Let $\\delta \\in (0,1)$ be such that $n> 2\\log (1/\\delta )$ .",
"Catoni's mean estimator $\\widehat{\\mu }_{n,\\alpha }$ with parameter $\\alpha = \\sqrt{\\frac{2\\log (1/\\delta )}{n\\sigma ^2\\left(1+\\frac{2\\log (1/\\delta )}{n-2\\log (1/\\delta )}\\right)}}$ satisfies that, with probability at least $1-2\\delta $ , $\\left| \\widehat{\\mu }_{n,\\alpha } - \\mu \\right| < \\sqrt{\\frac{2\\sigma ^2 \\log (1/\\delta )}{n-2\\log (1/\\delta )}}~.$ The theorem highlights that, with an appropriately chosen parameter $\\alpha $ , Catoni's estimator has a sub-Gaussian performance.",
"Quite remarkably, the constant $\\sqrt{2}$ is the best possible.",
"A disadvantage of Catoni's estimator with respect to median-of-means is that the estimator—at least in the form given in the theorem—depends on the variance $\\sigma ^2$ .",
"In general, it is unrealistic to assume knowledge of $\\sigma ^2$ .",
"If one substitutes $\\sigma ^2$ in the formula of $\\alpha $ by an upper bound $v$ , then the bound (REF ) still holds with $v$ replacing $\\sigma ^2$ .",
"In case no good upper bound for $\\sigma ^2$ is available, Catoni [14] shows how to use Lepski's method to select $\\alpha $ from the data that has near-optimal performance.",
"Huber [37] combines the median-of-means estimator with Catoni's estimator into a two-step procedure that to obtain an estimator with the optimal leading constant in the sub-Gaussian bound when $|\\sigma /\\mu |$ is bounded by a known constant.",
"Another problem—shared with the median-of-means estimator—is that Catoni's estimator also depends on the required confidence level $\\delta $ .",
"Such a dependence is necessary as it is shown in Section REF below.",
"A quick fix is to use the estimator with a $\\delta $ -independent parameter, though then the resulting estimate, naturally, cannot be sub-Gaussian.",
"One reasonable choice is $\\alpha =\\sqrt{2/(n\\sigma ^2)}$ .",
"In this case, it is easy to see that, whenever $n>2(1+\\log (1/\\delta ))$ , Catoni's estimator satisfies, with probability at least $1-2\\delta $ , $\\left| \\widehat{\\mu }_{n,\\alpha } - \\mu \\right| < \\sqrt{\\frac{\\sigma ^2}{2n}} \\cdot \\frac{1+ \\log (1/\\delta )}{1-\\frac{1+ \\log (1/\\delta )}{n}}~.$ This is not a sub-Gaussian bound because of an extra factor of $\\sqrt{\\log (1/\\delta )}$ but the “sub-exponential” tail probabilities are still non-trivial and useful."
],
[
"Trimmed mean",
"Perhaps the most natural attempt to improve the performance of the empirical mean is removing possible outliers using a truncation of $X$ .",
"Indeed, the so-called trimmed-mean (or truncated-mean) estimator is defined by removing a fraction of the sample, consisting of the $\\epsilon n$ largest and smallest points for some parameter $\\epsilon \\in (0,1)$ , and then averaging over the rest.",
"This idea is one of the most classical tools in robust statistics and we refer to Tukey and McLaughlin [78], Huber and Ronchetti [39], Bickel [8], Stigler [74] for early work on the theoretical properties of the trimmed-mean estimator.",
"However, it was only recently that the non-asymptotic sub-Gaussian property of the trimmed mean was established.",
"Indeed, Oliveira and Orenstein [70] proved that if $\\epsilon $ is chosen proportionally to $\\log (1/\\delta )/n$ , then the trimmed-mean estimator has a sub-Gaussian performance for all distributions with a finite variance.",
"To show how this works in the simplest way, here we analyze a simple variant of the trimmed-mean estimator.",
"The estimator splits the data in two equal parts.",
"One half is used to determine the correct truncation level.",
"The points from the other half are averaged, except for the data points that fall outside of the truncation region, which are ignored.",
"For convenience of the notation, we assume that the data consists of $2n$ independent copies of the random variable $X$ , denoted by $X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n$ .",
"For $\\alpha \\le \\beta $ , define the truncation function $\\phi _{\\alpha ,\\beta }(x) ={\\left\\lbrace \\begin{array}{ll}\\beta & \\mbox{if} \\ x > \\beta ,\\\\x & \\mbox{if} \\ x \\in [\\alpha ,\\beta ]~,\\\\\\alpha & \\mbox{if} \\ x < \\alpha ~,\\end{array}\\right.",
"}$ and for $x_1,\\ldots , x_m \\in \\mathbb {R}$ let $x_1^* \\le x_2^* \\le \\cdots \\le x_m^*$ be its non-decreasing rearrangement.",
"With this notation in place, the definition of the estimator is as follows: $(1)$ Given the confidence level $\\delta \\ge 8e^{-3n/16}$ , set $\\varepsilon =\\frac{16\\log (8/\\delta )}{3n}~.$ $(2)$ Let $\\alpha =Y_{\\varepsilon n}^*$ and $\\beta =Y_{(1-\\varepsilon ) n}^*$ (assuming, for simplicity, that $\\varepsilon n$ is an integer) and set $\\widehat{\\mu }_{2n} =\\frac{1}{n}\\sum _{i=1}^n \\phi _{\\alpha ,\\beta }(X_i)~.$ Theorem 6 Let $X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n$ be independent, identically distributed random variables with mean $\\mu $ and variance $\\sigma ^2$ .",
"Let $\\delta \\in (0,1)$ be such that $n> (16/3)\\log (8/\\delta )$ .",
"Then, with probability at least $1-\\delta $ , $|\\widehat{\\mu }_{2n}-\\mu | \\le 9 \\sigma \\sqrt{\\frac{\\log (8/\\delta )}{n}}~.$ Proof.",
"We start by showing that the truncation level is close to the appropriate quantiles of the distribution.",
"To this end, for $p\\in (0,1)$ , introduce the quantiles $Q_p = \\sup \\left\\lbrace M \\in \\mathbb {R}: \\mathbb {P}\\left\\lbrace X \\ge M\\right\\rbrace \\ge 1-p\\right\\rbrace ~.$ For ease of exposition, assume that $X$ has a nonatomic distribution.",
"(This assumption is not necessary, but simplifies notation.)",
"In that case $\\mathbb {P}\\lbrace X > Q_p\\rbrace = \\mathbb {P}\\lbrace X \\ge Q_p\\rbrace = 1-p$ .",
"By a straightforward application of Bernstein's inequality, with probability at least $1-2\\exp (-(3/16)\\varepsilon n)$ , we have both $\\left|\\lbrace i\\in [n]: Y_i \\ge Q_{1-2\\varepsilon }\\rbrace \\right| \\ge \\varepsilon n$ and $\\left|\\lbrace i\\in [n]: Y_i \\le Q_{1-\\varepsilon /2}\\rbrace \\right| \\ge (1-\\varepsilon ) n~.$ This implies that, with probability at least $1-2\\exp (-(3/16)\\varepsilon n)$ , $Q_{1-2\\varepsilon } \\le Y_{(1-\\varepsilon ) n}^* \\le Q_{1-\\varepsilon /2}~.$ By the same argument, with probability at least $1-2\\exp (-(3/16)\\varepsilon n)$ , $ Q_{\\varepsilon /2} \\le Y_{\\varepsilon n}^* \\le Q_{2\\varepsilon }~,$ From here, we simply need to show that $|\\mathbb {E}\\phi _{\\alpha ,\\beta }(X)- \\mu |$ is small and that $(1/n)\\sum _{i=1}^n \\phi _{\\alpha ,\\beta }(X_i)$ concentrates around its mean.",
"For the first step, consider the event $E$ that both (REF ) and (REF ) hold.",
"This event has probability at least $1-4\\exp (-(3/16)\\varepsilon n) =1-\\delta /2$ .",
"On the event $E$ , ${\\left| \\mathbb {E}\\left[\\phi _{\\alpha ,\\beta }(X) |Y_1,\\ldots ,Y_n\\right]- \\mu \\right|} \\\\& \\le & \\left|\\mathbb {E}\\left[(X-\\alpha )\\mathbb {1}_{X\\le \\alpha }|Y_1,\\ldots ,Y_n\\right] \\right|+\\left|\\mathbb {E}\\left[(X-\\beta )\\mathbb {1}_{X\\ge \\beta }|Y_1,\\ldots ,Y_n\\right] \\right|\\\\& \\le & |\\mathbb {E}(X-Q_{2\\varepsilon })\\mathbb {1}_{X\\le Q_{2\\varepsilon }}|+|\\mathbb {E}(X-Q_{1-2\\varepsilon })\\mathbb {1}_{X\\ge Q_{1-2\\varepsilon }}|~.$ To bound these two terms, forst notice that, by Chebyshev's inequality, $2\\varepsilon = \\mathbb {P}\\left\\lbrace X \\ge Q_{1-2\\varepsilon }\\right\\rbrace \\le \\frac{\\sigma _X^2}{(Q_{1-2\\varepsilon }-\\mu )^2}~,$ and in particular, $Q_{1-2\\varepsilon } \\le \\mu + \\frac{\\sigma }{\\sqrt{2\\varepsilon }}~.$ Hence, by the Cauchy-Schwarz inequality, $|\\mathbb {E}(X-Q_{1-2\\varepsilon })\\mathbb {1}_{X\\ge Q_{1-2\\varepsilon }}|& = &|\\mathbb {E}(X-\\mu )-(Q_{1-2\\varepsilon }-\\mu )) \\mathbb {1}_{X \\ge Q_{1-2\\varepsilon }}| \\\\& \\le &\\mathbb {E}|(X-\\mu )| \\mathbb {1}_{X \\ge Q_{1-2\\varepsilon }} + (Q_{1-2\\varepsilon }-\\mu )\\mathbb {P}\\lbrace X \\ge Q_{1-2\\varepsilon }\\rbrace \\\\& \\le &\\sigma \\sqrt{\\mathbb {P}\\left\\lbrace X \\ge Q_{1-2\\varepsilon }\\right\\rbrace } + 2\\varepsilon (Q_{1-2\\varepsilon }-\\mu ) \\\\& \\le & \\sigma \\sqrt{8\\varepsilon }~.$ A symmetric argument shows $|\\mathbb {E}(X-Q_{2\\varepsilon })\\mathbb {1}_{X\\le Q_{2\\varepsilon }}|\\le \\sigma \\sqrt{8\\varepsilon }$ , and therefore, on the event $E$ , we have $\\left| \\mathbb {E}\\left[\\phi _{\\alpha ,\\beta }(X) |Y_1,\\ldots ,Y_n\\right]- \\mu \\right| \\le \\sigma \\sqrt{32\\varepsilon } \\le 6\\sigma \\sqrt{\\frac{\\log (8/\\delta )}{n}}$ by our choice of $\\epsilon $ .",
"Next, let $Z=\\frac{1}{n} \\sum _{i=1}^n \\phi _{\\alpha ,\\beta }(X_i)-\\mathbb {E}\\left[ \\phi _{\\alpha ,\\beta }(X) |Y_1,\\ldots ,Y_n\\right]$ and observe that $Z= \\frac{1}{n} \\sum _{i=1}^n \\phi _{\\alpha -\\mu ,\\beta -\\mu }(X_i-\\mu )-\\mathbb {E}\\left[ \\phi _{\\alpha -\\mu ,\\beta -\\mu }(X-\\mu ) |Y_1,\\ldots ,Y_n\\right]~.$ Hence, on the event $E$ (that only depends on $Y_1,\\ldots ,Y_n$ ), $Z$ is an average of centered random variables that is bounded point-wise by $M=\\max \\lbrace |Q_{\\varepsilon /2}-\\mu |,|Q_{1-\\varepsilon /2}-\\mu |\\rbrace \\le \\sigma \\sqrt{2/\\varepsilon }$ and whose variance is at most $\\sigma ^2$ .",
"Therefore, by Bernstein's inequality, with probability at least $1-\\delta /2$ , $Z \\le \\sigma \\sqrt{\\frac{2\\log (2/\\delta )}{n}}+\\frac{\\log (2/\\delta )\\sigma \\sqrt{2/\\varepsilon }}{n}\\le 3\\sigma \\sqrt{\\frac{\\log (2/\\delta )}{n}}~.$ Putting the pieces together, we obtain the announced bound.",
"Besides its conceptual simplicity, an important advantage of the trimmed mean compared to other estimators with sub-Gaussian performance is that it is robust to adversarial contamination of the data.",
"This statement is formalized and proved in [55] where a multivariate extension is also introduced and analyzed."
],
[
"Multiple-$\\delta $ estimators",
"We have constructed various estimators–such as median-of-means and Catoni's estimator–that are sub-Gaussian under the only assumption that the underlying distribution has a finite second moment.",
"However, both estimators depend on the knowledge of the desired confidence parameter $\\delta $ .",
"We show next that is not a coincidence because without further information on the distribution, it is impossible to construct a single estimator that is sub-Gaussian for a nontrivial range of values of the confidence parameter $\\delta $ .",
"Next we reproduce a simplified version of an argument of Devroye, Lerasle, Lugosi, and Oliveira [22] who proved results of this kind.",
"The theorem below shows that it is impossible to construct an estimator that is $L$ -sub-Gaussian for some specified values of $\\delta _1$ and $\\delta _2$ , at the same time.",
"The particular values of $\\delta _1$ and $\\delta _2$ are of no special importance.",
"We present this result to show the basic ideas in a simple form.",
"For more general versions we refer to [22].",
"Theorem 7 For every $L\\ge 50$ and for every sample size $n$ , no estimator can be simultaneously $L$ -sub-Gaussian for both $\\delta _1=1/(2e\\sqrt{L^3+1})$ and $\\delta _2=2e^{-L^4/4}$ for all distributions with finite second moment.",
"Proof.",
"We show that not only it is impossible to construct a single $L$ -sub-Gaussian estimator for both $\\delta _1=1/(2e\\sqrt{L^3+1})$ and $\\delta _2=e^{-L^4/4}$ for all distributions with finite second moment but it is also the case for the restricted class of Poisson distributions.",
"Assume, on the contrary, that there exists an estimator $\\widehat{\\mu }_n$ that is $L$ -sub-Gaussian for both $\\delta _1$ and $\\delta _2$ for all Poisson distributions.",
"Let $X_1,\\ldots ,X_n$ be independent Poisson random variables with parameter $1/n$ and let $Y_1,\\ldots ,Y_n$ be independent Poisson random variables with parameter $c/n$ , where we set $c=L^3+1$ .",
"We assume, for the sake of simplicity, that $c$ is an integer.",
"By the sub-Gaussian property of $\\widehat{\\mu }_n$ , $\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(Y_1,\\ldots ,Y_n) < \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} \\right\\rbrace \\le \\delta _1~.$ Now note that the left-hand side of the inequality may be lower bounded as folows: ${\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(Y_1,\\ldots ,Y_n) < \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} \\right\\rbrace } \\\\& \\ge &\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(Y_1,\\ldots ,Y_n) < \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}}, \\sum _{i=1}^n Y_i = c \\right\\rbrace \\\\& \\ge &\\frac{1}{e\\sqrt{c}}\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(Y_1,\\ldots ,Y_n) < \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} \\Big {|} \\sum _{i=1}^n Y_i = c \\right\\rbrace \\\\& & \\text{(from the fact that $ \\sum _{i=1}^n Y_i$ is Poisson with parameter $c$ and Stirling's formula)}$ Next we use the fact that the conditional joint distribution of $n$ independent Poisson($\\lambda $ ) random variables, conditioned on the event that their sum equals $c$ , only depends on $c$ but not $\\lambda $ .",
"In particular, ${\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(Y_1,\\ldots ,Y_n) < \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} \\Big {|} \\sum _{i=1}^n Y_i = c \\right\\rbrace }\\\\& = & \\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) < \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} \\Big {|} \\sum _{i=1}^n X_i = c \\right\\rbrace ~.$ Thus, together with (REF ), and the choice $\\delta _1=1/(2e\\sqrt{c})$ .",
"we have that $\\frac{1}{2} &=& 1- e\\sqrt{c} \\delta _1 \\\\& \\le &\\frac{ \\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) \\ge \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} , \\sum _{i=1}^n X_i = c \\right\\rbrace }{\\mathbb {P}\\left\\lbrace \\sum _{i=1}^n X_i = c \\right\\rbrace }\\\\& \\le & e c!",
"\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) \\ge \\frac{c}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} \\right\\rbrace \\\\& \\le & e c!",
"\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) \\ge \\frac{1}{n} + \\frac{c-1}{n} -\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}}\\right\\rbrace \\\\& \\le & e c!",
"\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) \\ge \\frac{1}{n} + \\frac{c-1}{2n} \\right\\rbrace ~,$ where we used the fact that $\\frac{L}{n}\\sqrt{c\\log \\frac{1}{\\delta _1}} \\le \\frac{c-1}{2n}~,$ that follows from our choice of $\\delta _1$ whenever $L\\ge 10$ .",
"Now since $\\widehat{\\mu }_n$ is $L$ -sub-Gaussian for $\\delta _2=2e^{-L^4/4}$ , we have that $\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) \\ge \\frac{1}{n} +\\frac{c-1}{2n} \\right\\rbrace =\\mathbb {P}\\left\\lbrace \\widehat{\\mu }_n(X_1,\\ldots ,X_n) \\ge \\frac{1}{n} +\\frac{L}{n}\\sqrt{\\log (2/\\delta _2)} \\right\\rbrace \\le \\delta _2~.$ Summarizing, we have $1/2\\le e c!",
"\\delta _2 = 2e c!e^{-L^4/4}$ .",
"However, the expression on the right-hand side is less than $1/2$ for $L\\ge 50$ , leading to a contradiction.",
"We refer to [22] for a more complete version of Theorem REF and for an extensive discussion on constructing estimators that do not require knowledge of the desired confidence parameter (i.e., estimators that are sub-Gaussian for a wide range of values of $\\delta $ ).",
"In [22] it is shown how Lepski's method may be used to construct such estimators if some additional information, other than finiteness of the variance, is available on the underlying distribution.",
"In particular, if nontrivial upper and lower bounds on the variance are available, then such “$\\delta $ -independent” estimators exist for a wide range of values of $\\delta $ .",
"Existence of higher moments or certain weak symmetry assumptions may also be used."
],
[
"Estimating the mean of a random vector",
"In what follows, we discuss extensions of the mean estimation problem to the multivariate setting.",
"To set up the problem, let $X$ be a random vector taking values in $\\mathbb {R}^d$ .",
"Assume that the mean vector $\\mu = \\mathbb {E}X$ and covariance matrix $\\Sigma = \\mathbb {E}(X-\\mu ) (X-\\mu )^T$ exist.",
"Given $n$ independent, identically distributed samples $X_1,\\ldots ,X_n$ drawn from the distribution of $X$ , one wishes to estimate the mean vector.",
"Just like in the univariate case, a natural choice is the sample mean $\\overline{\\mu }_n=(1/n)\\sum _{i=1}^n X_i$ and it has a near-optimal behavior whenever the distribution is sufficiently light tailed.",
"However, as is the case in the univariate case, whenever heavy tails are a concern, the sample mean is to be avoided as it may have a sub-optimal performance."
],
[
"Sub-Gaussian performance",
"For the univariate problem, we constructed mean estimators with a sub-Gaussian performance.",
"In order to properly set up our goal for the $d$ -dimensional case, first we need to understand what “sub-Gaussian performance” means.",
"Just like in the univariate case, one would like to construct estimators that are “close” to the true mean $\\mu $ , with “high probability”.",
"The first question is how one measures distance in $\\mathbb {R}^d$ .",
"Arguably, the most natural distance measure is the Euclidean norm.",
"In this section we focus on this choice and we denote by $\\Vert \\cdot \\Vert $ the Euclidean norm.",
"We explore mean estimation of a random vector with respect to an arbitrary norm in Section REF .",
"If $X$ has a multivariate normal distribution with mean vector $\\mu $ and covariance matrix $\\Sigma $ , then the sample mean $\\overline{\\mu }_n$ is also multivariate normal with mean $\\mu $ and covariance matrix $(1/n)\\Sigma $ .",
"Thus, for all $t>0$ , $\\mathbb {P}\\left\\lbrace \\Vert \\overline{\\mu }_n-\\mu \\Vert \\ge \\mathbb {E}\\Vert \\overline{\\mu }_n-\\mu \\Vert + t \\right\\rbrace = \\mathbb {P}\\left\\lbrace \\left\\Vert \\overline{X} \\right\\Vert - \\mathbb {E}\\left\\Vert \\overline{X} \\right\\Vert \\ge t\\sqrt{n} \\right\\rbrace ~,$ where $\\overline{X}$ is a Gaussian vector in $\\mathbb {R}^d$ with zero mean and covariance matrix $\\Sigma $ .",
"A key property of Gaussian vectors is that $\\overline{X}$ has the same distribution as $\\Sigma ^{1/2}Y$ where $Y$ is a standard normal vector (i.e., with zero-mean and identity covariance matrix) and $\\Sigma ^{1/2}$ is the positive semidefinite square root of $\\Sigma $ .",
"Also, observe that for all $y,y^{\\prime }\\in \\mathbb {R}^d$ , $\\left| \\left\\Vert \\Sigma ^{1/2}y\\right\\Vert - \\left\\Vert \\Sigma ^{1/2}y^{\\prime }\\right\\Vert \\right| \\le \\left\\Vert \\Sigma ^{1/2}(y-y^{\\prime })\\right\\Vert \\le \\left\\Vert \\Sigma ^{1/2}\\right\\Vert _{2 \\rightarrow 2} \\cdot \\Vert y-y^{\\prime }\\Vert ~,$ where $\\left\\Vert \\Sigma ^{1/2}\\right\\Vert _{2 \\rightarrow 2}$ is the spectral norm of $\\Sigma ^{1/2}$ .",
"Thus, $\\Sigma ^{1/2}y$ is a Lipschitz function of $y\\in \\mathbb {R}^d$ with Lipschitz constant $\\Vert \\Sigma ^{1/2}\\Vert _{2 \\rightarrow 2}= \\sqrt{\\lambda _{\\text{max}}}$ , with $\\lambda _{\\text{max}}=\\lambda _{\\text{max}}(\\Sigma )$ denoting the largest eigenvalue of the covariance matrix $\\Sigma $ .",
"Now it follows from the Gaussian concentration inequality of Tsirelson, Ibragimov, and Sudakov [75] (see also Ledoux [51] and Boucheron, Lugosi, and Massart [10] for more information) that $\\mathbb {P}\\left\\lbrace \\left\\Vert \\overline{X} \\right\\Vert - \\mathbb {E}\\left\\Vert \\overline{X} \\right\\Vert \\ge t\\sqrt{n} \\right\\rbrace \\le e^{-nt^2/(2\\lambda _{\\text{max}})}~.$ Noting that $\\mathbb {E}\\left\\Vert \\overline{X} \\right\\Vert \\le \\sqrt{ \\mathbb {E}\\left\\Vert \\overline{X} \\right\\Vert ^2} = \\sqrt{\\mathrm {Tr}(\\Sigma )}~,$ the trace of the covariance matrix $\\Sigma $ , we have that, for $\\delta \\in (0,1)$ , with probability at least $1-\\delta $ , $\\Vert \\overline{\\mu }_n-\\mu \\Vert \\le \\sqrt{\\frac{\\mathrm {Tr}(\\Sigma )}{n}}+ \\sqrt{\\frac{2\\lambda _{\\text{max}}\\log (1/\\delta )}{n}}~.$ Thus, in the multivariate case, we will say that a mean estimator is sub-Gaussian if, with probability at least $1-\\delta $ , it satisfies an inequality of the form (REF ) (with possibly different constant factors).",
"Note that for any distribution with mean $\\mu $ and covariance matrix $\\Sigma $ , the mean-squared error of the empirical mean equals $\\mathbb {E}\\Vert \\overline{\\mu }_n-\\mu \\Vert ^2 = \\frac{\\mathrm {Tr}(\\Sigma )}{n}~.$ In particular, $\\mathbb {E}\\Vert \\overline{\\mu }_n-\\mu \\Vert \\le \\sqrt{\\frac{\\mathrm {Tr}(\\Sigma )}{n}}$ .",
"An important feature of the sub-Gaussian property (REF ) is that the random fluctuations are controlled by the spectral norm $\\lambda _{\\text{max}}$ of the covariance matrix, which is possibly much smaller than $\\mathrm {Tr}(\\Sigma )$ , the sum of all eigenvalues of $\\Sigma $ ."
],
[
"Multivariate median-of-means",
"For non-Gaussian and possibly heavy-tailed distributions, one cannot expect a sub-Gaussian behavior of the sample mean similar to (REF ).",
"As an alternative, one may try to extend the median-of-means estimator to the multivariate case.",
"An obvious idea is to divide the data into disjoint blocks, calculate the empirical mean within each block, and compute a multivariate median of the obtained empirical means.",
"However, there is no standard notion of a median for multivariate data, and it is not entirely clear what definition of a multivariate median works best for median-of-means mean estimators.",
"Among the numerous possibilities, we mention the coordinate-wise median, the geometric (or spatial) median, the Tukey (or halfspace) median, the Oja median, and the Liu median, see Small [73] for a survey and relevant references.",
"Regardless of what notion of a multivariate median we decide to adopt, we start by partitioning $[n]=\\lbrace 1,\\dots ,n\\rbrace $ into $k$ blocks $B_1,\\ldots ,B_k$ , each of size $|B_i|\\ge \\lfloor n/k\\rfloor \\ge 2$ .",
"Here $k$ is a parameter of the estimator to be chosen later.",
"For simplicity, we assume that $km=n$ for some positive integer $m$ .",
"Just like before, we compute the sample mean of the random vectors within each block: for $j=1,\\ldots ,k$ , let $Z_j=\\frac{1}{m}\\sum _{i\\in B_j}X_i~.$ Perhaps the most natural first try is to define $\\widehat{\\mu }_n$ as the vector of coordinate-wise medians of the $Z_j$ (i.e., the $\\ell $ -th component of the vector $\\widehat{\\mu }_n$ is the median of the $\\ell $ -th components of $Z_1,\\ldots ,Z_k$ , for $\\ell \\in [d]$ ).",
"Then Theorem REF and the union bound imply that, for any $\\delta \\in (0,1)$ , taking $k= \\left\\lceil 8 \\log (1/\\delta ) \\right\\rceil $ , with probability at least $1-\\delta $ , $\\left\\Vert \\widehat{\\mu }_n-\\mu \\right\\Vert \\le \\sqrt{\\frac{32\\mathrm {Tr}(\\Sigma )\\log (d/\\delta )}{n}}~,$ where we used the fact that $\\mathrm {Tr}(\\Sigma )=\\mathbb {E}\\Vert X-\\mathbb {E}X\\Vert ^2$ is the sum of the variances of the $d$ components of $X$ .",
"Clearly, this bound is far from the sub-Gaussian inequality (REF ) for several reasons.",
"First, it is not “dimension-free” as $d$ appears explicitly in the bound.",
"Perhaps more importantly, $\\log (1/\\delta )$ is multiplied by $\\mathrm {Tr}(\\Sigma )$ instead of $\\lambda _{\\text{max}}(\\Sigma )$ and that may be a major difference in high-dimensional problems, especially when one is interested in small failure probabilities.",
"An instructive example is when all eigenvalues of $\\Sigma $ are identical and equal to $\\lambda _{\\text{max}}$ .",
"If the dimension $d$ is large, (REF ) is of the order of $\\sqrt{(\\lambda _{\\text{max}}/n)(d+\\log (1/\\delta ))}$ while the bound above only gives the order $\\sqrt{(\\lambda _{\\text{max}}/n)(d\\log (d/\\delta ))}$ .",
"One may quite easily improve on this by using a different (non-standard) notion of median in the definition of the estimate: choose $\\widehat{\\mu }_n$ to be the point in $\\mathbb {R}^d$ with the property that the Euclidean ball centered at $\\widehat{\\mu }_n$ that contains more than $k/2$ of the points $Z_j$ has minimal radius.",
"Since $\\mathbb {E}\\Vert Z_j-\\mu \\Vert ^2 = \\mathrm {Tr}(\\Sigma )/m$ , by Chebyshev's inequality, $\\Vert Z_j-\\mu \\Vert \\le r \\stackrel{\\mathrm {def.",
"}}{=}2\\sqrt{\\mathrm {Tr}(\\Sigma )/m}$ with probability at least $3/4$ .",
"Thus, by choosing $k= \\left\\lceil 8 \\log (1/\\delta ) \\right\\rceil $ , we have that, with probability at least $1-\\delta $ , more than half of the points $Z_j$ satisfy $\\Vert Z_j-\\mu \\Vert \\le r~.$ Denote this event by $E$ .",
"(Thus, $\\mathbb {P}\\lbrace E\\rbrace \\ge 1-\\delta $ .)",
"On the event $E$ , this radius is at most $r$ .",
"Hence, at least one of the $Z_j$ is within distance $r$ to both $\\mu $ and $\\widehat{\\mu }_n$ .",
"Thus, by the triangle inequality, $\\Vert \\widehat{\\mu }_n-\\mu \\Vert \\le 2r$ .",
"We have obtained the following proposition.",
"Proposition 1 Let $X_1,\\ldots ,X_n$ be i.i.d.",
"random vectors in $\\mathbb {R}^d$ with mean $\\mu $ and covariance matrix $\\Sigma $ .",
"Let $\\delta \\in (0,1)$ and let $\\widehat{\\mu }_n$ be the estimator defined above with $k= \\left\\lceil 8 \\log (1/\\delta )\\right\\rceil $ .",
"Then, with probability at least $1-\\delta $ , $\\left\\Vert \\widehat{\\mu }_n-\\mu \\right\\Vert \\le 4\\sqrt{\\frac{\\mathrm {Tr}(\\Sigma ) (8 \\log (1/\\delta )+1)}{n}}~.$ The bound of Proposition REF is quite remarkable as it is “dimension-free” and no assumption other than the existence of the covariance matrix is made.",
"However, it still does not achieve a sub-Gaussian performance bound that resembles (REF ).",
"Moreover, the notion of median used here (i.e., the center of the smallest ball that contains at least half of the points) is problematic from a computational point of view, since computing such a median is a nontrivial problem.",
"An efficiently computable version of a multivariate median is the so-called geometric median, defined as $\\widehat{\\mu }_n = \\mathop {\\mathrm {argmin}}_{m\\in \\mathbb {R}^d} \\sum _{j=1}^k \\Vert Z_i-m\\Vert ~.$ This estimator was proposed by Minsker [65] and independently by Hsu and Sabato [36] (see also Lerasle and Oliveira [53]).",
"In particular, Minsker [65] proved that this version of the multivariate median-of-means estimator achieves a similar performance bound as Proposition REF .",
"Moreover, computing the geometric median—and therefore the multivariate median-of-means estimator—involves solving a convex optimization problem.",
"Thus, the geometric median may be approximated efficiently, see Cohen, Lee, Miller, Pachocki, and Sidford [20] for the most recent result and for the rich history of the problem.",
"We refer to Aloupis [2] for a survey of computational aspects of various other notions of multivariate medians.",
"For a quite different mean estimator based on the median-of-means idea with “almost” sub-Gaussian guarantees but with a serious computational burden, see Joly, Lugosi, and Oliveira [41].",
"In order to achieve a truly sub-Gaussian performance, we need to define a new estimator.",
"In what follows we define two that achieve the desired performance: the first, introduced in [56] is based on the idea of median-of-means tournaments and the second, from [57], is defined using the intersection of random slabs.",
"The former leads to an error estimate with respect to the Euclidean norm (see Section REF ), and the latter, described in Section REF holds with respect to an arbitrary norm.",
"However, before presenting these estimates, we recall a very different estimator introduced by Catoni and Giulini [16]."
],
[
"Thresholding the norm: the Catoni-Giulini estimator",
"In this section we briefly discuss a remarkably simple estimator, suggested and analyzed by Catoni and Giulini [16].",
"The Catoni-Giulini estimator is $\\widehat{\\mu }_n = \\frac{1}{n}\\sum _{i=1}^n X_i\\min \\left(1,\\frac{1}{\\alpha \\Vert X_i\\Vert } \\right)~,$ where $\\alpha >0$ is a (small) parameter.",
"Thus, $\\widehat{\\mu }_n$ is simply an empirical average of the $X_i$ , with the data points with large norm shrunk towards zero.",
"This estimate is trivial to compute, as opposed to the more complex estimators that we discuss in Sections REF and REF .",
"On the other hand, shrinking to zero is somewhat arbitrary and unnatural.",
"In fact, the estimator is not invariant under translations of the data in the sense that $\\widehat{\\mu }_n(X_1+a,\\ldots ,X_n+a)$ is not necessarily equal to $\\widehat{\\mu }_n(X_1,\\ldots ,X_n)+a$ when $a\\ne 0$ .",
"Catoni and Giulini prove that if one chooses the parameter as $\\alpha =\\sqrt{\\frac{c\\log (1/\\delta )}{v n}}~,$ where $v\\ge \\lambda _{\\text{max}}$ and $c>0$ is a numerical constant, then the estimator (REF ) satisfies, with probability at least $1-\\delta $ , $\\left\\Vert \\widehat{\\mu }_n - \\mu \\right\\Vert \\le C\\sqrt{\\frac{\\left(\\mathrm {Tr}(\\Sigma )+ v+\\Vert \\mu \\Vert ^2\\right)\\log (1/\\delta )}{n}}~,$ where $C$ is a constant depending on $c$ only.",
"This bound is similar to but weaker than that of Proposition REF , principally due to two facts.",
"First, the estimator requires prior knowledge of (a good upper bound of) $\\lambda _{\\text{max}}$ whereas the geometric median-of-means estimator assumes no such prior information.",
"Second, $\\Vert \\mu \\Vert ^2$ appears in the upper bound and a priori this can be arbitrarily large compared to $\\mathrm {Tr}(\\Sigma )$ .",
"The presence of this term is due to the lack of translation invariance of the estimator.",
"This second issue may be fixed by defining a two-stage estimator: first one may use a translation-invariant estimator like geometric-median-of-means defined in the previous section to get a rough estimate of the mean.",
"Then, using a new batch of independent data, one may center the data at the estimated mean and then use the Catoni-Giulini estimator for the centered data.",
"This new estimator is translation invariant, and the term $\\Vert \\mu \\Vert ^2$ may be replaced by the squared error of the estimator of the first step, that is, by $\\mathrm {Tr}(\\Sigma )\\log (1/\\delta )/n$ .",
"But even with this modification, the bound is not sub-Gaussian in the sense of (REF ).",
"Remarkably, however, the performance of the Catoni-Giulini estimator comes close to being sub-Gaussian in the desired sense under just a small extra assumption.",
"In particular, if $\\mathbb {E}\\Vert X\\Vert ^\\beta <\\infty $ for some $\\beta >2$ , then, with the same choice of $\\alpha $ as above, one has $\\left\\Vert \\widehat{\\mu }_n - \\mu \\right\\Vert \\le C\\left(\\sqrt{\\frac{v\\log (1/\\delta )}{n}} +\\sqrt{\\frac{\\left(\\mathrm {Tr}(\\Sigma )+v\\right)}{n}}+ \\frac{\\kappa _\\beta }{n^{(\\beta -1)/2}}\\right)~,$ where $\\kappa _\\beta $ is a constant that depends on $\\beta $ and the $\\beta $ -th raw moment of $\\Vert X\\Vert $ .",
"Thus, if the prior parameter $v$ is a good estimate of $\\lambda _{\\text{max}}$ in the sense that it is bounded by a constant multiple of it, then the first two terms of the bound are of the desired sub-Gaussian form.",
"The third term is of smaller order though again, it can be arbitrarily large if the mean is far from the origin, which may be remedied by making the estimator more complex.",
"We refer to Catoni and Giulini [15] for other estimates of a similar spirit and more discussion.",
"The proof techniques of [15], [16] rely on so-called PAC-Bayesian inequalities whose details go beyond the scope of this survey."
],
[
"Median-of-means tournaments",
"Here we introduce a mean estimator with a sub-Gaussian performance for all distributions whose covariance matrix exists, proposed by Lugosi and Mendelson [56].",
"The estimator presented below is the first and simplest instance of what we call median-of-means tournaments.",
"Recall that we are given an i.i.d.",
"sample $X_1,\\ldots ,X_n$ of random vectors in $\\mathbb {R}^d$ .",
"As in the case of the median-of-means estimator, we start by partitioning the set $\\lbrace 1,\\dots ,n\\rbrace $ into $k$ blocks $B_1,\\ldots ,B_k$ , each of size $|B_j|\\ge m \\stackrel{\\mathrm {def.",
"}}{=}\\lfloor n/k\\rfloor $ , where $k$ is a parameter of the estimator whose value depends on the desired confidence level, as specified below.",
"In order to simplify the presentation, we assume that $n$ is divisible by $k$ and therefore $|B_j|=m$ for all $j=1,\\ldots ,k$ .",
"Define the sample mean within each block by $Z_j=\\frac{1}{m}\\sum _{i\\in B_j}X_i~.$ For each $a\\in \\mathbb {R}^d$ , let $T_a=\\left\\lbrace x\\in \\mathbb {R}^d: \\exists J\\subset [k]: |J| \\ge k/2 \\ \\text{such that for all} \\ j\\in J, \\ \\Vert Z_j-x\\Vert \\le \\Vert Z_j-a\\Vert \\right\\rbrace $ and define the mean estimator by $\\widehat{\\mu }_n \\in \\mathop {\\mathrm {argmin}}_{a\\in \\mathbb {R}^d} {\\rm radius}(T_a)~,$ where ${\\rm radius}(T_a)=\\sup _{x\\in T_a} \\Vert x-a\\Vert $ .",
"Thus, $\\widehat{\\mu }_n$ is chosen to minimize, over all $a\\in \\mathbb {R}^d$ , the radius of the set $T_a$ defined as the set of points $x\\in \\mathbb {R}^d$ for which $\\Vert Z_j-x\\Vert \\le \\Vert Z_j-a\\Vert $ for the majority of the blocks.",
"If there are several minimizers, one may pick any one of them.",
"The set $T_a$ may be seen as the set of points in $\\mathbb {R}^d$ that are at least as close to the point cloud $\\lbrace Z_1,\\ldots ,Z_k\\rbrace $ as the point $a$ .",
"The estimator $\\widehat{\\mu }_n$ is obtained by minimizing the radius of $T_a$ .",
"Note that the minimum is always achieved.",
"This follows from the fact that ${\\rm radius}(T_a)$ is a continuous function of $a$ (since, for each $a$ , $T_a$ is the intersection of a finite union of closed balls, and the centers and radii of the closed balls are continuous in $a$ ).",
"One may interpret $\\mathop {\\mathrm {argmin}}_{a\\in \\mathbb {R}^d} {\\rm radius}(T_a)$ as yet another multivariate notion of the median of $Z_1,\\ldots ,Z_k$ .",
"Indeed, when $d=1$ , it is a particular choice of the median and the estimator coincides with the median-of-means estimator.",
"The following performance bound shows that the estimator has the desired sub-Gaussian performance.",
"Theorem 8 (Lugosi and Mendelson [56].)",
"Let $\\delta \\in (0,1)$ and consider the mean estimator $\\widehat{\\mu }_n$ with parameter $k= \\lceil 200\\log (2/\\delta )\\rceil $ .",
"If $X_1,\\ldots ,X_n$ are i.i.d.",
"random vectors in $\\mathbb {R}^d$ with mean $\\mu \\in \\mathbb {R}^d$ and covariance matrix $\\Sigma $ , then for all $n$ , with probability at least $1-\\delta $ , $\\left\\Vert \\widehat{\\mu }_n-\\mu \\right\\Vert \\le \\max \\left(960 \\sqrt{\\frac{\\mathrm {Tr}(\\Sigma )}{n}},240 \\sqrt{\\frac{\\lambda _{\\text{max}}\\log (2/\\delta )}{n}} \\right)~.$ Just like the performance bound of Proposition REF , Theorem REF is “infinite-dimensional” in the sense that the bound does not depend on the dimension $d$ explicitly.",
"Indeed, the same estimator may be defined for Hilbert-space valued random vectors and Theorem REF remains valid as long as $\\mathrm {Tr}(\\Sigma )=\\mathbb {E}\\Vert X-\\mu \\Vert ^2$ is finite.",
"Theorem REF is an outcome of the following observation.",
"Theorem 9 Using the same notation as above and setting $r=\\max \\left(960 \\sqrt{\\frac{\\mathrm {Tr}(\\Sigma )}{n}},240 \\sqrt{\\frac{\\lambda _{\\text{max}}\\log (2/\\delta )}{n}} \\right)~,$ with probability at least $1-\\delta $ , for any $a \\in \\mathbb {R}^d$ such that $\\Vert a-\\mu \\Vert \\ge r$ , one has $\\Vert Z_j-a\\Vert > \\Vert Z_j-\\mu \\Vert $ for more than $k/2$ indices $j$ .",
"In other words, $\\Vert a-\\mu \\Vert \\ge r$ implies that $a\\notin T_{\\mu }$ .",
"Theorem REF implies that for a `typical' collection $X_1,\\ldots ,X_n$ , $\\mu $ is closer to a majority of the $Z_j$ 's when compared to any $a \\in \\mathbb {R}^d$ that is sufficiently far from $\\mu $ .",
"Obviously, for an arbitrary collection $x_1,\\ldots ,x_n \\subset \\mathbb {R}^d$ such a point need not even exist, and it is surprising that for a typical i.i.d.",
"configuration, this property is satisfied by $\\mu $ .",
"The fact that Theorem REF implies Theorem REF is straightforward.",
"Indeed, the definition of $\\widehat{\\mu }_n$ and Theorem REF imply that, with probability at least $1-\\delta $ , ${\\rm radius}(T_{\\widehat{\\mu }_n}) \\le {\\rm radius}(T_\\mu ) \\le r$ .",
"Since either $\\mu \\in T_{\\widehat{\\mu }_n}$ or $\\widehat{\\mu }\\in T_\\mu $ , we must have $\\Vert \\widehat{\\mu }_n-\\mu \\Vert \\le r$ , as required.",
"The constants appearing in Theorem REF are certainly not optimal.",
"They were obtained with the goal of making the proof transparent.",
"The proof of Theorem REF is based on the following idea.",
"The mean $\\mu $ is the minimizer of the function $f(x)= \\mathbb {E}\\Vert X-x\\Vert ^2$ .",
"A possible approach is to use the available data to guess, for any pair $a,b\\in \\mathbb {R}^d$ , whether $f(a)< f(b)$ .",
"A natural choice is to use a median of means estimator to decide which of the two is better.",
"The “tournament\" is simply a way of comparing every such pair, as described next.",
"As we explain in what follows, it suffices to ensure that the comparison is correct between $\\mu $ and any point that is not too close to $\\mu $ .. To define the tournament, recall that $[n]$ is partitioned into $k$ disjoint blocks $B_1,\\ldots ,B_k$ of size $m=n/k$ .",
"For $a,b \\in \\mathbb {R}^d$ , we say that $a$ defeats $b$ if $ \\frac{1}{m} \\sum _{i \\in B_j} \\left(\\Vert X_i-b\\Vert ^2 - \\Vert X_i-a\\Vert ^2\\right) > 0$ on more than $k/2$ blocks $B_j$ .",
"The main technical lemma is the following.",
"Lemma 1 Let $\\delta \\in (0,1)$ , $k= \\lceil 200\\log (2/\\delta )\\rceil $ , and define $r=\\max \\left(960 \\sqrt{\\frac{\\mathrm {Tr}(\\Sigma )}{n}},240 \\sqrt{\\frac{\\lambda _{\\text{max}}\\log (2/\\delta )}{n}} \\right)~.$ With probability at least $1-\\delta $ , $\\mu $ defeats all $b\\in \\mathbb {R}^d$ such that $\\Vert b-\\mu \\Vert \\ge r$ .",
"The outcome of Lemma REF stands to reason: if $\\Vert b - \\mu \\Vert $ is large enough, that will be reflected in `typical values' of $(\\Vert X_i-\\mu \\Vert )_{i=1}^n$ and $(\\Vert X_i-b\\Vert )_{i=1}^n$ .",
"Comparing the values via (REF ) ensures `stability', and the fact that $b$ is far from $\\mu $ is exhibited with high probability.",
"We stress that the probability estimate has to be uniform in $b$ .",
"Such uniform estimates are a recurring theme in what follows.",
"Proof.",
"Note that $\\Vert X_i-b\\Vert ^2 - \\Vert X_i-\\mu \\Vert ^2 = \\Vert X_i-\\mu +\\mu -b\\Vert ^2 - \\Vert X_i-\\mu \\Vert ^2 = -2\\left\\langle X_i-\\mu ,b-\\mu \\right\\rangle +\\Vert b-\\mu \\Vert ^2~,$ set $\\overline{X}=X-\\mu $ and put $v=b-\\mu $ .",
"Thus, for a fixed $b$ that satisfies $\\Vert b-\\mu \\Vert \\ge r$ , $\\mu $ defeats $b$ if $-\\frac{2}{m}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,v \\right\\rangle +\\Vert v\\Vert ^2>0$ on the majority of blocks $B_j$ .",
"Therefore, to prove our claim we need that, with probability at least $1-\\delta $ , for every $v \\in \\mathbb {R}^d$ with $\\Vert v\\Vert \\ge r$ , $ -\\frac{2}{m}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,v \\right\\rangle +\\Vert v\\Vert ^2>0$ for more than $k/2$ blocks $B_j$ .",
"Clearly, it suffices to show that (REF ) holds when $\\Vert v\\Vert =r$ .",
"Consider a fixed $v \\in \\mathbb {R}^d$ with $\\Vert v\\Vert =r$ .",
"By Chebyshev's inequality, with probability at least $9/10$ , $\\left|\\frac{1}{m}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,v \\right\\rangle \\right| \\le \\sqrt{10} \\sqrt{\\frac{\\mathbb {E}\\left\\langle \\overline{X},v \\right\\rangle ^2}{m}} \\le \\sqrt{10} \\Vert v\\Vert \\sqrt{\\frac{\\lambda _{\\text{max}}}{m}}~,$ where recall that $\\lambda _{\\text{max}}$ is the largest eigenvalue of the covariance matrix $\\Sigma $ of $X$ .",
"Thus, if $ r=\\Vert v\\Vert \\ge 4\\sqrt{10} \\sqrt{\\frac{\\lambda _{\\text{max}}}{m}}$ then with probability at least $9/10$ , $ -\\frac{2}{m}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,v \\right\\rangle \\ge \\frac{-r^2}{2}.$ Applying Hoeffding's inequality ([32]), we see that (REF ) holds for a single $v$ with probability at least $1-\\exp (-k/50)$ on at least $8/10$ of the blocks $B_j$ .",
"Now we need to extend the above from a fixed vector $v$ to all vectors with norm $r$ .",
"In order to show that (REF ) holds simultaneously for all $v\\in r\\cdot S^{d-1}$ on at least $7/10$ of the blocks $B_j$ , we first consider a maximal $\\epsilon $ -separated set $V_1 \\subset r\\cdot S^{d-1}$ with respect to the $L_2(X)$ norm.",
"In other words, $V_1$ is a subset of $r\\cdot S^{d-1}$ of maximal cardinality such that for all $v_1,v_2\\in V_1$ , $\\Vert v_1-v_2\\Vert _{L_2(X)}= \\left\\langle v_1-v_2,\\Sigma (v_1-v_2) \\right\\rangle ^{1/2} \\ge \\epsilon $ .",
"We may estimate this cardinality by the “dual Sudakov” inequality (see Ledoux and Talagrand [52] and also Vershynin[83] for a version with the specific constant used here): the cardinality of $V_1$ is bounded by $\\log (|V_1|/2) \\le \\frac{1}{32} \\left(\\frac{\\mathbb {E}\\left[\\left\\langle G,\\Sigma G \\right\\rangle ^{1/2}\\right]}{\\epsilon /r}\\right)^2~,$ where $G$ is a standard normal vector in $\\mathbb {R}^d$ .",
"Notice that for any $a \\in \\mathbb {R}^d$ , $\\mathbb {E}_X \\left\\langle a,X \\right\\rangle ^2 = \\left\\langle a,\\Sigma a \\right\\rangle $ , and therefore, $\\mathbb {E}\\left[\\left\\langle G,\\Sigma G \\right\\rangle ^{1/2}\\right]& = & \\mathbb {E}_G \\left[\\left(\\mathbb {E}_X\\left[\\left\\langle G,\\overline{X} \\right\\rangle ^2\\right] \\right)^{1/2}\\right]\\le \\left(\\mathbb {E}_X \\mathbb {E}_G \\left[\\left\\langle G,\\overline{X} \\right\\rangle ^2 \\right] \\right)^{1/2} \\\\& = & \\left(\\mathbb {E}\\left[ \\left\\Vert \\overline{X}\\right\\Vert ^2\\right]\\right)^{1/2} = \\sqrt{\\mathrm {Tr}(\\Sigma )}~.$ Hence, by setting $ \\epsilon = 2 r \\left(\\frac{1}{k}\\mathrm {Tr}(\\Sigma )\\right)^{1/2}~,$ we have $|V_1|\\le 2e^{k/100}$ and by the union bound, with probability at least $1-2e^{-k/100}\\ge 1-\\delta /2$ , (REF ) holds for all $v\\in V_1$ on at least $8/10$ of the blocks $B_j$ .",
"Next we check that property (REF ) holds simultaneously for all $x$ with $\\Vert x\\Vert =r$ on at least $7/10$ of the blocks $B_j$ .",
"For every $x \\in r \\cdot S^{d-1}$ , let $v_x$ be the nearest element to $x$ in $V_1$ with respect to the $L_2(X)$ norm.",
"It suffices to show that, with probability at least $1-\\exp (-k/200)\\ge 1-\\delta /2$ , $ \\sup _{x \\in r\\cdot S^{d-1}} \\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{\\lbrace |m^{-1}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle | \\ge r^2/4\\rbrace } \\le \\frac{1}{10}~.$ Indeed, on that event it follows that for every $x \\in r\\cdot S^{d-1}$ , on at least $7/10$ of the blocks $B_j$ , both $-\\frac{2}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,v_x \\right\\rangle \\ge \\frac{-r^2}{2} \\ \\ \\ {\\rm and} \\ \\ \\ 2\\left|\\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x \\right\\rangle -\\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,v_x \\right\\rangle \\right| < \\frac{r^2}{2}$ hold and hence, on those blocks, $-\\frac{2}{m} \\sum _{i \\in B_j}\\left\\langle \\overline{X}_i,x \\right\\rangle +r^2>0$ as required.",
"It remains to prove (REF ).",
"Observe that $\\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{\\lbrace |m^{-1}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle | \\ge r^2/4\\rbrace } \\le \\frac{4}{r^2} \\frac{1}{k}\\sum _{j=1}^k \\left|\\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle \\right|~.$ Since $\\Vert x-v_x\\Vert _{L_2(X)} = \\sqrt{\\mathbb {E}\\left\\langle \\overline{X},x-v_x \\right\\rangle ^2} \\le \\epsilon $ it follows that for every $j$ $\\mathbb {E}\\left|\\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle \\right|\\le \\sqrt{\\frac{\\mathbb {E}\\left[\\left\\langle \\overline{X},x-v_x \\right\\rangle ^2\\right]}{m}}\\le \\frac{\\epsilon }{\\sqrt{m}}~,$ and therefore, ${\\mathbb {E}\\sup _{x \\in r\\cdot S^{d-1}} \\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{\\lbrace |m^{-1}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle | \\ge r^2/4\\rbrace } } \\\\& \\le &\\frac{4}{r^2} \\mathbb {E}\\sup _{x \\in r\\cdot S^{d-1}} \\frac{1}{k}\\sum _{j=1}^k \\left(\\left|\\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle \\right| - \\mathbb {E}\\left|\\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle \\right|\\right) + \\frac{4\\epsilon }{r^2\\sqrt{m}} \\\\& \\stackrel{\\mathrm {def.",
"}}{=}& (A)+(B)~.$ To bound $(B)$ , note that, by (REF ), $\\frac{4\\epsilon }{r^2\\sqrt{m}} = 8 \\left(\\frac{\\mathrm {Tr}(\\Sigma )}{n}\\right)^{1/2} \\cdot \\frac{1}{r} \\le \\frac{1}{60}$ provided that $r \\ge 480 \\left(\\frac{\\mathrm {Tr}(\\Sigma )}{n}\\right)^{1/2}~.$ We may bound $(A)$ by standard techniques of empirical processes such as symmetrization, contraction for Rademacher averages and de-symmetrization.",
"Indeed, let $\\sigma _1,\\ldots ,\\sigma _n$ be independent Rademacher random variables (i.e., $\\mathbb {P}\\lbrace \\sigma _i=1\\rbrace =\\mathbb {P}\\lbrace \\sigma _i=-1\\rbrace =1/2$ ), independent of all of the $X_i$ .",
"Then $(A) & \\le &\\frac{8}{r^2} \\mathbb {E}\\sup _{x \\in r\\cdot S^{d-1}} \\frac{1}{k}\\sum _{j=1}^k \\sigma _j \\left|\\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle \\right|\\\\& & \\text{(by a standard symmetrization inequality, see, e.g., \\cite [Lemma 2.3.6]{vaWe96})} \\\\& \\le &\\frac{8}{r^2} \\mathbb {E}\\sup _{x \\in r\\cdot S^{d-1}} \\left| \\frac{1}{k}\\sum _{j=1}^k \\sigma _j \\frac{1}{m} \\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle \\right|\\\\& & \\text{(by a contraction lemma for Rademacher averages, see \\cite {LeTa91})} \\\\& \\le &\\frac{16}{r^2} \\mathbb {E}\\sup _{x \\in r\\cdot S^{d-1}} \\left| \\frac{1}{n}\\sum _{i=1}^n \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle \\right|\\\\& & \\text{(see again \\cite [Lemma 2.3.6]{vaWe96})} \\\\& \\le &\\frac{32}{r} \\mathbb {E}\\sup _{\\lbrace t:\\Vert t\\Vert \\le 1\\rbrace } \\left|\\frac{1}{n} \\sum _{i=1}^n \\left\\langle \\overline{X}_i,t \\right\\rangle \\right| \\\\& & \\text{(noting that $\\Vert x-v_x\\Vert \\le 2r$)} \\\\& \\le & \\frac{32}{r} \\cdot \\frac{\\sqrt{\\mathbb {E}\\left\\Vert \\overline{X}\\right\\Vert ^2}}{\\sqrt{n}}= \\frac{32}{r} \\left(\\frac{\\mathrm {Tr}(\\Sigma )}{n} \\right)^{1/2}\\le \\frac{1}{30}$ provided that $r \\ge 960 \\left(\\frac{\\mathrm {Tr}(\\Sigma )}{n}\\right)^{1/2}.$ Thus, for $Y= \\sup _{x \\in r\\cdot S^{d-1}} \\frac{1}{k} \\sum _{j=1}^k \\mathbb {1}_{\\lbrace |m^{-1}\\sum _{i \\in B_j} \\left\\langle \\overline{X}_i,x-v_x \\right\\rangle | \\ge r^2/4\\rbrace }~,$ we have proved that $\\mathbb {E}Y \\le 1/60+1/30= 1/20$ .",
"Finally, in order to establish (REF ), it suffices to show that, $\\mathbb {P}\\lbrace Y > \\mathbb {E}Y + 1/20\\rbrace \\le e^{-k/200}$ , which follows from the bounded differences inequality (see, e.g., [10])."
],
[
"Proof of Theorem ",
"Theorem REF is easily derived from Lemma REF .",
"Fix a block $B_j$ , and recall that $Z_j=\\frac{1}{m}\\sum _{i \\in B_j}X_i$ .",
"Let $a,b \\in \\mathbb {R}^d$ .",
"Then $\\frac{1}{m}\\sum _{i \\in B_j} \\left(\\Vert X_i-a\\Vert ^2- \\Vert X_i -b\\Vert ^2 \\right)& = & \\frac{1}{m}\\sum _{i \\in B_j} \\left( \\Vert X_i-b-(a-b)\\Vert ^2- \\Vert X_i-b\\Vert ^2 \\right)\\\\& = & -\\frac{2}{m}\\sum _{i \\in B_j} \\left\\langle X_i-b,a-b \\right\\rangle + \\Vert a-b\\Vert ^2 = (*)$ Observe that $-\\frac{2}{m}\\sum _{i \\in B_j} \\left\\langle X_i-b,a-b \\right\\rangle = -2\\left\\langle \\frac{1}{m}\\sum _{i \\in B_j} X_i -b,a-b \\right\\rangle =-2\\left\\langle Z_j-b,a-b \\right\\rangle $ , and thus $(*) &= & -2\\left\\langle Z_j-b,a-b \\right\\rangle + \\Vert a-b\\Vert ^2 \\\\& = & -2\\left\\langle Z_j-b,a-b \\right\\rangle + \\Vert a-b\\Vert ^2 + \\Vert Z_j-b\\Vert ^2 - \\Vert Z_j-b\\Vert ^2 \\\\& = & \\Vert Z_j-b - (a-b)\\Vert ^2 - \\Vert Z_j-b\\Vert ^2 =\\Vert Z_j-a\\Vert ^2-\\Vert Z_j-b\\Vert ^2~.$ Therefore, $(*)>0$ (i.e., $b$ defeats $a$ on block $B_j$ ) if and only if $\\Vert Z_j-a\\Vert > \\Vert Z_j-b\\Vert $ .",
"Recall that Lemma REF states that, with probability at least $1-\\delta $ , if $\\Vert a-\\mu \\Vert \\ge r$ then on more than $k/2$ blocks $B_j$ , $\\frac{1}{m}\\sum _{i \\in B_j} \\left(\\Vert X_i-a\\Vert ^2- \\Vert X_i -\\mu \\Vert ^2 \\right) >0$ , which, by the above argument, is the same as saying that for at least $k/2$ indices $j$ , $\\Vert Z_j-a\\Vert > \\Vert Z_j-\\mu \\Vert $ .",
"Upon reflection it is clear that the ideas used in the proof of Theorem REF are rather general.",
"In fact, they are at the heart of the small-ball method introduced in Mendelson [61] (see also [63] for results of similar flavour).",
"The small-ball method holds in far more general situations than Theorem REF and will be repeated throughout this note.",
"To explain how the argument can be extended, let us outline again the three steps that allowed us to compare every $b$ and $\\mu $ : $(1)$ For any fixed $b \\in \\mathbb {R}^d$ we obtain a bound that holds with high probability; $(2)$ Then, thanks to the high probability estimate from $(1)$ , we invoke the union bound and control a large (yet finite) collection of points.",
"We have complete freedom to choose the collection as we want, and we select it as an $\\epsilon $ -net in the set in question.",
"$(3)$ The crucial part of the argument is passing from the control we have on every point in the net to the wanted uniform control on entire class; specifically, we show that if a `center', that is, an element of the net, is well-behavedin the proof of Theorem REF , `well-behaved' means that (REF ) holds for a majority of the blocks., then the same is true for any point close enough to the center.",
"To that end, we show that `random oscillations' do not destroy the good behaviour of a center on too many blocks."
],
[
"Computational considerations",
"An important issue that we have mostly swept under the rug so far is computational feasibility of mean estimators.",
"While the empirical mean is trivial to compute, many of the more sophisticated estimators discussed here are far from being so.",
"In particular, a basic requirement for any multivariate mean estimator for having a chance to being useful in practice is that it can be computed in polynomial time (i.e., in time that is a polynomial of the sample size $n$ and the dimension $d$ ).",
"As we already pointed it out, some of the estimators described above fall in this category.",
"For example, the geometric median-of-means estimator or the Catoni-Giulini estimator are both efficiently computable in this sense.",
"However, these estimators fall short from being sub-Gaussian.",
"The median-of-means tournament estimator is sub-Gaussian but its computation poses a highly nontrivial challenge.",
"In fact, the way the estimator is defined, it is likely to be computationally intractable (i.e., NP hard).",
"However, in a recent beautiful paper, Hopkins [33] defines a clever semi-definite relaxation of the median-of-means tournament estimator that can be computed in time $O(nd + d\\log (1/\\delta )^c)$ for a dimension-independent constant and, at the same time, achieves the desired sub-Gaussian guarantee under the only assumption that the covariance matrix exists.",
"This is the first efficiently computable sub-Gaussian multivariate mean estimator.",
"Even more recently, Cherapanamjeri, Flammarion, and Bartlett [18] improved the running time to $O(nd + d\\log (1/\\delta )^2 + \\log (1/\\delta )^4)$ by combining Hopkins' ideas with clever gradient-descent optimization.",
"This is likely not the last word on the subject as many exciting computational challenges arise in the context of mean estimation and regression.",
"In the theoretical computer science community there has been a recent important surge of results that address the problem of computationally efficient robust mean estimation.",
"In this context, an estimator is defined to be robust if it performs well in the presence of a small constant fraction of (possibly adversarial) outliers.",
"Various different models have been introduced, see Charikar, Steinhardt, and Valiant [17], Diakonikolas, Kamath, Kane, Li, Moitra, and Stewart [23], [24], [25], Diakonikolas, Kane, and Stewart [26], Diakonikolas, Kong, and Stewart [27], Hopkins and Li [34], Klivans, Kothari, and Meka [42], Kothari, Steinhardt, and Steurer [44], Lai, Rao, and Vempala [45], Loh and Tan [54], for a sample of this important growing body of literature.",
"Surveying this area goes beyond the scope of this paper."
],
[
"Uniform median-of-means estimators",
"The median-of-means tournament used in the previous section is an example of a uniform median-of-means estimator: given a class of functions ${\\mathcal {F}}$ , there is a high-probability event on which, for every $f$ in the class, the median of means estimator based on the data $f(X_1),\\ldots ,f(X_n)$ is close to the mean $\\mathbb {E}f(X)$ .",
"Indeed, the tournament is simply a median-of-means estimator that was used to check whether $a$ was closer to $\\mu $ than $b$ , or vice-versa, uniformly for every pair $a,b \\in \\mathbb {R}^n$ .",
"In what follows we present a general version of a uniform median-of-means estimator and turn our attention to two applications: estimating the mean of a random vector with respect to an arbitrary norm, and $L_2$ -distance oracles (the latter proves useful in regression problems, see Section REF and [59] for more details).",
"Formally, the question we consider is as follows: Let ${\\mathcal {F}}$ be a class of functions on a probability space $(\\Omega ,\\nu )$ .",
"Given an independent sample $(X_1,\\ldots ,X_n)$ distributed according to $\\nu $ , find an estimator $\\widehat{\\phi }_n(f)$ for each $f\\in {\\mathcal {F}}$ , such that, with high probability, for every $f \\in {\\mathcal {F}}$ , $|\\widehat{\\phi }_n(f) - \\mathbb {E}f(X) |$ is small.",
"A natural idea is to define $\\widehat{\\phi }_n(f)$ to be the median-of-means estimator based on $f(X_1),\\ldots ,f(X_n)$ .",
"It stands to reason that the bound established in Section REF for the performance of the median-of-means estimator cannot simply hold uniformly for every $f \\in {\\mathcal {F}}$ .",
"Rather, the uniform error consists of two terms: the `worst' individual estimate for a function $f \\in {\\mathcal {F}}$ , and a `global' error, taking into account the `complexity' of the class.",
"To analyze uniform median-of-means estimators, it is natural to follow the path of the small-ball method outlined in the previous section.",
"To this end, fix integers $k$ and $m$ and let $n=mk$ .",
"As always, we split the given sample into $k$ blocks, each one of cardinality $m$ , keeping in mind that the natural choice is $k \\sim \\log (2/\\delta )$ if one wishes a confidence of $1-\\delta $ .",
"For $0<\\eta <1$ set $p_m(\\eta ) = \\sup _{f \\in {\\mathcal {F}}} \\mathbb {P}\\left(\\left|\\frac{1}{m}\\sum _{i=1}^m f(X_i)-\\mathbb {E}f(X) \\right|\\ge \\eta \\right)~,$ denote by $D=\\left\\lbrace f: \\mathbb {E}f(X)^2\\le 1\\right\\rbrace $ the unit ball in $L_2(\\nu )$ and let ${\\cal M}({\\mathcal {F}},rD)$ be the maximal cardinality of a subset of ${\\mathcal {F}}$ that is $r$ -separated with respect to the $L_2(\\nu )$ norm.",
"The following bound was recently established in [57].",
"Theorem 10 There exist absolute constants $c_0,\\ldots ,c_4$ for which the following holds.",
"Set $\\eta _0,\\eta _1$ and $\\eta _2 \\ge c_0 \\eta _1/\\sqrt{m}$ that satisfy the following: $(1)$ $p_m(\\eta _0) \\le 0.05$ ; $(2)$ $\\log {\\cal M}({\\mathcal {F}},\\eta _1 D) \\le c_2 k \\log (e/p_m(\\eta _0))$ ; $(3)$ $\\mathbb {E}\\sup _{w \\in \\overline{W}} \\left|\\sum _{i=1}^n \\varepsilon _i w(X_i) \\right| \\le c_3 \\eta _2 n$ , where $\\varepsilon _1,\\ldots ,\\varepsilon _n$ are independent Rademacher random variables (i.e., $\\mathbb {P}\\lbrace \\varepsilon _i=1\\rbrace =\\mathbb {P}\\lbrace \\varepsilon _i=-1\\rbrace =1/2$ ) and $W=({\\mathcal {F}}-{\\mathcal {F}}) \\cap \\eta _1 D = \\lbrace f_1-f_2 : f_1,f_2 \\in {\\mathcal {F}}, \\ \\Vert f_1-f_2\\Vert _{L_2} \\le \\eta _1\\rbrace $ and $\\overline{W}=\\lbrace w -\\mathbb {E}w : w \\in W\\rbrace $ .",
"Let $r=\\eta _0+\\eta _2$ .",
"Then, with probability at least $1-2\\exp (-c_4k)$ , for all $f \\in {\\mathcal {F}}$ one has $\\left| \\frac{1}{m} \\sum _{i \\in B_j} f(X_i) - \\mathbb {E}f \\right| \\le r \\ \\ {\\rm for \\ at \\ least \\ } 0.6k \\ {\\rm blocks \\ } B_j~.$ The error $r$ in Theorem REF has the two terms we expected.",
"Indeed, $\\eta _0$ is error one would have if the goal were to obtain an individual mean estimator for a fixed function in ${\\mathcal {F}}$ : writing $\\sigma _f= \\sqrt{\\mathrm {Var}(f(X))}$ , by Chebyshev's inequality, for every $f \\in {\\mathcal {F}}$ , $\\mathbb {P}\\left(\\left|\\frac{1}{m}\\sum _{i=1}^m f(X_i)-\\mathbb {E}f \\right|\\ge \\eta _0 \\right) \\le \\frac{\\sigma _f^2}{m \\eta _0^2} \\le 0.05$ provided that $\\eta _0 \\gtrsim \\frac{\\sigma _f}{\\sqrt{m}} \\sim \\sigma _f \\sqrt{\\frac{\\log (2/\\delta )}{n}}~.$ As outlined in Section REF , this leads to the standard sub-Gaussian error estimate for the function $f \\in {\\mathcal {F}}$ .",
"On the other hand, $\\eta _2$ involves the Rademacher averages associated with ${\\mathcal {F}}-{\\mathcal {F}}$ , and captures the price one has to pay for the uniform control over the class ${\\mathcal {F}}$ .",
"The proof of Theorem REF follows the same path we outlined previously: the definition of $p_m$ allows us to show that the empirical mean of $f$ on a block $B_j$ of cardinality $m$ is close to the true mean with reasonable probability, say, larger than $0.95$ .",
"Thus, with probability $1-e^{-ck}$ , this property is satisfied on $0.9k$ blocks.",
"Next, the high-probability estimate combined with the union bound allow us to control all the elements in a finite set uniformly, as long as its cardinality is at most exponential in $k$ .",
"The set of choice is an appropriate net in ${\\mathcal {F}}$ and its mesh width $\\eta _1$ is selected to ensure that the cardinality of the net is small enough.",
"Finally, as always, the crucial component is to ensure that oscillations do not `corrupt' the good behaviour on too many blocks.",
"Since our interest is in the median of means, one can live with up to $0.4k$ of the blocks being corrupted, and the additional error of $\\eta _2$ suffices to guarantee that indeed no more than $0.4k$ blocks are affected.",
"The technical analysis can be found in [57], where Theorem REF is used for the study the problem of multivariate mean estimation with respect to a general norm, outlined in the next section.",
"We mention here that uniform estimators based on Catoni's mean estimator were studied by Brownlees, Joly, and Lugosi [11] in the context of regression function estimation.",
"Minsker [67] discusses uniform estimators in a similar spirit to those presented here, also for adversarially contaminated data."
],
[
"Multivariate mean estimation—the general case",
"To illustrate the power of the uniform median-of-means bounds established in the previous section, we now return to the problem of estimating the mean of a random vector.",
"As before, let $X_1,\\ldots ,X_n$ be independent, identically distributed random vectors in $\\mathbb {R}^d$ with mean $\\mu $ and covariance matrix $\\Sigma $ .",
"The question we seek to answer is to what extent one can estimate $\\mu $ when the error is measured by a given norm $\\Vert \\cdot \\Vert $ that is not necessarily the Euclidean norm.",
"An important example is the matrix operator norm, see Minsker [66], Catoni and Giulini [15], Mendelson and Zhivotovskiy [64].",
"One may now cast this general mean estimation problem in the framework of uniform median-of-means estimators outlined above.",
"The natural class of functions associated with the problem is the unit ball with respect to the dual of the norm $\\Vert \\cdot \\Vert $ (i.e., the set of norm-one linear functionals).",
"The natural choice of a measure $\\nu $ is the one induced by $X-\\mu $ .",
"Consider the event given by Theorem REF for this class of functions and denote the resulting error by $r$ .",
"It follows that for each norm-one functional $x^*$ , we have $\\mathbb {E}x^*(X-\\mu )=0$ and $\\left|\\frac{1}{m} \\sum _{i \\in B_j} x^*(X_j-\\mu )\\right| \\le r$ for a majority of the blocks $B_j$ .",
"Moreover, $\\frac{1}{m} \\sum _{i \\in B_j} x^*(X_i - \\mu ) = x^* \\bigl (\\frac{1}{m}\\sum _{i \\in B_j} X_i\\bigr )- x^*(\\mu )~.$ Thus, setting $Z_j = \\frac{1}{m}\\sum _{i \\in B_j} X_j$ , Theorem REF implies that for every norm-one functional $x^*$ , $|x^*(Z_j) - x^*(\\mu )| \\le r~.$ In other words, if we define the sets $\\mathbb {S}_{x^*} = \\left\\lbrace y \\in \\mathbb {R}^d : |x^*(Z_j) - x^*(y) | \\le r \\ \\ {\\rm for \\ the \\ majority \\ of \\ indices \\ } j \\right\\rbrace $ then on the event from Theorem REF one has that $\\mu \\in \\mathbb {S}= \\bigcap _{\\Vert x^*\\Vert =1} \\mathbb {S}_{x^*}~.$ From a geometric point of view, each set $\\mathbb {S}_{x^*}$ is the union of intersection of slabs: setting $\\alpha _j = x^*(Z_j)$ , $\\mathbb {S}_{x^*}= \\bigcup _{|I| \\ge [k/2]+1} \\bigcap _{i \\in I} \\lbrace y : |x^*(y) - \\alpha _j| \\le r\\rbrace ~,$ which is just a union of (potentially empty) slabs, defined by the functional $x^*$ .",
"The set $\\mathbb {S}$ is the resulting intersection of the sets $\\mathbb {S}_{x^*}$ .",
"Off hand, there is no reason why the intersection of the sets $S_{x^*}$ should not be empty.",
"The fact that it contains $\\mu $ is only due to the special nature of the $Z_j$ 's.",
"Since each set $\\mathbb {S}_{x^*}$ is data-dependent, so is $\\mathbb {S}$ .",
"With that in mind, the estimator we propose is obvious: set $\\widehat{\\mu }_n^{(r)}$ to be any point in $\\mathbb {S}$ .",
"To show that $\\Vert \\widehat{\\mu }_n^{(r)}-\\mu \\Vert \\le 2r$ , fix any norm-one functional $x^*$ .",
"Recall that if $ y \\in \\mathbb {S}$ then $|x^*(Z_j)-x^*(y)| \\le r$ on the majority of blocks.",
"Therefore, if $\\mu ,\\widehat{\\mu }_n^{(r)} \\in \\mathbb {S}$ there is some index $j$ such that, simultaneously, $|x^*(Z_j)-x^*(\\widehat{\\mu }_n^{(r)})| \\le r \\ \\ \\ {\\rm and} \\ \\ \\ |x^*(Z_j)-x^*(\\mu )| \\le r~,$ and therefore $|x^*(\\widehat{\\mu }^{(r)}-\\mu )| \\le 2r$ .",
"Thanks to Theorem REF , there is a high-probability event on which this is true for any norm-one functional, and, in particular, $\\Vert \\widehat{\\mu }_n^{(r)} - \\mu \\Vert = \\sup _{\\Vert x^*\\Vert =1} |x^*(\\widehat{\\mu }_n^{(r)}-\\mu )| \\le 2r~.$ Remark.",
"It is straightforward to verify that there is no need to select ${\\mathcal {F}}$ to be the set of all the norm-one linear functionals.",
"It is enough to define $\\mathbb {S}$ using the functionals that are extreme points of the unit ball in the dual space.",
"Thanks to Theorem REF and the argument we just outlined, the following was established in [57]: Theorem 11 Let $\\Vert \\cdot \\Vert $ be a norm on $\\mathbb {R}^d$ .",
"Suppose that the $X_i$ have mean $\\mu $ and covariance matrix $\\Sigma $ .",
"There exists a mean estimator $\\widehat{\\mu }_n$ such that, with probability at least $1-\\delta $ , $\\Vert \\widehat{\\mu }_n - \\mu \\Vert \\le \\frac{c}{\\sqrt{n}} \\left( \\max \\left\\lbrace \\mathbb {E}\\Vert \\zeta _n\\Vert , \\ \\ \\mathbb {E}\\Vert G\\Vert + R \\sqrt{\\log (2/\\delta )} \\right\\rbrace \\right)~,$ where $c$ is a numerical constant, $R=\\sup _{\\Vert x^*\\Vert =1} \\left(\\mathbb {E}(x^*(X-\\mu ))^2\\right)^{1/2}~,$ $\\zeta _n = \\frac{1}{\\sqrt{n}} \\sum _{i=1}^n \\varepsilon _i(X_i -\\mu )~,$ $\\lbrace \\epsilon _i\\rbrace $ is a sequence of i.i.d.",
"Rademacher random variables independent of $\\lbrace X_i\\rbrace $ , and $G$ is the centered Gaussian vector with covariance matrix $\\Sigma $ .",
"As is explained in [57], Theorem REF , there is a good reason to believe that the bound of the theorem is optimal in a rather strong sense.",
"We refer the reader to [57] for more details.",
"Remark.",
"Note that the error in Theorem REF has two types of terms: $\\frac{R}{\\sqrt{n}} \\sqrt{\\log (2/\\delta )}$ is the standard one-dimensional sub-Gaussian error, and its source is the marginal $x^*(X)$ with the largest variance.",
"At the same time, $\\mathbb {E}\\Vert G\\Vert $ and $\\mathbb {E}\\Vert \\zeta _n\\Vert $ are `global' parameters that calibrate the `complexity' of the norm.",
"When $\\Vert \\cdot \\Vert $ is the Euclidean norm, we recover the two terms on the right-hand side of (REF )."
],
[
"$L_2$ distance oracles",
"In this section we sketch how the ideas used in Theorem REF may be used to generate a median-of-means based (isomorphic) $L_2$ distance oracle.",
"A more accurate description of distance oracles may be found in [59].",
"Suppose ${\\mathcal {F}}$ is a class of real-valued functions on a probability space $(\\Omega ,\\nu )$ and let $X$ be a random variable distributed as $\\nu $ .",
"There are many natural situations in which, given an i.i.d.",
"sample $X_1,\\ldots ,X_n$ , one would like to have an accurate estimate on the $L_2$ distance $\\Vert f-h\\Vert _{L_2}\\stackrel{\\mathrm {def.",
"}}{=}\\sqrt{\\mathbb {E}(f(X)-h(X))^2}$ between any two class members $f,h\\in {\\mathcal {F}}$ .",
"In some cases, the estimates are required to be almost isometric, that is, with high probability, for all $f,h\\in {\\mathcal {F}}$ , the estimate should lie in the range $[(1-\\eta )\\Vert f-h\\Vert _{L_2}, (1+\\eta )\\Vert f-h\\Vert _{L_2}]$ for some small value of $\\eta $ .",
"However, in many situations (for example, in the regression problem we describe in Section ), a weaker property suffices: one would like to define a data-dependent functional $\\widehat{\\Psi }_n$ such that , with 'high' probability, for all $f,h \\in {\\mathcal {F}}$ and a `small' value $r$ , and some constants $0<\\alpha <1<\\beta $ , $\\bullet $ if $\\widehat{\\Psi }_n(f,h) \\ge r$ then $\\alpha \\Vert f-h\\Vert _{L_2}\\le \\widehat{\\Psi }_n(f,h) \\le \\beta \\Vert f-h\\Vert _{L_2}$ ; $\\bullet $ if $\\widehat{\\Psi }_n(f,h) \\le r$ then $\\Vert f-h\\Vert _{L_2} \\le r/\\alpha $ .",
"In other words, for every $f,h \\in {\\mathcal {F}}$ , based on the value of the data-dependent functional $\\widehat{\\Psi }_n(f,h)$ one may estimate $\\Vert f-h\\Vert _{L_2}$ in an isomorphic way—i.e., up to multiplicative constants.",
"We call such a functional a distance oracle.",
"For the sake of simplicity, instead of considering simultaneous estimation of pairwise distances of functions, we address the problem of estimating $L_2$ norms of functions.",
"In other words, given a class ${\\mathcal {F}}$ of functions as above, we are interested in constructing a data-dependent functional $\\widehat{\\Psi }_n$ such that if $\\widehat{\\Psi }_n(f)\\ge r$ then $\\alpha \\Vert f\\Vert _{L_2} \\le \\widehat{\\Psi }_n(f) \\le \\beta \\Vert f\\Vert _{L_2}$ , and if $\\widehat{\\Psi }_n(f) < r$ then $\\Vert f\\Vert _{L_2}\\lesssim r/\\alpha $ .",
"Such a functional may be called a norm oracle.",
"Given a norm oracle, one may construct a distance oracle in an obvious way.",
"In what follows we assume that there is some $q>2$ such that the $L_q$ and $L_2$ norms are equivalent on $\\lbrace f_1-f_2 : f_1,f_2 \\in {\\mathcal {F}}\\cup \\lbrace 0\\rbrace \\rbrace $ .",
"In other words, there is a constant $L$ such that $\\Vert f_1-f_2\\Vert _{L_q} \\le L \\Vert f_1-f_2\\Vert _{L_2}$ for all $f_1,f_2 \\in {\\mathcal {F}}\\cup \\lbrace 0\\rbrace $ .",
"Consider the set $H={\\rm star}({\\mathcal {F}},0) = \\lbrace \\lambda f : f \\in {\\mathcal {F}}, \\ 0 \\le \\lambda \\le 1\\rbrace $ and let $H_\\rho = H \\cap \\rho S(L_2)$ , where $\\rho S(L_2) = \\lbrace h : \\Vert h\\Vert _{L_2} =\\rho \\rbrace $ .",
"For every $h \\in H$ , set $Z_h(j) = \\frac{1}{m} \\sum _{i \\in B_j} |h(X_i)|$ and our estimator $\\widehat{\\Psi }_n(h)$ is the median of $Z_h(1),\\ldots ,Z_h(k)$ .",
"Recall that $D=\\left\\lbrace f: \\mathbb {E}f(X)^2\\le 1\\right\\rbrace $ denotes the unit ball in $L_2(\\nu )$ and let ${\\cal M}({\\mathcal {F}},rD)$ be the maximal cardinality of a subset of ${\\mathcal {F}}$ that is $r$ -separated with respect to the $L_2(\\nu )$ norm.",
"Theorem 12 There exist constants $c_1, A, B$ that depend on $q$ and $L$ , and absolute constants $c_2,\\ldots ,c_6$ such that the following holds.",
"Let $m=c_1(L,q)$ and set $k=n/m$ .",
"Under the $L_q-L_2$ norm equivalence condition, if $\\log {\\cal M}(H_\\rho , c_2 A \\rho D) \\le c_3k~,$ and $\\mathbb {E}\\sup _{w \\in (H_\\rho -H_\\rho ) \\cap c_2 A \\rho D} \\left|\\sum _{i=1}^n \\varepsilon _i w(X_i) \\right| \\le c_4 A \\rho n~,$ then with probability at least $1-2\\exp (-c_5k)$ , for all $h\\in H_\\rho $ , $\\bullet $ if $\\Vert h\\Vert _{L_2} \\ge \\rho $ then $A \\Vert h\\Vert _{L_2} \\le \\widehat{\\Psi }_n(h) \\le B \\Vert h\\Vert _2$ ; and $\\bullet $ if $\\Vert h\\Vert _{L_2} \\le \\rho $ then $\\widehat{\\Psi }_n(h) \\le c_6 B \\rho $ .",
"Note that Theorem REF shows that $\\widehat{\\Psi }_n$ is a desired norm oracle: if $\\widehat{\\Psi }_n(h) > c_6 B \\rho $ then it follows that $\\Vert h\\Vert _{L_2} \\ge \\rho $ , and thus $B^{-1} \\widehat{\\Psi }_n(h) \\le \\Vert h\\Vert _{L_2} \\le A^{-1} \\widehat{\\Psi }_n(h)~.$ On the other hand, if $\\widehat{\\Psi }_n(h) \\le c_6 B \\rho $ then one has two options: either $\\Vert h\\Vert _{L_2} \\le \\rho $ , or, $\\Vert h\\Vert _{L_2} \\ge \\rho $ , in which case $\\Vert h\\Vert _{L_2} \\le A^{-1} \\widehat{\\Psi }_n(h) \\le c_6(B/A) \\rho $ .",
"Thus, $\\Vert h\\Vert _{L_2} \\le \\rho \\max \\lbrace 1,c_6B/A\\rbrace $ .",
"The norm oracle is obtained by setting $r = c_6 B \\rho $ and choosing $\\alpha $ and $\\beta $ appropriately.",
"The proof of Theorem REF follows the small-ball method: we begin by showing that for a fixed $h \\in H_\\rho $ , and with high probability, $ \\left|\\left\\lbrace j : A \\rho \\le \\frac{1}{m} \\sum _{i \\in B_j} |h|(X_i) \\le B \\rho \\right\\rbrace \\right| \\ge 0.8k$ for some constants $A$ and $B$ .",
"Then, the high-probability estimate allows us to control a satisfactory net in $H_\\rho $ , and finally, one has to control `oscillations': a high-probability event such that if $h \\in H_\\rho $ and $\\pi h$ denotes the closest point to $h$ in the net, then $\\sup _{h \\in H_\\rho } \\left| \\left\\lbrace j : \\frac{1}{m}\\sum _{i \\in B_j} |h-\\pi h|(X_i) \\ge \\frac{A \\rho }{2}\\right\\rbrace \\right| \\le 0.2k~.$ With all three components in place, it is evident that for every $h \\in H_\\rho $ there are at least $0.6k$ blocks $B_j$ on which $A \\rho \\le \\frac{1}{m} \\sum _{i \\in B_j} |\\pi h|(X_i) \\le B \\rho \\ \\ \\ {\\rm and} \\ \\ \\ \\frac{1}{m}\\sum _{i \\in B_j} |h-\\pi h|(X_i) \\le \\frac{A \\rho }{2}~.$ On these blocks, $\\frac{1}{m} \\sum _{i \\in B_j} |h|(X_i) \\ge \\frac{1}{m} \\sum _{i \\in B_j} |\\pi h|(X_i) - \\frac{1}{m}\\sum _{i \\in B_j} |h-\\pi h|(X_i) \\ge \\frac{A \\rho }{2}~,$ and a similar estimate holds for the upper bound.",
"Once an isomorphic estimate is established in $H_\\rho ={\\rm star}({\\mathcal {F}},0) \\cap \\rho S(L_2)$ , the same estimate holds for any $h \\in H_r$ and any $r \\ge \\rho $ .",
"This is evident from the fact that $H={\\rm star}({\\mathcal {F}},0)$ is star-shaped around 0, implying that every $h \\in H_r$ has a `scaled down' version in $H_\\rho $ .",
"In particular, on the same event we have that if $f \\in {\\mathcal {F}}$ and $\\Vert f\\Vert _{L_2} \\ge \\rho $ , then $\\frac{A}{2} \\Vert f\\Vert _{L_2} \\le \\widehat{\\Psi }_n(f) \\le 2B \\Vert f\\Vert _{L_2}~.$ The second part of the claim follows the same lines (see [59] for more details).",
"The main question is how to ensure that (REF ) holds with high enough probability.",
"As it happens, (REF ) can be verified under minimal assumptions, as we now explain.",
"Assume, for example, that the given class ${\\mathcal {F}}$ satisfies a small-ball condition, namely, for every $\\epsilon >0$ there is a constant $\\kappa (\\epsilon )$ such that for every $f \\in {\\mathcal {F}}$ , $\\mathbb {P}(|f(X)| \\le \\kappa (\\epsilon )\\Vert f\\Vert _{L_2} ) \\le \\epsilon ~.$ Set $\\epsilon =0.05$ and let $\\kappa =\\kappa (0.05)$ .",
"Then with probability at least $1-2\\exp (-cn)$ there are at least $0.9n$ indices $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ such that $|f(X_i)| \\ge \\kappa \\Vert f\\Vert _{L_2}$ .",
"At the same time, $\\mathbb {P}(|f(X)| \\ge 10 \\Vert f\\Vert _{L_2}) \\le \\frac{1}{100}~,$ implying that with probability at least $1-2\\exp (-cn)$ , for at least $0.9n$ indices $1 \\le i \\le n$ , $|f(X_i)| \\le 10 \\Vert f\\Vert _{L_2}$ .",
"Thus, intersecting the two events (REF ) is established with probability at least $1-2\\exp (-c^\\prime n)$ for $m=1$ , $A =\\kappa $ and $B=10$ .",
"Of course it is true that not every random variable $f(X)$ satisfies the small-ball condition we use above.",
"However, there is an additional degree of freedom that has not been exploited yet: that the random variables one truly cares about are of the form $Z_f(j)=\\frac{1}{m}\\sum _{i \\in B_j} |f(X_i)|$ , leaving us some room to generate the necessary regularity.",
"Indeed, it is straightforward to verify that under minimal assumptions and for a small value of $m$ , the $Z_f(j)$ do satisfy a sufficiently strong small-ball condition.",
"This is an outcome of a Berry-Esseen type argument The case $q=3$ is the standard Berry-Esseen theorem while for $2<q<3$ one may use generalized Berry-Esseen bounds, see [71].",
": if there is some $q>2$ such that $\\Vert f\\Vert _{L_q} \\le L \\Vert f\\Vert _{L_2}$ then for $m=c(q,L)$ , $\\sqrt{m}(Z_f(j)-\\mathbb {E}|f|)$ is `close enough' to a Gaussian variable and it follows that $\\mathbb {P}( |Z_f(j)| \\le c_1\\Vert f\\Vert _{L_2}) \\le 0.05~.$"
],
[
"Median-of-means tournaments in regression problems",
"The problem of regression function estimation essentially amounts to estimating conditional expectations and as such, it is a natural candidate for extending ideas of mean estimation discussed in this paper.",
"In this section we explore some of the recent progress made in the study of regression problems driven by uniform median-of-means estimators.",
"The standard setup for regression function estimation may be formulated as follows.",
"Let $(X,Y)$ be a pair of random variables such that $X$ takes its values in some set $\\mathcal {X}$ while $Y$ is real valued.",
"Given a class ${\\mathcal {F}}$ of real-valued functions defined on $\\mathcal {X}$ , one's goal is to find $f\\in {\\mathcal {F}}$ for which $f(X)$ is a good prediction of $Y$ .",
"The performance of a predictor $f\\in {\\mathcal {F}}$ is measured by the mean-squared error $\\mathbb {E}(f(X)-Y)^2$ , also known as the risk.",
"The best performance in the class is achieved by the risk minimizer $f^*= \\mathop {\\mathrm {argmin}}_{f\\in {\\mathcal {F}}} \\mathbb {E}(f(X)-Y)^2~.$ We assume in what follows that the minimum is attained and $f^*\\in {\\mathcal {F}}$ exists and is unique.",
"The difficulty stems from the fact that the joint distribution of $(X,Y)$ is not known.",
"Instead, one is given an i.i.d.",
"sample $\\mathcal {D}_n=(X_i,Y_i)_{i=1}^n$ distributed according to the joint distribution of $X$ and $Y$ .",
"Given a sample size $n$ , a learning procedure is a map that assigns to each sample $\\mathcal {D}_n$ a (random) function in ${\\mathcal {F}}$ , which we denote by $\\widehat{f}$ .",
"The success of $\\widehat{f}$ is measured by the tradeoff between the accuracy $\\epsilon $ and the confidence $\\delta $ with which $\\widehat{f}$ attains that accuracy, that is, one would like to find $\\widehat{f}$ which satisfies that $\\mathbb {P}\\left( \\mathbb {E}\\left(\\left(\\widehat{f}(X)-Y\\right)^2 | \\mathcal {D}_n \\right) \\le \\inf _{f \\in {\\mathcal {F}}} \\mathbb {E}(f(X)-Y)^2 + \\epsilon \\right) \\ge 1-\\delta $ for values of $\\epsilon $ and $\\delta $ as small as possible.",
"Note that one has the freedom to select a function $\\widehat{f}$ that does not belong to ${\\mathcal {F}}$ ..",
"The question of this accuracy/confidence tradeoff has been the subject of extensive study, see, for example, the books [82], [21], [29], [81], [3], [80], [60], [43], [76], [13] for a sample of the large body of work devoted to this question.",
"The most standard and natural way of choosing $\\widehat{f}$ is by least squares regression, also known as empirical risk minimization: $\\widehat{f} = \\mathop {\\mathrm {argmin}}_{f\\in {\\mathcal {F}}} \\sum _{i=1}^n (f(X_i)-Y_i)^2~.$ A sample of the rich literature on the analysis of empirical risk minimization includes Györfi, Kohler, Krzyzak, Walk [29], van de Geer [80], Bartlett, Bousquet, and Mendelson [7], Koltchinskii [43], Massart [60].",
"The simple idea behind empirical risk minimization is that, for each $f\\in {\\mathcal {F}}$ , the empirical risk $(1/n) \\sum _{i=1}^n (f(X_i)-Y_i)^2$ is a good estimate of the risk $\\mathbb {E}(f(X)-Y)^2$ and the minimizer of the empirical risk should nearly match that of the “true” risk.",
"Naturally, when the empirical risks are not reliable estimates of their population counterparts, empirical risk minimization stands on shaky ground.",
"It should not come as a surprise that the performance of empirical risk minimization changes dramatically according to the tail behaviour of the functions involved in the given learning problem.",
"One may show (see, e.g., [49]) that if ${\\mathcal {F}}$ is convex and the random variables $\\lbrace f(X) : f \\in {\\mathcal {F}}\\rbrace $ and the target $Y$ have well-behaved sub-Gaussian tails, empirical risk minimization performed in ${\\mathcal {F}}$ yields good results: it essentially attains the optimal accuracy/confidence tradeoff for a certain range of the parameters.",
"However, the situation deteriorates considerably when either members of ${\\mathcal {F}}$ or $Y$ is heavy-tailed in some sense.",
"In such cases, the performance of empirical risk minimization may be greatly improved by employing more sophisticated mean estimation techniques.",
"For the analysis of least squares regression for some heavy-tailed situations, see Han and Wellner [31].",
"A growing body of recent work has addressed the problem of constructing regression function estimators that work well even when some of the $f(X)$ and $Y$ may be heavy tailed, see Audibert and Catoni [4], Brownlees, Joly, and Lugosi [11], Catoni and Giulini [15].",
"Chichignoud and Lederer [19], Fan, Li, and Wang [28], Hsu and Sabato [36], Lecué and Lerasle [46], [47], Lecué, Lerasle, and Mathieu [48], Lerasle and Oliveira [53], Lugosi and Mendelson [59], [58], Mendelson [62], and Minsker [65].",
"In this section we limit ourselves to sketching how median-of-means tournaments may be used in regression function estimation.",
"Median-of-means tournaments were introduced in [59] for the study of such regression problems when ${\\mathcal {F}}$ is a convex set.",
"It was shown that one may attain the optimal accuracy/confidence tradeoff in prediction problems in convex classes.",
"Similar methods were used in [58] and [47] to study the regularization framework.",
"In these papers the convexity of the underlying class ${\\mathcal {F}}$ played a central role in the analysis.",
"In fact, it is convexity that allows one to define an optimal $\\widehat{f}$ that takes values in ${\\mathcal {F}}$ .",
"In the general case, when ${\\mathcal {F}}$ need not be convex, selecting $\\widehat{f} \\in {\\mathcal {F}}$ can be a poor choice (see, e.g.",
"the discussion in [61]), and one has to adopt a totally different approach for naming an estimator.",
"An optimal choice of $\\widehat{f}$ for an arbitrary class ${\\mathcal {F}}$ was introduced by Mendelson [62], and that choice is also based on median-of-means tournament, though a different tournament than the one defined in [59].",
"Finally, we mention the general framework of $\\rho $ -estimators introduced by Baraud, Birgé, and Sart [6] and Baraud and Birgé [5].",
"The construction of their estimators bears certain similarities with the tournament procedures described here.",
"For the sake of simplicity, we will only consider the problem of regression in a closed and convex class ${\\mathcal {F}}$ .",
"We set $f^*={\\rm argmin}_{f \\in {\\mathcal {F}}} \\mathbb {E}(f(X)-Y)^2$ to be the minimizer in ${\\mathcal {F}}$ of the risk, and since ${\\mathcal {F}}$ is convex and closed, such a minimizer exists and is unique.",
"The excess risk of $f \\in {\\mathcal {F}}$ is defined to be $\\mathbb {E}{\\cal L}_f = \\mathbb {E}(f(X)-Y)^2-\\mathbb {E}(f^*(X)-Y)^2$ and the aim is to ensure that $\\mathbb {E}({\\cal L}_{\\widehat{f}}|D) \\le \\epsilon $ with probability at least $1-\\delta $ .",
"As one may expect from a median-of-means estimator, we select $k \\le n$ wisely, split the given sample $(X_i,Y_i)_{i=1}^n$ to $k$ blocks, each of cardinality $m=n/k$ , and compare the statistical performance of every pair of functions on each block.",
"Just as before, the belief is that because $\\mathbb {E}(f^*(X)-Y)^2$ is smaller than $\\mathbb {E}(f(X)-Y)^2$ this fact is exhibited by a median-of-means estimate, allowing us to prefer $f^*$ over $f$ .",
"Hence, if we can find a uniform median-of-means estimator, such a comparison would lead us to a function that has almost the same risk as $f^*$ .",
"With that in mind, the natural choice of a “match” in the tournament between two candidate functions $f$ and $h$ is counting the number of blocks on which $\\frac{1}{m}\\sum _{i \\in B_j} (f(X_i)-Y_i)^2$ is larger than $\\frac{1}{m}\\sum _{i \\in B_j} (h(X_i)-Y_i)^2$ .",
"The function that exhibits a superior performance (i.e., has a smaller empirical mean) on the majority of the blocks is the winner of the match.",
"In a perfect world, we would choose a function that won all of its matches.",
"However, the world is far from perfect and the outcomes of matches between functions that are `too close' are not reliable.",
"To address this issue, the tournament requires an additional component: a distance oracle, similar to the one presented in the previous section.",
"Thanks to the distance oracle one may verify in a data-dependent way when two functions are too close, and in such cases disregard the outcome of the match between them.",
"Let us describe some technical facts that are at the heart of the results in [59], [58].",
"Define the “quadratic” and “multiplier” processes $\\mathbb {Q}_{f,h}(j) = \\frac{1}{m}\\sum _{i \\in B_j} (f(X_i)-h(X_i))^2, \\ \\ \\ \\mathbb {M}_{f,h}(j)= \\frac{2}{m}\\sum _{i \\in B_j} (f(X_i)-h(X_i)) (h(X_i)-Y_i)$ and put $\\mathbb {B}_{f,h}(j) \\equiv \\frac{1}{m}\\sum _{i \\in B_j}(f(X_i)-Y_i)^2 - \\frac{1}{m}\\sum _{i \\in B_j}(h(X_i)-Y_i)^2 = \\mathbb {Q}_{f,h}(j)+\\mathbb {M}_{f,h}(j)~.$ Note that $\\mathbb {E}\\mathbb {B}_{f,h}(j)=\\mathbb {E}(f(X)-Y)^2 - \\mathbb {E}(h(X)-Y)^2$ .",
"Therefore, at least intuitively, if $\\mathbb {B}_{f,h}(j)>0$ for a majority of indices $1 \\le j \\le k$ , one would expect that $\\mathbb {E}(f(X)-Y)^2 > \\mathbb {E}(h(X)-Y)^2$ , making $h$ a better candidate to be a risk minimizer than $f$ .",
"When one is given a sample $(X_i,Y_i)_{i=1}^{3n}$ , the choice of $\\widehat{f}$ is carried out as follows: Step 1: $\\bullet $ Fix $r>0$ , corresponding to the desired accuracy parameter $\\epsilon \\sim r^2$ .",
"$\\bullet $ Let $\\widehat{\\Phi }_n$ be a distance oracle in ${\\mathcal {F}}$ similar to the one described in the previous section, which uses as data the first part of the sample $(X_i)_{i=1}^n$ .",
"Thus, for the right choice of parameters $\\alpha $ and $\\beta $ and with high probability the following holds: if $f,h \\in {\\mathcal {F}}$ and $\\widehat{\\Phi }_n(f,h) \\ge \\beta r$ then $\\Vert f-h\\Vert _{L_2} \\sim _{\\alpha ,\\beta } \\widehat{\\Phi }_n(f,h)$ , and if $\\widehat{\\Phi }_n(f,h) \\le \\beta r$ then $\\Vert f-h\\Vert _{L_2} \\le (\\beta /\\alpha )r$ .",
"Define ${\\cal DO}(f,h)=1$ if $\\widehat{\\Phi }_n(f,h) \\ge \\beta r$ and ${\\cal DO}(f,h)=0$ otherwise.",
"The binary valued functional ${\\cal DO}$ serves as the `referee' of the tournament.",
"Its role is to decide when a match between two functions is allowed to take place.",
"In a more mathematical language, when ${\\cal DO}(f,h)=1$ one has a good reason to expect that $f$ and $h$ are far enough to ensure that $(\\mathbb {B}_{f,h}(j))_{j=1}^k$ reflects the true value $\\mathbb {E}(f(X)-Y)^2 - \\mathbb {E}(h(X)-Y)^2$ .",
"Step 2: $\\bullet $ This round of the tournament consists of statistical matches between class members which are preformed using the second part of the sample $(X_i,Y_i)_{i=n+1}^{2n}$ .",
"A match is allowed to proceed only if ${\\cal DO}(f,h)=1$ ; otherwise, the match is drawn.",
"If a match does take place then $h$ defeats $f$ if $\\mathbb {B}_{f,h}(j)>0$ for a majority of indices $j$ , and $f$ defeats $h$ if the reverse inequality holds for a majority of the blocks.",
"$\\bullet $ A function $f$ qualifies from this round if it has has won or drawn all of its matches.",
"The crucial fact behind Step 2 is that, with high probability, the risk minimizer $f^*$ qualifies for the next round: if ${\\cal DO}(h,f^*)=1$ then $h$ and $f^*$ are far enough to ensure that $(\\mathbb {B}_{h,f^*}(j))_{j=1}^k$ reflects the true value $\\mathbb {E}(h(X)-Y)^2 - \\mathbb {E}(f^*(X)-Y)^2$ .",
"Since $f^*$ is the unique minimizer of the risk, the majority of values are positive.",
"Moreover, the same argument implies that if $h$ is a qualifier from Step 2, then $\\Vert h-f^*\\Vert _{L_2} \\le \\beta r$ .",
"Indeed, the match between $h$ and $f^*$ (or between any two qualifiers) must have been drawn by the referee's decision; thus $h$ must be `close' to $f^*$ .",
"Step 2 is not enough to identify a function with a small excess risk.",
"Indeed, all the qualifiers are close to $f^*$ , but the fact that $\\Vert f-f^*\\Vert _{L_2} \\le \\beta r$ does not imply that $\\mathbb {E}(f(X)-Y)^2-\\mathbb {E}(f^*(X)-Y)^2 \\lesssim r^2$ .",
"Therefore, the tournament has an additional step: the Champions League round, in which all the qualifiers play each other in a different type of match.",
"To find a function that does have an almost optimal risk one uses the third part of the sample $(X_i,Y_i)_{i=2n+1}^{3n}$ to define a `home and away' style matches: Step 3: $\\bullet $ Let $\\Psi _{h,f}=(h(X)-f(X))(f(X)-Y)$ and set $\\Psi _{h,f}(j) = \\frac{1}{m}\\sum _{i \\in B_j} \\Psi _{h,f}(X_i,Y_i)$ .",
"Let $\\alpha ,\\beta $ and $r$ be as above and put $r_1=2(\\beta /\\alpha )r$ .",
"$\\bullet $ $f$ wins its home match against $h$ if $\\Psi _{h,f}(j) \\ge -r_1^2/10$ for a majority of the indices $j$ .",
"$\\bullet $ A winner of the tournament is any qualifier that wins all of its home matches.",
"We set $\\widehat{f}$ to be any such winner.",
"To see the reason behind this choice of matches, recall that all the qualifiers $h$ satisfy that $\\Vert h-f^*\\Vert _{L_2} \\le \\beta r$ .",
"At the same time, the excess risk of $h$ is $\\mathbb {E}(h(X)-Y)^2 - \\mathbb {E}(f^*(X)-Y)^2 = \\Vert h-f^*\\Vert _{L_2}^2 + 2 \\mathbb {E}(h(X)-f^*(X)) \\cdot (f^*(X)-Y)~.$ Since $\\Vert h-f^*\\Vert _{L_2}^2$ is of the order of $r^2$ it is evident that if $\\mathbb {E}(h(X)-f^*(X)) \\cdot (f^*(X)-Y) \\lesssim r^2$ , then the excess risk of $h$ is also of the order of $r^2$ .",
"Observe that $\\mathbb {E}\\Psi _{h,f^*}=\\mathbb {E}(h(X)-f^*(X)) \\cdot (f^*(X)-Y)$ and that by the convexity of ${\\mathcal {F}}$ , $\\mathbb {E}\\Psi _{h,f^*} \\ge 0$ (this follows from the characterization of the nearest point map onto a closed, convex subset of a Hilbert space).",
"Moreover, $ \\mathbb {E}\\Psi _{h,f^*}=-\\Vert h-f^*\\Vert _{L_2}^2 - \\mathbb {E}\\Psi _{f^*,h}~.$ One shows that $\\Psi _{h,f^*}(j) \\gtrsim -r^2$ for a majority of indices $j$ .",
"This follows because the median of $(\\Psi _{h,f^*}(j))_{j=1}^k$ happens to be a uniform median-of-means estimator of the true mean $\\mathbb {E}\\Psi _{h,f^*}$ .",
"As a consequence, $f^*$ wins all of its home matches.",
"Also, if $h$ wins a home match against $f^*$ , (i.e., $\\Psi _{f^*,h}(j) \\gtrsim -r^2$ for a majority of indices $j$ ), then $\\mathbb {E}\\Psi _{f^*,h} \\gtrsim - r^2$ and by (REF ), $\\mathbb {E}\\Psi _{h,f^*} \\lesssim r^2$ .",
"That implies that every function that wins all of its home matches must have a small excess risk.",
"To conclude, all three components of the tournament procedure from [59] are derived using uniform median-of-means estimators (of different functionals) in the class ${\\mathcal {F}}$ .",
"Without going into technical details, at the heart of the analysis of Steps 2 and 3 of the tournament is the following fact: given a convex class ${\\mathcal {F}}$ that satisfies some minimal conditions, for the right choice of $k$ and $r$ (the choice of $r$ depends on the geometry of the class ${\\mathcal {F}}$ and on the parameters $\\gamma _1$ and $\\gamma _2$ appearing below), and for an absolute constant $c_1$ , we have that, with probability $1-2\\exp (-c_1k)$ , (1) for every $f \\in {\\mathcal {F}}$ such that $\\Vert f-f^*\\Vert _{L_2} \\ge r$ , one has $\\mathbb {B}_{f,f^*}(j) \\ge \\gamma _1 \\Vert f-f^*\\Vert _{L_2}^2$ for $0.99k$ of the blocks; (2) for every $f \\in {\\mathcal {F}}$ such that $\\Vert f-f^*\\Vert _{L_2} < r$ , one has $|\\mathbb {M}_{f,f^*}(j) - \\mathbb {E}\\mathbb {M}_{f,f^*}(j)| \\le \\gamma _2 r^2$ for $0.99k$ of the blocks.",
"These facts suffice for proving the validity of steps $(2)$ and $(3)$ in the tournament procedure.",
"A general bound for the performance of the procedure defined above was proven by Lugosi and Mendelson [59].",
"The achievable accuracy depends on the interaction between the geometry of the class ${\\mathcal {F}}$ and the distribution of $(X,Y)$ .",
"Instead of recalling the technical details in their full generality, we simply illustrate the performance on the canonical example of linear regression.",
"Let ${\\mathcal {F}}=\\lbrace \\left\\langle t,\\cdot \\right\\rangle : t \\in \\mathbb {R}^d\\rbrace $ be the class of linear functionals on $\\mathbb {R}^d$ .",
"Let $X$ be an isotropic random vector in $\\mathbb {R}^d$ (i.e., $\\mathbb {E}\\left\\langle t,X \\right\\rangle ^2 = 1$ for every $t$ in the Euclidean unit sphere) and assume that the distribution of $X$ is such that there are $q>2$ and $L>1$ for which, for every $t \\in \\mathbb {R}^d$ , $\\Vert \\left\\langle X,t \\right\\rangle \\Vert _{L_q} \\le L \\Vert \\left\\langle X,t \\right\\rangle \\Vert _{L_2}$ .",
"Assume that one is given $n$ noisy measurements of $\\left\\langle t_0,\\cdot \\right\\rangle $ for a fixed but unknown $t_0 \\in \\mathbb {R}^d$ , that is, assume that $Y=\\left\\langle t_0,X \\right\\rangle +W$ for some symmetric random variable $W$ that is independent of $X$ and has variance $\\sigma ^2$ .",
"One observes the “noisy\" data $(X_i,Y_i)_{i=1}^n$ and the aim is to approximate $t_0$ with a small error (accuracy) and with high probability (confidence).",
"Invoking standard methods as in [50], the best that one can guarantee using empirical risk minimization is a choice of $\\widehat{t} \\in \\mathbb {R}^d$ , for which $\\Vert \\widehat{t}-t_0\\Vert _2\\le \\frac{C }{\\delta } \\sigma \\sqrt{\\frac{d}{n}} \\ \\ \\ {\\rm with \\ probability \\ } 1-\\delta -2\\exp (-c_1 d)$ for some constant $C$ that depends on $q$ and $L$ .",
"Therefore, if one wishes for an error that is proportional to $\\sigma \\sqrt{d/n}$ , the best that one can hope for is a constant confidence $\\delta $ .",
"The median-of-means tournament procedure, when applied to this example, selects $\\widehat{t}$ for which $\\Vert \\widehat{t}-t_0\\Vert _2 \\le C \\sigma \\sqrt{\\frac{d}{n}} \\ \\ \\ {\\rm with \\ probability \\ } 1-2\\exp (-cd)$ for some numerical constants $c,C>0$ .",
"As it is argued in [59] that this is the optimal confidence at any level that is proportional to $\\sqrt{d/n}$ .",
"In fact, the median-of-means tournament procedure gives the optimal confidence for any accuracy $r \\ge c^\\prime \\sigma \\sqrt{d/n}$ .",
"Standard empirical risk minimization can only achieve such accuracy/confidence tradeoff for sub-Gaussian distributions.",
"Acknowledgements.",
"We thank Sam Hopkins, Stanislav Minsker, and Roberto Imbuzeiro Oliveira for illuminating discussions on the subject.",
"We also thank two referees for their thorough reports and insightful comments."
]
] | 1906.04280 |
[
[
"Efficiently escaping saddle points on manifolds"
],
[
"Abstract Smooth, non-convex optimization problems on Riemannian manifolds occur in machine learning as a result of orthonormality, rank or positivity constraints.",
"First- and second-order necessary optimality conditions state that the Riemannian gradient must be zero, and the Riemannian Hessian must be positive semidefinite.",
"Generalizing Jin et al.",
"'s recent work on perturbed gradient descent (PGD) for optimization on linear spaces [How to Escape Saddle Points Efficiently (2017), Stochastic Gradient Descent Escapes Saddle Points Efficiently (2019)], we propose a version of perturbed Riemannian gradient descent (PRGD) to show that necessary optimality conditions can be met approximately with high probability, without evaluating the Hessian.",
"Specifically, for an arbitrary Riemannian manifold $\\mathcal{M}$ of dimension $d$, a sufficiently smooth (possibly non-convex) objective function $f$, and under weak conditions on the retraction chosen to move on the manifold, with high probability, our version of PRGD produces a point with gradient smaller than $\\epsilon$ and Hessian within $\\sqrt{\\epsilon}$ of being positive semidefinite in $O((\\log{d})^4 / \\epsilon^{2})$ gradient queries.",
"This matches the complexity of PGD in the Euclidean case.",
"Crucially, the dependence on dimension is low.",
"This matters for large-scale applications including PCA and low-rank matrix completion, which both admit natural formulations on manifolds.",
"The key technical idea is to generalize PRGD with a distinction between two types of gradient steps: \"steps on the manifold\" and \"perturbed steps in a tangent space of the manifold.\"",
"Ultimately, this distinction makes it possible to extend Jin et al.",
"'s analysis seamlessly."
],
[
"Introduction",
"Machine learning has stimulated interest in obtaining global convergence rates in non-convex optimization.",
"Consider a possibly non-convex objective function $f \\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ .",
"We want to solve $\\min _{x \\in \\mathbb {R}^d} f(x).$ This is hard in general.",
"Instead, we usually settle for approximate first-order critical (or stationary) points where the gradient is small, or second-order critical (or stationary) points where the gradient is small and the Hessian is nearly positive semidefinite.",
"One of the simplest algorithms for solving (REF ) is gradient descent (GD): given $x_0$ , iterate $x_{t+1} = x_t - \\eta \\nabla f(x_t).$ It is well known that if $\\nabla f$ is Lipschitz continuous, with appropriate step-size $\\eta $ , GD converges to first-order critical points.",
"However, it may take exponential time to reach an approximate second-order critical point, thus, to escape saddle points [14].",
"There is an increasing amount of evidence that saddle points are a serious obstacle to the practical success of local optimization algorithms such as GD [25], [16].",
"This calls for algorithms which provably escape saddle points efficiently.",
"We focus on methods which only have access to $f$ and $\\nabla f$ (but not $\\nabla ^2 f$ ) through a black-box model.",
"Several methods add noise to GD iterates in order to escape saddle points faster, under the assumption that $f$ has $L$ -Lipschitz continuous gradient and $\\rho $ -Lipschitz continuous Hessian.",
"In this setting, an $\\epsilon $ -second-order critical point is a point $x$ satisfying $\\left\\Vert {\\nabla f(x)}\\right\\Vert \\le \\epsilon $ and $\\nabla ^2 f(x) \\succeq -\\sqrt{\\rho \\epsilon }I$ .",
"Under the strict saddle assumption, with $\\epsilon $ small enough, such points are near (local) minimizers [16], [17].",
"In 2015, Ge et al.",
"[16] gave a variant of stochastic gradient descent (SGD) which adds isotropic noise to iterates, showing it produces an $\\epsilon $ -second-order critical point with high probability in $O({\\text{poly}(d)}/{\\epsilon ^4})$ stochastic gradient queries.",
"In 2017, Jin et al.",
"[17] presented a variant of GD, perturbed gradient descent (PGD), which reduces this complexity to $O((\\log {d})^4/{\\epsilon }^2)$ full gradient queries.",
"Recently, Jin et al.",
"[18] simplified their own analysis of PGD, and extended it to stochastic gradient descent.",
"Jin et al.",
"'s PGD [18] works as follows: If the gradient is large at iterate $x_t$ , $\\left\\Vert {\\nabla f(x_t)}\\right\\Vert > \\epsilon $ , then perform a gradient descent step: $x_{t+1} = x_t - \\eta \\nabla f(x_t)$ .",
"If the gradient is small at iterate $x_t$ , $\\left\\Vert {\\nabla f(x_t)}\\right\\Vert \\le \\epsilon $ , perturb $x_t$ by $\\eta \\xi $ , with $\\xi $ sampled uniformly from a ball of fixed radius centered at zero.",
"Starting from this new point $x_t + \\eta \\xi $ , perform ${T}$ gradient descent steps, arriving at iterate $x_{t + {T}}$ .",
"From here, repeat this procedure starting at $x_{t + {T}}$ .",
"Crucially, Jin et al.",
"[18] show that, if $x_t$ is not an $\\epsilon $ -second-order critical point, then the function decreases enough from $x_t$ to $x_{t + {T}}$ with high probability, leading to an escape.",
"In this paper we generalize PGD to optimization problems on manifolds, i.e., problems of the form $\\min _{x \\in \\mathcal {M}} f(x)$ where $\\mathcal {M}$ is an arbitrary Riemannian manifold and $f \\colon \\mathcal {M}\\rightarrow \\mathbb {R}$ is sufficiently smooth [3].",
"Optimization on manifolds notably occurs in machine learning (e.g., PCA [35], low-rank matrix completion [12]), computer vision (e.g., [32]) and signal processing (e.g., [2])—see [4] for more.",
"See [29] and [26] for examples of the strict saddle property on manifolds.",
"Given $x\\in \\mathcal {M}$ , the (Riemannian) gradient of $f$ at $x$ , $\\mathrm {grad}\\,f(x)$ , is a vector in the tangent space at $x$ , $x\\mathcal {M}$ .",
"To perform gradient descent on a manifold, we need a way to move on the manifold along the direction of the gradient at $x$ .",
"This is provided by a retraction $\\mathrm {Retr}_x$ : a smooth map from $x\\mathcal {M}$ to $\\mathcal {M}$ .",
"Riemannian gradient descent (RGD) performs steps on $\\mathcal {M}$ of the form $x_{t+1} = \\mathrm {Retr}_{x_t}(-\\eta \\mathrm {grad}\\,f(x_t)).$ For Euclidean space, $\\mathcal {M}= {\\mathbb {R}}^d$ , the standard retraction is $\\mathrm {Retr}_x(s) = x + s$ , in which case (REF ) reduces to (REF ).",
"For the sphere embedded in Euclidean space, $\\mathcal {M}= S^{d} \\subset {\\mathbb {R}}^{d+1}$ , a natural retraction is given by metric projection to the sphere: $\\mathrm {Retr}_x(s) = (x+s)/\\left\\Vert {x+s}\\right\\Vert $ .",
"For $x\\in \\mathcal {M}$ , define the pullback $\\hat{f}_x = f \\circ \\mathrm {Retr}_x \\colon x\\mathcal {M}\\rightarrow {\\mathbb {R}}$ , conveniently defined on a linear space.",
"If $\\mathrm {Retr}$ is nice enough (details below), the Riemannian gradient and Hessian of $f$ at $x$ equal the (classical) gradient and Hessian of $\\hat{f}_x$ at the origin of $x\\mathcal {M}$ .",
"Since $x\\mathcal {M}$ is a vector space, if we perform GD on $\\hat{f}_x$ , we can almost directly apply Jin et al.",
"'s analysis [18].",
"This motivates the two-phase structure of our perturbed Riemannian gradient descent (PRGD), listed as Algorithm .",
"Our PRGD is a variant of RGD (REF ) and a generalization of PGD.",
"It works as follows: If the gradient is large at iterate $x_t \\in \\mathcal {M}$ , $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert > \\epsilon $ , perform an RGD step: $x_{t+1} = \\mathrm {Retr}_{x_t}({-\\eta \\mathrm {grad}\\,f(x_t)})$ .",
"We call this a “step on the manifold.” If the gradient at iterate $x_t$ is small, $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ , then perturb in the tangent space ${x_t}\\mathcal {M}$ .",
"After this perturbation, execute at most ${T}$ gradient descent steps on the pullback $\\hat{f}_{x_t}$ , in the tangent space.",
"We call these “tangent space steps.” We denote this sequence of ${T}$ tangent space steps by $\\lbrace s_j\\rbrace _{j\\ge 0}$ .",
"This sequence of steps is performed by ${TangentSpaceSteps}$ : a deterministic, vector-space procedure—see Algorithm .",
"By distinguishing between gradient descent steps on the manifold and those in a tangent space, we can apply Jin et al.",
"'s analysis almost directly [18], allowing us to prove PRGD reaches an $\\epsilon $ -second-order critical point on $\\mathcal {M}$ in $O((\\log {d})^4 / \\epsilon ^2)$ gradient queries.",
"Regarding regularity of $f$ , we require its pullbacks to satisfy Lipschitz-type conditions, as advocated in [11], [7].",
"The analysis is far less technical than if one runs all steps on the manifold.",
"We expect that this two-phase approach may prove useful for the generalization of other algorithms and analyses from the Euclidean to the Riemannian realm.",
"Recently, Sun and Fazel [30] provided the first generalization of PGD to certain manifolds with a polylogarithmic complexity in the dimension, improving earlier results by Ge et al.",
"[16] which had a polynomial complexity.",
"Both of these works focus on submanifolds of a Euclidean space, with the algorithm in [30] depending on the equality constraints chosen to describe this submanifold.",
"At the same time as the present paper, Sun et al.",
"[31] improved their analysis to cover any complete Riemannian manifold with bounded sectional curvature.",
"In contrast to ours, their algorithm executes all steps on the manifold.",
"Their analysis requires the retraction to be the Riemannian exponential map (i.e., geodesics).",
"Our regularity assumptions are similar but different: while we assume Lipschitz-type conditions on the pullbacks in small balls around the origins of tangent spaces, Sun et al.",
"make Lipschitz assumptions on the cost function directly, using parallel transport and Riemannian distance.",
"As a result, curvature appears in their results.",
"We make no explicit assumptions on $\\mathcal {M}$ regarding curvature or completeness, though these may be implicit in our regularity assumptions: see Section .",
"$\\text{PRGD}(x_0, \\eta , r, {T}, \\epsilon , T, b)$ [1] $t \\leftarrow 0$ $t \\le T$ $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert > \\epsilon $ $x_{t+1} \\leftarrow {TangentSpaceSteps}(x_t, 0, \\eta , b, 1)$ Riemannian gradient descent step $t \\leftarrow t+1$ $\\xi \\sim \\text{Uniform}(B_{x_t, r}(0))$ perturb $s_0 = \\eta \\xi $ $x_{t+{T}} \\leftarrow {TangentSpaceSteps}(x_t, s_0, \\eta , b, {T})$ perform ${T}$ steps in ${x_t}\\mathcal {M}$ $t \\leftarrow t + {T}$ ${TangentSpaceSteps}(x, s_0, \\eta , b, {T})$ $j = 0, 1, \\ldots , {T} - 1$ $s_{j+1} \\leftarrow s_j - \\eta \\nabla \\hat{f}_{x}(s_j)$ $\\left\\Vert {s_{j+1}}\\right\\Vert \\ge b$ if the iterate leaves the interior of the ball $B_{x,b}(0)$ $s_{{T}} \\leftarrow s_j - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)$ , where $\\alpha \\in (0,1]$ and $\\left\\Vert {s_j - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)}\\right\\Vert =b$ .",
"break return $\\mathrm {Retr}_x{(s_{{T}})}$"
],
[
"Main result",
"Here we state our result informally.",
"Formal results are stated in subsequent sections.",
"Theorem 1.1 (Informal) Let $\\mathcal {M}$ be a Riemannian manifold of dimension $d$ equipped with a retraction $\\mathrm {Retr}$ .",
"Assume $f \\colon \\mathcal {M}\\rightarrow \\mathbb {R}$ is twice continuously differentiable, and furthermore: $f$ is lower bounded.",
"The gradients of the pullbacks $f \\circ \\mathrm {Retr}_x$ uniformly satisfy a Lipschitz-type condition.",
"The Hessians of the pullbacks $f \\circ \\mathrm {Retr}_x$ uniformly satisfy a Lipschitz-type condition.",
"The retraction $\\mathrm {Retr}$ uniformly satisfies a second-order condition.",
"Then, setting $T = O((\\log {d})^4/\\epsilon ^2)$ , PRGD visits several points with gradient smaller than $\\epsilon $ and, with high probability, at least two-thirds of those points are $\\epsilon $ -second-order critical (Definition REF ).",
"PRGD uses $O((\\log {d})^4/\\epsilon ^2)$ gradient queries, and crucially no Hessian queries.",
"The algorithm requires knowledge of the Lipschitz constants defined below, which makes this a mostly theoretical algorithm—but see Appendix for explicit constants in the case of PCA."
],
[
"Other related work",
"Algorithms which efficiently escape saddle points can be classified into two families: first-order and second-order methods.",
"First-order methods only use function value and gradient information.",
"SGD and PGD are first-order methods.",
"Second-order methods also access Hessian information.",
"Newton's method, trust regions [24], [11] and adaptive cubic regularization [23], [7], [34] are second-order methods.",
"As noted above, Ge et al.",
"[16] and Jin et al.",
"[17] escape saddle points (in Euclidean space) by exploiting noise in iterations.",
"There has also been similar work for normalized gradient descent [20].",
"Expanding on [17], Jin et al.",
"[19] give an accelerated PGD algorithm (PAGD) which reaches an $\\epsilon $ -second-order critical point of a non-convex function $f$ with high probability in $O({(\\log {d})^6}/{\\epsilon ^{7/4}})$ iterations.",
"In [18], Jin et al.",
"show that a stochastic version of PGD reaches an $\\epsilon $ -second-order critical point in $O(d / \\epsilon ^4)$ stochastic gradient queries; only $O(\\text{poly}(\\log {d}) / \\epsilon ^4)$ queries are needed if the stochastic gradients are well behaved.",
"For an analysis of PGD under convex constraints, see [22].",
"There is another line of research, inspired by Langevin dynamics, in which judiciously scaled Gaussian noise is added at every iteration.",
"We note that although this differs from the first incarnation of PGD in [17], this resembles a simplified version of PGD in [18].",
"Sang and Liu [27] develop an algorithm (adaptive stochastic gradient Langevin dynamics, ASGLD), which provably reaches an $\\epsilon $ -second-order critical point in $O({\\log {d}}/{\\epsilon ^4})$ with high probability.",
"With full gradients, AGSLD reaches an $\\epsilon $ -second-order critical point in $O({\\log {d}}/{\\epsilon ^2})$ queries with high probability.",
"One might hope that the noise inherent in vanilla SGD would help it escape saddle points without noise injection.",
"Daneshmand et al.",
"[13] propose the correlated negative curvature assumption (CNC), under which they prove that SGD reaches an $\\epsilon $ -second-order critical point in $O({\\epsilon ^{-5}})$ queries with high probability.",
"They also show that, under the CNC assumption, a variant of GD (in which iterates are perturbed only by SGD steps) efficiently escapes saddle points.",
"Importantly, these guarantees are completely dimension-free.",
"A first-order method can include approximations of the Hessian (e.g., with a difference of gradients).",
"For example, Allen-Zhu's Natasha 2 algorithm [8] uses first-order information (function value and stochastic gradients) to search for directions of negative curvature of the Hessian.",
"Natasha 2 reaches an $\\epsilon $ -second-order critical point in $O({\\epsilon ^{-13/4}})$ iterations.",
"Many classical optimization algorithms have been generalized to optimization on manifolds, including gradient descent, Newton's method, trust regions and adaptive cubic regularization [15], [3], [1], [6], [11], [7], [9], [34].",
"Bonnabel [10] extends stochastic gradient descent to Riemannian manifolds and proves that Riemannian SGD converges to critical points of the cost function.",
"Zhang et al.",
"[33] and Sato et al.",
"[28] both use variance reduction to speed up SGD on Riemannian manifolds."
],
[
"Preliminaries: Optimization on manifolds",
"We review the key definitions and tools for optimization on manifolds.",
"For more information, see [3].",
"Let $\\mathcal {M}$ be a $d$ -dimensional Riemannian manifold: a real, smooth $d$ -manifold equipped with a Riemannian metric.",
"We associate with each $x \\in \\mathcal {M}$ a $d$ -dimensional real vector space $x\\mathcal {M}$ , called the tangent space at $x$ .",
"For embedded submanifolds of ${\\mathbb {R}}^n$ , we often visualize the tangent space as being tangent to the manifold at $x$ .",
"The Riemannian metric defines an inner product $\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _x$ on the tangent space $x\\mathcal {M}$ , with associated norm $\\left\\Vert {\\cdot }\\right\\Vert _x$ .",
"We denote these by $\\left\\langle {\\cdot },{\\cdot }\\right\\rangle $ and $\\left\\Vert {\\cdot }\\right\\Vert $ when $x$ is clear from context.",
"A vector in the tangent space is a tangent vector.",
"The set of pairs $(x, s_x)$ for $x\\in \\mathcal {M}, s_x\\in x\\mathcal {M}$ is called the tangent bundle $$ .",
"Define $B_{x,r}(s) = \\lbrace \\dot{s}\\in x\\mathcal {M}: \\left\\Vert {\\dot{s}-s}\\right\\Vert _x \\le r\\rbrace $ : the closed ball of radius $r$ centered at $s \\in x\\mathcal {M}$ .",
"We occasionally denote $B_{x,r}(s)$ by $B_r(s)$ when $x$ is clear from context.",
"Let $\\text{Uniform}(B_{x,r}(s))$ denote the uniform distribution over the ball $B_{x,r}(s)$ .",
"The Riemannian gradient $\\mathrm {grad}\\,\\!f(x)$ of a differentiable function $f$ at $x\\in \\mathcal {M}$ is the unique vector in $x\\mathcal {M}$ satisfying $f̥(x)[s] = \\left\\langle {\\mathrm {grad}\\,f(x)},{s}\\right\\rangle _x$ $\\forall s \\in x\\mathcal {M}$ , where $f̥(x)[s]$ is the directional derivative of $f$ at $x$ along $s$ .",
"The Riemannian metric gives rise to a well-defined notion of derivative of vector fields called the Riemannian (or Levi–Civita) connection $\\nabla $ .",
"The Hessian of $f$ is the derivative of the gradient vector field: $\\mathrm {Hess}\\,\\!f(x)[u] = \\nabla _{u}\\mathrm {grad}\\,\\!f (x)$ .",
"The Hessian describes how the gradient changes.",
"$\\mathrm {Hess}\\,\\!f(x)$ is a symmetric linear operator on $x\\mathcal {M}$ .",
"If the manifold is a Euclidean space, $\\mathcal {M}= {\\mathbb {R}}^d$ , with the standard metric $\\left\\langle {x},{y}\\right\\rangle = x^{T}y$ , the Riemannian gradient $\\mathrm {grad}\\,\\!f$ and Hessian $\\mathrm {Hess}\\,\\!f$ coincide with the standard gradient $\\nabla f$ and Hessian $\\nabla ^2 f$ (mind the overloaded notation $\\nabla $ ).",
"As discussed in Section , the retraction is a mapping which allows us to move along the manifold from a point $x$ in the direction of a tangent vector $s \\in x\\mathcal {M}$ .",
"Formally: Definition 2.1 (Retraction, from [3]) A retraction on a manifold $\\mathcal {M}$ is a smooth mapping $\\mathrm {Retr}$ from the tangent bundle $$ to $\\mathcal {M}$ satisfying properties 1 and 2 below.",
"Let $\\mathrm {Retr}_x \\colon x\\mathcal {M}\\rightarrow \\mathcal {M}$ denote the restriction of $\\mathrm {Retr}$ to $x\\mathcal {M}$ .",
"$\\mathrm {Retr}_x(0_x) = x$ , where $0_x$ is the zero vector in $x\\mathcal {M}$ .",
"The differential of $\\mathrm {Retr}_x$ at $0_x$ , $_x(0_x)$ , is the identity map.",
"(Our algorithm and theory only require $\\mathrm {Retr}$ to be defined in balls of a fixed radius around the origins of tangent spaces.)",
"Recall these special retractions, which are good to keep in mind for intuition: on $\\mathcal {M}= {\\mathbb {R}^{d}}$ , we typically use $\\mathrm {Retr}_x(s) = x+s$ , and on the unit sphere we typically use $\\mathrm {Retr}_x(s) = (x+s)/\\left\\Vert {x+s}\\right\\Vert $ .",
"For $x$ in $\\mathcal {M}$ , define the pullback of $f$ from the manifold to the tangent space by $\\hat{f}_x = f \\circ \\mathrm {Retr}_x \\colon x\\mathcal {M}\\rightarrow {\\mathbb {R}}.$ This is a real function on a vector space.",
"Furthermore, for $x \\in \\mathcal {M}$ and $s \\in x\\mathcal {M}$ , let $T_{x,s} = _x(s) \\colon x\\mathcal {M}\\rightarrow {\\mathrm {Retr}_x(s)}\\mathcal {M}$ denote the differential of $\\mathrm {Retr}_x$ at $s$ (a linear operator).",
"The gradient and Hessian of the pullback admit the following nice expressions in terms of those of $f$ , and the retraction.",
"Lemma 2.2 (Lemma 5.2 of [7]) For $f\\colon \\mathcal {M}\\rightarrow {\\mathbb {R}}$ twice continuously differentiable, $x\\in \\mathcal {M}$ and $s\\in x\\mathcal {M}$ , with $T_{x,s}^*$ denoting the adjoint of $T_{x,s}$ , $\\nabla \\hat{f}_x(s) & = T_{x,s}^*\\mathrm {grad}\\,f(\\mathrm {Retr}_x(s)), &\\nabla ^2 \\hat{f}_x(s) & = T_{x,s}^* \\mathrm {Hess}\\,f(\\mathrm {Retr}_x(s)) T_{x,s} + W_s,$ where $W_s$ is a symmetric linear operator on $x\\mathcal {M}$ defined through polarization by $\\left\\langle {W_s[\\dot{s}]},{\\dot{s}}\\right\\rangle = \\left\\langle {\\mathrm {grad}\\,f(\\mathrm {Retr}_x(s))},{\\gamma ^{\\prime \\prime }(0)}\\right\\rangle ,$ with $\\gamma ^{\\prime \\prime }(0) \\in {\\mathrm {Retr}_x(s)}\\mathcal {M}$ the intrinsic acceleration on $\\mathcal {M}$ of $\\gamma (t) = \\mathrm {Retr}_x(s + t\\dot{s})$ at $t=0$ .",
"The velocity of a curve $\\gamma \\colon {\\mathbb {R}}\\rightarrow \\mathcal {M}$ is $\\frac{d \\gamma }{d t} = \\gamma ^{\\prime }(t)$ .",
"The intrinsic acceleration $\\gamma ^{\\prime \\prime }$ of $\\gamma $ is the covariant derivative (induced by the Levi–Civita connection) of the velocity of $\\gamma $ : $\\gamma ^{\\prime \\prime } = \\frac{{d t} \\gamma ^{\\prime }.",
"When \\mathcal {M} is a Riemannian submanifold of {\\mathbb {R}}^n, \\gamma ^{\\prime \\prime }(t) does not necessarily coincide with \\frac{d^2 \\gamma }{d t^2}: in this case, \\gamma ^{\\prime \\prime }(t) is the orthogonal projection of \\frac{d^2 \\gamma }{d t^2} onto {\\gamma (t)} \\mathcal {M}.",
"}{}$"
],
[
"PRGD efficiently escapes saddle points",
"We now precisely state the assumptions, the main result, and some important parts of the proof of the main result, including the main obstacles faced in generalizing PGD to manifolds.",
"A full proof of all results is provided in the appendix."
],
[
"Assumptions",
"The first assumption, namely, that $f$ is lower bounded, ensures that there are points on the manifold where the gradient is arbitrarily small.",
"Assumption 1 $f$ is lower bounded: $f(x) \\ge f^*$ for all $x \\in \\mathcal {M}$ .",
"Generalizing from the Euclidean case, we assume Lipschitz-type conditions on the gradients and Hessians of the pullbacks $\\hat{f}_x = f \\circ \\mathrm {Retr}_x$ .",
"For the special case of $\\mathcal {M}= \\mathbb {R}^d$ and $\\mathrm {Retr}_x(s) = x + s$ , these assumptions hold if the gradient $\\nabla f(\\cdot )$ and Hessian $\\nabla ^2 f(\\cdot )$ are each Lipschitz continuous, as in [18] (with the same constants).",
"The Lipschitz-type assumptions below are similar to assumption A2 of [7].",
"Notice that these assumptions involve both the cost function and the retraction: this dependency is further discussed in [11], [7] for a similar setting.",
"Assumption 2 There exist $b_1 > 0$ and $L > 0$ such that $\\forall x \\in \\mathcal {M}$ and $\\forall s \\in x \\mathcal {M}$ with $\\left\\Vert {s}\\right\\Vert \\le b_1$ , $\\left\\Vert {\\nabla \\hat{f}_x(s) - \\nabla \\hat{f}_x(0)}\\right\\Vert \\le L\\left\\Vert {s}\\right\\Vert .$ Assumption 3 There exist $b_2 > 0$ and $\\rho > 0$ such that $\\forall x \\in \\mathcal {M}$ and $\\forall s \\in x \\mathcal {M}$ with $\\left\\Vert {s}\\right\\Vert \\le b_2$ , $\\left\\Vert {\\nabla ^2\\hat{f}_x(s) - \\nabla ^2\\hat{f}_x(0)}\\right\\Vert \\le \\rho \\left\\Vert {s}\\right\\Vert ,$ where on the left-hand side we use the operator norm.",
"More precisely, we only need these assumptions to hold at the iterates $x_0, x_1, \\ldots $ Let $b = \\min \\lbrace b_1, b_2\\rbrace $ (to reduce the number of parameters in Algorithm ).",
"The next assumption requires the chosen retraction to be well behaved, in the sense that the (intrinsic) acceleration of curves $\\gamma _{x,s}$ on the manifold, defined below, must remain bounded—compare with Lemma REF .",
"Assumption 4 There exists $\\beta \\ge 0$ such that, for all $x \\in \\mathcal {M}$ and $s \\in x\\mathcal {M}$ satisfying $\\left\\Vert {s}\\right\\Vert =1$ , the curve $\\gamma _{x,s}(t) = \\mathrm {Retr}_x(t s)$ has initial acceleration bounded by $\\beta $ : $\\left\\Vert {\\gamma _{x,s}^{\\prime \\prime }(0)}\\right\\Vert \\le \\beta $ .",
"If Assumption REF holds with $\\beta =0$ , $\\mathrm {Retr}$ is said to be second order [3].",
"Second-order retractions include the so-called exponential map and the standard retractions on ${\\mathbb {R}}^d$ and the unit sphere mentioned earlier—see [5] for a large class of such retractions on relevant manifolds.",
"Definition 3.1 A point $x \\in \\mathcal {M}$ is an $\\epsilon $ -second-order critical point of the twice-differentiable function $f \\colon \\mathcal {M}\\rightarrow \\mathbb {R}$ satisfying Assumption REF if $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert & \\le \\epsilon , & & \\text{ and } & \\lambda _{\\min }(\\mathrm {Hess}\\,f(x)) & \\ge -\\sqrt{\\rho \\epsilon },$ where $\\lambda _{\\min }(H)$ denotes the smallest eigenvalue of the symmetric operator $H$ .",
"For compact manifolds, all of these assumptions hold (all proofs are in the appendix): Lemma 3.2 Let $\\mathcal {M}$ be a compact Riemannian manifold equipped with a retraction $\\mathrm {Retr}$ .",
"Assume $f\\colon \\mathcal {M}\\rightarrow {\\mathbb {R}}$ is three times continuously differentiable.",
"Pick an arbitrary $b > 0$ .",
"Then, there exist $f^*, L > 0, \\rho >0$ and $\\beta \\ge 0$ such that Assumptions REF , REF , REF and REF are satisfied."
],
[
"Main results",
"Recall that PRGD (Algorithm ) works as follows.",
"If $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert > \\epsilon $ , perform a Riemannian gradient descent step, $x_{t+1} = \\mathrm {Retr}_{x_t}({-\\eta \\mathrm {grad}\\,f(x_t)})$ .",
"If $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ , then perturb, i.e., sample $\\xi \\sim \\text{Uniform}(B_{x_t, r}(0))$ and let $s_0 = \\eta \\xi $ .",
"After this perturbation, remain in the tangent space ${x_t}\\mathcal {M}$ and do (at most) ${T}$ gradient descent steps on the pullback $\\hat{f}_{x_t}$ , starting from $s_0$ .",
"We denote this sequence of ${T}$ tangent space steps by $\\lbrace s_j\\rbrace _{j\\ge 0}$ .",
"This sequence of gradient descent steps is performed by ${TangentSpaceSteps}$ : a deterministic procedure in the (linear) tangent space.",
"One difficulty with this approach is that, under our assumptions, for some $x = x_t$ , $\\nabla \\hat{f}_{x}$ may not be Lipschitz continuous in all of $x\\mathcal {M}$ .",
"However, it is easy to show that $\\nabla \\hat{f}_{x}$ is Lipschitz continuous in the ball of radius $b$ by compactness, uniformly in $x$ .",
"This is why we limit our algorithm to these balls.",
"If the sequence of iterates $\\lbrace s_j\\rbrace _{j\\ge 0}$ escapes the ball $B_{x,b}(0) \\subset {x}\\mathcal {M}$ for some $s_j$ , ${TangentSpaceSteps}$ returns the point between $s_{j-1}$ and $s_j$ on the boundary of that ball.",
"Following [18], we use a set of carefully balanced parameters.",
"Parameters $\\epsilon $ and $\\delta $ are user defined.",
"The claim in Theorem REF below holds with probability at least $1-\\delta $ .",
"Assumption REF provides parameter $f^*$ .",
"Assumptions REF and REF provide parameters $L, \\rho $ and $b = \\min \\lbrace b_1, b_2\\rbrace $ .",
"As announced, the latter two assumptions further ensure Lipschitz continuity of the gradients of the pullbacks in balls of the tangent spaces, uniformly: this defines the parameter $\\ell $ , as prescribed below.",
"Lemma 3.3 Under Assumptions REF and REF , there exists $\\ell \\in [L, L+\\rho b]$ such that, for all $x \\in \\mathcal {M}$ , the gradient of the pullback, $\\nabla \\hat{f}_x$ , is $\\ell $ -Lipschitz continuous in the ball $B_{x, b}(0)$ .",
"Then, choose $\\chi > 1/4$ (preferably small) such that $\\chi & \\ge 4 \\log _2\\left(2^{31}\\frac{\\ell ^2\\sqrt{d}(f(x_0)-f^*)}{\\delta \\sqrt{\\rho }\\epsilon ^{5/2}}\\right), $ and set algorithm parameters $\\eta & = \\frac{1}{\\ell }, &r & = \\frac{\\epsilon }{400 \\chi ^3}, &{T} & = \\frac{\\ell \\chi }{\\sqrt{\\rho \\epsilon }}, &$ where $\\chi $ is such that ${T}$ is an integer.",
"We also use this notation in the proofs: $ {F} & = \\frac{1}{50 \\chi ^3}\\sqrt{\\frac{\\epsilon ^3}{\\rho }}, & {L} & = \\frac{1}{4\\chi }\\sqrt{\\frac{\\epsilon }{\\rho }}.$ Theorem 3.4 Assume $f$ satisfies Assumptions REF , REF and REF .",
"For any $x_0 \\in \\mathcal {M}$ , with $0 < \\epsilon \\le b^2\\rho $ , $L \\ge \\sqrt{\\rho \\epsilon }$ , $\\epsilon ^{3/2} \\le 3\\sqrt{\\rho } \\left( f(x_0) - f^* \\right)$ and $ \\delta \\in (0,1)$ , choose $\\eta , r, {T}$ as in (REF ).",
"Then, setting $T = 8\\max \\left\\lbrace \\frac{{T}}{3}, \\frac{(f(x_0)-f^*){T}}{{F}}, \\frac{f(x_0)-f^*}{\\eta \\epsilon ^2}\\right\\rbrace = O\\bigg (\\frac{\\ell (f(x_0) - f^*)}{\\epsilon ^2} (\\log {d})^4\\bigg ),$ $PRGD(x_0, \\eta , r, {T}, \\epsilon , T, b)$ visits at least two iterates $x_t \\in \\mathcal {M}$ satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ .",
"With probability at least $1-\\delta $ , at least two-thirds of those iterates satisfy $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert & \\le \\epsilon & & \\text{ and } & \\lambda _{\\min }(\\nabla ^2 \\hat{f}_{x_t}(0)) & \\ge -\\sqrt{\\rho \\epsilon }.$ The algorithm uses at most $T + {T} \\le 2 T$ gradient queries (and no function or Hessian queries).",
"By Assumption REF and Lemma REF , $\\nabla ^2 \\hat{f}_{x_t}(0)$ is close to $\\mathrm {Hess}\\,f(x_t)$ , which allows us to conclude: Corollary 3.5 Assume $f$ satisfies Assumptions REF , REF , REF and REF .",
"For an arbitrary $x_0 \\in \\mathcal {M}$ , with $0 < \\epsilon \\le \\min \\lbrace \\rho / \\beta ^2,b^2\\rho \\rbrace $ , $L \\ge \\sqrt{\\rho \\epsilon }$ , $\\epsilon ^{3/2} \\le 3\\sqrt{\\rho } \\left( f(x_0) - f^* \\right)$ and $\\delta \\in (0,1)$ , choose $\\eta , r, {T}$ as in (REF ).",
"Then, setting $T$ as in (REF ), $PRGD(x_0, \\eta , r, {T}, \\epsilon , T, b)$ visits at least two iterates $x_t \\in \\mathcal {M}$ satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ .",
"With probability at least $1-\\delta $ , at least two-thirds of those iterates are $(4\\epsilon )$ -second-order points.",
"If $\\beta = 0$ (that is, the retraction is second order), then the same claim holds for $\\epsilon $ -second-order points instead of $4\\epsilon $ .",
"The algorithm uses at most $T + {T} \\le 2 T$ gradient queries.",
"Assume $\\mathcal {M}= {\\mathbb {R}}^d$ with standard inner product and standard retraction $\\mathrm {Retr}_x(s) = x+s$ .",
"As in [18], assume $f\\colon {\\mathbb {R}}^d \\rightarrow {\\mathbb {R}}$ is lower bounded, $\\nabla f$ is $L$ -Lipschitz in ${\\mathbb {R}}^d$ , and $\\nabla ^2 f$ is $\\rho $ -Lipschitz in ${\\mathbb {R}}^d$ .",
"Then, Assumptions REF , REF and REF hold with $b = +\\infty $ .",
"Furthermore, Assumption REF holds with $\\beta = 0$ so that $\\nabla ^2\\hat{f}_x(0) = \\mathrm {Hess}\\,f(x) = \\nabla ^2 f(x)$ (Lemma REF ).",
"For all $x\\in \\mathcal {M}$ , $\\nabla \\hat{f}_x(s)$ has Lipschitz constant $\\ell = L$ since $\\hat{f}_x(s) = f(x+s)$ .",
"Therefore, using $b = +\\infty $ , $\\ell = L$ and choosing $\\eta , r, {T}$ as in (REF ), PRGD reduces to PGD, and Theorem REF recovers the result of Jin et al.",
"[18]: this confirms that the present result is a bona fide generalization.",
"For the important special case of compact manifolds, Lemmas REF and REF yield: Corollary 3.6 Assume $\\mathcal {M}$ is a compact Riemannian manifold equipped with a retraction $\\mathrm {Retr}$ , and $f\\colon \\mathcal {M}\\rightarrow {\\mathbb {R}}$ is three times continuously differentiable.",
"Pick an arbitrary $b>0$ .",
"Then, Assumptions REF , REF , REF , REF hold for some $L>0, \\rho >0$ , $\\beta \\ge 0$ , so that Corollary REF applies with some $\\ell \\in [L, L+\\rho b]$ .",
"Remark 3.7 PRGD, like PGD (Algorithm 4 in [18]), does not specify which iterate is an $\\epsilon $ -second-order critical point.",
"However, it is straightforward to include a termination condition in PRGD which halts the algorithm and returns a suspected $\\epsilon $ -second-order critical point.",
"Indeed, Jin et al.",
"include such a termination condition in their original PGD algorithm [17], which here would go as follows: After performing a perturbation and ${T}$ (tangent space) steps in ${x_t}\\mathcal {M}$ , return $x_t$ if $\\hat{f}_{x_t}(s_{{T}}) - \\hat{f}_{x_t}(0) > -f_{\\mathrm {thres}}$ , i.e., the function value does not decrease enough.",
"The termination condition requires a threshold $f_{\\mathrm {thres}}$ which is balanced like the other parameters of PRGD in (REF )."
],
[
"Main proof ideas",
"Theorem REF follows from the following two lemmas which we prove in the appendix.",
"These lemmas state that, in each round of the while-loop in PRGD, if $x_t$ is not at an $\\epsilon $ -second-order critical point, PRGD makes progress, that is, decreases the cost function value (the first lemma is deterministic, the second one is probabilistic).",
"Yet, the value of $f$ on the iterates can only decrease so much because $f$ is bounded below by $f^*$ .",
"Therefore, the probability that PRGD does not visit an $\\epsilon $ -second-order critical point is low.",
"Lemma 3.8 Under Assumptions REF and REF , set $\\eta = 1/\\ell $ for some $\\ell \\ge L$ .",
"If $x\\in \\mathcal {M}$ satisfies $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert > \\epsilon $ with $\\epsilon \\le b^2 \\rho $ and $L \\ge \\sqrt{\\rho \\epsilon }$ , then, $f({TangentSpaceSteps}(x, 0, \\eta , b, 1)) - f(x) \\le -\\eta \\epsilon ^2/2.$ Lemma 3.9 Under Assumptions REF and REF , let $x\\in \\mathcal {M}$ satisfy both $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert \\le \\epsilon $ and $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_x(0)) \\le -\\sqrt{\\rho \\epsilon }$ with $\\epsilon \\le b^2\\rho $ and $L \\ge \\sqrt{\\rho \\epsilon }$ .",
"Set $\\eta , r, {T}, {F}$ as in (REF ) and (REF ).",
"Let $s_0 = \\eta \\xi $ with $\\xi \\sim \\text{Uniform}(B_{x,r}(0))$ .",
"Then, $\\mathbb {P}\\big [f({TangentSpaceSteps}(x, s_0, \\eta , b, {T})) - f(x) \\le -{F}/2\\big ] \\ge 1 - \\frac{\\ell \\sqrt{d}}{\\sqrt{\\rho \\epsilon }}2^{10-\\chi /2}.$ Lemma REF states that we are guaranteed to make progress if the gradient is large.",
"This follows from the sufficient decrease of RGD steps.",
"Lemma REF states that, with perturbation, GD on the pullback escapes a saddle point with high probability.",
"Lemma REF is analogous to Lemma 11 in [18].",
"Let $\\mathcal {X}_{\\mathrm {stuck}}$ be the set of tangent vectors $s_0$ in $B_{x,\\eta r}(0)$ for which GD on the pullback starting from $s_0$ does not escape the saddle point, i.e., the function value does not decrease enough after ${T}$ iterations.",
"Following Jin et al.",
"'s analysis [18], we bound the width of this “stuck region” (in the direction of the eigenvector $e_1$ associated with the minimum eigenvalue of the Hessian of the pullback, $\\nabla ^2\\hat{f}_x(0)$ ).",
"Like Jin et al., we do this with a coupling argument, showing that given two GD sequences with starting points sufficiently far apart, one of these sequences must escape.",
"This is formalized in Lemma REF of the appendix.",
"A crucial observation to prove Lemma REF is that, if the function value of GD iterates does not decrease much, then these iterates must be localized; this is formalized in Lemma REF of the appendix, which Jin et al.",
"call “improve or localize.” We stress that the stuck region concept, coupling argument, improve or local paradigm, and details of the analysis are due to Jin et al.",
"[18]: our main contribution is to show a clean way to generalize the algorithm to manifolds in such a way that the analysis extends with little friction.",
"We believe that the general idea of separating iterations between the manifold and the tangent spaces to achieve different objectives may prove useful to generalize other algorithms as well."
],
[
"About the role of curvature of the manifold",
"As pointed out in the introduction, concurrently with our work, Sun et al.",
"[31] have proposed another generalization of PGD to manifolds.",
"Their algorithm executes all steps on the manifold directly (as opposed to our own, which makes certain steps in the tangent spaces), and moves around the manifold using the exponential map.",
"To carry out their analysis, Sun et al.",
"assume $f$ is regular in the following way.",
"The Riemannian gradient is Lipschitz continuous in a Riemannian sense, namely, $\\forall x, y \\in \\mathcal {M}, && \\Vert \\mathrm {grad}\\,f(y) - \\Gamma _x^y \\mathrm {grad}\\,f(x)\\Vert & \\le L \\mathrm {dist}(x, y),$ where $\\Gamma _x^y \\colon x\\mathcal {M}\\rightarrow y\\mathcal {M}$ denotes parallel transport from $x$ to $y$ along any minimizing geodesic, and $\\mathrm {dist}$ is the Riemannian distance.",
"These notions are well defined if $\\mathcal {M}$ is a connected, complete manifold.",
"Similarly, they assume the Riemannian Hessian of $f$ is Lipschitz continuous in a Riemannian sense: $\\forall x, y \\in \\mathcal {M}, && \\Vert \\mathrm {Hess}\\,f(y) - \\Gamma _x^y \\circ \\mathrm {Hess}\\,f(x) \\circ \\Gamma _y^x \\Vert & \\le \\rho \\mathrm {dist}(x, y),$ in the operator norm.",
"Using (and improving) sophisticated inequalities from Riemannian geometry, they map the perturbed sequences back to tangent spaces for analysis, where they run an adapted version of Jin et al.",
"'s argument.",
"In so doing, it appears to be crucial to use the exponential map, owing to its favorable interplay with parallel transport along geodesics and Riemannian distance, providing a good fit with the regularity conditions above.",
"As they map sequences back from the manifold to a common tangent space through the inverse of the exponential map, the Riemannian curvature of the manifold comes into play.",
"Consequently, they assume $\\mathcal {M}$ has bounded sectional curvature (both from below and from above), and these bounds on curvature come up in their final complexity result: constants degrade if the manifold is more curved.",
"Since Riemannian curvature does not occur in our own complexity result for PRGD, it is legitimate to ask: is curvature supposed to occur?",
"If so, it must be hidden in our analysis, for example in the regularity assumptions we make, which are expressed in terms of pullbacks rather than with parallel transports.",
"And indeed, in several attempts to deduce our own assumptions from those of Sun et al., invariably, we had to degrade $L$ and $\\rho $ as a function of curvature—minding that these are only bounds.",
"On the other hand, under the assumptions of Sun et al., one can deduce that the regularity assumptions required in [11], [7] for the analysis of Riemannian gradient descent, trust regions and adaptive regularization by cubics hold with the exponential map, leading to curvature-free complexity bounds for all three algorithms.",
"Thus, it is not clear that curvature should occur.",
"We believe this poses an interesting question regarding the complexity of optimization on manifolds: to what extent should it be influenced by curvature of the manifold?",
"We intend to study this."
],
[
"Perspectives",
"To perform PGD (Algorithm 4 of [18]), one must know the step size $\\eta $ , perturbation radius $r$ and the number of steps ${T}$ to perform after perturbation.",
"These parameters are carefully balanced, and their values depend on the smoothness parameters $L$ and $\\rho $ .",
"In most situations, we do not know $L$ or $\\rho $ (though see Appendix for PCA).",
"An algorithm which does not require knowledge of $L$ or $\\rho $ but still has the same guarantees as PGD would be useful.",
"However, that certain regularity parameters must be known seems inevitable, in particular for the Hessian's $\\rho $ .",
"Indeed, the main theorems make statements about the spectrum of the Hessian, yet the algorithm is not allowed to query the Hessian.",
"GD equipped with a backtracking line-search method achieves an $\\epsilon $ -first-order critical point in $O(\\epsilon ^{-2})$ gradient queries without knowledge of the Lipschitz constant $L$ .",
"At each iterate $x_t$ of GD, backtracking line-search essentially uses function and gradient queries to estimate the gradient Lipschitz parameter near $x_t$ .",
"Perhaps PGD can perform some kind of line-search to locally estimate $L$ and $\\rho $ .",
"We note that if $\\rho $ is known and we use line-search-type methods to estimate $L$ , there are still difficulties applying Jin et al.",
"'s coupling argument.",
"Jin et al.",
"[18] develop a stochastic version of PGD known as PSGD.",
"Instead of perturbing when the gradient is small and performing ${T}$ GD steps, PSGD simply performs a stochastic gradient step and perturbation at each step.",
"Distinguishing between manifold steps and tangent space steps, we suspect it is possible to develop a Riemannian version of perturbed stochastic gradient descent which achieves an $\\epsilon $ -second-order critical point in $O(d / \\epsilon ^4)$ stochastic gradient queries, like PSGD.",
"This Riemannian version performs a certain number of steps in the tangent space, like PRGD.",
"More broadly, we anticipate that it should be possible to extend several classical optimization methods from the Euclidean case to the Riemannian case through this approach of running many steps in a given tangent space before retracting.",
"This ought to be particularly beneficial for algorithms whose computations or analysis rely intimately on linear structures, such as for coordinate descent algorithms, certain parallelized schemes, and possibly also accelerated schemes.",
"In preparing the final version of this paper, we found that this idea is also the subject of another paper at NeurIPS 2019, where it is called dynamic trivialization [21]."
],
[
"Acknowledgments",
"We thank Yue Sun, Nicolas Flammarion and Maryam Fazel, authors of [31], for numerous relevant discussions.",
"NB is partially supported by NSF grant DMS-1719558."
],
[
"Proof that assumptions hold for compact manifolds",
"[Proof of Lemma REF ] Since $\\mathcal {M}$ is compact and $f$ is continuous, $f$ is lower bounded by some $f^*$ .",
"Recall $\\hat{f}_x(s) = f \\circ \\mathrm {Retr}_x(s)$ .",
"Define $\\phi ,\\psi \\colon \\rightarrow {\\mathbb {R}}$ using operator norms by $\\phi (x,s) = \\left\\Vert {\\nabla ^2 \\hat{f}_x(s)}\\right\\Vert = \\left\\Vert {\\nabla _s^2 (f \\circ \\mathrm {Retr}(x,s))}\\right\\Vert ,$ $\\psi (x,s) = \\left\\Vert {\\nabla ^3 \\hat{f}_x(s)}\\right\\Vert = \\left\\Vert {\\nabla _s^3 (f \\circ \\mathrm {Retr}(x,s))}\\right\\Vert .$ Since $f$ is three times continuously differentiable and $\\mathrm {Retr}$ is smooth, $\\phi $ and $\\psi $ are each continuous on the tangent bundle $$ .",
"The set $S_b & = \\left\\lbrace (x,s) : x \\in \\mathcal {M},s\\in x\\mathcal {M}\\text{ with } \\left\\Vert {s}\\right\\Vert \\le b \\right\\rbrace $ is a compact subset of the tangent bundle $$ since $\\mathcal {M}$ is compact.",
"Thus, we may define $L & = \\max _{(x, s) \\in S_b}\\phi (x, s), & & \\textrm { and } & \\rho & = \\max _{(x, s) \\in S_b}\\psi (x, s),$ so that $\\left\\Vert {\\nabla ^2 \\hat{f}_x(s)}\\right\\Vert \\le L$ and $\\left\\Vert {\\nabla ^3 \\hat{f}_x(s)}\\right\\Vert \\le \\rho $ for all $x\\in \\mathcal {M}$ and $s\\in B_{x,b}(0)$ .",
"From here, it is clear that Assumptions REF and REF are satisfied, for we can just integrate as in eq.",
"(REF ) below.",
"Using the notation from Assumption REF , the map $\\upsilon \\colon \\rightarrow {\\mathbb {R}}$ given by $\\upsilon (x,s) = \\left\\Vert {\\gamma _{x,s}^{\\prime \\prime }(0)}\\right\\Vert $ is continuous since $\\mathrm {Retr}$ is smooth.",
"The set $V_b & = \\left\\lbrace (x,s) : x \\in \\mathcal {M}, s\\in x\\mathcal {M}\\text{ with } \\left\\Vert {s}\\right\\Vert = 1 \\right\\rbrace $ is also compact in $$ .",
"Hence, $\\beta = \\max _{(x, s) \\in V_b}\\upsilon (x, s)$ is a valid choice."
],
[
"Proofs for the main results",
"The proof follows that of Jin et al.",
"[18] closely, reusing many of their key lemmas: we repeat some here for convenience, while highlighting the specificities of the manifold case.",
"We consider it a contribution of this paper that, as a result of our distinction between manifold and tangent space steps, there is limited extra friction, despite the significantly extended generality.",
"In this section and the next, all parameters are chosen as in (REF ) and (REF ).",
"We assume $\\epsilon \\le b^2 \\rho $ .",
"We also assume $L \\ge \\sqrt{\\rho \\epsilon }$ because otherwise we can reach a point satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert \\le \\epsilon $ and $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_x(0)) \\ge - \\sqrt{\\rho \\epsilon }$ simply using RGD.",
"Indeed, RGD always finds a point $x\\in \\mathcal {M}$ satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert \\le \\epsilon $ , and Assumption REF implies $\\Vert \\nabla ^2\\hat{f}_x(0)\\Vert \\le L$ so that $\\lambda _{\\min }(\\nabla ^2\\hat{f}_x(0)) \\ge -L$ .",
"Thus, if $L < \\sqrt{\\rho \\epsilon }$ , every point $x\\in \\mathcal {M}$ satisfies $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_x(0)) \\ge - \\sqrt{\\rho \\epsilon }$ .",
"We want to prove Theorem REF .",
"This theorem follows from the following two lemmas (repeated from Lemmas REF and REF for convenience), which we prove in Appendix below.",
"Lemma REF is deterministic: it is a statement about the cost decrease produced by a single Riemannian gradient step, with bounded step size.",
"Lemma REF is probabilistic, and is analogous to Lemma 11 in [18].",
"Lemma 7.1 Under Assumptions REF and REF , set $\\eta = 1/\\ell $ for some $\\ell \\ge L$ .",
"If $x\\in \\mathcal {M}$ satisfies $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert > \\epsilon $ with $\\epsilon \\le b^2 \\rho $ and $L \\ge \\sqrt{\\rho \\epsilon }$ , then, $f({TangentSpaceSteps}(x, 0, \\eta , b, 1)) - f(x) \\le -\\eta \\epsilon ^2/2.$ Lemma 7.2 Under Assumptions REF and REF , let $x\\in \\mathcal {M}$ satisfy both $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert \\le \\epsilon $ and $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_x(0)) \\le -\\sqrt{\\rho \\epsilon }$ with $\\epsilon \\le b^2\\rho $ and $L \\ge \\sqrt{\\rho \\epsilon }$ .",
"Set $\\eta , r, {T}, {F}$ as in (REF ) and (REF ).",
"Let $s_0 = \\eta \\xi $ with $\\xi \\sim \\text{Uniform}(B_{x,r}(0))$ .",
"Then, $\\mathbb {P}\\big [f({TangentSpaceSteps}(x, s_0, \\eta , b, {T})) - f(x) \\le -{F}/2\\big ] \\ge 1 - \\frac{\\ell \\sqrt{d}}{\\sqrt{\\rho \\epsilon }}2^{10-\\chi /2}.$ [Proof of Theorem REF ] This proof is similar to Jin et al.",
"'s proof of Theorem 9 in [18].",
"Recall that we set $T = 8\\max \\left\\lbrace \\frac{{T}}{3}, \\frac{(f(x_0)-f^*){T}}{{F}}, \\frac{f(x_0)-f^*}{\\eta \\epsilon ^2}\\right\\rbrace .$ PRGD performs two types of steps: (1) if $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert > \\epsilon $ , an RGD step on the manifold, and (2) if $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ , a perturbation in the tangent space followed by GD steps in the tangent space.",
"There are at most $T/4$ iterates $x_t\\in \\mathcal {M}$ satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert > \\epsilon $ (i.e., iterates where an RGD step is performed), for otherwise Lemma REF and the definition of $T$ (REF ) would imply $f(x_T) < f(x_0) - T\\eta \\epsilon ^2/8 \\le f^*$ , which contradicts Assumption REF .",
"The variable $t$ in Algorithm is an upper bound on the number of gradient queries issued so far.",
"For each RGD step on the manifold, $t$ increases by exactly 1.",
"PRGD does not terminate before $t$ exceeds $T$ , and for every perturbation the counter increases by exactly ${T}$ .",
"Therefore, there are at least $3 T/(4{T})$ iterates $x_t\\in \\mathcal {M}$ satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ .",
"By the definition of $T$ (REF ), $3 T/(4{T}) \\ge 2$ .",
"Suppose PRGD visits more than $T/(4 {T})$ points $x_t\\in \\mathcal {M}$ satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ and $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_{x_t}(0)) \\le -\\sqrt{\\rho \\epsilon }$ .",
"Each of these iterates $x_t$ is followed by a perturbation and at most ${T}$ tangent space steps $\\lbrace s_j\\rbrace $ .",
"For at least one such $x_t$ , the sequence of tangent space steps does not escape the saddle point (that is, $f(x_{t+{T}}) - f(x_t)> -{F}/2$ ), for otherwise $f(x_T) < f(x_0) - T{F}/(8{T}) \\le f^*$ by the definition of $T$ (REF ).",
"Yet, by Lemma REF and a union bound, the probability that one or more of these sequences does not escape is at most $\\delta $ .",
"Indeed, factoring out the third term in the max, $T & = \\frac{8\\ell (f(x_0)-f^*)}{\\epsilon ^2} \\max \\bigg \\lbrace \\frac{1}{3}\\frac{\\chi }{\\sqrt{\\rho \\epsilon }}\\frac{\\epsilon ^2}{(f(x_0)-f^*)}, 50\\chi ^4, 1\\bigg \\rbrace \\\\& \\le \\frac{8\\ell (f(x_0)-f^*)}{\\epsilon ^2}\\max \\bigg \\lbrace \\chi , 50\\chi ^4, 1\\bigg \\rbrace = O \\left(\\frac{\\ell (f(x_0) - f^*)}{\\epsilon ^2}\\chi ^4\\right),$ where we used $\\epsilon ^{3/2} \\le 3\\sqrt{\\rho } \\left( f(x_0) - f^* \\right)$ .",
"Now using $\\max \\left\\lbrace \\chi , 50\\chi ^4, 1\\right\\rbrace \\le 2^{18 + \\chi /4}$ for all $\\chi > 1/4$ , and $\\chi \\ge 4 \\log _2\\left(2^{31} \\frac{\\ell ^2\\sqrt{d}(f(x_0)-f^*)}{\\delta \\sqrt{\\rho }\\epsilon ^{5/2}}\\right)$ , we find $T \\cdot \\frac{\\ell \\sqrt{d}}{\\sqrt{\\rho \\epsilon }}2^{10-\\chi /2} & \\le \\frac{\\ell ^2\\sqrt{d}}{\\sqrt{\\rho \\epsilon }}\\frac{(f(x_0)-f^*)}{\\epsilon ^2}2^{31-\\chi /4} \\le \\delta ,$ as announced.",
"Hence, with probability at least $1-\\delta $ , PRGD visits at most $T/(4 {T})$ points $x_t$ satisfying $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ and $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_{x_t}(0)) \\le -\\sqrt{\\rho \\epsilon }$ .",
"Using that there are at least $3 T/(4{T})$ iterates $x_t\\in \\mathcal {M}$ with $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ , we conclude that at least two-thirds of the iterates $x_t\\in \\mathcal {M}$ with $\\left\\Vert {\\mathrm {grad}\\,f(x_t)}\\right\\Vert \\le \\epsilon $ also satisfy $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_{x_t}(0)) \\ge -\\sqrt{\\rho \\epsilon }$ , with probability at least $1-\\delta $ .",
"Corollary REF follows directly from Theorem REF and the following lemma.",
"Lemma 7.3 For some $\\rho > 0$ (which would typically come from Assumption REF ), under Assumption REF on the retraction, let $x\\in \\mathcal {M}$ satisfy $\\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert \\le \\epsilon $ and $\\lambda _{\\min }(\\nabla ^2\\hat{f}_x(0)) \\ge -\\sqrt{\\rho \\epsilon }$ .",
"Then, $\\lambda _{\\min }(\\mathrm {Hess}\\,f(x)) \\ge -\\sqrt{\\rho \\epsilon } - \\beta \\epsilon $ .",
"In particular, if $\\epsilon \\le \\rho /\\beta ^2$ , then $\\lambda _{\\min }(\\mathrm {Hess}\\,f(x)) \\ge -\\sqrt{4\\rho \\epsilon }$ .",
"Considering $s = 0$ in Lemma REF , we may use $\\mathrm {Retr}_x(0) = x$ and that $T_{x, 0}$ is the identity (as per Definition REF ) to get $\\nabla ^2\\hat{f}_x(0) = \\mathrm {Hess}\\,f(x) + W_0$ , where $\\forall \\dot{s} \\in x\\mathcal {M}\\textrm { with } \\left\\Vert {\\dot{s}}\\right\\Vert = 1, \\qquad \\left\\langle {W_0[\\dot{s}]},{\\dot{s}}\\right\\rangle & \\le \\left\\Vert {\\gamma _{x, \\dot{s}}^{\\prime \\prime }(0)}\\right\\Vert \\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert \\le \\beta \\epsilon .$ Thus, $\\left\\Vert {W_0}\\right\\Vert \\le \\beta \\epsilon $ and we find $\\lambda _{\\min }(\\mathrm {Hess}\\,f(x)) \\ge -\\sqrt{\\rho \\epsilon } - \\beta \\epsilon $ .",
"For the last part, use $\\beta \\le \\sqrt{\\rho /\\epsilon }$ .",
"Corollary REF follows directly from Corollary REF and Lemma REF ."
],
[
"Proofs of key lemmas",
"The goal of this section is to prove Lemmas REF and REF .",
"All proofs deal with linear spaces, not manifolds.",
"The key ideas are due to Jin et al. [18].",
"The following lemma is needed because to apply Jin et al.",
"'s analysis we need the pullbacks not only to satisfy the restricted Lipschitz condition, Assumption REF , but also to have Lipschitz continuous gradient at least, uniformly in tangent space balls of fixed radius.",
"The lemma below implies Lemma REF .",
"Lemma 8.1 Let $f$ satisfy Assumptions REF and REF , and let $\\ell = L + \\rho b$ .",
"For all $x \\in \\mathcal {M}$ , it holds that $\\nabla \\hat{f}_x$ is $\\ell $ -Lipschitz continuous in the ball $B_{x,b}(0) \\subset x \\mathcal {M}$ .",
"By Assumption REF , $\\left\\Vert {\\nabla ^2\\hat{f}_x(0)}\\right\\Vert \\le L$ .",
"Hence, by Assumption REF , for all $s \\in B_{x,b}(0)$ , $\\left\\Vert {\\nabla ^2\\hat{f}_x(s)}\\right\\Vert \\le \\left\\Vert {\\nabla ^2\\hat{f}_x(0)}\\right\\Vert + \\left\\Vert {\\nabla ^2\\hat{f}_x(s) - \\nabla ^2\\hat{f}_x(0)}\\right\\Vert \\le L + \\rho \\left\\Vert {s}\\right\\Vert \\le L + \\rho b = \\ell .$ Let $s_1, s_2 \\in B_{x,b}(0)$ be arbitrary.",
"Then indeed, $\\left\\Vert {\\nabla \\hat{f}_x(s_2) - \\nabla \\hat{f}_x(s_1)}\\right\\Vert = \\left\\Vert {\\int _0^1 \\nabla ^2 \\hat{f}_x(s_1 + (s_2-s_1)\\tau )[s_2-s_1] d\\tau }\\right\\Vert \\le \\ell \\left\\Vert {s_2-s_1}\\right\\Vert ,$ where we used that the line segment from $s_1$ to $s_2$ is contained in $B_{x,b}(0)$ .",
"Together with the one above, the following standard lemma allows us to establish the sufficient decrease of $\\hat{f}_x$ in $B_{x,b}(0)$ upon taking a gradient step in the tangent space.",
"Lemma 8.2 Let $\\nabla \\hat{f}_x$ be $\\ell $ -Lipschitz continuous along the line segment connecting $s_j$ to $s_{j+1}$ , related by $s_{j+1} = s_j - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)$ with $\\eta = 1/\\ell $ and $\\alpha \\in [0, 1]$ .",
"Then, $\\hat{f}_{x}(s_{j+1}) - \\hat{f}_{x}(s_{j}) \\le - \\frac{\\alpha \\eta }{2}\\left\\Vert {\\nabla \\hat{f}_{x}(s_j)}\\right\\Vert ^2.$ It is a standard consequence of Lipschitz continuity of $\\nabla \\hat{f}_x$ along the line segment $\\tau \\mapsto (1-\\tau )s_j + \\tau s_{j+1}$ for $\\tau \\in [0, 1]$ that $\\hat{f}_x(s_{j+1}) \\le \\hat{f}_x(s_j) + \\left\\langle {\\nabla \\hat{f}_x(s_j)},{s_{j+1} - s_j}\\right\\rangle + \\frac{\\ell }{2} \\left\\Vert {s_{j+1} - s_j}\\right\\Vert ^2.$ Plugging in $s_{j+1} - s_j = - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)$ , we get $\\hat{f}_x(s_{j+1}) \\le \\hat{f}_x(s_j) + \\left[ - \\alpha \\eta + \\frac{\\ell \\alpha ^2 \\eta ^2}{2} \\right] \\left\\Vert {\\nabla \\hat{f}_x(s_j)}\\right\\Vert ^2.$ The coefficient between brackets is further equal to $\\left(-1 + \\frac{\\alpha }{2}\\right)\\alpha \\eta $ , which is at most $-\\alpha \\eta /2$ .",
"We are now ready to prove Lemma REF .",
"[Proof of Lemma REF ] The call to ${TangentSpaceSteps}(x, 0, \\eta , b, 1)$ produces a point $\\mathrm {Retr}_x(s_1)$ , with $s_1 = s_0 - \\alpha \\eta \\nabla \\hat{f}_x(s_0)$ , where $s_0 = 0$ , $\\alpha \\in [0, 1]$ and $\\left\\Vert {s_1}\\right\\Vert \\le b$ .",
"Owing to Assumption REF , we know that $\\nabla \\hat{f}_x$ is $L$ -Lipschitz continuous along the line segment connecting $s_0$ to $s_1$ .",
"Since $\\ell \\ge L$ , it is a fortiori $\\ell $ -Lipschitz continuous along that line segment: Lemma REF applies and yields $f(\\mathrm {Retr}_x(s_1)) = \\hat{f}_{x}(s_{1}) & \\le \\hat{f}_{x}(s_{0}) - \\frac{\\alpha \\eta }{2}\\left\\Vert {\\nabla \\hat{f}_{x}(s_0)}\\right\\Vert ^2 = f(x) - \\frac{\\alpha \\eta }{2} \\left\\Vert {\\nabla \\hat{f}_x(0)}\\right\\Vert ^2.$ If $\\alpha = 1$ , since $\\left\\Vert {\\nabla \\hat{f}_x(0)}\\right\\Vert = \\left\\Vert {\\mathrm {grad}\\,f(x)}\\right\\Vert > \\epsilon $ , we are done.",
"Owing to how ${TangentSpaceSteps}$ works, if $\\alpha < 1$ , then it must be that $\\Vert \\alpha \\eta \\nabla \\hat{f}_x(0)\\Vert = b$ , so that the inequality above yields $f(\\mathrm {Retr}_x(s_1)) & \\le f(x) - \\frac{b}{2}\\left\\Vert {\\nabla \\hat{f}_x(0)}\\right\\Vert \\le f(x) - \\frac{b\\epsilon }{2}.$ Using $\\epsilon \\le b^2\\rho $ and $\\ell \\ge L\\ge \\sqrt{\\rho \\epsilon }$ , $\\eta \\epsilon = \\frac{\\epsilon }{\\ell } \\le \\frac{\\sqrt{b^2\\rho \\epsilon }}{\\ell } = \\frac{\\sqrt{\\rho \\epsilon }}{\\ell }b \\le b.$ Hence, $f(\\mathrm {Retr}_x(s_1)) \\le f(x) - \\eta \\epsilon ^2/2$ , as desired.",
"(As a side note: Assumption REF is not truly necessary here; it is only convenient so that we can use the same definitions of $\\rho , b$ and $\\ell $ as in other parts of the paper.)",
"Lemma REF is Jin et al.",
"'s “improve or localize lemma” [18], with a tweak for variable step sizes.",
"The lemma states that if the function value does not decrease much, then the iterates are localized.",
"Lemma 8.3 Fix $j \\ge 0$ , $x \\in \\mathcal {M}$ and $s_0 \\in {x} \\mathcal {M}$ .",
"For all $0 \\le i \\le j-1$ , assume $0 \\le \\eta _i \\le \\eta = 1/\\ell $ , $s_{i+1} = s_i - \\eta _i \\nabla \\hat{f}_{x}(s_i)$ and $\\nabla \\hat{f}_x$ is $\\ell $ -Lipschitz continuous along the line segment connecting $s_i$ to $s_{i+1}$ .",
"Then, $\\left\\Vert {s_{j} - s_0}\\right\\Vert \\le \\sqrt{2\\eta j\\big (\\hat{f}_{x}(s_0) - \\hat{f}_{x}(s_j)\\big )}.$ Using a telescoping sum, triangle inequality, Cauchy–Schwarz and (to get to the last line) Lemma REF , we get: $\\left\\Vert {s_{j} - s_0}\\right\\Vert & = \\left\\Vert {\\sum _{i = 0}^{j-1} s_{i+1}-s_{i}}\\right\\Vert = \\left\\Vert {\\sum _{i = 0}^{j-1} - \\eta _{i} \\nabla \\hat{f}_{x}(s_{i})}\\right\\Vert \\le \\sum _{i = 0}^{j-1} \\sqrt{\\eta _{i}}\\left\\Vert {\\sqrt{\\eta _{i}} \\nabla \\hat{f}_{x}(s_{i})}\\right\\Vert \\\\& \\le \\sqrt{\\bigg (\\sum _{i = 0}^{j-1} \\eta _i \\left\\Vert {\\nabla \\hat{f}_{x}(s_{i})}\\right\\Vert ^2\\bigg )\\bigg (\\sum _{i = 0}^{j-1} \\eta _{i}\\bigg )} \\le \\sqrt{2\\eta j\\bigg (\\sum _{i = 0}^{j-1} \\frac{\\eta _{i}}{2} \\left\\Vert {\\nabla \\hat{f}_{x}(s_{i})}\\right\\Vert ^2\\bigg )} \\\\& \\le \\sqrt{2 \\eta j \\bigg (\\sum _{i = 0}^{j-1} \\hat{f}_{x}(s_{i}) - \\hat{f}_{x}(s_{i+1})\\bigg )} = \\sqrt{2 \\eta j \\big (\\hat{f}_{x}(s_{0}) - \\hat{f}_{x}(s_{j})\\big )}.",
"$ Lemma REF below and its proof are very similar to Jin et al.",
"'s Lemma 13 and its proof [18], except for a modification since $\\nabla \\hat{f}_x$ is only Lipschitz continuous in a ball.",
"This deterministic lemma formalizes the coupling sequence argument: if the Hessian of the pullback has a negative eigenvalue which is large in magnitude, upon initializing the tangent space steps at two appropriately chosen points $s_0^{}, s_0^{\\prime }$ , with certainty, one of them leads to significant decrease in the cost function.",
"As usual, we use parameters $\\eta , r, {T}$ as in (REF ) and ${F}, {L}$ as in (REF ).",
"Lemma 8.4 Under Assumptions REF and REF , let $x \\in \\mathcal {M}$ be such that $\\lambda _{\\min }(\\nabla ^2 \\hat{f}_x(0)) \\le - \\sqrt{\\rho \\epsilon }$ , with $\\epsilon \\le b^2\\rho $ and $L \\ge \\sqrt{\\rho \\epsilon }$ .",
"Let $s_0^{}, s_0^{\\prime } \\in {x}\\mathcal {M}$ be such that $\\left\\Vert {s_0}\\right\\Vert , \\left\\Vert {s_0^{\\prime }}\\right\\Vert \\le \\eta r$ , and $s_0^{} - s_0^{\\prime } = \\eta r_0 e_1$ , where $e_1$ is an eigenvector of unit norm associated with the minimum eigenvalue of $\\nabla ^2 \\hat{f}_x(0)$ , and $r_0 > \\omega = 2^{2-\\chi }\\ell {L}$ .",
"Let $s_{{T}}$ be defined by running ${TangentSpaceSteps}(x, s_0, \\eta , b, {T})$ (see Algorithm ).",
"Let $s_{{T}}^{\\prime }$ be similarly defined by running ${TangentSpaceSteps}(x, s_0^{\\prime }, \\eta , b, {T})$ .",
"Then, $\\min \\left\\lbrace \\hat{f}_{x}(s_{{T}})-\\hat{f}_{x}(s_{0}), \\hat{f}_{x}(s_{{T}}^{\\prime }) - \\hat{f}_{x}(s_{0}^{\\prime })\\right\\rbrace \\le - {F}.$ First, note that both sequences are initialized in the interior of the ball of radius $b$ .",
"Indeed, using $\\ell \\ge L, L \\ge \\sqrt{\\rho \\epsilon }$ , $\\epsilon \\le b^2\\rho $ and $\\chi > 1/4$ , $\\eta r = \\frac{1}{\\ell }\\frac{\\epsilon }{400 \\chi ^3} < \\frac{\\epsilon }{L}\\frac{64}{400} = b \\sqrt{\\frac{\\rho \\epsilon }{L^2}\\frac{\\epsilon }{b^2\\rho }}\\frac{64}{100} \\le b\\frac{64}{100} < b.$ The proof is by contradiction: assume $\\min \\left\\lbrace \\hat{f}_{x}(s_{{T}})-\\hat{f}_{x}(s_{0}), \\hat{f}_{x}(s_{{T}}^{\\prime }) - \\hat{f}_{x}(s_{0}^{\\prime })\\right\\rbrace > - {F}.$ Further assume, for the sake of contradiction, that one of the sequences $\\lbrace s_j\\rbrace _{j \\le {T}}, \\lbrace s_j^{\\prime }\\rbrace _{j \\le {T}}$ (defined in ${TangentSpaceSteps}$ ) escapes the interior of the ball $B_{x,b}(0)$ .",
"Without loss of generality, assume $\\lbrace s_j\\rbrace _{j\\le {T}}$ escapes.",
"Let $j \\le {T}-1$ be the minimum integer for which $\\left\\Vert {s_{j+1}}\\right\\Vert \\ge b$ .",
"Then, ${TangentSpaceSteps}(x, s_0, \\eta , b, {T})$ terminates with $s_j - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)$ for some $\\alpha \\in (0,1]$ satisfying $b = \\left\\Vert {s_j - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)}\\right\\Vert $ .",
"Using Lemma REF , $\\ell \\ge L \\ge \\sqrt{\\rho \\epsilon }$ and $\\chi > \\frac{1}{4}$ , $b & = \\left\\Vert {s_j - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)}\\right\\Vert \\le \\left\\Vert {s_j - \\alpha \\eta \\nabla \\hat{f}_{x}(s_j)-s_0}\\right\\Vert +\\left\\Vert {s_0}\\right\\Vert \\le \\sqrt{2\\eta (j+1){F}} + \\eta r \\\\ &\\le \\sqrt{2\\eta {T}{F}} + \\eta r \\le \\sqrt{\\frac{\\epsilon }{25\\chi ^2\\rho }} + \\frac{1}{400 \\chi ^3}\\sqrt{\\frac{\\epsilon }{\\rho }} \\le \\bigg (\\frac{1}{5\\chi } + \\frac{1}{400 \\chi ^3} \\bigg )\\sqrt{\\frac{\\epsilon }{\\rho }} \\le \\frac{1}{4\\chi }\\sqrt{\\frac{\\epsilon }{\\rho }} = {L}.$ Since $\\epsilon \\le b^2\\rho $ , we know that ${L} < b$ , which shows a contradiction.",
"Thus, neither of the sequences $\\lbrace s_j\\rbrace _{j \\le {T}}, \\lbrace s_j^{\\prime }\\rbrace _{j \\le {T}}$ leave the interior of $B_{x,b}(0)$ .",
"That is, $s_{j+1} = s_j - \\eta \\nabla \\hat{f}_{x}(s_j)$ and $\\left\\Vert {s_{j+1}}\\right\\Vert < b$ for $j = 0, 1, \\ldots , {T}-1$ , and similarly for $\\lbrace s_j^{\\prime }\\rbrace _{j \\le {T}}$ .",
"From here, we proceed exactly as in Lemma 13 of [18].",
"By Lemma REF , for all $j \\le {T}$ , $\\max \\left\\lbrace \\left\\Vert {s_j}\\right\\Vert , \\left\\Vert {s_j^{\\prime }}\\right\\Vert \\right\\rbrace \\le \\max \\left\\lbrace \\left\\Vert {s_j-s_0}\\right\\Vert , \\left\\Vert {s_j^{\\prime }-s_0^{\\prime }}\\right\\Vert \\right\\rbrace + \\eta r \\le \\sqrt{2\\eta {T}{F}} + \\eta r \\le {L}.$ Let $\\hat{s}_j = s_j - s_j^{\\prime }$ and $\\mathcal {H} = \\nabla ^2 \\hat{f}_{x}(0)$ .",
"Then, $\\hat{s}_{j+1} & = \\hat{s}_j - \\left(\\eta \\nabla \\hat{f}_{x}(s_j) - \\eta \\nabla \\hat{f}_{x}(s_j^{\\prime })\\right) = \\hat{s}_j - \\eta \\int _0^1 {\\nabla ^2 \\hat{f}_{x}\\left(s_j^{\\prime } + \\theta (s_j - s_j^{\\prime })\\right) [s_j-s_j^{\\prime }]} d\\theta \\\\ & = (I- \\eta \\mathcal {H})\\hat{s}_j - \\eta \\Delta _j \\hat{s}_j,$ where $\\Delta _j = \\int _0^1 {\\left(\\nabla ^2 \\hat{f}_{x}\\left(s_j^{\\prime } + \\theta (s_j - s_j^{\\prime })\\right) - \\mathcal {H}\\right)} d\\theta $ .",
"By Assumption REF , $\\left\\Vert {\\Delta _j}\\right\\Vert \\le \\int _0^1 \\rho \\left\\Vert {s_j^{\\prime } + \\theta (s_j - s_j^{\\prime })}\\right\\Vert d\\theta \\le \\int _0^1 \\rho \\max \\lbrace \\left\\Vert {s_j}\\right\\Vert , \\left\\Vert {s_j^{\\prime }}\\right\\Vert \\rbrace d\\theta \\le \\rho {L}.$ This will be useful momentarily.",
"It is easy to check by induction that $\\hat{s}_{j+1} & = p(j+1) - q(j+1),$ where $p(0) = \\hat{s}_0, q(0) = 0$ , and $p(j+1) & = (I - \\eta \\mathcal {H})^{j+1}\\hat{s}_0, & & \\text{ and } & q(j+1) & = \\eta \\sum _{i=0}^j(I-\\eta \\mathcal {H})^{j-i}\\Delta _{i}\\hat{s}_{i}.$ We use induction to show that $\\left\\Vert {q(j)}\\right\\Vert \\le \\left\\Vert {p(j)}\\right\\Vert /2$ .",
"The claim is clearly true for $j=0$ .",
"Suppose the claim is true for all $i \\le j$ .",
"We prove the claim for $j+1$ .",
"Let $-\\gamma = \\lambda _{\\min }(\\nabla ^2 \\hat{f}_x(0))$ .",
"Using $\\hat{s}_0 = \\eta r_0 e_1$ , notice in particular that $p(j) & = (I - \\eta \\mathcal {H})^j \\eta r_0 e_1 = (1+\\eta \\gamma )^j \\eta r_0 e_1,$ so that the norm of $p(j)$ grows with $j$ : $\\left\\Vert {p(j)}\\right\\Vert = (1+\\eta \\gamma )^j \\eta r_0$ .",
"Using the induction hypothesis, for all $i \\le j$ we have: $\\left\\Vert {\\hat{s}_i}\\right\\Vert \\le \\left\\Vert {p(i)}\\right\\Vert + \\left\\Vert {q(i)}\\right\\Vert \\le \\frac{3}{2}\\left\\Vert {p(i)}\\right\\Vert \\le 2(1+\\eta \\gamma )^i \\eta r_0.$ Furthermore, since $\\mathcal {H} \\preceq L I \\preceq \\ell I$ , it follows that $I - \\eta \\mathcal {H} \\succeq 0$ .",
"As a result, $\\left\\Vert {I - \\eta \\mathcal {H}}\\right\\Vert = \\lambda _{\\max }(I - \\eta \\mathcal {H}) = 1+\\eta \\gamma $ .",
"Therefore, also using $2\\eta \\rho {L}{T} = 1/2$ in the last step, $\\left\\Vert {q(j+1)}\\right\\Vert &= \\left\\Vert {\\eta \\sum _{i=0}^j(I-\\eta \\mathcal {H})^{j-i}\\Delta _i\\hat{s}_i}\\right\\Vert \\le \\eta \\rho {L}\\sum _{i=0}^j\\left\\Vert {(I-\\eta \\mathcal {H})^{j-i}}\\right\\Vert \\left\\Vert {\\hat{s}_i}\\right\\Vert \\\\ & \\le 2\\eta \\rho {L}\\sum _{i=0}^j(1+\\eta \\gamma )^{j-i}(1+\\eta \\gamma )^i\\eta r_0 \\\\ & \\le 2\\eta \\rho {L}{T}(1+\\eta \\gamma )^j\\eta r_0 = 2\\eta \\rho {L}{T}\\left\\Vert {p(j)}\\right\\Vert \\le \\left\\Vert {p(j+1)}\\right\\Vert /2,$ So we have proven $\\left\\Vert {q(j)}\\right\\Vert \\le \\left\\Vert {p(j)}\\right\\Vert /2$ for all $j$ .",
"Therefore, using the definition of $r_0$ in the last step, $\\max \\lbrace \\left\\Vert {s_{{T}}}\\right\\Vert ,\\left\\Vert {s^{\\prime }_{{T}}}\\right\\Vert \\rbrace & \\ge (\\left\\Vert {s_{{T}}}\\right\\Vert +\\left\\Vert {s_{{T}}^{\\prime }}\\right\\Vert )/2 \\ge \\left\\Vert {\\hat{s}_{{T}}}\\right\\Vert /2 \\ge (\\left\\Vert {p({T})}\\right\\Vert - \\left\\Vert {q({T})}\\right\\Vert )/2 \\\\ &\\ge \\left\\Vert {p({T})}\\right\\Vert /4 = (1+\\eta \\gamma )^{{T}}\\eta r_0 / 4 \\ge 2^{\\chi -2}\\eta r_0 > {L},$ which contradicts (REF ).",
"In the second to last step, we used $\\gamma \\ge \\sqrt{\\rho \\epsilon }$ and $\\sqrt{\\rho \\epsilon } \\le \\ell $ so that $\\frac{1}{\\chi }\\log _2\\left((1+\\eta \\gamma )^{{T}} \\right) \\ge \\frac{{T}}{\\chi } \\log _2\\left(1 + \\frac{\\sqrt{\\rho \\epsilon }}{\\ell } \\right) =\\frac{{T}}{\\chi } \\log _2\\left(1 + \\frac{\\chi }{{T}} \\right) \\ge 1,$ since $\\frac{1}{\\alpha } \\log _2(1+\\alpha ) \\ge 1$ for all $\\alpha \\in [0, 1]$ .",
"Except for the initial part, this proof is due to Jin et al. [18].",
"We are now ready to prove Lemma REF .",
"This proof is completely due to Jin et al.",
"[18]: we only somewhat modify how the proof is presented.",
"[Proof of Lemma REF ] Recall that $\\eta r < b$ (REF ), and define the stuck region $\\mathcal {X}_{\\mathrm {stuck}} = \\big \\lbrace s \\in B_{x,\\eta r}(0) : f({TangentSpaceSteps}(x,s,\\eta ,b,{T})) - f(x) > -{F}\\big \\rbrace .$ Running the tangent space steps with $s_0$ in that set does not yield sufficient improvement of the cost function despite the fact that the Hessian has a negative eigenvalue with large magnitude, hence the name.",
"We aim to show that this set has a small volume, so that it is unlikely to encounter it by random chance.",
"Let $S_{e_1}$ be the subspace of $x\\mathcal {M}$ orthogonal to $e_1$ .",
"Given $a \\in S_{e_1} \\cap B_{x,\\eta r}(0)$ , let $\\ell _a$ denote the line in $x\\mathcal {M}$ parallel to $e_1$ passing through $a$ .",
"Then, with $\\mathbb {1}$ denoting the indicator function, $\\text{Vol}(\\mathcal {X}_{\\mathrm {stuck}}) = \\int _{x\\mathcal {M}}\\mathbb {1}_{\\mathcal {X}_{\\mathrm {stuck}}}(y)dy = \\int _{S_{e_1} \\cap B_{x,\\eta r}(0)} \\left[ \\int _{\\ell _a}\\mathbb {1}_{\\mathcal {X}_{\\mathrm {stuck}}}(z) dz \\right] da.$ Lemma REF states any two points that are both on the line $\\ell _a$ and in $\\mathcal {X}_{\\mathrm {stuck}}$ must be close.",
"Specifically, for all $s, s^{\\prime } \\in \\ell _a\\cap \\mathcal {X}_{\\mathrm {stuck}}$ , we have $\\left\\Vert {s-s^{\\prime }}\\right\\Vert \\le \\eta \\omega $ , with $\\omega = 2^{2-\\chi }\\ell {L}$ .",
"Therefore, the set of problematic points on the line $\\ell _a$ is contained in a segment of length at most $\\eta \\omega $ and we deduce $\\int _{\\ell _a}\\mathbb {1}_{\\mathcal {X}_{\\mathrm {stuck}}}(z) dz \\le \\eta \\omega $ .",
"As a result, $\\text{Vol}(\\mathcal {X}_{\\mathrm {stuck}}) \\le \\eta \\omega \\int _{S_{e_1} \\cap B_{x,\\eta r}(0)} da = \\eta \\omega \\text{Vol}(\\mathbb {B}^{d-1}_{\\eta r}),$ where $\\mathbb {B}^k_R$ denotes a $k$ -dimensional (Euclidean) ball of radius $R$ .",
"Since $s_0 \\sim \\text{Uniform}(B_{x,\\eta r}(0))$ , $\\mathbb {P}(s_0 \\in \\mathcal {X}_{\\mathrm {stuck}}) &= \\frac{\\text{Vol}(\\mathcal {X}_{\\mathrm {stuck}})}{\\text{Vol}(\\mathbb {B}^d_{\\eta r})} \\le \\frac{\\eta \\omega \\text{Vol}(\\mathbb {B}^{d-1}_{\\eta r})}{\\text{Vol}(\\mathbb {B}^d_{\\eta r})} =\\frac{\\omega \\Gamma (1+d/2)}{r\\sqrt{\\pi }\\Gamma ((d+1)/2)} \\le \\frac{\\omega }{r}\\sqrt{\\frac{d}{\\pi }}\\ =\\frac{\\ell \\sqrt{d}}{\\sqrt{\\rho \\epsilon }}\\frac{400}{\\sqrt{\\pi }}2^{-\\chi }\\chi ^2 \\\\ &\\le \\frac{\\ell \\sqrt{d}}{\\sqrt{\\rho \\epsilon }}2^{10-\\chi /2},$ where we used the Gautschi inequality for the $\\Gamma $ function, and $\\chi > 1/4$ to bound $\\frac{400}{\\sqrt{\\pi }}2^{-\\chi }\\chi ^2 \\le 2^{10-\\chi /2}$ .",
"To conclude, note that if $s_0 \\notin \\mathcal {X}_{\\mathrm {stuck}}$ then $f({TangentSpaceSteps}(x,s_0,\\eta ,b,{T})) - f(x) \\\\ = f({TangentSpaceSteps}(x,s_0,\\eta ,b,{T})) - \\hat{f}_x(s_0) + \\hat{f}_x(s_0) - \\hat{f}_x(0) \\\\ \\le -{F} + \\epsilon \\eta r + \\ell \\eta ^2r^2/2 = -{F} + \\frac{\\sqrt{\\rho \\epsilon }}{\\ell }{F}\\big (50\\chi ^3\\big )\\bigg (\\frac{1}{400\\chi ^3} + \\frac{1}{2}\\Big (\\frac{1}{400\\chi ^3}\\Big )^2\\bigg ) \\\\ \\le -{F} + {F}\\bigg (\\frac{1}{8} + \\frac{25}{400^2\\chi ^3}\\bigg ) \\le -{F}/2,$ using $\\sqrt{\\rho \\epsilon } \\le \\ell $ and $\\chi > 1/4$ once more in the last step, and also $\\hat{f}_x(s_0) - \\hat{f}_x(0) \\le \\epsilon \\eta r + \\ell \\eta ^2r^2/2$ owing to the fact that $\\nabla \\hat{f}_x$ is $\\ell $ -Lipschitz continuous along the line segment connecting 0 and $s_0$ , $\\Vert s_0\\Vert \\le \\eta r$ and $\\Vert \\nabla \\hat{f}_x(0)\\Vert \\le \\epsilon $ ."
],
[
"Regularity constants for dominant eigenvector computation (PCA)",
"Computing the dominant eigenvector of a symmetric matrix $A \\in {\\mathbb {R}^{n\\times n}}$ (which notably comes up in PCA) comes down to solving $\\max _{x \\in {{S}^{n-1}}} f(x), & & f(x) = \\frac{1}{2} x^\\top \\!",
"A x,$ where ${{S}^{n-1}}= \\lbrace x \\in {\\mathbb {R}^n}: x^\\top \\!",
"x = 1 \\rbrace $ is the unit sphere.",
"If we use the retraction $\\mathrm {Retr}_x(s) = \\frac{x+s}{\\Vert x+s\\Vert }$ (where $\\Vert x\\Vert = \\sqrt{x^\\top \\!",
"x}$ )—for which Assumption REF holds with $\\beta = 0$ —then pullbacks are of the form $\\hat{f}_x(s) & = f(\\mathrm {Retr}_x(s)) = \\frac{1}{1+\\Vert s\\Vert ^2} \\frac{1}{2} (x+s)^\\top \\!",
"A (x+s),$ defined over the tangent spaces $x{{S}^{n-1}}= \\lbrace s \\in {\\mathbb {R}^n}: x^\\top \\!",
"s = 0 \\rbrace $ .",
"The gradient of $\\hat{f}_x$ at $s$ is given by $\\nabla \\hat{f}_x(s) & = \\mathrm {Proj}_x\\!\\left( \\frac{1}{1+\\Vert s\\Vert ^2} A(x+s) + \\frac{-1}{(1+\\Vert s\\Vert ^2)^2} (x+s)^\\top \\!",
"A (x+s) \\cdot s \\right) \\\\& = \\frac{1}{1+\\Vert s\\Vert ^2} \\left( \\mathrm {Proj}_x(A(x+s)) - 2\\hat{f}_x(s) \\cdot s \\right),$ where $\\mathrm {Proj}_x(s) = s - (x^\\top \\!",
"s) x$ is the orthogonal projector from ${\\mathbb {R}^n}$ to $x{{S}^{n-1}}$ .",
"It follows that $\\nabla \\hat{f}_x(s) - \\nabla \\hat{f}_x(0) & = \\frac{1}{1+\\Vert s\\Vert ^2} \\left( \\mathrm {Proj}_x(As) - 2 \\hat{f}_x(s) s - \\Vert s\\Vert ^2 \\mathrm {Proj}_x(Ax) \\right).$ Using $\\frac{1}{1+\\Vert s\\Vert ^2} \\le 1$ and $\\frac{\\Vert s\\Vert ^2}{1+\\Vert s\\Vert ^2} \\le \\frac{1}{2}\\Vert s\\Vert $ for all $s$ , and using the fact that an orthogonal projector can only reduce the norm of a vector, we find $\\Vert \\nabla \\hat{f}_x(s) - \\nabla \\hat{f}_x(0)\\Vert & \\le \\Vert As\\Vert + 2 \\left[\\sup _{s\\in x{{S}^{n-1}}} |\\hat{f}_x(s)| \\right] \\Vert s\\Vert + \\frac{1}{2} \\Vert Ax\\Vert \\Vert s\\Vert .$ Letting $\\Vert A\\Vert $ denote the operator norm of $A$ (largest singular value), we finally obtain $\\Vert \\nabla \\hat{f}_x(s) - \\nabla \\hat{f}_x(0)\\Vert & \\le \\frac{5}{2} \\Vert A\\Vert \\Vert s\\Vert .$ This shows that Assumption REF holds with $b_1 = \\infty $ and $L = \\frac{5}{2}\\Vert A\\Vert $ , or any larger number.",
"For example, the induced 1-norm of the matrix $A$ is straightforward to compute and is an upper-bound on $\\Vert A\\Vert $ .",
"Now aiming to control second-order derivatives, we compute a directional derivative of $\\nabla \\hat{f}_x(s)$ and obtain the Hessian of $\\hat{f}_x$ on the tangent space $x{{S}^{n-1}}$ : $\\nabla ^2 \\hat{f}_x(s)[\\dot{s}] & = -2 \\frac{\\left\\langle {s},{\\dot{s}}\\right\\rangle }{1+\\Vert s\\Vert ^2} \\nabla \\hat{f}_x(s) + \\frac{1}{1+\\Vert s\\Vert ^2} \\left[ \\mathrm {Proj}_x(A\\dot{s}) - 2 \\hat{f}_x(s) \\dot{s} - 2\\langle {\\nabla \\hat{f}_x(s)},{\\dot{s}}\\rangle s \\right],$ where $\\left\\langle {u},{v}\\right\\rangle = u^\\top \\!",
"v$ .",
"In particular, $\\nabla ^2 \\hat{f}_x(0)[\\dot{s}] = \\mathrm {Proj}_x(A\\dot{s}) - (x^\\top \\!",
"A x) \\dot{s}$ , so that $\\left\\langle {\\dot{s}},{\\left( \\nabla ^2 \\hat{f}_x(s) - \\nabla ^2 \\hat{f}_x(0) \\right)[\\dot{s}]}\\right\\rangle = -4 \\frac{\\left\\langle {s},{\\dot{s}}\\right\\rangle \\langle {\\nabla \\hat{f}_x(s)},{\\dot{s}}\\rangle }{1+\\Vert s\\Vert ^2} + \\left( \\frac{1}{1+\\Vert s\\Vert ^2} - 1 \\right) \\langle {\\dot{s}},{A \\dot{s}}\\rangle \\\\ - \\left( 2 \\frac{\\hat{f}_x(s)}{1+\\Vert s\\Vert ^2} - x^\\top \\!",
"A x \\right) \\Vert \\dot{s}\\Vert ^2.$ Using $\\frac{\\Vert s\\Vert }{1+\\Vert s\\Vert ^2} \\le \\frac{1}{2}$ , it is easy to see that $\\Vert \\nabla \\hat{f}_x(s)\\Vert \\le \\frac{3}{2}\\Vert A\\Vert $ and: $\\Vert \\nabla ^2 \\hat{f}_x(s) - \\nabla ^2 \\hat{f}_x(0)\\Vert & \\le 4 \\Vert \\nabla \\hat{f}_x(s)\\Vert \\Vert s\\Vert + \\frac{1}{2} \\Vert A\\Vert \\Vert s\\Vert \\\\ & \\quad + \\left[ \\sup _{s\\in x{{S}^{n-1}}} \\frac{\\left| (x+s)^\\top \\!",
"A (x+s) - (1+\\Vert s\\Vert ^2)^2 x^\\top \\!",
"A x \\right|}{(1+\\Vert s\\Vert ^2)^2 \\Vert s\\Vert } \\right] \\Vert s\\Vert \\\\& \\le (6+1/2)\\Vert A\\Vert \\Vert s\\Vert + \\left[ \\sup _{t > 0} \\frac{2t + 3t^2 + t^4}{(1+t^2)^2 t} \\right] \\Vert A\\Vert \\Vert s\\Vert \\\\& \\le 9 \\Vert A\\Vert \\Vert s\\Vert .$ This shows Assumption REF holds with $b_2 = \\infty $ and $\\rho = 9\\Vert A\\Vert $ ."
]
] | 1906.04321 |
[
[
"Magic entanglement renormalization for quantum fields"
],
[
"Abstract Continuous tensor networks are variational wavefunctions proposed in recent years to efficiently simulate quantum field theories (QFTs).",
"Prominent examples include the continuous matrix product state (cMPS) and the continuous multi-scale entanglement renormalization ansatz (cMERA).",
"While the cMPS can approximate ground states of a class of QFT Hamiltonians that are both local and interacting, cMERA is only well-understood for QFTs that are quasi-local and non-interacting.",
"In this paper we propose the magic cMERA, a concrete realization of cMERA for a free boson QFT that simultaneously satisfies four remarkable properties: (i) it is the exact ground state of a strictly local Hamiltonian; (ii) in the massless case, its spectrum of scaling operators is exactly soluble in real space; (iii) it has the short-distance structure of a cMPS; (iv) it is generated by a quasi-local entangler that can be written as a continuous matrix product operator.",
"None of these properties is fulfilled by previous cMERA proposals.",
"Properties (iii)-(iv) establish a firm connection between cMERA and cMPS wavefunctionals, opening the path to applying powerful cMPS numerical techniques, valid for interacting QFTs, also to cMERA calculations."
],
[
"Appendix I: Exact ground state of Hamiltonian $H^{\\Lambda }_m$",
"Let us consider a single bosonic quantum field $\\phi (x)$ in one spatial dimension, together with its conjugate momentum $\\pi (x)$ .",
"They obey the commutation relation $\\left[\\phi (x),\\pi (y) \\right] = i\\delta (x-y)$ .",
"Let us also introduce the annihilation operator $\\psi (x)$ and $\\psi (k)$ in real and momentum space $\\psi (x) &\\equiv & \\sqrt{\\frac{\\Lambda }{2}}\\phi (x)+\\frac{i}{\\sqrt{2\\Lambda }} \\pi (x),~~~~\\\\\\psi (k) &\\equiv & \\frac{1}{\\sqrt{2\\pi }}\\int \\!",
"dx~ e^{-ikx} \\psi (x)$ with $\\left[\\psi (x),\\psi (y)^{\\dagger } \\right]= \\delta (x-y)$ and $\\left[\\psi (k),\\psi (q)^{\\dagger } \\right]= \\delta (k-q)$ .",
"In this work we studied the Hamiltonian $H_m^{\\Lambda }$ , $H^{\\Lambda }_m &\\equiv & \\frac{1}{2}\\int \\!\\!",
"dx\\left(\\frac{1}{\\Lambda ^2}(\\partial _x \\pi (x))^2 +\\pi (x)^2 + (\\partial _x \\phi (x))^2 + m^2 \\phi (x)^2\\right) \\\\&=&\\frac{1}{\\Lambda }\\int \\!",
"dx \\left( \\partial _x \\psi ^{\\dagger }(x)\\partial _x \\psi (x) + \\frac{m^2 + \\Lambda ^2}{2} \\psi ^{\\dagger }(x)\\psi (x) + \\frac{m^2 - \\Lambda ^2}{4} \\left(\\psi (x)^2 + \\psi (x)^{\\dagger 2}\\right) \\right) \\\\&=&\\frac{1}{\\Lambda } \\int \\!\\!",
"dk\\left(k^2 + \\frac{m^2 + \\Lambda ^2}{2} \\right)\\psi ^{\\dagger }(k)\\psi (k) + \\frac{m^2-\\Lambda ^2}{4}\\left(\\psi (k)\\psi (-k) + \\psi (k)^{\\dagger }\\psi (-k)^{\\dagger }\\right) $ This Hamiltonian can be diagonalized by introducing the annihilation operators $a^{\\Lambda }_s(k) \\equiv \\sqrt{\\frac{\\alpha _s(k)}{2}} \\phi (k) + \\frac{i}{\\sqrt{2\\alpha _s(k)}}\\pi (k),$ where the function $\\alpha _s(k)$ is given by $\\alpha _s(k) \\equiv \\sqrt{\\frac{\\Lambda ^2k^2 + \\Lambda ^4 e^{-2s} }{k^2 + \\Lambda ^2}} = \\sqrt{\\frac{\\Lambda ^2k^2 + \\Lambda ^2 m^2 }{k^2 + \\Lambda ^2}} $ where $s = \\log (\\Lambda /m)$ .",
"Indeed, the Hamiltonian can be rewritten as $H^{\\Lambda }_m &=& \\int \\!\\!",
"dk~ E^{\\Lambda }_{m}(k) ~ a^{\\Lambda }_{s}(k)^{\\dagger } a^{\\Lambda }_s(k),~~\\\\E^{\\Lambda }_m(k) &\\equiv & \\sqrt{|k|^2+m^2} \\sqrt{1+\\left(\\frac{k}{\\Lambda }\\right)^2}$ as can be checked by direct replacement.",
"Notice that these expressions are valid for any value of the mass parameter $m \\in \\mathbb {R}$ , which we can take to be positive since only $m^2$ appears in the Hamiltonian.",
"Let us first consider two limit cases $m=\\Lambda $ and $m=0$ and then the full scale evolution.",
"For $m=\\Lambda $ (or $s=0$ ) the Hamiltonian simplifies to $H_{m=\\Lambda }^{\\Lambda } &=& \\frac{1}{\\Lambda } \\int \\!\\!",
"dk (k^2 + \\Lambda ^2 )\\psi ^{\\dagger }(k)\\psi (k)$ and $a^{\\Lambda }_{s=0}(k) \\equiv \\sqrt{\\frac{\\Lambda }{2}} \\phi (k) + \\frac{i}{\\sqrt{2\\Lambda }}\\pi (k),$ where the function $\\alpha _{s=0}(k)$ is given by $\\alpha _{s=0}(k) \\equiv \\sqrt{\\frac{\\Lambda ^2k^2 + \\Lambda ^4 }{k^2 + \\Lambda ^2}} = \\Lambda .",
"$ The ground state is the product state $|\\Lambda \\rangle $ that is annihilated by any $\\psi (k)$ , that is $\\psi (k)\\mbox{$| \\Lambda \\rangle $}=0~\\forall k$ , because the Hamiltonian is positive definite.",
"Through a Fourier transform, this condition is equivalent to $\\psi (x)\\mbox{$| \\Lambda \\rangle $}=0 ~~\\forall x$ , which we used in the main text to define the unentangled vacuum state $\\mbox{$| \\Lambda \\rangle $}$ ."
],
[
"2. Case $m=0$ : critical ground state ",
"For $m=0$ (or $s=\\infty $ ) the Hamiltonian is gapless.",
"This can be seen by the fact that $\\alpha (k)\\equiv \\alpha _{s=\\infty }(k)=\\sqrt{\\frac{k^2\\Lambda ^2}{k^2+\\Lambda ^2}}$ at small $k\\ll \\Lambda $ reduces to the CFT profile $\\alpha (k)=\\alpha ^{\\mbox{\\tiny CFT}}(k)-\\frac{|k|^3}{2\\Lambda ^2}+O(k^5),$ where $\\alpha ^{\\mbox{\\tiny CFT}}(k)=|k|.$ At large $k\\gg \\Lambda $ , the state approaches the unentangled state $\\mbox{$| \\Lambda \\rangle $}$ , in the sense that $\\alpha (k)=\\Lambda -\\frac{\\Lambda ^3}{k^2}+O(k^{-4}).$ The dispersion relation also reduces to the CFT dispersion $E^{\\mbox{\\tiny CFT}}(k)=|k|$ at small $k\\ll \\Lambda $ .",
"At large $k\\gg \\Lambda $ , the dispersion relation is dominated by the nonrelativistic kinetic energy, $E^{\\Lambda }(k)=\\frac{k^2}{\\Lambda }+O(|k|).$"
],
[
"3. Case $0 < m< \\Lambda $ : scale evolution",
"We will show that the ground state of Eq.",
"(REF ) with $m=\\Lambda e^{-s}$ is the cMERA state $\\mbox{$| \\Psi ^{\\Lambda }(s) \\rangle $} = e^{-is(L+K)} \\mbox{$| \\Lambda \\rangle $},$ where $K$ is the magic entangler $K=\\frac{-i}{2}\\int dk \\, g(k)(\\psi (k)\\psi (-k)-\\psi ^{\\dagger }(k)\\psi ^{\\dagger }(-k))$ with $g(k)=\\frac{\\Lambda ^2}{2(k^2+\\Lambda ^2)}.$ Clearly, at $s=0$ the cMERA state is $|\\Lambda \\rangle $ , the ground state of Eq.",
"(REF ) with $m=\\Lambda $ .",
"As shown in [50], the cMERA state Eq.",
"(REF ) is a Gaussian state annihilated by the $a^{\\Lambda }_s(k)$ of the form Eq.",
"(REF ), where $(\\partial _s-k\\partial _k)\\alpha _s(k)=-2\\alpha _s(k)g(k).$ Now we can substitute Eq.",
"(REF ) into Eq.",
"(REF ) and see that it holds for arbitrary $s\\in [0,\\infty )$ .",
"Since $\\alpha _s(k)$ uniquely determines a Guassian state, we have shown that $\\mbox{$| \\Psi ^{\\Lambda }(s) \\rangle $}$ is in the ground state of the massive free boson Hamiltonian with a UV cutoff $\\Lambda $ and mass $m=\\Lambda e^{-s}$ ."
],
[
"Appendix II: computation of correlations functions",
"Next we will compute correlation functions, involving the bosonic fields $\\psi (x),\\psi ^{\\dagger }(x)$ , of a Gaussian state annihilated by $a^{\\Lambda }(k) \\equiv \\sqrt{\\frac{\\alpha (k)}{2}} \\phi (k) + \\frac{i}{\\sqrt{2\\alpha (k)}}\\pi (k)$ with some function $\\alpha (k)$ .",
"First, we express the $\\psi (k)$ in terms of these annihilation operators: $\\psi (k)=\\sqrt{\\frac{\\Lambda }{2}}\\phi (k)+\\frac{i}{\\sqrt{2\\Lambda }}\\pi (k),$ where $\\phi (k)=\\frac{1}{\\sqrt{2\\alpha (k)}}(a^{\\Lambda }(k)+a^{\\Lambda \\dagger }(-k)) \\\\\\pi (k)=-i\\frac{\\sqrt{2\\alpha (k)}}{2}(a^{\\Lambda }(k)-a^{\\Lambda \\dagger }(-k)).$ The correlation functions can then be computed from the canonical commutation relations $[a^{\\Lambda }(k),a^{\\Lambda \\dagger }(k^{\\prime })]=\\delta (k-k^{\\prime })$ , $\\langle \\psi (k)\\psi (k^{\\prime })\\rangle &=& \\frac{1}{4} \\left(\\frac{\\Lambda }{\\alpha (k)}-\\frac{\\alpha (k)}{\\Lambda }\\right) \\delta (k+k^{\\prime }) \\\\&\\equiv & F(k) \\delta (k+k^{\\prime }), \\nonumber \\\\\\langle \\psi ^{\\dagger }(k)\\psi (k^{\\prime })\\rangle &=& \\frac{1}{4} \\left(\\frac{\\Lambda }{\\alpha (k)}+\\frac{\\alpha (k)}{\\Lambda }-2\\right) \\delta (k-k^{\\prime }) ~~~ \\\\&\\equiv & n(k) \\delta (k-k^{\\prime }).",
"\\nonumber $ Transforming them into real space, we obtain, $\\langle \\psi (x)\\psi (y) \\rangle = \\langle \\psi ^{\\dagger }(x)\\psi ^{\\dagger }(y) \\rangle = \\int \\frac{dk}{2\\pi } F(k) e^{ik(x-y)},$ $\\langle \\psi ^{\\dagger }(x)\\psi (y)\\rangle =\\int \\frac{dk}{2\\pi } n(k) e^{ik(x-y)}.$ In particular, the particle density $\\rho _0\\equiv \\langle \\psi ^{\\dagger }(x)\\psi (x)\\rangle $ can be computed analytically with the fixed point $\\alpha (k)$ in Eq.",
"(REF ), $\\frac{\\rho _0}{\\Lambda }=\\frac{1}{8\\pi }\\int dk\\, \\left(\\sqrt{\\frac{1+k^2}{k^2}}+\\sqrt{\\frac{k^2}{1+k^2}}-2\\right).$ The above expression has an IR divergence at $k=0$ .",
"Introducing a small mass $m\\ll \\Lambda $ , i.e., $\\alpha (k) = \\Lambda \\sqrt{\\frac{k^2+m^2}{k^2+\\Lambda ^2}}$ we find that $\\frac{\\rho _0}{\\Lambda } = \\frac{1}{8\\pi } \\log \\frac{\\Lambda ^2}{m^2}-\\frac{1.22741}{8\\pi }+O\\left(\\frac{m^2}{\\Lambda ^2}\\right).$ During the evolution Eq.",
"(REF ), the cMERA state is the ground state of the massive Hamiltonian with mass $m=\\Lambda e^{-s}$ .",
"We therefore expect that the particle density $\\rho _0$ increases linearly with $s$ for $s\\gg 1$ .",
"This is a direct consequence of the IR divergence, which is a feature of the free boson CFT in 1+1 dimensions.",
"Note that, however, the ground state energy density of the massless Hamiltonian $e_0 &\\equiv & \\langle \\partial _x\\psi ^{\\dagger }\\partial _x\\psi \\rangle +\\frac{\\Lambda ^2}{2}\\rho _0-\\frac{\\Lambda ^2}{4}\\langle \\psi ^2+\\psi ^{\\dagger 2}\\rangle \\\\&=& \\int \\frac{dk}{2\\pi } \\, \\left(k^2 n(k)+ \\frac{\\Lambda ^2}{2} (n(k)-F(k))\\right).$ is finite because the IR divergences in $n(k)$ and $F(k)$ cancel each other.",
"Its value only depends on the UV cutoff, $e_0=-\\frac{\\Lambda ^2}{6\\pi }.$ The fact that $e_0$ is finite is important in the context of numerical optimizations of cMERA through energy minimization [68]."
],
[
"Appendix III: cMERA as the ground state of a local Hamiltonian: generic case",
"Consider a Gaussian cMERA state annihilated by $a^{\\Lambda }(k) \\equiv \\sqrt{\\frac{\\alpha (k)}{2}} \\phi (k) + \\frac{i}{\\sqrt{2\\alpha (k)}}\\pi (k)$ with some function $\\alpha (k)$ .",
"Then it is the ground state of all Hamiltonians with the form $H = \\int dk\\, E(k) a^{\\Lambda }(k)^{\\dagger } a^{\\Lambda }(k),$ where $E(k)\\ge 0$ can be any dispersion relation.",
"In terms of original fields, the Hamiltonian is $H=\\frac{1}{2}\\int dk \\, \\left(\\frac{E(k)}{\\alpha (k)} \\pi (k)\\pi (-k)+E(k)\\alpha (k)\\phi (k)\\phi (-k)\\right).",
"~~$ This Hamiltonian is local (that is, it involves only a finite number of derivatives of the field operators) and invariant under spatial parity only if $\\frac{E(k)}{\\alpha (k)}&=&P_1(k^2) \\\\E(k)\\alpha (k)&=&P_2(k^2),$ where $P_1$ and $P_2$ are (finite-degree) polynomials.",
"Then we have $E(k)&=&\\sqrt{P_1(k^2)P_2(k^2)} \\\\\\alpha (k)&=&\\sqrt{\\frac{P_2(k^2)}{P_1(k^2)}}.$ Following Ref.",
"[50], we will also require that both the CFT dispersion relation and the CFT ground state be recovered at small $k$ , that is $E(k)=|k|+o(k),$ and $\\alpha (k)=|k|+o(k),$ and that the cMERA state approaches the product state in the UV, that is $\\lim _{k\\rightarrow \\infty } \\alpha (k)=\\Lambda .$ Expanding the polynomials $P_1$ and $P_2$ as $P_1(k^2) =\\sum _{l=0}^{l_m} a_l (k^2)^l,~~~~ P_2(k^2) =\\sum _{l=0}^{l^{\\prime }_m} b_l (k^2)^l,~~~$ Eqs.",
"(REF ), (REF ) imply $a_0=1,b_0=0,b_1=1$ .",
"Eq.",
"(REF ) forces $P_1$ and $P_2$ to have the same degree $2l^{\\prime }_m=2l_m$ and also that $b_{l_m}=\\Lambda ^2 a_{l_m}$ .",
"The most generic local quadratic Hamiltonian that has a cMERA ground state is therefore $H^{\\Lambda }=\\frac{1}{2}\\int dx \\, \\left(\\sum _{l=0}^{l_m} a_l (\\partial ^l_x \\pi (x))^2+ \\sum _{l=1}^{l_m} b_l (\\partial ^l_x \\phi (x))^2\\right)$ subject to the above constraints.",
"The magic cMERA in the main text corresponds to the simplest solution (polynomials of smallest degree), namely $l_m=1$ with $P_1=1+\\frac{k^2}{\\Lambda ^2}$ and $P_2=k^2$ .",
"Note that the degrees of $P_1$ and $P_2$ determine the order of derivatives appearing in the Hamiltonian.",
"In the main text, we have second order derivatives in $H$ , which are $(\\partial _x \\phi (x))^2$ and $(\\partial _x \\pi (x))^2$ , in accordance with the degree of $P_1$ and $P_2$ .",
"Choosing larger $l_m$ corresponds to regulating the CFT Hamiltonian with higher derivative terms.",
"The asymptotic behavior of $\\alpha (k)$ at large $k$ determines UV properties of the cMERA state.",
"For the set of cMERA states with $\\alpha (k)$ in Eq.",
"(REF ), that is, the set of cMERA states that can be the ground state of a local Hamiltonian, it is always true that $\\frac{\\alpha (k)}{\\Lambda }=1+O\\left(\\left(\\frac{\\Lambda ^2}{k^2}\\right)^n\\right)$ with some positive integer $n\\ge 1$ .",
"To determine $n$ , we first find the smallest $l_1$ such that $b_l=\\Lambda ^2 a_l$ for all $l_1\\le l\\le l_m$ , then $n=l_m-l_1+1$ is the number of such coefficients.",
"Since $b_{l_m}=\\Lambda ^2 a_{l_m}$ , it is clear that $1\\le n\\le l_m$ .",
"Now we can show that the cMERA in previous works [49], [50] cannot be the ground state of a local Hamiltonian.",
"Indeed, the previous cMERA proposals involve a function $\\alpha (k)$ that converges faster than any polynomial at large $k$ , contradicting Eq.",
"(REF ).",
"Eq.",
"(REF ) has various implications on the correlation functions.",
"First, Eq.",
"(REF ) implies that $n(k)=O\\left(\\left(\\frac{\\Lambda ^2}{k^2}\\right)^{2n}\\right).$ For $n=1$ , $n(k)\\sim 1/k^4$ which is compatible with a generic bosonic cMPS.",
"The minimial choice $n=l_m=1$ gives the magic cMERA state in the main text.",
"More generally, if $b_{l_m}=\\Lambda ^2 a_{l_m}$ but $b_{l_m-1}\\ne \\Lambda ^2 a_{l_m-1}$ , then $n=1$ and the ground state is compatible with the cMPS in the UV.",
"If $n>1$ , the state is compatible with a subclass of cMPS that satisfies certain regularity conditions, which imposes constraints on cMPS variational parameters.",
"Now consider the implication on the real space correlation function $n(x)\\equiv \\langle \\psi ^{\\dagger }(\\vec{x})\\psi (0)\\rangle .$ it has continous derivatives at $x=0$ up to $4n-2$ order.",
"For example, the expectation value of the non-relativistic kinetic term $\\langle \\partial _x\\psi ^{\\dagger }\\partial _x\\psi \\rangle =-\\partial ^2_x n(0)$ is always finite.",
"However, higher order derivatives diverge in the $n=1$ case.",
"We have therefore seen that, by asking the cMERA state to be the ground state of a local Hamiltonian, we automatically have correlation functions with finite orders of smoothness.",
"This is to be in contrast with previous cMERA proposals [49], [50], where correlation functions are infinite-order differentiable.",
"The entangler $K$ that generates this class of cMERA as the fixed point wavefunctionals also differs from previous works.",
"The fixed point $\\alpha (k)$ is related to $g(k)$ in Eq.",
"(REF ) by $g(k)=\\frac{k\\partial _k \\alpha (k)}{2\\alpha (k)}.$ Substituting Eq.",
"(REF ) into the equation above, we obtain $g(k)=\\frac{k^2}{2} \\frac{P_1(k^2)P^{\\prime }_2(k^2)-P^{\\prime }_1(k^2)P_2(k^2)}{P_1(k^2)P_2(k^2)}.$ Note that $g(k)$ decays no slower than $1/k^2$ at large $k$ because $P_1(k^2)P^{\\prime }_2(k^2)-P^{\\prime }_1(k^2)P_2(k^2)$ has a degree that is at most $2l_m-4$ .",
"We see that $g(k)$ decays polynomially.",
"This implies that its Fourier transform $g(x)$ at $x=0$ is not smooth.",
"For example, the magic cMERA corresponds to $g(x)\\propto e^{-\\Lambda |x|}$ , which does not have first-order derivative at $x=0$ .",
"This is in constrast with the Gaussian entangler $g(x)\\propto e^{-\\sigma (\\Lambda x)^2/4}$ which is smooth at $x=0$ .",
"At $k=0$ , $P_1(0)=a_0=1, P^{\\prime }_2(0)=b_1=1$ and $\\lim _{k\\rightarrow 0} P_2(k^2)/k^2=b_1=1$ , together give that $g(k)$ is smooth at $k=0$ .",
"To see this, let us rewrite $g(k)=\\frac{1}{2} \\left(\\frac{P^{\\prime }_2(k^2)}{P_2(k^2)/k^2}-\\frac{k^2P^{\\prime }_1(k^2)}{P_1(k^2)}\\right).$ Both $P_2(k^2)/k^2$ and $P_1(k^2)$ are polynomials which are nonvanishing at $k=0$ .",
"This ensures that $g(k)$ is infinite-order differentiable at $k=0$ .",
"The fact that $g(k)$ is smooth at $k=0$ implies that $g(x)$ decays at least exponentially at large $x$ , which keeps the entangler $K$ quasi-local.",
"We can also work out $g(k=0)=\\frac{1}{2},$ which ensures that the scaling dimensions (eigenvalues of $L+K$ ) come out correctly [50].",
"In conclusion, we have exhaustively determined the class of Gaussian bosonic cMERA states that can be the ground state of a local quadratic Hamiltonian.",
"They (i) are characterized by two polynomials, (ii) have correlation functions compatible with a cMPS or a subclass of cMPS in the UV, and (iii) are generated by a quasi-local entangler with $g(x)$ decaying at least exponentially at large $x$ but not smooth at $x=0$ ."
],
[
"1.Relation to conformal group",
"The scale invariant magic cMERA $\\mbox{$| \\Psi ^{\\Lambda } \\rangle $}$ in the main text is the exact ground state of any Hamiltonian of the form $H[E(k)] = \\int dk~~E(k) a^{\\Lambda }(k)^{\\dagger } a^{\\Lambda }(k)$ where the magic cMERA annihilation operators $a^{\\Lambda }(k)$ are fixed, namely $a^{\\Lambda }(k) &\\equiv & \\sqrt{\\frac{\\alpha (k)}{2}}\\phi (k) + \\frac{i}{\\sqrt{2\\alpha (k)}}\\pi (k), \\\\\\alpha (k) &\\equiv & \\sqrt{\\frac{k^2\\Lambda ^2}{k^2+\\Lambda ^2}},$ but where for the quasi-particle energies $E(k)$ we can choose any positive function.",
"Two specific choices of $E(k)$ stand up.",
"One makes $H$ strictly local, the other one makes $H$ part of a quasi-local representation of the conformal algebra.",
"In this work we wanted the Hamiltonian $H^{\\Lambda }$ to be local.",
"This requires the choice $E^{\\Lambda }(k) \\equiv \\sqrt{\\frac{k^2}{\\Lambda ^2}(k^2 + \\Lambda ^2)}.$ In Ref.",
"[50] we studied instead the dispersion relation $E^{\\mbox{\\tiny CFT}}(k)=|k|$ , in which case the Hamiltonian $H_{q.l.",
"}^{\\Lambda } \\equiv H[E^{\\mbox{\\tiny CFT}}(k)]$ is quasi-local, but by construction has the same spectrum as the local, relativistic CFT Hamiltonian $H^{\\mbox{\\tiny CFT}}$ in (REF ).",
"(Notice that in Ref.",
"[50], the quasi-local Hamiltonian $H_{q.l.",
"}^{\\Lambda }$ was denoted $H^{\\Lambda }$ ).",
"What makes $H_{q.l.",
"}^{\\Lambda }$ interesting is that it is part of a quasi-local realization of the conformal algebra, as described in Ref.",
"[50].",
"In particular, $D^\\Lambda \\equiv L+K$ is a quasi-local realization of the dilation operator, and we have that $D^{\\Lambda }$ and $H_{q.l.",
"}^{\\Lambda }$ obey the commutation relation $-i\\left[D^{\\Lambda },H_{q.l.",
"}^{\\Lambda } \\right] = H_{q.l.",
"}^{\\Lambda },$ which are the same as the commutation relation of CFT dilation operator $D^{\\mbox{\\tiny CFT}}$ and CFT Hamiltonian operator $H^{\\mbox{\\tiny CFT}}$ , namely $\\left[D^{\\mbox{\\tiny CFT}}, H^{\\mbox{\\tiny CFT}}\\right]$ .",
"That is, $H^{\\Lambda }_{q.l.",
"}$ is scale invariant (under the scale transformation generated by $D^{\\Lambda } = L + K$ ).",
"Instead, by requiring locality, which is of importance from a computational perspective [68], in this work we used a Hamiltonian $H^{\\Lambda }$ that is not scale invariant, that is $[D^{\\Lambda }, H^{\\Lambda }] \\ne 0$ .",
"We note, however, that since $H^{\\Lambda }$ and $H^{\\Lambda }_{q.l.",
"}$ have the same eigenvectors (indeed, by construction $\\left[ H^{\\Lambda }, H^{\\Lambda }_{q.l.",
"}\\right] =0$ ) and their dispersion relations $E^{\\Lambda }(k)$ of $E^{\\mbox{\\tiny CFT}}(k)$ are very similar at low energies $k \\ll \\Lambda $ , the violation of scale invariance is small at low energies."
],
[
"2.Derivation of scaling operators",
"Following Ref.",
"[50], the quasi-local scaling operators $\\phi ^{\\Lambda }(x)$ and $\\pi ^{\\Lambda }(x)$ are related to the sharp fields $\\phi (x)$ and $\\pi (x)$ by $\\phi ^{\\Lambda }(x) &=& \\int dy\\, \\mu _{\\phi }(x-y)\\phi (y)\\\\\\pi ^{\\Lambda }(x) &=& \\int dy\\, \\mu _{\\pi }(x-y)\\pi (y),$ where the Fourier transforms of the smearing functions are $\\mu _{\\phi }(k)&\\equiv & \\sqrt{\\frac{\\alpha (k)}{|k|}}=\\left(1+\\frac{k^2}{\\Lambda ^2}\\right)^{-1/2} \\\\\\mu _{\\pi }(k)&\\equiv & \\sqrt{\\frac{|k|}{\\alpha (k)}}=\\left(1+\\frac{k^2}{\\Lambda ^2}\\right)^{1/2}.$ They have distributional Fourier transforms [50] $\\mu _{\\phi }(x)&=&\\frac{2^{3/4}\\Lambda K_{1/4}(|\\Lambda x|)}{\\Gamma (1/4)|\\Lambda x|^{1/4}}\\\\\\mu _{\\pi }(x)&=&\\frac{2^{5/4}\\Lambda K_{3/4}(|\\Lambda x|)}{\\Gamma (-1/4)|\\Lambda x|^{3/4}}.$ Note that Eq.",
"(REF ) should be understood as the Hadamard finite-part integral $\\pi ^{\\Lambda }(0)=\\lim _{\\epsilon \\rightarrow 0} \\left(2 \\epsilon ^{-1/2}\\pi (0)+\\int _{\\mathcal {R}-(\\epsilon ,\\epsilon )} dx\\, \\mu _{\\pi }(x)\\pi (x) \\right).$ Other scaling operators include spatial derivatives $\\partial ^m_x \\phi ^{\\Lambda }(x)$ with scaling dimensions $m$ and $\\partial ^m_x \\pi ^{\\Lambda }(x)$ with scaling dimensions $m+1$ .",
"They can also be expressed as a distribution acting on the sharp fields $\\phi (x),\\pi (x)$ , with profiles $\\mu _{\\partial ^m_x\\phi }(x) &=& \\partial ^m_x \\mu _{\\phi }(x) \\\\\\mu _{\\partial ^m_x\\pi }(x) &=& \\partial ^m_x \\mu _{\\pi }(x).$ Some of the profile functions are plotted in the main text."
],
[
"1. Matrix product operator (MPO)",
"Consider a MPO made of matrices $A_m$ given by $A_m \\equiv \\left( \\begin{array}{ccc}\\mathbb {1} & E_m & 0 \\\\0 & \\lambda \\mathbb {1} & F_m \\\\0 & 0 & \\mathbb {1}\\end{array} \\right),$ where $E_m$ and $F_m$ are two operators and $\\mathbb {1}$ is the identity operator, all acting on the vector space of the lattice site $m$ .",
"The product of two contiguous MPO matrices $A_m$ and $A_{m+1}$ is $A_mA_{m+1} = \\left( \\begin{array}{ccc}\\mathbb {1} & E_m & 0 \\\\0 & \\lambda \\mathbb {1} & F_m \\\\0 & 0 & \\mathbb {1}\\end{array} \\right) \\left( \\begin{array}{ccc}\\mathbb {1} & E_{m+1} & 0 \\\\0 & \\lambda \\mathbb {1} & F_{m+1} \\\\0 & 0 & \\mathbb {1}\\end{array} \\right)= \\left( \\begin{array}{ccc}\\mathbb {1} & \\lambda E_m + E_{m+1} & E_mF_{m+1} \\\\0 & \\lambda ^2 \\mathbb {1} & F_m + \\lambda F_{m+1} \\\\0 & 0 & \\mathbb {1}\\end{array} \\right).$ Similarly, the product $A_{m}A_{m+1}A_{m+2}$ reads $A_{m}A_{m+1}A_{m+2} &=& \\left( \\begin{array}{ccc}\\mathbb {1} & \\lambda E_m + E_{m+1} & E_mF_{m+1} \\\\0 & \\lambda ^2 \\mathbb {1} & F_m + \\lambda F_{m+1} \\\\0 & 0 & \\mathbb {1}\\end{array} \\right)\\left( \\begin{array}{ccc}\\mathbb {1} & E_{m+2} & 0 \\\\0 & \\lambda \\mathbb {1} & F_{m+2} \\\\0 & 0 & \\mathbb {1}\\end{array} \\right)\\\\&=&\\left( \\begin{array}{ccc}\\mathbb {1} & ~~\\lambda ^2 E_m + \\lambda E_{m+1} + \\lambda E_{m+2}~~ &~~E_{m} F_{m+1} + E_{m+1}F_{m+2} + \\lambda E_m F_{m+2}~~\\\\0 & \\lambda ^3 \\mathbb {1} & F_m + \\lambda F_{m+1} +\\lambda ^2 F_{m+2} \\\\0 & 0 & \\mathbb {1}\\end{array} \\right),$ and by iteration we find that the product $A_1 A_2\\cdots A_N$ of $N$ such matrices reads $A_1 A_2\\cdots A_N =\\left( \\begin{array}{ccc}\\mathbb {1} & ~~~\\sum _{m=1}^N \\lambda ^{N-m} E_{m} ~~~& \\sum _{m=1}^N \\sum _{n=m+1}^{N} \\lambda ^{n-m-1} E_m F_n\\\\0 & \\lambda ^{N} \\mathbb {1} & \\sum _{m=1}^N \\lambda ^{m-1} F_m \\\\0 & 0 & \\mathbb {1}\\end{array} \\right).$ With the choice $E_m = F_m = \\sqrt{\\beta \\lambda \\epsilon }~ b_{m}$ and $\\lambda = e^{-\\epsilon \\Lambda }$ , the product becomes $A_1 A_2\\cdots A_N = \\left( \\begin{array}{ccc}\\mathbb {1} & ~~~ \\sqrt{\\beta \\epsilon } ~\\sum _{m=1}^N e^{-\\Lambda \\epsilon (N-m+\\frac{1}{2})} ~ b_{m} ~~~& \\beta \\epsilon ~ \\sum _{m=1}^N \\sum _{n=m+1}^{N} e^{-\\Lambda \\epsilon (n-m)} ~b_{m} b_{n}~~\\\\0 & e^{-\\Lambda \\epsilon N} \\mathbb {1} & \\sqrt{\\beta \\epsilon } ~\\sum _{m=1}^N \\lambda ^{-\\Lambda \\epsilon (m-\\frac{1}{2})} ~ b_{m} \\\\0 & 0 & \\mathbb {1}\\end{array} \\right).$ We are interested in the matrix element $(1,3)$ of this product, namely $\\mbox{$\\langle 1 |$} A_1 A_2 \\cdots A_N \\mbox{$| 3 \\rangle $} = \\frac{-i\\Lambda \\epsilon }{4} \\sum _{m<n} e^{-\\Lambda \\epsilon (n-m)} b_mb_n,$ which accounts for one half of the discrete version $K^{\\mbox{\\scriptsize lattice}}$ in the main text (the other half, quadratic in creation operators $b_m^{\\dagger } b_n^{\\dagger }$ , is obtained similarly)."
],
[
"2. Continuous matrix product operator (cMPO)",
"Next we introduce operators $\\psi (x_m) \\equiv b_m / \\sqrt{\\epsilon }$ , where $x_m \\equiv \\epsilon m$ , and expand the above matrix $A_m$ in powers of $\\epsilon $ , $A_m = \\left( \\begin{array}{ccc}\\mathbb {1} & E_m & 0 \\\\0 & \\lambda \\mathbb {1} & F_m \\\\0 & 0 & \\mathbb {1}\\end{array} \\right) = \\left( \\begin{array}{ccc}\\mathbb {1} &~~ \\epsilon ~\\sqrt{\\beta }e^{-\\Lambda \\epsilon /2} \\psi (x_m)& 0 \\\\0 & e^{-\\Lambda \\epsilon } \\mathbb {1} & ~~\\epsilon ~\\sqrt{\\beta }e^{-\\Lambda \\epsilon /2} \\psi (x_m) \\\\0 & 0 & \\mathbb {1}\\end{array} \\right) = \\mathbb {1} + \\epsilon \\mathcal {A}_m + O(\\epsilon ^2 ),$ where the cMPO matrix $\\mathcal {A}(x_m) = \\mathcal {A}_{m}$ reads $\\mathcal {A}(x_m) = \\left( \\begin{array}{ccc}0 &~~ \\sqrt{\\beta } \\psi (x_m) & 0 \\\\0 & -\\Lambda \\mathbb {1} & ~~\\sqrt{\\beta }\\psi (x_m) \\\\0 & 0 & 0\\end{array} \\right).$ We can now expressed the matrix product $A_1 A_2 \\cdots A_N$ in the double limit $\\epsilon \\rightarrow 0$ and $N\\rightarrow \\infty $ , with finite $L = N\\epsilon $ , as a path ordered exponential, $\\mathcal {P}\\exp \\left(\\int _0^L dx~\\mathcal {A}(x) \\right) \\equiv \\lim _{\\small {\\begin{array}{c} \\epsilon \\rightarrow 0\\\\N \\rightarrow \\infty \\end{array}}} \\left(1+\\epsilon \\mathcal {A}(x_1)\\right) \\left(1+\\epsilon \\mathcal {A}(x_2)\\right)\\cdots \\left(1+\\epsilon \\mathcal {A}(x_N)\\right),$ whose matrix element $(1,3)$ reads $\\mbox{$\\langle 1 |$}\\mathcal {P}\\exp \\left(\\int _0^L dx~\\mathcal {A}(x) \\right)\\mbox{$| 3 \\rangle $} &=& \\frac{-i\\Lambda }{4} \\lim _{\\small {\\begin{array}{c} \\epsilon \\rightarrow 0\\\\N \\rightarrow \\infty \\end{array}}}\\sum _{m=1}^N \\epsilon \\sum _{n=m+1}^N \\epsilon ~ e^{-\\Lambda \\epsilon (n-m)} \\psi (x_m) \\psi (x_n) \\\\&=& \\frac{-i\\Lambda }{4} \\int _0^L \\!\\!dx \\int _x^L \\!\\!dy ~ e^{- \\Lambda |x-y|} \\psi (x) \\psi (y),$ and thus accounts for half of the entangler $K$ in the main text.",
"We conclude that the entangler $K$ of the proposed magic cMERA can indeed be expressed in an extremely compact way using a cMPO.",
"In Ref.",
"[68] this observation, which also implies a compact cMPO representation for $e^{isK}$ for small $s$ , will be exploited as part of an efficient computational framework for cMERA, namely in order to numerically implement a scale evolution generated by $L+K$ ."
]
] | 1906.04218 |
[
[
"Altruism in populations at the extinction transition"
],
[
"Abstract We study the evolution of cooperation as a birth-death process in spatially extended populations.",
"The benefit from the altruistic behavior of a cooperator is implemented by decreasing the death rate of its direct neighbors.",
"The cost of cooperation is the increase of a cooperator's death rate proportional to the number of its neighbors.",
"When cooperation has higher cost than benefit, cooperators disappear.",
"Then the dynamics reduces to the contact process with defectors as the single particle type.",
"Increasing the benefit-cost ratio above 1, the extinction transition of the contact process splits into a set of nonequilibrium transitions between four regimes when increasing the baseline death rate $p$ as a control parameter: (i) defection only, (ii) coexistence, (iii) cooperation only, (iv) extinction.",
"We investigate the transitions between these regimes.",
"As the main result, we find that full cooperation is established at the extinction transition as long as benefit is strictly larger than cost.",
"Qualitatively identical phase diagrams are obtained for populations on square lattices and in pair approximation.",
"Spatial correlations with nearest neighbors only are thus sufficient for sustained cooperation."
],
[
"Introduction",
"Altruism or cooperativity [1] describe behavior that is more in favor of others than of the actor herself.",
"Alarm calls are an example of altruistic behavior: Increasing the risk of becoming prey itself first, one individual of a group warns the others of a predator approaching [2].",
"At first glance, the observation of altruism sustained over generations appears incompatible with Darwin's theory of natural selection, featuring the survival of the fittest [3], [4].",
"If non-altruists acting only to their own benefit have an advantage over altruists in terms of reproductive success, altruistic traits eventually disappear.",
"The question of sustained altruism and cooperativity has been addressed in the framework of evolutionary game theory, in particular by work on the Prisoner's Dilemma and Public Goods Games [5], [6].",
"In these and other games, the time evolution of the system is assumed to take place as a sequence of two elementary steps: (i) the combined behavioral choices of the participants lead to an assignment of a payoff to each player which (ii) determines the choice of their future strategies or roles.",
"In the simplest case, with two possible strategies, cooperation and defection, the dilemma arises as follows.",
"Regardless of the other agent's move, an agent's best (highest payoff) move is always defection.",
"On the other hand, the sum of all players payoffs is maximal when all cooperate.",
"Therefore, natural selection always favors defection [5], despite cooperation is the best global strategy.",
"The aforementioned social dilemma is frequently analyzed by means of the replicator equation [7], [8], [9] describing the time evolution of the fraction of players holding one of the two strategies.",
"If the fitness of an individual equals its payoff, the resulting replicator equation for the Prisoner's Dilemma has only two steady-state solutions, the only-defector and the only-cooperator solutions, the former being the only stable one.",
"Nevertheless, the prevalence of cooperation is still possible within the context of evolutionary games, provided appropriate reciprocity mechanisms are included in the dynamics [1], [6], [10], [11], [12]: Direct reciprocity, indirect reciprocity, kin selection, group selection, and network structure.",
"If compared to the well-mixed situation, the new mechanisms include update rules that favor the interactions among cooperators.",
"The network structure mechanism was one of the first reciprocity mechanisms studied in the literature.",
"It refers to the restriction of agents interactions among neighbors.",
"In a two-dimensional regular network, the survival of altruists was explained in terms of their ability of preventing the exploitation of defectors through the formation of clusters [13], [14], [15], [16].",
"Further progress in the field considered births and deaths: The second step of the dynamics, the one that allows a change of the strategy, is now interpreted as a death of a player followed by a birth.",
"The new ecologic perspective allowed to assess the importance of new relevant issues, such as the fluctuation of the population density [17], [18], [19], [20], the movement of agents [21], [22], [23], [24], [25], the spatial distribution of neighbors and their number [26], [27], among others.",
"Recent works also consider networks of interactions [7], [28], [29], [30], [31], [12], [32], focus on the critical properties of the system [33], [34], [35], include other novel dynamic rules [36], [37], [38], [39], [40], [41], [42], analyze the formation of patterns [13], [43], [44], [45], [46], [47], and evaluate the effect on the population growing as external pressure rises [48].",
"The latter aspect has been widely analyzed in the context of competing species [49], [50], but has not received much attention in relation with the prevalence of altruism.",
"Although general considerations about the prevalence of altruism in the context of the Public Goods Games can be inferred from the numerous studies on the topic [51], [52], the behavior of cooperation turns out to be very dependent on the specific dynamics considered [53], [54].",
"This is the case when trying to evaluate the importance of the spatial heterogeneity and the formation of clusters of cooperators: Many studies [14], [16], [55], [56], [18] explain the coexistence of cooperation and defection using the so called pair approximation, an approach that goes one step beyond mean field by tracking the dynamics of pairs of neighbors.",
"However, pair approximation still assumes spatial homogeneity of the system.",
"Hence, there is no need for the formation of clusters of cooperators for explaining their long-term survival.",
"Recent works on the evolution of cooperation suggest the need of giving up on certain common statements of evolutionary game theory [57], [58], [59], [60].",
"Particularly, some experiments on the dynamics of human cooperation show that people choose their strategy regardless the payoff of others [61].",
"Similar conclusions are given in the context of living beings [62].",
"See also recent experimental and numerical works on related topics [63], [64], [65], [66].",
"Here we study the evolution of cooperation in the framework of interacting particle systems.",
"We model birth and death in a spatially extended population as a contact process and ask the following: What is the phase diagram of the contact process with an additional — cooperative— type of particle that supports survival at neighboring sites?",
"Our approach provides a natural framework to assess the effects of different mechanisms on the behavior of the system and on the survival of cooperativity, such as the dynamics of interactions, the fluctuation in the population size, the presence or absence of cooperation clusters, and the spatial variation of parameters, among others.",
"The organization of the work is as follows.",
"In Sec.",
"we introduce the agent-based model of a population of cooperators and defectors living on a generic network.",
"For later sections the main focus is on the square lattice, where the system has only three relevant parameters: the total number of sites $N$ , the death parameter $p$ , and the cost-of-altruism parameter $\\epsilon $ .",
"Section includes stochastic simulations.",
"We obtain the phase diagram in the parameter space $(p,\\epsilon )$ showing the steady-state configurations of the system.",
"The effect of $p$ being spatially dependent is also addressed.",
"In Sec.",
"the system is described theoretically.",
"Three complementary formulations, using main-field or pair-approximation approaches, are given.",
"They aim at describing the system under different physical conditions.",
"Finally, a discussion and outlook of the main results are included in Sec.",
"."
],
[
"Definition of the model",
"The model describes the evolution of a population on an arbitrary network with $N$ nodes.",
"The set of neighbors of a node $i$ is denoted by $N_i$ .",
"The network is symmetric (undirected), so that $j \\in N_i$ implies $i\\in N_j$ ; also $i \\notin N_i$ (no self-loops).",
"Each agent in the population is either a cooperator $C$ or a defector $D$ , with $c_i$ and $d_i$ being their respective numbers at node $i$ .",
"A site or node of the network holds at most one agent ($C$ or $D$ ) but it may also be empty ($E$ ), hence $0\\le c_i+d_i\\le 1$ and $c_id_i=0$ .",
"Thus the state of the system $S$ is given by $S=\\lbrace c_i,d_i\\rbrace _{i=1}^N\\equiv \\lbrace x_i\\rbrace _{i=1}^N,\\quad x_i\\in \\lbrace c_i,d_i\\rbrace ,$ where $X$ is either a cooperator or a defector, and $x_i$ its number at site $i$ .",
"Moreover, the number $e_i=1-x_i$ gives 1 if site $i$ is empty and 0 if $x_i=1$ .",
"From condition $c_id_i=0$ we also have $e_ix_i=0$ .",
"A state transition is either the birth or the death of one agent at a site $i$ .",
"At the birth of a cooperator we set $c_i=1$ at a previously empty site $i$ , $e_i=1\\xrightarrow{} c_i=1$ Likewise for the birth of a defector, $d_i=1$ is set at an empty site $i$ , $e_i=1\\xrightarrow{} d_i=1$ These transitions occur at a rate proportional to the fraction of neighboring sites occupied by the agent type to be born, as $&\\pi _b(c_i,S) = e_i \\sum _{j \\in N_i}c_j/k_j \\equiv e_i \\tilde{c}_i,& \\\\&\\pi _b(d_i,S) = e_i \\sum _{j \\in N_i}d_j/k_j \\equiv e_i \\tilde{d}_i,&$ where $k_i = |N_i|$ is the degree (number of those neighbors) of node $i$ , and $\\tilde{x}_i\\equiv \\sum _{j \\in N_i} x_j/k_j$ .",
"The death of an agent is a state transition setting $c_i=0$ or $d_i=0$ at a previously occupied site $i$ , $&&c_i=1\\xrightarrow{} e_i=1, \\\\&&d_i=1\\xrightarrow{} e_i=1,$ with respective rates $\\nonumber \\pi _d(c_i,S)=&p\\left[c_i\\bar{e}_i+(1-\\epsilon )c_i\\bar{c}_i+(2-\\epsilon )c_i\\bar{d}_i\\right]& \\\\=& pc_i\\left\\lbrace 1- \\left[\\bar{c}_i - (1-\\epsilon ) (\\bar{c}_i+\\bar{d}_i)\\right] \\right\\rbrace ,& \\\\\\nonumber \\pi _d(d_i,S) =&p\\left(d_i\\bar{e}_i+d_i\\bar{d}_i\\right)& \\\\=&p d_i \\left(1- \\bar{c}_i\\right),&$ where now $\\bar{x}_i\\equiv k_i^{-1} \\sum _{j \\in N_i} x_j$ .",
"Agents die at a baseline rate $p$ .",
"This rate is reduced, however, by the fraction of adjacent sites occupied by a cooperator.",
"The death rate of a cooperator, on the other hand, has an additional positive term proportional (with factor $1-\\epsilon $ ) to the fraction of adjacent agents.",
"This way, the parameter $\\epsilon $ accounts for the cost of the altruistic act, the limit of $\\epsilon =0$ corresponding to maximum cost where the altruist definitely loses its life for saving that of its neighbor.",
"The other limit is costless altruism at $\\epsilon =1$ .",
"In the absence of cooperators, or in the absence of defectors with $\\epsilon =0$ , the model reduces to the contact process [67], [68] equivalent to the SIS (susceptible-infected-susceptible) model of epidemics [69], [70].",
"The equivalence is obtained by mapping each empty site to a susceptible individual and each site with a defector to an infected individual."
],
[
"Simulations",
"Let us first illustrate and numerically analyze the dynamics on periodic square lattices.",
"As defined above, the model features non-ergodicity.",
"Eventually both types of agents go extinct in a finite size system.",
"In the simulations in this section, a slightly modified version of the model is employed: We set to zero the death rate of an agent currently being the only one of its type (C or D).",
"This allows us to take long-term measurements of concentrations and distributions without having to restart the dynamics.",
"Given the rates, simulations are performed with a standard Gillespie algorithm [71], [72]."
],
[
"Square lattice with homogeneous parameters",
"Figure REF shows the parameter dependence of the stationary mean concentrations of agents.",
"At $\\epsilon =0$ , cooperators are absent in the whole range of $p$ , while the concentration of defectors is positive for $p < p_c \\approx 0.62$ and vanishes for $p>p_c$ .",
"Now fixing $0<\\epsilon <1$ and increasing $p$ from 0 to 1, the concentration of defectors $\\left\\langle d\\right\\rangle $ still decreases with $p$ .",
"Before $\\left\\langle d\\right\\rangle $ reaches zero, however, the concentration of cooperators $\\left\\langle c\\right\\rangle $ becomes positive.",
"Simulations on square lattices of smaller size ($N=20^2$ , $N=30^2$ ) and checks with $N=100^2$ yield results almost identical to those of Fig.",
"REF .",
"Figure: Total concentration of agents (a) on square lattices with N=50×50N=50 \\times 50 sites and (b) from the numerical solution of the pair approximation, Eqs ()-().",
"In both (a) and (b),the three curves are for parameter values ϵ=0.99,0.75,0.10\\epsilon = 0.99, 0.75, 0.10 (top to bottom).",
"The insets zoom in on the curves for ϵ=0.10\\epsilon =0.10.",
"The inset of (a) shows these curves for different system sizes N=30×30N=30 \\times 30 (dotted curve), N=50×50N=50\\times 50 (solid curve), and N=100×100N=100 \\times 100 (dashed curve).In the coexistence regime of cooperators and defectors (green area in Figure REF ), the growth of cooperation outweighs the decline of defection.",
"Here the total concentration of agents grows with $p$ , $\\frac{\\partial (\\left\\langle c\\right\\rangle + \\left\\langle d\\right\\rangle )}{\\partial p} >0~.$ Figure REF (a) explicitly shows this non-monotonicity by plotting $\\left\\langle c\\right\\rangle + \\left\\langle d\\right\\rangle $ versus $p$ for different choices of $\\epsilon $ .",
"Figure: Distributions of the number of agents on a square lattice with 50×5050 \\times 50 sites.",
"In the lower row, panels (c), (d), and (e), ϵ=0.75\\epsilon = 0.75.Each panel describes a transition between presence and absence of a type of agent.",
"The transitions are also marked in Fig.",
"with the panel identifiers (a)–(e).Let us now take a closer look at the transitions between the regimes observed in Figure REF .",
"To this end, we record the distributions in the number of agents (each type separately) and consider their changes under parameter variation.",
"Figure REF shows this analysis for five transitions (a)-(e), also marked in the bottom panel of Figure REF .",
"Transitions in Figs.",
"REF (a) and REF (e) are extinctions of one type of agent in the absence of the other type.",
"However, the transitions are distinguishable by the approximate exponents of the algebraic decay of distributions, giving $1/4$ for the extinction of defectors versus $3/7$ for cooperators.",
"This indicates that, even in the absence of defectors and close to the extinction transition (e), the dynamics of cooperators is essentially different from the contact process.",
"Differences in the distributions of the order parameter (Figure REF ), however, do not contradict transitions (a)-(e) belonging to the same universality class.",
"Transitions (a), (b) and (e) fulfill the premises of the directed percolation conjecture, cf.",
"section 3.3.6 in [73].",
"Transitions (c) and (d) do not fulfill the assumption of a unique absorbing state because only one type of agent goes extinct at the transition.",
"In preliminary numerical exploration (results not shown here), we have found the scaling of the order parameter (concentration of agents) compatible with the value $0.580(4)$ for exponent $\\beta $ in directed percolation in two dimensions [74].",
"We conjecture that all transitions (a)-(e) belong to the universality class of directed percolation."
],
[
"Spatially dependent parameter $p$",
"Let us study a variation of the model with a spatial dependence of the parameter $p$ , a way of mimicking ecological conditions [75], [76].",
"For an agent at lattice site $(x,y)$ , $x,y\\in \\lbrace 1,\\dots ,L\\rbrace $ , the death rate is based on the parameter value $p(x) = {\\left\\lbrace \\begin{array}{ll}\\frac{2x-1}{L} & \\text{if } x\\le L/2 \\\\\\frac{2(L-x)+1}{L} & \\text{otherwise.}\\end{array}\\right.",
"}$ For $L$ even, the minimum value $1/L$ is assumed by $p(x)$ at $x=1$ and $x=L$ ; its maximum value $1-1/L$ is obtained at $x=L/2$ and $x=L/2+1$ .",
"The parameter $\\epsilon $ remains spatially homogeneous, here $\\epsilon =0.75$ .",
"Figure REF (a) shows the concentration of agents as a function of lattice coordinate $x$ , i.e.",
"averaged over lattice coordinate $y$ and time.",
"We see that the effect of parameter $p$ is local.",
"The $p$ -dependence of $\\left\\langle c\\right\\rangle $ and $\\left\\langle d\\right\\rangle $ observed under spatially homogeneous $p$ in Section REF qualitatively matches that of the scenario with spatially dependent $p$ ."
],
[
"Analytic approximations",
"In this section, we derive three complementary theoretical descriptions of our model, defined in Sec. .",
"The first two ones are based on a mean-field approximation, while the third one uses the pair approximation.",
"As will be shown, the different approaches have different ranges of applicability and explain the prevalence/extinction and even the coexistence of altruism and defection under different physical and biological conditions.",
"In the case of the pair approximation, a very similar phase diagram to the numerical one shown in Fig.",
"REF is obtained.",
"Our starting point is the master equation for the probability $P(S,t)$ of finding the system in state $S$ at time $t$ .",
"By means of a probabilistic balance in the continuum time limit [77], and using the rates given by Eqs.",
"(REF )-(), the master equation reads as $\\begin{split}\\partial _tP(S,t)=\\sum _{i=1}^N&\\sum _{x_i\\in \\lbrace c_i,d_i\\rbrace }\\left\\lbrace (E^-_{x_i}-1)\\left[\\pi _b(x_i,S)P(S,t)\\right] \\right.",
"\\\\& \\left.",
"+(E^+_{x_i}-1)\\left[\\pi _d(x_i,S)P(S,t)\\right] \\right\\rbrace ,\\end{split}$ where the operators $E^\\pm _{x_i}$ act on a generic function $f(x_1,\\dots ,x_i,\\dots ,x_N)$ as $E^\\pm _{x_i}f(x_1,\\dots ,x_i,\\dots ,x_N)=f(x_1,\\dots ,x_i\\pm 1,\\dots ,x_N)$ , with $x_k\\in \\lbrace c_k,d_k\\rbrace $ , $k=1,\\dots ,N$ .",
"By taking moments of the master equation (REF ) we can derive equations for the mean numbers of cooperators and defectors in site $i$ , $\\left\\langle c_i\\right\\rangle $ and $\\left\\langle d_i\\right\\rangle $ .",
"After using the relation $e_i=1-c_i-d_i$ and some manipulations, we obtain $\\nonumber \\frac{d}{dt}\\left\\langle c_i\\right\\rangle =&& \\left\\langle \\pi _b(c_i)-\\pi _d(c_i)\\right\\rangle \\\\\\nonumber =&& \\left\\langle \\tilde{c}_i e_i\\right\\rangle -p\\left[\\left\\langle c_i\\bar{e}_i\\right\\rangle +(1-\\epsilon )\\left\\langle c_i\\bar{c}_i\\right\\rangle \\right.\\\\\\nonumber && \\left.+(2-\\epsilon )\\left\\langle c_i\\bar{d}_i\\right\\rangle \\right]\\\\\\nonumber =&&-p\\left\\langle c_i\\right\\rangle +\\left\\langle \\tilde{c}_i\\right\\rangle -\\left[\\left\\langle c_i\\tilde{c}_i\\right\\rangle -\\epsilon p \\left\\langle c_i\\bar{c}_i\\right\\rangle \\right.",
"\\\\&& \\qquad \\left.",
"+p(1-\\epsilon )\\left\\langle c_i\\bar{d}_i\\right\\rangle +\\left\\langle \\tilde{c}_i d_i\\right\\rangle \\right], \\\\\\nonumber \\frac{d}{dt}\\left\\langle d_i\\right\\rangle =&& \\left\\langle \\pi _b(d_i)-\\pi _d(d_i)\\right\\rangle \\\\\\nonumber =&& \\left\\langle \\tilde{d}_i e_i\\right\\rangle -p\\left[\\left\\langle d_i\\bar{e}_i\\right\\rangle +\\left\\langle d_i\\bar{d}_i\\right\\rangle \\right]\\\\\\nonumber =&&-p\\left\\langle d_i\\right\\rangle +\\left\\langle \\tilde{d}_i\\right\\rangle \\\\&&-\\left[\\left\\langle c_i\\tilde{d}_i\\right\\rangle -p\\left\\langle \\bar{c}_i d_i\\right\\rangle +\\left\\langle d_i\\bar{d}_i\\right\\rangle \\right],$ for $i=1,\\dots ,N$ .",
"Since the first moments are coupled to the second ones through correlations between neighbors, it is also convenient to derive equations for the two node correlations for neighboring sites, i.e.",
"$\\left\\langle x_ix_j\\right\\rangle $ with $j\\in N_i$ : $\\nonumber \\frac{d}{dt}\\left\\langle c_ic_j\\right\\rangle =&& \\left\\langle c_i\\pi _b(c_j)+\\pi _b(c_i)c_j-c_i\\pi _d(c_j)-\\pi _d(c_i)c_j\\right\\rangle \\\\\\nonumber = && \\left\\langle c_ie_j\\tilde{c}_j\\right\\rangle +\\left\\langle \\tilde{c}_i e_i c_j\\right\\rangle -p\\left\\langle c_ic_j(\\bar{e}_i+\\bar{e}_j)\\right\\rangle \\\\\\nonumber && -p(1-\\epsilon )\\left\\langle c_ic_j(\\bar{c}_i+\\bar{c}_j)\\right\\rangle \\\\&& -p(2-\\epsilon )\\left\\langle c_ic_j(\\bar{d}_i+\\bar{d}_j)\\right\\rangle , \\\\\\nonumber \\frac{d}{dt}\\left\\langle c_id_j\\right\\rangle =&& \\left\\langle c_i\\pi _b(d_j)+\\pi _b(c_i)d_j-c_i\\pi _d(d_j)-\\pi _d(c_i)d_j\\right\\rangle \\\\\\nonumber =&& \\left\\langle c_ie_j\\tilde{d}_j\\right\\rangle +\\left\\langle \\tilde{c}_i e_i d_j\\right\\rangle -p\\left\\langle \\bar{e}_ic_id_j\\right\\rangle \\\\\\nonumber && -p\\left\\langle c_id_j(\\bar{e}_j+\\bar{d}_j)\\right\\rangle -p(1-\\epsilon )\\left\\langle \\bar{c}_ic_id_j\\right\\rangle \\\\&& -p(2-\\epsilon )\\left\\langle \\bar{d}_ic_id_j\\right\\rangle , \\\\\\nonumber \\frac{d}{dt}\\left\\langle d_id_j\\right\\rangle =&& \\left\\langle d_i\\pi _b(d_j)+\\pi _b(d_i)d_j-d_i\\pi _d(d_j)-\\pi _d(d_i)d_j\\right\\rangle \\\\\\nonumber = && \\left\\langle d_ie_j\\tilde{d}_j\\right\\rangle +\\left\\langle \\tilde{d}_i e_i d_j\\right\\rangle -p\\left\\langle d_id_j(\\bar{e}_i+\\bar{e}_j)\\right\\rangle \\\\&& -p\\left\\langle d_id_j(\\bar{d}_i+\\bar{d}_j)\\right\\rangle ,$ where $\\tilde{x}_i$ and $\\bar{x}_i$ are defined just after Eqs.",
"() and (), respectively.",
"The two remaining moments, $\\left\\langle c_ie_j\\right\\rangle $ and $\\left\\langle d_ie_j\\right\\rangle $ can be obtained from the previous ones by means of the identity $1=e_i+c_i+d_i$ , as $\\left\\langle c_ie_j\\right\\rangle =\\left\\langle c_i\\right\\rangle -\\left\\langle c_ic_j\\right\\rangle -\\left\\langle c_id_j\\right\\rangle $ and $\\left\\langle d_ie_j\\right\\rangle =\\left\\langle d_i\\right\\rangle -\\left\\langle d_id_j\\right\\rangle -\\left\\langle d_ic_j\\right\\rangle $ .",
"Although the system of Eqs.",
"(REF )-() are exact and valid for any structure of neighbors (network), it is not closed, due to the presence of three nodes correlations.",
"Therefore, in order to have a closed set of equations, three approximations are explored.",
"The first two ones make use of the mean-field approximation, where two node correlations are ignored, and the third one uses pair approximation.",
"Furthermore, we restrict ourselves to regular networks where $k_i=k$ for all $i$ , so as to simplify the description (now $\\tilde{x}_i=\\bar{x}_i=k^{-1} \\sum _{j \\in N_i} x_j$ )."
],
[
"Exact relations",
"Before proceeding with the approximations, some exact relations will be derived.",
"They apply for homogeneous steady-state configurations.",
"Consider first the case of only defectors.",
"Since $\\left\\langle c\\right\\rangle =0$ , we also have $\\left\\langle cc\\right\\rangle =\\left\\langle cd\\right\\rangle =\\left\\langle ce\\right\\rangle =0$ .",
"Using Eq.",
"(), together with $\\left\\langle de\\right\\rangle =\\left\\langle d\\right\\rangle -\\left\\langle dd\\right\\rangle $ , we have $\\left\\langle dd\\right\\rangle =(1-p)\\left\\langle d\\right\\rangle ,$ and, with Eq.",
"() and the identity $\\left\\langle dde\\right\\rangle +\\left\\langle ddd\\right\\rangle =\\left\\langle dd\\right\\rangle $ , $p\\left[1-k(1-p)\\right]\\left\\langle d\\right\\rangle +(k-1)\\left\\langle ded\\right\\rangle =0.$ which implies, in order to have positive solutions, $1-k(1-p)\\le 0$ , that is $p\\le 1-\\frac{1}{k}.$ This is an overestimation of the extinction probability of defectors, for all $\\epsilon \\in [0,1]$ .",
"For $\\epsilon =0$ , where the model is the SIS model, and the square lattice ($k=4$ ), the previous estimation is $0.75$ while the one from the simulations is around $0.62$ [78], [79], see also Fig.",
"REF .",
"For the only-cooperator case, it is $\\left\\langle d\\right\\rangle =0$ and $\\left\\langle dd\\right\\rangle =\\left\\langle cd\\right\\rangle =\\left\\langle de\\right\\rangle =0$ .",
"Using Eq.",
"(REF ) together with $\\left\\langle ce\\right\\rangle =\\left\\langle c\\right\\rangle -\\left\\langle cc\\right\\rangle $ , we get $\\left\\langle cc\\right\\rangle =\\frac{1-p}{1-\\epsilon p}\\left\\langle c\\right\\rangle ,$ and, with Eq.",
"(REF ) and the identity $\\left\\langle cce\\right\\rangle +\\left\\langle ccc\\right\\rangle =\\left\\langle cc\\right\\rangle $ , $\\begin{split}&\\frac{p}{1-\\epsilon p}\\left[p(1-\\epsilon )-(k-1)(1-p)\\right]\\left\\langle c\\right\\rangle \\\\&\\qquad +(k-1)\\left(\\left\\langle cec\\right\\rangle +p\\epsilon \\left\\langle ccc\\right\\rangle \\right)=0,\\end{split}$ which now implies $p(1-\\epsilon )-(k-1)(1-p)\\le 0$ or $p\\le 1-\\frac{1-\\epsilon }{k-\\epsilon }\\ge 1-\\frac{1}{k}.$ Again, this is an overestimation of the critical probability extinction when there are only cooperators in the system.",
"The critical value here is bigger or equal to the one of Eq.",
"(REF ), as expected due to the altruistic benefit.",
"Equation (REF ) also provides an estimation of the dependence of the critical probability on $\\epsilon $ .",
"In particular, it tends to 1 for $\\epsilon \\rightarrow 1$ , in agreement with the numerical simulations of Fig.",
"REF .",
"Equations (REF ) and (REF ), and also the other relations, are the same for $\\epsilon =0$ provided we interchange the types of particles, because the model with only defectors and only cooperators coincide in this limit.",
"This can be seen from the rates defining the dynamics in Eqs.",
"(REF )-(): The rates for defectors in the absence of cooperators are the same as the rates for cooperators in the absence of defectors at $\\epsilon =0$ ."
],
[
"Global mean-field approximation",
"For the global mean-field case, equivalent to the dynamics on a complete graph in the limit of infinite system size, correlations among nodes are absent.",
"In general, assuming the mean-field approximation implies the following two approximations: $&& \\left\\langle x_i x_j\\right\\rangle \\simeq \\left\\langle x_i\\right\\rangle \\left\\langle x_j\\right\\rangle , \\qquad i\\ne j \\\\&& \\left\\langle x_i\\right\\rangle \\simeq \\left\\langle x_j\\right\\rangle \\equiv \\left\\langle x\\right\\rangle , \\qquad \\text{for all } i.$ This is also a good approximation when there is no correlation expected between the agents, for instance when there is one kind of agent and the distribution of empty sites is homogeneously distributed.",
"Then, the concentrations $\\left\\langle c\\right\\rangle $ and $\\left\\langle d\\right\\rangle $ of cooperators and defectors evolve, according to Eqs.",
"(REF ) and (), as $\\nonumber \\frac{d}{dt}\\left\\langle c\\right\\rangle =&&\\left\\langle c\\right\\rangle \\left\\lbrace (1-p)-(1-\\epsilon p)\\left\\langle c\\right\\rangle \\right.",
"\\\\&& \\qquad \\left.-\\left[1+p(1-\\epsilon )\\right]\\left\\langle d\\right\\rangle \\right\\rbrace , \\\\\\frac{d}{dt}\\left\\langle d\\right\\rangle =&&\\left\\langle d\\right\\rangle \\left[(1-p)-(1-p)\\left\\langle c\\right\\rangle -\\left\\langle d\\right\\rangle \\right].$ The system (REF ) and () can be used now to analyze the homogeneous steady-state solutions.",
"Requiring stationarity, $\\frac{d}{dt}\\left\\langle c\\right\\rangle =\\frac{d}{dt}\\left\\langle d\\right\\rangle =0$ , we find the trivial solution $\\left\\langle c\\right\\rangle =\\left\\langle d\\right\\rangle =0$ (all sites empty) and, two other, nontrivial ones, namely $&& \\left\\langle c\\right\\rangle =0\\; \\mathrm {and} \\; \\left\\langle d\\right\\rangle =1-p, \\\\&& \\left\\langle c\\right\\rangle =\\frac{1-p}{1-\\epsilon p}\\; \\mathrm {and} \\; \\left\\langle d\\right\\rangle =0.$ The trivial solution is clearly unstable, since the coefficient $1-p$ of the less degree terms in Eqs.",
"(REF ) and () is positive for $p< 1$ .",
"However, it is an absorbing state, and their presence becomes important for small system sizes, as already mentioned in Sec.",
".",
"In order to assess the stability of the solution with only defectors, consider the perturbation of Eq.",
"(REF ): $\\left\\langle c\\right\\rangle =0+\\left\\langle c\\right\\rangle _1$ and $\\left\\langle d\\right\\rangle =1-p+\\left\\langle d\\right\\rangle _1$ with $\\left\\langle c\\right\\rangle _1\\sim \\left\\langle d\\right\\rangle _1$ .",
"Then, up to linear order in the perturbations, we have $&&\\frac{d}{dt}\\left\\langle c\\right\\rangle _1\\simeq -p(1-p)(1-\\epsilon )\\left\\langle c\\right\\rangle _1, \\\\&&\\frac{d}{dt}\\left\\langle d\\right\\rangle _1\\simeq -(1-p)\\left[(1-p)\\left\\langle c\\right\\rangle _1+\\left\\langle d\\right\\rangle _1\\right].$ The first equation, and hence the second one, have $\\left\\langle c\\right\\rangle _1=\\left\\langle d\\right\\rangle _1=0$ as the steady solution, revealing the stable character of (REF ).",
"Proceeding similarly with the only-cooperators solution, Eq.",
"(), we obtain the system $\\frac{d}{dt}\\left\\langle c\\right\\rangle _1\\simeq && -\\frac{1-p}{1-\\epsilon p}\\left\\lbrace (1-\\epsilon p)\\left\\langle c\\right\\rangle _1 \\right.\\\\&& \\nonumber \\qquad \\left.",
"+\\left[1+p(1-\\epsilon )\\right]\\left\\langle d\\right\\rangle _1\\right\\rbrace , \\\\\\frac{d}{dt}\\left\\langle d\\right\\rangle _1\\simeq && p(1-p)\\frac{1-\\epsilon }{1-\\epsilon p}\\left\\langle d\\right\\rangle _1,$ which now reveals the unstable character of the solution, since the solution of Eq.",
"() increases exponentially with time.",
"According to this analysis, in well-mixed populations, cooperators go extinct."
],
[
"Local mean-field approximation",
"We can go one step beyond the global mean-field approximation by considering situations where the concentrations of cooperators and defectors change from site to site.",
"In particular, we suppose situations where the site dependence can be encoded through a vector $\\mathbf {r}$ , which is nothing but the vector of space position in a regular graph.",
"This way, we deduce in the sequel a macroscopic description that removes one of the approximation of the global mean field, namely that of Eq.",
"(), but still neglects correlations, Eq.",
"(REF ).",
"The procedure is similar to the one used in Ref.",
"[80].",
"By looking at the dynamics on a length scale $L$ much larger than the typical distance between sites $l$ , the relevant quantities become local concentrations: $&& \\kappa (\\mathbf {r}) \\equiv \\left\\langle c_i\\right\\rangle , \\\\&& \\delta (\\mathbf {r}) \\equiv \\left\\langle d_i\\right\\rangle .$ In a regular graph in $\\mathbb {R}^d$ , for example, $\\kappa (\\mathbf {r})$ and $\\delta (\\mathbf {r})$ give the number of cooperators and defectors inside a region of volume $l^d$ centered at position $\\mathbf {r}$ .",
"The new quantities are assumed to be smooth functions of $\\mathbf {r}$ , a property that allows us to relate any density of site $j\\in N_i$ and position $\\mathbf {l}$ , say $\\chi (\\mathbf {r}+\\mathbf {l})=\\kappa (\\mathbf {r}+\\mathbf {l})$ or $\\chi (\\mathbf {r}+\\mathbf {l})=\\delta (\\mathbf {r}+\\mathbf {l})$ at position $\\mathbf {r}$ , with that of site $i$ , $\\chi (\\mathbf {r})$ , as $&&\\chi (\\mathbf {r}+\\mathbf {l})\\simeq \\chi (\\mathbf {r})+\\nabla \\chi (\\mathbf {r})\\cdot \\mathbf {l}+\\frac{1}{2}\\nabla \\nabla \\chi (\\mathbf {r}): \\mathbf {l} \\mathbf {l},$ Hence, we have $\\left\\langle \\bar{x}_i\\right\\rangle =\\frac{1}{k_i}\\sum _{k\\in N_i}\\chi (\\mathbf {r}+\\mathbf {l}_k) \\simeq \\chi (\\mathbf {r})+\\nabla ^2_r\\chi (\\mathbf {r}).$ where we have assumed $\\sum _{k\\in N_i}\\mathbf {l}_k\\simeq 0$ , which is an exact expression for a regular square lattice and quiet a good approximation for isotropic configurations.",
"Moreover, $\\nabla ^2_r\\chi (\\mathbf {r})\\equiv \\frac{1}{2k_i}\\sum _{k\\in N_i}\\nabla \\nabla \\chi (\\mathbf {r}): \\mathbf {l}_k \\mathbf {l}_k\\simeq \\frac{l^2}{2d}\\nabla ^2\\chi (\\mathbf {r}),$ which is valid, again, under isotropic configurations of sites.",
"With approximations (REF ), (REF ), and (REF ), the exact system (REF ) and () becomes the following reaction-diffusion system $\\nonumber &&\\partial _t\\kappa =\\kappa \\left\\lbrace (1-p)-(1-\\epsilon p)\\kappa -\\left[1+p(1-\\epsilon )\\right]\\delta \\right\\rbrace \\\\&& \\quad +\\left[1-(1-\\epsilon p)\\kappa -\\delta \\right]\\nabla _r^2\\kappa -p(1-\\epsilon )\\kappa \\nabla _r^2\\delta , \\\\\\nonumber &&\\partial _t\\delta =\\delta \\left\\lbrace (1-p)-(1-p)\\kappa -\\delta \\right\\rbrace \\\\&& \\qquad +\\left[1-\\kappa -\\delta \\right]\\nabla _r^2\\delta +p\\delta \\nabla _r^2\\kappa .$ As expected, we recover the mean-field description for homogeneous solutions, hence we still have the solutions given in Eqs.",
"(REF ) and ().",
"However, an important benefit of the present description, if compared to that of the global mean-field approximation, is the possibility of studying the latter solutions under local perturbations, in contrast to homogeneous and global ones done in the previous subsection.",
"Consider the homogeneous solution of Eq.",
"(REF ), $\\kappa _0=0$ and $\\delta _0=1-p$ .",
"Following the standard linear stability analysis, we seek solutions of the form $\\kappa =\\kappa _0+\\kappa _1$ and $\\delta =\\delta _0+\\delta _1$ , with $\\kappa _1\\sim \\delta _1\\ll \\delta _0$ .",
"After linearizing and seeking solutions of the form $\\chi _1=\\tilde{\\chi }_1 e^{i\\xi \\cdot \\mathbf {r}}$ , system (REF ) and () becomes $&&\\partial _t\\tilde{\\kappa }_1=-p\\left[(1-p)(1-\\epsilon )+\\frac{l^2}{2d}\\xi ^2\\right]\\tilde{\\kappa }_1, \\\\&&\\partial _t\\tilde{\\delta }_1=-\\left[(1-p)+p\\frac{l^2}{2d}\\xi ^2\\right]\\left[(1-p)\\tilde{\\kappa }_1+\\tilde{\\delta }_1\\right].$ The steady state solution for any wavelength $\\mathbf {\\xi }$ is the trivial one, meaning that the solution of only defectors is linearly stable: any initial and small spatial perturbation in the number of defectors (and also cooperators) tends to zero as time increases.",
"Proceeding similarly with the solution of Eq.",
"(), we get $\\nonumber &&\\partial _t\\tilde{\\kappa }_1=-\\left[(1-p)(1-\\epsilon )+p\\frac{l^2}{2d}\\xi ^2\\right]\\tilde{\\kappa }_1, \\\\&& \\qquad -\\frac{1-p}{1-\\epsilon p}\\left[1+p(1-\\epsilon )\\left(1-\\frac{l^2}{2d}\\xi ^2\\right)\\right]\\tilde{\\delta }_1, \\\\&&\\partial _t\\tilde{\\delta }_1=\\frac{p(1-\\epsilon )}{1-\\epsilon p}\\left[(1-p)-\\frac{l^2}{2d}\\xi ^2\\right]\\tilde{\\delta }_1.$ In this case, the stability of the system depends on the value of $\\xi $ .",
"Setting $\\xi =2\\pi /L$ , the smallest allowed value for the given boundary conditions, the solution () is stable for $p<p_c^*$ with $p_c^*=1-\\frac{2\\pi ^2l^2}{dL^2}\\simeq 1-\\frac{2\\pi ^2}{dN^{\\frac{2}{d}}},$ where we have used the approximation $L/l\\simeq N^{1/d}$ .",
"This means that, under this approximation, the only-cooperators solution is stable for systems small enough.",
"For $N\\rightarrow \\infty $ it is $p_c^*\\rightarrow 1$ , and the solution is always unstable, and we recover the result of mean field.",
"Although the local mean-field approximation could in principle be seen as very crude, it shows the importance of taking into account the system size while describing altruism, as already pointed out in Ref.",
"[60].",
"In this case, the inclusion of spatial dependence, while still neglecting correlations, stabilizes the only-cooperators solution for $p<\\tilde{p}_c$ .",
"Moreover, the results suggest the existence of other solutions, spatially non-homogeneous ones, and the possibility of discontinuous (first-order) transitions among them.",
"This is because the only-defectors solution keeps always linearly stable, with no other stable solution close to it.",
"Figure: Phases in pair approximation.",
"The extinction transition between the regime of only cooperators (red area) to the empty system (white area) is given by the expression in pp and ϵ\\epsilon in Equation ().",
"The transition between coexistence (green area) and the regime of only cooperators is described by Equation () using expressions ()-().The transition between coexistence and the regime of only defectors (blue area) has an approximate description (dashed curve) in Equation () using expressions ()-().",
"The exact solution (boundary between blue and green area) has been obtained as well, details given elsewhere."
],
[
"Pair approximation",
"The previous mean-field approaches are expected to fail when the concentration of defectors and cooperators are locally correlated.",
"Since births occur among neighboring sites, correlations are expected to be important, in general.",
"Hence, we reconsider system (REF )-(), and try to express the three nodes moments as a function of the one and two nodes mean values.",
"Although different approaches are possible (see for instance [80]), we explore here the so-called pair approximation.",
"Pair approximation has been extensively applied to a variety of stochastic processes defined on a network, aiming at describing different situations as diverse as spin dynamics [81], [82], opinion dynamics [83], [84], [85], [86], [87], epidemics [88], [89], [90], and population dynamics [14], [16], [18], [55], [56].",
"In each of the cases, the pair approximation assumes that the probability of a given node quantity $x_i$ conditioned to the values of a neighboring site $x_j$ and to a next-neighboring site $x_k$ is independent of the latter [91]: $\\text{Prob}(x_i|x_jx_k)\\simeq \\text{Prob}(x_i|x_j)$ .",
"In other words, the state of a neighbor of a given node is considered to be independent of the state of another neighbor.",
"In our model, where $x_i\\in \\lbrace c,d,e\\rbrace $ takes the values 0 or 1, the mean values $\\left\\langle x_i x_j\\right\\rangle $ and $\\left\\langle x_i x_j x_k\\right\\rangle $ are essentially the respective probabilities of the given quantities, hence, under the pair approximation $\\left\\langle x_ix_jx_k\\right\\rangle =\\text{Prob}(x_i|x_jx_k)\\text{Prob}(x_jx_k)\\simeq P(x_i|x_j)\\text{Prob}(x_jx_k)$ , we have $\\left\\langle x_ix_jx_k\\right\\rangle \\simeq \\frac{\\left\\langle x_ix_j\\right\\rangle \\left\\langle x_jx_k\\right\\rangle }{\\left\\langle x_j\\right\\rangle }.$ Note that the order of appearance of the variables inside the brackets is important: $x_i$ refers to a node which is a neighbor of $x_j$ and $x_j$ is a neighbor of $x_k$ .",
"Observe that the pair approximation keeps the correlations regardless of the occupancy of the middle node, namely $\\sum _{x_j\\in \\lbrace c,d,e\\rbrace }\\left\\langle x_ix_jx_k\\right\\rangle =\\left\\langle x_i1x_k\\right\\rangle \\simeq \\sum _{x_j\\in \\lbrace c,d,e\\rbrace }\\frac{\\left\\langle x_ix_j\\right\\rangle \\left\\langle x_jx_k\\right\\rangle }{\\left\\langle x_j\\right\\rangle }\\ne \\left\\langle x_i\\right\\rangle \\left\\langle x_j\\right\\rangle $ , in general.",
"For simplicity, we consider homogeneous situations for which system (REF )-(), within the pair approximation of Eq.",
"(REF ), becomes $\\nonumber \\frac{d}{dt}\\left\\langle c\\right\\rangle &=&(1-p)\\left\\langle ce\\right\\rangle -p(1-\\epsilon )\\left\\langle cc\\right\\rangle \\\\&& -p(2-\\epsilon )\\left\\langle cd\\right\\rangle , \\\\\\frac{d}{dt}\\left\\langle d\\right\\rangle &=&(1-p)\\left\\langle de\\right\\rangle -p\\left\\langle dd\\right\\rangle , \\\\\\nonumber \\frac{k}{2}\\frac{d}{dt}\\left\\langle cc\\right\\rangle &=&\\left\\langle ce\\right\\rangle -p(1-\\epsilon )\\left\\langle cc\\right\\rangle \\\\\\nonumber &&+(k-1)\\left\\lbrace \\frac{\\left\\langle ce\\right\\rangle ^2}{\\left\\langle e\\right\\rangle }-p\\left[\\frac{\\left\\langle cc\\right\\rangle \\left\\langle ce\\right\\rangle }{\\left\\langle c\\right\\rangle } \\right.",
"\\right.",
"\\\\&&\\left.\\left.+(1-\\epsilon )\\frac{\\left\\langle cc\\right\\rangle ^2}{\\left\\langle c\\right\\rangle }+(2-\\epsilon )\\frac{\\left\\langle cc\\right\\rangle \\left\\langle cd\\right\\rangle }{\\left\\langle c\\right\\rangle }\\right]\\right\\rbrace , \\\\\\nonumber k\\frac{d}{dt}\\left\\langle cd\\right\\rangle &=&-p(2-\\epsilon )\\left\\langle cd\\right\\rangle +(k-1)\\left\\lbrace 2\\frac{\\left\\langle ce\\right\\rangle \\left\\langle ed\\right\\rangle }{\\left\\langle e\\right\\rangle } \\right.",
"\\\\\\nonumber &&-p\\left[\\frac{\\left\\langle ec\\right\\rangle {\\left\\langle cd\\right\\rangle }}{\\left\\langle c\\right\\rangle }+\\frac{\\left\\langle cd\\right\\rangle \\left\\langle de\\right\\rangle }{\\left\\langle d\\right\\rangle }+\\frac{\\left\\langle cd\\right\\rangle \\left\\langle dd\\right\\rangle }{\\left\\langle d\\right\\rangle } \\right.",
"\\\\&&\\left.\\left.+(1-\\epsilon )\\frac{\\left\\langle cc\\right\\rangle \\left\\langle cd\\right\\rangle }{\\left\\langle c\\right\\rangle }+(2-\\epsilon )\\frac{\\left\\langle cd\\right\\rangle ^2}{\\left\\langle c\\right\\rangle }\\right]\\right\\rbrace , \\\\\\nonumber \\frac{k}{2}\\frac{d}{dt}\\left\\langle dd\\right\\rangle &=&\\left\\langle de\\right\\rangle -p\\left\\langle dd\\right\\rangle +(k-1)\\left[\\frac{\\left\\langle de\\right\\rangle ^2}{\\left\\langle e\\right\\rangle } \\right.\\\\&&\\left.",
"-p\\left(\\frac{\\left\\langle dd\\right\\rangle \\left\\langle de\\right\\rangle }{\\left\\langle d\\right\\rangle }+\\frac{\\left\\langle dd\\right\\rangle ^2}{\\left\\langle d\\right\\rangle }\\right)\\right],$ where $\\left\\langle xy\\right\\rangle $ is for any two adjacent nodes with particles $x$ and $y$ .",
"Hence, $\\left\\langle xy\\right\\rangle =\\left\\langle yx\\right\\rangle $ ."
],
[
"Steady-state solutions",
"The system (REF )-() has several steady-state solutions.",
"The most obvious one it the trivial solution, without particles, $\\left\\langle c\\right\\rangle =\\left\\langle d\\right\\rangle =0$ .",
"This is the absorbing state we have already mentioned.",
"By setting all time derivatives of Eqs.",
"(REF )-() to zero and $c=0$ , we obtain the steady-state solution for defection only: $\\left\\langle d\\right\\rangle =\\frac{(1-p)k-1}{k-(1+p)},$ valid for $p\\le 1-\\frac{1}{k}.$ Observe that the previous inequality is the same as the one in Eq.",
"(REF ), derived using exact relations.",
"However, in this case, $p=1-1/k$ is the exact critical value for the extinction of defectors in the absence of cooperators, within the pair approximation.",
"The only-cooperators solution is obtained from Eqs.",
"(REF )-() as a steady-state solution with $d=0$ .",
"Now, $\\left\\langle c\\right\\rangle =\\frac{(1-p)k-(1-\\epsilon p^2)}{[k-(1+p)](1-\\epsilon p)},$ for $p\\le \\frac{k-\\sqrt{k^2-4\\epsilon (k-1)}}{2\\epsilon }\\ge 1-\\frac{1}{k}.$ The equality of the last relation holds for $\\epsilon \\rightarrow 0$ .",
"For the other limiting value of $\\epsilon $ , i.e.",
"$\\epsilon \\rightarrow 1$ , the upper allowed value of $p$ is 1, as we also obtained exactly.",
"Other steady-state solutions describing coexistence, but close to the previous ones, can also be found as follows.",
"First, we notice that the system (REF )-(), under the steady-state condition, can be reduced to a nonlinear system of only two equations with $\\left\\langle c\\right\\rangle $ and $\\left\\langle d\\right\\rangle $ as unknown quantities.",
"Second, we seek solutions close to the one-type ones, i.e.",
"$\\left\\langle c\\right\\rangle \\simeq \\frac{1-k(1-p)-\\epsilon p^2}{(1-k+p)(1-\\epsilon p)}$ and $\\left\\langle d\\right\\rangle \\simeq 0$ for the only-cooperators case and $\\left\\langle c\\right\\rangle \\simeq 0$ and $\\left\\langle d\\right\\rangle \\simeq \\frac{1-k(1-p)}{1-k+p}$ for the only-defectors.",
"For the former case, the resulting equations are linear and nontrivial solutions appear below the following line: $\\epsilon _c(p)=\\frac{A(p)B(p)}{C(p)+\\sqrt{C^2(p)-A^2(p)B(p)}},$ with $A(p)&=&2p(1-p)(k-1-kp) \\\\B(p)&=&(k+1-(k+2)p+2p^2)/[p(1-p)], \\\\\\nonumber C(p)&=&k(k-1)-(2k^2-3k+2)p \\\\&&+(k^2-3k+1)p^2+(k+2)p^3-2p^4.$ For the only-defectors case, the resulting set of equation is nonlinear, but one can still find a condition for a nontrivial solution to exist.",
"Now, the nontrivial solutions are above $\\epsilon _d(p)$ , which has the following approximate expression $\\epsilon _d(p)\\simeq \\frac{E(p)}{F(p)}\\left[1-\\sqrt{1-\\frac{2G(p)F(p)}{E^2(p)}}\\right],$ with $\\nonumber E(p)&=&(k-1)^3+(k-1)(4k^2-7k+8)p\\\\\\nonumber && -\\lbrace k[5k(k-5)+28]-3\\rbrace p^2\\\\&& -(13k-5)kp^3, \\\\\\nonumber F(p)&=&2p\\lbrace (k-1)(2k^2-3k+2)\\\\\\nonumber &&\\quad -(2k^3-14k^2+14k-1)p \\\\&& \\qquad -(9k-4)kp^2\\rbrace , \\\\\\nonumber G(p)&=&(k-1-kp)[(k-1)^2\\\\&&\\quad +(3k^2-7k+10)p+2(3k-1)p^2].$ Since $\\epsilon _c(p)\\ge \\epsilon _d(p)$ , the coexistence solutions are in the region in between the two lines, as shown in Fig.",
"REF for the square lattice ($k=4$ ).",
"Finally, there may be other solutions describing coexistence not necessarily close to the only-cooperator nor only-defectors ones.",
"This can be shown explicitly for $\\epsilon =1$ , for which we can obtain explicit expressions.",
"After some algebra, we get $&& \\left\\langle c\\right\\rangle =\\frac{2(k-1)(2k-3)(1-3p)}{(1-p)[4(k-1)(k-p-2)+p+1]}, \\\\&& \\left\\langle d\\right\\rangle =\\frac{(2k-3)[(4k-5)p-1]}{4(k-1)(k-p-2)+p+1}, \\\\&& \\left\\langle cd\\right\\rangle =\\frac{4(k-1)(k-p-2)+p+1}{2(k-1)(2k-3)}\\left\\langle c\\right\\rangle \\left\\langle d\\right\\rangle ,$ valid for $\\frac{1}{4k-5}\\le p\\le \\frac{1}{3}.$ Moreover, it can be seen that this solution is linearly unstable, with only one unstable mode.",
"However, the characteristic time of the unstable mode is much slower than the others, meaning that the system can stay close to the solution for a long time."
],
[
"Stability of the steady-state solutions",
"The stability of the only-defectors and only-cooperators solutions have been studied by means of a modified linear stability analysis of system (REF )-(), following several steps.",
"First, using the identities $\\left\\langle ce\\right\\rangle =\\left\\langle c\\right\\rangle -\\left\\langle cc\\right\\rangle -\\left\\langle cd\\right\\rangle $ and $\\left\\langle de\\right\\rangle =\\left\\langle d\\right\\rangle -\\left\\langle cd\\right\\rangle -\\left\\langle dd\\right\\rangle $ , all mean values are expressed in terms of $\\left\\langle c\\right\\rangle $ , $\\left\\langle d\\right\\rangle $ , $\\left\\langle cc\\right\\rangle $ , $\\left\\langle cd\\right\\rangle $ , and $\\left\\langle dd\\right\\rangle $ .",
"Second, the homogeneous solution is linearly perturbed as $\\mathbf {u}=\\mathbf {u}_0+\\gamma \\mathbf {u}_1,$ with $\\mathbf {u}=(\\left\\langle c\\right\\rangle ,\\left\\langle d\\right\\rangle ,\\left\\langle cc\\right\\rangle ,\\left\\langle cd\\right\\rangle ,\\left\\langle dd\\right\\rangle )$ the vector of the homogeneous solutions, $\\mathbf {u}_0$ is the vector of the unperturbed solutions, $\\mathbf {u}_1$ is the perturbation vector, and $\\gamma $ a perturbative parameter.",
"Third, the proposed solution is replaced in (REF )-() and the resulting system is expanded up to linear order in $\\gamma $ .",
"Contrary to the usual linear perturbation schemes, we obtain a nonlinear closed system of equations for the unknown perturbation quantities $u_{1,i},$ for $i=1,\\dots , 5$ .",
"For both, the only-cooperators and only-defectors solutions, the equation for the perturbation can be written as $\\frac{d}{dt}\\mathbf {u}_1=M(\\beta )\\mathbf {u}_1,$ with $M$ being a matrix and $\\beta $ a linear function of $\\left\\langle cc\\right\\rangle /\\left\\langle c\\right\\rangle $ , $\\left\\langle cd\\right\\rangle /\\left\\langle c\\right\\rangle $ , $\\left\\langle cd\\right\\rangle /\\left\\langle d\\right\\rangle $ , and $\\left\\langle dd\\right\\rangle /\\left\\langle d\\right\\rangle $ , whose explicit form depends on the solution considered.",
"In any case, $\\beta $ is a bounded function, since $0\\le \\left\\langle xy\\right\\rangle /\\left\\langle x\\right\\rangle \\le 1$ for $x,y\\in \\lbrace c,d\\rbrace $ .",
"Finally, the asymptotic behavior of $\\mathbf {u}_1(t)$ for $t\\rightarrow \\infty $ , hence the stable or unstable character of $\\mathbf {u}_0$ , can be determined from the spectra of $M(\\beta )$ for any $\\beta $ , using the following lemma.",
"Lemma: If all eigenvalues of $M(\\beta )$ have negative real parts for all values of $\\beta $ , then $\\mathbf {u}_0$ is linearly stable.",
"Proof: Given a time $t>0$ and an integer $n>0$ , we define $t_i=\\frac{t}{M}i$ for $i=0,\\dots ,n$ .",
"Thanks to the Mean Value Theorem, it is $\\mathbf {u}_1(t_i)=\\left(I+M_1\\frac{t}{n}\\right)\\mathbf {u}(t_{i-1})$ for $i\\ge 1$ , where $M_i$ is the value of $M$ for a time in $(t_{i-1},t_i)$ and use has been made of Eq.",
"(REF ) to evaluate the time derivative.",
"By iteration, $\\mathbf {u}_1(t_i)=\\left[\\prod _{k=1}^i\\left(I+M_k\\frac{t}{n}\\right)\\right]\\mathbf {u}_0$ .",
"Denoting by $\\Vert \\cdot \\Vert $ any vector norm, we have $\\Vert \\mathbf {u}_1(t)\\Vert = \\Vert \\left[\\prod _{k=1}^n(I+M_k\\frac{t}{n})\\right]\\mathbf {u}_0 \\Vert \\le \\Vert (I+\\tilde{M}\\frac{t}{n})^n\\mathbf {u}_0 \\Vert $ where $\\tilde{M}$ is such that $\\Vert (I+\\tilde{M}\\frac{t}{n})\\mathbf {u}_0 \\Vert =\\max _{k}\\Vert \\left(I+M_k\\frac{t}{n}\\right)\\mathbf {u}_0 \\Vert $ .",
"Taking $n\\rightarrow \\infty $ , $\\Vert \\mathbf {u}_1(t)\\Vert \\le \\Vert e^{\\tilde{M} t}\\mathbf {u}_0\\Vert $ which tends to zero as $t\\rightarrow \\infty $ , as all eigenvalues of $\\tilde{M}$ have negative real parts.",
"$\\square $ Using the lemma, we see that the only-cooperators solution is stable above line $\\epsilon _c(p)$ given by Eq.",
"(REF ), and the only-defectors solution is stable below line $\\epsilon _d(p)$ given approximately by Eq.",
"(REF ).",
"This implies that the instability of the one-type solutions is due to the presence of coexistence solutions which become stable.",
"Numerical evaluation of the time evolution of system (REF )-() confirms the theoretical analysis."
],
[
"Discussion",
"As the root of the present work, we introduce a basic stochastic model of a spatially extended population of altrustic and non-altruistic agents, called cooperators and defectors.",
"The population evolves by a birth-death process.",
"In line with the considerations by Huang and colleages [19], an agents' interaction with another agent influences the death rates only.",
"Agents' interactions are altruistic acts.",
"They lower the death rate of the recipient while increasing the death rate of the donor, relative to a baseline death rate $p$ for agents in isolation, $p$ being one of two parameters of the model.",
"The benefit-cost ratio of the altruistic is encoded in the second parameter $\\epsilon $ .",
"Results are obained as (1) stochastic simulations of finite systems and (2) stationary solutions and their stability in approximate descriptions by rate equations.",
"The pair approximation, neglecting all spatial correlations except those of nearest neighbors, yields our main result: For any benefit-cost ratio above 1, the stable stationary solutions in dependence of baseline death rate $p$ display (i) a regime of co-existence of cooperators and defectors and (ii) a regime of a population of cooperators only.",
"In the $(p,\\epsilon )$ parameter plane, these regimes and the related transitions appear as a continuation of the known extinction transition for a spatial population without cooperative interaction (also known as contact process, asynchronous SIS model).",
"The latter case corresponds to benefit-cost ratio of exactly 1 ($\\epsilon =0$ ).",
"The phase diagram from pair approximation is fully qualitatively consistent with that from stochastic simulation with finite square lattices.",
"Simulations of sufficiently large instances of $k$ -regular random graphs yield an equivalent phase diagram (results not shown here); this holds also for preliminary simulation results on other graphs, including scale-free [92] and small-world networks [93].",
"Thus we speculate that the observed type of $(p,\\epsilon )$ phase diagram is generic, holding for most types of connected sparse graphs.",
"For dense graphs, however, we expect mean-field behavior without stable cooperation seen in Sec.",
"REF .",
"Consider a spatially extended population subject to a decline in livability, which in reality may be a reduction of food resources or an increase of predators.",
"In our model, this scenario is represented by increasing $p$ and comes with the following prediction.",
"Initially without cooperators, the concentration of agents decreases until reaching a transition point with the onset of co-existence.",
"In this regime, the concentration of defectors further decreases; this decrease is overcompensated by the increase in cooperators.",
"Thus in the co-existence regime, there is net population growth under increasing $p$ [48].",
"Further increase of $p$ first leads to a regime with a population containing cooperators only and then an extinction phase where zero population size is the only stable solution.",
"From earlier studies, both theoretical [48] and experimental, increasing baseline death rate has been known to enhance cooperation.",
"Perturbing populations of yeast cells by dilution shocks, Sánchez and Gore find populations with larger fractions of cooperative cells (providing digestive enzyme to the population) more likely to survive [94].",
"Datta and co-authors observe cooperation promoted when a population expands the space it occupies, cooperators forming a wave of invaders [95].",
"According to the rule by Ohtsuki and colleagues [51], cooperation supersedes defection when the benefit-cost ratio is larger than the agent's number of neighbors $z$ .",
"While their theory assumes a population of constant size and each agent with a constant number $z$ of neighbors, we here see cooperation enhanced when the number of neighbors (occupied adjacent sites) is reduced dynamically due to a shrinking population density.",
"When the population most “needs” it, i.e.",
"at low density close to extinction, cooperation appears as a stable stationary solution for any benefit-cost ratio above 1.",
"Future work may check if a rule relating benefit-cost ratio and neighborhood size characterizes the appearance of stable cooperation also in the present model with varying population size.",
"For experimentally testing the present model's predictions, the unperturbed steady state of a population would have to be observed.",
"Giving up the spatial homogeneity of the baseline death rate $p$ , we have investigated the scenario of a gradient between low $p$ (high livability) and high $p$ (low livability).",
"The regimes encountered previously by tuning $p$ for the whole system are now found simultaneously at their corresponding spatial position.",
"In particular, high concentration of cooperators is found next to the region uninhabited due to large death rate $p$ .",
"There is a region of co-existence where total population concentration increases with $p$ also spatially.",
"Cooperation arises when and where needed to avoid extinction."
],
[
"Acknowledgments",
"Partial financial support has been received from the Agencia Estatal de Investigacion (AEI, Spain) and Fondo Europeo de Desarrollo Regional under Project PACSS RTI2018-093732-B-C21 (AEI/FEDER,UE) and the Spanish State Research Agency, through the Maria de Maeztu Program for units of Excellence in R&D (MDM-2017-0711).",
"KK acknowledges funding from MINECO through the Ramón y Cajal program and through project SPASIMM, FIS2016-80067-P (AEI/FEDER, EU)."
]
] | 1906.04202 |
[
[
"NoMoS: An $R \\times B$ Drift Momentum Spectrometer for Beta Decay\n Studies"
],
[
"Abstract The beta decay of the free neutron provides several probes to test the Standard Model of particle physics as well as to search for extensions thereof.",
"Hence, multiple experiments investigating the decay have already been performed, are under way or are being prepared.",
"These measure the mean lifetime, angular correlation coefficients or various spectra of the charged decay products (proton and electron).",
"NoMoS, the Neutron decay prOducts MOmentum Spectrometer, presents a novel method of momentum spectroscopy: it utilizes the $R \\times B$ drift effect to disperse charged particles dependent on their momentum in an uniformly curved magnetic field.",
"This spectrometer is designed to precisely measure momentum spectra and angular correlation coefficients in free neutron beta decay to test the Standard Model and to search for new physics beyond.",
"With NoMoS, we aim to measure inter alia the electron-antineutrino correlation coefficient $a$ and the Fierz interference term $b$ with an ultimate precision of $\\Delta a/a < 0.3\\%$ and $\\Delta b < 10^{-3}$ respectively.",
"In this paper, we present the measurement principles, discuss measurement uncertainties and systematics, and give a status update."
],
[
"Introduction",
"The Standard Model of particle physics (SM) is the basis of our current understanding of elementary particles and their fundamental interactions.",
"Although it describes a wide variety of phenomena and gives insights into various aspects of particle physics, current observations show its limitations (dark matter, baryon asymmetry, etc.).",
"A very sensitive test of the SM or new physics beyond is investigating the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix.",
"If one considers the first row of the CKM matrix, $V_{\\text{ud}}$ gives the most dominant contribution to the unitarity condition.",
"There are several measurement options to determine $V_{\\text{ud}}$ , including superallowed ($0^+ \\rightarrow 0^+$ ) nuclear, neutron, nuclear ($T=1/2$ ) mirror and pion beta decays [1].",
"The superallowed nuclear beta decays currently give the most stringent constraint as the other options' uncertainties are dominated experimentally.",
"Recently however, the inner radiative correction $\\Delta _R^V$ has been updated, resulting in a downward shift of $V_{\\text{ud}}$ extracted from superallowed beta decays and a 4 $\\sigma $ deviation from CKM unitarity [2], which strongly increases the motivation for further investigations.",
"Note that deviations from CKM unitarity can be used to perform indirect searches for physics beyond, e.g., for scalars or supersymmetry.",
"Neutron beta decay presents a compelling alternative to determine $V_{\\text{ud}}$ as it doesn't require nuclear corrections, in contrast to the superallowed decays."
],
[
"Neutron Beta Decay in the Standard Model",
"The beta decay of the free neutron is well described within the V-A theory of the SM.",
"Fermi's Golden Rule for the neutron's decay rate yields the following insightful correlation between the mean lifetime $\\tau _{\\text{n}}$ , the weak axial-vector coupling constant $g_\\text{A}$ and $V_\\text{ud}$ [3]: $\\frac{1}{\\tau _{\\text{n}}}=\\frac{G_{\\mu }^2 \\vert V_ {\\text{ud}}\\vert ^2}{2\\pi ^3}m_e^5 \\left(1+3g_A^2 \\right)(1+\\delta _R)(1+\\Delta _R^V)f$ with the Fermi coupling constant $G_\\mu $ , the electron's mass $m_e$ as well as the outer and inner radiative corrections $\\delta _R$ and $\\Delta _R^V$ respectively, and the phase space factor $f$ .",
"Hence, $V_{\\text{ud}}$ can be determined from independent measurements of $\\tau _{\\text{n}}$ and the ratio of axial-vector to vector coupling constant $\\lambda = g_{\\text{A}}/g_{\\text{V}}$ in neutron beta decay (the conserved vector current hypothesis requires $g_{\\text{V}}=1$ for zero momentum transfer).",
"As discussed in [4], the current discrepancies in the determination of $\\tau _{\\text{n}}$ (significant difference between single measurement techniques) and $\\lambda $ (time-dependent trend) present additional considerable motivation to further investigate this decay.",
"Up until now, $\\lambda $ is determined most precisely from measurements of the electron asymmetry parameter $A$ [5], [6], [7], [8], [9], [10] (current Particle Data Group accuracy $\\Delta A /A=0.84 \\,\\%$ [11], which doesn't include the most recent results).",
"Measurements of the electron-antineutrino angular correlation coefficient $a$ have not reached sub-percent accuracy yet [12], [13], [14] (currently $\\Delta a /a=2.6 \\,\\%$ [11]).",
"However, they offer an independent approach with significant potential for improvement.",
"Therefore, a number of experiments is currently putting effort into improving on it [15], [16], [17], [14], [18], [19], [20].",
"The data analysis of the aSPECT experiment is almost finished and will lead to a final uncertainty of $\\Delta a/a \\sim 1\\% $ [21].",
"aCORN continues data taking with an expected ultimate uncertainty of $\\sim 1 \\%$ [14].",
"The Nab experiment aims to determine $a$ with an ultimate precision of $ \\sim 0.1 \\%$ [20].",
"The coefficient $a$ can inter alia be determined from the proton momentum spectrum, which we plan to measure with NoMoS for a systematically independent determination of $\\lambda $ .",
"We aim to measure $a$ with an ultimate precision of $\\Delta a/a < 0.3 \\%$ .",
"In the weak interaction, extensions of the SM introduce additional couplings in beta decay [22], for example scalar and tensor couplings (see [23], [24], [25] for extensive discussions of possible extensions).",
"One observable with exceptional sensitivity to these exotic couplings is the Fierz interference term $b$ .",
"A non-zero measurement of the Fierz term with $10^{-3}$ absolute sensitivity is complementary to and competitive with searches for non-SM scalar and tensor couplings in pion and muon decay and with the LHC at full luminosity and energy [26], [27], [28].",
"The Fierz term can be measured in a variety of beta decays with different sensitivities to scalar and tensor couplings.",
"Despite their extraordinary precision, pure Fermi $0^+ \\rightarrow 0^+$ decays are only sensitive to scalar couplings.",
"Neutron beta decay is a mixed Fermi/Gamow-Teller transition and therefore sensitive to both scalar and tensor couplings, which further motivates a precise measurement of $b$ in neutron beta decay.",
"Recently, the UCNA collaboration extracted the Fierz term for the first time in neutron beta decay from existing beta asymmetry data ($b=0.067 \\pm 0.005_\\text{stat} \\, _{-0.061}^{+0.090} \\, _\\text{sys}$ ) [29].",
"Note that the result's error is dominated by systematics.",
"The Nab experiment aims to determine $b$ with an ultimate accuracy of $\\Delta b<10^{-3}$ [30].",
"With NoMoS we plan to measure the Fierz term with an ultimate accuracy of $\\Delta b < 10^{-3}$ via the electron's momentum spectrum.",
"Figure REF shows the sensitivity of the electron momentum spectrum on $b$ .",
"The further physics goals of NoMoS are presented in Ref.",
"[31].",
"Altogether, NoMoS is a promising tool to both test the SM and search for new physics beyond.",
"Figure: Top: The electron momentum spectrum in free neutron decay for the SM value of b=0b=0.",
"Bottom: Deviation for other value of b=0.001b=0.001."
],
[
"The $R \\times B$ Spectrometer",
"As introduced in Sec.",
", we present a novel technique of momentum spectroscopy which poses an independent approach for precision measurements in neutron beta decay.",
"NoMoS uses the $R \\times B$ drift effect to separate charged particles according to their momentum.",
"The drift velocity is given by [32], [33]: $\\vec{v}_{\\text{RxB}}=\\frac{p}{qRB}v_{\\parallel } f(\\theta )\\frac{\\vec{R}\\times \\vec{B}}{RB}+ \\frac{\\gamma m}{q B^2}\\left(\\dot{\\vec{v}}_{\\text{RxB}} \\times \\vec{B}\\right)$ with the particle's momentum $p$ and charge $q$ , the curvature radius $\\vec{R}$ and the magnetic field $\\vec{B}$ , the velocity component parallel to the $B$ -field $v_\\parallel $ , the relativistic factor $\\gamma $ , the particle's mass $m$ and the factor $f(\\theta )=\\left(\\cos (\\theta )+1/\\cos (\\theta )\\right)/2$ , which depends on the particle's incident angle $\\theta $ .",
"Equation (REF ) is implicit as the inertia drift (second summand) includes the time derivative of the drift velocity.",
"The inertia drift introduces higher order contributions, though the next order is already suppressed by $\\approx 10^{-3}$ .",
"Neglecting the inertia drift and assuming a constant curvature radius, a constant $B$ -field and $\\vec{B}$ always perpendicular to $\\vec{R}$ , one can integrate Eq.",
"(REF ) over time to obtain the drift distance in zeroth order [34] $D_{\\text{0}}=\\frac{p \\alpha }{q B} f(\\theta )$ where the angle of curvature $\\alpha =v_\\parallel T / R$ is used ($T$ is the time of travel during the drift).",
"A huge advantage of this method is, that protons ($q=e$ ) and electrons ($q=-e$ ) drift in opposite directions and can therefore be measured separately."
],
[
"Installation Sites",
"NoMoS is the first realization of an $R \\times B$ spectrometer and can be used on the one hand as a standalone experiment with Beta emitters: These are used for commissioning, calibration and characterization, and later to test the hypothesis of Lorentz invariance violation [35], or Neutrons: In this set-up (e.g., at the ILL), a beam of cold neutrons passes through a dedicated decay volume and the charged decay products are magnetically guided towards the RxB drift region (for details see next section and Fig.",
"REF ).",
"On the other hand, for high precision experiments, it can be coupled adiabatically to a magnetic field collecting the charged decay products from a long decay volume and afterwards filtering their incident angle using the magnetic mirror effect, as in the new facility PERC [36], [37], [38] or later at a PERC-like instrument at the proposed pulsed cold neutron beam facility ANNI [39] at the ESS.",
"NoMoS is divided into four areas: the experimental interface, the beam preparation area, the drift and the detection regions, as shown in Fig.",
"REF .",
"First, cold neutrons pass through a decay volume (either in situ or external - see subsection above) while some of them decay there.",
"Their charged decay products experience the local magnetic field $B_{\\text{DV}}$ and therefore gyrate around B-field lines until they reach either the upstream end of the experiment or a magnetic filter on the downstream side.",
"The filter has the field strength $B_\\text{F}>B_{\\text{DV}}$ .",
"Charged particles with an incident angle $\\theta \\ge \\theta _{\\text{max}}=\\arcsin (1/\\sqrt{r_\\text{F}})$ with $r_\\text{F}=B_\\text{F}/B_{\\text{DV}}$ (typical values are $r_\\text{F}=2$ or 4) are reflected from the filter by the magnetic mirror effect.",
"The magnetically transmitted particles are then guided towards an aperture, located in the beam preparation area.",
"The aperture defines geometrically the cross-section of the particle beam entering the drift region.",
"It has a magnetic field $B_\\text{A}<B_\\text{F}$ and typical values of the magnetic field ratio $r_\\text{A}=B_\\text{A}/B_{\\text{DV}}$ are 1 or 10.",
"The thus prepared beam enters the drift region where tilted $R \\times B$ coils establish a B-field with constant curvature (with absolute value $B_\\text{RxB} \\le B_\\text{A}$ , $r_\\text{RxB}=B_\\text{RxB}/B_{\\text{DV}}$ ).",
"Hence, the charged particles drift according to Eq.",
"(REF ) with $B=B_\\text{RxB}$ and $\\theta =\\theta _\\text{RxB}=\\arcsin \\left( \\sin {\\theta _\\text{DV} \\sqrt{r_\\text{RxB}}} \\right) $ .",
"Correction coils at both ends of the drift region serve to precisely define the angle of curvature $\\alpha $ .",
"After the charged particles passed the drift region, the electrons and protons are magnetically guided towards the spatial-resolving $R \\times B$ drift detector.",
"The detector is located in the detection region, which has a magnetic field of $B_\\text{Det}$ .",
"If protons are measured, post-acceleration to detectable energies is required.",
"Figure REF shows the preliminary shape of the magnetic field through the NoMoS magnet system along an exemplary particle trajectory in the standalone case.",
"Figure: Top: Shape of the magnetic field for the preliminary design of the NoMoS magnet system.",
"Shown are the decay volume (DV - indicated with shading), beam preparation area, drift and detection regions (from left to right).",
"Main component (B z B_z or tangential B tang B_\\text{tang}) along an examplary trajectory.",
"Bottom: The residuals represent the difference ΔB\\Delta B of the magnetic field at the particle's guiding center and its real position.",
"The oscillation in the drift region stems from the radial gradient ∂B φ /∂R\\partial B_\\phi / \\partial R (for details see Sec.",
").Several detectors will be installed in NoMoS, both to measure the $R \\times B$ drift distance and to investigate systematic effects: $R \\times B$ drift detector: The main detector for the drift distance measurement will have a spatial resolution of < 1 mm and ideally a surface area of $20 \\times 10$ $\\text{cm}^2$ .",
"Most probably, two independent detectors will be used side by side, one for electrons and one for protons.",
"The proton detector will be held at a high negative potential to post-accelerate them to detectable energies.",
"For electrons an additionally energy-resolving detector is envisaged to investigate false drift distances due to, e.g., backscattering-off of the detector itself or scattering out of the aperture (see next point).",
"Active aperture: The aperture in the beam preparation area has a finite thickness.",
"Through its inner face, some particles can be scattered out, potentially changing their angles and energies while still entering the drift region.",
"This alters not only the particle distributions entering the drift region but also the drift distance distribution.",
"To correct for this false effect, the energy-loss of the scattered particles will be measured by active surfaces at the inner face.",
"In this way, the active aperture functions as a veto detector for false drift distances at the $R \\times B$ drift detector.",
"Backscatter detector: Some of the decay products can be backscattered-off of the $R \\times B$ drift detector.",
"Then they deposit only part of their energy in it, and the angular distribution of the backscattered particles will range from 0 to 90$$ .",
"Hence, backscattered particles re-entering the drift region can hit the inner wall of NoMoS' vacuum vessel.",
"Therefore, we investigate to install an energy-resolving electron detector on the wall, along the drift tube, to detect their energy in coincidence with the $R \\times B$ drift detector.",
"Beam monitor: In the standalone case, an electron detector will be installed at the upstream end of the experimental interface to detect those electrons emitted towards the upstream end, reflected at the small magnetic field gradient in the decay volume or from the magnetic filter, and those backscattered-off of the aperture.",
"This detector will monitor the time stability of the particle beam and cross-check the angular selection (for details see the next section).",
"At PERC or later ANNI, the time stability is monitored by a small parasitic monitor (out of sight of the aperture)."
],
[
"Measurement Uncertainties and Systematics",
"The statistical sensitivity of an $a$ or $b$ measurement with NoMoS can be determined by spectral fitting of Monte Carlo generated data.",
"In Ref.",
"[40], the energy spectra have been investigated with the minimum variance bound estimator method.",
"Table REF shows the statistical sensitivity of the proton and electron energy spectra on $a$ and $b$ , respectively in comparison with the statistical uncertainty in the respective fit parameters for the drift distance spectra.",
"Obviously, the drift distance spectra are a little less sensitive to $a$ and $b$ .",
"At the ILL and PERC (unpulsed) we expect a detection rate of about 1 kHz.",
"Hence, assuming no additional fit parameters, one day of drift distance measurement yields a statistical uncertainty of $\\sigma _b \\approx 8 \\times 10^{-4}$ and $\\sigma _a /a \\approx 0.28 \\%$ , respectively.",
"Adding additional fit parameters accounting for systematic effects increases the required measurement time.",
"Table: Statistical sensitivity of the electron and proton energy spectra on aa and bb in comparison with the drift distance spectra in the standalone case.",
"NN represents the number of measured electrons or protons.Precision measurements of the proton and of the electron momentum spectrum require a thorough understanding of all systematic effects of the NoMoS spectrometer.",
"We aim to understand and describe the particle transport through NoMoS and how the spectra are thereby affected as precisely as possible by a transport function.",
"This transport function must describe not only the $R \\times B$ drift in the drift region but also the beam preparation and detection effects.",
"Then it can be used to fit the measured drift distance spectra.",
"Furthermore, it can be used to investigate the sensitivity of $a$ and $b$ to systematic effects.",
"In the following, we discuss the most important systematic effects.",
"The investigation of these effects is still in progress, hence the numbers given in this section are preliminary."
],
[
"Global Systematics",
" Magnetic field: The magnetic field's homogeneity and absolute value have to be checked through a thorough magnetic field map as both quantities affect the transport function.",
"The time stability will be monitored with magnetic sensors to correct for fluctuations.",
"Adiabatic motion: Adiabaticity should be conserved during the complete particle transport to prevent false effects in the particles' final position on the detector (and energy).",
"This is being investigated by particle tracking simulations.",
"Small non-adiabatic effects have to be estimated and suppressed.",
"Background: There are several potential sources of background in NoMoS measurements.",
"Sources for environmental background include the reactor, neighbouring experiments and cosmics.",
"Sources for beam-related background include the collimation system and residual gas.",
"The different contributions are disentangled by measurements with different neutron beam profiles (in the standalone case), magnetic field on/off and, for proton measurements, with post-acceleration on/off.",
"Doppler effect: In the standalone case, cold neutrons pass through NoMoS perpendicularly to the detection system, which suppresses the Doppler effect due to neutron motion.",
"This is not the case at PERC or later ANNI, where cold neutrons pass through NoMoS parallely to the detection system.",
"Investigations by the PERC collaboration have shown that the mean neutron energy has to be known with a precision of better than $10^{-2}$ [37].",
"Residual gas: Proton measurements impose tight restrictions on the residual gas [37] (and ref.",
"therein).",
"Therefore, a pressure level of $10^{-9}$ mbar is desired.",
"In addition, another neutron shutter will be implemented in order to enable automated background measurements.",
"Particle trapping: Local magnetic field minima must be avoided as they can give rise to potential traps, especially in the decay volume.",
"In addition, surface potential variations must be suppressed as they can lead to local field extrema and therefore give rise to potential penning traps.",
"By spectrometer design, charged particles that gyrate around a magnetic field line in the drift region experience an oscillating magnetic field.",
"To prevent particle trapping, the magnetic field $B_\\text{RxB}$ is superimposed by a small decreasing gradient towards the detection region.",
"All the features we use to prepare the particle beam introduce systematic effects which are being integrated into the transport function.",
"Beam characteristics: Potential inhomogeneities in the neutron density distribution can modify the particle spectrum entering the beam preparation area and therefore have to be determined and taken into account in the data analysis.",
"Angular selection: Due to small inhomogeneities of the filter field ($B_F(\\vec{r})$ ) or the decay volume field ($B_{DV}(\\vec{r})$ ), a position dependent $r_\\text{F}(\\vec{r})$ is obtained, making $\\theta _{\\text{max}}$ position dependent.",
"This dependency is included in the transport function and the position dependence of $r_\\text{F}$ will be determined through magnetic field mapping.",
"For a $b$ measurement on the $10^{-3}$ level, $r_\\text{F}$ has to be known at the level of $\\Delta r_\\text{F}/ r_\\text{F} \\approx 10^{-3}$ .",
"Edge effect: The transmission through the aperture is position, angle and momentum dependent, which modifies the particle spectra entering the drift region.",
"Parameters affecting this edge effect are the magnetic field ratio at the aperture $r_\\text{A}$ as well as the dimensions of the aperture and its proper alignment with respect to the neutron beam, the magnetic field lines and the $R \\times B$ drift detector.",
"Scattering at the aperture: The modification of the electron spectrum due to both the scattering-off of the aperture and the scattering out through its inner face require a correction.",
"In Ref.",
"[36] it has been shown that the errors due to these corrections can be suppressed by making the aperture active.",
"The drift systematics are defined by the magnetic field.",
"Some of the effects can be reduced while others are unavoidable due to the design of the magnet system: Absolute $R \\times B$ -field: For a $b$ measurement on the $10^{-3}$ level, the absolute $R \\times B$ -field value has to be known to $\\Delta B_\\text{RxB}/B_\\text{RxB} < 10^{-4}$ .",
"Note that adding an additional fit parameter for the $R \\times B$ -field value decreases the systematic uncertainty by about a factor of 10 while increasing the measuring time by a factor of about four.",
"B-field gradients: NoMoS has several B-field gradients.",
"Some are a natural consequence of the magnet design while others are artificially introduced, mainly in order to study systematic effects: Due to the design of the $R \\times B$ coils, there is an unavoidable gradient in the main component of the magnetic field, $\\partial B_\\phi / \\partial r $ ($\\phi $ is the direction along the curvature, $r$ is the radial direction).",
"Hence, particles passing through the drift region experience an oscillating local magnetic field.",
"However, the gradient is not linear ($\\propto 1/r$ ) and therefore the mean magnetic field experienced by the particles is not the same as the field at the guiding center of gyration.",
"Figure REF shows a schematic visualization of this effect.",
"$B_\\text{RxB}$ can change over the arc length of the drift region (e.g., because of the addition of a small gradient $\\partial B_\\phi / \\partial \\phi $ along the arc length to omit magnetic traps).",
"$r_\\text{RxB}$ and the local incident angle $\\theta $ are defined by this field.",
"The effect of a small gradient is being estimated using a mean magnetic field.",
"For a measurement of $b$ on the $10^{-3}$ level, $r_\\text{RxB}$ has to be known at the level of $\\Delta r_\\text{RxB}/r_\\text{RxB} \\approx 10^{-3}$ in the drift region.",
"Along the arc length, the particles drift further and further and thereby get closer to the coils in drift direction ($x$ ).",
"A very small gradient $\\partial B_\\phi / \\partial x$ representing the increase of magnetic field towards the coils is being included in the transport function via a position dependence.",
"Opening angle $\\alpha $ : The $R \\times B$ drift effect acts as soon as there is a curved magnetic field.",
"The point at which the drift gains/loses significant contribution defines the beginning/end of the curvature angle $\\alpha $ in Eq.",
"(REF ).",
"Deviations from the nominal value $\\alpha =180 $ at the beginning/end $\\Delta \\alpha _\\text{start}$ and $\\Delta \\alpha _\\text{end}$ can be position dependent.",
"The thorough magnetic field map will be input for particle tracking simulations, through which we will determine $\\alpha $ position-dependently.",
"Then this dependence will be integrated in the transport function.",
"Note that adding an additional fit parameter for the opening angle $\\alpha $ decreases the systematic uncertainty by a factor of about 10, while increasing the measuring time by a factor of about 2.3.",
"Furthermore we can investigate this systematic effect by varying $\\alpha $ by changing either the magnetic set-up (correction coils) or the position of the $R \\times B$ drift detector (inside the $R \\times B$ drift region).",
"For a measurement of $b$ on the $10^{-3}$ level, $\\alpha $ has to be known at the level of $\\Delta \\alpha / \\alpha \\approx 10^{-4}$ .",
"Figure: A visualization of the effect of the radial gradient ∂B φ /∂r\\partial B_\\phi / \\partial r in the drift region: The main component of the magnetic field is decreasing along the radial direction.",
"Particles which gyrate around their guiding center (GC) experience an oscillating B-field.",
"As the gradient is not linear, the mean magnetic field is not the same as the field at the guiding center.A thorough magnetic field map is vital for the determination of most systematic effects.",
"Hence, during commissioning of the spectrometer as well as before and after beam times, special attention will be paid to the field mapping.",
"Hall probes will serve to measure the shape of the B-field, while NMR probes are needed for the determination of the absolute height of the $R \\times B$ -field.",
"For the spatial-resolving $R \\times B$ drift detector, the following systematic effects have to be taken into account: Alignment: A proper alignment with respect to the neutron beam, the aperture and the magnetic field lines is crucial for a quantitative drift measurement.",
"Non-active detector surface: Normally, the area between individual detector strips ($\\mathcal {O}$ (10%)) is not active.",
"This type of binning effect is included in the transport function.",
"Detection efficiency: The detection efficiency may vary with the particles' energy.",
"This effect and possible corrections for it (limited fit range, calibration) have to be further investigated.",
"Backscattering: The backscattering of decay electrons and post-accelerated protons from the detector is being investigated using scattering software simulations.",
"For a thin detector dead layer and low detection threshold, undetected backscattering is substantially lower than the total backscattering probability.",
"For a typical Si detector with thin Al entrance window, this probability is in the order of several percent for electrons and of one percent for protons and, for protons, its energy and angle dependence is rather small.",
"Assuming that all backscattered particles are undetected, using additional fit parameters for the backscattering would result in a systematic uncertainty of $\\le 1\\times 10^{-4}$ on $b$ (absolute) and $< 7\\times 10^{-4}$ on $a$ (relative), respectively.",
"Post-acceleration of protons: On the one hand, the acceleration turns the protons' incident angles forward, which reduces the backscattering probability.",
"On the other hand, the high voltage electrode can be a source of field emission and can generate additional $\\vec{E} \\times \\vec{B}$ drift effects.",
"Both effects have to be suppressed and are being considered in the design of the entire detection system.",
"Edge effect: An additional edge effect perpendicular to the drift direction is not expected as the aperture height will be chosen such that the particle beam completely fits inside the height of the detector, including two gyration radii on both sides."
],
[
"Summary",
"We have presented a novel momentum spectrometer with an extensive physics program [31].",
"Precision measurements of $a$ and $b$ are planned for tests of the SM and searches for new physics beyond.",
"The majority of systematic effects is already included in the transport function (presented in this work), which enables a direct fit of the detected spectra.",
"A more detailed description of the transport function is under way and accompanied by the optimization of the magnet system.",
"In parallel, we are investigating detection systematics and are working on the design of the detection system.",
"It is envisioned that the construction of the magnet system will start in summer 2019, after a detailed technical design study.",
"Following a construction period of 12 to 18 months, the magnet system will be commissioned with beta emitters.",
"A first measurement with neutrons is intended to take place at the ILL. For high precision measurements, experimental campaigns at PERC and later ANNI are planned."
],
[
"Acknowledgement",
"We would like to thank Ferenc Glück (Karlsruhe Institute of Technology, Germany) for helpful discussions about adiabaticity, post-acceleration and the transport function, as well as Eberhard Widmann (SMI) for his support and helpful discussions.",
"Furthermore Daniel Moser would like to thank Waleed Khalid and Raluca Jiglau (SMI) for helpful discussions.",
"This work is supported by the Austrian Academy of Sciences within the New Frontiers Groups Programme NFP 2013/09, the Austrian Science Fund under contracts No.",
"W1252 (DK-PI) and P26636, the ILL under collaboration agreement No.",
"ILL-1519.1, the TU Wien and the SMI Wien."
]
] | 1906.04511 |
[
[
"Symmetric multisets of permutations"
],
[
"Abstract The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer.",
"For which multisubsets $B$ of the symmetric group $\\fS_n$ is the quasisymmetric function $$Q(B) = \\sum_{\\pi \\in B}F_{\\Des(\\pi), n}$$ a symmetric function?",
"Here $\\Des(\\pi)$ is the descent set of $\\pi$ and $F_{\\Des(\\pi), n}$ is Gessel's fundamental basis for the vector space of quasisymmetric functions.",
"The purpose of this paper is to provide a useful characterization of these multisets.",
"Using this characterization we prove a conjecture of Elizalde and Roichman.",
"Two other corollaries are also given.",
"The first is a short new proof that conjugacy classes are symmetric sets, a well known result first proved by Gessel and Reutenauer.",
"Our second corollary is a unified explanation that both left and right multiplication of symmetric multisets, by inverse $J$-classes, is symmetric.",
"The case of right multiplication was first proved by Elizalde and Roichman."
],
[
"Introduction",
"For integers $m\\le n$ set $[m,n] := \\lbrace m,m+1,\\ldots , n\\rbrace $ .",
"When $m=1$ we simply write $[n]$ instead of $[1,n]$ .",
"We denote by $_n$ the symmetric group on $[n]$ .",
"A multiset $B$ whose elements are taken from $_n$ is denoted by $B\\Subset _n$ .",
"The standard notation $B\\subseteq _n$ is reserved to indicate that $B$ is a set.",
"Additionally, given two multisets $A,B\\Subset _n$ we write $A\\sqcup B$ to denote their disjoint union.",
"For any $\\pi \\in _n$ its descent set is $(\\pi ):={i\\in [n-1]}{\\pi _i>\\pi _{i+1}}\\subseteq [n-1].$ For each $J\\subseteq [n-1]$ define the inverse $J$ -class as $R_J^{-1} = {\\pi ^{-1}\\in _n}{(\\pi )\\subseteq J}.$ For multisets $A,B\\Subset _n$ we write $A\\equiv B$ to indicate that there is descent-set-preserving bijection between $A$ and $B$ or, equivalently, in terms of generating functions, that $\\sum _{\\pi \\in A} \\textbf {x}^{(\\pi )} = \\sum _{\\pi \\in B} \\textbf {x}^{(\\pi )}.$ To indicate that $(\\pi ) = (\\tau )$ for $\\pi ,\\tau \\in _n$ we abuse this notation and sometimes write $\\pi \\equiv \\tau $ .",
"Further, define $AB$ to be the multiset of all products $\\pi \\tau $ where $\\pi \\in A$ and $\\tau \\in B$ .",
"In the case when $A= \\lbrace \\pi \\rbrace $ we simply write $\\pi B$ .",
"Playing a crucial role in this paper are the products $BR_J^{-1}$ and $R_J^{-1}B$ .",
"In the case that $B$ is such that $BR_J^{-1}\\equiv R_J^{-1}B$ for all $J\\subseteq [n-1]$ we say that $B$ is $D$ -commutative with all inverse $J$ -classes.",
"Next we establish some basic notions for the theory of symmetric functions.",
"Let $\\mathbf {x}= \\lbrace x_1,x_2,\\ldots \\rbrace $ be a countably infinite set of commuting variables.",
"We say a formal power series in $\\mathbb {Q}[[\\mathbf {x}]]$ is symmetric if it is of bounded degree and invariant under all permutations of its indices.",
"The vector space of all homogeneous symmetric functions of degree $n$ is denoted by $\\Lambda (n)$ .",
"We make use of two classical bases for $\\Lambda (n)$ .",
"Given an integer partition $\\lambda = (\\lambda _1 \\ge \\lambda _2\\ge \\cdots \\ge \\lambda _p )\\vdash n$ we set $m_\\lambda $ to be the symmetric function obtained by symmetrizing the monomial $x_{1}^{\\lambda _1}\\cdots x_{p}^{\\lambda _p}.$ The $m_\\lambda $ are called monomial symmetric functions and they constitute our first basis.",
"We also need the Schur functions which we denote by $\\lbrace s_\\lambda \\rbrace _{\\lambda \\vdash n}$ .",
"For a combinatorial definition of these functions and a proof that they are a basis for $\\Lambda (n)$ we direct the reader to [3] or [8].",
"We say that $f\\in \\Lambda (n)$ is Schur positive provided that it can be written as $f= \\sum _{\\lambda \\vdash n} c_\\lambda s_\\lambda $ where $c_\\lambda $ are nonnegative integer coefficients.",
"We next recall the quasisymmetric functions.",
"For each integer composition $\\alpha =(\\alpha _1,\\ldots , \\alpha _p)\\models n$ we define the monomial symmetric function as $M_\\alpha : = \\sum _{i_1<i_2<\\cdots <i_p} x_{i_1}^{\\alpha _1}x_{i_2}^{\\alpha _2}\\cdots x_{i_p}^{\\alpha _p}.$ We denote by $(n)$ the $\\mathbb {Q}$ -span of $\\lbrace M_\\alpha \\rbrace _{\\alpha \\models n}$ and call the elements of this space quasisymmetric functions.",
"As the monomial symmetric functions, which are indexed by compositions of $n$ , are a canonical basis for this space, we have $\\dim ((n)) = 2^{n-1}$ .",
"Lastly, since $m_\\lambda = \\sum _{\\alpha } M_\\alpha ,$ where the sum is over compositions $\\alpha $ obtained by permuting the parts of $\\lambda $ , it follows that $\\Lambda (n) \\subseteq (n)$ .",
"We shall also need Gessel's fundamental basis for $(n)$ .",
"To define this basis set $F_\\alpha = \\sum _{\\beta \\le \\alpha } M_\\beta ,$ where $\\beta \\le \\alpha $ indicates that $\\beta $ is a refinement of $\\alpha $ .",
"The collection $\\lbrace F_\\alpha \\rbrace _{\\alpha \\models n}$ is called the fundamental basis for $(n)$ .",
"To connect quasisymmetric functions to multisets of permutations we recall the well-worn bijection between subsets $J = \\lbrace j_1<j_2<\\cdots < j_s\\rbrace $ of $[n-1]$ and compositions of $n$ given by $J \\mapsto (j_1, j_2-j_1, \\ldots , j_{s}-j_{s-1}, n-j_s).$ We denote the image of $J$ under this bijection by $(J)$ .",
"Using this correspondence we also index the fundamental basis of $(n)$ by subsets of $[n-1]$ and write $F_{J,n}: = F_\\alpha $ where $\\alpha $ corresponds to $J\\subseteq [n-1]$ .",
"For any $B\\Subset _n$ we recall the quasisymmetric function $Q(B) := \\sum _{\\pi \\in B} F_{(\\pi ),n}\\in (n),$ first defined in [4].",
"We say that $B$ is symmetric if $Q(B)$ is symmetric.",
"If, moreover, $Q(B)$ is Schur-positive then, following the language first established by Adin and Roichman in [1], we say $B$ is fine.",
"Writing $J\\sim K$ whenever $(J)$ is a permutation of $(K)$ , the main result of this paper is a proof that the following are equivalent: $B$ is $D$ -symmetric (defined in Section ) $B$ is $D$ -commutative with all $J$ -classes $BR_J^{-1}\\equiv BR_K^{-1}$ whenever $J\\sim K$ $R_J^{-1}B\\equiv R_K^{-1}B$ whenever $J\\sim K$ $B$ is symmetric.",
"As an immediate consequence we prove Conjecture 10.4 in [2], due to Elizalde and Roichman, in which they hypothesizes that if $B$ is fine then $BD_J^{-1} \\equiv D_J^{-1}B$ where $D_J^{-1} = {\\pi ^{-1}\\in _n}{(\\pi ) = J}$ .",
"Two additional corollaries are proved.",
"The first gives a new proof of the fact that conjugacy classes are symmetric.",
"This was first established by Gessel and Reutenauer in [4] in which they proved the stronger result, by way of a more involved proof, that conjugacy classes are actually fine sets.",
"Our second corollary gives a unified explanation that the multiset obtained by either left or right multiplication of symmetric multisets by $R_J^{-1}$ is symmetric.",
"The case of right multiplication in the context of fine sets was first proved by Elizalde and Roichman in [2] although their techniques were unable to establish the case of left multiplication.",
"The paper is organized as follows.",
"In the next section we establish key definitions and lemmas used throughout.",
"In Section we formally state our main theorem and prove the aforementioned corollaries.",
"As the proof of our main theorem is involved, we break it into several propositions.",
"The propositions that follow immediately from definitions are in the main part of Section .",
"Statements that imply symmetry are contained in Subsection REF as they involve a careful analysis of the change of basis between the fundamental basis and the monomial quasisymmetric functions.",
"The proof that symmetry implies $D$ -symmetry involves techniques from the theory of tableaux and occupies Subsection REF ."
],
[
"Preliminaries",
"An ordered set partition of $[n]$ is a sequence $=(U_1,\\ldots , U_s)$ of nonempty disjoint sets $U_i$ , called blocks, whose union is $[n]$ .",
"We write $\\vdash [n]$ to indicate that $$ is an ordered set partition of $[n]$ and we denote by $\\Pi (n)$ the set of all ordered set partitions of $[n]$ .",
"For each composition $\\alpha \\models n$ we further define $\\Pi (\\alpha )$ to be the collection of all $\\vdash [n]$ where $|U_i| = \\alpha _i$ .",
"If $\\alpha $ corresponds to $J\\subseteq [n-1]$ then we also denote this set by $\\Pi (n,J)$ .",
"Example 2.1 The set partitions $=(\\lbrace 2,5,8\\rbrace ,\\lbrace 1,3,4\\rbrace ,\\lbrace 6,7,9\\rbrace )\\quad \\quad =(\\lbrace 2,5,8\\rbrace ,\\lbrace 6,7,9\\rbrace ,\\lbrace 1,3,4\\rbrace )$ are distinct ordered with $,\\in \\Pi (3,3,3)= \\Pi (\\lbrace 3,6\\rbrace ,9)$ .",
"For the remainder of this section fix $J=\\lbrace j_1<\\cdots <j_s\\rbrace \\subseteq [n-1]$ and adopt the convention that $j_0=0$ and $j_{s+1} = n$ .",
"To avoid repeating ourselves the symbol $$ will always denote an element of $\\Pi (n,J)$ and $U_i$ will always denote its $i$ th part.",
"Definition 2.2 Let $U$ and $V$ be disjoint alphabets and let $\\pi $ and $\\tau $ be permutations of $U$ and $V$ respectively.",
"We define the shuffle of $\\pi $ and $\\tau $ to be the set $\\pi \\tau $ consisting of all permutations of $U\\cup V$ where the letters in $U$ appear in the same order as in $\\pi $ and the letters in $V$ appear in the same order as in $\\tau $ .",
"Additionally, for $\\pi \\in _n$ and $\\tau \\in _m$ we define $\\pi \\tau : = \\pi \\tau ^{+n}$ where $\\tau ^{+n}$ is the word obtained by adding $n$ to each term of $\\tau $ .",
"So $\\pi \\tau \\subseteq _{n+m}$ .",
"Example 2.3 If $\\pi = 12$ and $\\tau = 21$ then $\\pi \\tau = \\lbrace 1243, 1423, 4123, 1432, 4132, 4312\\rbrace ,$ where we have dropped commas for readability.",
"As the main results in this paper involve the sets $R_J^{-1}$ we now look at multiple ways of describing such sets.",
"First note that $R_J^{-1} = (1\\ldots j_1) (j_1+1\\ldots j_2) \\cdots (j_s+1\\ldots n).$ Hence to every $\\in \\Pi (n,J)$ there corresponds the permutation $\\delta _\\in R_J^{-1}$ defined so that $U_1$ is the set of positions occupied by the subsequence $(1,2,\\ldots ,j_1)$ , $U_2$ is the set of positions occupied by the subsequence $(j_1+1,\\ldots , j_2)$ , etc.",
"So $R_J^{-1} = {\\delta _}{\\in \\Pi (n,J)}$ from which it follows that $\\Pi (n,J)$ is in correspondence with $R_J^{-1}$ .",
"Example 2.4 If $n=4$ and $J = \\lbrace 2\\rbrace \\subseteq [3]$ then we obtain the following correspondence: Table: NO_CAPTIONwhere, for example, we have written $(12, 34)$ instead of the more verbose $(\\lbrace 1,2\\rbrace , \\lbrace 3,4\\rbrace )$ .",
"Next we consider right multiplication by inverse $J$ -classes.",
"Recall that if $\\pi ,\\tau \\in _n$ then $\\pi \\cdot (\\tau _1\\tau _2\\ldots \\tau _n) = \\pi _{\\tau _1}\\ldots \\pi _{\\tau _n},$ where $\\cdot $ denotes group multiplication.",
"Applying this to (REF ) it follows that $\\pi R_J^{-1} = (\\pi _1\\ldots \\pi _{j_1}) (\\pi _{j_1+1}\\ldots \\pi _{j_2})\\cdots (\\pi _{j_s+1}\\ldots \\pi _{n}).$ Example 2.5 When $=(\\lbrace 2,5,8\\rbrace ,\\lbrace 1,3,4\\rbrace ,\\lbrace 6,7,9\\rbrace )$ we see that $\\delta _= {\\color {blue}4}\\ {\\color {red}1}\\ {\\color {blue}5}\\ {\\color {blue}6}\\ {\\color {red}2}\\ 7\\ 8\\ {\\color {red}3}\\ 9.$ and if we take $\\pi = {\\color {red} 9\\ 1\\ 4\\ }{\\color {blue} 5\\ 2\\ 8\\ }7\\ 3\\ 6$ then $\\pi \\cdot \\delta _= {\\color {blue}5}\\ {\\color {red}9}\\ {\\color {blue}2}\\ {\\color {blue}8}\\ {\\color {red}1}\\ 7\\ 3\\ {\\color {red}4}\\ 6.$ Note that the first three terms in $\\pi $ (colored red) are in positions $2,5,8$ , the next three (colored blue) are in positions $1,3,4$ and the last three (black) are in positions $6,7,9$ .",
"We now require the following consequence of Stanley's famous shuffling theorem (see, [9]).",
"Lemma 2.6 Assume $\\pi $ and $\\tau $ are permutations of disjoint alphabets as are $\\pi ^{\\prime }$ and $\\tau ^{\\prime }$ .",
"If $\\pi \\equiv \\pi ^{\\prime }$ and $\\tau \\equiv \\tau ^{\\prime }$ then $\\pi \\tau \\equiv \\pi ^{\\prime }\\tau ^{\\prime }.$ Using this lemma together with (REF ) we see that $\\pi R_J^{-1} \\equiv (\\pi _1\\ldots \\pi _{j_1})(\\pi _{j_{1}+1}\\ldots \\pi _{j_2})\\cdots (\\pi _{j_s}\\ldots \\pi _{s+1}), $ where $$ is standardization.",
"Although not apparent at this point it turns out that it is much easier to work with the elements on the right side of (REF ).",
"To do this we make the following definition.",
"Definition 2.7 For $\\in \\Pi (n,J)$ and $\\pi \\in _n$ we define $\\sigma _(\\pi )$ to be the element on the right side of (REF ) in which $U_1$ is the set of positions occupied by the subsequence $(\\pi _1\\ldots \\pi _{j_1})$ , $U_2$ is the set of positions occupied by the subsequence $(\\pi _{j_1+1}\\ldots \\pi _{j_2})^{+j_1}$ , etc.",
"The next lemma is now immediate and makes use of a standard convention.",
"For any function $f:_n \\rightarrow _n$ and $B\\Subset _n$ we denote by $f(B)$ the multiset obtained by applying $f$ to each element in $B$ .",
"Lemma 2.8 For any $B\\Subset _n$ we have $BR_J^{-1} \\equiv \\bigsqcup _{\\in \\Pi (n,J)} \\sigma _(B).$ Next consider left multiplication by inverse $J$ -classes.",
"Although it trivially follows by our definitions that $R_J^{-1}\\pi = {\\delta _\\pi }{\\in \\Pi (n,J)},$ a “twist\" is needed in order for us to compare left and right multiplication.",
"As such we make the following definition.",
"Definition 2.9 For each $\\pi \\in _n$ define $\\rho _(\\pi ) := \\delta _\\cdot \\pi $ where $$ is the ordered set partition in $\\Pi (n,J)$ whose $i$ th block is $\\pi (U_i)$ .",
"Since the mapping $\\pi :\\Pi (n,J) \\rightarrow \\Pi (n,J)$ given by $(U_1,U_2,\\ldots ) \\mapsto (\\pi (U_1),\\pi (U_2),\\ldots )$ is clearly bijective we obtain the next lemma.",
"Lemma 2.10 For any $B\\Subset _n$ we have $R_J^{-1}B = \\bigsqcup _{\\in \\Pi (n,J)} \\rho _(B).$ We end this section with an important description of how the descent structure of $\\rho _(\\pi )$ and $\\sigma _(\\pi )$ are related.",
"For any $S\\subseteq \\mathbb {Z}$ set $S^*={i\\in S}{i+1\\in S}$ .",
"Using this define for any $= (U_1,U_2,\\ldots ) \\vdash [n]$ the set $^* := U_1^* \\cup U_2^* \\cup \\cdots \\subseteq [n-1]$ Lemma 2.11 Let $\\pi ,\\tau \\in _n$ and $\\in \\Pi (n,J)$ .",
"Then $\\rho _(\\pi ) \\equiv \\sigma _(\\tau )$ if and only if for each $u\\in ^*$ we have $u \\in (\\pi ) \\iff \\delta _(u) \\in (\\tau ).$ Moreover we have $(\\rho _(\\pi )) = (\\delta _) \\cup \\left((\\pi )\\cap ^*\\right) $ and $(\\sigma _(\\tau )) = (\\delta _) \\cup {u\\in ^*}{\\delta _(u)\\in (\\tau )}.",
"$ As $\\delta _$ is increasing on the blocks of $$ , then $(\\delta _)$ and $(\\pi )\\cap ^*$ are disjoint.",
"Hence the two sets on the right side of (REF ) and (REF ) are, respectively, disjoint.",
"Therefore to prove the first claim it suffices to show that (REF ) and (REF ) hold.",
"Before continuing we make the following definition.",
"For finite sets of integers $A$ and $B$ we write $A<B$ provided that $\\max (A) < \\min (B)$ .",
"We first prove (REF ).",
"Take $\\in \\Pi (n,J)$ so that its $i$ th block is $W_i:=\\pi (U_i)$ .",
"By our definition $\\rho _(\\pi ) = \\delta _\\cdot \\pi $ .",
"Observe that $\\delta _$ is increasing on each block $W_i$ and $\\delta _(W_i)> \\delta _(W_j)$ whenever $i>j$ .",
"So $\\delta _(\\pi (u)) > \\delta _(\\pi (u+1))$ if and only if either a) $\\pi (u)> \\pi (u+1)$ and $u,u+1\\in U_i$ , or b) $\\pi (u) \\in W_i$ and $\\pi (u+1)\\in W_j$ with $i>j$ .",
"As the first condition is equivalent to $u\\in (\\pi )\\cap ^*$ and the second is equivalent to $u\\in U_i$ and $u+1\\in U_j$ with $i>j$ , i.e., $u\\in (\\delta _)$ , we arrive at (REF ).",
"Next we prove (REF ).",
"From the definitions we see that $\\sigma _(\\tau )(U_i) > \\sigma _(\\tau )(U_j) \\iff i>j \\iff \\delta _(U_i)>\\delta _(U_j).$ If $u,u+1$ are in distinct blocks of $$ then $u\\in (\\sigma _(\\tau ))\\iff u\\in (\\delta _).$ Next assume $u,u+1\\in U_i$ , i.e., $u\\in ^*$ .",
"Note that the subsequences of $\\sigma _(\\tau )$ and $\\delta _$ indexed by $U_i$ are $(\\tau _{j_{i-1}+1},\\ldots , \\tau _{j_i})^{+j_{i-1}}\\quad \\quad (j_{i-1}+1, j_{i-1}+2, \\ldots , j_i),$ respectively.",
"In this case we have $u\\in (\\sigma _(\\tau ))\\iff \\delta _(u)\\in (\\tau ).$ Together these cases prove that (REF ) holds.",
"Lemma 2.12 For any $A,B\\Subset _n$ if $A\\equiv B$ then $AR_J^{-1} \\equiv BR_J^{-1}\\quad \\quad R_J^{-1}A \\equiv R_J^{-1}B.$ Let $f:A\\rightarrow B$ be our descent-set-preserving bijection and let $\\pi \\in A$ .",
"By Equations (REF ) and (REF ) from the previous lemma, it follows that $\\sigma _(\\pi )\\equiv \\sigma _(f(\\pi ))$ and $\\rho _(\\pi )\\equiv \\rho _(f(\\pi ))$ .",
"This lemma now follow from Lemmas REF and REF , respectively."
],
[
"A characterization of symmetric sets",
"The purpose of this section is to state and prove a useful characterization of symmetric multisets.",
"Using this characterization we then simultaneously explain several well known results in the theory of symmetric sets as well as prove the aforementioned conjecture of Elizalde and Roichman.",
"To state our main theorem we require a couple of definitions.",
"Throughout this section $B\\Subset _n$ and $J,K\\subseteq [n-1]$ .",
"Notation 3.1 Let $\\alpha ,\\beta \\models n$ .",
"We write $\\alpha \\sim \\beta $ to indicate that the sequence $\\beta $ is a permutation of the sequence $\\alpha $ .",
"For $J, K\\subseteq [n-1]$ we also write $J\\sim K$ provided that $(J)\\sim (K)$ .",
"Definition 3.2 We say $B\\Subset _n$ is $D$ -symmetric if for every $\\vdash [n]$ there exists a bijection $\\Psi _:B\\rightarrow B$ so that for each $u \\in ^*$ and $\\pi \\in B$ we have $u\\in (\\pi ) \\iff \\delta _{}(u) \\in (\\Psi _(\\pi )).$ We now come to our main theorem.",
"Theorem 3.3 The following are equivalent: $B$ is $D$ -symmetric $B$ is $D$ -commutative with all inverse $J$ -classes $BR_J^{-1}\\equiv BR_K^{-1}$ whenever $J\\sim K$ $R_J^{-1}B\\equiv R_K^{-1}B$ whenever $J\\sim K$ $B$ is symmetric.",
"Before diving into the proof of this theorem we state and prove several corollaries.",
"In [2] Elizalde and Roichman hypothesis that if $B$ is fine then $BD_J^{-1} \\equiv D_J^{-1}B$ where $D_J^{-1} = {\\pi ^{-1}\\in _n}{(\\pi ) = J}$ .",
"By a straightforward application of inclusion-exclusion their conjecture is equivalent to showing that fine sets (which of course are symmetric) $D$ -commute with all inverse $J$ -classes.",
"As this is immediate from Theorem REF we record it as our first corollary.",
"Corollary 3.4 If $B\\Subset _n$ is fine then $B$ is $D$ -commutative with all inverse $J$ -classes.",
"Another immediate corollary of our theorem is the well known fact that conjugacy classes are symmetric.",
"Gessel and Reutenauer first proved the stronger fact in [6] that such sets are actually fine.",
"Their proof relies on ideas from representation theory.",
"Corollary 3.5 Let $C\\subseteq _n$ be a conjugacy class.",
"Then $C$ is symmetric.",
"As $C$ is a conjugacy class we know that $C\\pi = \\pi C$ for all $\\pi \\in _n$ .",
"So for any $S\\subseteq _n$ we have $CS = SC$ .",
"In particular $C$ is $D$ -commutative with all inverse $J$ -classes and our claim follows by Theorem REF .",
"Our next corollary simultaneously explains why the collection of symmetric multisets of $_n$ is closed under multiplication by inverse $J$ -classes on the right and on the left.",
"In [2] Elizalde and Roichman prove that right multiplication of fine multisets by inverse $J$ -classes yields fine multisets.",
"Although not explicitly done in their paper, one can easily extend this result to conclude that the same holds for symmetric multisets.",
"Their results again use ideas from representation theory.",
"That said, they were unable to obtain similar results in the context of left multiplication which, one can speculate, is the reason for their Conjecture 10.4.",
"We now provide a short uniform explanation for symmetric invariance under both left and right multiplication by inverse $J$ -classes.",
"Corollary 3.6 For any symmetric $B\\Subset _n$ the multisets $R_J^{-1}B$ and $BR_J^{-1}$ are also symmetric.",
"Take $B$ as stated and consider $K\\sim K^{\\prime }\\subseteq [n-1]$ .",
"As $B$ is symmetric Theorem REF tells us that $R_K^{-1}B \\equiv R_{K^{\\prime }}^{-1}B\\quad \\quad BR_K^{-1} \\equiv BR_{K^{\\prime }}^{-1}.$ Therefore for any $J\\subseteq [n-1]$ it follows from Lemma REF that $R_K^{-1}BR_J^{-1}\\equiv R_{K^{\\prime }}^{-1}BR_J^{-1}\\quad \\quad R_J^{-1}BR_K^{-1}\\equiv R_{J}^{-1}BR_{K^{\\prime }}^{-1}.$ By another application of Theorem REF we conclude that $BR_J^{-1}$ and $R_J^{-1}B$ are both symmetric.",
"We note that the above corollary does not hold if $R_J^{-1}$ is replaced by an arbitrary symmetric set $A$ .",
"For example if $A = \\lbrace 1324, 4132\\rbrace $ and $B = \\lbrace 2143, 2314\\rbrace $ then $Q(A) =Q(B)= m_{22} + m_{211} + 2m_{1111}$ but $Q(AB) =M_{31} + M_{22} + 2M_{112} + 2M_{121} + 2M_{211} + 4M_{1111}$ .",
"We now turn our attention to the proof of Theorem REF .",
"As the proof has several parts we start with an outline.",
"In this section we show that: a) $$ b) in Proposition REF a) $$ c) in Proposition REF .",
"In Subsection REF we establish the following sufficient conditions for symmetry: c) $$ e) and d) $$ e) in Proposition REF b) $$ e) in Proposition REF .",
"In Subsection REF we finally show: e) $$ a) in Proposition REF .",
"Carrying out the above agenda shows that a), b), c), and e) are equivalent and that d) implies e).",
"It remains to show that e) implies d).",
"Assuming, for the moment, that all but d) are equivalent we can give a short proof of this fact.",
"We do so next and then return to the agenda outlined above.",
"Proposition 3.7 If $B$ is symmetric then $R_J^{-1}B \\equiv R_K^{-1}B$ whenever $J\\sim K$ .",
"If $B$ is symmetric then we know that it is $D$ -commutative with all inverse $J$ -classes and that $BR_J^{-1} \\equiv BR_K^{-1}$ whenever $J\\sim K$ .",
"Therefore $R_J^{-1}B \\equiv BR_J^{-1}\\equiv BR_K^{-1} \\equiv R_K^{-1}B$ whenever $J\\sim K$ .",
"We now begin with a proof that a) implies b).",
"Proposition 3.8 If $B\\Subset _n$ is $D$ -symmetric then it is $D$ -commutative with all inverse $J$ -classes.",
"As $B$ is $D$ -symmetric there exist bijections $\\Psi _:B\\rightarrow B$ for each $\\in \\Pi (n,J)$ that satisfy (REF ).",
"It now follows by Lemma REF that $u\\in (\\rho _(\\pi )) &\\iff u\\in (\\delta _) \\cup ((\\pi )\\cap ^*)\\\\&\\iff u\\in (\\delta _) u \\in ^*\\delta _(u) \\in (\\Psi _(\\pi ))\\\\& \\iff u\\in (\\sigma _(\\Psi _(\\pi ))).",
"$ Using this we obtain our desired result since $R_J^{-1}B =\\bigsqcup _{\\pi \\in B\\atop \\in \\Pi (n,J)} \\lbrace \\rho _(\\pi ) \\rbrace \\equiv \\bigsqcup _{\\pi \\in B\\atop \\in \\Pi (n,J)}\\sigma _(\\Psi _(\\pi ))\\equiv \\bigsqcup _{\\pi \\in B\\atop \\in \\Pi (n,J)} \\lbrace \\sigma _(\\pi )\\rbrace \\equiv BR_J^{-1},$ where the first step follows from Lemma REF , the second step follows our previous calculation, the third since $\\Psi _:B\\rightarrow B$ is a bijection, and the last step because of Lemma REF .",
"Next we prove that a) implies c).",
"Proposition 3.9 If $B$ is $D$ -symmetric, then $BR_J^{-1} \\equiv BR_K^{-1}$ whenever $J\\sim K$ .",
"Assume $B$ is $D$ -symmetric and set $J=\\lbrace j_1<\\cdots <j_p\\rbrace $ .",
"It suffices to prove the proposition when $K$ is such that $(K)$ is the composition obtained by transposing the $k$ th and $(k+1)$ st blocks of $(J)$ .",
"In particular, if we set $s^{\\prime } = r+t-s$ where $r = j_{k-1},\\ s = j_k,\\ \\ t = j_{k+1},$ then $K = J\\setminus \\lbrace s\\rbrace \\cup \\lbrace s^{\\prime }\\rbrace $ .",
"Now define $= ([r], [s+1,t], [r+1,s], [t+1,n])$ noting that $\\delta _(u) = {\\left\\lbrace \\begin{array}{ll}u & \\textrm { if } u \\in [r] \\cup [t+1,n]\\\\u-(s-r) &\\textrm { if } u \\in [s+1,t]\\\\u+(t-s) &\\textrm { if } u \\in [t+1,s]\\end{array}\\right.",
"}$ As $B$ is $D$ -symmetric there exists a bijection $\\Psi :B\\rightarrow B$ , corresponding to $$ , that satisfies (REF ).",
"Letting $\\pi \\in B$ and $\\tau = \\Psi (\\pi )$ it follows that $\\pi _1\\ldots \\pi _{r} \\equiv \\tau _1\\ldots \\tau _{r}\\quad \\quad \\pi _{t+1}\\ldots \\pi _{n} \\equiv \\tau _{t+1}\\ldots \\tau _{n}$ that $\\pi _{s+1}\\ldots \\pi _{t} \\equiv \\tau _{r+1}\\ldots \\tau _{s^{\\prime }}\\quad \\quad \\pi _{r+1}\\ldots \\pi _{s} \\equiv \\tau _{s^{\\prime }+1}\\ldots \\tau _{t}.$ So Lemma REF together with (REF ) gives $\\pi R_J^{-1} \\equiv \\Psi (\\pi )R_K^{-1}$ .",
"As $\\Psi :B\\rightarrow B$ is bijective the lemma now follows.",
"To conclude this section we establish some needed properties of $D$ -symmetric sets.",
"Our first lemma follows directly from the definition of $D$ -symmetry.",
"In that lemma we make use of the following convention.",
"For any finite set of positive integers $S$ set $^S: =\\prod _{i\\in S}x_i.$ Our second lemma follows directly from the first lemma.",
"In both cases we omit formal proofs.",
"Lemma 3.10 Let $B\\Subset _n$ .",
"Then $B$ is $D$ -symmetric if and only if for all $\\vdash [n]$ we have $\\sum _{\\pi \\in B} ^{\\delta _((\\pi )\\ \\cap \\ ^*)} = \\sum _{\\pi \\in B} ^{(\\pi )\\ \\cap \\ \\delta _(^*)}.$ Lemma 3.11 Set $A,B\\Subset _n$ .",
"We then have If $B$ is $D$ -symmetric and $B\\equiv A$ then $A$ is also $D$ -symmetric.",
"If $\\lbrace B_i\\rbrace _{i\\in I}$ is a collection of $D$ -symmetric multisets then $\\bigsqcup _{i\\in I} B_i$ is $D$ -symmetric.",
"If $A,B$ are $D$ -symmetric with $A\\subseteq B$ then so is $B\\setminus A$ .",
"Lemma 3.12 A multiset $B\\Subset _n$ is $D$ -symmetric if and only if for each $=(U,V)\\vdash [n]$ there exists a bijection $\\Psi _:B\\rightarrow B$ satisfying (REF ).",
"The forward direction is trivial.",
"We concentrate on the reverse direction.",
"For the set partition $(\\lbrace 1,2,\\ldots , n\\rbrace )$ consisting of 1 block, observe that the identity function on $B$ satisfies (REF ).",
"Now assume, for an inductive proof, that for every set partition with $p$ blocks there exists a corresponding bijection that satisfies (REF ).",
"Consider a set partition $= (U_1,\\ldots , U_p, U_{p+1})$ with $p+1$ blocks and define $= (U_1,\\ldots , U_{p-1}, U_p\\cup U_{p+1}).$ As $$ has $p$ blocks we know by induction that there exists some bijection $\\Psi _:B\\rightarrow B$ that satisfies (REF ).",
"Define $= ([n]\\setminus W, W)\\vdash [n]$ where $W = \\delta _(U_{p+1})$ .",
"As this set partition has two blocks there exits a bijection $\\Psi _:B\\rightarrow B$ satisfying (REF ).",
"It now suffices to prove that $\\Psi _\\circ \\Psi _$ is a bijection corresponding to $$ that satisfies (REF ).",
"For $u\\in ^*\\subseteq ^*$ we know that $u,u+1$ are in the same block in $$ and hence $\\delta _(u)+1 = \\delta _(u+1)$ .",
"As the pair $u,u+1$ are also in the same block of $$ then $u,u+1\\in U_{p+1}$ or $u,u+1\\in [n]\\setminus U_{p+1}$ .",
"So $\\delta _(u),\\delta _(u+1)$ are in the same block of $$ , i.e., $\\delta _(u)\\in ^*$ .",
"Combining these pieces it now follows that if $u\\in ^*$ then $u\\in (\\pi ) &\\iff \\delta _(u)\\in (\\Psi _(\\pi )) \\\\&\\iff \\delta _\\circ \\delta _(u) \\in (\\Psi _\\circ \\Psi _(\\pi )) \\\\&\\iff \\delta _(u) \\in (\\Psi _\\circ \\Psi _(\\pi )),$ where the last equivalence follows by the easy-to-check fact that $\\delta _= \\delta _\\circ \\delta _$ ."
],
[
"Sufficiency",
"The goal of this subsection is to prove that each of b), c), and d) individually implies e).",
"We begin with some discussion and definitions.",
"Recall that $Q(B) = \\sum _{\\pi \\in B} F_{(\\pi ),n} = \\sum _{\\alpha \\models n}b_\\alpha M_\\alpha \\in (n) $ for some integers $b_\\alpha $ .",
"Next recall the correspondence $$ between subsets and compositions and the notation $\\alpha \\ge \\beta $ , for $\\alpha ,\\beta \\models n$ , meaning that $\\beta $ is a refinement of $\\alpha $ .",
"Now for each $J\\subseteq [n-1]$ the definition of the fundamental basis means that $b_{(J)} = |{\\pi \\in B}{((\\pi )) \\ge (J)}| = |{\\pi \\in B}{(\\pi ) \\subseteq J}|.$ Defining $B_J:= {\\pi \\in B}{(\\pi )\\subseteq J}$ we have $b_{(J)} = |B_J|$ .",
"Recall that the monomial symmetric function $m_\\lambda $ can be written as $m_\\lambda = \\sum _{\\alpha \\sim \\lambda } M_\\alpha .$ Therefore to show $Q(B) \\in \\Lambda (n)$ it suffices to prove that $|B_J| = |B_K|$ whenever $J\\sim K$ .",
"With this discussion in mind we turn to the proof that c) and d) each imply e).",
"Proposition 3.13 If $BR_J^{-1} \\equiv BR_K^{-1}$ whenever $J\\sim K$ or $R_J^{-1}B \\equiv R_K^{-1}B$ whenever $J\\sim K$ , then $B$ is symmetric.",
"We start with the first claim.",
"By Lemma REF we have $\\sum _{\\pi \\in BR_J^{-1}} ^{(\\pi )} = \\sum _{\\in \\Pi (n,J)\\atop \\pi \\in B} ^{(\\sigma _(\\pi ))}.$ Now observe that the coefficient on $^\\emptyset $ in this expression is $|{\\pi \\in B}{(\\pi )\\subseteq J}| = |B_J|.$ This can be seen by considering (REF ) of Lemma REF and noting that in order for $(\\sigma _(\\pi )) = \\emptyset $ we must have $\\delta _= 1$ .",
"By our assumption we know that if $J\\sim K$ then $BR_J^{-1} \\equiv BR_K^{-1}$ .",
"So $|B_J| = |B_K|$ which, in light of discussion above, proves our first claim.",
"To prove the second claim, we know by Lemma REF that $\\sum _{\\pi \\in R_J^{-1}B} ^{(\\pi )} = \\sum _{\\in \\Pi (n,J)\\atop \\pi \\in B} ^{(\\rho _(\\pi ))}.$ By appealing to (REF ) in Lemma REF , a similar proof to that in the first case establishes our second claim.",
"We now turn our attention to proving that if $B$ is $D$ -commutative with all inverse $J$ -classes then it is symmetric, i.e., that b) implies e) in our main theorem.",
"For reference and to set the stage note that Lemmas REF and REF imply that if $BR_J^{-1} \\equiv R_J^{-1}B$ then $\\sum _{\\in \\Pi (n,J)\\atop \\pi \\in B} \\textbf {x}^{(\\sigma _(\\pi ))}= \\sum _{\\in \\Pi (n,J)\\atop \\pi \\in B} \\textbf {x}^{(\\rho _(\\pi ))}.$ They key idea in the coming proofs is to consider the coefficient $c_i$ on $^{\\lbrace i\\rbrace }$ in this generating function.",
"To describe this coefficient we make the following definitions.",
"Definitions 3.14 For any $\\in \\Pi (n)$ define $r():= [n-1]\\setminus ^*\\quad \\quad s():=[n-1]\\setminus \\delta _(^*).$ Additionally, for any $\\alpha \\models n$ and $i\\in [n-1]$ set $\\Pi _i(\\alpha ) = {\\in \\Pi (\\alpha )}{(\\delta _) = \\lbrace i\\rbrace }$ and let $\\Gamma _i(\\alpha )$ consist of all $\\in \\Pi _i(\\alpha )$ such that all blocks in $$ are intervals.",
"We now have the following lemma.",
"Lemma 3.15 Fix $\\alpha \\models n$ .",
"Take $f_= \\rho _$ or $\\sigma _$ and define $c_i$ to be the coefficient on $^{\\lbrace i\\rbrace }$ in $\\sum _{\\in \\Pi (\\alpha )\\atop \\pi \\in B} ^{(f_(\\pi ))}.$ Then there is some $a\\ge 0$ so that $c_i = {\\left\\lbrace \\begin{array}{ll}a+ \\sum _{\\in \\Pi _i(\\alpha )}|B_{r()}|& \\textrm { if } f_= \\sigma _\\\\a+ \\sum _{\\in \\Pi _i(\\alpha )}|B_{s()}|& \\textrm { if } f_= \\rho _.\\end{array}\\right.",
"}$ First consider the case when $f_=\\rho _$ .",
"Appealing to (REF ) in Lemma REF we have $(\\rho _(\\pi ))=\\lbrace i\\rbrace $ if and only if $(\\delta _)=\\emptyset \\quad \\quad (\\pi )\\cap ^* = \\lbrace i\\rbrace $ or vice versa.",
"In the displayed case, note that $(\\delta _)=\\emptyset $ if and only if $$ is the unique partition $\\in \\Pi (\\alpha )$ whose first block is the interval $[\\alpha _1]$ and whose second block is the following $\\alpha _2$ integers, etc.",
"Set $a = |{\\pi \\in B}{(\\pi )\\cap ^*}|$ .",
"Now consider the other case when $(\\delta _)=\\lbrace i\\rbrace \\quad \\quad (\\pi )\\cap ^* = \\emptyset .$ This occurs if and only if $\\in \\Pi _i(\\alpha )$ and $(\\pi )\\subseteq [n-1]\\setminus ^* = r()$ .",
"This establishes the case when $f_= \\rho _$ .",
"Now consider the case when $f=\\sigma _$ .",
"Appealing to (REF ) in Lemma REF we have $(\\sigma _(\\pi )) = \\lbrace i\\rbrace $ if and only if $(\\delta _) = \\emptyset \\quad \\quad {u\\in ^*}{\\delta _(u)\\in (\\pi )} = \\lbrace i\\rbrace $ or vice versa.",
"As before, the displayed case can only occur when $= $ .",
"As $\\delta _= 1$ we further have $\\lbrace i\\rbrace = {u\\in ^*}{\\delta _(u)\\in (\\pi )} = (\\pi )\\cap ^*.$ so that we can take $a$ as above in this case as well.",
"Now consider the case when $(\\delta _) = \\lbrace i\\rbrace \\quad \\quad {u\\in ^*}{\\delta _(u)\\in (\\pi )} = \\emptyset .$ This occurs when $\\in \\Pi _i(\\alpha )$ and $(\\pi ) \\cap \\delta _(^*) = \\emptyset $ .",
"As the latter is equivalent to $(\\pi ) \\subseteq [n-1] \\setminus \\delta _(^*)\\subseteq s()$ , this explains our second term.",
"Lemma 3.16 Fix $\\alpha \\models n$ For any $\\vdash \\Pi (\\alpha )$ we have $r() \\sim s()$ .",
"First consider the case when all the blocks of $$ are intervals.",
"Let $J\\subseteq [n-1]$ be such that $(J) = \\alpha $ .",
"As the blocks in $$ are intervals we see that $\\delta _(^*) = [n-1]\\setminus J.$ This means that $s()=J$ .",
"We must now show that $(r()) \\sim \\alpha = (J)$ .",
"Again using the fact that the blocks of $$ are intervals we may define $=(W_1, W_2,\\ldots )$ to be the ordered set partition obtained by permuting the blocks of $$ so that $\\max (W_i) < \\min (W_{i+1})$ .",
"As $^* = ^*$ it follows that $r() = r()=\\lbrace \\max (W_1)<\\max (W_2)<\\cdots \\rbrace $ which, in turn, implies that $(r()) = (|W_1|, |W_2|, \\ldots )$ .",
"By our choice of $$ our claim now follows.",
"Now consider an arbitrary $\\in \\Pi (\\alpha )$ and let $$ be the refinement given by replacing each block $U_i$ of $$ with the sequence $(I_1,I_2,\\ldots )$ of maximal nonempty intervals in $U_i$ ordered so that $\\max (I_i)< \\min (I_{i+1})$ .",
"Observe that $^* = ^*$ and $\\delta _= \\delta _$ .",
"Consequently, $r() = r()$ and $s() = s()$ .",
"The general claim now follows from our first paragraph.",
"For the next few proofs, some additional terminology relating to permutations of compositions is required.",
"For any composition $\\alpha \\models n$ with $p$ parts and $I=\\lbrace i_1<i_2<\\cdots <i_s\\rbrace \\subseteq [p]$ we write $\\alpha (I):= (\\alpha _{i_1},\\alpha _{i_2},\\ldots , \\alpha _{i_s},\\alpha _{j_1},\\alpha _{j_2},\\ldots , \\alpha _{j_t})$ where $j_1<j_2<\\ldots <j_t$ are the elements in $[p]\\setminus I$ .",
"In particular for any $m\\le p$ we have $\\alpha ([m]) = \\alpha $ and, in general, $\\alpha \\sim \\alpha (I)$ .",
"We also define $S_k(\\alpha )$ to be the set of all $I\\subseteq [p]$ such that $\\sum _{i\\in I}\\alpha _i = k$ .",
"As all the parts of $\\alpha $ are positive integers there is at most one $I=[m]\\subseteq [p]$ with $I\\in S_k(\\alpha )$ .",
"In this case set $S_k^{\\prime }(\\alpha ) = S_k(\\alpha )\\setminus \\lbrace I\\rbrace $ otherwise set $S_k^{\\prime }(\\alpha ) = S_k(\\alpha )$ .",
"Lemma 3.17 Fix $\\lambda \\vdash n$ and assume for each composition $\\alpha \\sim \\lambda $ there exists some $C_\\alpha \\ge 0$ .",
"If for each such $\\alpha $ and $k\\le n$ we have $|S_k(\\alpha )|\\cdot C_\\alpha = \\sum _{I\\in S_k(\\alpha )} C_{\\alpha (I)},$ then the $C_\\alpha $ 's are equal.",
"For $\\alpha ,\\beta \\sim \\lambda $ there exists a sequence of compositions $\\alpha =\\alpha ^{(1)}, \\alpha ^{(1)}, \\ldots , \\alpha ^{(t)}=\\beta $ so that $\\alpha ^{(i+1)} = \\alpha ^{(i)}(\\lbrace j\\rbrace )$ for some $j$ .",
"Now choose $\\alpha $ so that $C_\\alpha $ is maximized.",
"By this choice of $\\alpha $ together with (REF ) and $k = \\alpha _j$ it follows that $C_\\alpha = C_{\\alpha (\\lbrace j\\rbrace )}$ .",
"This with our first observation implies our lemma.",
"Lemma 3.18 Set $k< n$ and $\\alpha \\models n$ .",
"There exists a bijection $f:S_k^{\\prime }(\\alpha )\\rightarrow \\Gamma _k(\\alpha )$ so that for each $I\\in S_k^{\\prime }(\\alpha )$ we have $(r(f(I))) = \\alpha (I)$ .",
"Fix $I \\in S_k^{\\prime }(\\alpha )$ and let $J\\subseteq [n-1]$ be such that $(J) = \\alpha (I)$ .",
"Define $f(I)=(U_1,\\ldots , U_p)\\vdash [n]$ by $(U_i)_{i\\in I} = ([j_1], [j_1+1,j_2],\\ldots , [j_{t-1},j_{t}]) \\vdash [k]$ and $(U_i)_{i\\in [p]\\setminus I}= [j_{t+1},j_{t+2}],\\ldots , [j_{p-1},n]) \\vdash [n]\\setminus [k].$ As $I \\ne [m]\\subseteq [p]$ for some form $m\\le p$ it follows that $f(I)\\in \\Gamma _k(\\alpha )$ .",
"Our map $f$ is certainly injective.",
"It also follows that every partition in $\\Gamma _k(\\alpha )$ can be constructed in this manner.",
"Hence $f$ is also surjective.",
"Continuing with the notation above we see that $r(f(I)) = [n-1]\\setminus f(I)^* = J$ .",
"As $(J) = \\alpha (I)$ this justifies our last claim.",
"Proposition 3.19 If $B\\Subset _n$ is $D$ -commutative with all inverse $J$ -classes then $B$ is symmetric.",
"Define $B_\\alpha := B_J$ where $(J) = \\alpha $ .",
"By recalling the discussion at the start of this subsection, it suffices to prove this proposition by showing that $|B_\\alpha | = |B_\\beta |$ whenever $\\alpha \\sim \\beta $ .",
"We proceed by induction on the number of parts in our compositions where the base case is when our composition has $n$ parts.",
"This case holds trivially since $(1^n)$ is the only composition with $n$ parts.",
"Take $\\alpha \\models n$ with $p$ parts.",
"As $B$ is $D$ -commutative with all inverse $J$ -classes, we know that (REF ) holds for this $\\alpha $ .",
"In light of this equation and Lemma REF it follows that $\\sum _{\\in \\Pi _k(\\alpha )}|B_{r()}|=\\sum _{\\in \\Pi _k(\\alpha )}|B_{s()}|$ holds for all $1\\le k<n$ .",
"Now consider a particular $\\in \\Pi _k(\\alpha )\\setminus \\Gamma _k(\\alpha )$ .",
"As $$ has $p$ blocks and at least one of them is not an interval, it follows that $|[n-1]\\setminus ^*|\\ge p$ .",
"Therefore the composition corresponding to $r()$ and $s()$ has at least $p+1$ parts and by Lemma REF we know that $r() \\sim s()$ .",
"It now follows by induction that $|B_{r()}|= |B_{s()}|$ .",
"Consequently $\\sum _{\\in \\Gamma _k(\\alpha )}|B_{r()}| = \\sum _{\\in \\Gamma _k(\\alpha )}|B_{s()}| = |\\Gamma _k(\\alpha )|\\cdot |B_\\alpha |,$ where the second equality follows from the fact that if $\\in \\Gamma _k(\\alpha )$ then $(s()) = \\alpha $ .",
"By appealing the bijection in Lemma REF we have for all $k< n$ $\\sum _{I\\in S_k^{\\prime }(\\alpha )} |B_{\\alpha (I)}| = |S_k^{\\prime }(\\alpha )|\\cdot |B_\\alpha |.$ In the case $I \\in S_k(\\alpha )\\setminus S_k^{\\prime }(\\alpha )$ then $\\alpha (I) = \\alpha $ .",
"By adding the term $|B_\\alpha |$ to both sides of (REF ) if necessary we have $\\sum _{I\\in S_k(\\alpha )} |B_{\\alpha (I)}| = |S_k(\\alpha )|\\cdot |B_\\alpha |.$ for all $k<n$ and compositions $\\alpha $ with $p$ parts.",
"As the case when $k=n$ yields a trivial equality we may further assume $k\\le n$ .",
"Appealing to Lemma REF with $C_\\alpha := |B_\\alpha |$ we conclude that $|B_\\alpha | = |B_\\alpha |$ for compositions $\\alpha \\sim \\beta $ with $p$ parts.",
"This completes our proof."
],
[
"Necessity: Symmetry implies $D$ -symmetry",
"The goal of this subsection is to prove that symmetric multisets are $D$ -symmetric, i.e., to prove Proposition REF .",
"We first show that this proposition holds for fine sets and, using this fact, “bootstrap” up to the general result.",
"As fine sets are intimately connected to the theory of tableaux we begin by introducing the needed ideas from this theory.",
"A standard Young tableaux of shape $\\mu \\vdash P$ is a filling of the Young digram of $\\mu $ with each number in $[n]$ used exactly once so that rows and columns are strictly increasing.",
"We denote by $(\\mu )$ the set of all standard Young tableaux of shape $\\mu $ and set $(n) = \\cup _{\\mu \\vdash n}(\\mu )$ .",
"For any $P\\in (n)$ and $m\\le n$ we define $P_{<m}$ to be the standard Young tableaux in $(m-1)$ given by the entries in $P$ that are $<m$ .",
"We refer to a coordinate location in a Young tableau as a box and the element in a box as a value.",
"All boxes are coordinatized using matrix coordinates.",
"For any $P\\in (n)$ we define its descent set by $P: = {i}{\\textrm {i+1 is on a row below i in P}}\\subseteq [n-1].$ mathmode, boxsize=2.3em Given $P\\in (n)$ we say a sequence of boxes $b_0,\\ldots , b_m$ in $P$ is a promotion path provided that $b_{i+1}$ is whichever of the boxes immediately below or to the right of $b_i$ that contain the smaller value for all $0\\le i<m$ .",
"So given an initial box $b_0$ the maximal promotion path starting at $b_0$ is uniquely determined.",
"Consequently it makes sense to define the $v$ -promotion path in $P$ to be the maximal promotion path that starts at the box containing the value $v$ .",
"For any $\\mu \\vdash n$ and $a,b\\in [n]$ with $a\\le b$ define the promotion operator $\\partial _a^b: (\\mu ) \\rightarrow (\\mu )$ as follows.",
"Fix some $P\\in (\\mu )$ and consider the skew tableau formed by the values in $[a,b]$ .",
"Let $c_0,\\ldots , c_m$ be the $a$ -promotion path in this skew tableau.",
"Next delete the entry in box $c_0$ and slide the value in $c_{i+1}$ into $c_i$ .",
"As $c_m$ is now empty, place in it the value $b+1$ .",
"Finally decrement each value in this skew tableau by 1 to obtain $\\partial _a^b P\\in (\\mu )$ .",
"Example 3.20 Take $P$ to be the tableau on the left then $\\partial _3^{12}P$ is the tableau on the right: mathmode, boxsize=2.2em $\\begin{ytableau}1& *(gray)3& *(lightgray)6& *(lightgray)7\\\\2& *(gray)5& *(gray)9& *(gray)11\\\\*(lightgray)4& *(lightgray)10& 13& 15\\\\*(lightgray)8& 14\\\\*(lightgray)12\\end{ytableau}\\qquad \\qquad \\qquad \\begin{ytableau}1& *(lightgray)4& *(lightgray)5& *(lightgray)6\\\\2& *(lightgray)8& *(lightgray)10& *(lightgray)12\\\\*(lightgray)3& *(lightgray)9& 13& 15\\\\*(lightgray)7& 14\\\\*(lightgray)11\\end{ytableau}\\ .$ Here the boxes corresponding to the skew tableau are in gray and the promotion path in $P$ is in dark gray.",
"We point out two important properties of the promotion operator.",
"First it is clear that $P_{<a}=(\\partial _a^b P)_{<a}.$ Second, we see that for each $\\mu \\vdash n$ the mapping $\\partial _a^b:(\\mu ) \\rightarrow (\\mu )$ is bijective as its inverse can be constructed as follows.",
"Let $c_0$ be the box containing $b$ and define the unique maximal sequence of boxes $c_0,\\ldots , c_m$ so that $c_{i+1}$ is whichever of the boxes immediately above or to the left of $c_i$ that contains the larger value.",
"Now delete the value $b$ in $c_0$ and slide the value in $b_{i+1}$ into box $b_{i}$ .",
"Next increment all the values in $[a,b]$ by 1 and place $a$ in the empty box $b_m$ .",
"We introduce more theory related to tableaux below as it is needed.",
"For now we have enough to prove our first few lemmas.",
"Lemma 3.21 For $Q\\in (n)$ and $a<u<b$ we have $u\\in Q\\ \\Longleftrightarrow \\ u-1 \\in (\\partial _a^b Q).$ First assume that $u\\in Q$ .",
"In the calculation of $\\partial _a^b Q$ the values in $[a+1,b]$ in $Q$ shift up one unit, shift left one unit, or remain fixed before being decremented by 1.",
"Hence the only way $u-1\\notin (\\partial _a^b Q)$ is if $Q$ contains one of the following configurations: mathmode, boxsize=2.2em *(lightgray) w z1 z u (lightgray) u+1 *(lightgray)w *(lightgray)u z1 *(lightgray)u+1 , where the promotion path is highlighted in gray.",
"In both cases a simple check shows that such promotion paths are impossible.",
"(E.g., in the second case $z_1<u$ .)",
"So neither of these two cases can occur.",
"We conclude that the forward direction of our lemma holds.",
"To prove the reverse direction, we need to show that if $u\\notin Q$ then $u-1\\notin (\\partial _a^bQ)$ .",
"Observe that if $Q^*$ denotes the conjugate of $Q$ then $u\\in Q^*$ .",
"Also note that the operation of taking the conjugate commutes with $\\partial _a^b$ .",
"So to prove this direction we need only apply the above argument to $Q^*$ .",
"The lemma now follows.",
"Lemma 3.22 Let $Q\\in (n)$ and $u\\le k+1<n$ .",
"Then $u\\in Q\\iff k+1\\in (\\partial _{u}^{k+1}\\circ \\partial _{u+1}^{k+2} Q ).$ Set $\\partial := \\partial _{u}^{k+1}\\circ \\partial _{u+1}^{k+2}$ .",
"Suppose for a contradiction that $u\\in (Q)$ and $k+1\\notin (\\partial Q)$ .",
"Let $B=(b_0,\\ldots , b_m)$ be the promotion path used by $\\partial _{u+1}^{k+2}$ and let $C=(c_0,\\ldots ,c_\\ell )$ be the promotion path used by $\\partial _u^{k+1}$ .",
"So in $Q$ boxes $b_0$ and $c_0$ contain $u+1$ and $u$ respectively, and in $\\partial Q$ boxes $b_m$ and $c_\\ell $ contain $k+2$ and $k+1$ respectively.",
"As $u\\in Q$ then $c_0$ is strictly above $b_0$ .",
"By our assumption that $k+1\\notin (\\partial Q)$ we must also have that $c_\\ell $ is weakly below $b_m$ .",
"In fact since $b_m$ contains $k+2$ in $\\partial _{u+1}^{k+2} Q$ then $c_\\ell $ is also strictly to the left of $b_m$ .",
"Now consider the first time a box in $C$ is weakly below and strictly to the left of some box in $B$ .",
"Certainly this box cannot be $c_0$ (since $c_0$ is above $b_0$ ) and in fact we must have the following configuration of values in $Q$ : mathmode, boxsize=2.2em *(blue) w (purple) *(red)u , where the blue and purple boxes are in $C$ and the red and purple boxes are in $B$ .",
"So $w<u$ .",
"Therefore in $\\partial _{u+1}^{k+2}Q$ the purple box contains $u-1$ and the white box contains $w^{\\prime } =w$ or $w-1$ .",
"Since the standard Young tableaux $\\partial _{u+1}^{k+2}Q$ contains exactly one occurrence of $w$ it follows that $u-1>w^{\\prime }$ .",
"It is now immediate that if the blue box is in $C$ then $C$ must contain the white box and not the purple box.",
"We arrive at our desired contradiction proving that the forward direction holds.",
"The converse can be proved by an argument similar to that used to prove the converse in the previous lemma.",
"Definition 3.23 For any $\\mu \\vdash n$ let $k\\le n$ be such that $V=\\lbrace v_1<v_2<\\cdots < v_{n-k}\\rbrace \\subseteq [n]$ .",
"Define the mapping $\\partial _V:(\\mu ) \\rightarrow (\\mu )$ by $\\partial _{V}:= \\partial _{v_1}^{k+1}\\circ \\partial _{v_2}^{k+2}\\circ \\cdots \\circ \\partial _{v_{n-k}}^n.$ Take $\\partial _{\\emptyset }$ to be the identity function.",
"We pause to comment on our choice of indexing above.",
"In what follows the operator $\\partial _V$ appears in the context where $V$ is the second block of an ordered partition where the first block has size $k$ .",
"As a result we have chosen to index the elements of $V$ as above.",
"Example 3.24 For example $V= \\lbrace 3,9,10\\rbrace \\subseteq [15]$ .",
"Then we have mathmode, boxsize=1.9em Table: NO_CAPTIONwhere the skew tableau used to obtain the following tableau is highlighted.",
"The next lemma follows immediately from the fact that each promotion operator is bijective.",
"Lemma 3.25 Fix $\\mu \\vdash n$ and some $V\\subseteq [n]$ .",
"Then $\\partial _V:(\\mu ) \\rightarrow (\\mu )$ is a bijection.",
"Lemma 3.26 Let $Q\\in (n)$ and fix $=(U,V)\\in \\Pi (n)$ .",
"For $u\\in ^*$ we have $u\\in Q\\iff \\delta _(u)\\in (\\partial _{V} Q).$ We prove this by induction on $| V|$ .",
"Observe that when $V=\\emptyset $ then $\\partial _VQ = Q$ and $\\delta _= 1$ so the lemma holds in this case.",
"Now take $|U|=k<n$ so that $|V|>0$ and let $v_1 = \\min (V)$ .",
"Define $V^{\\prime }=V\\setminus \\lbrace v_1\\rbrace $ and $U^{\\prime } = U \\cup \\lbrace v_1\\rbrace $ and set $= (U^{\\prime },V^{\\prime })$ .",
"For $x\\in [v_1]$ we then have $\\delta _(x) = x$ and by induction we know that if $u\\in ^*$ then $u\\in Q \\iff \\delta _(u) \\in ( \\partial _{V^{\\prime }} Q).$ Now define the function $f:[n]\\rightarrow [n]$ by setting $f(x) = {\\left\\lbrace \\begin{array}{ll}x - 1 & \\textrm { if } v_1<x\\le k+1\\\\x & \\textrm { otherwise}.\\end{array}\\right.",
"}$ Observe that for $x\\ne v_1$ we have $f\\circ \\delta _(x) =\\delta _(x)$ .",
"Next recall that the positions of values less than $v_1$ and greater than $k+1$ are invariant under the action of $\\partial _{v_1}^{k+1}$ .",
"This together with Lemma REF tells us that for any $T\\in (n)$ and $u\\ne v_1-1,v_1,k+1$ we have $u\\in T &\\iff f(u) \\in (\\partial _{v_1}^{k+1}T).$ We now consider the following two cases.",
"Case 1: $u\\in ^*$ and $u\\ne v_1$ As $u\\ne v_1$ then $\\delta _(u) \\ne v_1$ .",
"As $v_1=\\min (V)$ then $u\\ne v_1-1$ and so $\\delta _(u) \\ne v_1-1$ .",
"As $u\\ne \\max (U)$ then $\\delta _(u) \\ne k+1$ .",
"Computing we now have $u\\in Q &\\iff \\delta _(u) \\in ( \\partial _{V^{\\prime }} Q)\\\\& \\iff f\\circ \\delta _(u) \\in ( \\partial _{v_1}^{k+1}\\partial _{V^{\\prime }} Q)\\\\& \\iff \\delta _(u) \\in ( \\partial _V Q).$ So the lemma holds in this case.",
"Case 2: $u=v_1\\in ^*$ In this case $|V|\\ge 2$ and $v_1+1 = v_2$ .",
"Set $V^{\\prime \\prime } = V\\setminus \\lbrace v_1,v_2\\rbrace $ so that $V^{\\prime \\prime }$ is either empty or all its values are $>v_2$ .",
"By definition we have $\\partial _V Q = \\partial _{v_1}^{k+1}\\circ \\partial _{v_2}^{k+2}R$ where $R = \\partial _{V^{\\prime \\prime }}Q$ .",
"(Note that if $|V| = 2$ then $Q=R$ .)",
"As $Q_{<v_2+1} = R_{<v_2+1}$ and $v_1<v_2$ we have $v_1\\in Q \\iff v_1\\in R$ .",
"This together with Lemma REF gives $v_1\\in Q &\\iff v_1 \\in R \\\\&\\iff k+1\\in (\\partial _V Q) \\\\&\\iff \\delta _(v_1)\\in (\\partial _V Q).$ This completes our proof.",
"In order to continue we recall some additional well known results from the theory of tableaux.",
"We denote the Robinson-Schensted correspondence by $:_n \\rightarrow \\bigcup _{\\mu \\vdash n} (\\mu )\\times (\\mu ).$ If $\\pi \\mapsto (P,Q)$ under this bijection we call $P$ the insertion tableau.",
"We omit the definition of $$ as it is not needed here but we point out two key properties of this mapping that are well-known.",
"First $$ is a bijection.",
"The second well known property we state as a lemma.",
"For a reference see [7], Chapter 5, Exercise 22.",
"Lemma 3.27 If $(\\pi ) = (P,Q)$ , then $(\\pi ) = Q$ .",
"For each $P\\in (n)$ we define the corresponding Knuth class to be the set $K(P):= {\\pi \\in _n}{\\textrm {P is the insertion tableau for \\pi }}.$ Additionally we need the following well-known result due to Gessel.",
"Theorem 3.28 ([5]) For any $\\lambda \\vdash n$ we have $s_\\lambda = \\sum _{Q\\in (\\lambda )} F_{Q,n}.$ This result together with the Robinson-Schensted correspondence tells us that if $P$ has shape $\\lambda $ then $Q(K(P)) = s_\\lambda $ .",
"In fact we can say more.",
"Consider any fine multiset $B\\Subset _n$ with $Q(B) = \\sum _{\\lambda \\vdash n} c_\\lambda s_\\lambda $ where $c_\\lambda \\ge 0$ .",
"For each shape $\\lambda $ fix some $P_\\lambda $ of that shape.",
"Define the multiset $A = \\bigsqcup _{\\lambda \\vdash n}\\bigsqcup _{i=1}^{c_\\lambda } K(P_\\lambda )$ so that $Q(A) = Q(B)$ and hence $A\\equiv B$ .",
"We make use of this in our next lemma.",
"Lemma 3.29 If $B\\Subset _n$ is fine then it is $D$ -symmetric.",
"As $B$ is fine it follows from the discussion preceding the statement of this lemma that $B\\equiv \\bigsqcup K(P_i)$ where our disjoint union is over some finite index set $I$ and $P_i\\in (n)$ .",
"By parts a) and b) of Lemma REF it then suffices to show that individual Knuth classes are $D$ -symmetric.",
"To show this fix $P\\in (\\mu )$ and consider the Knuth class $K(P)$ .",
"Take $=(U,V)\\vdash [n]$ and consider the mapping $\\Psi _:K(P)\\rightarrow K(P)$ given by $\\pi \\mapsto ^{-1}(P,\\partial _V Q)$ where $RS(\\pi ) = (P,Q)$ .",
"The bijectivity of this mapping follows from Lemma REF and the fact that $$ is bijective.",
"Now take $u\\in ^*$ and fix some $\\pi \\in K(P)$ with $(\\pi ) = (P,Q)$ .",
"By Lemma REF and Lemma REF , since $=(U,V)$ , we obtain our desired result $u\\in (\\pi ) \\iff u\\in Q \\iff \\delta _(u) \\in (\\partial _V Q)\\iff \\delta _(u) \\in (\\Psi _(\\pi )).$ By Lemma REF we now conclude that $B$ is $D$ -symmetric.",
"Proposition 3.30 If $B\\Subset _n$ is symmetric then it is $D$ -symmetric.",
"As $B$ is assumed to be symmetric and the Schur functions $\\lbrace s_\\lambda \\rbrace _{\\lambda \\vdash n}$ are an integral basis for $\\Lambda (n)_\\mathbb {Z}$ , the ring of symmetric functions of degree $n$ with integral coefficients, we can write $Q(B) = \\sum _{\\lambda \\vdash n} c_\\lambda s_\\lambda $ where $c_\\lambda \\in \\mathbb {Z}$ .",
"Now define the multiset $A = \\bigsqcup _{\\lambda \\vdash n\\atop c_\\lambda <0}\\bigsqcup _{i = 1}^{|c_\\lambda |} K(\\lambda )$ so that by Theorem REF have $Q(A) = \\sum _{\\lambda \\vdash n \\atop c_\\lambda <0}|c_\\lambda |s_\\lambda ,$ so that $A$ is fine, and by our choice $A$ we also see that $Q(A\\sqcup B) = Q(A) + Q(B) = \\sum _{\\lambda \\vdash n\\atop c_\\lambda >0} c_\\lambda s_\\lambda ,$ so that $A\\sqcup B$ is also fine.",
"By Lemma REF we now see that $A$ and $A\\sqcup B$ are both $D$ -symmetric.",
"It now follows from Lemma REF that $B$ is $D$ -symmetric as well."
]
] | 1906.04399 |
[
[
"An Approximate Bayesian Approach to Model-assisted Survey Estimation\n with Many Auxiliary Variables"
],
[
"Abstract Model-assisted estimation with complex survey data is an important practical problem in survey sampling.",
"When there are many auxiliary variables, selecting significant variables associated with the study variable would be necessary to achieve efficient estimation of population parameters of interest.",
"In this paper, we formulate a regularized regression estimator in the framework of Bayesian inference using the penalty function as the shrinkage prior for model selection.",
"The proposed Bayesian approach enables us to get not only efficient point estimates but also reasonable credible intervals.",
"Results from two limited simulation studies are presented to facilitate comparison with existing frequentist methods."
],
[
"*"
],
[
" Model-assisted estimation with complex survey data is an important practical problem in survey sampling.",
"When there are many auxiliary variables, selecting significant variables associated with the study variable would be necessary to achieve efficient estimation of population parameters of interest.",
"In this paper, we formulate a regularized regression estimator in the framework of Bayesian inference using the penalty function as the shrinkage prior for model selection.",
"The proposed Bayesian approach enables us to get not only efficient point estimates but also reasonable credible intervals.",
"Results from two limited simulation studies are presented to facilitate comparison with existing frequentist methods.",
"Keywords: Generalized regression estimation; Regularization; Shrinkage prior; Survey Sampling"
],
[
"Introduction",
"Probability sampling is a scientific tool for obtaining a representative sample from the target population.",
"In order to estimate a finite population total from a target population, Horvitz-Thompson (HT) estimator obtained from a probability sample satisfies design-consistency and the resulting inference is justified from the randomization perspective [15].",
"However, the HT estimator uses the first-order inclusion probability only and does not fully incorporate all available information in the finite population.",
"To improve its efficiency, regression estimation is often used by incorporating auxiliary information in the finite population.",
"[9], [11], [18], and [4] present comprehensive overviews of variants of regression estimation in survey sampling.",
"There are also other directions of improvement on the HT estimator based on prediction using augmented models [39], [40], [38].",
"The regression estimation approaches in survey sampling assume a model for the finite population, i.e., the superpopulation model, as $y_i = {\\mbox{$x$}}_i^t {\\mbox{$\\beta $}}+ e_i,$ where $y_i$ is a response variable, ${\\mbox{$x$}}_i$ and ${\\mbox{$\\beta $}}$ are vectors of auxiliary variables and regression coefficients, respectively, and $e_i$ is an error term satisfying ${\\rm E}(e_i)=0$ and ${\\rm Var}(e_i)=\\sigma ^2$ .",
"The superpopulation model does not necessarily hold in the sample as the sampling design can be informative [29], [21].",
"Under the regression superpopulation model in (REF ), [16] show that the asymptotic variance of the regression estimator achieves the lower bound of [13].",
"Thus, the regression estimator is asymptotically efficient in the sense of achieving the minimum anticipated variance under the joint distribution of the sampling design and the superpopulation model in (REF ).",
"On the other hand, the dimension of the auxiliary variables ${\\mbox{$x$}}_i$ could be large in practice.",
"Even when the number of observed covariates is not necessarily large, the dimension of ${\\mbox{$x$}}_i$ could be very large once we include polynomial or interaction terms to achieve flexible modeling, as considered in Section .",
"However, in this case, the optimality of the regression estimator is untenable.",
"When there are many auxiliary variables, the asymptotic bias of the regression estimator using all the auxiliary variables is no longer negligible and the resulting inference can be problematic.",
"Simply put, including irrelevant auxiliary variables can introduce substantial variability in point estimation, but its uncertainty is not fully accounted for by the standard linearization variance estimation, resulting in misleading inference.",
"To overcome the problem, variable selection techniques for regression estimation have been considered in literatures [33], [32].",
"The classical selection approach is based on a step-wise method.",
"However, the step-wise methods will not necessarily produce the best model [8] although the potential effect on prediction could be limited.",
"Another approach is to employ regularized estimation of regression coefficients.",
"For example, [24] propose a regularized regression estimation approach based on the LASSO penalty of [34].",
"However, there are two main problems with the regularization approach in regression estimation.",
"First, the choice of the regularization parameter is not straightforward under survey sampling when the parameter is strongly related to the selection results.",
"Second, after model selection, the frequentist inference is notoriously difficult to make.",
"In this paper, to overcome the above difficulties, we adopt a Bayesian framework in the regularized regression estimation.",
"We first introduce an approximate Bayesian approach for regression estimation when $p+1= \\mbox{dim} ({\\mbox{$x$}})$ is fixed, using the approximate Bayesian approach considered in [36].",
"The proposed Bayesian method fully captures the uncertainty in parameter estimation for the regression estimator and has good coverage properties.",
"Second, the proposed Bayesian method is extended to the problem of large $p$ in regularized regression estimation.",
"By incorporating the penalty function for regularization into the prior distribution, the uncertainty associated with model selection and parameter estimation is fully captured in the Bayesian machinery.",
"Furthermore, the choice of the penalty parameter can be handled by using its posterior distribution.",
"Hence, the proposed method provides a unified approach to Bayesian inference with sparse model-assisted survey estimation.",
"The proposed method is a calibrated Bayesian [22] and it is asymptotically equivalent to the frequentist model-assisted approach for a fixed $p$ .",
"The paper is organized as follows.",
"In Section 2, the basic setup is introduced.",
"In Section 3, the approximate Bayesian inference using regression estimation is proposed under a fixed $p$ setup.",
"In Section 4, the proposed method is extended to high dimensional setup by developing sparse regression estimation using shrinkage prior distributions.",
"In Section 5, the proposed method is extended to non-linear regression models.",
"In Section 6, results from two limited simulation studies are presented.",
"The proposed method is applied to the real data example in Section 7.",
"Some concluding remarks are made in Section 8."
],
[
"Basic setup",
"Consider a finite population of a known size $N$ .",
"Associated with unit $i$ in the finite population, we consider measurement $\\lbrace {\\mbox{$x$}}_i, y_i\\rbrace $ where ${\\mbox{$x$}}_i$ is the vector of auxiliary variables with dimension $p$ and $y_i$ is the study variable of interest.",
"We are interested in estimating the finite population mean $\\bar{Y}= N^{-1} \\sum _{i=1}^N y_i$ from a sample selected by a probability sampling design.",
"Let $A$ be the index set of the sample and we observe $\\lbrace {\\mbox{$x$}}_i, y_i\\rbrace _{i\\in A}$ from the sample.",
"The HT estimator $\\hat{\\bar{Y}}_{HT}=N^{-1} \\sum _{i \\in A} \\pi _i^{-1} y_i$ , where $\\pi _i$ is the first-order inclusion probability of unit $i$ , is design unbiased but it is not necessarily efficient.",
"If the finite population mean $\\bar{{\\mbox{$X$}}}= N^{-1} \\sum _{i=1}^N {\\mbox{$x$}}_i$ is known, then we can improve the efficiency of $\\hat{\\bar{Y}}_{\\rm HT}$ by using the following regression estimator: $\\hat{\\bar{Y}}_{\\rm reg}= \\frac{1}{N} \\sum _{i=1}^N {\\mbox{$x$}}_i^t \\hat{{\\mbox{$\\beta $}}}$ where $\\hat{{\\mbox{$\\beta $}}}$ is an estimator of ${\\mbox{$\\beta $}}$ in (REF ).",
"Typically, we use $\\hat{{\\mbox{$\\beta $}}}$ obtained by minimizing the weighted quadratic loss $Q({\\mbox{$\\beta $}}) = \\sum _{i\\in A}\\pi _i^{-1}(y_i-{\\mbox{$x$}}_i^t{\\mbox{$\\beta $}})^2,$ motivated from the model (REF ).",
"If an intercept term is included in ${\\mbox{$x$}}_i$ such that ${\\mbox{$x$}}_i^t = (1, {\\mbox{$x$}}_{1i}^t )$ , we can express $ \\hat{\\bar{Y}}_{\\rm reg}= \\hat{\\beta }_0 + \\bar{{\\mbox{$X$}}}_1^t \\hat{{\\mbox{$\\beta $}}}_1 = \\hat{N}^{-1} \\sum _{i \\in A} \\pi _i^{-1} \\left( y_i - {\\mbox{$x$}}_{1i}^t \\hat{{\\mbox{$\\beta $}}}_1 \\right) + \\bar{{\\mbox{$X$}}}_1^t \\hat{{\\mbox{$\\beta $}}}_1$ where $\\hat{N} = \\sum _{i \\in A} \\pi _i^{-1}$ and $\\hat{{\\mbox{$\\beta $}}}_1$ is given by $\\hat{{\\mbox{$\\beta $}}}_1 = \\left\\lbrace \\sum _{i \\in A} \\pi _i^{-1} ({\\mbox{$x$}}_{1i} - \\hat{\\bar{{\\mbox{$X$}}}}_{1, \\pi } )^{\\otimes 2} \\right\\rbrace ^{-1} \\sum _{i \\in A} \\pi _i^{-1} ({\\mbox{$x$}}_{1i} - \\hat{\\bar{{\\mbox{$X$}}}}_{1, \\pi } ) y_i$ where $ \\hat{\\bar{{\\mbox{$X$}}}}_{1, \\pi } = \\hat{N}^{-1} \\sum _{i \\in A} \\pi _i^{-1} {\\mbox{$x$}}_{1i} $ and $B^{ \\otimes 2} = B B^{\\prime }$ for some matrix $B$ .",
"To discuss some asymptotic properties of $\\hat{\\bar{Y}}_{\\rm reg}$ in (REF ), we consider a sequence of finite populations and samples as discussed in Isaki and Fuller (1982), where $N$ increases with $n$ .",
"Note that $\\hat{\\bar{Y}}_{\\rm reg}- \\bar{Y}&= \\hat{\\bar{Y}}_{\\pi } - \\bar{Y}+ \\Big (\\bar{{\\mbox{$X$}}}_1 - \\hat{\\bar{{\\mbox{$X$}}}}_{1, \\pi } \\Big )^t \\hat{{\\mbox{$\\beta $}}}_1 \\\\&= \\hat{\\bar{Y}}_{\\pi } - \\bar{Y} + \\left(\\bar{{\\mbox{$X$}}}_1 - \\hat{\\bar{{\\mbox{$X$}}}}_{1, \\pi } \\right)^t{{\\mbox{$\\beta $}}}_1 + R_n$ where $\\hat{\\bar{Y}}_{\\pi } = \\hat{N}^{-1} \\sum _{i \\in A} \\pi _i^{-1} y_i$ and $R_n = \\left(\\bar{{\\mbox{$X$}}}_1 -\\hat{\\bar{{\\mbox{$X$}}}}_{1} \\right)^t\\left( \\hat{{\\mbox{$\\beta $}}}_1 - {{\\mbox{$\\beta $}}}_1 \\right)$ for any ${\\mbox{$\\beta $}}_{1}$ .",
"If we choose ${\\mbox{$\\beta $}}_{1} = p \\lim _{n\\rightarrow \\infty } \\hat{{\\mbox{$\\beta $}}}_1$ with respect to the sampling probability and $p=\\mbox{dim} ({\\mbox{$x$}}_1)$ is fixed in the asymptotic setup, then we can obtain $R_n = O_p (n^{-1})$ and safely use the main terms of (REF ) to describe the asymptotic behavior of $\\hat{\\bar{Y}}_{\\rm reg}$ .",
"To emphasize its dependence on $\\hat{{\\mbox{$\\beta $}}}_1$ in the regression estimator, we can write $\\hat{\\bar{Y}}_{\\rm reg}= \\hat{\\bar{Y}}_{\\rm reg}( \\hat{{\\mbox{$\\beta $}}}_1 )$ .",
"Roughly speaking, we can obtain $\\sqrt{n} \\left\\lbrace \\hat{\\bar{Y}}_{\\rm reg}( \\hat{{\\mbox{$\\beta $}}}_1 ) - \\hat{\\bar{Y}}_{\\rm reg}( {\\mbox{$\\beta $}}_1) \\right\\rbrace = O_p (n^{-1/2} p).$ and, if $p=o(n^{1/2})$ then we can safely ignore the effect of estimating ${\\mbox{$\\beta $}}_1$ in the regression estimator.",
"See Supplementary Material for a sketched proof of (REF ).",
"If, on the other hand, the dimension $p$ is larger than $O(n^{1/2})$ , then we cannot ignore the effect of estimating ${\\mbox{$\\beta $}}_1$ .",
"In this case, we can consider using some variable selection idea to reduce the dimension of ${\\mbox{$X$}}$ .",
"For variable selection, we may employ techniques of regularized estimation of regression coefficients.",
"The regularization method can be described as finding $(\\hat{\\beta }_0^{(R)},\\hat{{\\mbox{$\\beta $}}}_1^{(R)}) = \\mbox{argmin}_{\\beta _0,\\beta _1} \\lbrace Q( {\\mbox{$\\beta $}}) + p_\\lambda ( {\\mbox{$\\beta $}}_1) \\rbrace ,$ where $Q( {\\mbox{$\\beta $}})$ is defined in (REF ) and $p_{\\lambda } ( {\\mbox{$\\beta $}}_1)$ is a penalty function with parameter $\\lambda $ .",
"Some popular penalty functions are presented in Table 1.",
"Once the solution to (REF ) is obtained, then the regularized regression estimator is given by $ \\hat{\\bar{Y}}_{\\rm reg}( \\hat{{\\mbox{$\\beta $}}}_1^{(R)} ) = \\bar{{\\mbox{$X$}}}_1^t \\hat{{\\mbox{$\\beta $}}}_1^{(R)} + \\frac{1}{\\hat{N}} \\sum _{i \\in A} \\frac{1}{\\pi _i} \\left( y_i - {\\mbox{$x$}}_{1i}^t \\hat{{\\mbox{$\\beta $}}}_1^{(R)} \\right).$ Statistical inference with the regularized regression estimator in (REF ) is not fully investigated in the literature.",
"For example, [7] consider the regularized regression estimator using adaptive LASSO of [41], but they assume that the sampling design is non-informative and the uncertainty in model selection is not fully incorporated in their inference.",
"Generally speaking, making inference after model selection under superpopulation frequentist framework is difficult.",
"The approximated Bayesian method introduced in the next section will capture the full uncertainty in the Bayesian framework.",
"Table: Popular penalized regression methods"
],
[
"Approximate Bayesian survey regression estimation",
"Developing Bayesian model-assisted inference under complex sampling is a challenging problem in statistics.",
"[36] recently propose the so-called approximate Bayesian method for design-based inference using asymptotic normality of a design-consistent estimator.",
"Specifically, for a given parameter $\\theta $ with a prior distribution $\\pi ( \\theta )$ , if one can find a design-consistent estimator $\\hat{\\theta }$ of $\\theta $ , then the approximate posterior distribution of $\\theta $ is given by $p( \\theta \\mid \\hat{\\theta } ) = \\frac{ f ( \\hat{\\theta } \\mid \\theta ) \\pi (\\theta ) }{ \\int f ( \\hat{\\theta } \\mid \\theta ) \\pi (\\theta ) {\\rm d} \\theta } ,$ where $f ( \\hat{\\theta } \\mid \\theta )$ is the sampling distribution of $\\hat{\\theta }$ , which is often approximated by a normal distribution.",
"Drawing on this idea, one can develop an approximate Bayesian approach to capture the full uncertainty in the regression estimator.",
"Let $\\hat{{\\mbox{$\\beta $}}}= \\left( \\sum _{i \\in A} \\pi _i^{-1} {\\mbox{$x$}}_{i} {\\mbox{$x$}}_{i}^t \\right)^{-1} \\sum _{i \\in A} \\pi _i^{-1} {\\mbox{$x$}}_{i} y_i$ be the design-consistent estimator of ${\\mbox{$\\beta $}}$ and $\\hat{{\\mbox{$V$}}}_\\beta $ be the corresponding asymptotic variance-covariance matrix of $\\hat{{\\mbox{$\\beta $}}}$ , given by $\\hat{{\\mbox{$V$}}}_{\\beta } =\\left( \\sum _{i \\in A} \\pi _i^{-1} {\\mbox{$x$}}_{i} {\\mbox{$x$}}_{i}^t \\right)^{-1} \\left( \\sum _{i \\in A} \\sum _{j \\in A} \\frac{ \\Delta _{ij} }{ \\pi _{ij} } \\frac{\\hat{e}_i {\\mbox{$x$}}_{i} }{ \\pi _i} \\frac{ \\hat{e}_j {\\mbox{$x$}}_j^t}{ \\pi _j } \\right) \\left( \\sum _{i \\in A} \\pi _i^{-1} {\\mbox{$x$}}_{i} {\\mbox{$x$}}_{i}^t \\right)^{-1},$ where $\\hat{e}_i = y_i - {\\mbox{$x$}}_{i}^t \\hat{{\\mbox{$\\beta $}}}$ , $\\Delta _{ij} = \\pi _{ij} - \\pi _i \\pi _j$ and $\\pi _{ij}$ is the joint inclusion probability of unit $i$ and $j$ .",
"Under some regularity conditions, as discussed in Chapter 2 of [12], we can establish $\\hat{{\\mbox{$V$}}}_{\\beta 11}^{-1/2} \\left( \\hat{{\\mbox{$\\beta $}}}_1 - {\\mbox{$\\beta $}}_1 \\right) \\mid {\\mbox{$\\beta $}}\\stackrel{\\mathcal {L}}{ \\longrightarrow } N(0, I )$ as $n \\rightarrow \\infty $ , where $\\hat{{\\mbox{$V$}}}_{ \\beta 11} $ is the submatrix of $\\hat{{\\mbox{$V$}}}_{\\beta }$ with $\\hat{{\\mbox{$V$}}}_{\\beta } = \\begin{pmatrix}\\hat{V}_{\\beta 00} & \\hat{V}_{\\beta 01} \\\\\\hat{V}_{\\beta 10} & \\hat{V}_{\\beta 11}\\end{pmatrix}.$ Thus, using (REF ) and (REF ), we can obtain the approximate posterior distribution of ${\\mbox{$\\beta $}}$ as $p({\\mbox{$\\beta $}}_1|\\hat{{\\mbox{$\\beta $}}}_1)=\\frac{\\phi _p (\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11} )\\pi ({\\mbox{$\\beta $}}_1 )}{\\int \\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11} )\\pi ({\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1},$ where $\\phi _p$ denotes a $p$ -dimensional multivariate normal density and $\\pi ({\\mbox{$\\beta $}}_1)$ is a prior distribution for ${\\mbox{$\\beta $}}_1$ .",
"Now, we consider the conditional posterior distribution of $\\bar{Y}$ for a given ${\\mbox{$\\beta $}}_1$ .",
"First, define $\\hat{\\bar{Y}}_{\\rm reg}( {\\mbox{$\\beta $}}_1) = \\bar{{\\mbox{$X$}}}_1^t {\\mbox{$\\beta $}}_1 + \\frac{1}{\\hat{N}} \\sum _{i \\in A} \\frac{1}{\\pi _i} \\left( y_i - {\\mbox{$x$}}_{1i}^t {{\\mbox{$\\beta $}}}_1 \\right).$ Note that $\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1)$ is an approximately design-unbiased estimator of $\\bar{Y}$ , regardless of ${\\mbox{$\\beta $}}_1$ .",
"Under some regularity conditions, we can show that $\\hat{\\bar{Y}}_{\\rm reg}( {\\mbox{$\\beta $}}_1)$ follows a normal distribution asymptotically.",
"Thus, we obtain $\\frac{ \\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1)-\\bar{Y}}{\\sqrt{\\hat{V}_{e}({\\mbox{$\\beta $}}_1)}} \\mid \\bar{Y}, {\\mbox{$\\beta $}}_1 \\stackrel{\\mathcal {L}}{ \\longrightarrow } N(0,1),$ where $\\hat{V}_{e} ({\\mbox{$\\beta $}}_1)=\\frac{1}{N^2}\\sum _{i\\in A}\\sum _{j\\in A}\\frac{\\Delta _{ij}}{\\pi _{ij}} \\frac{1}{ \\pi _i} \\frac{1}{\\pi _j} (y_i-{\\mbox{$x$}}_{1i}^t{\\mbox{$\\beta $}}_1)(y_j-{\\mbox{$x$}}_{1j}^t{\\mbox{$\\beta $}}_1),$ is a design consistent variance estimator of $\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1)$ for given ${\\mbox{$\\beta $}}_1$ .",
"We then use $\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e} ({\\mbox{$\\beta $}}_1))$ as the density for the approximate sampling distribution of $\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1)$ in (REF ), where $\\phi (\\cdot ; \\mu ,\\sigma ^2)$ is the normal density function with mean $\\mu $ and variance $\\sigma ^2$ .",
"Thus, the approximate conditional posterior distribution of $\\bar{Y}$ given ${\\mbox{$\\beta $}}$ can be defined as $p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),{\\mbox{$\\beta $}}_1)\\propto \\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e} ({\\mbox{$\\beta $}}_1))\\pi (\\bar{Y}\\mid {\\mbox{$\\beta $}}_1),$ where $\\pi (\\bar{Y}\\mid {\\mbox{$\\beta $}}_1)$ is a conditional prior distribution of $\\bar{Y}$ given ${\\mbox{$\\beta $}}_1$ .",
"Without extra assumptions, we can use a flat prior distribution for $\\pi ( \\bar{Y} \\mid {\\mbox{$\\beta $}}_1)$ .",
"Therefore, combining (REF ) and (REF ), the approximate posterior distribution of $\\bar{Y}$ can be obtained as $\\begin{split}&p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),\\hat{{\\mbox{$\\beta $}}}_1)\\\\& \\ \\ =\\frac{\\int \\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_1))\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11} )\\pi ({\\mbox{$\\beta $}}_1)\\pi (\\bar{Y}\\mid {\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1}{\\iint \\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e} ({\\mbox{$\\beta $}}_1))\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11} )\\pi ({\\mbox{$\\beta $}}_1)\\pi (\\bar{Y}\\mid {\\mbox{$\\beta $}}_1 ){\\rm d}{\\mbox{$\\beta $}}_1 {\\rm d}\\bar{Y}}.\\end{split}$ Generating posterior samples from (REF ) can be easily carried out via the following two steps: 1.",
"Generate posterior sample ${\\mbox{$\\beta $}}_1^{\\ast }$ of ${\\mbox{$\\beta $}}_1$ from (REF ).",
"2.",
"Generate posterior sample of $\\bar{Y}$ from the conditional posterior (REF ) given ${\\mbox{$\\beta $}}_1^{\\ast }$ .",
"Based on the approximate posterior samples of $\\bar{Y}$ , we can compute the posterior mean as a point estimator as well as credible intervals for uncertainty quantification for $\\bar{Y}$ including the variability in estimating ${\\mbox{$\\beta $}}_1$ .",
"The following theorem presents an asymptotic property of the proposed approximate Bayesian method.",
"Theorem 1 Under the regularity conditions described in the Supplementary Material, conditional on the full sample data, $\\sup _{\\bar{Y}\\in \\Theta _Y}\\Big |p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),\\hat{{\\mbox{$\\beta $}}}_1)-\\phi (\\bar{Y};\\hat{\\bar{Y}}_{\\rm reg},\\hat{V}_{e} )\\Big |\\rightarrow 0,$ in probability as $n\\rightarrow \\infty $ and $n/N\\rightarrow f\\in [0,1)$ , where $\\Theta _Y$ is some Borel set for $\\bar{Y}$ and $p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),\\hat{{\\mbox{$\\beta $}}}_1)$ is given in (REF ).",
"Theorem REF is a special case of the Bernstein-von Mises theorem [35] in survey regression estimation, and its sketched proof is given in the Supplementary Material.",
"The proof is not necessarily rigorous but contains enough details to deliver the main ideas.",
"According to Theorem REF , the credible interval for $\\bar{Y}$ constructed from the approximated posterior distribution (REF ) is asymptotically equivalent to the frequentist confidence interval based on the asymptotic normality of the common survey regression estimator.",
"Therefore, the proposed Bayesian method implements the frequentist inference of the survey regression estimator at least asymptotically."
],
[
"Approximate Bayesian method with shrinkage priors",
"We now consider the case when there are many auxiliary variables in applying regression estimation.",
"When $p$ is large, it is desirable to select a suitable subset of auxiliary variables that are associated with the response variable to avoid inefficient regression estimation due to irrelevant covariates.",
"To deal with the problem in a Bayesian way, we may define the approximate posterior distribution of $\\bar{Y}$ given ${\\mbox{$\\beta $}}_1$ as similar to (REF ).",
"That is, we use the asymptotic distribution of the estimators $\\hat{{\\mbox{$\\beta $}}}_1$ of ${\\mbox{$\\beta $}}_1$ and assign a shrinkage prior for ${\\mbox{$\\beta $}}_1$ .",
"Let $\\pi _{\\lambda }({\\mbox{$\\beta $}}_1)$ be the shrinkage prior for ${\\mbox{$\\beta $}}_1$ with a structural parameter $\\lambda $ which might be multivariate.",
"Among the several choices of shrinkage priors, we specifically consider two priors for ${\\mbox{$\\beta $}}_1$ : Laplace [28] and horseshoe [5], [6].",
"The Laplace prior is given by $\\pi _\\lambda ({\\mbox{$\\beta $}}_1)\\propto \\exp (-\\lambda \\sum _{k=1}^p|\\beta _k|)$ , which is related to Lasso regression [34], so that the proposed approximated Bayesian method can be seen as the Bayesian version of a survey regression estimator with Lasso [24].",
"The horseshoe prior is a more advanced shrinkage prior of the form: $\\pi _{\\lambda }({\\mbox{$\\beta $}}_1)=\\prod _{k=1}^p\\int _0^{\\infty }\\phi (\\beta _k; 0,\\lambda ^2u_k^2)\\frac{2}{\\pi (1+u_k^2)}{\\rm d}u_k,$ where $\\phi (\\cdot ; a,b)$ denotes the normal density function with mean $a$ and variance $b$ .",
"It is known that the horseshoe prior enjoys more severe shrinkage for the zero elements of ${\\mbox{$\\beta $}}_1$ than the Laplace prior, thus allowing strong signals to remain large [5].",
"Similarly to (REF ), we can develop a posterior distribution of ${\\mbox{$\\beta $}}_1$ using the shrinkage prior $p_{\\lambda } ({\\mbox{$\\beta $}}_1| \\hat{{\\mbox{$\\beta $}}}_1)=\\frac{\\phi (\\hat{{\\mbox{$\\beta $}}}_1;{\\mbox{$\\beta $}}_1,\\hat{V}_{\\beta 11} )\\pi _{\\lambda }({\\mbox{$\\beta $}}_1)}{\\int \\phi (\\hat{{\\mbox{$\\beta $}}}_1;{\\mbox{$\\beta $}}_1,\\hat{V}_{\\beta 11} )\\pi _{\\lambda }({\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1},$ where $\\hat{V}_{\\beta 11} $ is the asymptotic variance-covariance matrix of $\\hat{{\\mbox{$\\beta $}}}_1$ , defined in (REF ).",
"Once ${\\mbox{$\\beta $}}_1$ are sampled from (REF ), we can use the same posterior distribution of $\\bar{Y}$ in (REF ) for a given ${\\mbox{$\\beta $}}$ .",
"Therefore, the approximate posterior distribution of $\\bar{Y}$ can be obtained as $\\begin{split}&p_{\\lambda }(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),\\hat{{\\mbox{$\\beta $}}}_1)\\\\&\\ \\ =\\frac{\\int \\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_1))\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11} )\\pi _{\\lambda }({\\mbox{$\\beta $}}_1)\\pi (\\bar{Y}\\mid {\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1}{\\iint \\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e} ({\\mbox{$\\beta $}}_1))\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11} )\\pi _{\\lambda } ({\\mbox{$\\beta $}}_1)\\pi (\\bar{Y}\\mid {\\mbox{$\\beta $}}_1 ){\\rm d}{\\mbox{$\\beta $}}_1 {\\rm d}\\bar{Y}}.\\end{split}$ Generating posterior samples from (REF ) can be easily carried out via the following two steps: 1.",
"For a given $\\lambda $ , generate posterior sample ${\\mbox{$\\beta $}}_1^{\\ast }$ of ${\\mbox{$\\beta $}}_1$ from $p_\\lambda (\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),\\hat{{\\mbox{$\\beta $}}}_1)$ in (REF ).",
"2.",
"Generate posterior sample of $\\bar{Y}$ from the conditional posterior (REF ) for given ${\\mbox{$\\beta $}}_1^{\\ast }$ .",
"Remark 1 Let $\\hat{\\beta }_0^{(R)} $ and $\\hat{{\\mbox{$\\beta $}}}_1^{(R)}$ be the estimator of $\\beta _0$ and ${\\mbox{$\\beta $}}_1$ defined as $(\\hat{\\beta }_0^{(R)},\\hat{{\\mbox{$\\beta $}}}_1^{(R)})={\\rm argmin}_{\\beta _0,\\beta _1} \\left\\lbrace \\sum _{i\\in A}\\frac{1}{\\pi _i}(y_i-\\beta _0-{\\mbox{$x$}}_{1i}^t{\\mbox{$\\beta $}}_1)^2+{\\rm P}_\\lambda ({\\mbox{$\\beta $}}_1)\\right\\rbrace ,$ where ${\\rm P}({\\mbox{$\\beta $}}_1)=-2\\log \\pi _{\\lambda }({\\mbox{$\\beta $}}_1)$ is the penalty (regularization) term for ${\\mbox{$\\beta $}}_1$ induced from prior $\\pi _{\\lambda }({\\mbox{$\\beta $}}_1)$ .",
"For example, the Laplace prior for $\\pi _{\\lambda }({\\mbox{$\\beta $}}_1)$ leads to the penalty term ${\\rm P}({\\mbox{$\\beta $}}_1)=2\\lambda \\sum _{k=1}^p|\\beta _k|$ , in which $\\hat{{\\mbox{$\\beta $}}}_1^{(R)}$ corresponds to the regularized estimator of ${\\mbox{$\\beta $}}_1$ used in [24].",
"Since the exponential of $-\\sum _{i\\in A}\\pi _i^{-1}(y_i-\\beta _0-{\\mbox{$x$}}_i^t{\\mbox{$\\beta $}}_1)^2$ is close to the approximated likelihood $\\phi _p((\\hat{\\beta }_0,\\hat{{\\mbox{$\\beta $}}}_1^t); (\\beta _0,{\\mbox{$\\beta $}}_1^t), \\hat{{\\mbox{$V$}}}_{\\beta } )$ used in the approximated Bayesian method when $n$ is large, the mode of the approximated posterior of $(\\beta _0,{\\mbox{$\\beta $}}_1^t)$ would be close to the frequentist estimator (REF ) as well.",
"Remark 2 By the frequentist approach, $\\lambda $ is often called the tuning parameter and can be selected via a data-dependent procedure such as cross validation as used in [24].",
"On the other hand, in the Bayesian approach, we assign a prior distribution on the hyperparameter $\\lambda $ and consider integration with respect to the posterior distribution of $\\lambda $ , which means that uncertainty of the hyperparameter estimation can be taken into account.",
"Specifically, we assign a gamma prior for $\\lambda ^2$ as the Laplace prior and a half-Cauchy prior for $\\lambda $ as the horseshoe prior (REF ).",
"They both lead to familiar forms of full conditional posterior distributions of $\\lambda $ or $\\lambda ^2$ .",
"The details are given in the Supplementary Material.",
"As in Section , we obtain the following asymptotic properties of the proposed approximate Bayesian method.",
"Theorem 2 Under the regularity conditions described in the Supplementary Material, conditional on the full sample data, $\\sup _{\\bar{Y}\\in \\Theta _Y}\\Big |p_{\\lambda } (\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),\\hat{{\\mbox{$\\beta $}}}_1)-\\phi (\\bar{Y};\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1^{(R)}),\\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1^{(R)} ))\\Big |\\rightarrow 0,$ in probability as $n\\rightarrow \\infty $ and $n/N\\rightarrow f\\in [0,1)$ , where $\\Theta _Y$ is some Borel set for $\\bar{Y}$ and $p_{\\lambda } (\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1),\\hat{{\\mbox{$\\beta $}}}_1)$ is given in (REF ).",
"The sketched proof is given in the Supplementary Material.",
"Theorem REF ensures that the proposed approximate Bayesian method is asymptotically equivalent to the frequentist version in which ${\\mbox{$\\beta $}}_1$ is estimated by the regularized method with penalty corresponding to the shrinkage prior used in the Bayesian method.",
"Moreover, the proposed Bayesian method can be extended to cases using general non-linear regression, as demonstrated in the next section."
],
[
"An Extension to non-linear models",
"The proposed Bayesian methods can be readily extended to work with non-linear regression.",
"Some extensions of the regression estimator to nonlinear models are also considered in [37], [2], and [26].",
"We consider a general working model for $y_i$ as ${\\rm E}(y_i\\mid {\\mbox{$x$}}_i)=m({\\mbox{$x$}}_i;{\\mbox{$\\beta $}})=m_i$ and ${\\rm Var}(y_i\\mid {\\mbox{$x$}}_i)=\\sigma ^2a(m_i)$ for some known functions $m(\\cdot ;\\cdot )$ and $a(\\cdot )$ .",
"The model-assisted regression estimator for $\\bar{Y}$ with ${\\mbox{$\\beta $}}$ known is then $\\hat{\\bar{Y}}_{\\rm reg, m}({{\\mbox{$\\beta $}}})=\\frac{1}{N}\\left\\lbrace \\sum _{i=1}^Nm({\\mbox{$x$}}_i;{{\\mbox{$\\beta $}}} )+\\sum _{i\\in A}\\frac{1}{\\pi _i}\\Big (y_i-m({\\mbox{$x$}}_i;{{\\mbox{$\\beta $}}} )\\Big )\\right\\rbrace ,$ and its design-consistent variance estimator is obtained by $\\hat{V}_{e,m} ({{\\mbox{$\\beta $}}} )=\\frac{1}{N^2}\\sum _{i\\in A}\\sum _{j\\in A}\\frac{\\Delta _{ij}}{\\pi _{ij}} \\frac{1}{ \\pi _i} \\frac{1}{\\pi _j} \\lbrace y_i- m( {\\mbox{$x$}}_i; {{\\mbox{$\\beta $}}} ) \\rbrace \\lbrace y_j- m( {\\mbox{$x$}}_j ; {{\\mbox{$\\beta $}}}) \\rbrace ,$ which gives the approximate conditional posterior distribution of $\\bar{Y}$ given ${\\mbox{$\\beta $}}$ .",
"That is, similarly to (REF ), we can obtain $p(\\bar{Y}| \\hat{\\bar{Y}}_{\\rm reg, m}({{\\mbox{$\\beta $}}}) ,{\\mbox{$\\beta $}})\\propto \\phi ( \\hat{\\bar{Y}}_{\\rm reg, m}({{\\mbox{$\\beta $}}}); \\bar{Y}, \\hat{V}_{e,m } ({\\mbox{$\\beta $}}))\\pi (\\bar{Y}\\mid {\\mbox{$\\beta $}}).$ To generate the posterior values of ${\\mbox{$\\beta $}}$ , we first find a design-consistent estimator $\\hat{{\\mbox{$\\beta $}}}$ of ${\\mbox{$\\beta $}}$ .",
"Note that a consistent estimator $\\hat{{\\mbox{$\\beta $}}}$ can be obtained by solving $\\hat{U}({\\mbox{$\\beta $}})\\equiv \\sum _{i\\in A}\\pi _i^{-1}\\lbrace y_i-m ({\\mbox{$x$}}_i; {\\mbox{$\\beta $}}) \\rbrace h({\\mbox{$x$}}_i; {\\mbox{$\\beta $}})=0,$ where $h({\\mbox{$x$}}_i; {\\mbox{$\\beta $}})=(\\partial m_i/\\partial {\\mbox{$\\beta $}})/a(m_i)$ .",
"For example, for binary $y_i$ , we may use a logistic regression model with $m({\\mbox{$x$}}_i;{\\mbox{$\\beta $}})=\\exp ({\\mbox{$x$}}_i^t{\\mbox{$\\beta $}})/\\lbrace 1+\\exp ({\\mbox{$x$}}_i^t{\\mbox{$\\beta $}})\\rbrace $ and ${\\rm Var}(y_i)=m_i(1-m_i)$ , which leads to $h({\\mbox{$x$}}_i;{\\mbox{$\\beta $}})={\\mbox{$x$}}_i$ .",
"Under some regularity conditions, we can establish the asymptotic normality of $\\hat{{\\mbox{$\\beta $}}}$ .",
"That is, $\\hat{{\\mbox{$V$}}}_{\\beta }^{-1/2} (\\hat{{\\mbox{$\\beta $}}}-{\\mbox{$\\beta $}}) \\mid {\\mbox{$\\beta $}}\\stackrel{\\mathcal {L}}{ \\longrightarrow } N(0, I) ,$ where $\\hat{{\\mbox{$V$}}}_{\\beta } =\\left\\lbrace \\sum _{i \\in A} \\frac{1}{\\pi _i} \\hat{{\\mbox{$h$}}}_i \\dot{m} ({\\mbox{$x$}}_i; \\hat{{\\mbox{$\\beta $}}})^{t} \\right\\rbrace ^{-1} \\left( \\sum _{i \\in A} \\sum _{j \\in A} \\frac{ \\Delta _{ij} }{ \\pi _{ij} } \\frac{\\hat{e}_i \\hat{{\\mbox{$h$}}}_i }{ \\pi _i} \\frac{ \\hat{e}_j \\hat{{\\mbox{$h$}}}_j^t}{ \\pi _j } \\right) \\left\\lbrace \\sum _{i \\in A} \\frac{1}{\\pi _i} \\hat{{\\mbox{$h$}}}_i\\dot{m} ({\\mbox{$x$}}_i; \\hat{{\\mbox{$\\beta $}}})^t \\right\\rbrace ^{-1},$ with $\\hat{e}_i=y_i-m({\\mbox{$x$}}_i;\\hat{{\\mbox{$\\beta $}}})$ , $\\hat{{\\mbox{$h$}}}_i = h( {\\mbox{$x$}}_i; \\hat{{\\mbox{$\\beta $}}})$ , and $\\dot{m} ({\\mbox{$x$}}; {\\mbox{$\\beta $}}) = \\partial m( {\\mbox{$x$}}; {\\mbox{$\\beta $}})/ \\partial {\\mbox{$\\beta $}}$ .",
"Note that $\\dot{m} ({\\mbox{$x$}}; {\\mbox{$\\beta $}}) =m_i(1-m_i){\\mbox{$x$}}_i$ under a logistic regression model.",
"Thus, the posterior distribution of ${\\mbox{$\\beta $}}$ given $\\hat{{\\mbox{$\\beta $}}}$ can be obtained by $p( {\\mbox{$\\beta $}}\\mid \\hat{{\\mbox{$\\beta $}}} ) \\propto \\phi ( \\hat{{\\mbox{$\\beta $}}}\\mid {\\mbox{$\\beta $}}, \\hat{{\\mbox{$V$}}}_\\beta ) \\pi ({\\mbox{$\\beta $}}) .$ We can use a shrinkage prior $\\pi ( {\\mbox{$\\beta $}})$ for ${\\mbox{$\\beta $}}$ in (REF ) if necessary.",
"Once ${\\mbox{$\\beta $}}^*$ is generated from (REF ), the posterior values of $\\bar{Y}$ are generated from (REF ) for a given ${\\mbox{$\\beta $}}^*$ .",
"This formula enables us to define the approximate posterior distribution of ${\\mbox{$\\beta $}}$ of the form (REF ), so that the approximate Bayesian inference for $\\bar{Y}$ can be carried out in the same way as in the linear regression case.",
"Note that Theorem REF still holds under the general setup as long as the regularity conditions given in the Supplementary Material are satisfied."
],
[
"Simulation",
"We investigate the performance of the proposed approximate Bayesian methods against standard frequentist methods using two limited simulation studies.",
"In the first simulation, we consider a linear regression model for a continuous $y$ variable.",
"In the second simulation, we consider a binary $y$ and apply the logistic regression model for the non-linear regression estimation.",
"In the first simulation, we generate $x_i=(x_{i1},\\ldots ,x_{ip^{\\ast }})^t$ , $i=1,\\ldots ,N$ , from a multivariate normal distribution with mean vector $(1,\\ldots ,1)^t$ and variance-covariance matrix $2R(0.2)$ , where $p^* = 50$ and the $(i,j)$ -th element of $R(\\rho )$ is $\\rho ^{|i-j|}$ .",
"The response variables $Y_i$ are generated from the following linear regression model: $Y_i=\\beta _0+\\beta _1x_{i1}+\\cdots +\\beta _{p^{\\ast }}x_{ip^{\\ast }}+\\varepsilon _i, \\ \\ \\ \\ i=1,\\ldots ,N,$ where $N=10,000$ , $\\varepsilon _i\\sim N(0,2)$ , $\\beta _1=1$ , $\\beta _4=-0.5$ , $\\beta _7=1$ , $\\beta _{10}=-0.5$ and the other $\\beta _k$ 's are set to zero.",
"For the dimension of the auxiliary information, we consider four scenarios for $p$ of $20, 30, 40$ and 50.",
"For each $p$ , we assume that we can access only $(x_{i1},\\ldots ,x_{ip})^t$ a subset of the full information $(x_{i1},\\ldots ,x_{ip^{\\ast }})^t$ .",
"Note that for all scenarios the auxiliary variables significantly related with $Y_i$ are included, and so only the amount of irrelevant information gets larger as $p$ gets larger.",
"We selected a sample size of $n=300$ from the finite population, using two sampling mechanism: (A) simple random sampling (SRS) and (B) probability-proportional-to-size sampling (PPS) with size measure $z_i=\\max \\lbrace \\log (1+|Y_i+e_i|) ,1\\rbrace $ with $e_i\\sim {\\rm Exp}(2)$ .",
"The parameter of interest is $\\bar{Y}=N^{-1}\\sum _{i=1}^NY_i$ .",
"We assume that $\\bar{X}_k=N^{-1}\\sum _{i=1}^Nx_{ik}$ is known for all $k=1,\\ldots ,p$ .",
"For the simulated dataset, we apply the proposed approximate Bayesian methods with the uniform prior $\\pi ({\\mbox{$\\beta $}}_1)\\propto 1$ , Laplace prior and horseshoe prior (REF ) for ${\\mbox{$\\beta $}}_1$ , which are denoted by AB, ABL and ABH, respectively.",
"For all the Bayesian methods, we use $\\pi (\\bar{Y})\\propto 1$ .",
"We generate 5,000 posterior samples of $\\bar{Y}$ after discarding the first 500 samples and compute the posterior mean of $\\bar{Y}$ as the point estimate.",
"As for the frequentist methods, we apply the original generalized regression estimator without variable selection (GREG) as well as the GREG method with Lasso regularization [24], ridge estimation of ${\\mbox{$\\beta $}}_1$ [31] and forward variable selection (GREG-V) using adjusted coefficient of determination.",
"We also adopted the mixed modeling approach to the GREG estimation [27] which is similar to GREG-R.",
"Moreover, the HT estimator is employed as a benchmark for efficiency comparison.",
"In GREG-L, the tuning parameter is selected via 10-fold cross validation, and we use the gamma prior ${\\rm Ga}(\\lambda _{\\ast }^2,1)$ for $\\lambda ^2$ in ABL, where $\\lambda _{\\ast }$ is the selected value for $\\lambda $ in GREG-L.",
"In ABH, we assign a prior for the tuning parameter and generate posterior samples.",
"Based on $1,000$ replications, we calculate the square root of mean squared errors (RMSE) and bias of point estimators which are reported in Table REF .",
"We also evaluated the performance of $95\\%$ confidence (credible) intervals using coverage probabilities (CP) and the average length (AL), which are shown in Table REF .",
"Table REF shows that RMSE and bias of AB and GREG are almost identical, which is consistent with the fact that AB is a Bayesian version of GREG.",
"Moreover, the results show that the existing shrinkage methods such as GREG-L and the proposed Bayesian methods ABL and ABH tend to produce smaller RMSEs and smaller absolute biases than GREG or AB as $p$ increases, which indicates the importance of suitable selection of auxiliary variables when $p$ is large.",
"From Table REF , it is observed that the CPs of GREG decreases as $p$ increases and are significantly smaller than the nominal level since GREG ignores the variability in estimating ${\\mbox{$\\beta $}}$ and the variability increases as $p$ increases.",
"On the other hand, the Bayesian version AB can take account of the variability estimating ${\\mbox{$\\beta $}}$ and the CPs are around the nominal level and ALs of AB are larger than those of GREG.",
"Although the performance of GREG-L is much better than GREG due to the shrinkage techniques, the CPs are not necessarily close to the nominal level.",
"Note that GREG-M takes account of the variability estimating ${\\mbox{$\\beta $}}$ , but not in other parameters, thereby the coverage performance is limited.",
"It is also confirmed that the proposed ABH and ABL methods produce narrower intervals than AB.",
"In the second simulation study, we consider the binary case for $y_i$ and apply the non-linear regression method discussed in Section 5.",
"The binary response variables $Y_i$ are generated from the following logistic regression model: $Y_i\\sim {\\rm Ber}(\\delta _i), \\ \\ \\ \\log \\left(\\frac{\\delta _i}{1-\\delta _i}\\right)=\\beta _0+\\beta _1x_{i1}+\\cdots +\\beta _{p}x_{ip}, \\ \\ \\ \\ i=1,\\ldots ,N,$ where $\\beta _0=-1$ and the other settings are the same as the linear regression case.",
"We selected a sample size of $n=300$ from the finite population, using two sampling mechanism: (A) simple random sampling and (B) probability-proportional-to-size sampling with size measure $z_i=\\max \\lbrace \\log (1+0.5Y_i+e_i) ,0.5\\rbrace $ with $e_i\\sim {\\rm Exp}(3)$ .",
"We again apply the three Bayesian methods and three frequents methods, GREG, GREG-L and GREG-R, based on a logistic regression model to obtain point estimates and confidence/credible intervals of the population mean $\\bar{Y}=N^{-1}\\sum _{i=1}^NY_i$ .",
"The obtained RMSE and bias of point estimates and CP and AL of intervals based on 1,000 replications are reported in Tables REF and REF , respectively, which also shows again the superiority of the proposed Bayesian approach to the frequentist approach in terms of uncertainty quantification.",
"Table: Square root of Mean squared errors (RMSE) and bias of point estimators under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with linear regression.All values are multiplied by 100.Table: Coverage probabilities (CP) and average lengths (AL) of 95%95\\% confidence/credible intervals under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with linear regression.All values are multiplied by 100.Table: Square root of Mean squared errors (RMSE) and bias of point estimators under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with logistic regression.All values are multiplied by 100.Table: Coverage probabilities (CP) and average lengths (AL) of 95%95\\% credible/confidence intervals under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with logistic regression.All values are multiplied by 100."
],
[
"Example",
"We applied the proposed methods to the synthetic income data available from the sae package [25] in R. In the dataset, the normalized annual net income is observed for a certain number of individuals in each province of Spain.",
"The dataset contains 9 covariates; four indicators of the four groupings of ages ($16-24$ , $25-49$ , $50-64$ and $\\ge 65$ denoted by ag1$,\\ldots ,$ag4, respectively), the indicator of having Spanish nationality na, the indicators of education levels (primary education ed1 and post-secondary education ed2), and the indicators of two employment categories (employed em1 and unemployed em2).",
"We also adopted 13 interaction variables: ag1*na, ag2*na, ag3*na, ag4*na, ag2*ed1, ag3*ed1, ag4*ed1, ag1*em1, ag2*em1, ag3*em1, ag4*em1, ed1*em1 and ed2*em1, as auxiliary variables, thereby $p=22$ in this example.",
"The dataset also contains information of survey weights, so that we used its inverse value as the sampling probability.",
"Since there is no information regarding the details of sampling mechanism, we approximate the joint inclusion probability as the product of two sampling probabilities.",
"In this example, we focus on estimating average income in three provinces, Palencia, Segovia and Soria, where the number of sampled units are 72, 58 and 20, respectively.",
"The number of non-sampled units were around $10^6$ .",
"It should be noted that the number of sample sizes are not so large compared with the number of auxiliary variables, especially in Soria.",
"Hence, the estimation error of regression coefficients would not be negligible and the proposed Bayesian methods would be appealing in this case.",
"In order to perform joint estimation and inference in the three provinces, we employed the following working model: $y_i = \\alpha + \\sum _{h\\in \\lbrace 1, 2,3\\rbrace } x_{0i}^{(h)} \\beta _{0}^{(h)} + {\\mbox{$x$}}_i^t {\\mbox{$\\beta $}}_1 + e_i,$ where $\\alpha $ is an intercept term, $x_{0i}^{(h)}=1$ if $i$ belong to province $h$ , where $h=1$ for Palencia, $h=2$ for Segovia, and $h=3$ for Soria, and ${\\mbox{$x$}}_i$ is the vector of auxiliary variables with dimension $p=22$ (9 auxiliary variables and 13 interaction variables).",
"Here $y_i$ is the log-transformed net income and $e_i$ is the error term.",
"Under the working model (REF ), the posterior distribution of $\\bar{Y}_h$ is $p\\lbrace \\bar{Y}_h \\mid \\hat{\\bar{Y}}_{h, \\text{reg}} ( \\beta _0^{(h)}, {\\mbox{$\\beta $}}_1),\\beta _0^{(h)}, {\\mbox{$\\beta $}}_1 \\rbrace \\propto \\phi ( \\hat{\\bar{Y}}_{h, \\text{reg}} ( \\beta _0^{(h)}, {\\mbox{$\\beta $}}_1) \\mid \\bar{Y}_h, \\hat{V}_{e,h} ({\\mbox{$\\beta $}}) ) \\pi ( \\bar{Y}_h ) ,$ where $\\hat{\\bar{Y}}_{h, \\text{reg}}= \\hat{\\beta }_0^{(h)} + \\bar{{\\mbox{$X$}}}_h^t \\hat{{\\mbox{$\\beta $}}}_1 + \\frac{1}{N_h} \\sum _{i \\in A_h} \\frac{1}{\\pi _i} \\left( y_i - \\hat{\\beta }_0^{(h)} - {\\mbox{$x$}}_i^t \\hat{{\\mbox{$\\beta $}}}_1 \\right) ,$ and $\\hat{V}_{e,h} ( {\\mbox{$\\beta $}}) = \\frac{1}{N_h^2} \\sum _{i \\in A_h} \\sum _{j \\in A_h} \\frac{ \\Delta _{ij} }{ \\pi _{ij} } \\frac{1}{\\pi _i } \\frac{1}{ \\pi _j} \\left( y_i - \\beta _0^{(h)} - {\\mbox{$x$}}_i^t {\\mbox{$\\beta $}}_1 \\right) \\left( y_j - \\beta _0^{(h)} - {\\mbox{$x$}}_j^t {\\mbox{$\\beta $}}_1 \\right).$ Based on the above formulas, we performed the proposed approximate Bayesian methods for $\\bar{Y}_h$ for each $h$ , and computed $95\\%$ credible intervals for the log-transformed average income with 5000 posterior samples after discarding the first 500 samples as burn-in period.",
"We considered three types of priors for ${\\mbox{$\\beta $}}_1$ , flat, Laplace and horseshoe priors as considered in Section .",
"We also calculated $95\\%$ confidence intervals of the log-transformed average income based on the two frequentist methods, GREG and GREG-L, using the working model (REF ).",
"In applying GREG-L, the tuning parameter in the Lasso estimator was selected via 10 fold cross validation.",
"The $95\\%$ credible intervals of ${\\mbox{$\\beta $}}_1$ based on the approximate posterior distributions under Laplace and horseshoe priors are shown in Figure REF , in which the design-consistent and Lasso estimates of ${\\mbox{$\\beta $}}_1$ are also given.",
"It is observed that the approximate posterior mean of ${\\mbox{$\\beta $}}_1$ shrinks the design-consistent estimates of ${\\mbox{$\\beta $}}_1$ toward 0 although exactly zero estimates are not produced as the frequentist Lasso estimator does.",
"The Lasso estimate selects only one variable among 22 candidates, and the variable is also significant in terms of the credible interval in both two priors.",
"Moreover, the two Bayesian methods detect one or two more variables to be significant judging from the credible intervals.",
"Comparing the results from two priors, the horseshoe prior provides narrower credible intervals than the Laplace prior.",
"In Figure REF , we show the resulting credible and confidence intervals of the average income in the three provinces.",
"It is observed that the proposed Bayesian methods, AB and ABL, tend to produce wider credible intervals than the confidence intervals of the corresponding frequencies methods, GREG and GREG-L, respectively, which is consistent with the simulation results in Section .",
"We can also confirm that the credible intervals of ABH are slightly narrower than those of ABL, which would reflect the differences of interval lengths of ${\\mbox{$\\beta $}}_1$ as shown in Figure REF .",
"Figure: 95%95\\% credible intervals of regression coefficients under Laplace (left) and horseshoe (right) priors.Figure: 95%95\\% confidence and credible intervals for average income based on five methods in three provinces in Spain."
],
[
"Concluding Remarks",
"We have proposed an approximate Bayesian method for model-assisted survey estimation using parametric regression models as working models.",
"The proposed method is justified under the frequentist framework and captures the full uncertainty in estimating regression parameters even when the number of the auxiliary variables is large.",
"A main advantage of the proposed method is that it uses a shrinkage prior for regularized regression estimation, which not only provides an efficient point estimator, but also fully captures the uncertainty associated with model selection and parameter estimation via a Bayesian framework.",
"Although we only consider two popular prior distributions, the Laplace prior and the horseshoe prior, other priors, such as the spike-and-slab prior [17], can be adopted in the same way.",
"Further investigation regarding the choice of the shrinkage prior distributions will be an important research topic in the future.",
"Although our working model is parametric, the proposed approximate Bayesian method can be applied to other semiparametric models such as local polynomial model [3], P-spline regression model [2], or a neural network model [26].",
"By finding suitable prior distributions for the semiparametric models, the model complexity parameters will be determined automatically and the uncertainty will be captured in the approximate Bayesian framework.",
"Finally, under more complicated sampling design such as multi-stage stratified cluster sampling, the main idea can be applied similarly since the proposed Bayesian method relies on the sampling distribution of the GREG estimator, which is asymptotically normal as shown by [20].",
"If the asymptotic normality is questionable, one can use a weighted likelihood bootstrap to approximate Bayesian posterior, as in [23].",
"Such extensions are beyond the scope of this paper and will be considered in the future."
],
[
"Supplementary Materials",
"Supplementary Material includes technical details for posterior computation, proofs of theorems and additional results of simulation studies.",
"We thank the AE and three anonymous referees for very constructive comments.",
"The first author was supported by Japan Society for the Promotion of Science KAKENHI grant number JP18K12757.",
"The second author was supported by US National Science Foundation (MMS-1733572).",
"Supplementary Material for `An Approximate Bayesian Approach to Model-assisted Survey Estimation with Many Auxiliary Variables' This Supplementary Material contains a proof of (5), details of posterior computation, proofs of theorems and results of additional simulation suites."
],
[
"Proof of (2.5)",
"We assume the same conditions in the proof of Theorem 1, given in Section .",
"From (2.4), we have $E( R_n) & = & - E\\left\\lbrace ( \\hat{\\bar{{\\mbox{$X$}}}}_{\\rm HT} - \\bar{{\\mbox{$X$}}}_N )^t ( \\hat{{\\mbox{$\\beta $}}}_1 - {\\mbox{$\\beta $}}_{1\\ast } ) \\right\\rbrace = - \\mbox{tr} \\left\\lbrace {\\rm Cov} \\left( \\hat{\\bar{{\\mbox{$X$}}}}_{\\rm HT}, \\hat{{\\mbox{$\\beta $}}}_1 \\right) \\right\\rbrace \\\\&=& - \\sum _{j=1}^p {\\rm Cov} \\left( \\hat{\\bar{x}}_{\\rm HT, j} , \\hat{\\beta }_j \\right)= O(p/n),$ where the expectation is taken with respect to the sampling distribution.",
"Also, we can show that $ V( R_n ) = O( p/ n^2 )$ .",
"Therefore, using Chebychev inequality, we have $R_n = O_p ( p/n) $ and result (2.5) follows."
],
[
"Posterior computation",
"We provide the algorithm for generating the approximate posterior distribution of ${\\mbox{$\\beta $}}_1$ given in (4.20) with two shrinkage priors, Laplace and horseshoe (4.18) priors.",
"Using the mixture representation of both priors, we get the following Gibbs sampling algorithm."
],
[
"Laplace prior",
"We consider the mixture representation of Laplace distribution: $\\beta _k|\\tau _k\\sim N(0,\\tau _k^2)$ and $\\tau _k^2\\sim {\\rm Exp}(\\lambda ^2/2)$ , independently, for $k=1,\\ldots ,p$ .",
"For $\\lambda ^2$ , we consider the conjugate prior ${\\rm Ga}(a,b)$ , where ${\\rm Ga}(a,b)$ is a gamma distribution with shape parameter $a$ and rate parameter $b$ .",
"The full conditional distribution of ${\\mbox{$\\beta $}}_1$ is multivariate normal with mean ${\\mbox{$A$}}^{-1}\\hat{{\\mbox{$V$}}}_{\\beta 11}^{-1}\\hat{{\\mbox{$\\beta $}}}_1$ and variance-covariance matrix ${\\mbox{$A$}}^{-1}$ where ${\\mbox{$A$}}=\\hat{{\\mbox{$V$}}}_{\\beta 11}^{-1}+\\mathbf {D}^{-1}$ with $\\mathbf {D}={\\rm diag}(\\tau _1^2,\\ldots ,\\tau _p^2)$ .",
"The full conditional distribution of $\\lambda ^2$ is ${\\rm Ga}(a+p,b+\\sum _{k=1}^p\\tau _k^2/2)$ , and $\\tau _1^2,\\ldots ,\\tau _p^2$ are conditionally independent, with $1/\\tau _j^2$ conditionally inverse-Gaussian with parameters $\\mu =\\sqrt{\\lambda /\\beta _j^2}$ in the parametrization of the inverse-Gaussian density given by $f(x)=\\sqrt{\\frac{\\lambda }{2\\pi }}x^{-3/2}\\exp \\left\\lbrace -\\frac{\\lambda (x-\\mu )^2}{2\\mu ^2x}\\right\\rbrace , \\ \\ x>0.$ The prior for ${\\mbox{$\\beta $}}_1$ can be expressed as a hierarchy: $\\beta _k|u_k\\sim N(0,\\lambda ^2u_k^2)$ and $u_k\\sim {\\rm HC}(0,1)$ independently for $k=1,\\ldots ,p$ , where ${\\rm HC}(0,1)$ is the standard half-Cauchy distribution.",
"Using the hierarchical expression of the half-Cauchy distribution, we obtain the following Gibbs sampling steps.",
"Let ${\\mbox{$A$}}=\\hat{{\\mbox{$V$}}}_{\\beta 11}^{-1}+\\mathbf {B}^{-1}$ , where $\\mathbf {B}=\\lambda ^2{\\rm diag}(u_1^2,\\ldots ,u_p^2)$ .",
"The full conditional distribution of ${\\mbox{$\\beta $}}_1$ is multivariate normal with mean ${\\mbox{$A$}}^{-1}\\hat{{\\mbox{$V$}}}_{\\beta 11}^{-1}\\hat{{\\mbox{$\\beta $}}}_1$ and variance-covariance matrix ${\\mbox{$A$}}^{-1}$ .",
"The full conditional distribution of $u_k^2$ and $\\lambda ^2$ are, respectively, give by ${\\rm IG}\\left(1,\\frac{1}{\\xi _k}+\\frac{\\beta _k^2}{2\\lambda ^2}\\right) \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ {\\rm IG}\\left(\\frac{p+1}{2},\\frac{1}{\\gamma }+\\frac{1}{2}\\sum _{k=1}^p\\frac{\\beta _k^2}{u_k^2}\\right),$ where ${\\rm IG}(a,b)$ denotes an inverse-Gamma distribution with shape parameter $a$ and rate parameter $b$ .",
"Here $\\xi _k$ and $\\gamma $ are additional latent variables, and their full conditional distributions are given by ${\\rm IG}(1,1+1/\\delta _k^2)$ and ${\\rm IG}(1,1+1/\\lambda ^2)$ , respectively."
],
[
"A sketched proof of Theorem 1",
"To discuss the asymptotic properties of the approximate Bayesian method, we first assume a sequence of finite populations and samples with finite fourth moments as in [16].",
"The finite population is a random sample from an unknown superpopulation model.",
"Let $\\bar{Y}_{\\ast }$ and ${\\mbox{$\\beta $}}_{1\\ast }$ be the true values of $\\bar{Y}$ and ${\\mbox{$\\beta $}}_1$ .",
"Let $B_n=(\\bar{Y}_{\\ast }-r_n,\\bar{Y}_{\\ast }+r_n)$ and $C_n$ be a ball with centre ${\\mbox{$\\beta $}}_{1\\ast }$ and radius $r_n\\sim n^{\\tau -1/2}$ for $0<\\tau <1/2$ .",
"We make the following regularity assumptions (C1) Assume that the sufficient conditions for the asymptotic normality of $\\hat{\\bar{Y}}_{\\rm reg}$ for $\\bar{Y}\\in B_n$ hold for the sequence of finite populations and samples.",
"(C2) Assume that the prior distribution $\\pi (\\bar{Y})$ is positive and satisfies a Lipschitz condition over its support $\\Theta _Y$ ; that is, there exists $C_1<\\infty $ such that $|\\pi (\\theta _1)-\\pi (\\theta _2)|\\le C_1|\\theta _1-\\theta _2|$ for $\\theta _1,\\theta _2\\in \\Theta _Y$ .",
"(C3) Assume that $\\hat{{\\mbox{$V$}}}_{\\beta 11}={\\mbox{$V$}}_{\\beta 11}\\lbrace 1+o_P(1)\\rbrace $ and $(\\hat{{\\mbox{$\\beta $}}}_1-{\\mbox{$\\beta $}}_1)^t\\hat{{\\mbox{$V$}}}_{\\beta 11}^{-1}(\\hat{{\\mbox{$\\beta $}}}_1-{\\mbox{$\\beta $}}_1)=(\\hat{{\\mbox{$\\beta $}}}_1-{\\mbox{$\\beta $}}_1)^t{\\mbox{$V$}}_{\\beta 11}^{-1}(\\hat{{\\mbox{$\\beta $}}}_1-{\\mbox{$\\beta $}}_1)\\lbrace 1+o_P(1)\\rbrace $ for any ${\\mbox{$\\beta $}}\\in C_n$ and $n\\rightarrow \\infty $ .",
"(C4) Assume that $\\pi ({\\mbox{$\\beta $}})$ is positive and finite over its support $\\Theta _\\beta $ .",
"Sufficient conditions for (C1) are discussed within various asymptotic structures [1], [30].",
"Conditions (C2) and (C4) are satisfied for common priors such as (multivariate) normal distribution .",
"Condition (C3) essentially requires that the design variance estimators be consistent and meet a certain continuity condition.",
"Let $g(\\bar{Y},{\\mbox{$\\beta $}})=\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}))\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11})\\pi ({\\mbox{$\\beta $}}_1)$ .",
"Then, the approximated posterior distribution is given by $p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{{\\mbox{$\\beta $}}}_1)&=\\frac{\\int g(\\bar{Y},{\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1}{\\iint g(\\bar{Y},{\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1{\\rm d}\\bar{Y}}.$ Note that $ \\int g(\\bar{Y},{\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1=\\int _{\\beta \\in C_n}g(\\bar{Y},{\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1+\\int _{\\beta \\in \\mathbb {R}^p\\setminus C_n}g(\\bar{Y},{\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1$ By the same argument in the proof of Theorem 1 in [36], we have $\\operatornamewithlimits{plim}_{n\\rightarrow \\infty }\\int _{\\beta \\in C_n}\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, \\hat{{\\mbox{$V$}}}_{\\beta 11}){\\rm d}{\\mbox{$\\beta $}}_1=1,$ so the second term in (REF ) is $o_P(1)$ .",
"On the other hand, under condition (C3), $\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1;{\\mbox{$\\beta $}}_1,\\hat{{\\mbox{$V$}}}_{\\beta 11})=\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1,{\\mbox{$V$}}_{\\beta 11})\\lbrace 1+o_P(1)\\rbrace $ as $n\\rightarrow \\infty $ , for any ${\\mbox{$\\beta $}}_1\\in C_n$ , thereby under condition (C4), $\\int _{\\beta \\in C_n}g(\\bar{Y},{\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1&=\\int _{\\beta \\in C_n}\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_1))\\phi _p(\\hat{{\\mbox{$\\beta $}}}_1; {\\mbox{$\\beta $}}_1, {\\mbox{$V$}}_{\\beta 11})\\pi ({\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1\\\\&=\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast }); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast }))\\pi ({\\mbox{$\\beta $}}_{1\\ast })\\lbrace 1+o_P(1)\\rbrace $ as $n\\rightarrow \\infty $ since $V\\rightarrow 0$ and $\\hat{{\\mbox{$\\beta $}}}_1\\rightarrow {\\mbox{$\\beta $}}_{1\\ast }$ as $n\\rightarrow \\infty $ .",
"Hence, we have $p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{{\\mbox{$\\beta $}}}_1)&=\\frac{\\pi ({\\mbox{$\\beta $}}_{1\\ast })\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast }); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast }))\\pi (\\bar{Y})\\lbrace 1+o_P(1)\\rbrace }{\\pi ({\\mbox{$\\beta $}}_{1\\ast })\\int \\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast }); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast }))\\pi (\\bar{Y})d\\bar{Y}\\lbrace 1+o_P(1)\\rbrace } \\\\&=\\frac{\\pi (\\bar{Y})}{\\pi (\\bar{Y}_{\\ast })}\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast }); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast }))\\lbrace 1+o_P(1)\\rbrace \\\\&=\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast }); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast }))\\lbrace 1+o_P(1)\\rbrace \\\\&=\\phi (\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1); \\bar{Y}, \\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1))\\lbrace 1+o_P(1)\\rbrace , $ for any $\\bar{Y}\\in B_n$ as $n\\rightarrow \\infty $ , where (REF ) follows from (C2), and () follows from the properties $\\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1)=\\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast })\\lbrace 1+o_P(1)\\rbrace $ and $\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1)=\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast })\\lbrace 1+o_P(1)\\rbrace $ under (C1).",
"Let $R_n=\\lbrace \\bar{Y}\\in \\Theta _Y : \\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1)^{-1}(\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1)-\\bar{Y})^2\\le \\chi ^2_1(q)\\rbrace $ , where $\\chi ^2_k(q)$ is the upper $100q\\%$ -quantile of the chi-squared distribution with $k$ degree of freedom.",
"Then, $\\operatornamewithlimits{plim}_{n\\rightarrow \\infty }P(R_n)=q$ .",
"Since $\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1)-\\bar{Y}_{\\ast }=O_p(n^{-1/2})$ and $r_n=n^{\\tau -1/2}$ , which is slower than $n^{-1/2}$ , it holds that $\\lim _{n\\rightarrow \\infty } P(R_n\\subset B_n)=1$ .",
"Then, $&\\lim _{n\\rightarrow \\infty }P\\left(\\int _{B_n}\\phi (\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1); \\bar{Y}, \\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1)){\\rm d}\\bar{Y}\\ge \\int _{R_n}\\phi (\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1); \\bar{Y}, \\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1)){\\rm d}\\bar{Y}\\right)=1,$ which means that $\\lim _{n\\rightarrow \\infty }P\\left(\\int _{B_n}\\phi (\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1); \\bar{Y}, \\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1)){\\rm d}\\bar{Y}\\ge q\\right)=1$ for any $q\\in (0,1)$ , implying $\\operatornamewithlimits{plim}_{n\\rightarrow \\infty }\\int _{B_n}\\phi (\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1); \\bar{Y}, \\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1)){\\rm d}\\bar{Y}=1.$ Then, $&\\sup _{\\bar{Y}\\in \\Theta _Y}\\Big |p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{{\\mbox{$\\beta $}}}_1)-\\phi (\\bar{Y};\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1))\\Big |\\\\\\le &\\sup _{\\bar{Y}\\in B_n}\\Big |p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{{\\mbox{$\\beta $}}}_1)-\\phi (\\bar{Y};\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1))\\Big |\\\\& +\\sup _{\\bar{Y}\\in \\Theta _Y\\setminus B_n}\\Big |p(\\bar{Y}|\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{{\\mbox{$\\beta $}}}_1)-\\phi (\\bar{Y};\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1),\\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1))\\Big |,$ which are both $o_P(1)$ from () and (REF ).",
"This completes the proof."
],
[
"A sketched proof of Theorem 2",
"The condition (C4) given in the proof of Theorem 1 may not be satisfied for shrinkage priors.",
"For example, the horseshoe prior diverge at the origin $\\beta _k=0$ .",
"In what follows, let ${\\mbox{$\\beta $}}=(\\beta _0,{\\mbox{$\\beta $}}_1^t)$ and define $\\hat{{\\mbox{$\\beta $}}}_1$ and $\\hat{{\\mbox{$\\beta $}}}_1^{(R)}$ in the same way.",
"We use the following alternative condition for the shrinkage prior $\\pi _{\\lambda }({\\mbox{$\\beta $}}_1)$ : (C5) The regularized estimator $\\hat{{\\mbox{$\\beta $}}}_1^{(R)}$ under penalty $-\\log \\pi _{\\lambda }({\\mbox{$\\beta $}}_1)$ is asymptotically normal, that is, $\\sqrt{n}(\\hat{{\\mbox{$\\beta $}}}_1^{(R)}-{\\mbox{$\\beta $}}_{1\\ast })\\rightarrow N(0,\\mathbf {C})$ , where $\\mathbf {C}$ is a positive definite matrix and $\\lambda $ is appropriately chosen.",
"Under the Laplace prior, $\\hat{{\\mbox{$\\beta $}}}_1^{(R)}$ is equivalent to the Lasso estimator, and the above property holds if $\\lambda =o(\\sqrt{n})$ [19], [24].",
"For general prior $\\pi _\\lambda ({\\mbox{$\\beta $}}_1)$ , this condition holds if the assumption regarding the penalty term $P_\\lambda ({\\mbox{$\\beta $}}_1)$ given in [10] is satisfied.",
"It is noted that $\\phi _p&((\\hat{\\beta }_0,\\hat{{\\mbox{$\\beta $}}}_1^t); (\\beta _0,{\\mbox{$\\beta $}}_1^t), \\hat{{\\mbox{$V$}}}_{\\beta 11})\\pi _\\lambda ({\\mbox{$\\beta $}}_1)\\\\&\\propto \\exp \\left\\lbrace -\\frac{1}{2}(\\hat{{\\mbox{$\\beta $}}}_1-{\\mbox{$\\beta $}})^t\\hat{{\\mbox{$V$}}}_{\\beta 11}^{-1}(\\hat{{\\mbox{$\\beta $}}}_1-{\\mbox{$\\beta $}})+\\log \\pi _\\lambda ({\\mbox{$\\beta $}}_1)\\right\\rbrace \\\\&=\\exp \\left\\lbrace -\\frac{1}{2}\\sum _{i\\in A}\\frac{1}{\\pi _i}(y_i-\\beta _0-x_i^t{\\mbox{$\\beta $}}_1)^2+\\log \\pi _\\lambda ({\\mbox{$\\beta $}}_1)\\right\\rbrace \\lbrace 1+o_P(1)\\rbrace \\\\&=\\exp \\left\\lbrace -\\frac{n}{2}(\\hat{{\\mbox{$\\beta $}}}_1^{(R)}-{\\mbox{$\\beta $}}_1)^t\\mathbf {C}^{-1}(\\hat{{\\mbox{$\\beta $}}}_1^{(R)}-{\\mbox{$\\beta $}}_1)\\right\\rbrace \\lbrace 1+o_P(1)\\rbrace .$ Define $g(\\bar{Y},{\\mbox{$\\beta $}}_1)=\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_1); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_1))\\phi (\\hat{{\\mbox{$\\beta $}}}_1;{\\mbox{$\\beta $}}_1,\\hat{{\\mbox{$V$}}}_{\\beta 11})\\pi _{\\lambda }({\\mbox{$\\beta $}}_1).$ Then, it holds that $\\int _{{\\mbox{$\\beta $}}_1\\in R_n}g(\\bar{Y},{\\mbox{$\\beta $}}_1){\\rm d}{\\mbox{$\\beta $}}_1=\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast }); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast })) \\lbrace 1+o_P(1)\\rbrace $ as $n\\rightarrow \\infty $ , where $R_n$ is a ball with center ${\\mbox{$\\beta $}}_{1\\ast }$ and radius $O(n^{\\tau -1/2})$ for $0<\\tau <1/2$ .",
"Hence, the statement can be proved in the same way as the proof of Theorem 1 since $\\phi (\\hat{\\bar{Y}}_{\\rm reg}({\\mbox{$\\beta $}}_{1\\ast }); \\bar{Y}, \\hat{V}_{e}({\\mbox{$\\beta $}}_{1\\ast }))=\\phi (\\hat{\\bar{Y}}_{\\rm reg}(\\hat{{\\mbox{$\\beta $}}}_1^{(R)}); \\bar{Y}, \\hat{V}_{e}(\\hat{{\\mbox{$\\beta $}}}_1^{(R)}))\\lbrace 1+o_P(1)\\rbrace $ ."
],
[
"Additional simulation results",
"We here provide additional simulation results.",
"We considered the same scenarios in the main document with $n=400$ .",
"The results are reported in Table S1$\\sim $ 4.",
"Table: Square root of Mean squared errors (RMSE) and bias of point estimators under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with linear regression.All values are multiplied by 100.Table: Coverage probabilities (CP) and average lengths (AL) of 95%95\\% confidence/credible intervals under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with linear regression.All values are multiplied by 100.Table: Square root of Mean squared errors (RMSE) and bias of point estimators under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with logistic regression.All values are multiplied by 100.Table: Coverage probabilities (CP) and average lengths (AL) of 95%95\\% credible/confidence intervals under p∈{20,30,40,50}p\\in \\lbrace 20, 30, 40, 50\\rbrace in scenarios (A) and (B) with logistic regression.All values are multiplied by 100."
]
] | 1906.04398 |
[
[
"Shear Alfv\\'en and acoustic continuum in general axisymmetric toroidal\n geometry"
],
[
"Abstract The equations describing the continuous spectrum of shear Alfv\\'en and ion sound waves propagating along magnetic field lines are introduced and solved in the ballooning space for general geometry in the ideal MHD limit.",
"This approach is equivalent to earlier analyses by Chu et al.",
"1992 [Phys.",
"Fluids B 4, 3713 (1992)] but the present formulation in the ballooning space allows to readily extend it to include gyrokinetic and three-dimensional equilibrium effects.",
"In particular, following Chen and Zonca 2017 [Phys.",
"Plasmas 24, 072511 (2017)], the MHD limit is adopted to illustrate the general methodology in a simple case, and the equations are solved within the framework of Floquet and Hill's equation theory.",
"The connection of shear Alfv\\'en and ion sound wave continuum structures to the generalized plasma inertia in the general fishbone like dispersion relation is also illustrated and discussed.",
"As an application, the continuous frequency spectrum is calculated for a reference equilibrium of the Divertor Tokamak Test facility.",
"The results are compared with those obtained by the MARS code adopting the standard methodology, demonstrating excellent agreement."
],
[
"Introduction",
"Computing the shear Alfvén wave (SAW) continuous frequency spectrum[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] in general toroidal geometry is important because it determines mode structures and dispersive properties of Alfvénic fluctuations excited by energetic particles (EPs) in fusion devices, which, in turn, play crucial role in determining the plasma collective behaviors and eventually may affect fusion performance[14].",
"Due to equilibrium magnetic field curvature[15], [16], [17], [18], [19], SAW and ion sound wave (ISW) continuous frequency spectrum are coupled and, thus, their structures should be considered self-consistently, taking into account realistic equilibrium magnetic field geometry and plasma nonuniformity.",
"In numerical applications adopting realistic equilibrium reconstruction, the typical approach to the coupled SAW and ISW spectra is to Fourier decompose the fluctuation structure in general poloidal and toroidal angles and to compute the null space (kernel) of the matrix of the highest order radial derivative[19], e.g., of the radial plasma displacement.",
"This is completely equivalent to solve for the radial singular structures of the coupled SAW and ISW propagating along the field lines, described by two second order coupled differential equations[20].",
"In particular, the latter method has been adopted to compute the SAW and ISW spectra in DIII-D[20], focusing on both shaping and finite pressure effects on the continuous spectrum structures at low frequency[20], [21], [22], [23].",
"If the ideal MHD plasma description is adopted, again, the two approaches described above are mathematically equivalent, although it has been pointed out in Ref.",
"chu92 that the second method has advantages when computing the high toroidal mode number continuous spectra.",
"Kinetic description becomes necessary at short wavelengths and/or low frequencies to account for finite parallel electric field and wave damping.",
"For the SAW-ISW coupling at low frequency and corresponding frequency gaps[20], [21], [22], [23] with possible discrete modes localized therein[24], [25], [26], the need of kinetic theory for the damping assessment was pointed out in Refs.",
"chavdarovski09,zonca10,chavdarovski14 and motivated gyrokinetic stability studies of low-frequency Alfvénic fluctuations exctited by EPs with realistic geometry and plasma profiles[30], [31], [32].",
"The radial singular structures characterizing the SAW and ISW continuous spectra are most easily isolated[33], [34] adopting the ballooning mode representation[35], or the mode structure decomposition approach[36], which applies for arbitrary mode numbers and reduces to the ballooning representation in the high mode number limit.",
"Within this framework, the general description of SAW and ISW using gyrokinetic theory has been given in Refs.",
"chen16,zonca14a,zonca14b, and allows to include thermal plasma[39], [40], [41], [42], [43], [44] as well as EPs induced[45], [41], [42], [38], [43] kinetic effects in the analysis of the continuous spectra.",
"This premise provides a motivation of the present work, which is to show that the coupled second order differential equations describing SAW and ISW propagation along equilibrium magnetic field lines for general geometry are most easily expressed and solved in the ballooning space.",
"In the ideal MHD limit, as already stated, this is equivalent to the approach adopted in Ref.",
"chu92.",
"The present methodology, however, is readily extended to include gyrokinetic description[14], [37], [38] and 3D geometry[46].",
"Furthermore, this formulation not only allows us to discuss the structures of SAW and ISW continuous spectra in general toroidal geometry, but it provides the simplest and most direct way of computing the generalized inertia as an ingredient of the general fishbone-like dispersion relation (GFLDR)[14], [37], [38]; that is, the unified framework for describing Alfvénic fluctuations excited by EPs in tokamaks.",
"In Sec.",
", in order to establish a link to prior work more closely, we take the ideal MHD limit[47] of the general description[14], [37], [38] and we write the coupled SAW and ISW equations for radial singular structures in general tokamak geometry.",
"Along with providing a synthetic summary of prior theoretical analyses, which are applied here to realistic plasma equilibria, we illustrate the link of the generalized inertia in the GFLDR to the structures of the continuous spectra and show how it is computed[14], [37], [38], [47].",
"Furthermore, the important role of polarization of physical fluctuations is analyzed with emphasis on its relevance for assessing their effective absorption by resonantly excited radial singular structures of the continuous spectrum.",
"We then analyze the general form of the obtained equations, collocating that within the framework of Floquet theory[48] and Hill's equation[49], [50], [51].",
"As original application, in Sec.",
"we solve for the SAW and ISW continuous spectra for a reference equilibrium of the Divertor Tokamak Test (DTT) facility[52], [53].",
"We discuss both (artificially) decoupled and (realistic) coupled cases to illustrate features of the obtained SAW and ISW spectra and directly compare results of the present approach with those obtained by the MARS[54] code, which is equivalent to computing the null space of the matrix of the highest order radial derivative[19] of the plasma radial displacement.",
"In particular, we are able to identify the role of various equilibrium geometry effects in SAW-SAW, ISW-ISW and SAW-ISW couplings and corresponding structures in the continuous spectra.",
"All these detailed features are naturally accounted for in the present formulation and can be included as boundary conditions for calculating parallel mode structures and GFLDR[37], [38] dispersion relations in general Tokamak geometry.",
"Doing so, however, is beyond the scope of the present work and will be done elsewhere.",
"Further extensions to gyrokinetic analyses and 3D geometry are discussed in Sec , where concluding remarks are provided along with considerations about future perspectives."
],
[
"Theoretical framework",
"The general theoretical framework is discussed in Ref.",
"chen16, based on the detailed derivations given by Zonca and Chen in Refs.",
"zonca14a,zonca14b.",
"There, the general equations governing SAW and ISW fluctuation in low-$\\beta $ magnetized fusion plasmas are given by quasineutrality condition and vorticity equation, which are derived from the gyrokinetic equation[55].",
"Meanwhile, Ref.",
"chen16 also provides a thorough discussion of how gyrokinetic vorticity equation and quasineutrality condition yield the proper reduced MHD limit when appropriate [56], [57], [58].",
"The reduced MHD limiting form is also derived by the standard approach in Ref.",
"chen17.",
"In the present application to general tokamak equilibria, we merely assume the aforementioned theoretical framework without derivation and only provide the essential elements for introducing the notation and solving the relevant equations.",
"The general features of the mode structure decomposition of fluctuations in toroidal plasmas[36] are also discussed in Refs.",
"zonca14a,zonca14b along with its usefulness to generally describe continuous spectra."
],
[
"Notation and fundamental equations",
"In this work we assume an axisymmetric equilibrium magnetic field $B_{0}$ expressed in flux coordinates $(r,\\theta ,\\varphi )$ : $ B _ { 0 } = F ( \\psi ) \\nabla \\varphi + \\nabla \\varphi \\times \\nabla \\psi \\;,$ where $r(\\psi )$ is a radial-like flux coordinate, $\\varphi $ is the physical toroidal angle and the angular coordinate $\\theta $ can be chosen such that the Jacobian $J = ( \\nabla \\psi \\times \\nabla \\theta \\cdot \\nabla \\varphi ) ^ { - 1 }$ has a convenient expression.",
"Following the common practice, we also introduce the straight field line toroidal angle $\\zeta $ , defined such that the safety factor $q = \\frac{\\mathbf {B}_0 \\cdot \\mathbf {\\nabla }\\zeta }{\\mathbf {B}_0 \\cdot \\mathbf {\\nabla }\\theta } = q(r)$ is a flux function.",
"We introduce the leading order perpendicular plasma displacement as usual, $\\delta \\mathbf {\\xi }_{\\perp } = \\frac{c}{B_0} \\mathbf {b} \\times \\mathbf {\\nabla }\\Phi _s \\; , $ with $\\Phi _s$ being the perturbed stream function[47].",
"The fluctuating field $\\nonumber \\Phi _s (r,\\theta ,\\zeta ) = \\sum _{m} \\exp ( i n \\zeta - i m \\theta ) \\Phi _{s m}(r)$ expressed as a Fourier series, where $m$ and $n$ represent the poloidal and toroidal mode numbers, respectively, can be decomposed as[36]: $\\Phi _s (r,\\theta ,\\zeta ) & = & 2\\pi \\sum _{\\ell \\in \\mathbb {Z}} e^{in\\zeta -inq(\\theta -2\\pi \\ell )} \\hat{\\Phi }_s (r,\\theta -2\\pi \\ell )=\\nonumber \\\\ & = & \\sum _{m \\in \\mathbb {Z}} e^{in\\zeta -i m\\theta } \\int d\\vartheta e^{i(m-nq)\\vartheta } \\hat{\\Phi }_s (r,\\vartheta ) \\; .",
"$ Here, time dependences are left implicit for simplicity of notation, and $\\vartheta $ represents the extended poloidal angle[35] coordinate following equilibrium magnetic field lines.",
"Periodic poloidal angle-$\\theta $ dependences of equilibrium quantities are replaced by the same periodic dependences on $\\vartheta $ .",
"However, while fluctuations must be periodic in $\\theta $ space for physical reasons, they are generally not periodic in $\\vartheta $ , as shown in Eq.",
"().",
"The radial singular structures corresponding to the continuous spectra are obtained from the limiting forms of, respectively, vorticity and pressure equations for $|\\vartheta |\\rightarrow \\infty $[37], [38], [47].",
"Following Refs.",
"chen16,zonca14a,zonca14b, it is possible to derive the expressions describing the structure of the continuum modes in flux coordinates.",
"In particular, assuming $JB_0^2$ constant on a flux surface, yielding the so called “Boozer coordinates”, we obtain[47]: $\\left( \\partial _ { \\vartheta } ^ { 2 } - \\frac{ \\partial _ { \\vartheta } ^ { 2 } | \\mathbf {\\nabla }r | }{ | \\mathbf {\\nabla }r | } + \\frac{ \\omega ^ { 2 } J ^ { 2 } B _ { 0 } ^ { 2 } }{ v _ { A } ^ { 2 } } \\right) y _ { 1 } &= ( 2 \\Gamma \\overline{ \\beta } ) ^ { 1 / 2 } \\kappa _ { g } \\frac{ J ^ { 2 } B _ { 0 } \\overline{ B } _ { 0 } }{ q R _ { 0 } } \\frac{ s \\vartheta }{ | s \\vartheta | } y _ { 2 }\\;, \\\\\\left( 1 + \\frac{ c _ { s } ^ { 2 } }{ \\omega ^ { 2 } } \\frac{ 1 }{ J ^ { 2 } B _ { 0 } ^ { 2 } } \\partial _ { \\vartheta } ^ { 2 } \\right) y _ { 2 }&= ( 2 \\Gamma \\overline{ \\beta } ) ^ { 1 / 2 } \\kappa _ { g } \\frac{ \\overline{ B } _ { 0 } }{ B _ { 0 } } q R _ { 0 } \\frac{ s \\vartheta }{ | s \\vartheta | } y _ { 1 } \\;,$ where $y _ { 1 } \\equiv \\frac{ \\hat{ \\phi } _ { s } }{ \\left( \\overline{\\beta } q ^ { 2 } \\right) ^ { 1 / 2 } } \\frac{ c k_{\\vartheta } }{ \\bar{B}_{0} R _ { 0 } }\\;, \\qquad y_{2} \\equiv i \\frac{\\delta \\hat{P}_{comp}}{(2 \\Gamma )^{1/2}P_{0}}\\;,$ $\\hat{\\phi }_s (r,\\vartheta ) \\equiv |s \\vartheta | |\\mathbf {\\nabla }r| \\hat{\\Phi }_s(r,\\vartheta )$ , $s=rq^{\\prime }/q$ is the magnetic shear, $k_{\\vartheta } = - n q/r$ , and $\\delta \\hat{P}_{comp}(r, \\vartheta )$ is the representation of the compressional component of the pressure perturbation according to Eq.",
"()[47].",
"Furthermore, $\\kappa _g$ is the geodesic curvature in Boozer coordinates: $\\kappa _ { g } = - \\frac{ 1 }{ J B _ { 0 } } \\frac{ F }{ | \\mathbf {\\nabla }\\psi | } \\frac{ \\partial }{ \\partial \\vartheta } \\ln {B _ { 0 }}\\;,$ $\\bar{\\beta }= 8\\pi \\Gamma P_0/\\bar{B}_0^2$ , $\\bar{B}_{0}$ is the magnetic field on axis, $r = a \\rho _{tor}$ , where $\\rho _{tor}$ is defined, as usual, as a function of the normalized toroidal magnetic flux and $a$ is the minor radius of the torus.",
"Note, again, that periodic equilibrium field dependences on $\\theta $ are replaced by the same periodic dependences on $\\vartheta $ in the ballooning space.",
"These equations describe SAWs and ISW continuous spectra coupled by equilibrium non-uniformities and geodesic curvature.",
"Both type of waves propagate along equilibrium magnetic field lines and, therefore, radial equilibrium nonuniformities are accounted for by $r$ , which enters in Eqs.",
"(REF ) as a parameter.",
"Note also that the continuous spectrum does not depend on magnetic shear although it formally appears in Eqs.",
"(REF ) and (REF ).",
"This can be understood since the continuous spectrum is a local feature of singular radial fluctuation structures.",
"More in depth discussion about this point and what happens for vanishing magnetic shear can be found in Refs.",
"chen16,zonca14a,zonca14b.",
"Equations (REF ), obtained in the MHD limit[47], are valid for general tokamak equilibrium geometry and can be used to calculate the coupled SAW and ISW continuous spectra for cases of practical interest, such as the DTT reference scenario[52], [53] introduced in Sec.",
"REF .",
"These equations are readily extended to 3D plasma equilibria, such as stellarators, thanks to the generality of ballooning representation[46].",
"The same formulation can be adopted within a linear gyrokinetic description of coupled SAW and ISW continua, in which case Eqs.",
"(REF ) are replaced by the short radial scale gyrokinetic vorticity equation, i.e., Eq.",
"(6) of Ref.",
"zonca14b, and the quasineutrality condition, i.e., Eq.",
"(36) of the same work.",
"This theoretical framework is similar to the description of electron wave-packets in a 1D periodic lattice, see e.g.",
"Ref.",
"kittel1976introduction.",
"The interesting analogy[14] is due to the fact that the governing equations of low-frequency plasma waves in magnetized plasmas describe the propagation of wave-packets along magnetic field lines with periodicity given by the plasma connection length.",
"Therefore, this correspondence is independent of the axisymmetry of the equilibrium magnetic field and/or the fluid/kinetic nature of the theory.",
"In particular, Eqs.",
"(REF ), are Schrödinger-like and produce structures analogue to electronic bands in solid-state physics.",
"Further discussion of this interesting analogy is given in Refs.",
"chen16,zonca14a,zonca14b.In the following, we will focus on the solution of Eqs.",
"(REF ).",
"By direct inspection, it is readily noted that, adopting Boozer coordinates, ISW continuous spectrum is more simply represented than SAW continuous spectrum, due to the modulation term $\\propto \\partial _\\vartheta ^2 |\\mathbf {\\nabla }r|/ |\\mathbf {\\nabla }r|$ .",
"Since there is usually more interest in the SAW than in the ISW branch due to the stronger kinetic damping of the latter, we can rewrite Eqs.",
"(REF ), introducing an appropriate angular coordinate to express the SAW frequency continuum more transparently; i.e., $\\eta \\equiv 2 \\pi \\frac{\\int _ { 0 } ^ { \\vartheta } d \\vartheta ^ { \\prime } |\\mathbf {\\nabla }r|^{-2}}{\\int _ { 0 } ^ { 2 \\pi } d \\vartheta ^ { \\prime } |\\mathbf {\\nabla }r|^{-2}}\\;.$ By construction, the Jacobian $J_{\\eta }$ of this new set of coordinates, dubbed continuum flux coordinates (CFC), is such that $J_{\\eta } B_{0}^{2}/|\\mathbf {\\nabla }r|^{2}$ is a flux function.",
"We obtain the following expression for the derivative along $\\eta $ (at constant $\\psi $ ): $| \\mathbf {\\nabla }r | ^ { 2 } \\mathcal {C} \\left.",
"\\frac{ \\partial }{ \\partial \\vartheta } \\right| _ { \\psi } = \\left.",
"\\frac{ \\partial }{ \\partial \\eta } \\right| _ { \\psi }\\;,$ where $\\mathcal {C} = (2 \\pi )^{-1} \\int _ { 0 } ^ { 2 \\pi } d \\vartheta ^ { \\prime } \\left| \\mathbf {\\nabla }r \\right| ^ { - 2 }$ .",
"The Jacobians in the two set of coordinates are related by the following expression: $J_{\\eta } = J \\mathcal {C} | \\mathbf {\\nabla }r | ^ { 2 }\\;.$ Introducing the normalizations: $g_{1} = \\frac{1}{|\\mathbf {\\nabla }r | }y_{1}, \\quad g_{2} = \\frac{c_{s}}{\\omega q R_{0} | \\mathbf {\\nabla }r |} \\frac{|s \\vartheta |}{s \\vartheta }y_{2}\\;,$ we can re-write Eqs.",
"(REF ) as $\\left( \\partial _ { \\eta } ^ { 2 } + \\frac{ \\omega ^ { 2 } J _ { \\eta } ^ { 2 } B _ { 0 } ^ { 2 } }{ v _ { A } ^ { 2 } } \\right) g _ { 1 } &= ( 2 \\Gamma \\overline{ \\beta } ) ^ { 1 / 2 } \\kappa _ { g } \\frac{ \\omega J _ { \\eta } ^ { 2 } B _ { 0 } \\overline{ B } _ { 0 } }{ c _ { s } } g _ { 2 }\\;, \\\\\\left( \\partial _ { \\eta } ^ { 2 } - | \\mathbf {\\nabla }r | \\partial _ { \\eta } ^ { 2 } | \\mathbf {\\nabla }r | ^ { - 1 } + \\frac{ \\omega ^ { 2 } J _ { \\eta } ^ { 2 } B _ { 0 } ^ { 2 } }{ c _ { s } ^ { 2 } } \\right) g _ { 2 } &= ( 2 \\Gamma \\overline{ \\beta } ) ^ { 1 / 2 } \\kappa _ { g } \\frac{ \\omega J _ { \\eta } ^ { 2 } B _ { 0 } \\overline{ B } _ { 0 } }{ c _ { s } } g _ { 1 }\\;,$ respectively, where the geodesic curvature in this new set of coordinates reads: $\\kappa _ { g } = - \\frac{ 1 }{ J _ { \\eta } B_{0}^{2}} \\frac{ F }{ | \\mathbf {\\nabla }\\psi | } \\frac{ \\partial }{ \\partial \\eta } B _ { 0 }\\;.$ Equations (REF ) are completely equivalent to Eqs.",
"(REF ) and will be solved in the following, illustrating the properties of coupled SAW and ISW continuous frequency spectra in general tokamak plasma equilibria.",
"They are in the form of two coupled second order differential equations with periodic coefficients.",
"Therefore, all known results of Floquet theory[48] can be applied.",
"These are summarized in the remaining part of Sec.",
"for the reader's convenience.",
"Equations (REF ) are a linear system of second order differential equations with periodic coefficients.",
"Therefore, after re-writing it as four first order differential equations, we know from Floquet theory[48] that it must have solutions in the form: $\\mathbf {x}_{i} =e ^ {i \\nu _{i} \\eta } \\mathbf { P }_{i} ( \\eta )\\;,$ where $\\mathbf {P}_{i}$ is a $2 \\pi $ -periodic function, $i = 1,2,3,4$ and the $\\nu _{i}$ are complex numbers called characteristic Floquet exponents of Eqs.",
"(REF ).",
"Finding the solutions of Eqs.",
"(REF ) for a given value of $r$ and, therefore, calculating $\\nu _{i}$ for each $\\omega $ value is equivalent to obtaining the (local) dispersion curves of the system: $\\nu _{i} = \\nu _{i}(\\omega ,r)\\;.$ We emphasize that this relation involves only local quantities and describes wave packets propagating along magnetic field on a given flux surface.",
"Recalling the physical meaning of the derivative along the $\\vartheta $ coordinate[14], [37], [38], we can relate the characteristic Floquet exponents to $\\nabla _{\\parallel }\\equiv (JB_0)^{-1} \\partial _\\vartheta = (J_\\eta B_0)^{-1} \\partial _\\eta $ , and, in particular, to the toroidal and poloidal mode numbers $n,m$ : $\\nu _{i} ^ { 2 } ( \\omega , r ) = ( n q ( r ) - m ) ^ { 2 }\\;.$ Equation (REF ) readily follows from the definition of $\\nabla _\\parallel $ and the mode structure decomposition of Eq.",
"()[14], [37], [38], [47].",
"Therefore, integrating Eqs.",
"(REF ) for different $\\omega $ values and using Eqs.",
"(REF ) and (REF ), it is possible to calculate the continuous frequency spectrum for every value of the toroidal mode number.",
"In particular, Eq.",
"(REF ) explicitly shows that continuous spectra can be computed for arbitrary $n$ , once the dispersion curves $\\nu _i = \\nu _i (\\omega , r)$ are given.",
"Extensions to 3D/stellarator geometry and gyrokinetic theory would be obtained in the same way from the more general governing equations as briefly discussed in Sec.",
"REF .",
"For given $\\nu _i(\\omega ,r)$ ; i.e., for given dispersion curves of Eqs.",
"(REF ), it is possible to calculate the corresponding vector solutions; i.e., eigenvectors of the fundamental matrix $\\mathbf {X}(\\vartheta )$ , that is a matrix-valued function whose columns are linearly independent solutions of Eqs.",
"(REF ).",
"Furthermore, since the considered problem is linear, we can generally write the normalized eigenvector solution corresponding to $\\nu _i(\\omega ,r)$ as $g_1^{(i)}(\\eta ;\\nu _i,r) & = & e_1^{(i)}(\\nu _i,r) \\hat{g}_1^{(i)}(\\eta ;\\nu _i,r) \\; , \\nonumber \\\\g_2^{(i)}(\\eta ;\\nu _i,r) & = & e_2^{(i)}(\\nu _i,r) \\hat{g}_2^{(i)}(\\eta ;\\nu _i,r) \\; .",
"$ Here, without loss of generality, we can impose $\\int d \\eta |\\hat{g}_1^{(i)}|^2 = \\int d\\eta |\\hat{g}_2^{(i)}|^2 = 1 \\; , $ where the $\\eta $ -space integration domain may be chosen as any convenient multiple of $2\\pi $ due to the periodicity of Eqs.",
"(REF ).",
"Meanwhile, $|e_1^{(i)}(\\nu _i,r)|^2 + |e_2^{(i)}(\\nu _i,r)|^2 = 1 \\; .",
"$ Thus, the $(e_1^{(i)}, e_2^{(i)})$ vector can be considered as definition of fluctuation polarization in the linear (Hilbert) space of Floquet solutions $[g_1^{(i)}(\\eta ;\\nu _i,r), g_2^{(i)}(\\eta ;\\nu _i,r)]$ , corresponding to continuous spectrum fluctuations for a given $\\nu _i$ .",
"Polarization is a necessary element that uniquely identifies the continuous spectrum, described by solutions of Eqs.",
"(REF ), together with the characteristic Floquet exponent, $\\nu _i(\\omega , r)$ , and the corresponding parallel fluctuation structures, $\\hat{g}_1^{(i)}(\\eta ;\\nu _i,r)$ and $\\hat{g}_2^{(i)}(\\eta ;\\nu _i,r)$ .",
"A pictorial representation of parallel propagation of radial singular structures of the continuous spectrum and of their polarization in the orthogonal Hilbert space is given in FIG.REF .",
"Figure: Parallel propagation of radial singular structures of the continuous spectrum and their polarization in the orthogonal Hilbert space, which is a “virtual” (non-physical) linear space where Floquet solutions [g 1 (i) (η;ν i ,r),g 2 (i) (η;ν i ,r)][g_1^{(i)}(\\eta ;\\nu _i,r), g_2^{(i)}(\\eta ;\\nu _i,r)] of Eqs.",
"() are represented.Here, it is also worthwhile pointing out the connection of the Floquet characteristic exponent to the generalized plasma inertia, $\\Lambda $ , which is an essential element of the general fishbone-like dispersion relation (GFLDR)[14], [37], [38]: $i |s| \\Lambda = \\delta \\hat{W}_f + \\delta \\hat{W}_k \\;\\; , $ where $\\delta \\hat{W}_f$ and $\\delta \\hat{W}_k$ account for, respectively, the generalized potential energy due to fluid-like plasma response and the kinetic plasma behavior due to, e.g., EPs.",
"At a given radial location, the short scale (large-$\\eta $ ) radial structure of any considered physical fluctuation (antenna driven and/or eigenmode) satisfies the same Eqs.",
"(REF ) describing the continuous spectrum.",
"Thus, the large-$|\\eta |$ local physical solution must be a linear superposition of the solutions of Eq.",
"(REF ).",
"Among them, generally, only two satisfy boundary conditions (outgoing/decaying wave in $\\eta $ -space[14], [37], [38]).",
"Renumbering them as $i=1,2$ , without loss of generality, we have $\\mathbf {x} = w_1 \\mathbf {x}_1 + w_2 \\mathbf {x}_2 \\; ,$ where $w_1$ and $w_2$ are weights of the linear combination, which we can generally assume satisfying the normalization condition $w_1^2 + w_2^2 = 1$ .",
"The actual value of $(w_1,w_2)$ is determined consistently with the mode structure and fluctuation dispersion relation.",
"Adopting the GFLDR theoretical framework[14], [37], [38], it is possible to show[37], [38] $i \\Lambda = \\left.\\frac{\\left(i \\nu _{1} P_{1}(\\eta )+P_{1}^{\\prime }(\\eta )\\right) w_{1}+\\left(i \\nu _{2} P_{2}(\\eta )+P_{2}^{\\prime }(\\eta )\\right) w_{2}}{P_{1}(\\eta ) w_{1}+P_{2}(\\eta ) w_{2}}\\right|_{\\eta =2 \\ell \\pi }\\;\\; , $ where $P_i(\\eta )|_{\\eta = 2\\ell \\pi } (i = 1,2)$ is referred to the $g_1$ solution of Eqs.",
"(REF ) represented as in Eq.",
"(REF ) for $\\nu _i (i=1,2)$ , respectively, and, for simplicity, we have assumed that the equilibrium is general but still up-down symmetric.",
"Since the continuum fluctuation structures, Eq.",
"(REF ), bear the information of mode polarization, this relation shows the importance of mode polarization to assess the actual coupling of any physical fluctuation to the continuous spectrum; and further illuminates the profound connection between the structures of SAW and ISW continuous spectra and the GFLDR, and the usefulness of the present approach when extended to 3D/stellarator geometry and/or gyrokinetic descriptions.",
"More on this important point is discussed in Section below.",
"In order to calculate $\\nu _{i}$ as a function of $\\omega $ for each $r$ , we note that the fundamental matrix $\\mathbf {X}(\\vartheta )$ of the first order system associated to Eqs.",
"(REF ) must satisfy the following relation: $\\mathbf { X } ( \\vartheta + 2 \\pi ) = \\mathbf { X } ( \\vartheta ) \\mathbf { M }\\;.$ It can be shown[48] that $\\mathbf {M}$ does not depend on $\\vartheta $ and, therefore, is conveniently calculated for $\\vartheta =0$ : $\\mathbf { M } = \\mathbf { X } ^ { - 1 } ( 0 ) \\mathbf { X } ( 2 \\pi )\\;.$ Choosing the initial conditions such that $\\mathbf { X } ( 0 ) = \\mathbf { I }$ , the 4x4 identity matrix in the present case, we obtain: $\\mathbf { M } = \\mathbf { X } ( 2 \\pi )\\;.$ The eigenvalues of $\\mathbf {M}$ are the characteristic multipliers of the system and can be expressed in terms of the characteristic Floquet exponents by the following relation: $\\rho _ { i } = e ^ {i 2 \\pi \\nu _ { i } }\\;.$ Corresponding eigenvectors can be used to calculate parallel mode structures and polarization vectors, Eq.",
"(REF ); e.g., to calculate $P_i^{\\prime }(2\\pi )/P_i(2\\pi )$ and the value of the generalized inertia from Eq.",
"(REF ).",
"The same information can be adopted as boundary conditions for the calculation of fluid and kinetic potential energies in the GFLDR[14], [37], [38].",
"However, doing so is outside the scope of the present work."
],
[
"Hill's equation",
"In the decoupled case ($\\kappa _g = 0$ ), Eqs.",
"(REF ) become a system of two independent differential equations in the form: $\\frac{ d ^ { 2 } x }{ d \\eta ^ { 2 } } + V ( \\eta ; \\omega ) x = 0\\;,$ where $x=g_{1},g_{2}$ ; which are a particular cases of Hill's equation[49], [50], [51].",
"Note that, consistent with previous notations, parametric dependences on $r$ are left implicit for notation simplicity, while the dependence on $\\omega $ of the potential function $V(\\eta ; \\omega )$ is indicated explicitly.",
"Reducing each one of these equations to a first order system and using the results summarized in the previous subsection, we know that they will have Floquet solutions satisfying the following property: $x_{i} ( \\eta + 2 \\pi ; \\omega ) = \\rho _{i} x_{i} ( \\eta ;\\omega )\\;,$ where $i=1,2$ .",
"We can show that the following relation holds[60]: $\\rho _{i} ^ { 2 } - 2 A ( \\omega ) \\rho _{i} + 1 = 0\\;,$ with: $A ( \\omega ) = \\frac{ 1 }{ 2 } \\left(x _ {1 0 } ( 2 \\pi ; \\omega ) + x_ { 01 } ^ { \\prime } ( 2 \\pi ; \\omega ) \\right)\\;,$ where $x_{10}$ and $x_{01}$ are the particular solutions of the first order system associated to each Hill's equation such that $x_{10}(0) = (1,0)$ and $x_{01}(0) = (0,1)$ .",
"In the particular case of even $V(\\eta ; \\omega )$ , it can be shown that $A(\\omega ) = x_{10}(2\\pi ;\\omega )$ .",
"Solving Eq.",
"(REF ) gives the following expression for the multipliers: $\\rho _ { 1,2 } = \\frac{ + A \\pm \\sqrt{ A ^ { 2 } - 4 } }{ 2 }\\;.$ If $|A^{2}|<2$ , roots are complex conjugates with unity absolute value while, if $|A^{2} = 2|$ , they coincide and are equal to 1.",
"Finally, if $|A^{2} > 2|$ , roots are real and reciprocal to each other.",
"In particular, the first case describes the frequency continuum while the third the frequency gaps.",
"From this expression is possible to calculate the characteristic exponent from the value of $A(\\omega )$ .",
"In the first case we obtain: $\\cos {(\\nu 2 \\pi )} = \\frac{A(\\omega )}{2}\\;,$ where $\\nu $ is real.",
"From this analysis we expect up to two (opposite) values of $\\nu $ for each $\\omega $ in the decoupled case, corresponding to waves propagating in opposite directions along the field lines.",
"This result can be generalized to higher dimensionality, e.g.",
"see Ref.Denk1995, to describe the coupled system, where we expect up to two (pairs of opposite) $\\nu $ values for each $\\omega $ corresponding, again, to waves propagating in opposite directions along the field lines."
],
[
"Numerical results",
"In this Section, in order to show the generality of the present approach, we calculate the frequency continuum of SAW and ISW waves in a Divertor Tokamak Test Facility (DTT) reference scenario.",
"DTT is an Italian project [52], [53] sponsored by the EUROfusion consortium with the goal of designing a new machine capable of eventually integrating all relevant physics and technological issues concerning power exhaust solutions for DEMO.",
"The main DTT parameter are reported in FIG.REF .",
"In order to isolate the role of various equilibrium geometry effects in SAW-SAW, ISW-ISW and SAW-ISW couplings and discuss the corresponding structures in the continuous spectra, we illustrate first the decoupled SAW and ISW continuous spectra by artificially letting $\\kappa _g=0$ (Section REF ).",
"The realistic situation with coupled SAW and ISW spectra is then addressed in Section REF .",
"Numerical results obtained with the present approach are compared with those by the MARS code adopting the conventional method, showing excellent agreement."
],
[
"Normalized equations",
"The set of equations describing SAWs coupled with ISW can be re-written in terms of the following dimensionless quantities: $\\Omega = \\frac{ \\omega R_ { 0 } }{ \\overline{ v } _ { A 0 } }, \\quad \\hat{J}_{\\eta }^{2} \\equiv \\frac{J_{\\eta }^{2}\\bar{B}_{0}^{2}}{R_{0}^{2}}, \\quad \\hat{\\rho }_{m0}=\\frac{\\rho _{m0}}{\\bar{\\rho }_{m0}}, \\quad \\hat{B}_{0}=\\frac{B_{0}}{\\bar{B}_{0}}, \\quad \\hat{\\kappa }_{g}= \\kappa _{g}R_{0}\\;,$ where $\\overline{ v } _ { A 0 }$ is the Alfvén velocity on the magnetic axis.",
"The non-dimensional form of Eqs.",
"(REF ) then reads now $\\left( \\partial _ { \\eta } ^ { 2 } + \\hat{ J } _ { \\eta } ^ { 2 } \\hat{\\rho }_{m0} \\Omega ^ { 2 } \\right) g _ { 1 } &= 2 \\hat{B} _ { 0 } \\hat{\\rho }_{m0}^ { 1 / 2 } \\hat{ J } _ { \\eta } ^ { 2 } \\hat{\\kappa } _ { g } \\Omega \\, g _ { 2 }\\;, \\\\\\left( \\partial _ { \\eta } ^ { 2 } - |\\mathbf {\\nabla }r | \\partial _ { \\eta } ^ { 2 } | \\mathbf {\\nabla }r | ^ { - 1 } + \\frac{ 2 }{ \\Gamma \\beta } \\hat{\\rho }_{m0} \\hat{ J } _ { \\eta } ^ { 2 } \\Omega ^ { 2 } \\right) g _ { 2 } &= 2 \\hat{B} _ { 0 } \\hat{\\rho }_{m0}^ { 1 / 2 } \\hat{ J } _ { \\eta } ^ { 2 } \\hat{\\kappa } _ { g } \\Omega \\, g _ { 1 } \\;.$ Eqs.",
"(REF ) are studied numerically in the remaining part of this work for the DTT reference scenario illustrated in the following subsection."
],
[
"DTT reference scenario",
"The magnetic equilibrium has been originally calculated by means of the free bounddary equilibrium evolution code CREATE-NL [61] and further refined and mapped on flux coordinates by using the high-resolution equilibrium solver CHEASE [62].",
"We consider a double null setting whose basic profiles are depicted in FIG.REF as a function of the normalized toroidal radius $r/a$ .",
"We note that, for the analyzed case, the kinetic pressure on axis is $1.0768\\times 10^{6}$ Pa, the flux function $F$ , see Eq.",
"(REF ), on axis is $12.45$ Vs, while the density is $10^{19}$ m$^{-3}$ .",
"Figure: Plots of the main profiles for the reference double null scenario as a function of r/ar/a.",
"Every quantity except qq is normalized to its value on the magnetic axis: flux function (left-hand panel), kinetic pressure, density together with the safety factor qq (right-hand panel).Moreover, the normalized density profile has been obtained by studies of plasma scenario formation using the fast transport simulation code METIS [63].",
"For the addressed scenario, the $(\\psi ,\\vartheta )$ and $(\\psi ,\\eta )$ isosurfaces are depicted in FIG.REF .",
"Figure: (Color online) Contour lines of ψ\\psi and isolines of the ϑ\\vartheta coordinates (left-hand panel, red lines) and η\\eta coordinates (right-hand panel, green lines).",
"RR and ZZ are expressed in meters."
],
[
"SAW continuous spectrum",
"The equation describing SAW continuous spectrum can be obtained from Eqs.",
"(REF ) by neglecting the coupling terms with the ISW branch.",
"Thus, we have $\\left( \\partial _{\\eta }^{2}+ \\hat{J}_{\\eta }^{2} \\hat{\\rho }_{m0} \\Omega ^{2}\\right)g_{1}=0\\;.$ In order to illustrate with a practical application the procedure introduced in Sec.",
"REF , we first show the local dispersion curves $\\nu (\\Omega )$ obtained integrating Eq.",
"(REF ) at fixed $r/a$ , i.e., on a given $\\psi $ isosurface, for different $\\Omega $ values.",
"These curves are plotted in FIG.REF for the reference value $r/a=0.58$ , where, for convenience, $\\nu $ is chosen as abscissa showing different branches and gaps.",
"Figure: Plot of the local dispersion curves ν(Ω)\\nu (\\Omega ) for fixed r/a=0.58r/a=0.58.Noting Eq.",
"(REF ) and assuming a Fourier decomposition of the fluctuations mapped back from ballooning to real space[36], we can write $\\hat{J}_\\eta \\hat{B}_0 R_0 k_{\\parallel m, n} = nq - m \\; .",
"$ Thus, two counter-propagating SAWs can form a standing wave and cause the formation of a frequency gap in the continuous spectrum[13], [18], [64], [65], when the Bragg reflection condition is satisfied; that is: $k_{\\parallel m+p, n} = - k_{\\parallel m, n}\\;\\;\\;\\;\\; \\Rightarrow \\;\\;\\;\\;\\; k_{\\parallel m, n} = \\frac{p}{2\\hat{J}_\\eta \\hat{B}_0 R_0} \\; , \\;\\; p \\in \\mathbb {Z} \\; .",
"$ Due to the periodicity of Floquet solutions, Eq.",
"(REF ), Eq.",
"(REF ) can be effectively reduced to the first Brillouin zone, $0\\le \\nu \\le 1/2$ , and frequency gaps are expected to appear at[37], [38] (cf.",
"Appendix ) $\\Omega ^2 = \\frac{p^2/4}{\\hat{\\rho }_{m0} \\left\\langle \\hat{J}_\\eta ^2 \\right\\rangle _{g_1}} \\; , \\;\\; p \\in \\mathbb {Z}\\; ; $ with $\\left\\langle \\hat{J}_\\eta ^2 \\right\\rangle _{g_1} \\equiv \\frac{\\int d \\eta \\hat{J}_\\eta ^2 |g_1|^2 }{\\int d \\eta |g_1|^2 } \\; .$ consistent with Eqs.",
"(REF ) and (REF ).",
"As anticipated above, dispersion curves $\\nu = \\nu (\\Omega ,r/a)$ , i.e., the dimensionless form of Eq.",
"(REF ), include all the relevant information for the construction of continuous spectra.",
"Using Eq.",
"(REF ) and combining the results of different flux surfaces, we obtain the SAW continuous spectrum as a function of $r/a$ as depicted in FIG.REF , where continua for various toroidal mode numbers are shown in different colors and illustrate the formation of toroidicity- as well as ellipticity-induced frequency gaps at, respectively, $\\Omega \\simeq 1/2$ (TAE[19]) and $\\Omega \\simeq 1$ (EAE[66], [67]).",
"Higher frequency gaps could also be readily plotted by means of the dispersion curves reported in FIG.REF but are omitted here to illustrate results more clearly.",
"Figure: (Color online) SAW continuous spectrum as a function of the normalized radial position for several values of the toroidal mode number as indicated in the plot.We now compare the results with the spectrum calculated by MARS code[54].",
"The continuous spectrum is obtained assigning a guess real frequency $\\omega _{c} = \\omega _{\\mathrm {guess}}$ and looking for converged solutions at $\\omega = \\omega _{\\mathrm {c}}$ , which correspond to a perturbed velocity whose toroidal contravariant $(m,n)$ Fourier components $v^{\\phi }_{m,n}(r)$ exhibit a divergent radial dependence of the type $v^{\\phi }_{m,n}(r) \\propto 1/(r-r_{\\mathrm {c}})$ .",
"When this condition is recognized, the points $(r_{\\mathrm {c}}/a,\\omega _{\\mathrm {c}})$ in the plane $(r/a,\\omega )$ are considered as the location of the continuous spectrum.",
"The comparison for the toroidal mode number $n=2$ is depicted in FIG.REF , showing good agreement between the two approaches.",
"Figure: (Color online) Comparison of the decoupled SAW continuous spectrum calculated within the proposed approach (solid lines) and the results obtained by means of the MARS code (red bullets), in the case n=2n=2.As anticipated above, one of the peculiar features of the approach proposed by the present work is that the dispersion curves are independent of $n$ and, therefore, the evaluation of the continuous spectrum for different toroidal mode numbers merely requires the numerical solution of an algebraic equation, Eq.",
"(REF ).",
"Another interesting property of the present methodology is the freedom in the choice of the radial mesh due to the (radial) local nature of the problem that we are solving.",
"This feature can be exploited to more accurately describe regions where the continuous spectrum is more convoluted.",
"Finally, the present approach is readily extended to 3D geometries and kinetic analyses, as noted in the Introduction and Section .",
"The same analysis can be repeated for the ISW continuum.",
"In fact, Eq.",
"() can be cast in the same form as Eq.",
"REF using Boozer coordinates and normalizing the frequency to the local sound speed rather than the Alfvén speed on axis.",
"Results, thus, would be exactly the same as in FIG.REF upon a suitable rescaling of ISW continuum frequencies.",
"In particular, a toroidicity induced gap would appear for $p=1$ in Eq.",
"(REF ); an ellipticity induced gap would occur for $p=2$ and so on.",
"These frequency gaps have not received significant attention in the literature since, as already pointed out, the ISW branch is typically affected by strong Landau damping[27], [28], [29], [31], [30], [32], [42] and, therefore, discrete modes that are possibly located in these gaps are characterized by high excitation thresholds[14], [37], [38], [47].",
"Meanwhile, the features of ISW-ISW and SAW-SAW couplings due to equilibrium geometry are identical, as are the corresponding structures of continuous spectra.",
"Therefore, a detailed calculation of the uncoupled ISW continuum is omitted here."
],
[
"Coupled system",
"In the general case, we cannot neglect the SAW-ISW coupling due to equilibrium non-uniformities and we need to study the coupled system of Eqs.",
"(REF ) and ().",
"In order to calculate the structures of SAW and ISW continuous spectra, we apply the procedure described in Section .",
"For the reference value $r/a=0.58$ , the result is depicted in FIG.REF , reproducing the expected behavior.",
"Figure: Local dispersion curves ν(Ω)\\nu (\\Omega ) for the SAW-ISW coupled continuum at r/a=0.58r/a=0.58.",
"Colored circles denote the interaction of ISW-ISW (green), SAW-SAW (red) and SAW-ISW (orange).",
"Solid circles corresponding to interactions between counter-propagating waves, while dashed circles to interactions between co-propagating waves.A change of the topology emerges with respect to the uncoupled case in the vicinity of the intersection points between SAW and ISW dispersion curves.",
"In particular, it is worthwhile noting that standing waves and corresponding frequency gap formation can occur not only at the Bragg reflection condition, Eq.",
"(REF ), due to the interaction of two counter-propagating SAWs or ISWs (cf.",
"green and red solid circles in FIG.REF ), but also when $k_{\\parallel m+p, n} \\left( \\frac{\\Gamma \\hat{\\beta }}{2} \\right)^{1/2}= - k_{\\parallel m, n}\\;\\;\\;\\;\\; \\Rightarrow \\;\\;\\;\\;\\; k_{\\parallel m, n} = \\frac{p}{\\hat{J}_\\eta \\hat{B}_0 R_0} \\frac{(\\Gamma \\hat{\\beta }/2)^{1/2}}{1+(\\Gamma \\hat{\\beta }/2)^{1/2}}\\; , \\;\\; p \\in \\mathbb {Z} \\; ,$ where $\\hat{\\beta }\\simeq \\bar{\\beta }$ is rigorously defined in Eq.",
"(REF ).",
"In FIG.REF , regions where Eq.",
"(REF ) is satisfied are denoted by the full orange circles, corresponding to a forward propagating ISW, modulated by the equilibrium non-uniformity within the magnetic flux surface, forming a standing wave by interaction with a backward scattered SAW.",
"Modulated forward propagating ISW can also interact with a forward propagating SAW, when $k_{\\parallel m-p, n} \\left( \\frac{\\Gamma \\hat{\\beta }}{2} \\right)^{1/2}= k_{\\parallel m, n}\\;\\;\\;\\;\\; \\Rightarrow \\;\\;\\;\\;\\; k_{\\parallel m, n} = \\frac{p}{\\hat{J}_\\eta \\hat{B}_0 R_0} \\frac{(\\Gamma \\hat{\\beta }/2)^{1/2}}{1-(\\Gamma \\hat{\\beta }/2)^{1/2}}\\; , \\;\\; p \\in \\mathbb {Z} \\; .$ In this case, however, a standing wave cannot be formed, as it is clearly shown by the dashed orange circles in FIG.REF .",
"The gap formation at low frequency for $0.1<\\nu <0.2$ is due to the interaction denoted by the full orange circle and satisfying Eq.",
"(REF ) for $p=1$ .",
"Combining the results of different flux surfaces, we finally get the behavior of the continuous spectra as shown in FIG.REF for $n=2$ and in FIG.REF for different toroidal mode numbers.",
"Figure: (Color online) SAW-ISW coupled continuum as a function of r/ar/a for n=2n=2.Figure: (Color online) SAW-ISW coupled continuum as a function of r/ar/a for three toroidal wave numbers as indicated in the plot.",
"For the sake of clarity we plot only the first four branches.The richness of the continuous spectrum at low frequency[20], [21], [22], [23] is readily noted, with the frequency gaps due to SAW-ISW and/or ISW-ISW couplings, dubbed Beta induced Alfvén Acoustic Eigenmode (BAAE) gap[26], [24], [25], as well as SAW-ISW and/or SAW-SAW couplings, dubbed Beta induced Alfvén Eigenmode (BAE) gap[20], [68], [69], shown in FIG.REF .",
"The complex structures of the continuous spectrum are further illustrated by adding $n=10$ and $n=20$ toroidal mode numbers in FIG.REF .",
"For example, the weak dependence of upper branch of the continuous spectrum near the BAAE gap in the vicinity of mode rational surface is a reflection of the weaker plasma response to excitations by either fluid or EP effects[47].",
"By direct comparison between plots of the continuous spectrum and local dispersion curves, it is possible to identify the most important physical processes.",
"As an example, comparing FIG.REF and FIG.REF , we could deduce that the first three gaps observed in FIG.REF with $r/a \\sim 0.58$ are produced by the interaction of two ISWs.",
"In fact, for $n=2$ and the given safety factor profile, we obtain $\\nu \\sim 0.5$ from Eq.",
"(REF ).",
"Furthermore, the wave poloidal mode numbers producing the gaps in FIG.REF could be obtained by means of Eqs.",
"(REF ) and (REF ).",
"All this information could be routinely included in continuum spectrum plots.",
"Nonetheless, for the sake of clarity, in this work we prefer to show only the frequency plot.",
"Note, however, that the presence of continuous spectrum structures in a certain region of the $(r/a,\\Omega )$ plane does not automatically imply continuum damping by resonant excitation[8], [7], [6] of those same structures.",
"As anticipated in Section , a crucial role is played by fluctuation polarization, Eq.",
"(REF ), which must be computed self-consistently accounting for fluctuation structure and dispersion relation, often affected non-perturbatively by EPs[14], [30], [31], [32], [37], [38], [42], [47].",
"Without solving for self-consistent mode structure and dispersion relation, a qualitative information about the effective role of resonant excitation of continuum structures can still be obtained from $|e_1|^2$ adopted as Alfvénicity parameter ${\\cal A} = \\left| e_1 (\\nu , r) \\right|^2 \\; .$ With this definition, consistent with that adopted in Ref.",
"chen17 for quantifying mode polarization in the considered MHD limit, ${\\cal A} \\simeq 1$ for SAW continuum, since $g_2 \\sim {\\cal O} (\\beta ^{1/2}) g_1$ at low frequency.",
"Meanwhile, noting $g_1 \\sim {\\cal O} (\\beta ^{1/2}) g_2$ for ISW, ${\\cal A} \\sim {\\cal O}(\\beta )$ for the acoustic continuum.",
"In general, $\\mathcal {A}$ can be calculated for each point of the local dispersion curve and, therefore, for each point of the continuous spectrum plot.",
"In fact, this corresponds to evaluate not only the Floquet characteristic exponent at each radial location, but the corresponding parallel fluctuation structure, $g_1^{(i)}(\\eta ; \\nu _i,r)$ (playing the role of “eigenvector”), and the corresponding polarization $e_1^2(\\nu ,r) = \\frac{\\int (g_1^{(i)}(\\eta ; \\nu _i,r))^2 d \\eta }{\\int [ (g_1^{(i)}(\\eta ; \\nu _i, r))^2+(g_2^{(i)} (\\eta ; \\nu _i, r))^2] d \\eta }\\,,$ consistent with Eqs.",
"(REF )-(REF ).",
"This allows, in principle, to represent the Alfvénicity by means of a color-map in these graphs.",
"A detailed analysis along these lines is beyond the scope of this work, which mainly aims at the illustration of the methodology and the generality of its applications.",
"As a particular case, we have computed the polarization and Alfvénicity at $r/a=0.58$ for the local dispersion curves represented in FIG.REF .",
"For reference, we have assumed $n=2$ such that $\\nu = 0.477$ corresponds to the value on $nq(0.58)$ reduced to the first Brillouin zone (cf.",
"Sec.",
"REF ).",
"FIG.REF illustrates the relevant continuum frequencies, for which we have computed polarization and the values of Alfvénicity, ${\\cal A}$ , according to Eqs.",
"(REF )-(REF ).",
"These are, ${\\cal A}=0.02, 0.06, 0.59, 0.28, 0.61, 0.34$ and $0.98$ , from low to high frequencies, respectively.",
"The first two values are very small, in consistency with the acoustic nature of the fluctuation that is predominant as expected from the green marking of the circle denoting SSW-SSW interaction in FIG.REF .",
"The mixed polarization nature (orange circle, denoting SSW-SAW coupling) is also clearly indicated by the Alfvénicity, which grows for the third to the sixth branch.",
"We could actually state that ${\\cal A}$ is a proper quantitative definition of the mixed polarization in these cases.",
"Finally, the highest frequency branch is clearly Alfvénic, as suggested by both ${\\cal A} = 0.98$ and the red circle in FIG.REF denoting SAW-SAW coupling.",
"In general, a SAW polarized fluctuation, characterized by $(e_1,e_2) = (1,0)$ in the uncoupled case, has weak interaction with the acoustic polarized continuum, which has $(e_1, e_2) = (0,1)$ in the uncoupled limit.",
"Figure: Local dispersion curves ν(Ω)\\nu (\\Omega ) for the SAW-ISW coupled continuum at r/a=0.58r/a=0.58.",
"The green vertical line indicates ν=nq(r)mod1\\nu = n q(r) \\bmod 1, with n=2n=2.",
"Continuum frequencies are marked with red dashed lines.Meanwhile, an ISW polarization weakly interacts with the SAW continuous spectrum.",
"This point is of particular importance when inspecting the structures of continuous spectra approaching the plasma edge, where, due to the decreasing ratio of sound to Alfvén speed, high order ISW sidebands may be generated by strong shaping equilibrium modulation effects and interact with SAW.",
"This interaction generates a thick web of continuous spectrum structures, shown in FIG.REF again for $n=2$ , where they are compared with numerical results obtained by MARS, showing excellent agreement also in this case.",
"Clearly, not all structures of this web are physically relevant, because of the important role of wave polarization.",
"More detailed illustration of this point requires a separate analysis computing the actual Alfvén wave fluctuation spectrum in DTT for the reference scenario (see, e.g., Refs.",
"Wang20181,Wang2019).",
"Figure: (Color online) Comparison of the outputs of MARS code (red bullet) and the SAW-ISW coupled continuum (solid lines), for n=2n=2.This work, as an application of the general theoretical framework discussed in Refs.",
"chen16,zonca14a,zonca14b, illustrates the calculation of continuous spectrum structures in realistic tokamak geometry for the MHD limiting case of governing equations[47].",
"The use of ballooning formalism allows the calculation of local dispersion curves, isolating the physics information concerning the plasma response to radial local singular fluctuations, constituting the continuous spectrum.",
"Meanwhile, toroidal and poloidal mode numbers are treated explicitly for arbitrary values, simplifying the numerical calculation of the complicated continuous spectrum structures at high mode number and/or near the plasma boundary.",
"The present approach clearly illustrates the importance of fluctuation polarization for the assessment of damping by resonant absorption of radial singular continuous spectrum structures.",
"As a qualitative estimate of the coupling strength of Alfvénic fluctuations to the acoustic continuum, without solving for the actual fluctuation structures, an Alfvénicity parameter is introduced, which is related to the fluctuation polarization and can be usefully adopted to visualize the importance of acoustic couplings.",
"Besides the simplicity of the present approach, based on the solution of two coupled second order ordinary differential equations for the local (singular) plasma response, the interesting aspect is that the present methodology could be readily extended to 3D plasma equilibria, such as stellarators, thanks to the generality of ballooning representation[46].",
"Following the the general theoretical framework of Refs.",
"chen16,zonca14a,zonca14b, the present formulation could be employed within a linear gyrokinetic description of coupled SAW and ISW continua.",
"Relevant equations, in this case, would be the short radial scale gyrokinetic vorticity equation (Eq.",
"(6) of Ref.",
"zonca14b) and the quasineutrality condition (Eq.",
"(36) of the same work).",
"Although the uderlying equations are different, the fundamental structure would be that of two second order periodic ordinary differential equations for the local (singular) plasma response, to be analyzed according to the well-known Floquet theory.",
"Within the present framework, the calculation of continuous spectrum, corresponding parallel fluctuation structures and polarization vectors provides the boundary conditions for computing any physical mode structures and corresponding dispersion relations in arbitrary geometry."
],
[
"Acknowledgment",
"This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053.",
"The views and opinions expressed herein do not necessarily reflect those of the European Commission.",
"We thank DTT Team for the reference scenario."
],
[
"Derivation of useful integral quadratic forms",
"When we take the decoupled ($\\kappa _g=0$ ) limit of the system of differential equations, Eqs.",
"(REF ) and (), it is possible to derive the following integral quadratic forms $& & \\int d \\eta \\left| \\partial _\\eta g_1 \\right|^2 = \\hat{\\rho }_{m0} \\Omega ^2 \\int d \\eta \\hat{J}_\\eta ^2 \\left| g_1 \\right|^2 \\; , \\nonumber \\\\& & \\int d \\eta \\left| \\partial _\\eta \\left( | \\mathbf {\\nabla }r | g_2 \\right) \\right|^2 \\frac{1}{| \\mathbf {\\nabla }r |^2} = \\frac{ 2 \\hat{\\rho }_{m0} \\Omega ^2}{\\Gamma \\bar{\\beta }} \\int d \\eta \\hat{J}_\\eta ^2 \\hat{B}_0^2 \\left| g_2 \\right|^2 \\; .",
"$ In the neighborhood of a given point of intersecting SAW and ISW continua, identified by a given $\\nu (\\Omega )$ for fixed $r/a$ , we can write $ \\int d \\eta \\left| \\partial _\\eta g_1 \\right|^2 = (nq - m)^2 \\int d \\eta \\left| g_1 \\right|^2 \\; ;$ and, similarly, $ \\int d \\eta \\left| \\partial _\\eta \\left( | \\mathbf {\\nabla }r | g_2 \\right) \\right|^2 \\frac{1}{| \\mathbf {\\nabla }r |^2} = (nq - m - p)^2 \\int d \\eta \\left| g_2 \\right|^2 \\; ,$ with $p \\in \\mathbb {Z}$ .",
"Here, we have noted that, near the given point of intersecting SAW and ISW continua, the $g_1$ and $g_2$ solutions are modulated by the equilibrium non-uniformity within the flux surface.",
"The first of Eqs.",
"(REF ) is what allows deriving Eq.",
"(REF ), while the combination of the two is what readily yields Eqs.",
"(REF ) and (REF ), with the following definition of $\\hat{\\beta }$ : $\\hat{\\beta }\\equiv \\bar{\\beta }\\frac{ \\int d \\eta \\hat{J}_\\eta ^2 \\left| g_1 \\right|^2 }{ \\int d \\eta \\left| g_1 \\right|^2 } \\frac{ \\int d \\eta \\left| g_2 \\right|^2}{ \\int d \\eta \\hat{J}_\\eta ^2 \\hat{B}_0^2 \\left| g_2 \\right|^2 } \\; .",
"$"
]
] | 1906.04451 |
[
[
"A characterisation of Baer subplanes"
],
[
"Abstract Let $K$ be a set of $q^2+2q+1$ points in $PG(4,q)$.",
"We show that if every 3-space meets $K$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then $K$ is a ruled cubic surface.",
"Moreover, $K$ corresponds via the Bruck-Bose representation to a tangent Baer subplane of $PG(2,q^2)$.",
"We use this to prove a characterisation in $PG(2,q^2)$ of a set of points $B$ as a tangent Baer subplane by looking at the intersections of $B$ with Baer-pencils."
],
[
"Introduction",
"A ruled cubic surface in ${\\mbox{\\textrm {P}G}}(4,q)$ is a variety $3_2$ which rules a line and a conic according to a projectivity (see Section REF for more details).",
"By [8], a hyperplane of ${\\mbox{\\textrm {P}G}}(4,q)$ meets a ruled cubic surface in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic.",
"We show that the converse holds, and prove the following characterisation.",
"Theorem 1 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ .",
"Then ${\\cal K}$ is a ruled cubic surface if and only if every 3-space meets ${\\cal K}$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic.",
"Using known results on the Bruck-Bose representation gives the following corollary.",
"Corollary 2 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space meets ${\\cal K}$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic.",
"Then there is a regular spread ${\\cal S}$ in a 3-space $\\Sigma _\\infty $ so that in the corresponding Bruck-Bose plane ${\\cal P}({\\cal S})\\cong {\\mbox{\\textrm {P}G}}(2,q^2)$ , the set of points corresponding to ${\\cal K}$ form a Baer subplane of ${\\mbox{\\textrm {P}G}}(2,q^2)$ which is tangent to $\\ell _\\infty $ .",
"We further prove a characterisation in ${\\mbox{\\textrm {P}G}}(2,q^2)$ of a tangent Baer subplane by the intersection types with $\\ell _\\infty $ -Baer pencils.",
"A Baer pencil in ${\\mbox{\\textrm {P}G}}(2,q^2)$ is the cone of $q+1$ lines joining a vertex point $P$ to a Baer subline base $b$ , so contains $q^2(q+1)+1$ points.",
"Every line of ${\\mbox{\\textrm {P}G}}(2,q^2)$ not through the vertex of a Baer pencil meets the Baer pencil in a Baer subline.",
"An $\\ell _\\infty $ -Baer pencil is a Baer pencil containing $\\ell _\\infty $ , so has vertex in $\\ell _\\infty $ , and base $b$ meeting $\\ell _\\infty $ in a point.",
"We will prove the following characterisation.",
"Theorem 3 Let ${\\cal B}$ be a set of $q^2+q+1$ points in ${\\mbox{\\textrm {P}G}}(2,q^2)$ , $q\\ge 5$ , with $T={\\cal B}\\cap \\ell _\\infty $ a point.",
"Then ${\\cal B}$ is a Baer subplane if and only if every $\\ell _\\infty $ -Baer pencil meets ${\\cal B}$ in either a point; one or two Baer sublines; or a non-degenerate $\\mathbb {F}_q$ -conic.",
"Section contains background material.",
"Section contains a proof of Theorem REF and Corollary REF .",
"Section contains a proof of Theorem REF ."
],
[
"A ruled cubic surface of ${\\mbox{\\textrm {P}G}}(4,q)$",
"We consider a scroll in ${\\mbox{\\textrm {P}G}}(4,q)$ that rules a line and a non-degenerate conic according to a projectivity.",
"That is, in ${\\mbox{\\textrm {P}G}}(4,q)$ , let $\\ell $ be a line and let $ be a non-degenerate conic in a plane $$ with $ =$.",
"Let $ ,$ be the non-homogeneous coordinates of $$, $ respectively.",
"That is, without loss of generality, we can write $\\ell =\\lbrace (1,\\theta ,\\ 0,0,0)\\,|\\,\\theta \\in \\mathbb {F}_q\\cup \\lbrace \\infty \\rbrace \\rbrace $ and $\\lbrace (0,0,\\ 1,\\phi ,\\phi ^2)\\,|\\,\\phi \\in \\mathbb {F}_q\\cup \\lbrace \\infty \\rbrace \\rbrace $ .",
"Let $\\sigma $ be a projectivity in ${\\mbox{PGL}}(2,q)$ that maps $(1,\\theta )$ to $(1,\\phi )$ .",
"Consider the set of $q+1$ lines of ${\\mbox{\\textrm {P}G}}(4,q)$ joining a point of $\\ell $ to the corresponding point of $ according to $$.",
"Then the points on these lines form a \\emph {scroll ruling a line and a conic according to a projectivity}.The line $$ is called the \\emph {line directrix}.",
"The ruling lines are called \\emph {generators} and are pairwise disjoint.",
"Further, the line directrix and the generators are the only lines on the scroll.The scroll contains $ q2$ conics, called conic directrices, which pairwise meet in a point.",
"Each conic directrix contains one point of each generator and is disjoint from $$.This scroll is a variety $ 32$ of dimension 2 and order 3, so is a {\\em ruled cubic surface}.",
"For more details, see \\cite {BV}, in particular, all ruled cubic surfaces are projectively equivalent.$"
],
[
"The Bruck-Bose representation of ${\\mbox{\\textrm {P}G}}(2,q^2)$ in {{formula:4fc23d1a-b43f-4b65-a8a6-5ed6d964e588}}",
"The Bruck-Bose representation of ${\\mbox{\\textrm {P}G}}(2,q^2)$ in ${\\mbox{\\textrm {P}G}}(4,q)$ was introduced independently by André [1] and Bruck and Bose [5], [6].",
"Let $\\Sigma _\\infty $ be a hyperplane of ${\\mbox{\\textrm {P}G}}(4,q)$ and let ${\\cal S}$ be a regular line spread of $\\Sigma _\\infty $ .",
"Consider the incidence structure ${\\cal P}({\\cal S})$ with points the points of ${\\mbox{\\textrm {P}G}}(4,q)\\setminus \\Sigma _\\infty $ and the lines of ${\\cal S}$ ; lines the planes of ${\\mbox{\\textrm {P}G}}(4,q)\\setminus \\Sigma _\\infty $ that contain an element of ${\\cal S}$ , and a line at infinity $\\ell _\\infty $ whose points correspond to the lines of ${\\cal S}$ ; and incidence induced by incidence in ${\\mbox{\\textrm {P}G}}(4,q)$ .",
"Then ${\\cal P}({\\cal S})\\cong {\\mbox{\\textrm {P}G}}(2,q^2)$ .",
"Associated with a regular spread ${\\cal S}$ in ${\\mbox{\\textrm {P}G}}(3,q)$ are a unique pair of transversal lines in the quadratic extension ${\\mbox{\\textrm {P}G}}(3,q^2)$ .",
"These transversal lines are disjoint from ${\\mbox{\\textrm {P}G}}(3,q)$ , and are conjugate with respect to the map $X=(x_0,\\ldots ,x_3)\\mapsto X^q=(x_0^q,\\ldots ,x_3^q)$ .",
"We denote these transversal lines by $g,g^q$ .",
"The spread ${\\cal S}$ is the set of $q^2+1$ lines $XX^q\\cap {\\mbox{\\textrm {P}G}}(3,q)$ for $X\\in g$ , in particular, the points of $\\ell _\\infty $ are in 1-1 correspondence with the points of $g$ .",
"If $ {\\cal X}$ is a set of points in ${\\mbox{\\textrm {P}G}}(2,q^2)$ , then we denote the corresponding set of points in the Bruck-Bose representation in ${\\mbox{\\textrm {P}G}}(4,q)$ by $[{\\cal X}]$ .",
"For more details on this representation, see [2].",
"In particular, the representation of Baer sublines and Baer subplanes is well known.",
"In particular, a Baer subplane of ${\\mbox{\\textrm {P}G}}(2,q^2)$ tangent to $\\ell _\\infty $ corresponds to a ruled cubic surface of ${\\mbox{\\textrm {P}G}}(4,q)$ .",
"The following result characterises the ruled cubic surfaces of ${\\mbox{\\textrm {P}G}}(4,q)$ which correspond to Baer subplanes of ${\\mbox{\\textrm {P}G}}(2,q^2)$ .",
"Result 4 [7] Let ${\\cal K}$ be a ruled cubic surface in ${\\mbox{\\textrm {P}G}}(4,q)$ with line directrix $b$ .",
"Let $\\Sigma _\\infty $ be a 3-space that meets ${\\cal K}$ in the line $b$ and let ${\\cal S}$ be a regular spread in $\\Sigma _\\infty $ containing $b$ .",
"Then ${\\cal K}$ corresponds via the Bruck-Bose representation to a Baer subplane tangent to $\\ell _\\infty $ in the Bruck-Bose plane ${\\cal P}({\\cal S})\\cong {\\mbox{\\textrm {P}G}}(2,q^2)$ if and only if in the extension to ${\\mbox{\\textrm {P}G}}(4,q^2)$ , ${\\cal K}$ contains the transversal lines of the regular spread ${\\cal S}$ ."
],
[
"Proof of Theorem ",
"We begin by labelling the five different types of 3-spaces given in Theorem REF .",
"Definition 5 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ .",
"Let $\\Pi $ be a 3-space, we say $\\Pi $ has type $T_1$ : if $\\Pi \\cap {\\cal K}$ is 1 line, $T_2$ : if $\\Pi \\cap {\\cal K}$ is 2 lines, $T_3$ : if $\\Pi \\cap {\\cal K}$ is 3 lines, $T_4$ : if $\\Pi \\cap {\\cal K}$ is a line and a non-degenerate conic, $T_5$ : if $\\Pi \\cap {\\cal K}$ is a twisted cubic.",
"Throughout this section, we let ${\\cal K}$ be a set of $q^2+2q+1$ points of ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_1$ , $T_2$ , $T_3$ , $T_4$ or $T_5$ , that is, ${\\cal K}$ satisfies the assumptions of Theorem REF .",
"We prove Theorem REF with a series of lemmas, the structure of these lemmas is as follows.",
"In Lemma REF we show that ${\\cal K}$ can be partitioned into $q+1$ mutually skew lines which we call sticks.",
"In Lemma REF we show that ${\\cal K}$ contains a base line which meets each stick.",
"In Lemma REF , we show that ${\\cal K}$ contains $q^2$ non-degenerate conics.",
"In Lemma REF , we show that the sticks rule the base line and a non-degenerate conic of ${\\cal K}$ according to a projectivity.",
"That is, ${\\cal K}$ is a ruled cubic surface as defined in Section REF , with line directrix the base line, and generators the sticks.",
"We then use [7] to conclude that ${\\cal K}$ corresponds to a tangent Baer subplane of ${\\mbox{\\textrm {P}G}}(2,q^2)$ .",
"Lemma 6 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Let $\\Pi $ be a 3-space of Type $T_4$ , so $\\Pi \\cap {\\cal K}$ is a line $\\ell $ and a non-degenerate conic $.",
"Then $ is a point and $\\langle \\ell ,=\\Pi $ .",
"Proof Let $\\Pi $ be a 3-space meeting ${\\cal K}$ in a line $\\ell $ and a non-degenerate conic $, and let $$ be the plane containing $ .",
"Suppose $\\ell $ is contained in $\\pi $ .",
"Each 3-space containing $\\pi $ has type $T_i$ for some $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ , so has type $T_4$ and hence contains no further point of ${\\cal K}$ .",
"Every point of ${\\mbox{\\textrm {P}G}}(4,q)$ lies in a 3-space containing $\\pi $ , so if $\\ell \\subset \\pi $ , then $|{\\cal K}|=|{\\cal K}\\cap \\pi |\\le 2(q+1)$ , a contradiction.",
"Hence $\\ell $ meets $\\pi $ in a point.",
"Suppose the point $P=\\ell \\cap \\pi $ is not in $.As the $ q+1$ $ 3$-spaces through $$, contain anon-degenerate conic of $ K$, they are of Type $ T4$.Hence each 3-space about $$ also contains a line of $ K$, which necessarily contains the point $ P$.",
"Therefore the number of points of $ K$ is$ |+1+(q+1)q=q2+2q+2$, a contradiction.",
"Hence the line $$ meets the non-degenerate conic $ in a point.",
"$\\square $ Lemma 7 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Then ${\\cal K}$ contains at least one non-degenerate conic.",
"Proof Suppose that ${\\cal K}$ does not contain a non-degenerate conic, that is, there are no 3-spaces of type $T_4$ .",
"We work to a contradiction.",
"There are four cases for the lines of ${\\cal K}$ : (a) ${\\cal K}$ contains no lines; (b) ${\\cal K}$ contains exactly one line; (c) ${\\cal K}$ contains two lines which meet; (d) ${\\cal K}$ contains $s\\ge 2$ lines which are pairwise skew.",
"Case (a): suppose ${\\cal K}$ does not contain any lines, then every 3-space has type $T_5$ .",
"Let $\\Pi $ be a 3-space, so there is a plane $\\pi $ contained in $\\Pi $ that contains 3 points of ${\\cal K}$ .",
"Each 3-space through $\\pi $ has type $T_5$ , so $|{\\cal K}|=3+(q+1) (q-2)=q^2-q+1$ , a contradiction.",
"Hence case (a) does not occur.",
"Case (b): suppose that ${\\cal K}$ contains exactly one line $\\ell $ .",
"Let $P\\in {\\cal K}\\setminus \\ell $ and consider the plane $\\alpha =\\langle P,\\ell \\rangle $ .",
"The 3-spaces through $\\alpha $ meet ${\\cal K}$ in at least $P,\\ell $ , and so are of type $T_2$ or $T_3$ , contradicting our assumption that ${\\cal K}$ contains exactly one line.",
"Hence case (b) does not occur.",
"Case (c): suppose ${\\cal K}$ contains two lines $\\ell ,\\ell ^{\\prime }$ which meet in a point.",
"So $\\pi =\\langle \\ell ,\\ell ^{\\prime } \\rangle $ is a plane.",
"The 3-spaces containing $\\pi $ contain at least 2 lines of ${\\cal K}$ , so are of type $T_2$ or $T_3$ .",
"Suppose there is a 3-space $\\Pi $ of type $T_3$ that contains $\\pi $ .",
"If the points of $\\Pi \\cap {\\cal K}$ all lie in $\\pi $ , then every 3-space containing $\\pi $ has type $T_3$ and hence contains no further point of ${\\cal K}$ .",
"So $|{\\cal K}|=|\\pi \\cap {\\cal K}|\\le 3q+1$ , a contradiction as $q\\ge 3$ .",
"Hence $\\Pi $ contains a line $m$ which is not contained in $\\pi $ .",
"Consider the point $m\\cap \\pi $ .",
"If $m\\cap \\pi $ is not in $\\ell $ or $\\ell ^{\\prime }$ , then every 3-space that contains $\\pi $ contains $\\ell ,\\ell ^{\\prime }$ and a further point of ${\\cal K}$ , so has type $T_3$ .",
"In this case, the number of points of ${\\cal K}$ is $|{\\cal K}|= (2q+1)+1+(q+1).q=q^2+3q+2$ , a contradiction.",
"Hence the point $m\\cap \\pi $ lies in $\\ell \\cup \\ell ^{\\prime }$ .",
"Let $x$ be the number of 3-spaces of type $T_2$ containing $\\pi $ and let $y$ be the number of 3-spaces of type $T_3$ containing $\\pi $ .",
"As every 3-space containing $\\pi $ has type $T_2$ or $T_3$ , $x+y=q+1$ .",
"Further, counting the points of ${\\cal K}$ in planes about $\\pi $ we have $2q+1+y.q=q^2+2q+1$ .",
"Hence $y=q$ and $x=1$ .",
"Hence the points of ${\\cal K}$ consist of the points on $q+2$ lines $\\lbrace \\ell ,\\ell ^{\\prime },m_1,\\ldots ,m_q\\rbrace $ where each $m_i$ meets $\\pi $ in a point of $\\ell \\cup \\ell ^{\\prime }$ , and no two $m_i$ lies in a common 3-space about $\\pi $ .",
"Suppose the lines $m_1,\\ldots ,m_q$ are concurrent at the point $\\ell \\cap \\ell ^{\\prime }$ .",
"Let $\\Pi $ be a 3-space not through $\\ell \\cap \\ell ^{\\prime }$ , so $\\Pi $ meets each of the $q+2$ lines $\\lbrace m_1,\\ldots ,m_q,\\ell ,\\ell ^{\\prime }\\rbrace $ in distinct points, and $|{\\cal K}\\cap \\Pi |=q+2$ , contradicting $\\Pi $ having type $T_i$ for some $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Thus at least one of the lines $m_1,\\ldots ,m_q$ meets $\\pi $ in a point of $\\ell \\cup \\ell ^{\\prime }$ which is not the point $\\ell \\cap \\ell ^{\\prime }$ .",
"Without loss of generality, suppose $m_1$ meets $\\pi $ in a point of $\\ell $ distinct from $\\ell \\cap \\ell ^{\\prime }$ .",
"Let $X\\in {\\cal K}$ with $X\\notin \\pi $ and $X\\notin m_1$ .",
"Label the points on $\\ell ^{\\prime }$ by $Y_0=\\ell ^{\\prime }\\cap \\ell ,Y_1,\\ldots ,Y_q$ .",
"So for $i=1,\\ldots ,q$ , the line $Y_iX$ is disjoint from $m_1$ .",
"Consider the 3-spaces $\\Pi _i=\\langle Y_i,X,m_1\\rangle $ , $i=0,\\ldots ,q$ .",
"As there are no type $T_4$ 3-spaces; and $\\Pi _i\\cap {\\cal K}$ contains the line $m_1$ and the two points $X$ , $Y_i$ ; each $\\Pi _i$ has type $T_2$ or $T_3$ .",
"Now $\\Pi _0\\cap {\\cal K}$ contains $\\ell ,m_1, X$ , so $\\Pi _0$ has type $T_3$ .",
"Hence $\\Pi _0$ contains at least $q-1$ points of ${\\cal K}$ not in $\\ell \\cup \\ell ^{\\prime }\\cup m_1\\cup X$ .",
"If some $\\Pi _i$ has type $T_2$ , then the two lines in $\\Pi _i\\cap {\\cal K}$ are $XY_i$ and $m_1$ .",
"As these two lines are disjoint, $\\Pi _i\\cap {\\cal K}$ contains $q-1$ points not in $\\ell \\cup \\ell ^{\\prime }\\cup m_1\\cup X$ .",
"If some $\\Pi _i$ , $i\\ne 0$ , has type $T_3$ , then the three lines in $\\Pi _i\\cap {\\cal K}$ are either $m_1,XY_i$ and a further line; or $m_1$ and distinct lines $m^{\\prime } $ through $X$ and $m^{\\prime \\prime }$ through $Y_i$ .",
"In the latter case, if $m^{\\prime }$ , $m^{\\prime \\prime }$ meet, then they meet in a point of $m_1$ .",
"As $XY_i$ and $m_1$ are disjoint, $\\Pi _i\\cap {\\cal K}$ contains at least $2q-2$ points not in $\\ell \\cup \\ell ^{\\prime }\\cup m_1\\cup X$ .",
"Let $a$ be the number of type $T_2$ 3-spaces in $\\lbrace \\Pi _1,\\ldots ,\\Pi _q\\rbrace $ and let $b$ be the number of type $T_3$ 3-spaces in $\\lbrace \\Pi _1,\\ldots ,\\Pi _q\\rbrace $ , so $a+b=q$ .",
"Counting the points of ${\\cal K}$ gives $|{\\cal K}|\\ge (q-1)+a(q-1)+b(2q-2)+(2q+1+q+1)$ .",
"Simplifying using $a+b=q$ gives $0\\ge b(q-1)+q+2$ , contradicting $b\\ge 0$ .",
"Hence case (c) does not occur.",
"Case (d): suppose ${\\cal K}$ contains at least two lines, and the lines contained in ${\\cal K}$ are pairwise skew.",
"Let $\\ell ,m$ be two skew lines of ${\\cal K}$ , then the 3-space $\\langle \\ell ,m\\rangle $ has type $T_2$ or $T_3$ .",
"Let $X\\in \\ell $ and consider the plane $\\pi =\\langle X,m\\rangle $ .",
"By assumption, the lines of ${\\cal K}$ are pairwise skew, so there is a unique line of ${\\cal K}$ through $X$ , namely $\\ell $ .",
"Let $\\Sigma $ be a 3-space through $\\pi $ , with $\\ell $ not in $\\Sigma $ .",
"As $\\Sigma \\cap {\\cal K}$ contains the line $m$ and another point $X$ , $\\Sigma $ has type $T_2$ or type $T_3$ .",
"Hence $\\Sigma \\cap {\\cal K}$ contains a line through $X$ , so contains the unique line $\\ell $ of ${\\cal K}$ through $X$ , a contradiction.",
"That is, case (d) does not occur.",
"We have shown that if ${\\cal K}$ does not contain a non-degenerate conic, then none of the cases (a), (b), (c), (d) can occur, a contradiction.",
"Thus ${\\cal K}$ contains at least one non-degenerate conic.",
"$\\square $ Lemma 8 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Then ${\\cal K}$ can be partitioned into a set of $q+1$ lines called sticks.",
"Further, no three sticks lie in a 3-space.",
"Proof By Lemma REF , ${\\cal K}$ contains a non-degenerate conic $.",
"Denote the plane containing $ by $\\pi $ .",
"Each 3-space which contains $\\pi $ has type $T_4$ and so contains a further line of ${\\cal K}$ meeting $.",
"Moreover, by Lemma~\\ref {lem:line-conic-not-plane}, this line meets $ and is not contained in $\\pi $ .",
"Thus the $q+1$ 3-spaces containing $\\pi $ give rise to a set of $q+1$ distinct lines of ${\\cal K}$ , we call these lines sticks.",
"Suppose that two sticks $m,m^{\\prime }$ meet in a point, which is necessarily a point of $.",
"As the 3-spaces containing $$ partition the points of $ PG(4,q)$, there is a point $ X through which there is no line of ${\\cal K}$ .",
"Consider the 3-space $\\Pi =\\langle m,m^{\\prime },X\\rangle $ .",
"As $\\Pi $ contains two lines and a further point of ${\\cal K}$ , $\\Pi $ has type $T_3$ and so contains a line of ${\\cal K}$ through $X$ , a contradiction.",
"Thus the $q+1$ sticks meets $ in distinct points, and lie in distinct 3-spaces containing $$, hence they are pairwise skew, and so partition the $ (q+1)2$ points of $ K$.$ Finally suppose $\\Pi $ is a 3-space containing three sticks $m_1,m_2,m_3$ .",
"Then $\\Pi $ has type $T_3$ , so does not contain a non-degenerate conic, so does not contain $\\pi $ .",
"So $\\Pi $ meets $\\pi $ in a line $\\ell $ .",
"Further, each stick meets $\\pi $ in a point of $, so $$ contains the three collinear points $ mi, $i=1,2,3$ , a contradiction.",
"$\\square $ Lemma 9 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Then ${\\cal K}$ contains exactly $q+2$ lines; consisting of the $q+1$ sticks, and a baseline which intersects every stick.",
"Proof By Lemma REF , ${\\cal K}$ contains at least $q+1$ lines, namely the sticks.",
"Suppose ${\\cal K}$ contains no further line.",
"Let $m,m^{\\prime }$ be two sticks and let $X$ be any point on $m$ .",
"Consider the plane $\\pi =\\langle X,m^{\\prime }\\rangle $ .",
"Let $\\Pi $ be a 3-space containing $\\pi $ , as $\\Pi $ meets ${\\cal K}$ in at least a line and a point, $\\Pi $ must be type $T_2$ , $T_3$ or $T_4$ .",
"If $\\Pi $ is type $T_3$ , then $\\Pi $ contains three lines, which by assumption are all sticks, contradicting Lemma REF .",
"As there is a unique line of ${\\cal K}$ through $X$ , there is a unique 3-space of type $T_2$ containing $\\pi $ .",
"So the remaining $q$ 3-spaces containing $\\pi $ have type $T_4$ .",
"Recalling Lemma REF , such a 3-space meets ${\\cal K}$ in $2q+1$ points, $q+2$ of which lie in $\\pi $ .",
"Hence counting the points of ${\\cal K}$ yields $(q+2)+q+q(q-1)=q^2+q+2$ , a contradiction.",
"Hence ${\\cal K}$ contains at least one line which is not a stick.",
"Now suppose ${\\cal K}$ contained two lines, $\\ell ,\\ell ^{\\prime }$ which are not sticks.",
"As the sticks partition ${\\cal K}$ , and $\\ell ,\\ell ^{\\prime }$ lie in ${\\cal K}$ but are not sticks, they each meet every stick in a point.",
"If $\\ell ,\\ell ^{\\prime }$ do not meet, then the $q+1$ sticks all lie in the subspace $\\Sigma =\\langle \\ell ,\\ell ^{\\prime }\\rangle $ .",
"If $\\ell ,\\ell ^{\\prime }$ meet in a point, then the $q$ sticks not containing the point $\\ell \\cap \\ell ^{\\prime }$ lie in the subspace $\\Sigma =\\langle \\ell ,\\ell ^{\\prime }\\rangle $ .",
"In both cases, $\\Sigma $ is either a 3-space, or is contained in a 3-space, that does not have type $T_i$ , $i=1,\\ldots ,5$ , a contradiction.",
"Hence there is at most one line of ${\\cal K}$ which is not a stick.",
"Thus ${\\cal K}$ contains exactly one line which is not a stick, and this line meets every stick.",
"$\\square $ Lemma 10 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Let $P,Q$ be any two points on distinct sticks, not lying on the baseline of ${\\cal K}$ .",
"Then there exists a unique non-degenerate conic $ contained in $ K$ through $ P,Q$.",
"Moreover, $ meets each stick of ${\\cal K}$ in a unique point, and does not meet the baseline.",
"Proof If there were two non-degenerate conics of ${\\cal K}$ containing two points $P$ and $Q$ of ${\\cal K}$ , then the two conics lie in a 3-space which is not type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ , a contradiction.",
"Hence $P,Q$ lie in at most one non-degenerate conic of ${\\cal K}$ .",
"Let $m,m^{\\prime }$ be sticks of ${\\cal K}$ , and denote the baseline of ${\\cal K}$ by $b$ (recall the baseline meets every stick).",
"Let $P\\in m$ , $Q\\in m^{\\prime }$ , with $P,Q\\notin b$ .",
"We construct a non-degenerate conic containing $P$ and $Q$ .",
"Consider the plane $\\pi =\\langle m,Q\\rangle $ .",
"As $\\pi $ contains at least the line $m$ and point $Q$ of ${\\cal K}$ , each 3-space containing $\\pi $ is type $T_2$ , $T_3$ or $T_4$ .",
"The 3-space $\\Pi _0=\\langle \\pi ,b\\rangle $ contains $m,m^{\\prime },b$ and so has type $T_3$ .",
"As the sticks partition the points of ${\\cal K}$ , there are no type $T_2$ 3-spaces containing $\\pi $ , and a unique type $T_3$ 3-space containing $\\pi $ , namely $\\Pi _0$ .",
"So the $q$ remaining 3-spaces $\\Pi _1,\\ldots ,\\Pi _q$ through $\\pi $ are type $T_4$ .",
"That is, $\\Pi _i$ , $i=1,\\ldots ,q$ , meets ${\\cal K}$ in the stick $m$ and a non-degenerate conic $i$ which contains $Q$ .",
"Further, by Lemma REF , $i$ meets the stick $m$ in a point, but $i$ is not contained in $\\pi $ .",
"Suppose for some $i\\ne j$ , the non-degenerate conics $i,j$ meet $m$ in the same point $X$ , then $i,j$ both contain the points $P,X$ , contradicting the first paragraph.",
"Hence each $i$ meets $m$ in a distinct point.",
"Now suppose some $i$ contains the point $m\\cap b$ , Let $\\Pi ^{\\prime }$ be the 3-space containing $i$ and $b$ .",
"As $\\Pi ^{\\prime }$ contains two points of the stick $m^{\\prime }$ (namely $m^{\\prime }\\cap b$ and $Q\\in i$ ), $\\Pi ^{\\prime }$ contains $m^{\\prime }$ .",
"Hence $\\Pi ^{\\prime }$ is a 3-space and $\\Pi ^{\\prime }\\cap {\\cal K}$ contains two lines $b,m^{\\prime }$ and a non-degenerate conic $i$ , a contradiction.",
"Hence each $i$ meets $m$ in a distinct point which is not $m\\cap b$ .",
"One of these points of $m$ is $P$ , so one of the $i$ contains both $P$ and $Q$ .",
"That is, $P,Q$ lie in at least one non-degenerate conic of ${\\cal K}$ .",
"We showed above that $P,Q$ lie in at most one non-degenerate conic of ${\\cal K}$ , hence they lie in a unique non-degenerate conic of ${\\cal K}$ .",
"$\\square $ Lemma 11 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Then ${\\cal K}$ is a scroll ruling the base line $b$ and a non-degenerate conic according to a projectivity.",
"Proof Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"By Lemma REF , ${\\cal K}$ contains $q+1$ sticks which we denote $\\lbrace m_0,\\ldots ,m_q\\rbrace $ .",
"These sticks partition the points of ${\\cal K}$ and are no three in a 3-space.",
"By Lemma REF , ${\\cal K}$ contains a base line $b$ which meets each stick in a distinct point.",
"By Lemma REF , ${\\cal K}$ contains a non-degenerate conic $ that meets each stick in a point, and does not meet $ b$.",
"So the points of $ K$ lie on $ q+1$ lines $ {m0,...,mq}$ that are pairwise skew, each joining a point of $ b$ to a point of $ .",
"That is, ${\\cal K}$ is a set of lines that rules $b$ and $.",
"We show that this ruling is a projectivity by: in step 1 we showthat we can project $ K$ from a point of $ K$ onto a hyperbolic quadric in a 3-space; then in step 2 we use this hyperbolic quadric to demonstrate the required projectivity.$ Step 1: Let $P$ be a point on the stick $m_0$ with $P\\notin b$ .",
"By Lemma REF , there is a unique non-degenerate conic of ${\\cal K}$ through $P$ and any point of ${\\cal K}$ not on $m_0$ or $b$ ; moreover, this conic meets $m_0$ in $P$ and does not meet $b$ .",
"As $|{\\cal K}|=q^2+2q+1$ , we obtain a set $\\lbrace 1,\\ldots ,q\\rbrace $ of non-degenerate conics of ${\\cal K}$ which each contain $P$ , such that each point in ${\\cal K}\\backslash \\lbrace m_0\\cup b\\rbrace $ lies in exactly one of these non-degenerate conics.",
"Denote the plane containing $i$ by $\\pi _i$ .",
"If $\\pi _i$ contains a point $X\\in {\\cal K}$ with $X\\notin i$ , then $\\pi _i$ contains the stick through $X$ , contradicting Lemma REF .",
"So $\\pi _i\\cap {\\cal K}=i$ , $i=1,\\ldots ,q$ , so in particular, $b\\cap \\pi _i=\\emptyset $ .",
"Let $t_1$ be any line of $\\pi _1$ not through $P$ , so $t_1$ is not a line of ${\\cal K}$ and does not meet $b$ .",
"Consider the 3-space $\\Pi =\\langle b,t_1\\rangle $ .",
"If $P\\in \\Pi $ , then $\\Pi $ contains the plane $\\pi _1$ , and so $b$ meets $\\pi _1$ , a contradiction, hence $P\\notin \\Pi $ .",
"Let $\\phi $ denote the projection of ${\\cal K}$ from $P$ onto $\\Pi $ .",
"We have $\\phi (b)=b$ .",
"The stick $m_0$ contains $P$ and so $\\phi (m_0)$ is the point $\\phi (m_0)=M_0=m_0\\cap b$ .",
"For $j=1,\\ldots ,q$ , the stick $m_j$ is projected onto a line $s_j=\\phi (m_j)$ .",
"For $j=1,\\ldots ,q$ , let $M_j=m_j\\cap b$ , and note that for $i\\ne j$ , we have $M_i\\ne M_j$ .",
"Further, let $P_j=1\\cap m_j$ and $Q_j=PP_j\\cap t_1$ , so we have $s_j=M_jQ_j$ .",
"Suppose two lines $s_i,s_j$ , $i\\ne j$ meet in a point $Y$ .",
"As $M_i\\ne M_j$ , we have $Y\\notin b$ .",
"If $Y\\in t_1$ , then the line $PY$ contains three points of 1, namely $P,P_i,P_j$ , a contradiction as 1 is non-degenerate.",
"So $Y\\notin t_1$ , and so $Q_i\\ne Q_j$ .",
"Hence $\\langle s_i,s_j\\rangle $ is a plane containing two points of $b$ (namely $M_i,M_j$ ) and two points of $t_1$ (namely $Q_i,Q_j$ ).",
"Hence $b,t_1$ meet, a contradiction.",
"Thus $s_1,\\ldots , s_q$ are pairwise skew lines.",
"Hence $\\phi ({\\cal K})$ is a set of $q^2+q+1$ points, namely the points on the lines $s_1,\\ldots , s_q$ , and the point $M_0$ .",
"Recall that the base line $b$ lies in $\\phi (K)$ and contains the point $M_0$ and meets each of $s_1,\\ldots ,s_q$ in a distinct point $M_i=s_i\\cap b=m_i\\cap b$ .",
"Now consider the $q$ -arcs $i\\setminus \\lbrace P\\rbrace $ , $i=1,\\ldots ,q$ .",
"We have $\\phi (i\\setminus \\lbrace P\\rbrace )=\\langle P, \\ i\\setminus \\lbrace P\\rbrace \\rangle \\cap \\Pi $ , and so $\\phi (i\\setminus \\lbrace P\\rbrace )$ is a set of $q$ points on the line $t_i=\\pi _i\\cap \\Pi $ .",
"By Lemma REF , $i$ , $i=1,\\ldots ,q$ , meets the stick $m_j$ , $j=1,\\ldots ,q$ in point not on $b$ .",
"Hence in $\\phi ({\\cal K})$ , the line $t_i$ meets the line $s_j$ in a point not on $b$ , $i,j=1,\\ldots ,q$ .",
"The $q+1$ lines $\\lbrace b,t_1,\\ldots ,t_q\\rbrace $ are mutually skew, and contain the $q^2+q+1$ points of $\\phi ({\\cal K})$ , and $q$ additional points, one on each line $t_i$ .",
"The $q$ lines $\\lbrace s_1,\\ldots ,s_q\\rbrace $ are mutually skew, and cover the $q^2+q$ points of $\\phi ({\\cal K})\\setminus \\lbrace M_0\\rbrace $ .",
"Hence ${\\cal R}=\\lbrace b,t_1,\\ldots ,t_q\\rbrace $ is a regulus, with opposite regulus ${\\cal R}^{\\prime }$ containing $s_1,\\ldots ,s_q$ .",
"As $q\\ge 3$ , there is a unique line $s_0$ in ${\\cal R}^{\\prime }$ through $M_0$ which meets $t_1,\\ldots ,t_q$ .",
"Thus we can project ${\\cal K}$ onto $q^2+q+1$ points of the hyperbolic quadric ${\\cal R}\\cup {\\cal R}^{\\prime }$ .",
"Step 2: We showed in step 1 that we can project ${\\cal K}$ onto $q^2+q+1$ points of the hyperbolic quadric ${\\cal R}\\cup {\\cal R}^{\\prime }$ .",
"For $i\\in \\lbrace 1,\\ldots ,q\\rbrace $ , consider the stick $m_i$ which joins the point $M_i\\in b$ with the point $m_i\\cap 1$ .",
"This maps under $\\phi $ to the line $\\phi (m_i)=s_i$ , joining the point $\\phi (M_i)=M_i $ with the point $\\phi (m_i\\cap 1)=s_i\\cap t_1$ .",
"A hyperbolic quadric is ruled by a projectivity, in particular, it is ruled by a projectivity from $b$ to $t_1$ .",
"That is, there is a projectivity that maps the point $M_i\\in b$ to the point $s_i\\cap t_1$ , $i=0,\\ldots ,q$ .",
"The projection of ${\\cal K}$ onto $\\Pi $ preserves this projectivity, so we have a projectivity mapping the points of $b\\setminus \\lbrace M_0\\rbrace $ to the points of $1\\setminus \\lbrace P\\rbrace $ which maps $m_i\\cap b$ to $m_i\\cap 1$ , $i=1,\\ldots ,q$ .",
"This projectivity will also map $M_0$ to $P$ .",
"That is, the sticks of ${\\cal K}$ are lines of a scroll ruling the line $b$ and the non-degenerate conic 1 according to a projectivity.",
"$\\square $ Proof of Theorem REF Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"By Lemma REF , ${\\cal K}$ is a scroll that rules a line and a non-degenerate conic according to a projectivity.",
"Hence by [4], ${\\cal K}$ is a ruled cubic surface.",
"$\\square $ Proof of Corollary REF Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 3$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ , then by Theorem REF , ${\\cal K}$ is a ruled cubic surface.",
"Let $\\Sigma _\\infty $ be a 3-space that meets ${\\cal K}$ in exactly the line directrix (baseline) $b$ .",
"For each conic $i$ , $i=1,\\ldots ,q^2$ contained in ${\\cal K}$ , let $\\ell _i$ be the line in $\\Sigma _\\infty $ that lies in the plane containing $i$ .",
"By [7], the lines $\\ell _i$ , $i=1,\\ldots ,q^2$ together with $b$ form a regular spread ${\\cal S}$ , moreover ${\\cal K}$ corresponds via the Bruck-Bose representation to a tangent Baer subplane of ${\\cal P}({\\cal S})\\cong {\\mbox{\\textrm {P}G}}(2,q^2)$ .",
"$\\square $"
],
[
"Proof of Theorem ",
"An $\\ell _\\infty $ -Baer pencil of ${\\mbox{\\textrm {P}G}}(2,q^2)$ has the following nice representation in the Bruck-Bose setting.",
"Result 12 [3] An $\\ell _\\infty $ -Baer pencil of ${\\mbox{\\textrm {P}G}}(2,q^2)$ with vertex $L$ corresponds in the ${\\mbox{\\textrm {P}G}}(4,q)$ Bruck-Bose representation to a 3-space of ${\\mbox{\\textrm {P}G}}(4,q)$ containing the spread line $[L]$ .",
"Conversely if $\\Pi $ is a 3-space in ${\\mbox{\\textrm {P}G}}(4,q)$ distinct from $\\Sigma _\\infty $ , then $\\Pi $ corresponds in ${\\mbox{\\textrm {P}G}}(2,q^2)$ to an $\\ell _\\infty $ -Baer pencil with vertex corresponding to the unique spread line in $\\Pi $ .",
"We first show how an $\\ell _\\infty $ -Baer pencil meets a Baer subplane of ${\\mbox{\\textrm {P}G}}(2,q^2)$ tangent to $\\ell _\\infty $ .",
"Lemma 13 Let ${\\cal B}$ be a Baer subplane in ${\\mbox{\\textrm {P}G}}(2,q^2)$ tangent to $\\ell _\\infty $ at the point $ T={\\cal B}\\cap \\ell _\\infty $ .",
"Then an $\\ell _\\infty $ -Baer pencil with vertex $L\\in \\ell _\\infty $ meets ${\\cal B}$ in either: the point $T$ ; one or two Baer sublines containing $T$ ; two Baer sublines with one containing $T$ and one contained in a line through $L$ ; or a non-degenerate $\\mathbb {F}_q$ -conic of ${\\cal B}$ .",
"Proof Let ${\\cal B}$ be a Baer subplane in ${\\mbox{\\textrm {P}G}}(2,q^2)$ tangent to $\\ell _\\infty $ at the point $ T={\\cal B}\\cap \\ell _\\infty $ , so by Result REF , in ${\\mbox{\\textrm {P}G}}(4,q)$ , $[{\\cal B}]$ is a ruled cubic surface with line directrix the spread line $[T]$ .",
"By Result REF , an $\\ell _\\infty $ -Baer pencil with vertex $L$ corresponds to a 3-space that meets $\\Sigma _\\infty $ in the spread line $[L]$ .",
"By [8], a hyperplane of ${\\mbox{\\textrm {P}G}}(4,q)$ meets a ruled cubic surface in one of the following: (a) 1 line; (b) 2 lines, (c) 3 lines, (d) a line and a non-degenerate conic, (e) a twisted cubic.",
"In case (a), the line is $[T]$ , and so the corresponding $\\ell _\\infty $ -Baer pencil meets ${\\cal B}$ in the point $T$ .",
"In case (b), the lines are $[T]$ and one generator which meets $[T]$ , so the corresponding $\\ell _\\infty $ -Baer pencil meets ${\\cal B}$ in a Baer subline through $T$ .",
"In case (c), the lines are $[T]$ and two generators which meet $[T]$ , so the corresponding $\\ell _\\infty $ -Baer pencil meets ${\\cal B}$ in two Baer sublines through $T$ .",
"In case (d), the line is a generator, and by [8], the non-degenerate conic corresponds to a Baer subline disjoint from $\\ell _\\infty $ , so the corresponding $\\ell _\\infty $ -Baer pencil meets ${\\cal B}$ in two Baer sublines, one containing $T$ , and one contained in a line through $L$ .",
"In case (e), by [8], the twisted cubic corresponds to a non-degenerate conic in ${\\cal B}$ which contains the point $T$ .",
"$\\square $ We now prove that the converse holds.",
"Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ .",
"Recall that in Definition REF we defined five types of 3-spaces with respect to ${\\cal K}$ .",
"Here we define three further types of 3-spaces.",
"Definition 14 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ .",
"Let $\\Pi $ be a 3-space, we say $\\Pi $ has type $T_6$ : if $\\Pi \\cap {\\cal K}$ is one point, $T_7$ : if $\\Pi \\cap {\\cal K}$ is a non-degenerate conic and two lines.",
"$T_8$ : if $\\Pi \\cap {\\cal K}$ is a line and a twisted cubic.",
"Lemma 15 Let ${\\cal K}$ be a set of $q^2+2q+1$ points in ${\\mbox{\\textrm {P}G}}(4,q)$ , $q\\ge 5$ , such that every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,8\\rbrace $ , then every 3-space has type $T_i$ , $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Proof Suppose there is a 3-space $\\Pi $ of type $T_6$ , and let $P=\\Pi \\cap {\\cal K}$ .",
"Let $\\pi $ be a plane contained in $\\Pi $ with $P\\notin \\pi $ , so $|\\pi \\cap {\\cal K}|=0$ .",
"Denote the $q+1$ 3-spaces containing $\\pi $ by $\\Pi ,\\Pi _1,\\ldots ,\\Pi _q$ .",
"As $|\\pi \\cap {\\cal K}|=0$ , $\\Pi _i\\cap {\\cal K}$ cannot contain a line.",
"Hence each $\\Pi _i$ , $i=1,\\ldots ,q$ has type $T_5$ or $T_6$ .",
"Let $x$ be the number of type $T_5$ 3-spaces containing $\\pi $ and let $y$ be the number of type $T_6$ 3-spaces containing $\\pi $ , so $y\\ge 1$ .",
"Then $x+y=q+1$ , and counting the points in ${\\cal K}$ we have $x(q+1)+y=q^2+2q+1$ .",
"These equations have solution $x=q+1$ , $y=0$ , a contradiction, hence there is no 3-space of type $T_6$ .",
"Suppose there is a 3-space $\\Pi $ of type $T_7$ , so $\\Pi \\cap {\\cal K}$ is a non-degenerate conic $ and two lines $ ,m$.Let $$ be the plane of $$ containing the non-degenerate conic $ .",
"If $\\pi $ contains both $\\ell $ and $m$ , then every 3-space containing $\\pi $ is of type $T_7$ , and contains no further point of ${\\cal K}$ , hence $|{\\cal K}|\\le (q+1)+(q+1)+q$ , a contradiction.",
"If $\\pi $ contains $\\ell $ and meets $m$ in a point, then every 3-space containing $\\pi $ is of type $T_7$ , and contains $q$ points of ${\\cal K}$ not in $\\pi $ , so $|{\\cal K}|\\ge (q+1)+(q-1)+q(q+1)$ , a contradiction.",
"Hence both $\\ell $ and $m$ meet $\\pi $ in a point, and $|\\pi \\cap {\\cal K}|=q+1+t$ for some $t\\in \\lbrace 0,1,2\\rbrace $ .",
"Denote the $q+1$ 3-spaces containing $\\pi $ by $\\Pi ,\\Pi _1,\\ldots ,\\Pi _q$ .",
"As $\\pi $ contains a non-degenerate conic of ${\\cal K}$ , each of these 3-spaces has type $T_4$ or $T_7$ .",
"Let $x$ be the number of 3-spaces containing $\\pi $ of type $T_4$ , $y$ the number of 3-spaces containing $\\pi $ of type $T_7$ , so $x+y=q+1$ and $y\\ge 1$ .",
"A 3-space of type $T_4$ containing $\\pi $ contains $q$ points of ${\\cal K}$ that are not in $\\pi $ .",
"A 3-space of type $T_7$ containing $\\pi $ contains at least $2q-1$ points of ${\\cal K}$ that are not in $\\pi $ .",
"Counting points of ${\\cal K}$ gives $|{\\cal K}|\\ge xq+y(2q-1)+(q+1+t)$ , hence $y(q-1)+t\\le 0$ .",
"As $q\\ge 3$ , we have $y=t=0$ , contradicting $y\\ge 1$ .",
"Hence there is no 3-space of type $T_7$ .",
"Suppose there is a 3-space $\\Pi $ of type $T_8$ , so $\\Pi \\cap {\\cal K}$ is a line $\\ell $ and a twisted cubic ${\\cal N}_3$ .",
"The line $\\ell $ contains at most two points of ${\\cal N}_3$ , and each plane of $\\Pi $ contains at most three points of ${\\cal N}_3$ .",
"As $q\\ge 4$ , there are at least three points of ${\\cal N}_3$ not on $\\ell $ .",
"Let $\\pi $ be the plane containing these three points, so $\\pi \\cap \\ell \\notin {\\cal N}_3$ .",
"As $\\pi \\cap \\ell $ is a point in ${\\cal K}$ , we have $|\\pi \\cap {\\cal K}|=4$ .",
"Let $\\Sigma $ be any 3-space containing $\\pi $ .",
"As $|\\pi \\cap {\\cal K}|=4$ , $\\Sigma $ has type $T_7$ or $T_8$ .",
"As we have shown that there are no type $T_7$ 3-spaces, each 3-space containing $\\pi $ has type $T_8$ .",
"Hence $|\\Sigma \\cap {\\cal K}|\\ge 2q$ .",
"Counting points of ${\\cal K}$ in the $q+1$ 3-spaces containing $\\pi $ gives $(q^2+2q+1)\\ge 4+(q+1)(2q- 4)$ , and so $q(q-4)\\le 1$ , a contradiction as $q\\ge 5$ .",
"Hence there is no 3-space of type $T_8$ .",
"$\\square $ Lemma 16 Let ${\\cal B}$ be a set of $q^2+q+1$ points in ${\\mbox{\\textrm {P}G}}(2,q^2)$ , $q\\ge 5$ , with $T={\\cal B}\\cap \\ell _\\infty $ a point.",
"Suppose every $\\ell _\\infty $ -Baer pencil meets ${\\cal B}$ in either a point; one or two Baer sublines; or a non-degenerate $\\mathbb {F}_q$ -conic.",
"Then ${\\cal B}$ is a Baer subplane.",
"Proof Let ${\\cal B}$ be a set of $q^2+q+1$ points in ${\\mbox{\\textrm {P}G}}(2,q^2)$ , $q\\ge 5$ , with $T={\\cal B}\\cap \\ell _\\infty $ a point.",
"We work in the Bruck-Bose representation.",
"So ${\\mbox{\\textrm {P}G}}(2,q^2)$ is represented in ${\\mbox{\\textrm {P}G}}(4,q)$ , and the representation involves a regular 1-spread ${\\cal S}$ in the hyperplane at infinity $\\Sigma _\\infty $ .",
"In the Bruck-Bose representation in ${\\mbox{\\textrm {P}G}}(4,q)$ , $[{\\cal B}]$ is a set of $q^2+q$ affine points and the spread line $[T]$ , so $[{\\cal B}]$ is a set of $q^2+2q+1$ points.",
"Let $[$ be any 3-space distinct from $\\Sigma _\\infty $ .",
"Then $[$ contains a unique spread line $[P]$ .",
"By [3], $[$ corresponds to an $\\ell _\\infty $ -Baer pencil $ in $ PG(2,q2)$ with vertex $ P$ (possibly $ P=T$).As $ contains $\\ell _\\infty $ , $T\\in , so $ TB.",
"So by assumption ${\\cal B}\\cap is either a point; one or two Baer sublines; or a non-degenerate conic.As each $$-Baer pencil contains $ T$, we have four cases: a$$-Baer pencil meets $ B$ in either (a) the point $ T$; (b) one Baer subline containing $ T$; (c) two Baer sublines (at least one containing $ T$); or (d) a non-degenerate$ Fq$-conic containing $ T$.",
"We show that in each case, $ [B][$ is type $ Ti$ for some $ i{1,...,8}$.$ Case (a): suppose ${\\cal B}\\cap is the point $ T$.",
"If $ P=T$, then in $ PG(4,q)$, $ [B]$ meets $ [$ in the spread line $ [T]$, and so $ [$ is a 3-space of type $ T1$.If $ PT$, then $ [B]$ meets $ [$ in one point of $ [T]$, and so $ [$ is a 3-space of type $ T6$.$ Case (b): suppose ${\\cal B}\\cap is a Baer subline $ n$ through $ T$.",
"Then in $ PG(4,q)$, $ n$ corresponds to a line $ [n]$ that meets $$ in a point of $ [T]$.",
"Hence if $ P=T$, then$ [B]$ meets $ [$ in two lines, namely the spread line $ [T]$ and the line $ [n]$.",
"That is, $ [$ is a 3-space of type $ T2$.",
"However, if $ PT$, then$ [B]$ meets $ [$ in one line, namely $ [n]$, and so $ [$ is a 3-space of type $ T1$.$ Case (c): suppose ${\\cal B}\\cap is two Baer sublines $ m,n$, with $ Tm$.The line $ m$ contains $ T$ and so corresponds to a line $ [m]$ of $ PG(4,q)$ that meets $$ in a point.",
"If $ Tn$, then $ [n]$ is a line of $ PG(4,q)$ that meets $$ in a point.",
"If $ Tn$, then $ [n]$ is a conic of $ PG(4,q)$.Hence if $ PT$, then $ [B]$ meets $ [$ in either two lines $ [n],[m]$ (and so $ [$ is a 3-space of type $ T2$); or in a conic and a line $ [m]$ (and so $ [$ is a 3-space of type $ T4$).If $ P=T$, then$ [B]$ meets $ [$ in: either three lines, namely the spread line $ [T]$ and the lines $ [n],[m]$, and $ [$ is a 3-space of type $ T3$;or in a conic and two lines $ [T]$, $ [m]$, and $ [$ is a 3-space of type $ T7$.$ Case (d): suppose ${\\cal B}\\cap is a non-degenerate $ Fq$-conic $ .",
"Note that as $T\\in , we have $ T. By [8], in ${\\mbox{\\textrm {P}G}}(4,q)$ , $[$ is a twisted cubic that meets $[T]$ in a point.",
"Hence in ${\\mbox{\\textrm {P}G}}(4,q)$ , if $P=T$ , $[{\\cal B}]$ meets $[$ in a line and a twisted cubic.",
"That is, $[$ is a 3-space of type $T_8$ .",
"If $P\\ne T$ , then $[{\\cal B}]$ meets $[$ in a twisted cubic, and so $[$ is a 3-space of type $T_5$ .",
"Hence we have shown that every 3-space of ${\\mbox{\\textrm {P}G}}(4,q)$ has type $T_i$ for some $i\\in \\lbrace 1,\\ldots ,8\\rbrace $ .",
"So by Lemma REF , every 3-space has type $T_i$ for some $i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"Hence by Theorem REF , $[{\\cal B}]$ is a ruled cubic surface of ${\\mbox{\\textrm {P}G}}(4,q)$ .",
"By Result REF , $[{\\cal B}]$ corresponds to a Baer subplane of ${\\mbox{\\textrm {P}G}}(2,q^2)$ if and only if the extension of $[{\\cal B}]$ to ${\\mbox{\\textrm {P}G}}(4,q^2)$ contains the transversal lines of the regular spread ${\\cal S}$ .",
"Let $[L]$ be a spread line distinct from $[T]$ , and $[$ a 3-space containing $[L]$ .",
"Denote the transversals lines of ${\\cal S}$ by $g,g^q$ , so the extension of $[L]$ meets the transversal lines $g$ and $g^q$ in points we denote $g\\cap [L]$ , $g^q\\cap [L]$ .",
"We need to show that these two points lie in the extension of $[{\\cal B}]$ .",
"By assumption, $[$ does not contain $[T]$ .",
"So by [8], $[$ meets the ruled cubic surface $[{\\cal B}]$ in either (i) a non-degenerate conic disjoint from $[T]$ and a generator line of $[{\\cal B}]$ , or (ii) a twisted cubic with a point in $[T]$ .",
"Correspondingly in ${\\mbox{\\textrm {P}G}}(2,q^2)$ , $ is an $$-Baer pencil with vertex $ LT$.$ First we look at case (i).",
"The generator line in $[\\cap [{\\cal B}]$ corresponds in ${\\mbox{\\textrm {P}G}}(2,q^2)$ to a Baer subline through $T$ .",
"So by assumption, ${\\cal B}$ is two Baer sublines.",
"The conic in $[\\cap [{\\cal B}]$ is a set of $q+1$ affine points, no two on a line with $T$ , and so corresponds to a Baer subline $b$ that lies in a line $m$ with $m\\cap \\ell _\\infty \\ne T$ .",
"In ${\\mbox{\\textrm {P}G}}(4,q)$ , $[b]$ is a non-degenerate conic lying in a plane about a spread line.",
"As the only spread line in $[$ is $[L]$ , the conic $[b]$ lies in a plane about $[L]$ .",
"Moreover, the extension of the conic $[b]$ to ${\\mbox{\\textrm {P}G}}(4,q^2)$ contains the two points $g\\cap [L]$ , $g^q\\cap [L]$ , see [2].",
"Hence the extension of the ruled cubic surface $[{\\cal B}]$ contains the two points $g\\cap [L]$ , $g^q\\cap [L]$ .",
"In case (ii), the twisted cubic in $[\\cap [{\\cal B}]$ contains a point of $[T]$ , so contains $q$ affine points no two on a line through $T$ .",
"Hence by assumption, ${\\cal B}$ is an $\\mathbb {F}_q$ -conic ${\\cal O}$ which contains the point $T$ .",
"By [3], $[{\\cal O}]$ corresponds in ${\\mbox{\\textrm {P}G}}(4,q)$ to a twisted cubic $[{\\cal O}]$ whose extension to ${\\mbox{\\textrm {P}G}}(4,q^2)$ contains the two points $g\\cap [L]$ , $g^q\\cap [L]$ .",
"Hence the extension of the ruled cubic surface contains the two points $g\\cap [L]$ , $g^q\\cap [L]$ .",
"Hence for each line $[L]$ in the spread ${\\cal S}$ distinct from $[T]$ , the points $g\\cap [L]$ , $g^q\\cap [L]$ lie in the extension of the ruled cubic surface $[{\\cal B}]$ , and so $g$ , $q^q$ are lines of the extended ruled cubic surface.",
"Thus by Result REF , ${\\cal B}$ is a tangent Baer subplane of ${\\mbox{\\textrm {P}G}}(2,q^2)$ , as required.",
"$\\square $ Theorem REF now follows immediately from Lemmas REF and REF ."
]
] | 1906.04318 |
[
[
"Maximally Rotating Supermassive Stars at the Onset of Collapse: Effects\n of Gas Pressure"
],
[
"Abstract The \"direct collapse\" scenario has emerged as a promising evolutionary track for the formation of supermassive black holes early in the Universe.",
"In an idealized version of such a scenario, a uniformly rotating supermassive star spinning at the mass-shedding (Keplerian) limit collapses gravitationally after it reaches a critical configuration.",
"Under the assumption that the gas is dominated by radiation pressure, this critical configuration is characterized by unique values of the dimensionless parameters $J/M^2$ and $R_p/M$, where $J$ is the angular momentum, $R_p$ the polar radius, and $M$ the mass.",
"Motivated by a previous perturbative treatment we adopt a fully nonlinear approach to evaluate the effects of gas pressure on these dimensionless parameters for a large range of masses.",
"We find that gas pressure has a significant effect on the critical configuration even for stellar masses as large as $M \\simeq 10^6 M_{\\odot}$.",
"We also calibrate two approximate treatments of the gas pressure perturbation in a comparison with the exact treatment, and find that one commonly used approximation in particular results in increasing deviations from the exact treatment as the mass decreases, and the effects of gas pressure increase.",
"The other approximation, however, proves to be quite robust for all masses $M \\gtrsim 10^4 M_{\\odot}$."
],
[
"Introduction",
"Supermassive black holes (SMBHs) reside at the centers of galaxies.",
"The most recent observational confirmation was provided by the spectacular images of the Event Horizon Telescope Collaboration , showing radiation emitted by material accreting onto the SMBH at the center of the galaxy M87 and shadowing by the black hole's event horizon.",
"Accreting SMBHs are also believed to power quasars and active galactic nuclei, which have been observed out to large cosmological distances , .",
"Examples of quasars at large distances include J1342+0928, at a redshift of $z \\simeq 7.5$ , and powered by a SMBH with mass of approximately $7.8 \\times 10^8 M_\\odot $ , J1120-0641, at a redshift of $z \\simeq 7.1$ and with a black-hole mass of approximately $2.0 \\times 10^9 M_{\\odot }$ , as well as the ultra-luminous quasar J0100+2802 at a redshift of $z = 6.3$ and with a mass of about $1.2 \\times 10^{10} M_\\odot $ .",
"The existence of such massive black holes at so early an age in the Universe poses an important question , , , – namely, how could they have formed in such a short time?",
"One possible evolutionary scenario involves the collapse of first-generation – i.e.",
"Population III (Pop III) – stars to form seed black holes, which then grow through accretion and/or mergers.",
"Growth by merger may be limited by recoil speeds .",
"Growth by accretion depends in part on the efficiency of the conversion of matter to radiation, and is usually limited by the Eddington luminosity , .",
"While this already constrains the formation of SMBHs from stellar-mass black holes , including the effects of photoionization and heating appears to reduce the accretion rate to just a fraction of the Eddington limit (see [2], ; see also for how natal kicks affect the accretion rate, as well as for recent simulations in a cosmological context).",
"It is difficult to see, therefore, how seed black holes with masses of Pop III stars, about 100 $M_{\\odot }$ , could grow to the masses of SMBHs by $z \\simeq 7$ .",
"In fact, argue that the existence of the objects J1342+0928, J1120-0641, and J0100+2802 “is at odds with early black hole formation models that do not involve either massive ($\\gtrsim 10^4 M_{\\odot }$ ) seeds or episodes of hyper-Eddington accretion\" (see also their Fig. 2).",
"The observation of these distant quasars therefore suggests the direct collapse of objects with masses of $M \\gtrsim 10^{4-5} M_{\\odot }$ as a plausible alternative scenario for the formation of SMBHs , , , , , , , , , , [1], .",
"The progenitor object in such a “direct collapse\" scenario is often referred to as a supermassive star (SMS).",
"The properties of SMSs have been the subject of an extensive body of literature (see, e.g., , , , , , [3], , for some early references, as well as , , and for textbook treatments).",
"Numerous authors and groups have studied possible avenues for their formation (see, e.g., , , , , , , ; see also for recent simulations in the context of cosmological evolutions) as well as their ability to avoid fragmentation , , , , , , .",
"In , we considered an idealized evolutionary scenario for rotating SMSs.",
"We assumed that SMSs are dominated by radiation pressure, and that they cool and contract while maintaining uniform rotation.",
"Since the star will spin up during the contraction, it will ultimately reach mass-shedding, i.e.",
"the Kepler limit, and will subsequently evolve along the mass-shedding limit .",
"Ultimately, the SMS reaches a critical configuration at which it becomes radially unstable to collapse to a black hole.",
"The critical configuration is characterized by unique values of the dimensionless parameters $J/M^2$ and $R_p/M$ , where $J$ is the angular momentum, $R_p$ the polar radius, $M$ the mass, and where we have adopted geometrized units with $G = c = 1$ .",
"We computed the values of these parameters both from numerical models of fully relativistic, rotating $n = 3$ polytropes, and from an approximate but analytical energy functional approach that accounts for the stabilizing effects of rotation and the destabilizing effects of relativistic gravity with leading-order terms only.",
"Both approaches result in similar values for the critical parameters (see Table 2 in Paper I).",
"The uniqueness of the parameters characterizing the critical configuration implies that the subsequent evolution, namely the collapse to a black hole, as well as the gravitational wave signal emitted in the collapse, is unique as well.",
"Numerical simulations have shown that this collapse will lead to a spinning black hole with mass $M_{\\rm BH} / M \\simeq 0.9$ and angular momentum $J_{\\rm BH}/M_{\\rm BH}^2 \\simeq 0.7$ , surrounded by a disk with mass $M_{\\rm disk}/M \\simeq 0.1$ , , , , , , , .",
"Given the importance of the critical parameters, we examined in to what degree they depend on some of the assumptions made, and computed leading-order corrections due to gas pressure, magnetic fields, dark matter and dark energy.",
"We determined these corrections using a perturbative treatment based on the energy functional approach mentioned above.",
"As one might expect, the largest corrections by far are those caused by gas pressure.",
"We treated the effects of gas pressure using two different approximations: one based on a formal expansion (“Approximation I\", see Section 17.3 in ST, as well as Section REF below), and the other by adjusting the polytropic index $n$ (“Approximation II\", see, e.g., Exercise 17.3 in ST, Problem 2.26 in , as well as Section REF below).",
"The latter approach, Approximation II, is very simple to implement, and is therefore quite commonly used in numerical simulations , .",
"While it results in expressions for the non-dimensional parameters discussed above that are identical to those from Approximation I, at least to leading order, expressions for some dimensional quantities differ even at leading order.",
"Motivated by this observation, we revisit in this paper the effects of gas pressure on maximally rotating SMSs at the onset of collapse.",
"We improve on our treatment in Paper II in two ways.",
"First, we use the “Rotating neutron star\" (RNS) code of to construct fully relativistic models of rotating SMSs, rather than relying on a perturbative treatment within the energy functional approach.",
"Second, we employ exact expressions for a mixture of radiation and gas pressure in addition to the two approximate treatments of gas pressure described above.",
"As a result, we can treat these stars accurately even for less massive models, for which the gas pressure becomes increasingly important, and can calibrate the accuracy of the two approximate treatments and their impact on this idealized direct-collapse scenario.",
"We note, though, that we ignore other effects that may become important for smaller masses, including electron-positron pair production or nuclear reactions.",
"Our findings are summarized in Fig.",
"REF below, which shows the dimensionless parameters $R_p/M$ and $J/M^2$ for the critical configuration as a function of stellar mass for a large range of stellar masses.",
"We find good agreement between the exact and approximate treatments of the gas pressure, as well as with the perturbative results of Paper II, for large masses with $M \\gtrsim 10^6 M_{\\odot }$ .",
"This confirms our finding of Paper II that, even for these large masses, gas pressure has an important effect on the above parameters.",
"For smaller masses both approximations lead to deviations from the exact treatment of gas pressure, but those stemming from Approximation II are significantly larger than those from Approximation I.",
"This paper is organized as follows.",
"In Section we derive the equation of state (EOS) for a SMS supported by a combination of radiation and gas pressure.",
"We model this EOS in three different ways: exactly, assuming that the star is isentropic (Section REF ), as well as using the two Approximations I and II described above (Sections REF and REF ).",
"In Section we use these three treatments of the EOS to explore their effects on equilibrium models of nonrotating, spherically-symmetric SMSs.",
"In Section we consider rotating SMSs and determine the parameters characterizing their critical configurations at the onset of collapse to a black hole.",
"We conclude in Section with a brief summary."
],
[
"Equation of State",
"In this Section we use thermodynamic relationships to derive the EOS for a SMS supported by both radiation and gas pressure.",
"We first treat the gas pressure terms exactly, assuming that the star is isentropic, and then introduce two different approximations.We closely follow the treatment of Paper II in this discussion.",
"We close this Section with a description of our numerical implementation of the different EOSs."
],
[
"Exact Approach to Handling Gas Pressure",
"We begin by finding expressions for the total pressure, total internal energy density, and total entropy per baryon.",
"We then introduce dimensionless variables, collect the key expressions, and discuss our approach to generating a tabulated EOS, leaving the numerical details to Section REF .",
"The total pressure, $P$ , is the sum of the radiation and gas pressures, $P = P_{r} + P_{g}.$ The radiation pressure $P_r$ is given by $P_{r} = \\frac{1}{3}aT^{4},$ where $T$ is the temperature and $a \\equiv \\frac{8\\pi ^{5}k_{B}^{4}}{15h^{3}}$ the radiation constant in geometrized units.",
"In (REF ) we have also introduced the Boltzmann constant $k_{B}$ and Planck's constant $h$ .",
"Assuming a fully ionized hydrogen gas for simplicity, the gas pressure $P_{g}$ is $P_{g} = 2n_{B}k_{B}T,$ where $n_{B} = \\frac{\\rho _{0}}{m_{B}}$ is the baryon number density, $\\rho _{0}$ is the rest-mass density, and $m_{B}$ is the baryon rest mass.",
"The total pressure is then given by $P = P_{r} + P_{g} = \\frac{1}{3}aT^{4} + 2n_{B}k_{B}T.$"
],
[
"Total Internal Energy Density",
"Similarly, the total internal energy density $\\epsilon $ is the sum of contributions from the radiation, $\\epsilon _{r} = aT^{4},$ and the (nonrelativistic) plasma, $\\epsilon _{g} = 3n_{B}k_{B}T,$ where we have again assumed a fully ionized hydrogen gas.",
"We then have $\\epsilon = \\epsilon _{r} + \\epsilon _{g} = aT^{4} + 3n_{B}k_{B}T.$ The total (energy) density $\\rho $ is the sum of the rest-mass density and the total internal energy density, i.e.",
"$\\rho = \\rho _{0} + \\epsilon .$"
],
[
"Total Entropy per Baryon",
"The total entropy per baryon, $s$ , is again the sum of contributions from the radiation and the gas, $s = s_{r} + s_{g},$ and is related to the internal energy density and pressure through the first law of thermodynamics, $Tds = d\\left(\\frac{\\epsilon }{n_{B}}\\right)+Pd\\left(\\frac{1}{n_{B}}\\right).$ The photon entropy per baryon, $s_{r}$ , is $s_{r} = \\frac{4am_{B}T^{3}}{3\\rho _{0}},$ and the gas entropy per baryon, $s_{g}$ , is $s_{g} = k_{B}\\ln \\left(\\frac{4m_{e}^{3/2}m_{B}^{7/2}}{\\rho _{0}^{2}}\\left(\\frac{k_{B}T}{2\\pi \\hbar ^{2}}\\right)^{3}\\right) + 5k_{B},$ where $m_{e}$ is the electron mass and $\\hbar \\equiv h/2\\pi $ .",
"Substituting eqs.",
"(REF ) and (REF ) into eq.",
"(REF ), we find that the total entropy per baryon is $s = \\frac{4am_{B}T^{3}}{3\\rho _{0}} + k_{B}\\ln \\left(\\frac{4m_{e}^{3/2}m_{B}^{7/2}}{\\rho _{0}^{2}}\\left(\\frac{k_{B}T}{2\\pi \\hbar ^{2}}\\right)^{3}\\right) + 5k_{B}.$"
],
[
"Collecting Equations",
"In geometrized units the pressure and the various energy densities all have the same units of $length^{-2}$ .",
"Therefore, we can nondimensionalize them using the same constant, which proves to be convenient for later numerical work.",
"Defining a constant $K$ with units of $length^{2/3}$ , $K \\equiv \\frac{a}{3}\\left(\\frac{3s}{4m_{B}a}\\right)^{4/3},$ we define dimensionless pressure, rest-mass density, internal energy density, and total density as P K3 P, 0 K3 0, K3 , and $\\bar{\\rho } \\equiv K^{3}\\rho ,$ respectively.",
"In terms of these dimensionless variables, eqs.",
"(REF ), (REF ), (REF ), and (REF ) now take the form P = 13aK3T4 + 20mBkBT, = aK3T4 + 30mBkBT, = 0 + aK3T4 + 30mBkBT, and $s = \\frac{4a}{3}\\frac{T^{3}}{\\bar{\\rho _{0}}}K^{3}m_{B} + k_{B}\\ln \\left(\\frac{4m_{e}^{3/2}m_{B}^{7/2}}{\\bar{\\rho _{0}}^{2}}K^{6}\\left(\\frac{k_{B}T}{2\\pi \\hbar ^{2}}\\right)^{3}\\right) + 5k_{B}.$ Given a pressure $\\bar{P}$ and an entropy per baryon $s$ , we solve eqs.",
"(REF ) and (REF ) simultaneously for $T$ and $\\bar{\\rho _{0}}$ , which we then substitute into eqs.",
"(REF ) and (REF ) to calculate $\\bar{\\epsilon }$ and $\\bar{\\rho }$ , respectively.",
"The result is a tabulated EOS that we use in numerical calculations in Sections and .",
"We discuss the construction of these tabulated EOSs in more detail in Section REF below.",
"Instead of adopting an exact description of radiation and gas pressure, it is also common to use approximate treatments.",
"We introduce two different approximations, “Approximation I\" and “Approximation II\", in Sections REF and REF below.",
"For the purpose of comparing these approximate treatments with the exact solution it is convenient to define a small dimensionless parameter $\\beta $ , $\\beta \\equiv 8k_{B}/s \\approx P_{g}/P_{r}.$ We note that slightly different definitions of $\\beta $ are used in the literature.",
"In Paper II, in particular, we defined $\\beta $ in terms of the radiation entropy $s_r$ rather than the total entropy $s$ .",
"To linear order, however, the two definitions are equivalent, so that all linear-order expressions in Paper II can be used without modification.",
"With the definition (REF ) a constant $\\beta $ now means constant total entropy per baryon throughout a star, instead of constant radiative entropy per baryon.",
"Constant total entropy per baryon is the more realistic assumption, and is made plausible for SMSs because they are expected to be convective (see, e.g.",
"the Appendix of )."
],
[
"Approximation I",
"Approximation I is based on a formal expansion, and takes into account the effects of the gas to leading order only.",
"We refer the reader to Section 17.3 in ST for a detailed treatment, but review the main results here.",
"If $s_{g} \\ll s_{r}$ , we can approximate $s_{r}$ with $s$ and write the temperature as $T \\approx \\left(\\frac{3s\\rho _{0}}{4m_{B}a}\\right)^{1/3}\\left(1-\\frac{s_{0}}{3s}-\\frac{k_{B}}{3s}\\ln \\left(\\frac{3s}{4m_{b}a\\rho _{0}}\\right)\\right),$ where $s_{0}$ is defined as $s_{0} \\equiv \\left(3\\ln \\left(\\frac{k_{B}}{2\\pi \\hbar ^{2}}\\right)+\\frac{3}{2}\\ln m_{e}+\\frac{7}{2}\\ln m_{B}+2\\ln 2 + 5\\right)k_{B}.$ (see eq.",
"(17.3.4) in ST).",
"The natural scale factor, which we called $K_{I}$ in paper II, is the same as $K$ defined in (REF ), $K_I= K$ .",
"Defining the auxiliary functions $\\bar{\\lambda } \\equiv -\\frac{4s_{0}}{s} + \\frac{12k_{B}}{s} - \\frac{4k_{B}}{s}\\ln \\frac{3s}{4m_{B}a}$ and $\\bar{\\mu } \\equiv \\frac{4k_{B}}{s},$ we write the internal energy density as $\\epsilon \\approx K\\rho _{0}^{4/3}\\left(3 + \\bar{\\lambda } + \\bar{\\mu }\\ln \\rho _{0}\\right).$ The functions $\\bar{\\lambda }$ and $\\bar{\\mu }$ are decorated with bars because they are dimensionless versions of the corresponding functions $\\lambda $ and $\\mu $ defined in eqs.",
"(17.3.11) and (17.3.12) of ST.",
"In terms of $\\beta $ , eq.",
"(REF ) becomes $\\epsilon \\approx K\\rho _{0}^{4/3}\\left(3 - \\beta \\left(1 - \\frac{5}{2}\\ln \\beta - \\frac{1}{2}\\ln \\left(K^{3}\\rho _{0}\\right)+\\frac{1}{2}\\ln \\eta \\right)\\right),$ where $\\eta \\equiv \\frac{2^{4}3^{4}5^{2}}{\\pi ^{7}}\\left(\\frac{m_{e}}{m_{B}}\\right)^{3/2} \\approx 1.367\\times 10^{-4}.$ The pressure can be found in terms of $\\epsilon $ as $P \\approx \\frac{1}{3}\\frac{1+\\beta }{1+\\beta /2}\\epsilon .$ As in Section REF , we define dimensionless fluid variables using the scaling relations (REF ) through (REF ).",
"Given the pressure and the entropy per baryon, we can solve eq.",
"(REF ) for the internal energy density $\\epsilon $ .",
"Substitution into eq.",
"(REF ) then allows us to find a numerical solution for the rest-mass density $\\rho _{0}$ , which can simply be added to the internal energy density to find the total density $\\rho $ .",
"From these, we construct another tabulated EOS (see also Section REF below)."
],
[
"Approximation II",
"A pure radiation gas is an $n = 3$ , or $\\Gamma = 1 + 1/n = 4/3$ polytrope.",
"In Approximation II, the EOS is still taken to be of polytropic form, with the effects of the gas pressure approximated by a small change in the polytropic index (see, e.g., Exercise 17.3 in ST, and Problem 2.26 in ).",
"We compute the adiabatic exponent from $\\Gamma _{1} \\equiv \\left(\\frac{d\\ln P}{d\\ln \\rho _{0}}\\right)_{s} = \\frac{4}{3} + \\frac{\\beta \\left(4+\\beta \\right)}{3\\left(1+\\beta \\right)\\left(8+\\beta \\right)} \\approx \\frac{4}{3}+\\frac{\\beta }{6},$ and require the pressure $P$ to obey $P = K_{II}\\rho _{0}^{\\Gamma _{1}}.$ We find that $K_{II}$ is $K_{II} = (1+\\beta )K\\rho _{0}^{-\\beta /6},$ which is not truly constant.",
"Approximating $K_{II}$ as independent of $\\rho _{0}$ for small $\\beta $ , we can find the internal energy density to be $\\epsilon = n_{1}P,$ where the approximate polytropic index is $n_{1} = \\frac{1}{\\Gamma _{1} -1} = \\frac{3}{1+\\beta /2}.$ The scale factor used to define dimensionless quantities is now $K_{II}\\approx (1+\\beta )K$ .",
"Given a pressure and an entropy per baryon, we can use eq.",
"(REF ) to calculate the internal energy density $\\epsilon $ and eq.",
"(REF ) to calculate the rest-mass density $\\rho _{0}$ .",
"The total density $\\rho $ is again the sum of $\\rho _{0}$ and $\\epsilon $ ."
],
[
"Numerical Implementation",
"Given an EOS, a pressure $\\bar{P}$ , and a total entropy per baryon $s$ , we would like to calculate the remaining thermodynamic variables.",
"For all three approaches, we first compute $\\beta $ from $s$ using (REF ).",
"For Approximation II (Section REF ), we can then compute all quantities analytically.",
"For the exact approach (Section REF ) and Approximation I (Section REF ), however, we need to find roots of equations numerically.",
"In practice, we use the Numerical Recipes () routines rtsafe and mnewt for one-dimensional and two-dimensional iterative root-finding, respectively.",
"These routines require analytical derivatives.",
"For the exact EOS, for example, we use eqs.",
"(REF ) and (REF ) to define F1(T,0) P - 13 a K3 T4 - 20kBTmB, F2(T,0) s - 4 a mB K3 T33 0 - kB(4 me3/2mB7/2K602(kBT22)3) - 5kB.",
"To solve eqs.",
"(REF ) and (REF ) simultaneously for $T$ and $\\bar{\\rho }_{0}$ , the required analytical derivatives are the Jacobian matrix elements J11 TF1 = -43 a K3 T3 - 20kBmB, J12 0F1 = -2kBTmB, J21 TF2 = -4amBK3T20-3kBT, J22 0F2 = 4amBK3T3302+2kB0.",
"Note that $J_{21}$ is also the derivative needed for the numerical solution of eq.",
"(REF ) for $T$ when given $s$ and $\\bar{\\rho }_{0}$ .",
"In addition to derivative information, mnewt needs a good initial guess for the solutions $T$ and $\\bar{\\rho }_{0}$ .",
"Because we expect the addition of gas terms to make only a small difference, we can assume a polytropic solution with $n=3$ and use Tguess = (3Pa K3)1/4, 0,guess = 4 a mB K3 Tguess33s as initial guesses.",
"Once the solutions for $T$ and $\\bar{\\rho }_{0}$ have been found, $\\bar{\\epsilon }$ and $\\bar{\\rho }$ can be found from eqs.",
"(REF ) and (REF ).",
"Our EOS class is called directly from our TOV-solver for the calculations in Section .",
"A separate driver routine calls our EOS class to generate tabulated EOSs suitable for input to the RNS code () used for calculations in Section ."
],
[
"Nonrotating Supermassive Stars",
"As a first experiment we explore the effects of the different treatments of the EOS on the structure of nonrotating SMS.",
"To do so, we solve the Tolman-Oppenheimer-Volkoff equations , $\\frac{dm}{dr} = 4\\pi \\left(\\rho _{0}+\\epsilon \\right)r^{2},$ and $\\frac{dP}{dr} = -\\left(\\rho _{0}+\\epsilon +P\\right)\\frac{m + 4\\pi Pr^{3}}{r^2\\left(1-2m/r\\right)},$ where $m(r)$ is the mass inside areal radius $r$ .",
"The stellar radius $R$ is defined as the value of $r$ at which the pressure $P$ first vanishes.",
"The stellar mass is then given by $M = m(R)$ .",
"Eqs.",
"(REF ) and (REF ) can be non-dimensionalized as previously discussed for the EOSs.",
"For each of our three approaches to handling gas pressure we pick a value for the total entropy per baryon and numerically integrate the TOV equations at fixed entropy for a variety of central rest-mass densities.",
"At a critical central rest-mass density $\\rho _{0,c}$ the mass $M$ of the star along a sequence of constant entropy is maximized, marking the onset of radial instability (“turning-point\" criterion).",
"We call this mass $M_{\\rm crit}$ .",
"As motivated in Paper II, we combine these critical parameters into a single dimensionless parameter $x_{\\rm crit}$ $x_{\\rm crit} \\equiv \\bar{M}_{\\rm crit}^{2/3}\\bar{\\rho }_{0,c}^{1/3} = M_{\\rm crit}^{2/3}\\rho _{0,c}^{1/3}.$ For a SMS supported by radiation pressure alone, i.e.",
"pure $n=3$ polytropes, we have $\\rho _{0,c}=0$ and hence $x_{\\rm crit} = 0$ , indicating that all of these stars are unstable in relativistic gravity.",
"The maximum mass is then given by the Newtonian value $\\bar{M}_{\\rm crit}=\\bar{M}_{0}^{{\\rm sph}} = 4.555$ .",
"In the presence of gas pressure, the mass will take a maximum at some finite central density $\\rho _{0,c} > 0$ , thereby stabilizing those configurations with central densities smaller than this critical value.",
"In the following we parametrize the critical configurations for our EOSs by the values of $x_{\\rm crit}$ and by the relative mass differences $\\delta _{M}^{{\\rm sph}}$ , defined through $\\bar{M}_{{\\rm crit}} = \\bar{M}_{0}^{{\\rm sph}}\\left(1+\\delta _{M}^{{\\rm sph}}\\right).$ In Figs.",
"REF and REF we show results for $x_{\\rm crit}$ and $\\delta _{M}^{{\\rm sph}}$ as a function of $\\beta =8k_{B}/s$ .",
"Figure: The dimensionless variable x crit =M crit 2/3 ρ 0,c 1/3 x_{\\rm crit} = M^{2/3}_{\\rm crit}\\rho ^{1/3}_{0, c} as a function of β=8k B /s\\beta =8k_{B}/s fornonrotating SMS solutions to the TOV equations.",
"Crosses (redonline) denote the numerical results for the exact treatment of theEOS from Section .",
"The solid line (blue online)represents the analytical, leading-order perturbative prediction() from Section (which is identicalfor Approximations I and II to the EOS).",
"The open circles (outlinedin black online) and filled squares (green online) denote thenumerical results from Section , using Approximations Iand II to the EOS, respectively.",
"For finite entropy (nonzeroβ\\beta ) a SMS is partially supported by gas pressure, and nonzerox crit x_{\\rm crit} indicates that this stabilizes it against collapsefor central densities below ρ 0,c \\rho _{0,c}.",
"Compare with Fig.",
"1 ofPaper II.",
"As suggested in Paper II, Approximation I is indeedcloser to the exact solution, despite Approximation II agreeingbetter with the leading-order perturbative prediction.Figure: The relative change in the mass δ M sph \\delta ^{\\rm sph}_{M} (see()) as a function of β=8k B /s\\beta =8k_{B}/s for nonrotatingSMS solutions.",
"Crosses (red online) denote the numerical resultsusing the exact EOS from Section .",
"The solid and dashedlines (blue online) represent the analytical, leading-orderperturbative predictions () and() derived using the energy functional method withApproximations I and II to the EOS, respectively.",
"The open circles(outlined in black online) and filled squares (green online) denotethe numerical results from Section .",
"The triangles(purple online) labeled SUS represent numerical results of, who adopted Approximation II.",
"The relative changein the critical mass increases in magnitude as β\\beta increases.As in Fig.",
", we find that Approximation I is closer tothe exact treatment of the EOS than Approximation II.",
"Compare withFig.",
"2 of Paper II.Both Fig.",
"REF and Fig.",
"REF also include perturbative results, labeled “Pert.",
"\", that are computed from analytical, leading-order perturbative expressions derived from a simple energy functional approach (see Paper II for details).",
"Both Approximation I and II lead to identical expressions for $x_{\\rm crit}$ , $x_{{\\rm crit}} = \\frac{k_{2}}{4k_{4}}\\beta $ (see eqs.",
"(49) and (56) in Paper II, hereafter (II.49) and (II.56)), where $k_{2}=0.63899$ and $k_{4}=0.918294$ .",
"The two approximations differ, however, in their predictions for the corrections to the mass.",
"For Approximation I, this correction is $\\delta _{M}^{{\\rm sph,I}} = \\left(\\frac{3}{4}\\ln \\frac{k_{2}}{4k_{4}}+2\\ln \\beta +\\frac{3}{2}C\\right)\\beta ,$ (see (II.51)) with $C = \\frac{k_{\\tau }}{2} - \\frac{1}{3}\\ln \\bar{M}_0^{\\rm sph} - \\frac{1}{6}\\ln \\eta - \\frac{1}{3},$ $k_{\\tau } = -0.45928$ , and $\\eta $ given by (REF ), while for Approximation II it is $\\delta _{M}^{{\\rm sph,II}} = \\left(\\frac{3}{4}\\ln \\frac{k_{2}}{4k_{4}}+\\frac{3}{4}\\ln \\beta -\\frac{1}{2}\\ln \\bar{M}_{0}^{{\\rm sph}}\\right)\\beta $ (see (II.60)).",
"Fig.",
"REF (compare with Fig.",
"1 of Paper II) shows that when a SMS is partially supported by gas pressure ($\\beta >0$ ) it is stabilized against collapse ($x_{\\rm crit}>0$ ) for central densities below $\\rho _{0,c}$ .",
"The numerical solution using the exact EOS falls between the solutions using Approximations I and II.",
"As suggested in Paper II, Approximation I is closer to the exact solution, despite Approximation II agreeing better with the perturbative prediction.",
"Fig.",
"REF (compare with Fig.",
"2 of Paper II) shows that the relative change in the critical mass increases in magnitude as $\\beta $ increases.",
"As in Fig.",
"REF , the numerical solution from handling the EOS exactly falls between the solutions using Approximations I and II, but is much closer to the results of Approximation I.",
"Also included in this plot are numerical results of , labeled SUS, who adopted Approximation II.",
"Not surprisingly, their results agree very well with our corresponding results."
],
[
"Rotating Supermassive Stars",
"As discussed in Paper I, rotation can stabilize a SMS even when it is supported by a pure radiation gas, i.e.",
"an $n=3$ polytrope.",
"In fact, for maximally rotating SMS, i.e.",
"stars rotating uniformly at the mass-shedding limit, the critical configuration marking the onset of a radial instability is characterized by unique values of the dimensionless parameters x0 5.97 10-3, j0 0.919, where $j = J / M^2$ is the dimensionless angular momentum, and $\\bar{M}_0 \\simeq 4.56$ (see Section REF below).",
"In this Section we evaluate how changes in these parameters due to the presence of gas pressure are affected by the different treatments of the gas pressure.",
"Specifically, we will compute changes $\\delta _x$ , $\\delta _j$ and $\\delta _M$ , defined by x = x0(1+x), j = j0(1+j), M = M0(1+M), using the exact and approximate treatments of the gas pressure.",
"As in Section we will also compare these changes with the perturbative expressions of Paper II."
],
[
"Numerical Method",
"We use a version of the RNS code (see ) slightly modified for use with SMSs.",
"We use the tabulated EOS option with the EOSs discussed in Section and tables assembled using code discussed in Section REF .",
"We change the default surface values for energy density, pressure, and enthalpy in the example main.c to zero for tabulated EOSs.",
"We also make a radial step size in the RNS code's TOV-solver in equil.c six orders of magnitude larger.",
"Both changes are needed because SMSs are far less dense than neutron stars, and much larger.",
"We add a high-resolution grid option to the makefile for these calculations, increasing the number of angular gridpoints to 801 and the number of radial gridpoints to 1601.",
"Given an EOS and a central energy density the example RNS code spins up a TOV solution until the star reaches mass-shedding, finding many intermediate configurations along the way.",
"For a given EOS we consider many different central densities, allowing us to compute the data displayed in Figs.",
"REF and REF .",
"The curves of constant $\\bar{J}$ in Figs.",
"REF and REF are constructed by interpolation.",
"Stable and unstable configurations are separated by locating the maximum mass $\\bar{M}$ along curves of constant $\\bar{J}$ (see and discussion in ).",
"We mark these turning points in Figs.",
"REF and REF with black dots.",
"The turning point corresponding to the maximally rotating configuration is marked separately as the critical point."
],
[
"Pure Radiation Fluid",
"We start our analysis for a SMS supported by a pure radiation fluid, i.e.",
"for an $n=3$ polytrope, essentially reproducing the numerical analysis of Paper I.",
"Our results are shown in Fig.",
"REF .",
"In particular, we identify the critical configuration as the mass-shedding configuration at the onset of radial instability.",
"The dimensionless parameters $x_0$ , $j_0$ and $\\bar{M}_0$ characterizing this critical configuration are given in eqs.",
"() through (REF ) above, see also Table REF .",
"We note that these values differ slightly from those computed in Paper I; we believe that the differences are due to the significantly higher numerical resolution used in our work here.",
"In the following Sections we evaluate how gas pressure, treated both exactly and approximately, affects these critical parameters."
],
[
"Exact Approach",
"We start with the exact treatment of the EOS, as described in Section REF .",
"To do so, we run the RNS code with the corresponding tabulated EOS for different values of $\\beta = 8 k_B /s$ .",
"For each value of $\\beta $ we again choose a number of different values for the central density, and let the RNS code spin the star up to mass shedding.",
"Results from these calculations are shown in the left column of Fig.",
"REF .",
"We determine the critical configurations, marked by the red dots in Fig.",
"REF , as before, and compute their physical parameters (see Table REF ).",
"Finally, we compute the corresponding changes from eqs.",
"() —(), and plot these changes in Fig.",
"REF .",
"We summarize our results for critical configurations of maximally rotating SMSs partially supported by gas pressure in Figs.",
"REF and REF .",
"In Fig.",
"REF we show the dimensionless parameters $R_p/M$ and $J/M^2$ as a function of $\\beta $ , as well as plotted against each other, while in Fig.",
"REF we show the parameters as a function of mass.",
"For the exact treatment of the EOS we compute these physical masses by rescaling the dimensionless masses $\\bar{M}$ computed in the code according to $M =K^{3/2} \\bar{M}$ , with $K$ given by (REF ).",
"Figure: The dimensionless parameters R p /MR_p/M and J/M 2 J/M^2 of thecritical configuration, both as a function of β\\beta (top left andtop right panels) and plotted against each other (bottom panel).",
"Asexpected from Fig.",
", our numerical resultsagree well with perturbative results for small values of β\\beta .For larger values of β\\beta , deviations between the exact treatmentof the EOS and the two approximations increase as well, withApproximation I performing better than Approximation II.",
"Whenplotted against each other (bottom panel), values of R p /MR_p/M andJ/M 2 J/M^2 appear to lie on a single line, so that deviations in radiusappear to be compensated for by deviations in the angular momentum.Note, however, that, according to different approaches, individualconfigurations on this line correspond to different values ofβ\\beta ."
],
[
"Approximation I",
"For Approximation I we compute and analyze models of rotating SMS in the same way as for the exact approach, except that we now run the RNS code with EOS tables computed as discussed in Section REF .",
"We show results from these calculations in the middle column of Fig.",
"REF .",
"We again identify the critical configurations for different values of $\\beta $ , compute the corresponding changes from eqs.",
"() —(), and plot these changes in Fig.",
"REF .",
"We also graph the parameters $R_p/M$ and $J/M^2$ in Fig.",
"REF and REF , where, for Approximation I, we have again computed the mass in Fig.",
"REF from $M = K^{3/2} \\bar{M}$ , with $K$ given by (REF ).",
"In Paper II we adopted a perturbative approach within a simple energy functional model to compute leading-order corrections to the critical parameters.",
"Applying these methods for rotating SMSs and adopting Approximation I, these changes are given by $\\delta _{j} = -\\frac{1}{2}\\delta {x} = -\\frac{k_{1}}{8k_{3}}\\frac{1}{\\bar{M}_{0}^{2/3}\\left(2j_{0}^{2} - j_{{\\rm min}}^{2}\\right)x_{0}}\\beta $ (see (II.93)) and $\\delta _{M}^{I} = \\left(\\frac{3}{4}\\ln x_{0} + \\frac{5}{4}\\ln \\beta + \\frac{3}{2}C + \\frac{9k_{5}}{4k_{3}}\\frac{x_{0}}{2j_{0}^{2}-j_{{\\rm min}}^{2}}\\right)\\beta ,$ (see (II.96)), where $k_{3}=1.2041$ , and $k_{5}=0.331211$ (J. C. Lombardi Jr., 1997, priv. comm.).",
"We also find $j_{\\rm min}=0.886$ for our $n=3$ polytrope simulations.",
"We use these expressions to calculate the perturbative curves labeled “Pert.",
"I\" in Figure REF .",
"From $\\delta _j$ , we can compute changes in the dimensionless ratios $R_p/M$ and $J/M^2$ from $\\left(\\frac{R_{p}}{M}\\right)_{\\rm crit} = \\left(\\frac{R_{p}}{M}\\right)_{\\rm crit, 0}\\left(1+2\\delta _{j}\\right)$ and $\\left(\\frac{J}{M^{2}}\\right)_{\\rm crit} = \\left(\\frac{J}{M^{2}}\\right)_{\\rm crit, 0}\\left(1+\\delta _{j}\\right)$ (see (II.87) and (II.88)).",
"These equations are plotted as the solid lines in Figs.",
"REF and REF , using eq.",
"(REF ) and the leading-order relationship between $\\beta $ and $M$ $\\beta \\approx 8.46\\left(\\frac{M}{M_{}}\\right)^{-1/2}$ (see, e.g., (II.40))."
],
[
"Approximation II",
"Finally we repeat the same exercise with EOS tables computed from Approximation II, as discussed in Section REF .",
"We show numerical results in the right column of Fig.",
"REF .",
"As before, critical configurations are marked by red dots.",
"We identify the physical parameters for these critical configurations, compute changes from eqs.",
"() – (), and plot these changes in Fig.",
"REF .",
"As before, we also graph the polar radius and the angular momentum in Figs.",
"REF and REF .",
"In the latter, we now compute the mass from $M = K_{II}^{n_1/2} \\bar{M}$ , with $K_{II} \\approx (1 + \\beta ) K$ and $n_1$ given by Eq.",
"(REF ) (see Section REF ).",
"Adopting Approximation II in the perturbative treatment of the energy functional approach leads to the same $\\delta _j$ as Approximation I, given by (REF ), while $\\delta _M$ is now given by $\\delta _{M}^{II} = \\left(\\frac{3}{4}-\\frac{1}{2}\\ln \\bar{M}_{0}+\\frac{3}{4}\\ln x_{0}-\\frac{3}{4}\\frac{j_{0}^{2}}{2j_{0}^{2}-j_{{\\rm min}}^{2}}\\right)\\beta .$ (see (II.105)).",
"We use these expressions to calculate the perturbative curves labeled “Pert.",
"II\" in Fig.",
"REF .",
"Since expressions for $R_p/M$ and $J/M^2$ are the same in Approximation I and II, both are represented by the same perturbative line in Figs.",
"REF and REF ."
],
[
"Comparison",
"Figs.",
"REF , REF , and REF show that, for small $\\beta $ , corresponding to large masses, all approaches, including the perturbative treatment, lead to similar predictions for dimensionless quantities, including the dimensionless parameters $R_p/M$ and $J/M^2$ characterizing the critical configuration.",
"In particular, our numerical results confirm our perturbative finding of Paper II that, even for masses as large as $M \\simeq 10^6 M_{\\odot }$ , gas pressure has a significant effect on these parameters.",
"The reason for this behavior is the fact that, to leading order, corrections to the parameters scale with $M^{-1/2}$ , and therefore decrease only slowly as the mass increases (see Eqs.",
"(II.142) and (II.142)).",
"For moderate values of $\\beta $ , or stellar masses $\\lesssim 10^6 M_{\\odot }$ , the analytic perturbative treatment starts to deviate from the numerical results, while both approximations implemented numerically continue to agree with each other up to larger values of $\\beta $ , and smaller masses.",
"Ultimately, Approximation II in particular shows increasing deviations from the exact treatment as well.",
"As we had noted in Paper II, Approximation II results in predictions for changes in the mass that differ from Approximation I and the exact treatment even at leading order, both in the numerical and perturbative treatments (see the right panel in Fig.",
"REF ).",
"We believe that at least some of these deviations may be related to the different scaling used for the different approaches: for the exact treatment and Approximation I, we rescale dimensional quantities with $K^{3/2}$ (see Eq.",
"(REF )), while for Approximation II we rescale with $K_{II}^{n_1/2}$ .",
"Since $K_{II}$ is only approximately constant, this approximation may well be responsible for deviations that we find in dimensional quantities.",
"In the left two panels of Fig.",
"REF we show the dimensionless parameters $R_p/M$ and $J/M^2$ as a function of $\\beta $ .",
"As before, we find that, as the gas pressure becomes more important, Approximation II leads to larger deviations from the exact treatment than Approximation I.",
"In the right panel of Fig.",
"REF we plot $R_p/M$ versus $J/M^2$ .",
"Remarkably, all three approaches lead to results that appear to follow a single line, even though, according to the different treatments of the gas pressure, individual configurations on this line would be identified with different values of $\\beta $ .",
"Finally we graph $R_p/M$ and $J/M^2$ against stellar masses $M$ in Fig.",
"REF .",
"The deviations between the exact treatment of the EOS and Approximation II now appear slightly larger than in the left two panels of REF , which we attribute to the scaling of the mass, as discussed above."
],
[
"Summary and Discussion",
"Recent observations of increasingly young quasars have heightened interest in the direct-collapse scenario for the formation of SMBHs, in which a SMS becomes unstable and collapses gravitationally.",
"A number of groups have studied possible avenues for the formation of SMSs , , , , , , , .",
"In this paper we continue the study of an idealized version of a direct-collapse scenario, involving uniformly rotating SMSs evolving along the mass-shedding limit until they reach a critical configuration marking the onset of radial instability (see Papers I and II).",
"Identifying this critical configuration is important since it determines the dynamics of the subsequent collapse to a SMBH, including the accompanying gravitational wave signal and the properties of the remnant.",
"In fact, many fully relativistic simulations of this collapse have adopted models of the critical configuration as initial data , , , , , .",
"In this paper we study the effects of gas pressure on the critical configuration.",
"While we believe that our findings are interesting in their own right, we also hope that they will help improve future dynamical simulations of the collapse of SMSs to SMBHs.",
"In Paper I we found that the critical configuration is characterized by unique values of $R_p/M$ and $J/M^2$ as long as the star is dominated by radiation pressure.",
"In Paper II we computed leading-order corrections to these values when some of the assumptions of Paper I were relaxed; in particular we considered two different approximations to estimate the effects of gas pressure.",
"Approximation I is based on a formal expansion, while Approximation II accounts for the effects of gas pressure by simply adjusting the polytropic index in a polytropic EOS.",
"The latter is therefore simple to implement and has been used quite commonly.",
"Somewhat surprisingly, we found that some predictions stemming from these two approximations differed.",
"In this paper we apply the turning-point criterion to study more systematically the effects of gas pressure on the critical configuration of maximally rotating SMSs, and determine the critical configuration and its parameters for a large range of stellar masses.",
"We also evaluate differences stemming from different treatments of the gas pressure.",
"To do so, we expand on our treatment in Paper II in two ways.",
"Instead of employing a perturbative analysis within a simple analytic energy functional model, we now compute fully relativistic numerical models of rotating SMSs.",
"We also include a fully self-consistent, exact treatment of the EOS, in addition to the two approximations discussed above, so that we can calibrate the two approximations in the context of this idealized direct-collapse scenario.",
"As expected, all methods agree well for large masses, $M \\gtrsim 10^6M_{\\odot }$ , corresponding to large entropies, and hence to small $\\beta $ and small effects of the gas pressure.",
"In particular, our numerical results confirm the perturbative results of Paper II that, even for these large masses, the effects of gas pressure are important.",
"Not surprisingly, the perturbative treatment starts to deviate from the exact results first as $\\beta $ increases and the mass decreases.",
"Below $M \\simeq 10^5 M_{\\odot }$ , both approximations lead to increasing deviations from the exact treatment of gas pressure, but Approximation I remains much closer to the exact results than Approximation II."
],
[
"Acknowledgments",
"This work was supported in part by NSF grant PHYS-1707526 to Bowdoin College, NSF grant PHY-1662211 and NASA grant 80NSSC17K0070 to the University of Illinois at Urbana-Champaign, as well as through sabbatical support from the Simons Foundation (Grant No.",
"561147 to TWB)."
]
] | 1906.04190 |
[
[
"Learning Powerful Policies by Using Consistent Dynamics Model"
],
[
"Abstract Model-based Reinforcement Learning approaches have the promise of being sample efficient.",
"Much of the progress in learning dynamics models in RL has been made by learning models via supervised learning.",
"But traditional model-based approaches lead to `compounding errors' when the model is unrolled step by step.",
"Essentially, the state transitions that the learner predicts (by unrolling the model for multiple steps) and the state transitions that the learner experiences (by acting in the environment) may not be consistent.",
"There is enough evidence that humans build a model of the environment, not only by observing the environment but also by interacting with the environment.",
"Interaction with the environment allows humans to carry out experiments: taking actions that help uncover true causal relationships which can be used for building better dynamics models.",
"Analogously, we would expect such interactions to be helpful for a learning agent while learning to model the environment dynamics.",
"In this paper, we build upon this intuition by using an auxiliary cost function to ensure consistency between what the agent observes (by acting in the real world) and what it imagines (by acting in the `learned' world).",
"We consider several tasks - Mujoco based control tasks and Atari games - and show that the proposed approach helps to train powerful policies and better dynamics models."
],
[
"Introduction",
"Reinforcement Learning consists of two fundamental problems: learning and planning.",
"Learning comprises of improving the agent's current policy by interacting with the environment while planning involves improving the policy without interacting with the environment.",
"These problems evolve into the dichotomy of model-free methods (which primarily rely on learning) and model-based methods (which primarily rely on planning).",
"Recently, model-free methods have shown many successes, such as learning to play Atari games with pixel observations [19], [20] and learning complex motion skills from high dimensional inputs [26], [27].",
"But their high sample complexity is still a major criticism of the model-free approaches.",
"In contrast, model-based reinforcement learning methods have been introduced in the literature where the goal is to improve the sample efficiency by learning a dynamics model of the environment.",
"But model-based RL has several caveats.",
"If the policy takes the learner to an unexplored state in the environment, the learner's model could make errors in estimating the environment dynamics, leading to sub-optimal behavior.",
"This problem is referred to as the model-bias problem [9].",
"In order to make a prediction about the future, dynamics models are unrolled step by step which leads to “compounding errors” [32], [3], [15]: an error in modeling the environment at time $t$ affects the predicted observations at all subsequent time-steps.",
"This problem is much more challenging for the environments where the agent observes high-dimensional image inputs (and not compact state representations).",
"On the other hand, model-free algorithms are not limited by the accuracy of the model, and therefore can achieve better final performance by trial and error, though at the expense of much higher sample complexity.",
"In the model-based approaches, the dynamics model is usually trained with supervised learning techniques and the state transition tuples (collected as the agent acts in the environment) become the supervising dataset.",
"Hence the process of learning the model has no control over what kind of data is produced for its training.",
"That is, from the perspective of learning the dynamics model, the agent just observes the environment and does not “interact” with it.",
"On the other hand, there's enough evidence that humans learn the environment dynamics not just by observing the environment but also by interacting with the environment [7], [8].",
"Interaction is useful as it allows the agent to carry out experiments in the real world to determine causality, which is clearly a desirable characteristic when building dynamics models.",
"This leads to an interesting possibility.",
"The agent could consider two possible pathways: (i) Interacting with the environment by taking actions in the real world to generate new observations and (ii) Interacting with the learned dynamics models by imagining to take actions and predicting the new observations.",
"Consider the humanoid robot from the MuJoCo environment [21].",
"In the first case, the humanoid agent takes an action in the real environment, observes the change in its position (and location), takes another step and so on.",
"In the second case, the agent imagines taking a step, predicts what the observation would look like, imagines taking another step and so on.",
"The first case is the closed-loop setup, where the humanoid observes the state of the world, takes an action, gets the true observation from the environment, which it uses to choose the next action, and so on.",
"The second case is the open-loop setup, where the agent predicts subsequent states for multiple time steps into the future (with the help of the dynamics model) without interacting with the environment (see figure REF ).",
"As such, the two pathways may not be consistent given the challenges in learning a multi-step dynamics model.",
"By consistent, we mean the behavior of state transitions along the two paths should be indistinguishable.",
"Had the predictions from the open loop been similar to the predictions from the closed loop over a long time horizon, the two pathways would be consistent and we could say that the learner's dynamics model is grounded in reality.",
"To that end, our contributions are the following: We propose to ensure consistency by using an auxiliary loss which explicitly matches the generative behavior (from the open loop) and the observed behavior (from the closed loop) as closely as possible.",
"We show that the proposed approach helps to simultaneously train more powerful policies as well as better dynamics models, by using a training objective that is not solely focused on predicting the next observation.",
"We consider various tasks - 7 Mujoco based continuous control tasks and 4 Atari games - from OpenAI Gym suite [4], and RLLab [11] and show that using the proposed auxiliary loss consistently helps in achieving better performance across all the tasks.",
"We compare our proposed approach to the state-of-the-art state space models [5] and show that the proposed method outperforms the sophisticated baselines despite being very straightforward.",
"We also evaluate our approach on the pixel-based Half-Cheetah task from the OpenAI Gym suite [4].",
"The task is difficult for the “baseline” state-space models as only the position (and not the velocity) can be inferred from the images, making the task partially observable.",
"Our implementation of the paper is available at https://github.com/shagunsodhani/consistent-dynamics."
],
[
"Prelimaries",
"A finite time Markov decision process $\\mathcal {M}$ is generally defined by the tuple $(\\mathcal {S}, \\mathcal {A}, f, R, \\gamma )$ .",
"Here, $\\mathcal {S}$ is the set of states, $\\mathcal {A}$ the action space, $f(s_{t+1}|s_t, a_t)$ the transition distribution, $r: \\mathcal {S} \\times \\mathcal {A} \\rightarrow R$ is the reward function and $\\gamma $ the discount factor.",
"We define the return as the discounted sum of rewards $r(s_t, a_t)$ along a trajectory $\\tau := (s_{0}, a_{0}, ..., s_{T-1}, a_{T-1}, s_{T})$ , where $T$ refers to the effective horizon of the process.",
"The goal of reinforcement learning is to find a policy $\\pi _\\phi $ that maximizes the expected return.",
"Here $\\phi $ denotes the parameters of the policy $\\pi $ .",
"Model-based RL methods learn the dynamics model from the observed transitions.",
"This is usually done with a function approximator parameterized as a neural network $\\hat{f}_{\\theta }(s_{t+1}|s_t, a_t)$ .",
"In such a case, the parameters $\\theta $ of the dynamics model are optimized to maximize the log-likelihood of the state transition distribution."
],
[
"Environment Model",
"Consider a learning agent training to optimize an expected returns signal in a given environment.",
"At a given timestep $t$ , the agent is in some state $s_t \\in S$ .",
"It takes an action $a_t \\in A$ according to its policy $ a_{t} \\sim \\pi _t(a_{t}|s_{t})$ , receives a reward $r_t$ (from the environment) and transitions to a new state $s_{t+1}$ .",
"The agent is trying to maximize its expected returns and has two pathways for improving its behaviour: Closed-loop path: The learning agent interacts with the environment by taking actions in the real world at every step.",
"The agent starts in state $s_0$ and is in state $s_t$ at time $t$ .",
"It chooses an action $a_t$ to perform (using its policy $\\pi _t$ ), performs the chosen action, and receives a reward $r_t$ .",
"It then observes the environment to obtain the new state $s_{t+1}$ , uses this state to decide which action $a_{t+1}$ to perform next and so on.",
"Open-loop path: The learning agent interacts with the learned dynamics model by imagining to take actions and predicts the future observations (or future belief state in case of state space models).",
"The agent starts in state $s_0$ and is in state $s_t$ at time $t$ .",
"Note that the agent “imagines” itself to be in state $s_t^I$ and can not access the true state of the environment.",
"It chooses an action $a_t$ to perform (using its policy $\\pi _t$ ), performs the action in the “learner's” world (dynamics model) and imagines to transition to the new state $s_{t+1}^I$ .",
"Thus the current “imagined” state is used to predict the next “imagined” state.",
"During these “imagined” roll-outs, the agent does not interact with the environment but interacts with its “learned” version of the environment which we call the dynamics model or the learner's “world”.",
"As an alternative, the agent could use both the pathways simultaneously.",
"The agent could, in parallel, (i) build a model of the environment (dynamics model) and (ii) engage in interaction with the real environment as shown in Figure REF .",
"We propose to make the two pathways consistent with each other so as to ensure that the predictions from the learner's dynamics model are grounded in the observations from the environment.",
"We show that such a consistency constraint helps the agent to learn a powerful policy and a better dynamics model of the environment."
],
[
"Consistency Constraint",
"We want the “imagined” behavior (from the open loop) to be consistent with the observed behavior (from the closed loop) to ensure that the predictions from the learner's dynamics model are similar to the actual observations from the environment.",
"The dynamics model could either be in the observation space (pixel space) or in the state space.",
"State space models are generally more efficient as they model dynamics at a higher level of abstraction.",
"In that case, the learner predicts transitions in the state space by first encoding the actual observation (from the environment) into the state space of the learner and then imposing the consistency constraint in the (learned) state space.",
"At a given timestep $t$ , the learner is in some environment state $s_t$ while it imagines to be in state $s_t^I$ .",
"It takes an action $a_t$ according to its policy $ a_{t} \\sim \\pi _t(a_{t}|s_{t})$ .",
"Now the learner can make transition in two ways.",
"It could execute the action in the environment and transition to state $s_{t+1}$ (as governed by f, the dynamics of the environment).",
"Alternatively, it could execute the action in the learned dynamics environment $\\hat{f}_{\\theta }$ and imagine to transition to the state $s_{t+1}^I = \\hat{f}_{\\theta }(s_{t}^I, a_t)$ .",
"Note that the state $s_t$ is not used by the learner's dynamics model when making state transitions during the open-loop setup.",
"Many possibilities exist for imposing the consistency constraint.",
"In this work, we encode the state transitions (during both open-loop and closed-loop) into fixed-size real vectors using recurrent networks and enforce the output of the recurrent networks to be similar in the two cases.",
"Encoding the sequence can be seen as abstracting out the per-step state transitions into how the dynamics of the environment evolve over time.",
"This way, we do not focus on mimicking each state but the high-level dynamics of the state transitions.",
"We encourage the dynamics model to only focus on information that makes the multi-step predictions (from the open-loop) indistinguishable from the actual future observations from the environment (figure REF ).",
"Given the predicted state transitions and the real state transitions, we minimize the $L2$ error between the encoding of predicted future observations as coming from the learner's dynamics model (during open-loop) and the encoding of the future observations as coming from the environment (during closed loop).",
"Let us assume that the agent started in state $s_0$ and that $a_{0:T-1}$ denote the sequence of actions that the agent takes in the environment from time $t=0$ to $T-1$ resulting in state sequence $s_{1:T}$ that the agent transitions through.",
"Alternatively, the agent could have “imagined” a trajectory of state transitions by performing the actions $a_{0:T-1}$ in the learner's dynamics model.",
"This would result in the sequence of states $s^{I}_{1:T}$ .",
"The consistency loss is computed as follows: $enc(s_{1:T})) = RNN([s_1, s_2, ..., s_T])$ $enc(s^I_{1:T})) = RNN([s_1^I, s_2^I, ..., s_T^I])$ $l_{cc}(\\theta , \\phi ) = \\Vert enc(s_{1:T})) - enc(s^I_{1:T})) \\Vert $ where $\\Vert \\Vert $ denotes the L2 norm.",
"The agent which is trained with the consistency constraint is referred to as the consistent dynamics agent.",
"The overall loss for such a learning agent can be written as follows: $l_{total}(\\theta , \\phi ) = l_{rl}(\\phi ) + \\alpha l_{cc}(\\theta , \\phi )$ where $\\theta $ refers to the parameters of the agent's transition model $\\hat{f}$ and $\\phi $ refers to the parameters of the agent's policy $\\pi $ .",
"The first component of the loss function, $l_{rl}(\\theta , \\phi )$ , corresponds to the standard RL objective of maximizing the expected return and is referred to as the RL loss.",
"The second component of the loss, $l_{cc}(\\theta , \\phi )$ , corresponds to the loss associated with the consistency constraint and is referred to as consistency loss.",
"$\\alpha $ is a hyper-parameter to scale the consistency loss component with respect to the RL loss."
],
[
"Observation Space Model",
"For the observation space models, we represent the environment as a Markov Decision Process $\\mathcal {M}$ with an unknown state transition function $f: \\mathcal {S} \\times \\mathcal {A} \\rightarrow \\mathcal {S}$ .",
"At time $t$ , the agent is in state $s_t \\in \\mathcal {S}$ , learns a policy function $\\pi _{t}$ and a dynamics model $\\hat{f}$ to predict the next state $s_{t+1}$ given a state-action pair ($s_t$ , $a_t$ ).",
"We use the hybrid Model-based and Model-free (Mb-Mf) algorithm [22] as the baseline to design and learn the transition function and the policy.",
"[22] propose to use a trained, deep neural network based dynamics model to initialize a model free learning agent to combine the sample efficiency of model-based approaches with the high task-specific performance of model-free methods.",
"Both the transition function and the policy are parameterized using neural networks (Gaussian outputs) as $\\hat{f}_{\\theta }(s_t, a_t)$ and $\\pi _{\\phi }(s_t)$ where $\\theta $ and $\\phi $ denote the parameters of the dynamics model and the policy respectively.",
"The details about model and policy implementation are provided in the appendix.",
"In the closed loop setup, the agent starts in a state $s_0$ .",
"At time $t$ , it is in state $s_t$ , it chooses an action $a_t \\sim \\pi _t(a_t | s_t)$ , receives a reward $r_t$ and observes the next state $s_{t+1}$ which it uses to choose the next action $a_{t+1}$ .",
"In the open loop setup, the agent starts in a state $s_0$ .",
"At time $t$ , it is in state $s_t$ , while it imagines to be in state $s_t^I$ .",
"It chooses an $a_t \\sim \\pi _t(s_t)$ , imagines the next state $s^{I}_{t+1} = \\hat{f}(s_t^I, a_t)$ .",
"Simultaneously, the action $a_t$ is simulated in the environment to obtain the next environment state $s_{t+1}$ .",
"These environment states are needed to compute the consistency loss for training the agent.",
"As described in equation REF , we encode the two state transition sequences into fixed size vectors using recurrent models and then minimize the L2 norm between them."
],
[
"State Space Model",
"If the observation space is high dimensional, as in case of pixel-space observations(from high dimensional image data), state space models may be used to model the dynamics of the environment.",
"These models can be computationally more efficient than the pixel-space models as they make predictions at a higher level of abstraction and learn a compact representation of the observation.",
"Further, it may be easier to model the environment dynamics in the latent space as compared to the high dimensional pixel space.",
"We use the state-of-the-art Learning to Query model [5] as our state space model.",
"Consider a learning agent operating in an environment that produces an observation $o_t$ at every time-step $t$ .",
"These observations can be high-dimensional and highly redundant (for modelling the dynamics of the environment).",
"The agent learns to encode these observations $(o_t)$ into compact state-space representations ($s_t$ ) using an encoder $e$ and learns a policy function $\\pi $ to choose actions $a_t\\sim \\pi (a_t | s_t)$ .",
"The environment dynamics is given by an unknown observation transition function $f: \\mathcal {O} \\times \\mathcal {A}\\rightarrow \\mathcal {O}$ and the agent aims to learn the model dynamics in state-space representation using a state transition function $\\hat{f}$ .",
"Both the policy and state transition functions are parameterized using neural networks as $\\pi _{\\phi }$ and $\\hat{f}_{\\theta }$ , where $\\phi $ and $\\theta $ represent the parameters of the policy and the transition function respectively.",
"A latent variable $z_t$ is introduced per timestep to introduce stochasticity in the state transition function.",
"The observation space decoding $o_{t+1}$ can be obtained from the state space encoding as $o_{t+1} \\sim p(o_{t+1} | s_t, z_t)$ .",
"We now describe the steps in the closed loop and open loop setup."
],
[
"Closed Loop:",
"The agent starts in some state $s_0$ and receives an observation $o_1$ from the environment.",
"At time $t$ , the agent is in a state $s_{t-1}$ and receives an observation $o_t$ from the environment.",
"It samples a latent state vector $z_t \\sim q(z_t | e(o_t), s_{t-1}, a_{t-1})$ and transition to a new state, $s_t = {f}_\\theta (z_t, s_{t-1}, a_{t-1})$ .",
"It selects an action $a_t = \\pi (a_t | s_t)$ and decodes the state $s_t$ into observation $o_{t+1} \\sim p(o_{t+1} | s_t, z_t)$ ."
],
[
"Open Loop:",
"The agent starts in some state $s_0$ .",
"At time $t$ , the agent is in an imagined state $s^I_{t-1}$ .",
"It samples a latent state vector $z_t \\sim p(z_t | s^I_{t-1}, a_{t-1})$ and transitions to a new imaginary state $s^I_t = \\hat{f}_\\theta (z_t, s^I_{t-1}, a_{t-1})$ .",
"When the agent performs the action $a_t$ in the dynamics model, the action is simultaneously simulated in the external environment to obtain the next true observation $o_{t}$ .",
"These environment observations are then encoded into the latent state and are needed to ensure consistency between the learner's imagined state transition and the actual state transitions in the real environment.",
"$s^{I}_{1:T}$ denotes the sequence of states that the agent imagines and $o_{1:T}$ denotes the sequence of observations that the agent obtains from the environment.",
"These observations are encoded into the state space to yield a sequence of encoded environment observations $s_{1:T}$ .",
"We want to make the behavior of sequence $s_{1:T}$ indistinguishable from $s^{I}_{1:T}$ .",
"We follow the same approach as observation space models where we encode the two state-transition sequences into fixed length vectors using recurrent models and then minimize the L2 norm between them (as described in equation REF ).",
"The agent is trained by imitation learning using trajectories sampled using an expert policy.",
"The details about the model and policy implementation are provided in the appendix.",
"While stochasticity is useful for capturing long term dependencies, most of the latent space models (with stochastic dynamics) are trained with one step ahead predictions and they tend to produce inconsistent predictions when predicting multiple time steps into the future.",
"By using the proposed consistency loss in the latent space, we can enforce that the multi-step predictions be grounded in the observations from the actual environment.",
"Hence, the use of the proposed consistency loss, to improve the long term predictions (as demonstrated empirically), can also be seen as a regularizer."
],
[
"Rationale Behind Using Consistency Loss",
"Our goal is to provide a mechanism for the agent to have a direct “interaction” between the agent's policy and its dynamics model.",
"This interaction is different from the standard RL approaches where the trajectories sampled by the policy are used to train the dynamics model.",
"In those cases, the model has no control over what kind of data is produced for its training and there is no (“direct\") mechanism for the dynamics model to affect the policy, hence a “direct interaction” between the policy and the model is missing.",
"A practical instantiation of this idea is the consistency loss where we ensure consistency between the predictions (from the dynamics model) and the actual observations (from the environment).",
"This simple baseline works surprisingly well compared to the state-of-the-art methods (as demonstrated by our experiments).",
"Applying the consistency constraint means we have two learning signals for the policy: The one from the reinforcement learning loss (to maximize return) and the other due to the consistency constraint.",
"Our approach is different from just learning a k-step prediction model as in our case, the agent's behavior (i.e the agent's policy) is directly dependent on its dynamics model too.",
"The model and the policy are trained jointly to ensure that the predictions from the dynamics model are consistent with the observation from the environment.",
"This provides a mechanism where learning a model can itself change the policy (thus “interacting” with the policy).",
"In the standard case, the policy is optimized only using the RL gradient which aims at maximizing expected reward.",
"The state transition pairs (collected as the agent acts in the environment) become the supervising dataset for learning the model, and hence the policy is not affected when the model is being updated and there is no feedback from the model learning process to the policy.",
"Hence, the data used for training the model is coming from a policy which is trained independently of how well the model performs on the collected trajectories and the process of learning the model has no control over what kind of data is produced for its training."
],
[
"Related Work",
"Model based RL A large portion of the literature in policy search relies on the model-free methods, where no prior knowledge of the environment is required to find an optimal policy, through either policy improvement (value-based methods, [25], [18]), or direct policy optimization (policy gradient methods, [20], [26]).",
"Although conceptually simple, these algorithms have a high sample complexity.",
"To improve their sample-efficiency, one can learn a model of the environment alongside the policy, to sample experience from.",
"PILCO [9] is a model-based method that learns a probabilistic model of the dynamics of the environment and incorporates the uncertainty provided by the model for planning on long-term horizons.",
"This model of the dynamics induces a bias on the policy search though.",
"Previous work has tried to address the model-bias issue of model-based methods, by having a way to characterize the uncertainty of the models, and by learning a more robust policy [9], [24], [17].",
"Model Predictive Control [16] has also been proposed in the literature to account for imperfect models by re-planning at each step, but it suffers from a high computational cost.",
"There is no sharp separation between model-free and model-based reinforcement learning, and often model-based methods are used in conjunction with model-free algorithms.",
"One of the earliest examples of this interaction is the classic Dyna algorithm [30], which takes advantage of the model of the environment to generate simulated experiences, which get included in the training data of a model-free algorithm (like Q-learning, with Dyna-Q).",
"Extensions of Dyna have been proposed [29], [31], including deep neural-networks as function approximations.",
"Recently, the Model-assisted Bootstrapped DDPG [14] was proposed to incorporate model-based rollouts into a Deep Deterministic Policy Gradient method.",
"Recently, [33] used a predictive model in Imagination-Augmented Agents to provide additional context to a policy network.",
"We propose to ensure consistency between the open-loop and the closed-loop pathways as a means to learn a stronger policy, and a better dynamics model.",
"As such, our approach can be applied to a wide range of existing RL setups.",
"Several works have incorporated auxiliary loses which results in representations which can generalize.",
"[13] considered pseudo reward functions which help to generalize effectively across different Atari games.",
"In this work, we propose to use the consistency loss for improving both the policy and the dynamics model in the context of reinforcement learning."
],
[
"Experimental Results",
"Our empirical protocol is designed to evaluate how well the proposed Consistent Dynamics model compares against the state-of-the-art approaches for observation space models and state space models - in terms of both the sample complexity and the asymptotic performance.",
"We consider Mujoco based environments (observation space models) from RLLab with [22] as the baseline, Mujoco based tasks from OpenAI gym (state space models) with [5] as the baseline and Atari games from OpenAI gym with A2C as the baseline.",
"All the results are reported after averaging over 3 random seeds.",
"Note that even though [5] is a state-of-the-art model, our simplistic approach outperforms it."
],
[
"Observation Space Models",
"We use the hybrid Model-based and Model-free (Mb-Mf) algorithm [22] as the baseline model for the observation space models.",
"In this setup, the policy and the dynamics model are learned jointly.",
"The implementation details for these models have been described in the appendix and how to add the consistency loss to the baseline has been described in section REF .",
"We quantify the advantage of using consistency constraint by considering 4 classical Mujoco environments from RLLab [11]: Ant ($S \\in R^{41}$ , $A\\in R^{8}$ ), Humanoid ($S \\in R^{142}$ , $A\\in R^{21}$ ), Half-Cheetah ($S \\in R^{23}$ , $A\\in R^{6}$ ) and Swimmer ($S \\in R^{17}$ , $A\\in R^{3}$ ).",
"For computing the consistency loss, the learner's dynamics model is unrolled for $k=20$ steps.",
"The imagined state transitions and the actual state transitions are encoded into fixed length real vectors using GRU [6].",
"We report the effect of changing the unrolling length $k$ ."
],
[
"Average Episodic Return",
"The average episodic return (and the average discounted episodic return) is a good estimate of the effectiveness of the jointly trained dynamics model and policy.",
"To show that the consistency constraint helps in learning a more powerful policy and a better dynamics model, we compare the average episodic rewards for the baseline Mb-Mf model (which does not use the consistency loss) and the proposed consistent dynamics model (Mb-Mf model + consistency loss).",
"We expect that using consistency would either lead to higher rewards or improve sample efficiency.",
"Figure REF compares the average episodic returns for the agents trained with and without consistency.",
"We observe that using consistency helps to learn a better policy in fewer updates for all the four environments.",
"A similar trend is obtained for the average discounted returns (as shown in the appendix.",
"Since we are learning both the policy and the model of the environment at the same time, these results indicate that using the consistency constraint helps to jointly learn a more powerful policy and a better dynamics model."
],
[
"Effect of changing $k$",
"During the open-loop setup, the dynamics model is unrolled for $k$ steps.",
"The choice of $k$ could be an important hyper-parameter to control the effect of consistency constraint.",
"We study the effect of changing $k$ (during training) on the average episodic return for the Ant and Humanoid tasks, by training the agents with $k \\in \\lbrace 5, 20\\rbrace $ .",
"As an ablation, we also include the case of training the policy without using a model, in a fully model-free fashion.",
"We would expect that a smaller value of $k$ would push the average episodic return of the consistent dynamics model closer to the Mb-Mf case.",
"Figure REF shows that a higher value of $k$ ($k=20$ ) leads to better returns for both tasks.",
"Figure: Average episodic return on Ant and Humanoid environments, for a model-free agent, the Mb-Mf agent without any consistency constraint, and the Consistent Dynamics (Mb-Mf + consistency constraint) that are trained with a consistency constraint over time horizons of length 5 and 20.",
"Note that the results are averaged over 100 batches for both Ant and Humanoid."
],
[
"State Space Models",
"We use the state-of-the-art Learning to Query model [5] as the baseline state space model.",
"We train an expert policy for sampling high-reward trajectories from the environment.",
"The trajectories are used to train the policy $\\pi _\\phi $ using imitation learning and the dynamics model by maximum likelihood.",
"The details about the training setup are described in the appendix and how to add the consistency loss to the baseline has been described in section REF .",
"We consider 3 continuous control tasks from the OpenAI Gym suite [4]: Half-Cheetah, Fetch-Push [23] and Reacher.",
"During the open loop, the dynamics model is unrolled for $k=10$ steps for Half-Cheetah and $k=5$ for Fetch-Push and Reacher."
],
[
"Evaluating Dynamics Models",
"We want to show that the consistency constraint helps to learn a better dynamics model of the environment.",
"Since we learn a dynamics model over the states, we also need to jointly learn an observation model (decoder, see appendix) conditioned on the states.",
"We can then compute the log-likelihood of trajectories in the real environment (sampled with the expert policy) under this observation model.",
"We compare the log-likelihoods corresponding to these observations for the Learning to Query agent (trained without the consistency loss) and Consistent Dynamics agent (trained with the consistency loss).",
"We expect that the Consistent Dynamics agent would achieve a higher log likelihood.",
"Figure REF shows that in terms of imagination log likelihood, Consistent Dynamics agent (ie Learning to Query agent with consistency loss) outperforms the Learning to Query agent for all the 3 environments indicating that the agent learns a more powerful dynamics model of the environment.",
"Note that in the case of Fetch-Push and Reacher, we see improvements in the log-likelihood, even though the dynamics model is unrolled for just 5 steps.",
"Figure: Frostbite"
],
[
"Robustness to Compounding Errors",
"We also investigate the robustness of the proposed approach in terms of compounding errors.",
"When we use the recurrent dynamics model for prediction, the ground-truth sequence is not available for conditioning.",
"This leads to problems during sampling as even small prediction errors can compound when sampling for a large number of steps.",
"We evaluate the proposed model for robustness by predicting the future for much longer timesteps (50 timesteps) than it was trained on (10 timesteps).",
"More generally, in figure REF , we demonstrate that this auxiliary cost helps to learn a better model with improved long-term dependencies by using a training objective that is not solely focused on predicting the next observation, one step at a time.",
"Figure: Comparison of the imagination log likelihood for the Consistent Dynamics agent and Learning to Query agent for Half-Cheetah.",
"The agents were trained with sequence length of 10 but during testing, the dynamics models were evaluated for length 50.",
"The bars represents the values corresponding to the trained agent, averaged over the last 50 batches of training.",
"Using consistency constraint leads to an improved dynamics model (as it achieves better log-likelihood)"
],
[
"Atari Environment",
"We also evaluate our proposed consistency loss on a number of Atari games [2] using A2C as the baseline model and by adding consistency loss to A2C to obtain the Consistent Dynamics model.",
"Specifically, we consider four environments - Seaquest, Breakout, MsPacman, and Frostbite and show that in all the 4 environments, the proposed approach is more sample efficient as compared to a vanilla A2C approach thus demonstrating the applicability of our approach to different environments and learning algorithms."
],
[
"Conclusion",
"In this paper, we formulate a way to ensure consistency between the predictions of a dynamics model and the real observations from the environment thus allowing the agent to learn powerful policies, as well as better dynamics models.",
"The learning agent, in parallel, (i) builds a model of the environment and (ii) engages in an interaction with the environment.",
"This results in two sequences of state transitions: one in the real environment where the agent actually performs actions and other in the agent's dynamics model (or the “world”) where it imagines taking actions.",
"We apply an auxiliary loss which encourages the behavior of state transitions across the two sequences to be indistinguishable from each other.",
"We evaluate our proposed approach for both observation space models, and state space models and show that the agent learns a more powerful policy and a better generative model.",
"Future work would consider how these two interaction pathways could lead to more targeted exploration.",
"Furthermore, having more flexibility over the length over which we unroll the model could allow the agent to take these decisions over multiple timescales."
],
[
"Acknowledgements",
"The authors acknowledge the important role played by their colleagues at Mila throughout the duration of this work.",
"The authors would like to thank Bhairav Mehta, Gautham Swaminathan, Koustuv Sinha and Jonathan Binas for their feedback on the initial manuscript.",
"The authors are grateful to NSERC, CIFAR, Google, Samsung, Nuance, IBM, Canada Research Chairs, Canada Graduate Scholarship Program, Nvidia for funding, and Compute Canada for computing resources.",
"We are very grateful to Google for giving Google Cloud credits used in this project.",
"Appendix Environment Model Observation Space Model We use the experimental setup, environments and the hybrid model-based and model-free (Mb-Mf) algorithm as described in [22]Code available here: https://github.com/nagaban2/nn_dynamics.",
"We consider two training scenarios: training a model-based learning agent with and without the consistency constraint.",
"The consistency constraint is applied by unrolling the model for multiple steps using the observations predicted by the learner's dynamics model (closed-loop setup).",
"We train an on-policy RL algorithm for Cheetah, Humanoid, Ant and Swimmer tasks from RLLab [11] control suite.",
"We report both the average discounted and average un-discounted reward obtained by the learner in the two cases: with and without the use of consistency constraint.",
"The model and policy architectures for the observation space models are as follows: Transition Model: The transition model $\\hat{f}_{\\theta }(s_t, a_t)$ has a Gaussian distribution with diagonal covariance, where the mean and covariance are parametrized by MLPs [26], which maps an observation vector $s_t$ and an action vector $a_t$ to a vector $\\mu $ which specifies a distribution over observation space.",
"During training, the log likelihood $p(s|\\mu )$ is maximized and state-representations can be sampled from $p(s|\\mu )$ .",
"Policy: The learner's policy $\\hat{\\pi }_{\\phi }(s_t)$ is also a Gaussian MLP which maps an observation vector $s$ to a vector $\\mu _{policy}$ which specifies a distribution over action space.",
"Like before, the log-likelihood $p(a|\\mu )$ is maximized and actions can be sampled from $p(a|\\mu )$ .",
"Learner's policy and the dynamics model are implemented as Gaussian policies with MLPs as function approximations and are trained using TRPO [26].",
"Following the hybrid Mb-Mf approach [22], we normalize the states and actions.",
"The dynamics model is trained to predict the change in state $\\Delta s_t$ as it can be difficult to learn the state transition function when the states $s_t$ and $s_{t+1}$ are very similar and the action $a_t$ has a small effect on the output.",
"Figure: Open-loop and closed-loop pathways in the Observation Space Models.",
"The consistency constraint aims to make the behaviour of the open loop predictions indistinguishable from the close loop behaviourFigure: Open-loop and closed-loop pathways in the State Space Models.",
"The consistency constraint aims to make the behaviour of the open loop predictions indistinguishable from the close loop behaviour State Space Model We use the state-of-the-art Learning to Query model [5] as our state space model.",
"The model and policy architecture for the state space models are as follows: Encoder: The learner encodes the pixel-space observations ($64 \\times 64 \\times 3 $ ) from the environment into state-space observations (256 dimensional vectors) with a convolutional encoder (4 convolutional layers with $4 \\times 4$ kernels, stride 2 and 64 channels).",
"To model the velocity information, a stack of the latest 4 frames is used as the observation.",
"The pixel-space observation at time $t-1$ is denoted as $o_{t-1}$ , and is encoded into state-space observation $s_{t-1}$ .",
"Transition Model: The transition model is a Long Short-Term Memory model [12], that predicts the transitions in the state space.",
"For every time-step $t$ , latent variables $z_t$ are introduced, whose distribution is a function of previous state-space observation $s_{t-1}$ and previous action $a_{t-1}$ .",
"ie $z_t \\sim p(z_t | s_{t-1}, a_{t-1})$ .",
"The output of the transition model is then a deterministic function of $z_t, s_{t-1},$ and $a_{t-1}$ .",
"ie $s_t = f(z_t, s_{t-1}, a_{t-1})$ .",
"Stochastic Decoder: The learner can decode the state-space observations back into the pixel-space observations by use of stochastic convolutional decoder.",
"The decoder takes as input the current state-space observation $s_t$ and the current latent variable $z_t$ and generates the current observation-space distribution from which the learner could sample an observation.",
"ie $o_{t+1} \\sim p(o_{t+1} | s_t, z_t)$ .",
"This observation model is Gaussian, with a diagonal covariance.",
"In the closed-loop trajectory, when the learner cannot interact with the environment, the latent variables are sampled from the prior distribution $p(z_t | s_{t-1}, a_{t-1})$ .",
"The latent variables are sampled from Normal distributions with diagonal covariance matrices.",
"Since we cannot compute the log-likelihood ${L(\\theta )}$ in a closed form for the latent variable models, we minimize the evidence lower bound $\\textrm {ELBO}(p_{posterior}) \\le L(\\theta )$ .",
"As discussed previously, the consistency constraint is applied between the open-loop and closed-loop predictions with the aim of making their behavior as similar as possible.",
"Figure REF shows a graphical representation of the open-loop and close-loop pathways in the state-space model.",
"Figure: Comparison of the average episodic discounted rewards, for agents trained with and without consistency for the Ant, Humanoid, Half-Cheetah and Swimmer environments (respectively).",
"Using consistency constraint leads to better rewards in a fewer number of updates for all the cases.",
"Vertical lines in the rightmost figure show the points of saturation with an equal return.",
"Note that the results are averaged over 100 batches for Ant, Humanoid and Half-Cheetah and 10 batches for Swimmer.Expert policy Having access to some policy trained on a large number of experience is required to sample high-quality trajectories with pixel-observations.",
"To train these expert policies, we used policy-based methods such as Proximal Policy Optimization [28] for the half-cheetah and reacher environments, or Deep Deterministic Policy Gradient with Hindsight Experience Replay [1] for the pushing task.",
"The architectures and hyper-parameters used are similar to the ones given by the Baselines library [10].",
"Note that these expert policies were trained on the state representation of the agents (ie.",
"the positions and velocities of their joints), while the trajectories were generated with pixel-observations captured from a view external to the agent.",
"Figure: Comparison of the imagination log likelihood for the open loop setup for Consistent Dynamics agent and Learning to Query agent.",
"The plots correspond toHalf-Cheetah, Reacher and Fetch-Push environments respectively.",
"The bars represent the values corresponding to the trained agent, averaged over the last 50 batches of training.",
"Using consistency constraint leads to a better dynamics model for all the 3 environments.Figure: Comparison of the imitation learning loss for the Consistent Dynamics agent and Learning to Query agent.",
"The plots correspond toHalf-Cheetah, Reacher and Fetch-Push environments (respectively).",
"The bars represents the values corresponding to the trained agent, averaged over the last 50 batches of training.",
"Using consistency constraint leads to a more powerful policy.",
"Results Observation Space Models Figure REF compares the average discounted episodic returns for the agents trained with and without consistency for the observation space models.",
"We observe that using consistency helps to learn a better policy in fewer updates for all the four environments.",
"Since we are learning both the policy and the model of the environment at the same time, these results indicate that using the consistency constraint helps to jointly learn a more powerful policy and a better dynamics model.",
"State Space Models Figure REF shows that in terms of imagination log likelihood, Consistent Dynamics agent (ie Learning to Query agent with consistency loss) outperforms the Learning to Query agent for all the 3 environments indicating that the agent learns a more powerful dynamics model of the environment.",
"Note that in the case of Fetch-Push and Reacher, we see improvements in the log-likelihood, even though the dynamics model is unrolled for just 5 steps.",
"For the state-space models, we use the expert trajectories to train our policy $\\pi _\\phi $ via imitation learning.",
"To show that consistency constraint helps to learn a more powerful policy, we compare the imitation learning loss for the Consistent Dynamics agent (Learning to Query agent with consistency loss) and the baseline (Learning to Query agent) in figure REF and observe that the proposed model has a lower imitation learning loss as compared to the baseline model."
],
[
"Observation Space Model",
"We use the experimental setup, environments and the hybrid model-based and model-free (Mb-Mf) algorithm as described in [22]Code available here: https://github.com/nagaban2/nn_dynamics.",
"We consider two training scenarios: training a model-based learning agent with and without the consistency constraint.",
"The consistency constraint is applied by unrolling the model for multiple steps using the observations predicted by the learner's dynamics model (closed-loop setup).",
"We train an on-policy RL algorithm for Cheetah, Humanoid, Ant and Swimmer tasks from RLLab [11] control suite.",
"We report both the average discounted and average un-discounted reward obtained by the learner in the two cases: with and without the use of consistency constraint.",
"The model and policy architectures for the observation space models are as follows: Transition Model: The transition model $\\hat{f}_{\\theta }(s_t, a_t)$ has a Gaussian distribution with diagonal covariance, where the mean and covariance are parametrized by MLPs [26], which maps an observation vector $s_t$ and an action vector $a_t$ to a vector $\\mu $ which specifies a distribution over observation space.",
"During training, the log likelihood $p(s|\\mu )$ is maximized and state-representations can be sampled from $p(s|\\mu )$ .",
"Policy: The learner's policy $\\hat{\\pi }_{\\phi }(s_t)$ is also a Gaussian MLP which maps an observation vector $s$ to a vector $\\mu _{policy}$ which specifies a distribution over action space.",
"Like before, the log-likelihood $p(a|\\mu )$ is maximized and actions can be sampled from $p(a|\\mu )$ .",
"Learner's policy and the dynamics model are implemented as Gaussian policies with MLPs as function approximations and are trained using TRPO [26].",
"Following the hybrid Mb-Mf approach [22], we normalize the states and actions.",
"The dynamics model is trained to predict the change in state $\\Delta s_t$ as it can be difficult to learn the state transition function when the states $s_t$ and $s_{t+1}$ are very similar and the action $a_t$ has a small effect on the output.",
"Figure: Open-loop and closed-loop pathways in the Observation Space Models.",
"The consistency constraint aims to make the behaviour of the open loop predictions indistinguishable from the close loop behaviourFigure: Open-loop and closed-loop pathways in the State Space Models.",
"The consistency constraint aims to make the behaviour of the open loop predictions indistinguishable from the close loop behaviour"
],
[
"State Space Model",
"We use the state-of-the-art Learning to Query model [5] as our state space model.",
"The model and policy architecture for the state space models are as follows: Encoder: The learner encodes the pixel-space observations ($64 \\times 64 \\times 3 $ ) from the environment into state-space observations (256 dimensional vectors) with a convolutional encoder (4 convolutional layers with $4 \\times 4$ kernels, stride 2 and 64 channels).",
"To model the velocity information, a stack of the latest 4 frames is used as the observation.",
"The pixel-space observation at time $t-1$ is denoted as $o_{t-1}$ , and is encoded into state-space observation $s_{t-1}$ .",
"Transition Model: The transition model is a Long Short-Term Memory model [12], that predicts the transitions in the state space.",
"For every time-step $t$ , latent variables $z_t$ are introduced, whose distribution is a function of previous state-space observation $s_{t-1}$ and previous action $a_{t-1}$ .",
"ie $z_t \\sim p(z_t | s_{t-1}, a_{t-1})$ .",
"The output of the transition model is then a deterministic function of $z_t, s_{t-1},$ and $a_{t-1}$ .",
"ie $s_t = f(z_t, s_{t-1}, a_{t-1})$ .",
"Stochastic Decoder: The learner can decode the state-space observations back into the pixel-space observations by use of stochastic convolutional decoder.",
"The decoder takes as input the current state-space observation $s_t$ and the current latent variable $z_t$ and generates the current observation-space distribution from which the learner could sample an observation.",
"ie $o_{t+1} \\sim p(o_{t+1} | s_t, z_t)$ .",
"This observation model is Gaussian, with a diagonal covariance.",
"In the closed-loop trajectory, when the learner cannot interact with the environment, the latent variables are sampled from the prior distribution $p(z_t | s_{t-1}, a_{t-1})$ .",
"The latent variables are sampled from Normal distributions with diagonal covariance matrices.",
"Since we cannot compute the log-likelihood ${L(\\theta )}$ in a closed form for the latent variable models, we minimize the evidence lower bound $\\textrm {ELBO}(p_{posterior}) \\le L(\\theta )$ .",
"As discussed previously, the consistency constraint is applied between the open-loop and closed-loop predictions with the aim of making their behavior as similar as possible.",
"Figure REF shows a graphical representation of the open-loop and close-loop pathways in the state-space model.",
"Figure: Comparison of the average episodic discounted rewards, for agents trained with and without consistency for the Ant, Humanoid, Half-Cheetah and Swimmer environments (respectively).",
"Using consistency constraint leads to better rewards in a fewer number of updates for all the cases.",
"Vertical lines in the rightmost figure show the points of saturation with an equal return.",
"Note that the results are averaged over 100 batches for Ant, Humanoid and Half-Cheetah and 10 batches for Swimmer."
],
[
"Expert policy",
"Having access to some policy trained on a large number of experience is required to sample high-quality trajectories with pixel-observations.",
"To train these expert policies, we used policy-based methods such as Proximal Policy Optimization [28] for the half-cheetah and reacher environments, or Deep Deterministic Policy Gradient with Hindsight Experience Replay [1] for the pushing task.",
"The architectures and hyper-parameters used are similar to the ones given by the Baselines library [10].",
"Note that these expert policies were trained on the state representation of the agents (ie.",
"the positions and velocities of their joints), while the trajectories were generated with pixel-observations captured from a view external to the agent.",
"Figure: Comparison of the imagination log likelihood for the open loop setup for Consistent Dynamics agent and Learning to Query agent.",
"The plots correspond toHalf-Cheetah, Reacher and Fetch-Push environments respectively.",
"The bars represent the values corresponding to the trained agent, averaged over the last 50 batches of training.",
"Using consistency constraint leads to a better dynamics model for all the 3 environments.Figure: Comparison of the imitation learning loss for the Consistent Dynamics agent and Learning to Query agent.",
"The plots correspond toHalf-Cheetah, Reacher and Fetch-Push environments (respectively).",
"The bars represents the values corresponding to the trained agent, averaged over the last 50 batches of training.",
"Using consistency constraint leads to a more powerful policy."
],
[
"Observation Space Models",
"Figure REF compares the average discounted episodic returns for the agents trained with and without consistency for the observation space models.",
"We observe that using consistency helps to learn a better policy in fewer updates for all the four environments.",
"Since we are learning both the policy and the model of the environment at the same time, these results indicate that using the consistency constraint helps to jointly learn a more powerful policy and a better dynamics model."
],
[
"State Space Models",
"Figure REF shows that in terms of imagination log likelihood, Consistent Dynamics agent (ie Learning to Query agent with consistency loss) outperforms the Learning to Query agent for all the 3 environments indicating that the agent learns a more powerful dynamics model of the environment.",
"Note that in the case of Fetch-Push and Reacher, we see improvements in the log-likelihood, even though the dynamics model is unrolled for just 5 steps.",
"For the state-space models, we use the expert trajectories to train our policy $\\pi _\\phi $ via imitation learning.",
"To show that consistency constraint helps to learn a more powerful policy, we compare the imitation learning loss for the Consistent Dynamics agent (Learning to Query agent with consistency loss) and the baseline (Learning to Query agent) in figure REF and observe that the proposed model has a lower imitation learning loss as compared to the baseline model."
]
] | 1906.04355 |
[
[
"Quantifying Intrinsic Uncertainty in Classification via Deep Dirichlet\n Mixture Networks"
],
[
"Abstract With the widespread success of deep neural networks in science and technology, it is becoming increasingly important to quantify the uncertainty of the predictions produced by deep learning.",
"In this paper, we introduce a new method that attaches an explicit uncertainty statement to the probabilities of classification using deep neural networks.",
"Precisely, we view that the classification probabilities are sampled from an unknown distribution, and we propose to learn this distribution through the Dirichlet mixture that is flexible enough for approximating any continuous distribution on the simplex.",
"We then construct credible intervals from the learned distribution to assess the uncertainty of the classification probabilities.",
"Our approach is easy to implement, computationally efficient, and can be coupled with any deep neural network architecture.",
"Our method leverages the crucial observation that, in many classification applications such as medical diagnosis, more than one class labels are available for each observational unit.",
"We demonstrate the usefulness of our approach through simulations and a real data example."
],
[
"Introduction",
"Deep neural networks have been achieving remarkable success in a wide range of classification tasks in recent years.",
"Accompanying increasingly accurate prediction of the classification probability, it is of equal importance to quantify the uncertainty of the classification probability produced by deep neural networks.",
"Without a careful characterization of such an uncertainty, the prediction of deep neural networks can be questionable, unusable, and in the extreme case incur considerable loss [22].",
"For example, deep reinforcement learning suffers from a strikingly low reproducibility due to high uncertainty of the predictions [7].",
"Uncertainty quantification can be challenging though; for instance, [6] argued that modern neural networks architectures are poor in producing well-calibrated probability in binary classification.",
"Recognizing such challenges, there have been recent proposals to estimate and quantify the uncertainty of output from deep neural networks, and we review those methods in Section REF .",
"Despite the progress, however, uncertainty quantification of deep neural networks remains relatively underdeveloped [10].",
"In this paper, we propose deep Dirichlet mixture networks to produce, in addition to a point estimator of the classification probabilities, an associated credible interval (region) that covers the true probabilities at a desired level.",
"We begin with the binary classification problem and employ the Beta mixture model to approximate the probability distribution of the true but random probability.",
"We then extend to the general multi-class classification using the Dirichlet mixture model.",
"Our key idea is to view the classification probability as a random quantity, rather than a deterministic value in $[0,1]$ .",
"We seek to estimate the distribution of this random quantity using the Beta or the Dirichlet mixture, which we show is flexible enough to model any continuous distribution on $[0,1]$ .",
"We achieve the estimation by adding an extra layer in a typical deep neural network architecture, without having to substantially modify the overall structure of the network.",
"Then based on the estimated distribution, we produce both a point estimate and a credible interval for the classification probability.",
"This credible interval provides an explicit quantification of the classification variability, and can greatly facilitate our decision making.",
"For instance, a point estimate of high probability to have a disease may be regarded as lack of confidence if the corresponding credible interval is wide.",
"By contrast, a point estimate with a narrow credible interval may be seen as a more convincing diagnosis.",
"The feasibility of our proposal is built upon a crucial observation that, in many classification applications such as medical diagnosis, there exist more than one class labels.",
"For instance, a patient's computed tomography image may be evaluated by two doctors, each giving a binary diagnosis of existence of cancer.",
"In Section , we illustrate with an example of diagnosis of Alzheimer's disease (AD) using patients' anatomical magnetic resonance imaging.",
"For each patient, there is a binary diagnosis status as AD or healthy control, along with additional cognitive scores that are strongly correlated with and carry crucial information about one's AD status.",
"We thus consider the dichotomized version of the cognitive scores, combine them with the diagnosis status, and feed them together into our deep Dirichlet mixture networks to obtain a credible interval of the classification probability.",
"We remark that, existence of multiple labels is common rather than an exception in a variety of real world applications.",
"Our proposal provides a useful addition to the essential yet currently still growing inferential machinery to deep neural networks learning.",
"Our method is simple, fast, effective, and can couple with any existing deep neural network structure.",
"In particular, it adopts a frequentist inference perspective, but produces a Bayesian-style outcome of credible intervals."
],
[
"Related Work",
"There has been development of uncertainty quantification of artificial neural networks since two decades ago.",
"Early examples include the delta method [9], and the Bootstrap methods [4], [8], [2].",
"However, the former requires computing the Hessian matrix and is computationally expensive, whereas the latter hinges on an unbiased prediction.",
"When the prediction is biased, the total variance is to be underestimated, which would in turn result in a narrower credible interval.",
"Another important line of research is Bayesian neural networks [14], [15], which treat model parameters as distributions, and thus can produce an explicit uncertainty quantification in addition to a point estimate.",
"The main drawback is the prohibitive computational cost of running MCMC algorithms.",
"There have been some recent proposals aiming to address this issue, most notably, [5], [13] that used the dropout tricks.",
"Our proposal, however, is a frequentist solution, and thus we have chosen not to numerically compare with those Bayesian approaches.",
"Another widely used uncertainty quantification method is the mean variance estimation (MVE) approach [16].",
"It models the data noise using a normal distribution, and employs a neural network to output the mean and variance.",
"The optimization is done by minimizing the negative log-likelihood function.",
"It has mainly been designed for regression tasks, and is less suitable for classification.",
"There are some more recent proposals of uncertainty quantification.",
"One is the lower and upper bound estimation (LUBE) [11], [19].",
"LUBE has been proven successful in numerous applications.",
"However, its loss function is non-differentiable and gradient descent cannot be applied for optimization.",
"The quality-driven prediction interval method (QD) has recently been proposed to improve LUBE [17].",
"It is a distribution-free method by outputting prediction's upper bound and lower bound.",
"The uncertainty can be estimated by measuring the distance between the two bounds.",
"Unlike LUBE, the objective function of QD can be optimized by gradient descent.",
"But similar to MVE, it is designed for regression tasks.",
"Confidence network is another method to estimate confidence by adding an output node next to the softmax probabilities [3].",
"This method is suitable for classification.",
"Although its original goal was for out-of-distribution detection, its confidence can be used to represent the intrinsic uncertainty.",
"Later in Section REF , we numerically compare our method with MVE, QD, and confidence network.",
"We also clarify that our proposed framework is different from the mixture density network [1].",
"The latter trains a neural network to model the distribution of the outcome using a mixture distribution.",
"By contrast, we aim to learn the distribution of the classification probabilities and to quantify their variations."
],
[
"Dirichlet Mixture Networks",
"In this section, we describe our proposed Dirichlet mixture networks.",
"We begin with the case of binary classification, where the Dirichlet mixture models reduce to the simpler Beta mixture models.",
"Although a simpler case, the binary classification is sufficient to capture all the key ingredients of our general approach and thus loses no generality.",
"At the end of this section, we discuss the extension to the multi-class case."
],
[
"Loss Function",
"We begin with a description of the key idea of our proposal.",
"Let $\\lbrace 1, 2\\rbrace $ denote the two classes.",
"Given an observational unit $\\mathbf {x}$ , e.g., an image, we view the probability $p_{\\mathbf {x}}$ that $\\mathbf {x}$ belongs to class 1 as a random variable, instead of a deterministic value in $[0, 1]$ .",
"We then seek to estimate the probability density function $f(p; \\mathbf {x})$ of $p_{\\mathbf {x}}$ .",
"This function encodes the intrinsic uncertainty of the classification problem.",
"A point estimate of the classification probability only focuses on the mean, $\\int _0^1 f(p; \\mathbf {x}) \\mathrm {d}p$ , which is not sufficient for an informed decision making without an explicit quantification of its variability.",
"For example, it can happen that, for two observational units $\\mathbf {x}$ and $\\mathbf {x}^{\\prime }$ , their mean probabilities, and thus their point estimates of the classification probability, are the same.",
"However, the densities are far apart from each other, leading to completely different variabilities, and different interpretations of the classification results.",
"Figure REF shows an illustration.",
"Our proposal then seeks to estimate the density function $f(p; \\mathbf {x})$ for each $\\mathbf {x}$ .",
"Figure: Illustration of the setting.",
"Two or more labels are generated with the same probability p 𝐱 p_{\\mathbf {x}}, which is randomly drawn from a distribution that we wish to estimate.A difficulty arising from this estimation problem is that $f$ in general can be any density function on $[0, 1]$ .",
"To address this, we propose to simplify the problem by restricting to the case where $f$ is a Beta mixture; i.e., $f(p; \\mathbf {x}) = \\sum _{k=1}^K w^k \\frac{p^{\\alpha ^k_1 -1} (1-p)^{\\alpha ^k_2 -1}}{\\mathrm {Beta}(\\alpha ^k_1, \\alpha ^k_2)},$ where $\\mathrm {Beta}(\\cdot , \\cdot )$ is the Beta function, and the parameters $w^k, \\mathbf {\\alpha }^k = (\\alpha ^k_1, \\alpha ^k_2)$ are smooth functions of $\\mathbf {x}$ , $k = 1, \\ldots , K$ .",
"The weights $w^k$ satisfy that $w^1 + \\cdots + w^K = 1$ .",
"Later we show that this Beta mixture distribution is flexible enough to adequately model almost any distribution $f$ on $[0,1]$ .",
"With the form of density function (REF ) in place, our goal then turns to estimate the positive parameters $\\alpha _1^k, \\alpha _2^k$ , and $w^k$ .",
"To do so, we derive the loss function that is to be minimized by deep neural networks.",
"We employ the negative log-likelihood function from (REF ) as the loss function.",
"For the $j$ th observational unit of the training data, $j = 1, \\ldots , n$ , let $\\mathbf {x}_j$ denote the input, e.g., the subject's image scan, and $\\mathbf {y}_j = \\left( y_j^{(1)}, \\ldots , y_j^{(m_j)} \\right)$ denote the vector of labels taking values from $\\lbrace 1, 2\\rbrace $ .",
"Here we assume $m_j \\ge 2$ , reflecting that there are more than one class label for each observational unit.",
"Write $\\mathbf {w}= (w_1, \\ldots , w_K)$ and $\\mathbf {\\alpha }= (\\mathbf {\\alpha }^1, \\ldots , \\mathbf {\\alpha }^K)$ .",
"By integrating out $p$ , the likelihood function for the observed pair $(\\mathbf {x}_j, \\mathbf {y}_j)$ is $\\begin{aligned}&L_j(\\mathbf {w}, \\mathbf {\\alpha }; \\mathbf {x}_j, \\mathbf {y}_j) \\\\&= \\int _0^1 p^{\\sum _{l=1}^{m_j} \\mathbf {1}(y_j^{(l)}= 1)} (1-p)^{\\sum _{l=1}^{m_j} \\mathbf {1}(y_j^{(l)}=2)} f(p; \\mathbf {x}_j) \\mathrm {d}p.\\end{aligned}$ Write $S_{ij} = \\sum _{l=1}^{m_j} \\mathbf {1} \\left( y_j^{(l)} = i \\right)$ , where $\\mathbf {1}(\\cdot )$ is the indicator function, $i = 1, 2$ , $j = 1, \\ldots , n$ , and this term quantifies the number of times $\\mathbf {x}_j$ is labeled $i$ .",
"Plugging (REF ) into $L_j$ , we get $\\begin{aligned}& L_j(\\mathbf {w}, \\mathbf {\\alpha }; \\mathbf {x}_j, \\mathbf {y}_j) \\\\&= \\int _0^1 p^{S_{1j}} (1-p)^{S_{2j}} \\sum _{k=1}^K \\frac{w^k p^{\\alpha ^k_1 -1} (1-p)^{\\alpha ^k_2 -1}}{\\mathrm {Beta}(\\alpha ^k_1, \\alpha ^k_2)} \\mathrm {d}p\\\\&= \\sum _{k=1}^K \\int _0^1 \\frac{w^k}{\\mathrm {Beta}(\\alpha ^k_1, \\alpha ^k_2)} p^{\\alpha ^k_1 -1 + S_{1j}} (1-p)^{\\alpha ^k_2 -1 + S_{2j}} \\mathrm {d}p.\\end{aligned}$ By a basic property of Beta functions, we further get $L_j(\\mathbf {w}, \\mathbf {\\alpha }; x_j, \\mathbf {y}_j) = \\sum _{k=1}^K \\frac{w^k\\mathrm {Beta}(\\alpha ^k_1 + S_{1j}, \\alpha ^k_2 + S_{2j})}{\\mathrm {Beta}(\\alpha ^k_1, \\alpha ^k_2)}.$ Aggregating all $n$ observational units, we obtain the full negative log-likelihood function, $\\begin{aligned}& -\\ell (\\mathbf {w}, \\mathbf {\\alpha }; \\mathbf {x}_1, \\mathbf {y}_1, \\ldots , \\mathbf {x}_n, \\mathbf {y}_n) \\\\& = -\\sum _{j=1}^n \\log \\left[\\sum _{k=1}^K \\frac{w^k\\mathrm {Beta}(\\alpha ^k_1 + S_{1j}, \\alpha ^k_2 + S_{2j})}{\\mathrm {Beta}(\\alpha ^k_1, \\alpha ^k_2)} \\right].\\end{aligned}$ We then propose to employ a deep neural network learner to estimate $\\mathbf {w}$ and $\\mathbf {\\alpha }$ ."
],
[
"Credible Intervals",
"To train our model, we simply replace the existing loss function of a deep neural network, e.g., the cross-entropy, with the negative log-likelihood function given in (2).",
"Therefore, we can take advantage of current deep learning framework such as PyTorch for automatic gradient calculation.",
"Then we use the mini-batch gradient descent to optimize the entire neural network’s weights.",
"Once the training is finished, we obtain the estimate of the parameters of the mixture distribution, $\\lbrace \\mathbf {w}, \\mathbf {\\alpha }\\rbrace $ .",
"One implementation detail to notice is that the Beta function has no closed form derivative.",
"To address this issue, we used fast log gamma algorithm to obtain an approximation of the Beta function, which is available in PyTorch.",
"Also, we applied the softmax function to the weights of the mixtures to ensure that $w_1 + ... + w_K = 1$ , and took the exponential of $\\mathbf {\\alpha }^1, \\ldots \\mathbf {\\alpha }^K$ to ensure that these parameters remain positive as required.",
"Given the estimated parameters $\\widehat{\\mathbf {w}}, \\widehat{\\mathbf {\\alpha }}$ from the deep mixture networks, we next construct the credible interval for explicit uncertainty quantification.",
"For a new observation $\\mathbf {x}_0$ , the estimated distribution of the classification probability $p_{\\mathbf {x}_0}$ takes the form $\\hat{f}(p; \\mathbf {x}_0) = \\sum _{k=1}^K \\hat{w}^k(\\mathbf {x}_0) \\frac{p^{\\hat{\\alpha }^k_1(\\mathbf {x}_0) -1} (1-p)^{\\hat{\\alpha }^k_2(\\mathbf {x}_0) -1}}{\\mathrm {Beta}(\\hat{\\alpha }^k_1(\\mathbf {x}_0), \\hat{\\alpha }^k_2(\\mathbf {x}_0))},$ where we write $\\hat{w}^k, \\hat{\\alpha }^k_1, \\hat{\\alpha }^k_2$ in the form of explicit functions of $\\mathbf {x}_0$ .",
"The expectation of this estimated density $\\int _0^1 \\hat{f}(p; \\mathbf {x}_0) \\mathrm {d}p$ is an approximately unbiased estimator of $p_{\\mathbf {x}_0}$ .",
"Meanwhile, we can construct the two-sided credible interval of $p_{\\mathbf {x}_0}$ with the nominal level $\\alpha \\in (0,1)$ as $\\left[ \\widehat{Q}_{\\frac{\\alpha }{2}}, \\widehat{Q}_{1 - \\frac{\\alpha }{2}} \\right],$ where $\\widehat{Q}_{\\frac{\\alpha }{2}}$ and $\\widehat{Q}_{1 - \\frac{\\alpha }{2}}$ are the $\\alpha /2$ and $1 - \\alpha /2$ quantiles of the estimated density $\\hat{f}(p; \\mathbf {x}_0)$ .",
"Similarly, we can construct the upper and lower credible intervals as $\\left[0, \\widehat{Q}_{1 - \\alpha }\\right], \\text{ and } \\left[\\widehat{Q}_{\\alpha }, 1\\right],$ respectively, where $\\widehat{Q}_{\\alpha }$ and $\\widehat{Q}_{1 - \\alpha }$ are the $\\alpha $ and $1 - \\alpha $ quantiles of the estimated density $\\hat{f}(p; \\mathbf {x}_0)$ .",
"Next we justify our choice of Beta mixture for the distribution of classification probability, by showing that any density function under certain regularity conditions can be approximated well by a Beta mixture.",
"Specifically, denote by $\\mathcal {P}$ the set of all probability density functions $f$ on $[0, 1]$ with at most countable discontinuities that satisfy $\\nonumber \\int _0^1 f(p) \\left|\\log f(p)\\right| \\mathrm {d}p < \\infty .$ It is shown in [20] that any $f \\in \\mathcal {P}$ can be approximated arbitrarily well by a sequence of Beta mixtures.",
"That is, for any $f \\in \\mathcal {P}$ and any $\\epsilon > 0$ , there exists a Beta mixture distribution $f_{\\mathrm {Beta}}$ such that $\\nonumber \\mathrm {D}_{\\mathrm {KL}} \\left(f \\Vert f_{\\mathrm {Beta}} \\right) \\le \\epsilon ,$ where $\\mathrm {D}_{\\mathrm {KL}} (\\cdot \\Vert \\cdot )$ denotes the Kullback-Leibler divergence.",
"This result establishes the validity of approximating a general distribution function using a Beta mixture.",
"The proof of this result starts by recognizing that $f$ can be accurately approximated by piecewise constant functions on $[0,1]$ due to a countable number of discontinuities.",
"Next, each constant piece is a limit of a sequence of Bernstein polynomials, which are infinite Beta mixtures with integer parameters [21], [18]."
],
[
"Multiple-class Classification",
"We next extend our method to the general case of multi-class classification.",
"It follows seamlessly from the prior development except that now the labels $\\mathbf {y}_j = \\left( y_j^{(1)}, \\ldots , y_j^{(m_j)} \\right)$ take values from $\\lbrace 1, 2, \\ldots , d\\rbrace $ , where $d$ is the total number of classes.",
"Given an observation $\\mathbf {x}$ , the multinomial distribution over $\\lbrace 1, 2, \\ldots , d\\rbrace $ is represented by $\\mathbf {p}= (p_1, \\ldots , p_d)$ , which, as a point in the simplex $\\Delta = \\lbrace (c_1, \\ldots , c_d): c_i \\ge 0, c_1 + \\cdots + c_d = 1\\rbrace $ , is assumed to follow a Dirichlet mixture $f(\\mathbf {p}; \\mathbf {x}) = \\sum _{k=1}^K w^k \\frac{1}{\\mathrm {Beta}(\\mathbf {\\alpha }^k)} \\prod _{i=1}^d p_i^{\\alpha ^k_i -1},$ where the generalized Beta function takes the form $\\mathrm {Beta}(\\mathbf {\\alpha }) = \\frac{\\prod _{i=1}^d \\Gamma (\\alpha _i)}{\\Gamma (\\alpha _1 + \\cdots + \\alpha _d)}.$ The likelihood of the $j$ th observation is $L_j = \\int _{\\Delta } \\left( \\prod _{i=1}^d p_i^{S_{ij}} \\right) \\sum _{k=1}^K w^k \\frac{1}{\\mathrm {Beta}(\\mathbf {\\alpha }^k)} \\prod _{i=1}^d p_i^{\\alpha ^k_i -1} \\mathrm {d}\\mathbf {p},$ where $S_{ij} = \\sum _{l=1}^{m_j} \\mathbf {1}\\left( y_j^{(l)}= i \\right)$ .",
"Accordingly, the negative log-likelihood function is $\\footnotesize \\begin{aligned}&-\\ell (\\mathbf {w}, \\mathbf {\\alpha }; \\mathbf {x}_1, \\mathbf {y}_1, \\ldots , \\mathbf {x}_n, \\mathbf {y}_n) \\\\&= -\\sum _{j=1}^n \\log \\left[\\sum _{k=1}^K \\frac{w^k\\mathrm {Beta}\\left( \\alpha ^k_1 + S_{1j}, \\ldots , \\alpha ^k_d + S_{dj} \\right)}{\\mathrm {Beta}(\\mathbf {\\alpha }^k)} \\right].\\end{aligned}\\normalsize $ This is the loss function to be minimized in the Dirichlet mixture networks."
],
[
"Simulations on Coverage Proportion",
"We first investigate the empirical coverage of the proposed credible interval.",
"We used the MNIST handwritten digits data, and converted the ten outcomes (0-9) first to two classes (0-4 as Class 1, and 5-9 as Class 2), then to three classes (0-2 as Class 1, 3-6 as Class 2, and 7-9 as Class 3).",
"In order to create multiple labels for each image, we trained a LeNet-5 [12] to output the classification probability $p_i$ , then sampled multiple labels for the same input image based on a binomial or multinomial distribution with $p_i$ as the parameter.",
"We further divided the simulated data into training and testing sets.",
"We calculated the empirical coverage as the proportion in the testing set that the corresponding $p_i$ falls in the constructed credible interval.",
"We assessed the coverage performance by examining how close the empirical coverage is to the nominal coverage between the interval of 75% and 95%.",
"Ideally, the empirical coverage should be the same as the nominal level.",
"Figure: Empirical coverage of the estimated credible interval for a two-class classification task, with the two-label setting shown in (a), and the three-label setting in (b).",
"The blue line represents the empirical coverage of the estimated credible interval.",
"The orange 45-degree line represents the ideal estimation.",
"The closer the two lines, the better the estimation.Figure REF reports the simulation results for the two-class classification task, where panel (a) is when there are two labels available for each input, and panel (b) is when there are three labels available.",
"The orange 45-degree line represents the ideal coverage.",
"The blue line represents the empirical coverage of the credible interval produced by our method.",
"It is seen that our constructed credible interval covers 98.19% of the truth with the 95% nominal level for the two-label scenario, and 98.17% for the three-label scenario.",
"In general, the empirical coverage is close or slightly larger than the nominal value, suggesting that the credible interval is reasonably accurate.",
"Moreover, the interval becomes more accurate with more labels on each input.",
"Figure: Empirical coverage of the estimated credible interval for a three-class classification task, with the two-label setting shown in (a) and (b), and the three-label setting in (c) and (d).",
"The blue line represents the empirical coverage of the estimated credible interval.",
"The orange 45-degree line represents the ideal estimation.",
"The closer the two lines, the better the estimation.",
"For each graph, the probability is calculated in the one-vs-all fashion; e.g., (a) represents the credible interval of Class 1 versus Classes 2 and 3 combined.Figure REF reports the simulation results for the three-class classification task, where panels (a) and (b) are when there are two labels available, and panels (c) and (d) are when there are three labels available.",
"A similar qualitative pattern is observed in Figure REF as in Figure REF , indicating that our method works well for the three-class classification problem."
],
[
"Comparison with Alternative Methods",
"We next compare our method with three alternatives that serve as the baselines, the confidence network [3], the mean variance estimation (MVE) [16], and the quality-driven prediction interval method (QD) [17].",
"We have chosen those methods as baselines, as they also targeted to quantify the intrinsic variability and represented the most recent state-of-the-art solutions to this problem.",
"Figure: Data is generated from a Bernoulli distribution whose parameter is sampled from a Beta \\mathrm {Beta} distribution with parameter(f 1 f_1, f 2 f_2).",
"(a) and (b) show the 3D landscapes.",
"(c) shows 1,000 samples from this distribution with two labels for each data point.",
"Green means all labels are 1.",
"Red means all labels are 2.",
"Yellow means that labels are a mix of 1 and 2.To facilitate graphical presentation of the results, we simulated the input data $\\mathbf {x}$ from two-dimensional Gaussian mixtures.",
"Specifically, we first sampled $\\mathbf {x}$ from a mixture of two Gaussians with means at $(-2, 2)$ and $(2, -2)$ , and denote its probability density function as $\\psi _1$ .",
"We then sampled $\\mathbf {x}$ from another mixture of two Gaussians with means at $(2, 2)$ and $(-2, -2)$ , and denote its probability density function as $\\psi _2$ .",
"For each Gaussian component, the variance is set at 0.7.",
"We then sampled the probability $p$ of belonging to Class 1 from a Beta distribution with the parameters $\\psi _1 / \\psi _2 + 1$ and $\\psi _2 / \\psi _1 + 1$ .",
"Finally, we sampled the class labels from a Bernoulli distribution with the probability of success $p$ .",
"At each input sample $\\mathbf {x}$ , we sampled two class labels.",
"For a fair comparison, we duplicate the data for the baseline methods that only use one class label.",
"Figure REF (c) shows a scatter plot of 1,000 samples, each with two labels.",
"The green dots correspond to the samples whose class labels are 0 in both replications, the red dots are 1 in both replications, and the yellow dots are those samples whose class labels are different in two replications.",
"Most of the yellow dots are located along the two axis that separate the four quadrants.",
"Figure: Variance contour plots of our approach and baselines.",
"(a) shows the ideal variance plot.",
"(b) is the result of our approach.",
"(c), (d), (e) are the results of baselines.",
"Blue means low data-noise, and yellow means high data-noise.",
"From the results, (b) our approach looks most similar to the ideal.Figure REF reports the contour of the estimated variance.",
"Panel (a) is the true variance contour for the simulated data, obtained numerically from the data generation.",
"It shows that the largest variance occurs along the two axises that separate the four quadrants.",
"Panel (b) is the result of our approach.",
"We used ten mixtures here.",
"The predicted mean and variance were calculated using the law of total expectation and total variance.",
"Our method achieved a 98.4% classification accuracy.",
"More importantly, it successfully captured the variability of the classification probability and produced a variance contour that looks similar to (a).",
"Panel (c) is the result of the mean variance estimation [3].",
"It also achieved a 98.4% classification accuracy, but it failed to correctly characterize the variability.",
"This is partly due to that it models the variability as Gaussian.",
"(d) is the result of the quality-driven prediction interval method [17].",
"It only obtained a 89.1% classification accuracy.",
"As a distribution-free method, it predicted a higher variability in the center, but ignored other highly variable regions.",
"(e) is the result of the confidence network [3].",
"It achieved a 98.1% classification accuracy, a reasonably well variability estimation.",
"Overall, our method achieved the best performance while maintaining a high classification accuracy.",
"Figure: Beta mixture density functions outputted by the neural network.",
"(a) is the result at point (0,0).",
"(b) is the result at point (1,1).",
"Point (0,0) clearly has a higher variance.Figure REF shows the density function of the outputted distributions.",
"At point (0,0), it indeed has a higher variance."
],
[
"Data Description ",
"We illustrate our proposed method on a medical imaging diagnosis application.",
"We remark that, although the example dataset is small in size, with only thousands of image scans, our method is equally applicable to both small and large datasets.",
"Alzheimer's Disease (AD) is the leading form of dementia in elderly subjects, and is characterized by progressive and irreversible impairment of cognitive and memory functions.",
"With the aging of the worldwide population, it has become an international imperative to understand, diagnose, and treat this disorder.",
"The goal of the analysis is to diagnose patients with AD based on their anatomical magnetic resonance imaging (MRI) scans.",
"Being able to provide an explicit uncertainty quantification for this classification task, which is potentially challenging and of a high-risk, is especially meaningful.",
"The dataset we analyzed was obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI).",
"For each patient, in addition to his or her diagnosis status as AD or normal control, two cognitive scores were also recorded.",
"One is the Mini-Mental State Examination (MMSE) score, which examines orientation to time and place, immediate and delayed recall of three words, attention and calculation, language and vision-constructional functions.",
"The other is the Global Clinical Dementia Rating (CDR-global) score, which is a combination of assessments of six domains, including memory, orientation, judgment and problem solving, community affairs, home and hobbies, and personal care.",
"Although MMSE and CDR-global are not used directly for diagnosis, their values are strongly correlated with and carry crucial information about one's AD status.",
"Therefore, we took the dichotomized cognitive scores, and used them as labels in addition to the diagnosis status.",
"We used the ADNI 1-Year 1.5T dataset, with totally 1,660 images.",
"We resized all the images to the dimension $96\\times 96\\times 80$ .",
"The diagnosis contains three classes: normal control (NC), mild cognitive impairment (MCI), and Alzheimer disease (AD).",
"Among them, MCI is a prodromal stage of AD.",
"Since the main motivation is to identify patients with AD, we combined NC and MCI as one class, referred as NC+MCI, and AD as the other class, and formulated the problem as a binary classification task.",
"We used three types of assessments to obtain the three classification labels: the doctor's diagnostic assessment, the CDR-global score, and the MMSE score.",
"For the CDR-global score, we used 0 for NC, 0.5 for MCI, and 1 for AD.",
"For the MMSE score, we used 28-30 as NC, 24-27 as MCI, and 0-23 as AD.",
"Table REF summarizes the number of patients in each class with respect to the three different assessments.",
"Table: Detailed patient statistics"
],
[
"Classifier and Results",
"Figure REF describes the architecture of our neural network based classifier.",
"We used two consecutive 3D convolutional filters followed by max pooling layers.",
"The input is $96\\times 96\\times 80$ image.",
"The first convolutional kernel size is $ 5\\times 5\\times 5$ , and the max pooling kernel is $5\\times 5\\times 5$ .",
"The second convolutional kernel is $3\\times 3\\times 3$ , and the following max pooling kernel is $2\\times 2\\times 2$ .",
"We chose sixteen as the batch size, and $1e-6$ as the learning rate.",
"We chose a $K=3$ -component Beta mixture.",
"Figure: Architecture of the neural network used in the real data experiment.We randomly selected 90% of the data for training and the remaining 10% for testing.",
"We plotted the credible interval of all the 166 testing subjects with respect to their predicted probability of having AD or not in Figure REF (a).",
"We then separated the testing data into three groups: the subjects with their assessments unanimously labeled as NC+MCI (green dots), the subjects with their assessments unanimously labeled as AD (red dots), and the subjects with their assessments with a mix of NC+MCI and AD (blue dots, and referred as MIX).",
"Figure: Credible intervals constructed in the real data experiment.We observe that, for the patients in the NC+MCI category, 95% of them were estimated to have a smaller than 0.1 probability of being AD, and a tight credible interval with the width smaller than 0.15.",
"We further randomly selected five patients in the NC+MCI category and plotted their credible intervals in Figure REF (b).",
"Each has a close to 0 probability of having AD, and each with a tight credible interval.",
"For patients in the AD category, most exhibit the same pattern of having a tight credible interval, with a few potential outliers.",
"For the patients in the MIX category, we randomly selected five patients and plotted their predicted classification probability with the associated credible interval in Figure REF (c).",
"We see that Subject 4 was classified as AD with only 0.45 probability but has a large credible interval of width 0.3.",
"We took a closer look at this subject, and found that the wide interval may be due to inaccurate labeling.",
"The threshold value we applied to dichotomize the MMSE score was 23, in that a subject with the MMSE below or equal to 23 is classified as AD.",
"Subject 4 happens to be on the boundary line of 23.",
"This explains why the classifier produced a wide credible interval.",
"In Figure REF (a), we also observe that the classifier is less confident in classifying the patients in the MIX category, in that almost all the blue dots are above the 0.15 credible interval.",
"We again randomly selected five patients in the MIX category and plotted their predicted classification probabilities with the corresponding credible intervals in Figure REF (d).",
"Comparing to Figure REF (b), the credible intervals for patients in the MIX category are much wider than those in the unanimous NC+MCI category."
],
[
"Conclusion",
"We present a new approach, deep Dirichlet mixture networks, to explicitly quantify the uncertainty of the classification probability produced by deep neural networks.",
"Our approach, simple but effective, takes advantage of the availability of multiple class labels for the same input sample, which is common in numerous scientific applications.",
"It provides a useful addition to the inferential machinery for deep neural networks based learning.",
"There remains several open questions for future investigation.",
"Methodologically, we currently assume that multiple class labels for each observational sample are of the same quality.",
"In practice, different sources of information may have different levels of accuracy.",
"It is warranted to investigate how to take this into account in our approach.",
"Theoretically, Petrone and Wasserman [18] obtained the convergence rate of the Bernstein polynomials.",
"Our Dirichlet mixture distribution should at least have a comparable convergence rate.",
"This rate can guide us theoretically on how many distributions in the mixture should we need.",
"We leave these problems as our future research."
]
] | 1906.04450 |
[
[
"Exploiting the Sign of the Advantage Function to Learn Deterministic\n Policies in Continuous Domains"
],
[
"Abstract In the context of learning deterministic policies in continuous domains, we revisit an approach, which was first proposed in Continuous Actor Critic Learning Automaton (CACLA) and later extended in Neural Fitted Actor Critic (NFAC).",
"This approach is based on a policy update different from that of deterministic policy gradient (DPG).",
"Previous work has observed its excellent performance empirically, but a theoretical justification is lacking.",
"To fill this gap, we provide a theoretical explanation to motivate this unorthodox policy update by relating it to another update and making explicit the objective function of the latter.",
"We furthermore discuss in depth the properties of these updates to get a deeper understanding of the overall approach.",
"In addition, we extend it and propose a new trust region algorithm, Penalized NFAC (PeNFAC).",
"Finally, we experimentally demonstrate in several classic control problems that it surpasses the state-of-the-art algorithms to learn deterministic policies."
],
[
"Introduction",
"Model-free reinforcement learning combined with neural networks achieved several recent successes over a large range of domains [9], [7], [12].",
"Yet those methods are still difficult to apply without any expert knowledge, lack robustness and are very sensitive to hyperparameter optimization [2], [1].",
"In this context, we focus in this paper on improving methods that learn deterministic policies.",
"Such policies have three main advantages during the learning phase: 1) they usually require less interactive data because fewer parameters need to be learned, 2) their performances are less costly to estimate during testing phases because randomness only comes from the environment (as opposed to randomized policies), and 3) they are also less sensitive to the premature convergence problem, because they cannot directly control exploration.",
"Moreover, deterministic policies are preferred in some domains (e.g., robotics), because we do not want the agent to act stochastically after the learning phase.",
"In continuous state and action space domains, solution methods require function approximation.",
"Neural control architectures are excellent representations for policies because they can handle continuous domains, are easily scalable, and have a high degree of expressiveness.",
"The weights of such neural networks are usually updated with a policy gradient method.",
"As vanilla policy gradient suffers from high variance, it is generally implemented in an actor-critic architecture where an estimated value function helps to reduce the variance at the cost of introducing some bias [6].",
"In this architecture, the parameters (e.g., weights of neural networks) of the policy (i.e., actor) and its value function (i.e., critic) are updated simultaneously.",
"The basic version of an actor-critic architecture for learning deterministic policies in continuous domains is the deterministic policy gradient (DPG) method [13].",
"Learning the value function is crucial but also difficult, which is why several extensions of DPG have been proposed.",
"Deep Deterministic Policy Gradient (DDPG) [7] brings batch normalization [3], target networks and replay buffer [9] to DPG and is one of the most used actor-critic methods for learning continuous deterministic policies.",
"However, it has several limitations: 1) the critic learns the state-action value function (Q function), which is difficult to estimate, 2) it relies on the fact that non-biased estimates of the gradient of the Q function are accessible, which is not the case in the model-free setting, 3) it does not use compatible functions: the policy gradient might be poorly estimated.",
"In this work, we focus on an alternative method that estimates the state value function (V function) instead of the Q function to learn continuous deterministic policies.",
"[17] VanHasselt2007 were the first to propose to reinforce the policy toward an action with a positive temporal difference.",
"They experimentally showed that using such a method, in an incremental actor-critic algorithm, called Continuous Actor Critic Learning Automaton (CACLA), provided better results than both the stochastic and the deterministic policy gradientsIn their paper the deterministic policy gradient algorithm was called ADHDP [10].",
"in the Mountain Car and the Acrobot environments.",
"[18] zimmer2016,zimmer2018developmental validated those results in higher-dimensional environments, Half-Cheetah and Humanoid in Open Dynamic Engine [14], and proposed several extensions with the Neural Fitted Actor Critic (NFAC) algorithm.",
"However, no theoretical explanation for their good performance, nor a clear discussion about which objective function those methods optimize were given.",
"Providing such an explanation would help understand better why those algorithms work well, what are their properties and limitations, and how to further improve them.",
"We first show that CACLA and NFAC can be viewed as policy gradient methods and that they are closely related to a specific form of the stochastic policy gradient (SPG) [15].",
"Then we discuss some of their properties and limitations.",
"Moreover, we extend them with trust region updates and call the new algorithm Penalized Neural Fitted Actor Critic (PeNFAC).",
"Finally, we experimentally show that PeNFAC performs well on three high-dimensional continuous environments compared to the state-of-the-art methods."
],
[
"Background",
"A continuous Markov Decision Process (MDP) [16] is a tuple $(\\mathcal {S}, \\mathcal {A}, T,R,T_0)$ where $\\mathcal {S}$ is a continuous state space, $\\mathcal {A}$ is a continuous action space with $m$ dimensions, $T:\\mathcal {S} \\times \\mathcal {A} \\times \\mathcal {S} \\rightarrow [0, 1]$ is a transition function, $R: \\mathcal {S} \\times \\mathcal {A} \\rightarrow \\mathbb {R}$ is a reward function, $T_0$ is a distribution over initial states.",
"In the model-free setting, it is assumed that the transition function $T$ and the reward function $R$ are unknown and can only be sampled at specific states according to the interaction between the agent and the environment.",
"The following notations are used: $\\mu $ represents a deterministic policy and $\\pi $ a stochastic one.",
"Thus, for a given state $s \\in \\mathcal {S}$ , $\\mu (s)$ is an action, $\\pi (a|s)$ is the probability of sampling action $a$ from the policy $\\pi $ , and $\\pi (\\cdot |s)$ is a distribution over the action space $\\mathcal {A}$ .",
"For a policy $\\pi $ , we denote the discounted state distribution by: $d^\\pi _\\gamma (s) = \\int _{\\mathcal {S}} T_0(s_0)\\sum ^\\infty _{t=0} \\gamma ^{t} p(s|s_0,t,\\pi ) ds_0$ where $\\gamma \\in [0, 1)$ is a discount factor and $p(s|s_0,t,\\pi )$ is the probability of being in state $s$ after applying policy $\\pi $ $t$ timesteps from state $s_0$ .",
"Its state value function is defined by $V^\\pi (s)$ $=$ $\\mathbb {E}_\\pi \\big [ \\sum _{t=0}^{\\infty } \\gamma ^{t} R(S_t, A_t) \\big | S_0 = s \\big ]$ where $\\mathbb {E}_\\pi $ is the expectation induced by $T$ and $\\pi $ , and for all $t$ , $S_t$ and $A_t$ are random variables.",
"Its action value function is given by $Q^\\pi (s,a) = \\mathbb {E}_\\pi \\big [ \\sum _{t=0}^{\\infty } \\gamma ^{t} R(S_t, A_t) \\big | S_0 = s, A_0 = a \\big ],$ and its advantage function by $A^\\pi (s,a)=Q^\\pi (s,a)-V^\\pi (s)$ .",
"In reinforcement learning, the goal is to find a policy that optimizes the expectation of the discounted rewards: $J(\\pi ) = \\mathbb {E}_\\pi \\Big [ \\sum _{t=0}^\\infty \\gamma ^t R(S_t, A_t) \\big | S_0 \\sim T_0 \\Big ].$ Due to the continuity of the state/action spaces, this optimization problem is usually restricted to a class of parametrized policies, which we denote $\\pi _{\\theta }$ (stochastic case) or $\\mu _\\theta $ (deterministic case).",
"To simplify notations, we may write $\\pi $ or $\\mu $ instead of $\\pi _{\\theta }$ or $\\mu _\\theta $ .",
"The stochastic policy gradient (SPG) in the continuous case can be written as [15]: $\\nabla _\\theta J(\\pi ) = \\int _{\\mathcal {S}} d^\\pi _\\gamma (s) \\int _{\\mathcal {A}} A^\\pi (s,a) \\nabla _\\theta \\pi _\\theta (a|s) da ds.$ The DPG is defined as [13]: $\\nabla _\\theta J(\\mu ) &= \\int _{\\mathcal {S}} d^\\mu _\\gamma (s) \\Delta _{\\text{DPG}}(s,\\mu _\\theta ) ds, \\\\\\text{where} \\\\\\Delta _{\\text{DPG}}(s,\\mu _\\theta ) &= \\nabla _a A^\\mu (s,a) \\big |_{a=\\mu _\\theta (s)} \\nabla _\\theta \\mu _\\theta (s).$ Policy gradient methods usually take a step according to those directions: $\\theta _{t+1} \\leftarrow \\theta _t + \\alpha \\nabla _\\theta J$ .",
"However, it is difficult to select a proper learning rate $\\alpha $ to control the step size.",
"If $\\alpha $ is too big, the method may diverge.",
"If it is too low, the learning will converge slowly (thus requiring more samples).",
"To overcome this difficulty, a trust region method can be used to control the step size [11].",
"Indeed, one can guarantee monotonic gradient updates by exploiting an approximation of the policy advantage function [4] of $\\tilde{\\pi }$ with respect to $\\pi $ , which measures the difference of performance between the two policies: $J(\\tilde{\\pi }) =& J(\\pi ) + \\int _{\\mathcal {S}} d^{\\tilde{\\pi }}_\\gamma (s) \\int _{\\mathcal {A}} \\tilde{\\pi }(a|s) A^\\pi (s, a) da ds, \\\\\\approx & J(\\pi ) + \\int _{\\mathcal {S}} d^{\\pi }_\\gamma (s) \\int _{\\mathcal {A}} \\tilde{\\pi }(a|s) A^\\pi (s, a) da ds.",
"$ The latter approximation holds when the two policies are close, which can be enforced by a KL divergence constraint in trust region policy optimization [11]."
],
[
"Algorithms",
"In this section, we recall three related algorithms (CACLA, CAC, NFAC) that we discuss later."
],
[
"Continuous Actor Critic Learning Automaton",
"Continuous Actor Critic Learning Automaton (CACLA) [17] is an actor-critic method that learns a stochastic policy $\\pi $ and its estimated value function $\\hat{V}^\\pi $ .",
"We assume in this paper that CACLA uses isotropic Gaussian exploration, which implies that $\\pi $ can be written as follows: $\\pi _{\\theta ,\\sigma }(\\cdot |s) = \\mathcal {N}\\big (\\mu _\\theta (s), \\sigma ^2 I)$ where $I$ is the identity matrix and $\\sigma >0$ possibly annealed during learning.",
"CACLA alternates between two phases: 1) a hill climbing step in the action space using a random optimization (RO) algorithm [8], 2) a gradient-like update in the policy parameter space.",
"RO consists in repeating the following two steps: i) sample a new action $a^{\\prime }$ , which is executed in the environment in current state $s$ , by adding a normally distributed noise to the current action $a=\\mu (s)$ , ii) if $R(s, a^{\\prime }) + \\gamma \\hat{V}^\\pi (s^{\\prime }) > \\hat{V}^\\pi (s)$ then $a \\leftarrow a^{\\prime }$ else $a$ does not change.",
"Phase 2) is based on following update: $ \\text{If } \\delta (s,a) > 0: \\tilde{\\theta } \\leftarrow \\theta - \\alpha \\big (\\mu _\\theta (s) - a\\big ) \\nabla _\\theta \\mu _\\theta (s),$ where $\\delta (s,a) = R(s, a) + \\gamma \\hat{V}^\\pi (s^{\\prime }) - \\hat{V}^\\pi (s)$ is the temporal difference (TD) error.",
"As the expectation of the TD error is equal to the advantage function, this update can be interpreted as follows: if an exploratory action $a$ has a positive advantage then policy $\\mu $ should be updated towards $a$ .",
"Note that although CACLA executes a stochastic policy $\\pi $ , it can be seen as learning a deterministic policy $\\mu $ .",
"[17] VanHasselt2007 state that when learning in continuous action space, moving away from a bad action could be meaningless.",
"Indeed, while for stochastic policies, the probability of a bad action can be decreased, for deterministic policies, moving in the action space in the opposite direction of an action with a negative advantage may not necessarily lead to better actions.",
"Thus, CACLA's update is particularly appropriate for learning continuous deterministic policies."
],
[
"Continuous Actor Critic",
"In our discussion, we also refer to a slightly different version of CACLA, Continuous Actor Critic (CAC) [17].",
"The only difference between CAC and CACLA is that the update in CAC is scaled by the TD error: $\\text{If } \\delta (s,a) > 0: \\tilde{\\theta } \\leftarrow \\theta - \\alpha \\delta (s,a) \\big (\\mu _\\theta (s) - a\\big ) \\nabla _\\theta \\mu _\\theta (s),$ Thus an action with a larger positive advantage (here, estimated by the TD error) will have a bigger impact over the global objective."
],
[
"Neural Fitted Actor Critic",
"The Neural Fitted Actor Critic (NFAC) [18], [19] algorithm is an efficient instantiation of the CACLA update, which integrates the following techniques: batch normalization, $\\lambda $ -returns for both the critic and the actor, and batch learning with Adam[5].",
"In this algorithm, the update of the parameters is not done anymore at each time step, but at the end of a given number of episodes."
],
[
"Discussions",
"In this section, we discuss the algorithms to provide some theoretical explanation for their good performance."
],
[
"CACLA",
"We first explain the relationship between an algorithm based on stochastic policy gradient (SPG) and CACLA.",
"For this discussion, we assume that SPG is applied to parametrized policies that are Gaussian policies $\\pi _{\\theta , \\sigma }$ (i.e., Gaussian around $\\mu _\\theta $ ).",
"Then the first common feature between the two algorithms is that the distributions over states they induce during learning are the same (i.e., $d^{\\pi }_\\gamma (s)$ ) because they both use the same exploratory policy to interact with the environment.",
"Moreover, SPG can be written as follows: $&\\nabla _\\theta J(\\pi _{\\theta ,\\sigma }) \\\\&= \\int _{\\mathcal {S}} d^\\pi _\\gamma (s) \\int _{\\mathcal {A}} \\pi _{\\theta ,\\sigma }(a|s) A^\\pi (s,a) \\nabla _\\theta \\text{ log } \\pi _\\theta (a|s) da ds, \\\\&= \\frac{1}{\\sigma ^2} \\int _{\\mathcal {S}} d^\\pi _\\gamma (s) \\int _{\\mathcal {A}} \\pi _{\\theta ,\\sigma }(a|s) A^\\pi (s,a) \\big (a - \\mu _\\theta (s)\\big ) \\cdot \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\nabla _\\theta \\mu _\\theta (s) da ds .$ For CACLA, we interpret update (REF ) as a stochastic update in the following direction: $& \\int _{\\mathcal {S}} d^{\\pi }_\\gamma (s) \\Delta _{\\text{CACLA}}(s, \\mu _\\theta ) ds,\\\\ \\text{with } & \\Delta _{\\text{CACLA}}(s,\\mu _\\theta ) =\\int _{\\mathcal {A}} \\pi _{\\theta , \\sigma }(a|s)H\\big (A^\\pi (s,a)\\big ) \\times \\\\& \\hspace{120.0pt}\\big (\\mu _\\theta (s) - a\\big ) \\nabla _\\theta \\mu _\\theta (s) da ,$ where $H$ is the Heaviside function.",
"Indeed, the inner integral is estimated using a single Monte Carlo sample during the run of CACLA.",
"Under this form, it is easy to see the similarity between SPG and CACLA.",
"The constant factor $\\frac{1}{\\sigma ^2}$ can be neglected because it may be integrated into the learning rate.",
"The sign difference of the term $(a-\\mu _\\theta (s))$ is because SPG performs gradient ascent and CACLA gradient descent.",
"So the main difference between SPG and CACLA is the replacement of $A^\\pi (s,a)$ by $H(A^\\pi (s,a))$ .",
"Therefore CACLA optimizes its exploratory stochastic policy through an approximation of SPG hoping to improve the underlying deterministic policy (for a fixed state, the direction of CACLA and SPG are the same up to a scalar).",
"Moreover, relating CACLA's update with (REF ) also brings to light two main limitations.",
"The first one concerns the inner integral over the action space which has a high variance.",
"Therefore, we expect CACLA to be less and less data efficient in high-dimension action space (which is the main theoretical justification of DPG over SPG - see Appendix REF ).",
"The second limitation that appears is that over one update, CACLA does not share the same exact optimal solutions as DPG or SPG.",
"Indeed, if we define $\\theta ^*$ such as $\\nabla _\\theta J(\\mu _{\\theta })\\big |_{\\theta =\\theta ^*} = 0$ it is not possible to prove that (REF ) will also be 0 (because of the integral over the state space).",
"It means that CACLA could decrease the performance of this local optimal solution."
],
[
"CAC",
"Similarly, the update in CAC can be seen as a stochastic update in the following direction: $& \\int _{\\mathcal {S}} d^{\\pi }_\\gamma (s) \\Delta _{\\text{CAC}}(s, \\mu _\\theta ) ds, \\\\ \\text{with } & \\Delta _{\\text{CAC}}(s,\\mu _\\theta ) =\\int _{\\mathcal {A}} \\pi _{\\theta , \\sigma }(a|s) A^\\pi (s,a) H\\big (A^\\pi (s,a)\\big ) \\times \\\\ & \\hspace{100.0pt} \\big (\\mu _\\theta (s) - a\\big ) \\nabla _\\theta \\mu _\\theta (s) da .$ This shows that CAC is even closer to SPG than CACLA and provides a good theoretical justification of this update at a local level (not moving in potentially worst action).",
"However, there is also a justification at a more global level.",
"Lemma 4.1 For a fixed state, when the exploration tends to zero, CAC maintains the sign of the DPG update with a scaled magnitude: $\\lim _{\\sigma \\rightarrow 0} \\Delta _{\\text{CAC}} (s, \\mu _\\theta ) \\leftarrow g^+(s,\\pi ) \\circ \\Delta _{\\text{DPG}} (s, \\mu _\\theta ),$ where $g^+(s,\\pi )$ is a positive function between $[0; 1]^{n}$ with $n$ as the number of parameters of the deterministic policy and $\\circ $ is the Hadamard product (element-wise product).",
"The proof is provided in Appendix REF .",
"The consequence of this lemma is that, for a given state and low exploration, a local optimal solution for DPG will also be one for CAC.",
"However it is still not the case for the overall update because of the integral over the different states.",
"The weights given to each direction over different states are not the same in CAC and DPG.",
"One might think that in such a case, it would be better to use DPG.",
"However, in practice, the CAC update may in fact be more accurate when using an approximate advantage function.",
"Indeed, there exist cases where DPG with an approximate critic might update towards a direction which could decrease the performance.",
"For instance, when the estimated advantage $\\hat{A}\\big (s,\\mu (s) \\big )$ is negative, the advantage around $\\mu (s)$ is therefore known to be poorly estimated.",
"In such a case, thanks to the Heaviside function, CAC will not perform any update for actions $a$ in the neighborhood of $\\mu (s)$ such that $\\hat{A}(s, a) \\le 0$ .",
"However, in such a case, DPG will still perform an update according to this poorly estimated gradient."
],
[
"Extension to Trust Region",
"In this section, we extend the approach to use a trust region method."
],
[
"Trust Region for Deterministic Policies",
"We now introduce a trust region method dedicated to continuous deterministic policies.",
"Given current deterministic policy $\\mu $ , and an exploratory policy $\\pi $ defined from $\\mu $ , the question is to find a new deterministic policy $\\tilde{\\mu }$ that improves upon $\\mu $ .",
"Because a deterministic policy is usually never played in the environment outside of testing phases, a direct measure between two deterministic policies (i.e., a deterministic equivalent of Equation $\\ref {eq:stochperfmeasure}$ ) is not directly exploitable.",
"Instead we introduce the following measure: Lemma 5.1 The performance $J(\\tilde{\\mu })$ of a deterministic policy $\\tilde{\\mu }$ can be expressed by the advantage function of another stochastic policy $\\pi $ built upon a deterministic policy $\\mu $ as: $ J(\\tilde{\\mu }) = J(\\mu ) + \\int _{\\mathcal {S}} d_\\gamma ^{\\pi }(s) \\int _\\mathcal {A} \\pi (a|s) A^\\mu (s, a) da ds + \\\\\\int _{\\mathcal {S}} d_\\gamma ^{\\tilde{\\mu }}(s) A^\\pi \\big (s, \\tilde{\\mu }(s)\\big ) ds.$ See Appendix REF for the proof.",
"The first two quantities in the RHS of (REF ) are independent of ${\\tilde{\\mu }}$ .",
"The second one represents the difference of performance from moving from the deterministic policy $\\mu $ to its stochastic version $\\pi $ .",
"Because $d^{\\tilde{\\mu }}_\\gamma $ would be too costly to estimate, we approximate it with the simpler quantity $d_\\gamma ^\\pi $ , as done by [11] Schulman2015 for TRPO, a predecessor to PPO.",
"Theorem 5.2 Given two deterministic policies $\\mu $ and $\\tilde{\\mu }$ , a stochastic Gaussian policy $\\pi $ with mean $\\mu (s)$ in state $s$ and independent variance $\\sigma $ , if the transition function $T$ is L-Lipschitz continuous with respect to the action from any state then: $&\\Big | \\int _{\\mathcal {S}} d^{\\tilde{\\mu }}(s) A^\\pi \\big (s, \\tilde{\\mu }(s)\\big ) - \\int _{\\mathcal {S}} d^{\\pi }(s) A^\\pi \\big (s, \\tilde{\\mu }(s)\\big ) \\Big | \\le \\\\ & \\frac{\\epsilon L}{1-\\gamma } \\underset{t>0}{\\operatorname{max\\ }} \\Big ( \\big |\\big | \\tilde{\\mu }(s) - \\mu (s)\\big |\\big |_{2,\\infty } + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} \\Big )^t,$ where $\\epsilon = \\text{max}_{s,a} |A^\\pi (s,a)| $ .",
"The proof is available in Appendix REF .",
"Thus, to ensure a stable improvement at each update, we need to keep both $|| \\mu - \\tilde{\\mu } ||_{2,\\infty }$ and $\\sigma $ small.",
"Note that the Lipschitz continuity condition is natural in continuous action spaces.",
"It simply states that for a given state, actions that are close will produce similar transitions."
],
[
"Practical Algorithm",
"To obtain a concrete and efficient algorithm, the trust region method can be combined with the previous algorithms.",
"Its integration to NFAC with a CAC update for the actor is called Penalized Neural Fitted Actor Critic (PeNFAC).",
"[17] VanHasselt2007 observed that the CAC update performs worse that the CACLA update in their algorithms.",
"In their setting where the policy and the critic are updated at each timestep, we believe this observation is explained by the use of the TD error (computed from a single sample) to estimate the advantage function.",
"However, when using variance reduction techniques such as $\\lambda $ -returns and learning from a batch of interactions, or when mitigating the update with a trust region constraint, we observe that this estimation becomes better (see Figure REF ).",
"This explains why we choose a CAC update in PeNFAC.",
"In order to ensure that $|| \\mu - \\tilde{\\mu } ||_{2,\\infty }$ stays small over the whole state space, we approximate it with a Euclidean norm over the state visited by $\\pi $ .",
"To implement this constraint, we add a regularization term to the update and automatically adapts its coefficient, for a trajectory $(s_0, s_1, \\ldots , s_h)$ : $\\sum _{t=0}^{h-1} \\Delta _{\\text{CAC}}(s_t, \\mu _\\theta ) + \\beta \\nabla _\\theta \\big |\\big | \\mu _{\\text{old}}(s_t) - \\mu _\\theta (s_t) \\big |\\big |^2_2,$ where $\\beta $ is a regularization coefficient.",
"Similarly to the adaptive version of Proximal Policy Optimization (PPO) [12], $\\beta $ is updated in the following way (starting from $\\beta \\leftarrow 1$ ): if $\\hat{d}(\\mu ,\\mu _{\\text{old}}) < d_{\\text{target}} / 1.5$ : $\\beta \\leftarrow \\beta / 2 $ , if $\\hat{d}(\\mu ,\\mu _{\\text{old}}) > d_{\\text{target}} \\times 1.5$ : $\\beta \\leftarrow \\beta \\times 2 $ , where $\\hat{d}(\\mu ,\\mu _{\\text{old}}) = \\frac{1}{\\sqrt{m L}} \\sum _{s \\sim \\pi } || \\mu _{\\text{old}}(s) - \\mu _\\theta (s) ||_2$ with $L$ being the number of gathered states.",
"Those hyper-parameters are usually not optimized because the learning is not too sensitive to them.",
"The essential value to adapt for the designer is $d_\\text{target}$ .",
"Note that the introduction of this hyperparameter mitigates the need to optimize the learning rate for the update of the policy, which is generally a much harder task."
],
[
"Experiments",
"We performed two sets of experiments to answer the following questions: 1) How does PeNFAC compare with state-of-the-art algorithms for learning deterministic policies?",
"2) Which components of PeNFAC contribute the most to its performance?",
"The experiments were performed on environments with continuous state and action spaces in a time-discretized simulation.",
"We chose to perform the experiments on OpenAI Roboschool [12], a free open-source software, which allows anyone to easily reproduce our experiments.",
"In order to evaluate the performance of an algorithm, deterministic policies $\\mu $ obtained during learning are evaluated at a constant interval during testing phases: policy $\\mu $ is played in the environment without exploration.",
"The interactions gathered during this evaluation are not available to any algorithms.",
"The source code of the PeNFAC algorithm is available at github.com/matthieu637/ddrl.",
"The hyperparameters used are reported in Appendix as well as the considered range during the grid search."
],
[
"Performance of PeNFAC",
"We compared the performance of PeNFAC to learn continuous deterministic policies with two state-of-the-art algorithms: PPO and DDPG.",
"A comparison with NFAC is available in the ablation study (Section REF ) and in Appendix .",
"Because PPO learns a stochastic policy, for the testing phases, we built a deterministic policy as follows $\\mu (s) = \\mathbb {E}[a | a \\sim \\pi _\\theta (\\cdot ,s)]$ .",
"We denote this algorithm as \"deterministic PPO\".",
"In Appendix , we experimentally show that this does not penalize the comparison with PPO, as deterministic PPO provides better results than standard PPO.",
"For PPO, we used the OpenAI Baseline implementation.",
"To implement PeNFAC and compare it with NFAC, we use the DDRL library [19].",
"Given that DDPG is present in those two libraries, we provided the two performances for it.",
"The OpenAI Baseline version uses an exploration in the parameter space and the DDRL version uses n-step returns.",
"Figure: Comparison of PeNFAC, DDPG and deterministic PPO over 60 different seeds for each algorithm in Hopper.Figure: Comparison of PeNFAC, DDPG and deterministic PPO over 60 different seeds for each algorithm in HalfCheetah.Figure: Comparison of PeNFAC, DDPG and deterministic PPO over 60 seeds for each algorithm in Humanoid.We performed learning experiments over three high-dimensional domains: Hopper, HalfCheetah and Humanoid.",
"Dimensions of $\\mathcal {S} \\times \\mathcal {A}$ are $15 \\times 3$ (Hopper), $26 \\times 6$ (HalfCheetah) and $44 \\times 17$ (Humanoid).",
"The neural network architecture is composed of two hidden layers of 64 units for either the policy or the value function.",
"The choice of the activation function in the hidden units was optimized for each algorithm: we found that ReLU was better for all of them except for PPO (where tanh was better).",
"The output activation of the critic is linear and the output activation of the actor is tanh.",
"In Figures REF -REF , the lighter shade depicts one standard deviation around the average, while the darker shade is the standard deviation divided by the square root of the number of seeds.",
"In Figures REF -REF , PeNFAC outperforms DDPG and deterministic PPO during the testing phase.",
"On Humanoid, even after optimizing the hyperparameters, we could not obtain the same results as those of [12] PPO.",
"We conjecture that this may be explained as follows: 1) the RoboschoolHumanoid moved from version 0 to 1, 2) deterministic PPO Figure: Comparison of the different components (λ\\lambda -returns, fitted value-iteration, CAC vs CACLA update, batch normalization) of the PeNFAC algorithm during the testing phase over the HalfCheetah environment and 60 seeds for each version.might be less efficient than PPO, 3) neither LinearAnneal for the exploration, nor adaptive Adam step size is present in the OpenAI Baseline implementation.",
"However, we argue that the comparison should still be fair since PeNFAC also does not use those two components.",
"On Humanoid, we did not find a set of hyperparameters where DDPG could work correctly with both implementations."
],
[
"Components of PeNFAC",
"In Figure REF , we present an ablation analysis in the HalfCheetah domain to understand which components of the PenFAC algorithm are the most essential to its good performance.",
"From top to bottom plots of Figure REF , we ran PenFAC with or without trust region, with or without $\\lambda $ -returns, with or without fitted value iteration, with CACLA update or CAC update, and finally with or without batch normalization.",
"It appears that $\\lambda $ -returns and fitted value iteration are the most needed, while the effect of batch normalization is small and mostly helps in the beginning of the learning.",
"We also tried updating the actor every timestep without taking into account the sign of the advantage function (i.e., using SPG instead of CAC), but the algorithm was not able to learn at all.",
"This also demonstrates that the CAC update is an essential component of PenFAC."
],
[
"Conclusion",
"In the context of learning deterministic policies, we studied the properties of two not very well-known but efficient updates, Continuous Actor Critic Learning Automaton (CACLA) and Continuous Actor Critic (CAC).",
"We first showed how closely they both are related to the stochastic policy gradient (SPG).",
"We explained why they are well designed to learn continuous deterministic policies when the value function is only approximated.",
"We also highlighted the limitations of those methods: a potential poor sample efficiency when the dimension of the action space increases and no guarantee that the underlying deterministic policy will converge toward a local optimum of $J(\\mu _\\theta )$ even with a linear approximation.",
"In the second part, we extended Neural Fitted Actor Critic (NFAC), itself an extension of CACLA, with a trust region constraint designed for deterministic policies and proposed a new algorithm, Penalized NFAC (PeNFAC).",
"Finally, we tried our implementation on various high-dimensional continuous environments and showed that PeNFAC performs better than DDPG and PPO to learn continuous deterministic policies.",
"As future work, we plan to consider off-policy learning and the combination of the updates of CAC and DPG together to ensure the convergence toward a local optimum while benefiting from the good updates of CAC."
],
[
"Acknowledgments",
"This work has been supported in part by the program of National Natural Science Foundation of China (No.",
"61872238).",
"Experiments presented in this paper were carried out using the Grid’5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000.fr)."
],
[
"Relation between DPG and CAC update for a given state",
"For simplification, the proof of a single dimension $k$ of the parameter space is provided.",
"To denote the $k$th dimension of a vector $x$ , we write $x_k$ .",
"If $x$ is a matrix, $x_{:,k}$ represents the $k$th column vector.",
"We will use the following result from [13] Silver2014: $ \\lim _{\\sigma \\rightarrow 0} \\nabla _\\theta J(\\pi _{\\theta ,\\sigma }) = \\nabla _\\theta J(\\mu _\\theta ).$ Thus, the following standard regularity conditions are required: $T, T_0, R, \\mu , \\pi , \\nabla _a T, \\nabla _a R, \\nabla _\\theta \\mu $ are continuous in all variables and bounded.",
"From this result, we derive the following equation for a fixed state $s$ : $\\lim _{\\sigma \\rightarrow 0} \\int _{\\mathcal {A}} A^\\pi (s,a) \\nabla _\\theta \\pi _{\\theta ,\\sigma }(a|s) da = \\nabla _a A^\\mu (s,a) \\big |_{a=\\mu _\\theta (s)} \\nabla _\\theta \\mu _\\theta (s).$ We first study the special case of $\\Delta _{\\text{DPG}}(s, \\mu _\\theta )_k = 0$ and want to show that ${\\lim _{\\sigma \\rightarrow 0} \\Delta _{\\text{CAC}} (s, \\mu _\\theta )}_k$ is also zero: ${\\Delta _{\\text{DPG}}(s, \\mu _\\theta )}_k = 0 \\Rightarrow & \\nabla _a A^\\mu (s,a) \\big |_{a=\\mu _\\theta (s)} {\\nabla _\\theta \\mu _\\theta (s)}_{:,k} = 0,\\\\\\Rightarrow & \\lim _{\\sigma \\rightarrow 0} \\int _{\\mathcal {A}} A^\\pi (s,a) {\\nabla _\\theta \\pi _{\\theta ,\\sigma }(a|s)}_{:,k} da = 0, \\\\\\Rightarrow & \\lim _{\\sigma \\rightarrow 0} \\frac{1}{\\sigma ^2} \\int _{\\mathcal {A}} \\pi _{\\theta , \\sigma }(a|s) A^\\pi (s,a) \\big (a - \\mu _\\theta (s) \\big ) {\\nabla _\\theta \\mu _\\theta (s)}_{:,k} da = 0,\\\\\\Rightarrow & \\lim _{\\sigma \\rightarrow 0} \\frac{1}{\\sigma ^2} \\int _{\\mathcal {A}} \\pi _{\\theta , \\sigma }(a|s) H\\big (A^\\pi (s,a)\\big ) A^\\pi (s,a) \\big (a - \\mu _\\theta (s) \\big ) {\\nabla _\\theta \\mu _\\theta (s)}_{:,k} da = 0,\\\\\\Rightarrow & \\lim _{\\sigma \\rightarrow 0} {\\Delta _{\\text{CAC}} (s, \\mu _\\theta )}_k = 0.$ Now, we study the more general case ${\\Delta _{\\text{DPG}}(s, \\mu _\\theta )}_k \\ne 0$ : $g_k^+(s, \\mu _\\theta ) =& \\frac{\\lim _{\\sigma \\rightarrow 0} \\Delta _{\\text{CAC}} (s, \\mu _\\theta )_k}{\\Delta _{\\text{DPG}}(s, \\mu _\\theta )_k}, \\\\=& \\frac{\\lim _{\\sigma \\rightarrow 0} \\int _{\\mathcal {A}} A^\\pi (s,a) H(A^\\pi (s,a)) \\nabla _\\theta {\\pi _{\\theta ,\\sigma }(a|s)}_{:,k} da}{ \\lim _{\\sigma \\rightarrow 0} \\int _{\\mathcal {A}} A^\\pi (s,a) \\nabla _\\theta {\\pi _{\\theta ,\\sigma }(a|s)}_{:,k} da }, \\\\= & \\lim _{\\sigma \\rightarrow 0} \\frac{\\int _{\\mathcal {A}} A^\\pi (s,a) H(A^\\pi (s,a)) \\nabla _\\theta {\\pi _{\\theta ,\\sigma }(a|s)}_{:,k} da}{ \\int _{\\mathcal {A}} A^\\pi (s,a) {\\nabla _\\theta \\pi _{\\theta ,\\sigma }(a|s)}_{:,k} da },\\\\& \\Rightarrow 0 \\le g_k^+(s, \\mu _\\theta ) \\le 1.$"
],
[
"Performance of a deterministic policy expressed from a Gaussian stochastic policy",
"The proof is very similar to [4], [11] and easily extends to mixtures of stochastic and deterministic policies: $& \\int _{\\mathcal {S}} d_\\gamma ^{\\pi }(s) \\int _\\mathcal {A} \\pi (a|s) A^\\mu (s, a) da ds + \\int _{\\mathcal {S}} d_\\gamma ^{\\tilde{\\mu }}(s) A^\\pi (s, \\tilde{\\mu }(s)) ds = \\\\& \\int _{\\mathcal {S}} d_\\gamma ^{\\pi }(s) \\int _\\mathcal {A} \\pi (a|s) \\Big ( R(s,a) + \\gamma \\mathbb {E}\\big [V^\\mu (s^{\\prime }) | a\\big ] - V^\\mu (s) \\Big ) da +\\int _{\\mathcal {S}} d_\\gamma ^{\\tilde{\\mu }}(s) \\Big ( R(s,\\tilde{\\mu }(s)) + \\gamma \\mathbb {E}\\big [V^\\pi (s^{\\prime }) | \\tilde{\\mu }(s) \\big ] - V^\\pi (s) \\Big ) ds = \\\\& J(\\pi ) + J(\\tilde{\\mu }) + \\int _{\\mathcal {S}} d_\\gamma ^{\\pi }(s) \\int _\\mathcal {A} \\pi (a|s) \\Big ( \\gamma \\mathbb {E}\\big [V^\\mu (s^{\\prime }) | a\\big ] - V^\\mu (s) \\Big ) da + \\int _{\\mathcal {S}} d_\\gamma ^{\\tilde{\\mu }}(s) \\Big ( \\gamma \\mathbb {E}\\big [V^\\pi (s^{\\prime }) | \\tilde{\\mu }(s) \\big ] - V^\\pi (s) \\Big ) ds = \\\\& J(\\pi ) + J(\\tilde{\\mu }) + \\int _{\\mathcal {S}} d_\\gamma ^{\\pi }(s) \\Big ( - V^\\mu (s) + \\gamma \\int _\\mathcal {A} \\pi (a|s) \\mathbb {E}\\big [V^\\mu (s^{\\prime }) | a\\big ]da \\Big ) - J(\\pi ) = \\\\& J(\\tilde{\\mu }) - J(\\mu ).$"
],
[
"Trust region for continuous deterministic policies",
"For this theorem we also use the following standard regularity conditions: $I(\\mathcal {S}) = \\int _\\mathcal {S} ds < \\infty $ and $\\Big |\\Big | \\tilde{\\mu }(s) - \\mu (s))\\Big |\\Big |_{2,\\infty } < \\infty $ .",
"$m$ denotes the number of dimension of the action space.",
"We start from the two terms we want to bound: $&\\Big | \\int _{\\mathcal {S}} d^{\\tilde{\\mu }}_\\gamma (s) A^\\pi (s, \\tilde{\\mu }(s)) - \\int _{\\mathcal {S}} d^{\\pi }_\\gamma (s) A^\\pi (s, \\tilde{\\mu }(s)) \\Big | = \\\\& \\Big | \\int _{\\mathcal {S}} \\big ( d^{\\tilde{\\mu }}_\\gamma (s) - d^{\\pi }_\\gamma (s) \\big ) A^\\pi (s, \\tilde{\\mu }(s)) \\Big | \\le \\\\& \\int _{\\mathcal {S}} \\Big | d^{\\tilde{\\mu }}_\\gamma (s) - d^{\\pi }_\\gamma (s) \\Big | .",
"\\Big | A^\\pi (s, \\tilde{\\mu }(s)) \\Big | \\le \\\\& \\epsilon \\int _{\\mathcal {S}} \\Big | d^{\\tilde{\\mu }}_\\gamma (s) - d^{\\pi }_\\gamma (s) \\Big |, $ where $\\epsilon = \\text{max}_{s,a} |A^\\pi (s,a)| $ .",
"So, we need to bound the difference between $d^{\\tilde{\\mu }}$ and $d^{\\pi }$ for a given state $s$ : $& \\Big | d^{\\tilde{\\mu }}_\\gamma (s) - d^{\\pi }_\\gamma (s) \\Big | = \\\\& \\Big | \\int _{\\mathcal {S}} T_0(s_0) \\Big ( \\sum ^\\infty _{t=0} \\gamma ^{t} p(s|s_0,t,\\tilde{\\mu }) - \\sum ^\\infty _{t=0} \\gamma ^{t} p(s|s_0,t,\\pi ) \\Big ) ds_0 \\Big | = \\\\& \\Big | \\int _{\\mathcal {S}} T_0(s_0) \\sum ^\\infty _{t=0} \\gamma ^{t} \\Big ( p(s|s_0,t,\\tilde{\\mu }) - p(s|s_0,t,\\pi ) \\Big ) ds_0 \\Big | \\le \\\\& \\int _{\\mathcal {S}} \\Big | T_0(s_0) \\Big | \\sum ^\\infty _{t=0} \\gamma ^{t} \\Big | p(s|s_0,t,\\tilde{\\mu }) - p(s|s_0,t,\\pi ) \\Big | ds_0 \\le \\\\& \\int _{\\mathcal {S}} \\sum ^\\infty _{t=0} \\gamma ^{t} \\Big | p(s|s_0,t,\\tilde{\\mu }) - p(s|s_0,t,\\pi ) \\Big | ds_0 \\le \\\\& \\int _{\\mathcal {S}} \\sum ^\\infty _{t=0} \\gamma ^{t} \\underset{t^{\\prime }>0}{\\operatorname{max}} \\Big | p(s|s_0,t^{\\prime },\\tilde{\\mu }) - p(s|s_0,t^{\\prime },\\pi ) \\Big | ds_0 = \\\\& \\frac{1}{1-\\gamma } \\int _{\\mathcal {S}} \\underset{t>0}{\\operatorname{max}} \\Big | p(s|s_0,t,\\tilde{\\mu }) - p(s|s_0,t,\\pi ) \\Big | ds_0.",
"$ Finally, we have to bound the difference between $ p(s|s_0,t,\\tilde{\\mu })$ and $ p(s|s_0,t,\\pi ) $ .",
"To do so, we define $\\tau = \\lbrace s_1, ..., s_t=s\\rbrace $ , and $\\mathcal {D}_\\tau $ all the possible path from the state $s_1$ to the state $s_t=s$ .",
"$& \\Big | p(s|s_0,t,\\tilde{\\mu }) - p(s|s_0,t,\\pi ) \\Big | = \\\\& \\Big | \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\Big ( T(s_k | s_{k-1}, \\tilde{\\mu }(s_{k-1})) - \\int _\\mathcal {A} \\pi (a|s_{k-1}) T( s_k | s_{k-1}, a ) da \\Big ) d\\tau \\Big | \\le \\\\& \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\Big | T(s_k | s_{k-1}, \\tilde{\\mu }(s_{k-1})) - \\int _\\mathcal {A} \\pi (a|s_{k-1}) T( s_k | s_{k-1}, a ) da \\Big | d\\tau = \\\\& \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\Big | \\int _\\mathcal {A} \\pi (a|s_{k-1}) \\big ( T(s_k | s_{k-1}, \\tilde{\\mu }(s_{k-1})) - T( s_k | s_{k-1}, a ) \\big ) da \\Big | d\\tau \\le \\\\& \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\int _\\mathcal {A} \\pi (a|s_{k-1}) \\Big | T(s_k | s_{k-1}, \\tilde{\\mu }(s_{k-1})) - T( s_k | s_{k-1}, a ) \\Big | da d\\tau \\le $ $& L \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\int _\\mathcal {A} \\pi (a|s_{k-1}) \\Big |\\Big | \\tilde{\\mu }(s_{k-1}) - a\\Big |\\Big |_2 da d\\tau = \\\\& L \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\int \\frac{1}{(\\sigma \\sqrt{2 \\pi })^m} e^{-\\frac{1}{2\\sigma ^2} ||b||_2^2} \\Big |\\Big | \\tilde{\\mu }(s_{k-1}) - \\mu (s_{k-1}) + b\\Big |\\Big |_2 db d\\tau \\le \\\\& L \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\int \\frac{1}{(\\sigma \\sqrt{2 \\pi })^m} e^{-\\frac{1}{2\\sigma ^2} ||b||_2^2} \\Big ( \\Big |\\Big | \\tilde{\\mu }(s_{k-1}) - \\mu (s_{k-1})\\Big |\\Big |_2 + \\Big |\\Big | b\\Big |\\Big |_2 \\Big ) db d\\tau \\le \\\\& L \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\Big ( \\Big |\\Big | \\tilde{\\mu }(s_{k-1}) - \\mu (s_{k-1})\\Big |\\Big |_2 + \\int \\frac{1}{(\\sigma \\sqrt{2 \\pi })^m} e^{-\\frac{1}{2\\sigma ^2} ||b||_2^2} \\Big |\\Big | b\\Big |\\Big |_1 \\Big ) db d\\tau = \\\\& L \\int _{\\mathcal {D}_\\tau } \\prod _{k=1}^t \\Big ( \\Big |\\Big | \\tilde{\\mu }(s_{k-1}) - \\mu (s_{k-1})\\Big |\\Big |_2 + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} \\Big ) d\\tau \\le \\\\& L \\int _{\\mathcal {D}_\\tau } \\Big ( \\underset{s_k \\in \\tau }{\\operatorname{max\\ }} \\Big |\\Big | \\tilde{\\mu }(s_{k}) - \\mu (s_{k})\\Big |\\Big |_2 + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} \\Big )^t d\\tau \\le \\\\& I(\\mathcal {S})^t L \\Big ( \\underset{s_k \\in \\mathcal {S}}{\\operatorname{max\\ }} \\Big |\\Big | \\tilde{\\mu }(s_{k}) - \\mu (s_{k})\\Big |\\Big |_2 + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} \\Big )^t.",
"$ To obtain (REF ), we use the assumption that the transition function is L-Lipschitz continuous with respect to the action and the L2 norm.",
"To obtain (), we use (REF ).",
"Equation does no longer depend on $s$ and $s_0$ , thus added to (REF ) and (REF ) it gives: $& \\frac{\\epsilon L}{1-\\gamma } \\underset{t>0}{\\operatorname{max\\ }} I(\\mathcal {S})^{t+2} \\Big ( \\Big |\\Big | \\tilde{\\mu }(s) - \\mu (s)\\Big |\\Big |_{2,\\infty } + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} \\Big )^t \\le \\\\&\\frac{\\epsilon L}{1-\\gamma } \\underset{t>0}{\\operatorname{max\\ }} \\Big ( \\Big |\\Big | \\tilde{\\mu }(s) - \\mu (s)\\Big |\\Big |_{2,\\infty } + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} \\Big )^t.",
"$ To obtain (REF ), we suppose that $I(\\mathcal {S})$ is smaller than 1.",
"We can make this assumption without losing in generality: it would only affect the magnitude of the Lipschitz constant.",
"Thus if $ \\big |\\big | \\tilde{\\mu }(s) - \\mu (s)\\big |\\big |_{2,\\infty } + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} $ stays smaller than 1, the optimal $t$ will be 1, and (REF ) could be reduced to: $ \\frac{\\epsilon L}{1-\\gamma } \\Big ( \\Big |\\Big | \\tilde{\\mu }(s) - \\mu (s)\\Big |\\Big |_{2,\\infty } + \\frac{2m\\sigma }{\\sqrt{2 \\pi }} \\Big ).",
"$"
],
[
"Additional experiments on CACLA's update",
"In those two experiments, we want to highlight the good performance of CACLA compared to SPG and DPG without neural networks.",
"The main argument to use DPG instead of SPG is its efficiency when the action dimensions become large.",
"In the first experiment, we study if CACLA suffers from the same variance problem as SPG.",
"The second experiment supports our claim that CACLA is more robust than SPG and DPG when the approximation made by the critic is less accurate."
],
[
"Sensitivity to action space dimensionality",
"We used a setup similar to that of [13] Silver2014: those environments contain only one state and the horizon is fixed to one.",
"They are designed such that the dimensionality of the action space can easily be controlled but there is only little bias in the critic approximation.",
"The policy parameters are directly representing the action: $\\mu _\\theta (\\cdot ) = \\theta $ .",
"Compatible features are used to learn the Q value function for both SPG and DPG.",
"For CACLA, the value function V is approximated through a single parameter.",
"The Gaussian exploration noise and the learning rate of both the critic and actor have been optimized for each algorithm on each environment.",
"In Figure REF , similarly to [13] Silver2014, we observe that SPG is indeed more sensitive to larger action dimensions.",
"CACLA is also sensitive to this increase in dimensionality but not as much as SPG.",
"Finally, we also note that even if the solution of CACLA and DPG are not exactly the same theoretically, they are very similar in practice.",
"Figure: Comparison of DPG, SPG and CACLA over three domains with 100 seeds for each algorithm.",
"On the left, the action dimensions is 5 and 50 on the right."
],
[
"Robustness to the critic approximation errors",
"Compared to the previous experience, we introduce a bigger bias in the approximation of the critic by changing the application domains: the horizon is deeper and there is an infinite number of states.",
"The policy is represented as $\\mu _\\theta (s)=\\phi (s) \\cdot \\theta $ where $\\phi (s)$ are tiles coding features.",
"Figure: Comparison of CACLA, DPG and SPG over two environments of OpenAI Gym and one environment of Roboschool (60 seeds are used for each algorithm).In Figure REF , we observe that as soon as value functions become harder to learn, CACLA performs better than both SPG and DPG."
],
[
"Broader comparison between PeNFAC and NFAC",
"To avoid overloading previous curves, we did not report the performance of NFAC (except in the ablation study on the HalfCheetah environment).",
"In Figure REF , we extend this study to two other domains of Roboschool: Hopper and Humanoid.",
"Figure: Comparison of PeNFAC and NFAC over RoboschoolHopper and RoboschoolHumanoid with 60 seeds for each algorithm.We observe that PeNFAC is significantly better than NFAC which demonstrates the efficiency of the trust region update combined with CAC."
],
[
"Impact of evaluating PPO with a deterministic policy",
"In Figure REF , we observe that using a deterministic policy to evaluate the performance of PPO is not penalizing.",
"This is the only experiment of the paper where deterministic policies and stochastic policies are compared."
],
[
"Hyperparameters",
"For the sake of reproducibility [2], the hyperparameters used during the grid search are reported here.",
"In Tables REF -REF , \"ho\", \"ha\" and \"hu\" stand respectively for Hopper, HalfCheetah, and Humanoid Roboschool environments.",
"Table: Set of hyperparameters used during the training with every algorithm.Table: Set of hyperparameters used during the training with PeNFAC.Table: Set of hyperparameters used during the training with PPO.Table: Set of hyperparameters used during the training with DDPG (DDRL implementation).Table: Set of hyperparameters used during the training with DDPG (OpenAI baselines implementation)."
]
] | 1906.04556 |
[
[
"Proximity magnetoresistance in graphene induced by magnetic insulators"
],
[
"Abstract We demonstrate the existence of Giant proximity magnetoresistance (PMR) effect in a graphene spin valve where spin polarization is induced by a nearby magnetic insulator.",
"PMR calculations were performed for yttrium iron garnet (YIG), cobalt ferrite (CFO), and two europium chalcogenides EuO and EuS.",
"We find a significant PMR (up to 100%) values defined as a relative change of graphene conductance with respect to parallel and antiparallel alignment of two proximity induced magnetic regions within graphene.",
"Namely, for high Curie temperature (Tc) CFO and YIG insulators which are particularly important for applications, we obtain 22% and 77% at room temperature, respectively.",
"For low Tc chalcogenides, EuO and EuS, the PMR is 100% in both cases.",
"Furthermore, the PMR is robust with respect to system dimensions and edge type termination.",
"Our findings show that it is possible to induce spin polarized currents in graphene with no direct injection through magnetic materials."
],
[
"Introduction",
"Graphene is a two-dimensional (2D) material [1], [2] that has attracted a lot of interest in view of its unique physical properties and applications potential in diverse fields such as electronics, spintronics and quantum computing [3], [4], [5].",
"Due to its weak spin orbit coupling [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] graphene possesses a long spin relaxation time and lengths even at room temperature [16].",
"While these characteristics offer an optimal platform for spin manipulation, it remains however a challenge to achieve robust spin polarization efficiently at room temperature.",
"Several methods have been proposed in order to introduce ferromagnetic order on graphene, among which functionalization with adatoms [17], addition of defects [18], [19], and by means of proximity effect via an adjacent ferromagnet[20], [21], [22], [23], [24], [25].",
"The latter approach attracted a lot of interest using magnetic insulators (MI) as a substrate to induce exchange splitting in graphene.",
"When a material is placed on top of a magnetic insulator, it can acquire proximity induced spin polarization and exchange splitting [20] resulting from the hybridization between $p_z$ orbitals with those of the neighboring magnetic insulator.",
"For practical purposes, the implementation in spintronic devices of this kind of materials could lead to lower power consumption since no current injection across adjacent ferromagnet (FM) is required as in case of traditional spin injection techniques.",
"Experimentally, the existence of proximity exchange splitting via magnetic insulator in graphene have been demonstrated with exchange fields up to 100 T using the coupling between graphene and EuS [23].",
"For yittrium irog garnet/graphene (YIG/Gr) based system, using non-local spin transport measurements, Leutenantsmeyer et al.",
"[24] demonstrated exchange field strength of 0.2 T. Another possibility of inducing exchange splitting in graphene using FM metal, by separating them by alternative 2D material such as hexa-boron nitride (hBN), was also proposed theoretically [21].",
"Recent studies have suggested the creation of graphene-based devices where EuO-graphene junction can act as a spin filter and spin valve simultaneously by gating the system [26].",
"It was also demonstrated [27] that a double EuO barrier on top of a graphene strip can exhibit negative differential resistance making this system a spin selective diode.",
"However, the drawback of using EuO is its low Curie temperature and the predicted strong electron doping [20].",
"It was proposed therefore using high Curie temperature materials such as YIG or cobalt ferrite (CFO) [28].",
"Indeed, a large change in the resistance of a graphene-based spintronic device has been reported recently where the heavy doping induced by YIG could be treated by gating [29].",
"In this Letter we demonstrate the existence of Proximity Magnetoresistance (PMR) effect in graphene for four different magnetic insulators (MI), YIG, CFO, europium oxide (EuO) and europium sulfide (EuS).",
"Using ab initio parameters reported in Ref.",
"[ali2017], we show that for YIG and CFO based lateral graphene-based devices with armchair edges, PMR values could reach 77% and 22% at room temperature (RT), respectively.",
"With chalcogenides, EuS and EuO, PMR values can reach 100% at 16 K and 70 K, respectively.",
"In addition, we demonstrate the robustness of this effect with respect to system dimensions and edge type termination.",
"Furthermore, our calculations with spin-orbit coupling (SOC) included does not significantly affect the PMR.",
"These findings will stimulate experimental investigations of the proposed phenomenon PMR and development of other proximity effect based spintronic devices."
],
[
"Methodology",
"In order to calculate conductances and PMR, we employed the tight-binding approach with scattering matrix formalism conveniently implemented within the KWANT package [30].",
"The system modeled is shown in Fig.",
"REF and comprises two identical proximity induced magnetic regions of width $W$ and length $L$ resulting from insulators with magnetizations M$_1$ and M$_2$ , separated by a distance $d$ of nonmagnetic region of graphene sheet with armchair edges.",
"Both magnetic graphene regions are separated from the leads $L_1$ and $L_2$ by a small pure graphene region.",
"In order to take into account the magnetism arising in graphene from the proximity effects induced by the MI's, in the Hamiltonian are used the parameters obtained for different MI's in Ref.",
"[ali2017].",
"It is important to note that the magnetic regions do not affect the linear dispersion of graphene bands, except breaking the valley and electron-hole symmetry resulting in spin-dependent band splitting and doping.",
"The discretized Hamiltonian for the magnetic graphene regions can be expressed as: $H = \\sum _{i\\sigma } \\sum _l t_{l\\sigma } c^\\dagger _{(i+l)1\\sigma } c_{i0\\sigma } +h.c.+ \\sum _{i\\sigma \\sigma ^{\\prime }}\\sum _{\\mu =0}^1\\left[\\delta +(-1)^\\mu \\Delta _\\delta \\right] c^\\dagger _{i\\mu \\sigma } [\\vec{m} .",
"\\vec{\\sigma }] c_{i\\mu \\sigma ^{\\prime }}+ \\sum _{i\\sigma }\\sum _{\\mu =0}^1\\left[E_D+(-1)^\\mu \\Delta _s\\right] c^\\dagger _{i\\mu \\sigma } c_{i\\mu \\sigma }$ where $c^\\dagger _{i\\mu \\sigma }$ ($c^\\dagger _{i\\mu \\sigma }$ ) creates (annihilates) an electron of type $\\mu =0$ for A sites and $\\mu =1$ for B sites on the unit cell $i$ with spin $\\sigma =\\uparrow (\\downarrow )$ for up (down) electrons.",
"$\\vec{m}$ and $\\vec{\\sigma }$ respectively represent a unit vector that points in the direction of the magnetization and the vector of Pauli matrices, so that $\\vec{m}.\\vec{\\sigma }= m_x \\sigma _x+m_y\\sigma _y+m_z\\sigma _z$ .",
"The anisotropic hopping $t_{l\\sigma }$ connects unit cells $i$ to their nearest neighbor cells $i+l$ .",
"Parameters $\\delta $ , $\\Delta _\\delta $ , $\\Delta _s$ are defined via exchange spin-splittings ${{\\delta }_{e}}$ (${{\\delta }_{h}}$ ) of the electrons (holes) and spin-dependent band gaps $\\Delta _{\\sigma }$ defined in Ref. [ali2017].",
"$E_D$ indicates the Dirac cone position with respect to the Fermi level.",
"The Hamiltonian for the whole device is obtained by making aforementioned parameters spatially dependent.",
"Figure: (color online) Band structure obtained using tight-binding Hamiltonian defined by Eq.",
"() (solid lines) fitted to the band structure from DFT spin majority (green open circles) and spin minority (black filled circles) data for the cases with (a) YIG, (b) CFO, (c) EuS and (d) EuO from Ref. [ali2017].",
"The inset in (b) shows the anisotropic hoppings reported in Table To obtain hopping parameters of Hamiltonian (REF ), we fitted tight-binding bands to those obtained from first principles calculations in Ref. [ali2017].",
"The results of the fitting procedure in case of graphene magnetized by YIG, CFO, EuS and EuO are shown in Fig.",
"REF (a), (b), (c) and (d), respectively.",
"The corresponding hopping parameters are given in Table REF .",
"As one can see, the graphene bands obtained with tight-binding Hamiltonian given by Eq.",
"REF are in good agreement with those obtained using Density Functional Theory (DFT) confirming suitability of our model for transport calculations.",
"Of note, due to the presence of superficial tension at the interface between CFO and graphene, hopping parameters in this case are anisotropic as they depend on direction to the nearest neighbor as specified in the inset of Fig.",
"REF (b).",
"Table: hopping parameters used in equation for each magnetic insulator considered.The conductance for parallel and antiparallel configurations of magnetizations M$_1$ and M$_2$ in the linear response regime is then obtained according to: $G_{P(AP)}=\\frac{e}{h}\\sum _{\\sigma }\\int T_{P(AP)}^\\sigma \\left(\\frac{-\\partial f}{\\partial E} \\right)dE ,$ where $T_{P(AP)}^\\sigma $ indicates spin-dependent transmission probability for parallel(antiparallel) magnetizations configurations and $f=1/(e^{(E-\\mu )/k_BT}+ 1)$ represents the Fermi-Dirac distribution with $\\mu $ and $T$ being electrochemical potential (Fermi level) and temperature, respectively.",
"It is important to note that temperature smearing has been taken into account using the Curie temperature of each MI.",
"The PMR amplitude has been defined according to following expression: $\\textrm {PMR}=\\left(\\frac{G_P -G_{AP}}{G_P +G_{AP}}\\right)\\times 100 \\% ,$ In order to determine the impact of the system dimensions on the PMR, several calculations were carried out for different lengths, widths and separations of the magnetic regions.",
"Furthermore, we checked the robustness of PMR on edge type termination by calculating the PMR for systems with zigzag, armchair and rough edges.",
"The latter were created by removing atoms and bounds randomly and deleting the dangling atoms at the new edges."
],
[
"Results",
"In Fig.",
"REF we present the PMR curves for lateral device structures based on YIG, CFO, EuS and EuO on top of a graphene sheet with armchair edges.",
"Taking into account Curie temperatures for these materials, the curves were smeared out using 16 K (70 K) for EuS (EuO), and 300 K for YIG and CFO cases.",
"For system with YIG we found a maximum PMR value of 77% while for CFO the value obtained was 22%.",
"In case of chacolgenides EuS and EuO used, the maximum PMR values reach 100%.",
"Among the materials studied, YIG represents the most suitable candidate for lateral spintronic applications due to both high Curie temperature and considerably large PMR value.",
"Figure: (color online) Proximity magnetoresistance defined by Eq.",
"as a function of energy in respect to the Fermi level for YIG (blue circles), CFO(red squares), EuS(black diamonds) and EuO(green triangles) using temperature smeared conductances at T=300 K, 300 K, 16 K and 70 K, respectively.",
"System dimensions are L=49.2L=49.2 nm, W=39.6W=39.6 nm and d=1.5d=1.5 nm.In order to elucidate the underlying physics behind these PMR results, let us analyze details of the conductance behaviour.",
"In Fig.",
"REF (a)-(b) we reproduce the graphene bands in proximity of YIG and corresponding transmission probabilities resolved in spin for P and AP configurations at $T=0$ K for a system with dimensions $L=49.2$ nm, $W=39.6$ nm and $d=1.5$ nm.",
"One can see that for energies between -0.88 eV and -0.78 eV there is no majority spin states present and the only contribution to transmission $T_{P}^\\downarrow $ is from minority spin channel (Fig.",
"REF (b), red solid line).",
"In other words, the situation within this energy range is half-metallic giving rise to maximum PMR values of 100% using “pessimistic\" definition given by Eq.",
"(REF ).",
"The similar situation is for energy ranges between -0.72 eV and -0.75 eV but this time the only contribution $T_{P}^\\uparrow $ is from majority spin channel (Fig.",
"REF (b), red dashed line).",
"One should point out here that the conduction profile here is due combining both magnetic and nonmagnetic regions into one scattering region.",
"The conductance of a pure graphene nanoribbon sheet represents quantized steps due to transverse confinement with no conductivity at zero energy depending on its edges.",
"Inducing magnetism within graphene sheet leads to symmetry breaking with the shift of exchange splitted gaps in the vicinity of Dirac cone region below the Fermi level.",
"This leads to characteristic conductance profile with two minima at around -0.8 eV and 0 eV (not shown here) due to the Dirac cone regions of magnetized and the pure graphene.",
"The corresponding conductances for the parallel ($G_P$ ) and for the antiparallel ($G_{AP}$ ) magnetic configurations at $T=300$ K are shown in Fig REF (c).",
"Interestingly, even at room temperature the PMR for YIG based structure preserves a very high value of about 77% as already pointed above, a behavior that is very encouraging for future experiments on PMR.",
"As a guide to the eye with dashed lines we highlight the energy value where the PMR has a maximum in Fig.",
"REF .",
"Figure: (color online) (a) Band structure reproduced using the DFT parameters from Ref.",
"[ali2017] for graphene in proximity of YIG.",
"(b) Transmission probabilities for majority (dash lines) and minority (solid) spin channel for parallel (red) and antiparallel (blue) magnetization configurations at T=0T=0 K for a system with dimensions L=49.2L=49.2 nm, W=39.6W=39.6 nm and d=1.5d=1.5 nm.",
"(c) Resulting conductance for parallel (red circles) and antiparallel (blue squares) magnetization configurations at 300 K. (d) PMR for device with armchair (blue circles), rough (red squares) and zigzag (black triangles) edge termination of graphene.",
"PMR profiles as a function of (e) LL, (f) WW and (g) dd.",
"(h) Dependence of PMR for the energy outlined by dashed line in (e), (f) and (g) as a function of LL (black circles), WW (red squares) and dd (blue triangles).",
"The green square highlights the region where PMR becomes independent of system dimensions.Since the edges may strongly influence the aforementioned properties of the system, we next explore the robustness of PMR against different edge types of the graphene channel of the proposed device.",
"It is well known that electric field can trigger half-metallicity in zigzag nanoribbons due to the antiferromagnetic interaction of the edges [31].",
"On the other hand, graphene nanoribbons with armchair edges can display insulating or metallic behaviour depending on graphene nanoribbon (GNR) width [32], [33].",
"Armchair and zigzag edges are particular cases and the most symmetric edge directions in graphene.",
"But one can cut GNR at intermediate angular direction between these two limiting cases giving rise to an intermediate direction characterized by a chirality angle $\\theta $[34].",
"Graphene band structure is highly dependent on $\\theta $ .",
"When the angle is increased, the length of the edge states localized at the Fermi level decrease and eventually disappear in the limiting case when $\\theta =30^{\\circ }$ , i. e. when acquires armchair edge.",
"In the laboratory conditions, graphene sheets are finite and have imperfections that influence their transport properties.",
"For defects at the edges, it has been demonstrated that rough edges can diminish the conductance of a graphene nanoribbon as was shown in Ref.",
"[libisch2012-rough] or may exhibit a nonzero spin conductance as reported in Ref. [wimmer2008-rough].",
"In order to demonstrate the robustness of PMR with respect to the edge type, we thus performed calculations with the same system setup (Fig.",
"REF ) but this time for various edge terminations.",
"The resulting PMR behavior for the cases with armchair, rough edges and zigzag are shown in Fig.",
"REF (d).",
"The former have been modeled by creating extended vacancies distributed randomly.",
"It is clear that the maximum PMR value does not present a significant variation maintaining for all cases PMR values around 75%.",
"With this results in hand we can claim that the PMR is indeed robust with respect to edge termination type.",
"As a next step, we checked the dependence of the PMR on different system dimensions, i.e.",
"the length of the magnetic region $L$ , system width $W$ and the separation between the magnetic regions $d$ .",
"The corresponding dependences are presented respectively in Fig.",
"REF (e),(f) and (g).",
"One can see that for all energy ranges the PMR ratio has a tendency to increase as a function of $L$ approaching limiting value of 77% at energies around -0.81 eV indicated by a dashed line Fig.",
"REF (e).",
"As for dependence of the PMR as a function of GNR width $W$ , clear oscillations due to quantum well states formation are present with a tendency to vanish as system widens (Fig.",
"REF (f)).",
"On a contrary, the PMR shows almost constant behavior as a function of separation between the magnets $d$ (Fig.",
"REF (g)) due to the fact that transport is in ballistic regime.",
"For convenience, we summarize all these dependencies in Fig.",
"REF (h) at energy -0.81 eV as a function of $L$ , $W$ and $d$ .",
"One can clearly see that the PMR saturates as system dimensions are increased.",
"At the same time, it shows the oscillations in the PMR for small $W$ as well as the invariance of the PMR with respect to $d$ .",
"For large dimensions highlighted by the green box in Fig.",
"REF (h), we can claim that the PMR is indeed robust, and the maximum PMR value would be eventually limited only by the magnitude of the spin diffusion length in the system.",
"Finally, we consider the impact of spin-orbit coupling on the PMR.",
"Despite weak SOC within graphene, the proximity of adjacent materials can induce the interfacial Rashba SOC [7].",
"Rashba type SOC is included into our tight-binding approach adding the following term: $H_{SO} = i \\lambda _R \\sum _{i\\sigma \\sigma ^{\\prime }} \\sum _lc^\\dagger _{(i+l)1\\sigma } [\\sigma ^x_{\\sigma \\sigma ^{\\prime }} d_l^x - \\sigma ^y_{\\sigma \\sigma ^{\\prime }} d_l^y] c_{i0\\sigma ^{\\prime }} +h.c.$ where the vector $\\vec{d}_l=(d_l^x,d_l^y)$ connects the two nearest neighbours, $\\lambda _R$ indicates the SOC strength.",
"The values of $\\lambda _R$ are generally lie in the range between 1-10 meV (see, for instance, in Ref. [macdonald2014-prl]).",
"Keeping in mind this information, we present in in Fig.",
"REF the PMR dependences for three values of spin-orbit interaction.",
"One can see that increasing the strength of SOC $\\lambda _R$ lower the PMR.",
"This behavior is expected and could be attributed to the fact that spin-orbit interaction mixes the spin channels.",
"These dependencies allows us to conclude that PMR is quite robust also against SOC and even in the worst scenario remains of the order of 50 % (cf.",
"black triangles and blue circles in Fig.",
"REF ).",
"Figure: (color online) PMR dependencies for three values of Rashba spin-orbit interaction parameter λ R \\lambda _R defined by Eq.",
"() for YIG-based system with armchair edges and of dimensions L=49.2L=49.2 nm, W=39.6W=39.6 nm and d=1.5d=1.5 nm.",
"The dashed line is a guide to the eye that shows the maximum value when λ R =0\\lambda _R=0 eV."
],
[
"Conclusions",
"In this paper we introduced the proximity induced magnetoresistance phenomenon in graphene based lateral system comprising regions with proximity induced magnetism by four different magnetic insulators.",
"For YIG and CFO based devices we found PMR ratios of 77% and 22% at room temperature, respectively.",
"For chalcogenide based systems, i.e.",
"with EuS and EuO, we found PMR values of 100% for both at 16 K and 70 K, respectively.",
"Very importantly, it is demonstrated that the PMR is robust with respect to system dimensions and edge type termination.",
"Furthermore, the PMR survives in case of the presence of SOC decreasing only by about a half even in the case of considerably big SOC strength values.",
"We hope this work will encourage further experimental research and will be useful for the development of novel generation of spintronic devices based on generation and exploring spin currents without passing charge currents across ferromagnets."
],
[
"acknowledgments",
"We thank J. Fabian and S. Roche for fruitful discussions.",
"This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreements No.",
"696656 and 785219 (Graphene Flagship).",
"X.W.",
"acknowledge support by ANR Gransport."
]
] | 1906.04469 |
[
[
"Twisted and untwisted negativity spectrum of free fermions"
],
[
"Abstract A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion.",
"Such criterion is based on the observation that the spectrum of the partially transposed density matrix of an entangled state contains negative eigenvalues, in turn, used to define an entanglement measure called the logarithmic negativity.",
"Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the operation of PT to fermionic systems remained a technical challenge until recently when a more natural definition of PT for fermions that accounts for the Fermi statistics has been put forward.",
"In this paper, we study the many-body spectrum of the reduced density matrix of two adjacent intervals for one-dimensional free fermions after applying the fermionic PT.",
"We show that in general there is a freedom in the definition of such operation which leads to two different definitions of PT: the resulting density matrix is Hermitian in one case, while it becomes pseudo-Hermitian in the other case.",
"Using the path-integral formalism, we analytically compute the leading order term of the moments in both cases and derive the distribution of the corresponding eigenvalues over the complex plane.",
"We further verify our analytical findings by checking them against numerical lattice calculations."
],
[
"Abstract",
"A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion.",
"Such criterion is based on the observation that the spectrum of the partially transposed density matrix of an entangled state contains negative eigenvalues, in turn, used to define an entanglement measure called the logarithmic negativity.",
"Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the operation of PT to fermionic systems remained a technical challenge until recently when a more natural definition of PT for fermions that accounts for the Fermi statistics has been put forward.",
"In this paper, we study the many-body spectrum of the reduced density matrix of two adjacent intervals for one-dimensional free fermions after applying the fermionic PT.",
"We show that in general there is a freedom in the definition of such operation which leads to two different definitions of PT: the resulting density matrix is Hermitian in one case, while it becomes pseudo-Hermitian in the other case.",
"Using the path-integral formalism, we analytically compute the leading order term of the moments in both cases and derive the distribution of the corresponding eigenvalues over the complex plane.",
"We further verify our analytical findings by checking them against numerical lattice calculations."
],
[
"Introduction",
"Entanglement is an intrinsic property of quantum systems beyond classical physics.",
"Having efficient frameworks to compute entanglement between two parts of a system is essential not only for fundamental interests such as characterizing phases of matter [1], [2], [3], [4] and spacetime physics [5] but also for application purposes such as identifying useful resources to implement quantum computing processes.",
"For a bipartite Hilbert space $\\mathcal {H}_A\\otimes \\mathcal {H}_B$ , it is easy to determine whether a pure state $\\mathinner {|{\\Psi }\\rangle }$ is entangled or not: a product state, i.e.",
"any state of the form $\\mathinner {|{\\Phi _A}\\rangle }\\otimes \\mathinner {|{\\Phi _B}\\rangle }$ , is unentangled (separable), while a superposition state $\\mathinner {|{\\Psi }\\rangle }=\\sum _i \\alpha _i \\mathinner {|{\\Phi _A^{(i)}}\\rangle }\\otimes \\mathinner {|{\\Phi _B^{(i)}}\\rangle }$ , where $\\mathinner {|{\\Phi _{A/B}^{(i)}}\\rangle }$ is a set of local orthogonal states, is entangled.",
"The amount of entanglement in a given state can be quantified by the entropy of information within either subsystem $A$ or $B$ , in the form of the von Neumann entropy $S(\\rho _A)= -\\text{Tr}(\\rho _A \\ln \\rho _A)= -\\sum _i \\alpha _i^2 \\ln \\alpha _i^2,$ or the Rényi entanglement entropies (REEs) ${\\cal R}_n(\\rho _A)= \\frac{1}{1-n} \\ln \\text{Tr}(\\rho _A^n)= \\frac{1}{1-n} \\ln \\sum _i \\alpha _i^{2n},$ where $\\rho _A=\\text{Tr}_B(\\mathinner {|{\\Psi }\\rangle }\\mathinner {\\langle {\\Psi }|})=\\sum _i \\alpha _i^2 \\mathinner {|{\\Phi _A^{(i)}}\\rangle } \\mathinner {\\langle {\\Phi _A^{(i)}}|}$ is the reduced density matrix acting on $\\mathcal {H}_A$ .",
"Notice that $S(\\rho _A)=S(\\rho _B)$ and ${\\cal R}_n(\\rho _A)={\\cal R}_n(\\rho _B)$ and clearly, $S,{\\cal R}_n\\ge 0$ where the equality holds for a product state.",
"For analytical calculations, $S$ is usually obtained from ${\\cal R}_n$ via $S=\\lim _{n\\rightarrow 1}{\\cal R}_n$ .",
"It is well-known that eigenvalues of density matrices, i.e.",
"the entanglement spectrum, contains more information than merely the entanglement entropies.",
"The entanglement spectrum has been studied and utilized toward better understanding of the phases of matter [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], broken-symmetry phases [24], [25], [26], [27], [28], and more exotic phases such as many-body localized states [29], [30], [31].",
"In particular, in the context of conformal field theories (CFTs) in (1+1)d the distribution of eigenvalues was analytically derived [32] and was shown to obey a universal scaling function which depends only on the central charge of the underlying CFT.",
"The obtained scaling function for the distribution of the entanglement spectrum at criticality was further substantiated numerically [33], [34], especially for matrix product state representation at critical points [35], [36], [37].",
"It turned out that extending the above ideas to mixed states where the system is described by a density matrix $\\rho $ is not as easy as it may seem.",
"A product state $\\rho =\\rho _A\\otimes \\rho _B$ is similarly unentangled.",
"However, a large class of states, called separable states, in the form of $\\rho =\\sum _i p_i \\rho _A^{(i)} \\otimes \\rho _B^{(i)}$ with $p_i\\ge 0$ are classically correlated and do not contain any amount of entanglement.",
"Hence, the fact that superposition implies entanglement in pure states does not simply generalize to the entanglement in mixed states.",
"The positive partial transpose (PPT) [38], [39], [40], [41], [42], [43], [44] is a test designed to diagnose separable states based on the fact that density matrices are positive semi-definite operators.",
"The PT of a density matrix $\\rho =\\sum _i \\rho _{ijkl} \\mathinner {|{e_A^{(i)}, e_B^{(j)}}\\rangle } \\mathinner {\\langle {e_A^{(k)}, e_B^{(l)}}|}$ written in a local orthonormal basis $\\lbrace \\mathinner {|{e_A^{(k)}}\\rangle }, \\mathinner {|{e_B^{(j)}}\\rangle } \\rbrace $ is defined by exchanging the indices of subsystem $A$ (or $B$ ) as in $ \\rho ^{T_A}=\\sum _i \\rho _{ijkl} \\mathinner {|{e_A^{(k)}, e_B^{(j)}}\\rangle } \\mathinner {\\langle {e_A^{(i)}, e_B^{(l)}}|}.$ Note that $\\rho ^{T_A}$ is a Hermitian operator and the PPT test follows by checking whether or not $\\rho ^{T_A}$ contains any negative eigenvalue.",
"A separable state passes the PPT test, i.e.",
"all the eigenvalues of $\\rho ^{T_A}$ are non-negative, whereas an inseparable (i.e., entangled) state yields negative eigenvalues after PTA technical point is that there exists a set of inseparable states which also pass the PPT test [45].",
"They are said to contain bound entanglement which cannot be used for quantum computing processes such as teleportation [46].",
"This issue is beyond the scope of our paper and we do not elaborate further here..",
"Hence, the PPT criterion can be used to decide whether a given density matrix is separable or not.",
"Similar to the entropic measures of pure-state entanglement in (REF ) and (REF ), the (logarithmic) entanglement negativity associated with the spectrum of the partially transposed density matrix is defined as a candidate to quantify mixed-state entanglement [47], [48], [49], $ {\\cal N}(\\rho ) &= \\frac{\\left\\Vert {\\rho ^{T_A}} \\right\\Vert -1}{2}, \\\\{\\cal E}(\\rho ) &= \\ln \\left\\Vert {\\rho ^{T_A}} \\right\\Vert ,$ where $\\left\\Vert A\\right\\Vert = \\text{Tr}\\sqrt{A A^\\dag }$ is the trace norm.",
"When $A$ is Hermitian, the trace norm is simplified into the sum of the absolute value of the eigenvalues of $A$ .",
"Hence, the above quantities measure the negativity of the eigenvalues of $\\rho ^{T_A}$ .",
"It is also useful to define the moments of the PT (aka Rényi negativity) via $ {\\cal N}_n(\\rho ) =\\ln \\text{Tr}({\\rho ^{T_A}} )^n,$ where the logarithmic negativity is obtained from analytic continuation $ {\\cal E}(\\rho )= \\lim _{n\\rightarrow 1/2} {\\cal N}_{2n}(\\rho ).$ Note that for a pure state $\\rho =\\mathinner {|{\\Psi }\\rangle }\\mathinner {\\langle {\\Psi }|}$ , we have ${\\cal E}(\\rho )={\\cal R}_{1/2}(\\rho _A)$ where $\\rho _A$ is the reduced density matrix on $\\mathcal {H}_A$ .",
"The entanglement negativity has been used to characterize mixed states in various quantum systems such as in harmonic oscillator chains [50], [51], [52], [53], [54], [55], [56], [57], [58], quantum spin models [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], (1+1)d conformal and integrable field theories [69], [70], [71], [72], [73], [74], [75], [76], topologically ordered phases of matter in (2+1)d [77], [78], [79], [80], [81], and in out-of-equilibrium situations [82], [83], [84], [85], [86], [87], as well as holographic theories [88], [89], [90], [91], [92], [93] and variational states [94], [95], [96], [97].",
"Moreover, the PT was used to construct topological invariants for symmetry protected topological (SPT) phases protected by anti-unitary symmetries [98], [99], [100], [101] and there are experimental proposals to measure it with cold atoms [102], [103].",
"Unlike the entanglement spectrum which has been studied extensively, less is known about the spectrum of partially transposed density matrices in many-body systems.",
"It is true that the PPT test which predates the entanglement spectrum is based on the eigenvalues of the PT, but the test only uses the sign of the eigenvalues.",
"Therefore, studying the spectrum of the PT could be useful in characterizing quantum phases of matter.",
"Furthermore, the fact that PT is applicable to extract entanglement at finite-temperature states and that the eigenvalues have a sign structure (positive/negative) may help unravel some new features beyond the entanglement spectrum.",
"Recently, the distribution of eigenvalues of the PT, dubbed as the negativity spectrum, was studied for CFTs in (1+1)d [72].",
"It was found that the negativity spectrum is universal and depends only on the central charge of the CFT, similar to the entanglement spectrum, while the precise form of the spectrum depends on the sign of the eigenvalues.",
"This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges.",
"In this paper, we would like to study the negativity spectrum in fermionic systems.",
"The PT of fermionic density matrices however involves some subtleties due to the Fermi statistics (i.e., anti-commutation relation of fermion operators).",
"Initially, a procedure for the PT of fermions based on the fermion-boson mapping (Jordan-Wigner transformation) was proposed [104] and was also used in the subsequent studies [105], [106], [107], [108], [109], [110], [111].",
"However, this definition turned out to cause certain inconsistencies within fermionic theories such as violating the additivity property and missing some entanglement in topological superconductors, and give rise to incorrect classification of time-reversal symmetric topological insulators and superconductors.",
"Additionally, according to this definition it is computationally hard to find the PT (and calculate the entanglement negativity) even for free fermions, since the PT of a fermionic Gaussian state is not Gaussian.",
"This motivates us to use another way of implementing a fermionic PT which was proposed recently by some of us in the context of time-reversal symmetric SPT phases of fermions [100], [101], [112].",
"This definition does not suffer from the above issues and at the same time the associated entanglement quantity is an entanglement monotone [113].",
"From a practical standpoint, the latter definition has the merit that the partially transposed Gaussian state remains Gaussian and hence can be computed efficiently for free fermions.",
"A detailed survey of differences between the two definitions of PT from both perspectives of quantum information and condensed matter theory (specifically, topological phases of fermions) is discussed in Refs.",
"[112], [113].",
"Before we get into details of the fermionic PT in the coming sections, let us finish this part with a summary of our main findings.",
"We study the distribution of the many-body eigenvalues $\\lambda _i$ of the partially transposed reduced density matrix, $ P(\\lambda )=\\sum _i \\delta (\\lambda -\\lambda _i)$ for one-dimensional free fermions.",
"As a lattice realization, we consider the hopping Hamiltonian on a chain $ \\hat{H}=- \\sum _{j} [t ( f_{j+1}^\\dag f_j + \\text{H.c.}) +\\mu f_j^\\dag f_j ],$ where the fermion operators $f_j$ and $f_j^\\dag $ obey the anti-commutation relation $\\lbrace f_i,{f_j}^\\dag \\rbrace =f_i{f_j}^\\dag +{f_j}^\\dag f_i=\\delta _{ij}$ and $\\lbrace f_i,f_j\\rbrace =\\lbrace {f_i}^\\dag ,{f_j}^\\dag \\rbrace =0$ .",
"Recall that using the regular (matrix) PT – we will refer to it as the bosonic PT –, which applies to generic systems where local operators commute, the obtained PT density matrix is a Hermitian operator and its eigenvalues are either negative or positive.",
"However, it turned out that for fermions a consistent definition of PT involves a phase factor as we exchange indices in (REF ) and in general one can define two types of PT operation.",
"As we will explain in detail, these two types correspond to the freedom of spacetime boundary condition for fermions associated with the fermion-number parity symmetry.",
"We reserve ${\\rho ^{T_A}} $ and $\\rho ^{\\widetilde{T}_A}$ to denote the fermionic PT which leads to anti-periodic (untwisted) and periodic (twisted) boundary conditions along fundamental cycles of the spacetime manifold, respectively.",
"We should note that ${\\rho ^{T_A}} $ is pseudo-HermitianA pseudo-Hermitian operator $H$ is defined by $\\eta H^\\dag \\eta ^{-1}=H$ with $\\eta ^2=1$ where $\\eta $ is a unitary Hermitian operator satisfying $\\eta ^\\dag \\eta =\\eta \\eta ^\\dag =1$ and $\\eta =\\eta ^\\dag $ .",
"Essentially, pseudo-Hermiticity is a generalization of Hermiticity, in that it implies Hermiticity when $\\eta =1$ .",
"and may contain complex eigenvalues, while $\\rho ^{\\widetilde{T}_A}$ is Hermitian and its eigenvalues are real.",
"We use the spacetime path integral formulation to analytically calculate the negativity spectrum.",
"In the case of $\\rho ^{\\widetilde{T}_A}$ , we obtain results very similar to those of previous CFT work [72], where the distribution of positive and negative eigenvalues are described by two universal functions.",
"In the case of ${\\rho ^{T_A}} $ , we observe that the eigenvalues are complex but they have a pattern and fall on six branches in complex plane with a quantized complex phase of $\\angle \\lambda =2\\pi n/6$ .",
"We show that the spectrum is reflection symmetric with respect to the real axis and the eigenvalue distributions are described by four universal functions along $\\angle \\lambda =0, \\pm 2\\pi /6, \\pm 4\\pi /6, \\pi $ branches.",
"We further verify our findings by checking them against numerical lattice simulations.",
"The rest of our paper is organized as follows: in Section we provide a brief review of partial transpose for fermions, in Section we discuss the spacetime path-integral formulation of the moments of partially transposed density matrices.",
"The spectrum of the twisted and untwisted partial transpose is analytically derived in Section for different geometries, where numerical checks with free fermions on the lattice are also provided.",
"We close our discussion by some concluding remarks in Section .",
"In several appendices, we give further details of the analytical calculations and make connections with other related concepts."
],
[
"Preliminary remarks",
"In this section, we review some basic materials which we use in the next sections: the definition of PT for fermions, how to extract the distribution of the eigenvalues of an operator from its moments, and some properties of partially transposed Gaussian states."
],
[
"Twisted and untwisted partial transpose for fermions",
"In this part, we briefly discuss some background materials on our definitions of PT for fermions.",
"More details can be found in Refs.",
"[112], [113].",
"We consider a fermionic Fock space ${\\cal H}$ generated by $N$ local fermionic modes $f_j$ , $j=1,\\cdots ,N$ .",
"The Hilbert space is spanned by $\\mathinner {|{n_1,n_2,\\cdots ,n_N}\\rangle }$ which is a string of occupation numbers $n_j=0,1$ .",
"We define the Majorana (real) fermion operators in terms of canonical operators as $c_{2j-1}:=f^{\\dag }_j+f_j, \\quad c_{2j}:=i(f_j-f_j^{\\dag }), \\quad j=1, \\dots , N.$ These operators satisfy the commutation relation $\\lbrace c_j, c_k \\rbrace = 2 \\delta _{jk}$ and generate a Clifford algebra.",
"Any operator $X$ acting on $\\mathcal {H}$ can be expressed in terms of a polynomial of $c_j$ 's, $X = \\sum _{k=1}^{2N} \\sum _{p_1<p_2 \\cdots <p_k} X_{p_1 \\cdots p_k} c_{p_1} \\cdots c_{p_k},$ where $X_{p_1 \\dots p_k}$ are complex numbers and fully antisymmetric under permutations of $\\lbrace 1, \\dots , k\\rbrace $ .",
"A density matrix has an extra constraint, i.e., it commutes with the total fermion-number parity operator, $[\\rho ,(-1)^F]=0$ where $F=\\sum _j f_j^\\dag f_j$ .",
"This constraint entails that the Majorana operator expansion of $\\rho $ only contains even number of Majorana operators, i.e., $k$ in the above expression is even.",
"To study the entanglement, we consider a bipartite Hilbert space $\\mathcal {H}_A\\otimes \\mathcal {H}_B$ spanned by $f_j$ with $j=1,\\cdots ,N_A$ in subsystem $A$ and $j=N_A+1,\\cdots ,N_A+N_B$ in subsystem $B$ .",
"Then, a generic density matrix on $\\mathcal {H}_A\\otimes \\mathcal {H}_B$ can be expanded in Majorana operators as $ \\rho = \\sum _{k_1,k_2}^{k_1+k_2 = {\\rm even}} \\rho _{p_1 \\cdots p_{k_1}, q_1 \\cdots q_{k_2}} a_{p_1} \\cdots a_{p_{k_1}} b_{q_1} \\cdots b_{q_{k_2}},$ where $\\lbrace a_j \\rbrace $ and $\\lbrace b_j \\rbrace $ are Majorana operators acting on $\\mathcal {H}^A$ and $\\mathcal {H}^B$ , respectively, and the even fermion-number parity condition is indicated by the condition $k_1+k_2 = {\\rm even}$ .",
"Our definition of the PT for fermions is given by [112], [101] ${\\rho ^{T_A}} := \\sum _{k_1,k_2}^{k_1+k_2 = {\\rm even}} \\rho _{p_1 \\cdots p_{k_1}, q_1 \\cdots q_{k_2}} i^{k_1} a_{p_1} \\cdots a_{p_{k_1}} b_{q_1} \\cdots b_{q_{k_2}},$ and similarly for $ \\rho ^{T_B}$ .",
"It is easy to see that the subsequent application of the PT with respect to the two subsystems leads to the full transpose $(\\rho ^{T_A})^{T_B}=\\rho ^{T}$ , i.e.",
"reversing the order of Majorana fermion operators.",
"In addition, the definition (REF ) implies that $({\\rho ^{T_A}} )^\\dag &=(-1)^{F_A} {\\rho ^{T_A}} (-1)^{F_A},\\\\({\\rho ^{T_A}} )^{ T_A} &=(-1)^{F_A} \\rho (-1)^{F_A},$ where $(-1)^{F_A}$ is the fermion-number parity operator on $H_A$ , i.e.",
"$F_A=\\sum _{j\\in A} f_j^\\dag f_j$ .",
"The first identity, namely the pseudo-Hermiticity, can be understood as a consequence of the fact that $({\\rho ^{T_A}} )^\\dag $ is defined the same as (REF ) by replacing $i^{k_1}$ with $(-i)^{k_1}$ .",
"The second identity reflects the fact that the fermionic PT is related to the action of time-reversal operator of spinless fermions in the Euclidean spacetime [101].",
"We should note that the matrix resulting from the PT is not necessarily Hermitian and may have complex eigenvalues, although $\\text{Tr}{\\rho ^{T_A}} =1$ .",
"The existence of complex eigenvalues is a crucial property which was used in the context of SPT invariants to show that the complex phase of $\\text{Tr}(\\rho {\\rho ^{T_A}} )$ , which represents a partition function on a non-orientable spacetime manifold, is a topological invariant.",
"For instance, $\\text{Tr}(\\rho {\\rho ^{T_A}} )=e^{i2\\pi \\nu /8}$ for time-reversal symmetric topological superconductors (class BDI) which implies the $\\mathbb {Z}_8$ classification.",
"(Here $\\nu \\in \\mathbb {Z}_8$ is the topological invariant).",
"Nevertheless, we may still use Eq.",
"() to define an analog of entanglement negativity for fermions and calculate the trace norm in terms of square root of the eigenvalues of the composite operator $\\rho _\\times =[({\\rho ^{T_A}} )^\\dag {\\rho ^{T_A}} ]$ , which is a Hermitian operator with real positive eigenvalues.",
"On the other hand, from Eq.",
"(REF ) we realize that $\\rho _\\times = (\\rho ^{\\widetilde{T}_A})^2$ where we introduce the twisted PT by $\\rho ^{\\widetilde{T}_A}:={\\rho ^{T_A}} (-1)^{F_A}.$ It is easy to see from Eq.",
"(REF ) that this operator is Hermitian and then similar to the bosonic PT always contains real eigenvalues.",
"It is worth noting that $ (\\rho ^{\\widetilde{T}_A})^{\\widetilde{T}_A}=\\rho ,$ in contrast with the untwisted PT ().",
"As we will see shortly, this difference between ${\\rho ^{T_A}} $ and $\\rho ^{\\widetilde{T}_A}$ in the operator formalism will show up as anti-periodic and periodic boundary conditions across the fundamental cycles of spacetime manifold in the path-integral formalism.",
"The central result of our paper is to report analytical results for the spectrum of ${\\rho ^{T_A}} $ and $\\rho ^{\\widetilde{T}_A}$ ."
],
[
"The moment problem",
"In the replica approach to logarithmic negativity () and negativity spectrum, one first has to calculate the moments of PT, aka Rényi negativity (RN), ${\\cal N}_n^{({\\rm ns})}(\\rho )=\\ln \\text{Tr}[({\\rho ^{T_A}} )^n], \\qquad {\\cal N}_n^{({\\rm r})}(\\rho )=\\ln \\text{Tr}[(\\rho ^{\\widetilde{T}_A})^n],$ which are fermionic counterparts of the bosonic definition in Eq.",
"(REF ).",
"The superscripts $(\\textrm {ns})$ and $(\\textrm {r})$ stand for Neveu-Schwarz and Ramond respectively (the reason for this will be clear from the path integral representation of such quantities, see Section below).",
"Thus, the analog of analytic continuation (REF ) to obtain the logarithmic negativity is $ {\\cal E}(\\rho )= \\lim _{n\\rightarrow 1/2} {\\cal N}_{2n}^{({\\rm r})}.$ In the following, we review a general framework to analytically obtain the distribution of eigenvalues of density matrix (or its transpose) from the moments.",
"This method was originally used to derive the entanglement spectrum of (1+1)d CFTs [32].",
"Suppose we have an operator ${\\cal O}$ whose moments are of the form $ R_n := \\text{Tr}[{\\cal O}^n].$ In terms of the eigenvalues of ${\\cal O}$ , $\\lbrace \\lambda _j \\rbrace $ , we have $R_n=\\sum _j \\lambda _j^n = \\int P(\\lambda ) \\lambda ^n d\\lambda $ , where $P(\\lambda )$ is the associated distribution function (see Eq.",
"(REF )).",
"The goal is to find $P(\\lambda )$ by making use of the specific form of $R_n$ in (REF ).",
"The essential idea is to compute the Stjilties transform $ f (s) := \\frac{1}{\\pi } \\sum _{n=1}^{\\infty } R_n s^{-n}= \\frac{1}{\\pi } \\int d\\lambda \\frac{\\lambda P(\\lambda )}{s-\\lambda }.$ Assuming that the eigenvalues are real, the distribution function can be easily read off from the relation $ P (\\lambda ) = \\frac{1}{\\lambda } \\lim _{\\epsilon \\rightarrow 0} \\text{Im}f (\\lambda - i \\epsilon ).$ In the following we are going to focus on the complementary cumulative distribution function or simply the tail distribution, being a very simple object to be accessed for numerical comparison $n(\\lambda )=\\int _\\lambda ^{\\lambda _\\text{max}} d\\lambda P(\\lambda ).$ For specific types of operators such as the density matrices and their PT in (1+1)d CFTs, the moments can be cast in the form, $ R_n= r_n \\exp \\left(- b n + \\frac{a}{n} \\right), \\qquad \\forall n,$ where $a, b\\in \\mathbb {R}$ , $b>0$ and $r_n$ are non-universal constant.",
"In such cases, the distribution function is found to be [63] $P (\\lambda ; a, b) &=\\left\\lbrace \\begin{array}{ll}\\frac{a \\, \\theta (e^{-b} - \\lambda )}{\\lambda \\sqrt{a \\ln (e^{-b}/\\lambda )}} I_1 (2 \\sqrt{a \\ln (e^{-b}/\\lambda )})+ \\delta (e^{-b}- \\lambda ), & a>0,\\\\\\frac{- |a| \\, \\theta (e^{-b} - \\lambda )}{\\lambda \\sqrt{|a| \\ln (e^{-b}/\\lambda )}} J_1 (2 \\sqrt{|a| \\ln (e^{-b}/\\lambda )}) + \\delta (e^{-b}- \\lambda ) , & a<0,\\end{array}\\right.$ and the corresponding tail distribution is given by $n (\\lambda ; a, b) &=\\left\\lbrace \\begin{array}{ll}I_0 (2 \\sqrt{a \\ln (e^{-b}/\\lambda )}) , & a>0,\\\\J_0 (2 \\sqrt{|a| \\ln (e^{-b}/\\lambda )}), & a<0,\\end{array}\\right.$ where $J_\\alpha (x)$ and $I_\\alpha (x)$ are the regular Bessel functions and modified Bessel functions of the first kind, respectively.",
"Note that (REF ) and (REF ) are derived by ignoring the presence of the constants $r_n$ in (REF ).",
"This relies on the assumption that they do not change significantly upon varying $n$ , i.e., $\\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\ln r_n<\\infty $ .",
"The very same assumption has been adopted for the entanglement and bosonic negativity spectrum in Refs.",
"[32] and [72] where the derived distribution functions agree with the numerically obtained spectra."
],
[
"Partial transpose of Gaussian states",
"Here, we discuss how to compute the spectrum of the PT of a Gaussian state from the corresponding covariance matrix.",
"The idea is similar to that of the entanglement spectrum, while there are some differences as the covariance matrix associated with the partially transposed density matrix may contain complex eigenvalues.",
"Before we continue, let us summarize the structure of the many-body spectrum of ${\\rho ^{T_A}} $ and $\\rho ^{\\widetilde{T}_A}$ for free fermions, $\\text{Spec}[{\\rho ^{T_A}} ]&: \\left\\lbrace \\begin{array}{lll}(\\lambda _i,\\lambda _i^\\ast ) & \\text{Im}[\\lambda _i] \\ne 0, &\\\\(\\lambda _i,\\lambda _i) & \\text{Im}[\\lambda _i] = 0, & \\lambda _i<0, \\\\\\lambda _i & \\text{Im}[\\lambda _i] = 0, & \\lambda _i>0, \\\\\\end{array}\\right.\\\\\\text{Spec}[\\rho ^{\\widetilde{T}_A}]&:\\left\\lbrace \\begin{array}{ll}(\\lambda _i, \\lambda _i) & \\lambda _i<0,\\\\\\lambda _i & \\lambda _i>0,\\\\\\end{array}\\right.$ where repeating values mean two fold degeneracy.",
"We should note that the pseudo-Hermiticity of ${\\rho ^{T_A}} $ (REF ) ensures that the complex-valued subset of many-body eigenvalues of ${\\rho ^{T_A}} $ appear in complex conjugate pairs.",
"This property is general and applicable to any density matrix beyond free fermions.",
"An immediate consequence of this property is that any moment of ${\\rho ^{T_A}} $ is guaranteed to be real-valued.",
"A Gaussian density matrix in the Majorana fermion basis (REF ) is defined by $ \\rho _\\Omega = \\frac{1}{{\\cal Z}(\\Omega )} \\exp \\left(\\frac{1}{4} \\sum _{j,k=1}^{2N} \\Omega _{jk} c_j c_k \\right),$ where $\\Omega $ is a pure imaginary antisymmetric matrix and ${\\cal Z}(\\Omega )=\\pm \\sqrt{\\det \\left( 2\\cosh \\frac{\\Omega }{2}\\right)}$ is the normalization constant.",
"We should note that the spectrum of $\\Omega $ is in the form of $\\pm \\omega _j,\\ j=1,\\dots ,N$ and the $\\pm $ sign ambiguity in ${\\cal Z}(\\Omega )$ is related to the square root of determinant where we need to choose one eigenvalue for every pair $\\pm \\omega _j$ .",
"The sign is fixed by the Pfaffian.",
"This density matrix can be uniquely characterized by its covariance matrix, $\\Gamma _{jk}=\\frac{1}{2}\\text{Tr}(\\rho _\\Omega [c_j,c_k]),$ which is a $2N\\times 2N$ matrix.",
"These two matrices are related by $ \\Gamma = \\tanh \\left(\\frac{\\Omega }{2} \\right), \\qquad e^\\Omega =\\frac{\\mathbb {I}+\\Gamma }{\\mathbb {I}-\\Gamma }.$ Furthermore, one can consider a generic Gaussian operator which is also defined through Eq.",
"(REF ), but without requiring that the spectrum is pure imaginary.",
"An equivalent description in terms of the covariance matrix is also applicable for such operators.",
"The only difference is that the eigenvalues do not need to be real.",
"Let us recall how Rényi entropies (REF ) are computed for Gaussian states.",
"The density matrix (REF ) can be brought into a diagonal form $\\rho _\\Omega = {\\cal Z}^{-1} \\exp \\left(\\frac{i}{2} \\sum _{n} \\omega _n d_{2n} d_{2n-1} \\right)$ , where $\\omega _n$ is obtained from an orthogonal transformation of $\\Omega $ .",
"In terms of the eigenvalues of $\\Gamma $ , denoted by $\\pm \\nu _j$ , we have $\\rho _\\Omega = \\prod _n (1+ i \\nu _n d_{2n} d_{2n-1} )/2$ , leading to ${\\cal R}_n(\\rho )=\\frac{1}{1-n} \\sum _{j=1}^N \\ln \\left[ \\left(\\frac{1-\\nu _j}{2}\\right)^n+\\left(\\frac{1+\\nu _j}{2}\\right)^n \\right].$ We consider a density matrix on a bipartite Hilbert space (REF ) where the covariance matrix takes a block matrix form as $\\Gamma = \\left( \\begin{array}{cc}\\Gamma _{AA} & \\Gamma _{AB} \\\\\\Gamma _{BA} & \\Gamma _{BB}\\end{array} \\right).$ Here, $\\Gamma _{AA}$ and $\\Gamma _{BB}$ denote the reduced covariance matrices of subsystems $A$ and $B$ , respectively; while $\\Gamma _{AB}=\\Gamma _{BA}^\\dag $ describes the correlations between them.",
"We define the covariance matrix associated with a partially transposed Gaussian state by $ \\Gamma _{\\pm } = \\left( \\begin{array}{cc}-\\Gamma _{AA} & \\pm i \\Gamma _{AB} \\\\\\pm i \\Gamma _{BA} & \\Gamma _{BB}\\end{array} \\right),$ where $[\\Gamma _+]_{ij}=\\frac{1}{2}\\text{Tr}({\\rho ^{T_A}} [c_i,c_j])$ and $[\\Gamma _-]_{ij}=\\frac{1}{2}\\text{Tr}({\\rho ^{T_A}} ^\\dag [c_i,c_j])$ .",
"We should note that $\\Gamma _+$ and $\\Gamma _-$ have identical eigenvalues while they do not necessarily commute $[\\Gamma _+,\\Gamma _-]\\ne 0$ .",
"In general, the eigenvalues of $\\Gamma _+$ appear in quartets $(\\pm \\nu _k, \\pm \\nu _k^\\ast )$ when $\\text{Re}[\\nu _k]\\ne 0$ and $\\text{Im}[\\nu _k]\\ne 0$ or doublet $\\pm \\nu _k$ when $\\text{Re}[\\nu _k]= 0$ (i.e., pure imaginary) or $\\text{Im}[\\nu _k]= 0$ (i.e., real) .",
"$\\pm $ is because of skew symmetry $\\Gamma ^T_\\pm =-\\Gamma _\\pm $ .",
"In addition, the pseudo-Hermiticity of ${\\rho ^{T_A}} $ (REF ) implies that $ \\Gamma _\\pm ^\\dag =U_1 \\Gamma _\\pm U_1,$ where $U_1=(-\\mathbb {I}_A \\oplus \\mathbb {I}_B )$ is the matrix associated with the operator $(-1)^{F_A}$ .",
"This means that for every eigenvalue $\\nu _k$ its complex conjugate $\\nu _k^\\ast $ is also an eigenvalue.",
"As a result, the moments of PT can be written as ${\\cal N}_n^{({\\rm ns})}= \\sum _{j=1}^{N} \\ln \\left| \\left(\\frac{1-\\nu _j}{2}\\right)^n+\\left(\\frac{1+\\nu _j}{2}\\right)^n \\right|.$ Note that the sum is now over half of the eigenvalues (say in the upper half complex plane), due to the structure discussed above.",
"For $\\rho ^{\\widetilde{T}_A}$ we use the multiplication rule for the Gaussian operators where the resulting Gaussian matrix is given by $ e^{\\widetilde{\\Omega }_\\pm }= \\frac{\\mathbb {I}+\\Gamma _\\pm }{\\mathbb {I}-\\Gamma _\\pm } U_1,$ which is manifestly Hermitian due to the identity (REF ).",
"Using Eq.",
"(REF ), the normalization factor is found to be ${\\cal Z}_{\\widetilde{T}_A}= \\text{Tr}(\\rho ^{\\widetilde{T}_A})=\\text{Tr}[\\rho (-1)^{F_A}]=\\sqrt{\\det \\Gamma _{AA}}$ .",
"From (REF ) we construct the covariance matrix $\\widetilde{\\Gamma }_\\pm =\\tanh (\\widetilde{\\Omega }/2)$ and compute the moments of $\\rho ^{\\widetilde{T}_A}$ by ${\\cal N}_n^{({\\rm r})}= \\sum _{j=1}^{N} \\ln \\left| \\left(\\frac{1-\\tilde{\\nu }_j}{2}\\right)^n+\\left(\\frac{1+\\tilde{\\nu }_j}{2}\\right)^n \\right| + n\\ln {\\cal Z}_{\\widetilde{T}_A},$ where $\\pm \\tilde{\\nu }_j$ are eigenvalues of $\\widetilde{\\Gamma }_\\pm $ which are guaranteed to be real.",
"Consequently, the logarithmic negativity (REF ) is given by ${\\cal E}= \\sum _{j=1}^{N} \\ln \\left[ \\left|\\frac{1-\\tilde{\\nu }_j}{2}\\right| +\\left|\\frac{1+\\tilde{\\nu }_j}{2}\\right| \\right] + \\ln {\\cal Z}_{\\widetilde{T}_A},$ For particle-number conserving systems such as the lattice model in (REF ), the covariance matrix is simplified into the form $\\Gamma =\\sigma _2 \\otimes \\gamma $ where $\\gamma =(\\mathbb {I}-2C)$ and $C_{ij}=\\text{Tr}(\\rho f_i^\\dag f_j)$ is the correlation matrix and $\\sigma _2$ is the second Pauli matrix acting on the even/odd indices of Majorana operators $(c_{2j},c_{2j-1})$ .",
"In this case, the transformed correlation matrix for ${\\rho ^{T_A}} $ is given by $ \\gamma _{\\pm } = \\left( \\begin{array}{cc}-\\gamma _{AA} & \\pm i \\gamma _{AB} \\\\\\pm i \\gamma _{BA} & \\gamma _{BB}\\end{array} \\right).$ The eigenvalues can be divided to two categories: complex eigenvalues $\\nu _k$ , $\\text{Im}[\\nu _k]\\ne 0$ and real eigenvalues $u_k$ , $\\text{Im}[u_k]= 0$ .",
"The pseudo-Hermiticity property leads to the identity $\\gamma _{\\pm }^\\dag = U_1 \\gamma _\\pm U_1$ which implies that complex eigenvalues appear in pairs $(\\nu _k,\\nu _k^\\ast )$ .",
"Therefore, the many-body eigenvalues follow the form, $ \\lambda _{\\sigma ,\\sigma ^{\\prime }}&=\\prod _{\\sigma _l}\\frac{1+\\sigma _l u_l}{2}\\prod _{\\sigma _k=\\sigma _k^{\\prime }}\\frac{1+|\\nu _k|^2 + 2\\sigma _k \\text{Re}[\\nu _k]}{4}\\prod _{\\sigma _k=-\\sigma _k^{\\prime }} \\frac{1-|\\nu _k|^2 + 2 \\sigma _k i \\text{Im}[\\nu _k]}{4},$ where $\\sigma =\\lbrace \\sigma _k=\\pm \\rbrace $ is a string of signs.",
"Clearly, the many-body eigenvalues appear in two categories as well: complex conjugate pairs $(\\lambda _j,\\lambda _j^\\ast )$ and real eigenvalues which are not necessarily degenerate.",
"We can also derive a simple expression for the correlation matrix $\\widetilde{C}=(\\mathbb {I}-\\widetilde{\\gamma })/2$ associated with $\\rho ^{\\widetilde{T}_A}$ , $ \\widetilde{\\gamma }=\\left( \\begin{array}{cc}-\\gamma _{AA}^{-1}(\\mathbb {I}_A+\\gamma _{AB}\\gamma _{BA}) & i\\gamma _{AA}^{-1} \\gamma _{AB} \\gamma _{BB}\\\\i\\gamma _{BA} & \\gamma _{BB}\\end{array} \\right).$ In the following two sections, we compute the moments of the partially transposed density matrix and ultimately the logarithmic negativity.",
"First, we develop a general method using the replica approach [69], [70], [110] and provide an equivalent spacetime picture of the Rényi negativity.",
"Before we proceed, let us briefly review the replica approach to find the entanglement entropy.",
"Next, we make connections to our construction of PT.",
"A generic density matrix can be represented in the fermionic coherent state as $\\rho = \\int d\\alpha d\\bar{\\alpha }\\ d\\beta d\\bar{\\beta }\\ \\rho (\\bar{\\alpha },\\beta ) \\mathinner {|{\\alpha }\\rangle }\\mathinner {\\langle {\\bar{\\beta }}|} e^{-\\bar{\\alpha }\\alpha - \\bar{\\beta }\\beta } ,$ where $\\alpha $ , $\\bar{\\alpha }$ , $\\beta $ and $\\bar{\\beta }$ are independent Grassmann variables and we omit the real-space (and possibly other) indices for simplicity.",
"The trace formula then reads $Z_{{\\cal R}_n}=\\text{Tr}[\\rho ^n] =& \\int \\prod _{i=1}^n d\\psi _i d\\bar{\\psi }_i \\ \\prod _{i=1}^n \\left[ \\rho (\\bar{\\psi }_i,\\psi _i)\\right] e^{\\sum _{i,j} \\bar{\\psi }_i T_{ij} \\psi _j },$ where the subscripts in $\\psi _i$ and $\\bar{\\psi }_i$ denote the replica indices and ${T}$ is called the twist matrix, $ T=\\left(\\begin{array}{cccc}0 & -1 & 0 & \\dots \\\\0 & 0 & -1 & 0 \\\\\\vdots & \\vdots & \\ddots & -1 \\\\1 & 0 & \\cdots & 0 \\\\\\end{array} \\right).$ The above expression can be viewed as a partition function on a $n$ -sheet spacetime manifold where the $n$ flavors (replicas) $\\psi _i$ are glued in order along the cuts.",
"Alternatively, one can consider a multi-component field $\\Psi = (\\psi _1,\\cdots ,\\psi _n)^T$ on a single-sheet spacetime.",
"This way when we traverse a close path through the interval the field gets transformed as $\\Psi \\mapsto T \\Psi $ .",
"Hence, each interval can be represented by two branch points ${\\cal T}_n$ and ${\\cal T}_n^{-1}$ –the so-called twist fields– and the REE of one interval can be written as a two-point correlator [114], $Z_{{\\cal R}_n}=\\mathinner {\\langle {{\\cal T}_n(u) {\\cal T}_n^{-1} (v)}\\rangle },$ where $u$ and $v$ denote the real space coordinates of the two ends of the interval defining the subsystem $A$ .",
"Figure: (a) Spacetime manifold associated with Z 𝒩 n (α){Z}_{{\\cal N}_n}(\\alpha ), Eq.",
"(), for n=4n=4.",
"The operator e iαF A e^{i\\alpha F_A} twists the boundary condition of the cycles between two successive sheets, shown as the green path with dashed lines.",
"(b) Equivalent picture in terms of twist field where we define a multi-component field on a single spacetime sheet.Let us now derive analogous relations for the moments of partially transposed density matrix.",
"Using the definition of the PT in the coherent state basis [112] $( \\mathinner {|{\\psi _A,\\psi _B }\\rangle } \\mathinner {\\langle {\\bar{\\psi }_A,\\bar{\\psi }_B}|} )^{T_A} = \\mathinner {|{ i\\bar{\\psi }_A,\\psi _B }\\rangle } \\mathinner {\\langle {i\\psi _A,\\bar{\\psi }_B}|},$ we write the general expression for the moments of ${\\rho ^{T_A}} $ as ${Z}_{{\\cal N}_n}^{({\\rm ns})}=\\text{Tr}[({\\rho ^{T_A}} )^n] =& \\int \\prod _{i=1}^n d\\psi _i d\\bar{\\psi }_i \\ \\prod _{i=1}^n \\left[ \\rho (\\bar{\\psi }_i,\\psi _i)\\right]e^{\\sum _{i,j} \\bar{\\psi }_{iA} [ T^{-1}]_{ij} \\psi _{jA} }e^{\\sum _{i,j} \\bar{\\psi }_{iB} T_{ij} \\psi _{jB} },$ where $\\psi _{js}$ and $\\bar{\\psi }_{js}$ refer to the field defined within the $s=A, B$ interval of $j$ th replica.",
"Here, we are dealing with two intervals where the twist matrices are $T$ and $T^{-1}$ as shown in Fig.",
"REF (a).",
"Therefore, it can be written as a four-point correlator (REF (b)) ${Z}_{{\\cal N}_n}^{({\\rm ns})}=\\mathinner {\\langle {{\\cal T}_n^{-1}(u_A) {\\cal T}_n(v_A) {\\cal T}_n(u_B) {\\cal T}_n^{-1} (v_B)}\\rangle }.$ Note that the order of twist fields are reversed for the first interval.",
"From the coherent state representation, we can also write the moments of $\\rho ^{\\widetilde{T}_A}$ ${Z}_{{\\cal N}_n}^{({\\rm r})}=\\text{Tr}[(\\rho ^{\\widetilde{T}_A})^n]=& \\int \\prod _{i=1}^n d\\psi _i d\\bar{\\psi }_i \\ \\prod _{i=1}^n \\left[ \\rho (\\bar{\\psi }_i,\\psi _i)\\right]e^{\\sum _{i,j} \\bar{\\psi }_i^{A} \\widetilde{T}_{ij} \\psi _j^{A} }e^{\\sum _{i,j} \\bar{\\psi }_i^{B} T_{ij} \\psi _j^{B} }.$ The twist matrix for interval A is modified to be $ \\widetilde{T}=\\left(\\begin{array}{cccc}0 & \\cdots & 0 & -1 \\\\1 & \\ddots & \\vdots & \\vdots \\\\0 & 1 & 0 & 0 \\\\\\cdots & 0 & 1 & 0 \\\\\\end{array} \\right),$ which can be viewed as a gauge transformed twist matrix $T^{-1}$ .",
"Analogously, Eq.",
"(REF ) can be written in terms of a four-point correlator ${Z}_{{\\cal N}_n}^{({\\rm r})}=\\mathinner {\\langle {\\widetilde{\\cal T}_n^{-1}(u_A) \\widetilde{\\cal T}_n(v_A) {\\cal T}_n(u_B) {\\cal T}_n^{-1} (v_B)}\\rangle },$ where $\\widetilde{\\cal T}_n$ and $\\widetilde{\\cal T}^{-1}_n$ are twist fields associated with $\\widetilde{T}$ .",
"For fermions with a global $U(1)$ gauge symmetry (i.e., particle-number conserving systems) there is a freedom to twist boundary condition along the fundamental cycles (e.g.",
"the dashed-line path in Fig.",
"REF (a)) of the spacetime manifold by a $U(1)$ phase (or holonomy).",
"The boundary conditions are independent and in principle can be different for different pairs of sheets.",
"If we assume a replica symmetry (i.e.",
"uniform boundary conditions) $\\psi _i \\mapsto e^{i\\alpha } \\psi _i $ , the expression for the PT moments in the operator formalism is given by ${Z}_{{\\cal N}_n}(\\alpha )=\\text{Tr}[({\\rho ^{T_A}} e^{i\\alpha F_A})^n].$ Let us mention that some related quantities such as $\\text{Tr}[(\\rho \\ e^{i\\alpha F})^n]$ were previously introduced and dubbed charged entanglement entropies [115].",
"They were further used to determine symmetry resolved entanglement entropies which is the contribution from the density matrix to the entanglement entropies when projected onto a given particle-number sector [116], [117].",
"From (REF ), we get a family of RN parametrized by $\\alpha $ .",
"However, for a generic fermionic system (including superconductors), the $U(1)$ symmetry is reduced to $\\mathbb {Z}_2$ fermion-parity symmetry.",
"Hence, the two quantities of general interest would be ${Z}_{{\\cal N}_n}(\\alpha =\\pi ) &={Z}_{{\\cal N}_n}^{({\\rm r})}= \\text{Tr}[(\\rho ^{\\widetilde{T}_A})^n], \\\\{Z}_{{\\cal N}_n}(\\alpha =0) &= {Z}_{{\\cal N}_n}^{({\\rm ns})}=\\text{Tr}[({\\rho ^{T_A}} )^n].$ We should reemphasize that either quantities are described by a partition function on the same spacetime manifold (Fig.",
"REF ) as in the case of bosonic systems [70], while they differ in the boundary conditions for fundamental cycles of the manifold.",
"In other words, ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ and ${Z}_{{\\cal N}_n}^{({\\rm r})}$ correspond to anti-periodic (i.e., Neveu-Schwarz in CFT language) and periodic (Ramond) boundary conditions, respectively.",
"This can be readily seen by comparing $T^{-1}$ and $\\widetilde{T}$ .",
"These boundary conditions correspond to two replica-symmetric spin structures for the spacetime manifold.",
"This is different from bosonic PT of fermionic systems [107], [110], where RN is given by sum over all possible spin structures.",
"Essentially, the RNs associated with the two types of fermionic PT are identical to two terms in the expansion of bosonic PT in Ref. [107].",
"In what follows, we compute the two RNs for two partitioning schemes: Two adjacent intervals which is obtained by fusing the fields in $v_A$ and $u_B$ .",
"Hence, the RNs are given in terms of three-point correlators $ {Z}_{{\\cal N}_n}^{({\\rm ns})}= \\mathinner {\\langle {{\\cal T}_n^{-1}(u_A) {\\cal T}_n^2(v_A) {\\cal T}_n^{-1}(v_B) }\\rangle },$ and $ {Z}_{{\\cal N}_n}^{({\\rm r})}= \\mathinner {\\langle {\\widetilde{\\cal T}_n^{-1}(u_A) {\\cal Q}_n^2(v_A) {\\cal T}_n^{-1}(v_B) }\\rangle },$ where we introduce the fusion of unlike twist fields, ${\\cal Q}_n^2:= {\\cal T}_n\\widetilde{\\cal T}_n.$ Bipartite geometry where the two intervals together form the entire system which is in the ground state.",
"This time the RNs are obtained by further fusing the fields in $u_A$ and $v_B$ and the final expressions are therefore given by the two-point correlators $ {Z}_{{\\cal N}_n}^{({\\rm ns})}= \\mathinner {\\langle {{\\cal T}_n^{-2}(u_A) {\\cal T}_n^2(v_A) }\\rangle },$ and $ {Z}_{{\\cal N}_n}^{({\\rm r})}= \\mathinner {\\langle {{\\cal Q}_n^{-2}(u_A) {\\cal Q}_n^2(v_A) }\\rangle }.$"
],
[
"The spectrum of partial transpose",
"As mentioned, the first step to compute the tail distribution of the eigenvalues of partially transposed density matrix is to find its moments.",
"To this end, it is more convenient to work in a new basis where the twist matrices are diagonal and decompose the partition function of multi-component field $\\Psi $ to $n$ decoupled partition functions.",
"For REE, this leads to $Z_{{\\cal R}_n}= \\prod _{k=-(n-1)/2}^{(n-1)/2} Z_{k,n}$ , where $ Z_{k,n} = \\mathinner {\\langle {{\\cal T}_{k,n}(u) {\\cal T}_{k,n}^{-1} (v)}\\rangle }.$ The monodromy condition for the field around ${\\cal T}_{k,n}$ and ${\\cal T}_{k,n}^{-1}$ are given by $\\psi _k\\mapsto e^{\\pm i2\\pi k/n} \\psi _k$ .",
"The calculation of the above partition function can be further simplified in terms of correlators of vertex operators using the bosonization technique in (1+1)d. For instance, in the case of REE, (REF ) can be evaluated by [114] $Z_{k,n}= \\left\\langle V_k(u) V_{-k}(v)\\right\\rangle ,$ where $V_k(x) =e^{-i\\frac{k}{n} \\phi (x)}$ is the vertex operator and the expectation values is understood on the ground state of the scalar-field theory ${\\cal L}_{\\phi }=\\frac{1}{8\\pi }\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi $ .",
"The correlation function of the vertex operators is found by $ &\\mathinner {\\langle {V_{e_1}(z_1)\\cdots V_{e_N}(z_N)}\\rangle }\\propto \\prod _{i<j} \\left|z_j-z_i \\right|^{2e_i e_j}$ where $V_e(z)=e^{i e\\phi (z)}$ is the vertex operator and $\\sum _j e_j=0$ .",
"Hence, we can write for the partition function $ Z_{{\\cal R}_n}\\propto & \\left| u-v \\right|^{-2\\sum _k \\frac{k^2}{n^2}},$ leading to the familiar result ${\\cal R}_n = \\frac{n+1}{6n} \\ln |u-v| + \\cdots $ for the REE of 1d free fermions.",
"Note that ellipses come from the proportionality constant in (REF ) which show sub-leading terms and may depend on microscopic details.",
"In what follows, we apply the bosonization technique to evaluate ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ and ${Z}_{{\\cal N}_n}^{({\\rm r})}$ similar to what we did for the REE.",
"The scaling behavior of RNs in the lattice model is compared with the analytically predicted values of slopes (derived below) for various exponents $n=1,\\cdots ,7$ in Fig.",
"REF , where the agreement is evident.",
"We should note that the slope does not depend on the chemical potential $\\mu $ in the Hamiltonian (REF )."
],
[
"Spectrum of ${\\rho ^{T_A}} $ ",
"In the case of RN (REF ), we can carry out a similar momentum decomposition as ${Z}_{{\\cal N}_n}^{({\\rm ns})}= \\prod _{k=-(n-1)/2}^{(n-1)/2} Z^{(\\text{ns})}_{k,n},$ where $ Z^{(\\text{ns})}_{k,n} = \\mathinner {\\langle {{\\cal T}_{k,n}^{-1}(u_A) {\\cal T}_{k,n}(v_A) {\\cal T}_{k,n}(u_B) {\\cal T}_{k,n}^{-1} (v_B)}\\rangle }$ is the partition function in the presence of four twist fields.",
"We then use (REF ) to compute the above correlator for various subsystem geometries.",
"We should note that the following results only include the leading order term in the scaling limit, $\\ell _1,\\ell _2\\rightarrow \\infty $ , where $\\ell _1$ and $\\ell _2$ are the length of $A$ and $B$ subsystems, respectively.",
"Figure: Comparison of numerical (dots) and analytical (solid lines) results for the scaling behavior of the moments of partial transpose () in the up row and () in the down row for two subsystem geometries:(a) two adjacent intervals, and(b) bipartite geometry.",
"In (a), intervals have equal lengths ℓ 1 =ℓ 2 =ℓ\\ell _1=\\ell _2=\\ell and 20≤ℓ≤20020\\le \\ell \\le 200 on an infinite chain.In (b), the total system size is L=400L=400 and 20≤ℓ≤10020\\le \\ell \\le 100.Different colors correspond to different moments nn.Here, we consider adjacent intervals (c.f.",
"upper panel of Fig.",
"REF (a)).",
"The final result is given by $Z_{k,n}^{(\\text{ns})}=\\left\\lbrace \\begin{array}{lll}\\ell _1^{-4\\frac{k^2}{n^2}} \\ell _2^{-4\\frac{k^2}{n^2}} (\\ell _1+\\ell _2)^{2\\frac{k^2}{n^2}} & & \\left|k/n\\right|< 1/3 \\\\f(\\ell _1,\\ell _2; |k/n|) \\cdot (\\ell _1+\\ell _2)^{2|\\frac{k}{n}|(|\\frac{k}{n}|-1)} & & \\left|k/n\\right|> 1/3\\end{array} \\right.$ where $f(x,y;q) =\\frac{1}{2}\\left[x^{2(q-1)(-2q+1)} y^{2q(-2q+1)} + x\\leftrightarrow y \\right]$ .",
"Notice that the exponents change discontinuously as a function of $k$ .",
"This can be understood as a consequence of the $2\\pi $ ambiguity of the $U(1)$ phase that the Fermi field acquires as it goes around the twist fields.",
"Essentially, we need to find the dominant term with the lowest scaling dimension in the mode expansion (see Appendix for more details).",
"Adding up the terms in the ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ expansion, the final expression in the limit of two equal-length intervals $\\ell _1=\\ell _2$ is simplified into ${\\cal N}_n^{({\\rm ns})}= c_n \\ln \\ell + \\cdots $ where $c_{n}=\\left\\lbrace \\begin{array}{ll}-\\frac{1}{3} \\left( n- \\frac{3}{2n} \\right) &\\ \\ \\ \\ \\ n=6N, \\\\-\\frac{1}{3} \\left( n- \\frac{1}{n} \\right) & \\ \\ \\ \\ \\ n=6N+1, 6N+5, \\\\-\\frac{1}{3} \\left( n+ \\frac{1}{2n} \\right) &\\ \\ \\ \\ \\ n=6N+2,6N+4, \\\\-\\frac{1}{3} \\left( n+ \\frac{3}{n} \\right) &\\ \\ \\ \\ \\ n=6N+3,\\end{array}\\right.$ where $N$ is a non-negative integer.",
"It is worth recalling that for the bosonic systems, the spectrum of PT contains only positive and negative eigenvalues.",
"As a result, we see even/odd effect for the moments.",
"Here, however, the moments ${Z}_n^{(\\text{ns})}$ have a cyclic behavior with a periodicity of six, which signals the possibility for the eigenvalues to appear with a multiple of $2\\pi /6$ complex phase.",
"As we will see below, this is indeed the case in our numerical calculations.",
"We should also note that the above result can be obtained from the adjacent limit $v_A\\rightarrow u_B$ of two disjoint intervals (REF ) as explained in Appendix .",
"Taking this limit is a bit tricky and was previously overlooked in Ref.",
"[110], where it was incorrectly deduced that ${Z}_{{\\cal N}_n}^{({\\rm ns})}=0$ for two adjacent intervals.",
"We now discuss the spectrum of ${\\rho ^{T_A}} $ for two adjacent intervals.",
"It is instructive to look at the many-body eigenvalues as obtained in (REF ) from the single-body eigenvalues of the covariance matrix (REF ).",
"From the numerical observation that $\\text{Im}(\\nu _k)\\ne 0$ , we may drop the $u_l$ factor in (REF ).",
"Hence, the many-body spectrum simplifies to $\\lambda _{\\sigma ,\\sigma ^{\\prime }}=\\prod _{\\sigma _k=\\sigma _k^{\\prime }}\\omega _{R\\sigma _k}\\prod _{\\sigma _k=-\\sigma _k^{\\prime }}\\omega _{I\\sigma _k^{\\prime }},$ where $\\omega _{R\\sigma _k}&= \\frac{1+|\\nu _k|^2 + 2\\sigma _k \\text{Re}[\\nu _k]}{4},\\\\\\omega _{I\\sigma _k}&= \\frac{1-|\\nu _k|^2 + 2 \\sigma _k i \\text{Im}[\\nu _k]}{4},$ and $\\sigma _{k}=\\pm $ is a sign factor.",
"We should note that the complex and negative real eigenvalues come from product of $\\omega _{I\\sigma _k}$ .",
"This fact immediately implies that for every complex eigenvalue $\\lambda _j$ , $\\lambda _j^\\ast $ is also in the spectrum, since $\\omega _{I-\\sigma _k}=\\omega _{I\\sigma _k}^\\ast $ .",
"Moreover, the negative eigenvalues are at least two-fold degenerate.",
"In the case of free fermions, we numerically observe that $\\omega _{I\\pm }\\rightarrow |\\omega _{I\\pm }|e^{\\pm i\\frac{2\\pi }{6}}$ as we go towards the thermodynamic limit $N_A=N_B\\rightarrow \\infty $ .",
"As a result, the many-body eigenvalues are divided into two groups: first, real positive eigenvalues, and second, the complex or negative eigenvalues which take a regular form $\\lambda _j\\approx |\\lambda _j| e^{\\pm i\\frac{\\pi }{3} s_j}$ where $s_j=1,2,3$ .",
"Figure REF (a) shows the numerical spectrum of ${\\rho ^{T_A}} $ .",
"To explicitly demonstrate the quantization of the complex phase of eigenvalues, we plot a histogram of the complex phase in Fig.",
"REF (b) where sharp peaks at integer multiples of $\\pi /3$ are evident.",
"Due to this special structure of the eigenvalues, the moments of ${\\rho ^{T_A}} $ can be written as ${Z}_{{\\cal N}_n}^{({\\rm ns})}&= \\sum _k |\\lambda _k|^n e^{ \\frac{i\\pi ns_k }{3} } \\nonumber \\\\&= \\sum _j \\lambda _{0j}^n+2\\cos \\left(\\frac{\\pi n}{3}\\right) \\sum _j |\\lambda _{1j}|^n+2\\cos \\left(\\frac{2\\pi n}{3}\\right) \\sum _j |\\lambda _{2j}|^n+ \\cos \\left(n\\pi \\right) \\sum _j |\\lambda _{3j}|^n,$ where $\\lbrace \\lambda _{\\alpha j}\\rbrace , \\alpha =0,1,2,3$ denote the eigenvalues along $\\angle \\lambda = \\alpha \\pi /3$ branches.",
"Note that $\\lbrace \\lambda _{0j}\\rbrace ,\\lbrace \\lambda _{3j}\\rbrace $ , i.e., positive and negative real eigenvalues, are treated separately, while $\\lbrace \\lambda _{1j}\\rbrace $ and $\\lbrace \\lambda _{2j}\\rbrace $ represent the eigenvalues for both $\\angle \\lambda =\\pm \\pi /3$ and $\\angle \\lambda =\\pm 2\\pi /3$ branches.",
"A consequence of Eq.",
"(REF ) is that there are four linearly independent combinations of the eigenvalues in ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ .",
"This exactly matches the four possible scaling behaviors of ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ from our continuum field theory calculations (REF ).",
"Figure: Spectral properties of ρ T A {\\rho ^{T_A}} for two adjacent intervals with length ℓ\\ell on an infinite chain.",
"(a) Many-body eigenvalues are plotted over the complex plane.",
"The solid gray lines are guides for the eyes and a hint for the phase quantization.",
"(b) Histogram of complex phases of eigenvalues which indicates nearly quantized phases in units of π/3\\pi /3.",
"(c) Tail distribution function of modulus of eigenvalues.",
"The solid line is the analytical result ().To compute the many-body spectrum, we truncate the single-particle spectrum with the first 28 largest (in euclidean distance from ±1\\pm 1 on the complex plane) eigenvalues.As a first characterization of the negativity spectrum, we compute the distribution of modulus of eigenvalues.",
"To this end, it is sufficient to consider ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ for multiples of $n=6N$ which is ${Z}_{{\\cal N}_n}^{({\\rm ns})}=\\sum _k |\\lambda _k|^n$ .",
"Substituting (REF ) for $b$ and $a$ in (REF ) and (REF ), we get $P(|\\lambda |) &=\\delta (\\lambda _M-|\\lambda |) + \\sqrt{\\frac{3}{2}} \\frac{b\\theta (\\lambda _M-|\\lambda |)}{|\\lambda | \\xi } I_1 (\\sqrt{6} \\xi ), \\\\n(|\\lambda |) &= I_0 (\\sqrt{6} \\xi ),$ where $ \\xi =\\sqrt{b\\ln |\\lambda _M/\\lambda |},$ and $\\lambda _M$ is the largest eigenvalue given by $b=-\\ln \\lambda _M= \\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\ln \\text{Tr}({\\rho ^{T_A}} )^n= \\frac{1}{3} \\ln \\ell .",
"$ Figure REF (c) shows a good agreement between the analytical formula () and the numerically obtained spectra for various subsystem sizes.",
"We should note that there is no fitting parameter in () and we only plug in $\\lambda _M$ from numerics.",
"Figure: Spectrum of eigenvalues of ρ T A {\\rho ^{T_A}} with a certain complex phase (c.f. Fig.",
"(a)) for two equal intervals on an infinite chain.Solid lines are the prediction in Eq.",
"().Dots are numerics, with different colors corresponding to different subsystem sizes.",
"We use the same numerical procedure as in Fig.",
"to obtain few thousand largest (in modulus) many-body eigenvalues from a truncated set of single particle eigenvalues.We can further derive the distribution of eigenvalues along different branches in Fig.",
"REF (a).",
"The idea is to analytically continue ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ with $n=6N+m$ to arbitrary $n$ and solve the resulting four linearly independent equations generated by (REF ) to obtain the moments $\\sum _{j} |\\lambda _{s_j}|^n$ for each $s=0,\\cdots ,3$ .",
"This calculation relies on the assumption that $\\lim _{n\\rightarrow \\infty } \\frac{c_n}{n}$ does not depend on $m$ , which is indeed the case in (REF ).",
"Hence, we arrive at $P_\\alpha (\\lambda )& =\\delta (\\lambda _M-\\lambda )\\delta _{\\alpha 0} + \\frac{b \\theta (\\lambda _M-|\\lambda |)}{6|\\lambda |\\xi } \\sum _{\\beta =1}^2[M_{\\alpha \\beta } a_\\beta I_1(2a_\\beta \\xi )- \\widetilde{M}_{\\alpha \\beta } \\tilde{a}_\\beta J_1(2 \\tilde{a}_\\beta \\xi )],\\\\n_\\alpha (\\lambda )& =\\frac{1}{6} \\left[ \\sum _{\\beta =1}^2 M_{\\alpha \\beta } I_0(2 a_\\beta \\xi )+ \\widetilde{M}_{\\alpha \\beta } J_0(2\\tilde{a}_\\beta \\xi ) \\right],$ where $P_\\alpha (\\lambda )$ and $n_\\alpha (\\lambda )$ , $\\alpha =0,\\cdots , 3$ describe the distribution of eigenvalues along the $\\angle \\lambda =\\alpha \\pi /3$ branch.",
"Here, $M$ and $\\widetilde{M}$ encapsulate the coefficients $(M| \\widetilde{M})=\\left(\\begin{array}{cc|cc}1 & 2 &2 & 1 \\\\1 & 1 &-1 & -1 \\\\1 & -1 & -1 & 1 \\\\1 & -2 & 2 & -1\\end{array}\\right),$ $(a_1,a_2, \\tilde{a}_1,\\tilde{a}_2)=(\\sqrt{\\frac{3}{2}},1 , \\frac{1}{\\sqrt{2}}, \\sqrt{3})$ , and $\\xi $ and $b$ are defined in Eqs.",
"(REF ) and (REF ), respectively.",
"Several comments regarding the phase-resolved distributions (REF ) and () are in order.",
"The largest eigenvalue $\\lambda _M>0$ is located on the real axis and hence only appears in $P_0(\\lambda )$ .",
"The distribution of modulus is found by $(P_0+2P_1+2P_2+P_3)$ which reproduces (REF ).",
"It is easy to check that the distribution is normalized and consistent with the identity $\\text{Tr}{\\rho ^{T_A}} =1$ , $\\int \\lambda P(\\lambda ) d\\lambda &=\\int \\lambda [P_0(\\lambda )+P_1(\\lambda )-P_2(\\lambda )-P_3(\\lambda )] d\\lambda \\nonumber \\\\&= \\int _0^{\\lambda _M} \\lambda [\\delta (\\lambda _M-\\lambda )+\\frac{a_2}{\\lambda \\xi _2} I_1(2\\xi _2)] d\\lambda =1.$ It is also possible to study the scaling of the maximum eigenvalue (in modulus) $|\\lambda _M|$ along each branch.",
"For the bosonic negativity, there are only two branches (positive and negative real axis) and it was found that the scaling of the maxima is the same in the thermodynamic limit [72].",
"In our case, for a given branch (labeled by $\\alpha $ ) the maximum $|\\lambda _M^{\\alpha }|$ (with $|\\lambda ^{0}_M| \\equiv \\lambda _M$ ) can be extracted as $\\ln |\\lambda _{M}^{(\\alpha )}| = \\lim _{n \\rightarrow \\infty } \\frac{1}{n} \\ln \\sum _{j} |\\lambda _j^{(\\alpha )}|^n = -b$ where the result is independent of $\\alpha $ .",
"This again implies the same scaling along each branch, up to a possible unknown constant due to non-universal coefficient that we are dropping in the above formulas (see Eq.",
"(REF )).",
"We compare the analytical results with the numerical simulations for each branch in Fig.",
"REF .",
"As expected, the numerical spectra reach the continuum field theory calculations as we make the system larger.",
"We should point out that in contrast with the bosonic negativity spectrum and the entanglement spectrum which are given solely in terms of $I_\\alpha (x)$ , the modified Bessel function of the first kind, here the fermionic negativity spectrum contains the Bessel functions $J_\\alpha (x)$ as well.",
"Recall that unlike $I_\\alpha (x)$ which is strictly positive for $x>0$ , $J_\\alpha (x)$ does oscillate between positive and negative values.",
"Nevertheless, there is no issue in $P_\\alpha (\\lambda )$ which has to be non-negative, as the linear combinations of $I_\\alpha $ and $J_\\alpha $ in () are such that they are strictly positive over their range of applicability within each branch."
],
[
"Bipartite geometry",
"Here, we consider two intervals which make up the entire system as shown in the upper panel of Fig.",
"REF (b).",
"In this case, the branch points are identified pairwise as $u_A=v_B$ and $v_A=u_B$ , where $\\ell _1=v_A-u_A$ .",
"The partition functions in momentum space are found to be $Z_{k,n}^{(\\text{ns})}=\\left\\lbrace \\begin{array}{lll}\\ell _1^{-8\\frac{k^2}{n^2}} & & \\left|k/n\\right|< 1/4, \\\\\\ell _1^{-2(2|\\frac{k}{n}|-1)^2} & & \\left|k/n\\right|> 1/4.\\end{array} \\right.$ Similar to the adjacent intervals, the discontinuity in the $k$ -dependence comes from the $2\\pi $ ambiguity of the $U(1)$ monodromy (Appendix ).",
"As a result, we have ${\\cal N}_n^{({\\rm ns})}= c_n \\ln (\\ell _1)+\\cdots $ where $c_{n}=\\left\\lbrace \\begin{array}{ll}-\\frac{1}{6} \\left( n- \\frac{4}{n} \\right) &\\ \\ \\ \\ \\ n=4N, \\\\-\\frac{1}{6} \\left( n- \\frac{1}{n} \\right) & \\ \\ \\ \\ \\ n=2N+1, \\\\-\\frac{1}{6} \\left( n+ \\frac{8}{n} \\right) &\\ \\ \\ \\ \\ n=4N+2.",
"\\ \\end{array}\\right.$ A benchmark of these expressions against the scaling of RN in numerical simulations is shown in Fig.",
"REF (c).",
"Because of the cyclic analyticity of the ${\\cal N}_n^{({\\rm ns})}$ modulo four, we expect to have the many-body eigenvalues along the real and imaginary axes.",
"In other words, the complex phase of eigenvalues are multiples of $2\\pi /4$ .",
"We now derive the complex phase structure of many-body eigenvalues from the single particle spectrum.",
"In the current case, the density matrix is pure leading to the identity $\\gamma ^2=\\mathbb {I}$ for the covariance matrix.",
"This property implies that the spectrum of the transformed covariance matrix (REF ) can be fully determined by the covariance matrix associated to the subsystem A, i.e., $\\gamma _{AA}$ in Eq.",
"(REF ).",
"Hence, the single particle eigenvalues are given by $\\nu _k= \\mu _k + i \\sqrt{1-\\mu _k^2},$ and its Hermitian conjugate for $\\nu _k^\\ast $ , where $\\mu _k$ 's ($k=0,\\cdots , N_A$ ) denote the eigenvalues of $\\gamma _{AA}$ [104].",
"Using (REF ), the many-body eigenvalues can be written as $\\lambda _{\\sigma ,\\sigma ^{\\prime }}= \\prod _{\\sigma _k=\\sigma _k^{\\prime }}\\frac{1+\\sigma _k \\mu _k}{2}\\prod _{\\sigma _k=-\\sigma _k^{\\prime }} \\frac{\\sigma _k i \\sqrt{1-\\mu _k^2}}{2}.$ This decomposition has two types of factors: real positive and pure imaginary.",
"Therefore, the many-body eigenvalues manifestly lie on the real and imaginary axes.",
"Moreover, the many-body spectrum contains pairs of pure imaginary eigenvalues $\\pm i\\lambda _j$ .",
"The real negative eigenvalues are also two-fold degenerate since they are obtained from the product of even number of pure imaginary factors.",
"In contrast, the real positive eigenvalues are not necessarily degenerate.",
"As a result, the moments of ${\\rho ^{T_A}} $ take now the following form ${Z}_{{\\cal N}_n}^{({\\rm ns})}&= \\sum _j \\lambda _{0j}^n+2\\cos \\left(\\frac{\\pi n}{2}\\right) \\sum _j |\\lambda _{1j}|^n+ \\cos \\left(n\\pi \\right) \\sum _j |\\lambda _{2j}|^n,$ where $\\lbrace \\lambda _{\\alpha j}\\rbrace , \\alpha =0,1,2$ denote the eigenvalues along $\\angle \\lambda = \\alpha \\pi /2$ .",
"This expression in turn implies that there are three types of combinations of different branches for all $n$ , which is again consistent with (REF ).",
"By analytically continuing the three cases, we derive the moment $\\sum _j |\\lambda _{\\alpha j}|^n$ for each branch.",
"The resulting distributions are found to be $P_\\alpha (\\lambda )& =\\delta (\\lambda _M-\\lambda )\\delta _{\\alpha 0} + \\frac{ b \\theta (\\lambda _M-|\\lambda |)}{4|\\lambda |\\xi } \\left[ \\sum _{\\beta =1}^2 M_{\\alpha \\beta } a_\\beta I_1(2a_\\beta \\xi ) - \\widetilde{M}_{\\alpha } \\tilde{a} J_1(2 \\tilde{a}\\xi ) \\right],\\\\n_\\alpha (\\lambda )& =\\frac{1}{4} \\left[ \\sum _{\\beta =1}^2 M_{\\alpha \\beta } I_0(2 a_\\beta \\xi )+ \\widetilde{M}_{\\alpha } J_0(2\\tilde{a} \\xi ) \\right],$ where $M$ and $\\widetilde{M}$ encode the coefficients $(M| \\widetilde{M})=\\left(\\begin{array}{cc|c}1 & 2 & 1 \\\\1 & 0 & -1 \\\\1 & -2 & 1\\end{array}\\right),$ $(a_1, a_2, \\tilde{a})= (2, 1, 2 \\sqrt{2}) $ and $\\xi $ is defined in (REF ) with $b=-\\ln \\lambda _M=\\frac{1}{6}\\ln \\ell $ .",
"As shown in Fig.",
"REF , the above formulas are in decent agreement with numerical results.",
"Also in this case the maximum (in modulus) $|\\lambda _M^{\\alpha }|$ along the different branches can be evaluated through Eq.",
"(REF ), giving (up to an unknown non-universal constant) $\\ln |\\lambda _M^{\\alpha }| = -b$ independent of $\\alpha $ .",
"Finally, also for the bipartite geometry, a consistency check is obtained from ${\\rm Tr} \\rho ^{T_A}=1$ , which simply follows from a calculation analogous to Eq.",
"(REF ).",
"Again, the first step to find the moments is the momentum decomposition of (REF ), yielding ${Z}_{{\\cal N}_n}^{({\\rm r})}= \\prod _{k=-(n-1)/2}^{(n-1)/2} Z^{(\\text{r})}_{k,n},$ where the partition function $ Z^{(\\text{r})}_{k,n} = \\mathinner {\\langle { {\\widetilde{\\cal T}}_{k,n} ^{-1}(u_A) {\\widetilde{\\cal T}}_{k,n} (v_A) {\\cal T}_{k,n}(u_B) {\\cal T}_{k,n}^{-1} (v_B)}\\rangle }.$ is subject to modified monodromy conditions for the $ {\\widetilde{\\cal T}}_{k,n} $ and $ {\\widetilde{\\cal T}}_{k,n} ^{-1}$ , which are $\\psi _k\\mapsto e^{\\pm i(2\\pi k/n-\\pi )} \\psi _k$ .",
"This monodromy is different from the supersymmetric trace [118] (see Appendix for the definition and more details)."
],
[
"Adjacent intervals",
"In this case, we find that $Z^{(\\text{r})}_{k,n}\\propto & \\ \\ell _1^{-2(|\\frac{k}{n}|-\\frac{1}{2})(|\\frac{2k}{n}|-\\frac{1}{2})}\\cdot \\ell _2^{-2|\\frac{k}{n}|(|\\frac{2k}{n}|-\\frac{1}{2})}\\cdot (\\ell _1+\\ell _2)^{2|\\frac{k}{n}|(|\\frac{k}{n}|-\\frac{1}{2})}.$ It is important to note that for $k<0$ , we modified the flux at $u_1$ and $v_1$ by inserting additional $2\\pi $ and $-2\\pi $ fluxes, respectively, where the scaling exponent takes its minimum value (c.f. Appendix ).",
"Summing up $Z^{(\\text{r})}_{k,n}$ terms, we get $ {\\cal N}_n^{({\\rm r})}&= c^{(1)}_n \\ln (\\ell _1)+ c^{(2)}_n \\ln (\\ell _2)+ c^{(3)}_n \\ln (\\ell _1+\\ell _2) + \\cdots $ where $c^{(1)}_{n_o} &= -\\frac{1}{12} \\left( n_o+ \\frac{5}{n_o} \\right), \\\\c^{(2)}_{n_o} =c^{(3)}_{n_o} &= -\\frac{1}{12} \\left( n_o- \\frac{1}{n_o} \\right),$ for odd $n=n_o$ , and $c^{(1)}_{n_e}=c^{(2)}_{n_e} & = -\\frac{1}{6} \\left( \\frac{{n_e}}{2}- \\frac{2}{{n_e}} \\right), \\\\c^{(3)}_{n_e} &= -\\frac{1}{6} \\left( \\frac{{n_e}}{2}+ \\frac{1}{{n_e}} \\right),$ for even $n=n_e$ .",
"As a consistency check, we show in Appendix that the above formulae can be derived from two disjoints intervals as the distance between the intervals is taken to be zero.",
"Notice that the even $n$ case is identical to the general CFT results [70].",
"Also, from (REF ) we arrive at the familiar result for the logarithmic negativity, $ {\\cal E} = \\frac{1}{4} \\ln \\left( \\frac{\\ell _1\\ell _2}{\\ell _1+\\ell _2} \\right) + \\cdots $ For equal length intervals, we may write ${\\cal N}_n^{({\\rm r})}=c_n \\ln \\ell + \\cdots $ where $c_{n}=\\left\\lbrace \\begin{array}{ll}-\\frac{1}{4} \\left( n_o+ \\frac{1}{n_o} \\right) & \\ \\ \\ \\ \\ n=n_o\\quad \\text{odd}, \\\\-\\frac{1}{2} \\left( \\frac{{n_e}}{2}- \\frac{1}{{n_e}} \\right) &\\ \\ \\ \\ \\ n=n_e\\quad \\text{even}.",
"\\ \\end{array}\\right.$ Figure: Tail distribution function for the spectrum of ρ T ˜ A \\rho ^{\\widetilde{T}_A} of two equal adjacent intervals on an infinite chain.Solid lines are the analytical distributions from Eq.",
"().Dots are numerics, with different colors corresponding to different subsystem sizes.We use the same numerical procedure as in Fig.",
"to obtain few thousand largest (in modulus) many-body eigenvalues from a truncated set of single particle eigenvalues.As expected for Hermitian operator $\\rho ^{\\widetilde{T}_A}$ , here the moments ${\\cal N}_n^{({\\rm r})}$ only depend on parity of $n$ , i.e., whether $n$ is odd or even.",
"This means that the eigenvalues are real positive or negative.",
"We can also see this from the fact that the single particle spectrum is real.",
"The many-body eigenvalues follow the form $\\lambda _{\\sigma }= \\prod _{\\sigma _k=\\pm }(1+\\sigma _k \\nu _k)/2$ , where $\\nu _k$ are single-particle eigenvalues of the covariance matrix (REF ).",
"As discussed in the previous section, we carry out the same procedure to derive the distribution from analytic continuation of moments (in this case there are only two branches).",
"The final result reads $P(\\lambda ) &=\\delta (\\lambda _M-\\lambda )+ \\frac{b\\theta (\\lambda _M-|\\lambda |)}{2|\\lambda |\\xi } [- J_1 (2 \\xi ) \\text{sgn}(\\lambda )+\\sqrt{2} I_1 (2\\sqrt{2} \\xi ) ], \\\\n(\\lambda ) &= \\frac{1}{2} [J_0 (2 \\xi ) \\text{sgn}(\\lambda )+I_0 (2\\sqrt{2} \\xi ) ],$ where $\\xi $ obeys the same form as Eq.",
"(REF ) with a slight difference that $b=-\\ln \\lambda _M=\\frac{1}{4}\\ln \\ell $ .",
"We present a comparison of the above expression with numerical spectrum of free fermions on the lattice of different lengths in Fig REF .",
"There is a good agreement between analytical and numerical results.",
"We further find that, as it was the case for the bosonic negativity, the scaling of the minimum and maximum eigenvalue is the same.",
"Finally, we confirm that the distribution probability is properly normalized such that $\\int \\lambda P(\\lambda ) d\\lambda =\\text{Tr}[\\rho (-1)^{F_A}]$ and it is consistent with $\\mathcal {E} = \\frac{1}{4}\\ln \\ell $ , Eq.",
"(REF ), which follows from $\\mathcal {E} = \\ln \\int d \\lambda \\, |\\lambda | P (\\lambda ) = \\ln \\left[ \\lambda _M + \\int _{0}^{\\lambda _M} d \\lambda \\, \\frac{b \\sqrt{2}}{\\xi } I_1 (2 \\sqrt{2} \\xi ) \\right] =\\frac{1}{4}\\ln \\ell .$"
],
[
"Bipartite geometry",
"In this case, we start by computing the correlator $ Z^{(\\text{r})}_{k,n} = \\mathinner {\\langle { {\\cal Q}_{k,n}^{-2}(u_A) {\\cal Q}_{k,n}^2(v_A) }\\rangle } \\propto \\ell _1^{-2(|\\frac{2k}{n}|-\\frac{1}{2})^2}.$ Here again, we have to minimize the scaling exponent for $k<0$ by inserting additional $2\\pi $ fluxes (c.f. Appendix ).",
"The RN is then found to be ${\\cal N}_n^{({\\rm r})}= c_n \\ln (\\ell _1) + \\cdots $ where $c_{n}=\\left\\lbrace \\begin{array}{ll}-\\frac{1}{6} \\left( n_o+ \\frac{2}{n_o} \\right) & \\ \\ \\ \\ \\ n=n_o\\quad \\text{odd}, \\\\-\\frac{1}{3} \\left( \\frac{{n_e}}{2}- \\frac{2}{{n_e}} \\right) &\\ \\ \\ \\ \\ n=n_e\\quad \\text{even}.",
"\\ \\end{array}\\right.$ From this, we derive the distribution of many-body eigenvalues to be $P(\\lambda ) &=\\delta (\\lambda _M-\\lambda )+\\frac{b\\theta (\\lambda _M-|\\lambda |)}{2|\\lambda |\\xi } [-\\sqrt{2} J_1 (2\\sqrt{2} \\xi ) \\text{sgn}(\\lambda )+ 2 I_1 (4 \\xi ) ], \\\\n(\\lambda ) &= \\frac{1}{2} [ J_0 (2\\sqrt{2} \\xi ) \\text{sgn}(\\lambda )+ I_0 (4 \\xi ) ],$ where $\\xi $ is given in (REF ) and $b=-\\ln \\lambda _M=\\frac{1}{6}\\ln \\ell $ .",
"We finish this part by a remark about the covariance matrix.",
"Using the fact that $\\gamma ^2=\\mathbb {I}$ for pure states, the covariance matrix (REF ) can be further simplified into $\\widetilde{\\gamma }=\\left( \\begin{array}{cc}\\gamma _{AA}-2\\gamma _{AA}^{-1} & -i\\gamma _{AB} \\\\i\\gamma _{BA} & \\gamma _{BB}\\end{array} \\right).$ Similar to the adjacent intervals, we can calculate the many-body spectrum out of eigenvalues of the above covariance matrix.",
"We confirm that the numerical results and analytical expressions match.",
"However, we avoid showing the plots here as they look quite similar to Fig.",
"REF ."
],
[
"Conclusions",
"In summary, we study the distribution of the eigenvalues of partially transposed density matrices, aka the negativity spectrum, in free fermion chains.",
"Taking the PT of fermionic density matrices is known to be a difficult task even for free fermions (or Gaussian states).",
"However, recent studies [100], [101], [112] suggest that this difficulty could be circumvented if we use a different definition for partial transpose which is closely related to time-reversal transformation.",
"In a matrix representation of a fermionic density matrix, e.g.",
"in Fock space basis, the latter operation involves multiplying a $\\mathbb {Z}_4$ complex phase factor in addition to the matrix transposition where the phase factor solely depends on the fermion-number parity of the state of subsystems to be exchanged in the transpose process.",
"It turned out that the phase factor in the fermionic partial transpose lead to two types of partial transpose operation ${\\rho ^{T_A}} $ and $\\rho ^{\\widetilde{T}_A}$ .",
"The difference is that ${\\rho ^{T_A}} $ is pseudo-Hermitian and may contain complex eigenvalues, while $\\rho ^{\\widetilde{T}_A}$ is Hermitian and its eigenvalues are real.",
"This is in contrast with the fact that the standard partial transpose ${\\rho ^{T_A}} $ is always a Hermitian operator which implies a real spectrum.",
"In this paper, we presented analytical and numerical results for the negativity spectra using both types of fermionic partial transpose.",
"In the case of $\\rho ^{\\widetilde{T}_A}$ , we find that the negativity spectra share a lot of similarities with those found in a previous CFT work [72].",
"However, in the case of ${\\rho ^{T_A}} $ , we realize that the eigenvalues form a special pattern on the complex plane and fall on six branches with a quantized phase of $2\\pi n/6$ .",
"The spectrum in the latter case is mirror symmetric with respect to the real axis, and there are four universal functions which describe the distributions along the six branches.",
"The sixfold distribution of eigenvalues is not specific to complex fermion chain (described by the Dirac Hamiltonian) with $c=1$ and also appears in the critical Majorana chain with $c=1/2$ .",
"We further confirmed that our analytical expressions are applicable to the Majorana chain upon modifying the central charge $c$ .",
"Given our free fermion results in one dimension, there are several avenues to pursue for future research.",
"A natural extension is to explore possible structures in the negativity spectrum of free fermions in higher dimensions.",
"It would also be interesting to understand the effect of disorder and spin-orbit coupling on this distribution.",
"In particular, the random singlet phase (RSP) [119], which can be realized in the strongly disordered regime of one-dimensional free fermions, is characterized by logarithmic entanglement entropy [120], [121], [63] that is a hallmark of $(1+1)$ d critical theories.",
"An interesting question is how the negativity spectrum of critical RSP differs from the clean limit which was studied in this paper.",
"Another direction could be studying strongly correlated fermion systems and specially interacting systems which have a description in terms of projected free fermions such as the Haldane-Shastry spin chain [122], [123].",
"Furthermore, it is worth investigating how thermal fluctuations affect the negativity spectrum in finite-temperature states.",
"Finally, the negativity spectrum may be useful in studying the quench dynamics and shed light on thermalization."
],
[
"Acknowledgments",
"The authors would like to acknowledge insightful discussions with Erik Tonni, David Huse, Zoltan Zimboras."
],
[
"Funding information",
"SR and HS were supported in part by the National Science Foundation under Grant No.",
"DMR-1455296, and under Grant No.",
"NSF PHY-1748958.",
"PC and PR acknowledge support from ERC under Consolidator grant number 771536 (NEMO).",
"We all thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work.",
"H.S.",
"acknowledges the support from the ACRI fellowship (Italy) and the KITP graduate fellowship program.",
"SR is supported by a Simons Investigator Grant from the Simons Foundation."
],
[
" Twist fields, bosonization, etc.",
"The Rényi entanglement entropy (REE) of a reduced density matrix $\\rho $ is defined in Eq.",
"(REF ).",
"For non-interacting systems with conserved $U(1)$ charge, we can transform the trace formulas into a product of $n$ decoupled partition functions.",
"Let us first illustrate this idea for REE [114].",
"We can diagonalize the twist matrix $T$ in Eq.",
"(REF ) and rewrite the REE in terms of $n$ -decoupled copies, $Z_{{\\cal R}_n}=& \\int \\prod _k d\\psi _k d\\bar{\\psi }_k \\ \\prod _k \\left[ \\rho (\\bar{\\psi }_k,\\psi _k)\\right] e^{\\sum _{k} \\lambda _k \\bar{\\psi }_{k} \\psi _k },$ where $\\lambda _k=e^{i 2\\pi \\frac{k}{n}}$ for $k=(n-1)/2,\\cdots ,(n-1)/2$ are eigenvalues of the twist matrix.",
"In this new basis, the transformation rule $\\Psi \\rightarrow T\\Psi $ for the field passing through the interval becomes a phase twist, i.e., $\\psi _k \\mapsto \\lambda _k \\psi _k$ .",
"Therefore, the REE can be decomposed into product of separate factors as $Z_{{\\cal R}_n}= \\prod _{k=-(n-1)/2}^{(n-1)/2} Z_{k,n},$ where $Z_{k,n}$ is the partition function containing an interval with the twisting phase $2\\pi k/n$ .",
"We reformulate the partition function in the presence of phase twisting intervals in terms of a theory subject to an external gauge field which is a pure gauge everywhere (except at the points $u_{i}$ and $v_{i}$ where it is vortex-like).",
"This is obtained by a singular gauge transformation $\\psi _{k}(x)\\rightarrow e^{i\\int _{x_{0}}^{x}dx^{{\\prime }\\mu }A_{\\mu }^{k}(x^{{\\prime }})}\\psi _{k}\\left( x\\right),$ where $x_{0}$ is an arbitrary fixed point.",
"Hence, for a subsystem made of $p$ intervals, $A = \\bigcup _{i=1}^p [u_i, v_i]$ , we can absorb the boundary conditions across the intervals into an external gauge field and the resulting Lagrangian density becomes $ {\\cal L}_{k}=\\bar{\\psi }_{k}\\gamma ^{\\mu }\\left( \\partial _{\\mu }+i\\,A^k_{\\mu }\\right) \\psi _{k}.$ where the $U(1)$ flux is given by $\\epsilon ^{\\mu \\nu }\\partial _{\\nu }A_{\\mu }^{k}(x)=2\\pi \\frac{k}{n}\\sum _{i=1}^{p}\\big [ \\delta (x-u_{i})-\\delta (x-v_{i})\\big ] \\,.",
"$ Note that there is an ambiguity in the flux strength, namely, $2\\pi m$ (integer $m$ ) fluxes may be added to the right hand side of the above expression, while the monodromy for the fermion fields does not change.",
"To preserve this symmetry (or redundancy), $Z_k$ must be written as a sum over all representations [124], [125], [126], [127].",
"The asymptotic behavior of each term in this expansion is a power law $\\ell ^{-\\alpha _m}$ in thermodynamic limit (large (sub-)system size).",
"Here, we are interested in the leading order term which corresponds to the smallest exponent $\\alpha _m$ .",
"As we will see in the case of entanglement negativity, we need to consider $m\\ne 0$ for some values of $k$ .",
"Let us first discuss this expansion for a generic case.",
"Let ${\\cal S}_n$ be a partition function on a multi-sheet geometry (for either Rényi entropy or negativity).",
"As mentioned, after diagonalizing the twist matrices ${\\cal S}_n$ can be decomposed as ${\\cal S}_n= \\sum _{k} \\ln Z_k,$ where $Z_k$ is the partition function in the presence of $2p$ flux vortices at the two ends of $p$ intervals between $u_{2i-1}$ and $u_{2i}$ , that is $Z_{k}=\\left\\langle e^{i\\int A_{k,\\mu } j_{k}^{\\mu }d^{2}x}\\right\\rangle \\,,$ in which $\\epsilon ^{\\mu \\nu }\\partial _{\\nu }A_{k,\\mu }(x)=2\\pi \\sum _{i=1}^{2p} \\nu _{k,i} \\delta (x-u_{i}) \\,,$ and $2\\pi \\nu _{k,i}$ is vorticity of gauge flux determined by the eigenvalues of the twist matrix.",
"The total vorticity satisfies the neutrality condition $\\sum _i \\nu _{k,i}=0$ for a given $k$ .",
"In order to obtain the asymptotic behavior, one needs to take the sum over all the representations of $Z_k$ (i.e., flux vorticities mod $2\\pi $ ), ${Z}_k= \\sum _{\\lbrace m_i\\rbrace } Z_k^{(m)}$ where $\\lbrace m_i\\rbrace $ is a set of integers and $Z^{(m)}_{k}=\\left\\langle e^{i\\int A_{k,\\mu }^{(m)}j_{k}^{\\mu }d^{2}x}\\right\\rangle \\,,$ is the partition function for the following fluxes, $\\epsilon ^{\\mu \\nu }\\partial _{\\nu }A_{\\mu }^{(m), k}(x)=2\\pi \\sum _{i=1}^{2p} \\widetilde{\\nu }_{k,i} \\delta (x-u_{i}),$ and $\\widetilde{\\nu }_{k,i}=\\nu _{k,i}+ m_i$ are shifted flux vorticities.",
"The neutrality condition requires $\\sum _{i} m_{i}=0$ .",
"Using the bosonization technique, we obtain $Z_k^{(m)} = C_{\\lbrace m_i\\rbrace } \\prod _{i<j} |u_i-u_j|^{2\\widetilde{\\nu }_{k,i}\\widetilde{\\nu }_{k,j}},$ where $C_{\\lbrace m_i\\rbrace }$ is a constant depending on cutoff and microscopic details.",
"We make use of the neutrality condition $-2\\sum _{i<j} \\widetilde{\\nu }_{k,i}\\widetilde{\\nu }_{k,j} = \\sum _{i} \\widetilde{\\nu }_{k,i}^2$ and rewrite $Z_k &\\sim \\sum _{\\lbrace m_i\\rbrace } C_{\\lbrace m_i\\rbrace }\\ \\ell ^{2\\sum _{i<j} \\widetilde{\\nu }_{k,i}\\widetilde{\\nu }_{k,j} }= \\sum _{\\lbrace m_i\\rbrace } C_{\\lbrace m_i\\rbrace }\\ \\ell ^{-\\sum _{i} \\widetilde{\\nu }_{k,i}^2 }$ where $\\ell $ is a length scale.",
"From this expansion, the leading order term in the limit $\\ell \\rightarrow \\infty $ is clearly the one(s) which minimizes the quantity $\\sum _{i} \\widetilde{\\nu }_{k,i}^2$ .",
"This is identical to the condition derived from the generalized Fisher-Hartwig conjecture [128], [129].",
"A careful determination of the leading order term for REE by a similar approach was previously discussed in Ref.",
"[125], [126], [127].",
"We now carry out this process for ${Z}_{{\\cal N}_n}^{({\\rm ns})}$ in Eq.",
"(REF ) for two adjacent intervals.",
"Here, we need to minimize the quantity $f_{m_1m_2m_3}(\\nu )=(\\nu +m_1)^2+(\\nu +m_3)^2+(-2\\nu +m_2)^2$ for a given $\\nu =k/n=-(n-1)/2n,\\cdots ,(n-1)/2n$ by finding the integers $(m_1,m_2,m_3)$ constrained by $\\sum _i m_i=0$ .",
"For instance, let us compare $(0,0,0)$ with $(-1,1,0)$ , $f_{000}(\\nu ) &= 6\\nu ^2, \\\\f_{-110}(\\nu ) &= 6\\nu ^2-6\\nu +2.$ So, we have $f_{000}(\\nu ) > f_{-110}(\\nu ) \\qquad \\text{for} \\quad \\nu >\\frac{1}{3}.$ Similarly, we find that $f_{000}(\\nu ) > f_{1-10}(\\nu ) \\qquad \\text{for} \\quad \\nu <-\\frac{1}{3}.$ In summary, we resolve the flux ambiguity by adding the triplet $(m_1,m_2,m_3)$ as follows $\\left\\lbrace \\begin{array}{lll}(0,0,0) & & |\\nu |\\le 1/3 \\\\(-1,1,0), (0,1,-1) & & \\nu >1/3 \\\\(1,-1,0), (0,-1,1) & & \\nu <-1/3 .\\end{array} \\right.$ This leads us to write Eq.",
"(REF ).",
"Finally, similar derivation can be carried out to arrive at Eqs.",
"(REF ), (REF ) and (REF )."
],
[
"Rényi negativity for disjoint intervals",
"In this appendix, we derive the RN associated with ${\\rho ^{T_A}} $ and $\\rho ^{\\widetilde{T}_A}$ for two disjoint intervals and show that upon taking the distance between the intervals to zero, we recover the results for two adjacent intervals as discussed in the main text."
],
[
"Moments of $\\rho ^{{T}_A}$",
"This geometry is characterized by $v_A-u_A=\\ell _1$ , $u_B-v_A=d$ , and $v_B-u_B=\\ell _2$ (c.f. Fig.",
"REF (a)).",
"The leading order term of the momentum decomposed partition function in the case of disjoint intervals is given by $ Z_{k,n}^{(\\text{ns})}= c_{k0} \\left(\\frac{x}{\\ell _1\\ell _2}\\right)^{2k^2/n^2} +\\cdots $ where $ x =\\frac{(\\ell _1+\\ell _2+d)d}{(\\ell _1+d)(\\ell _2+d)}.$ Consequently, the RN is found to be ${\\cal N}_n^{({\\rm ns})}&= \\left(\\frac{n^2-1}{6n}\\right) \\ln \\left(\\frac{x}{\\ell _1\\ell _2}\\right) + \\cdots $ We compare the above formula with the scaling behavior of the numerical results in Fig.",
"REF (b), where we find that they match.",
"As a consistency check, we show that the RN between adjacent intervals can be derived as a limiting behavior of the disjoint intervals.",
"However, we realize from (REF ) that $\\lim _{d\\rightarrow 0} Z_{k,n}^{(\\text{ns})}=0$ (as is done also in Ref. [110]).",
"A more careful treatment goes by considering higher order terms coming from different representations in (REF ) $Z_{k,n}^{(\\text{ns})}&=c_{k0} \\left(\\frac{x}{\\ell _1\\ell _2}\\right)^{2\\frac{k^2}{n^2}}+ c_{k1} \\left(\\frac{x}{\\ell _1\\ell _2}\\right)^{2|\\frac{k}{n}|(|\\frac{k}{n}|-1)} \\left[g(\\ell _1,\\ell _2;k/n)+ g(\\ell _1+d,\\ell _2+d;k/n)\\right] + \\cdots $ where $g(x,y;q)=x^{-2} \\left(x/y\\right)^{2|q|}+x\\leftrightarrow y$ and $c_{ki}$ are coefficients dependent on the microscopic details.",
"Next, we obtain the leading order term in the coincident limit $d=\\varepsilon $ , where $\\varepsilon \\ll \\ell _1,\\ell _2$ .",
"To this end, we rewrite the above expansion (REF ) as $Z_{k,n}^{(\\text{ns})}=\\varepsilon ^{2k^2/n^2} Z_{k,n}^{(0)}+\\varepsilon ^{2|k/n|(|k/n|-1)} Z_{k,n}^{(1)} + \\cdots $ where the scaling dimensions are $[Z_{k,n}^{(0)}] &\\sim L^{-6N^2/n^2}, \\\\[Z_{k,n}^{(1)}] &\\sim L^{-2(3k^2/n^2-3|k/n|+1)}.$ As we see, for $|k/n|>1/3$ , the second term is dominant.",
"This immediately implies that upon taking $(\\ell _i+d)\\sim \\ell _i$ , we recover the original result (REF )."
],
[
"Moments of $\\rho ^{\\widetilde{T}_A}$",
"Similarly, we find the $k$ -th contribution to the $n$ -th moment of $\\rho ^{\\widetilde{T}_A}$ to be $Z_{k,n}^{(\\text{r})} =x^{2|k/n|(|k/n|-1/2)} \\frac{1}{\\ell _1^{2(|k/n|-1/2)^2} \\ell _2^{2k^2/n^2}} +\\cdots $ which gives rise to the following form for the RN, ${\\cal N}_n^{({\\rm r})}&=c^{(1)}_n \\ln (\\ell _1)+ c^{(2)}_n \\ln (\\ell _2)+ c^{(3)}_n \\ln (x)+ \\cdots $ where $c^{(1)}_{n}&=\\left\\lbrace \\begin{array}{ll}-\\frac{1}{6} \\left( n_o+ \\frac{2}{n_o} \\right) & \\ \\ \\ \\ \\ n=n_o\\quad \\text{odd}, \\\\-\\frac{1}{6} \\left( {n_e}- \\frac{1}{{n_e}} \\right) &\\ \\ \\ \\ \\ n=n_e\\quad \\text{even}, \\ \\end{array}\\right.\\\\c^{(2)}_{n} & = - \\left( \\frac{n^2-1}{6n} \\right), \\\\c^{(3)}_{n}& =\\left\\lbrace \\begin{array}{ll}-\\frac{1}{12} \\left( n_o- \\frac{1}{n_o} \\right) & \\ \\ \\ \\ \\ n=n_o\\quad \\text{odd}, \\\\-\\frac{1}{6} \\left( \\frac{{n_e}}{2}+ \\frac{1}{{n_e}} \\right) &\\ \\ \\ \\ \\ n=n_e\\quad \\text{even}.",
"\\ \\end{array}\\right.$ We compare the scaling behaviors of analytical expressions and numerical results in Fig.",
"REF (c).",
"As we see, they are in good agreement.",
"It is easy to verify that taking the adjacent limit $d= \\varepsilon $ of two disjoint intervals in Eq.",
"(REF ) leads to Eq.",
"(REF ).",
"We should note that in this case the leading order term in the momentum expansion (REF ) remains always the same (REF ) in contrast with the previous case (REF ).",
"Figure: Comparison of numerical (dots) and analytical (solid lines) results for the scaling behavior of the moments of partial transpose () and () for two disjoint intervals (the geometry is shown in panel (a)).",
"Here, d=40d=40 and intervals have equal lengths ℓ 1 =ℓ 2 =ℓ\\ell _1=\\ell _2=\\ell where 20≤ℓ≤20020\\le \\ell \\le 200 on an infinite chain.",
"The analytical results are given by Eq.",
"() in panel (b) and Eq.",
"() in panel(c).",
"Different colors correspond to different moments nn."
],
[
"Partial transpose with supersymmetric trace",
"Let ${\\cal N}_n^{({\\rm susy})}(\\rho )=\\ln \\widetilde{\\text{Tr}}[(\\rho ^{\\widetilde{T}_A})^n],$ where $\\widetilde{\\text{Tr}}$ is the supersymmetric (susy) trace for the interval $A$ .",
"The susy trace is distinct from the regular trace in that the $T$ matrix which glues together $\\rho ^{\\widetilde{T}_A}$ for fermions is given by (REF ), while the susy trace is similar to a bosonic trace (even though applied to a fermionic density matrix) where the $T$ matrix is given by (REF ) (see below).",
"It is easy to see that $T^n=(-1)^{n-1}$ for the regular trace of fermionic density matrices whereas $T^{n}=1$ for the susy trace.",
"Clearly, there is no difference between the susy and regular traces for even $n$ when considering $(\\rho ^{\\widetilde{T}_A})^n$ .",
"The susy trace was used previously to define the susy entanglement entropies [118].",
"(See Refs.",
"[130], [131], [132] for related works.)",
"In terms of the partial transpose (REF ), the susy trace is simplified into $ {\\cal N}_n^{({\\rm susy})}(\\rho ) =\\left\\lbrace \\begin{array}{ll}\\ln \\text{Tr}(\\rho ^{T_A} \\rho ^{T_A\\dag } \\cdots \\rho ^{T_A}\\rho ^{T_A\\dag })& \\ \\ n\\ \\text{even}, \\\\\\ln \\text{Tr}(\\rho ^{T_A} \\rho ^{T_A\\dag } \\cdots \\rho ^{T_A}) & \\ \\ n\\ \\text{odd},\\end{array} \\right.$ which was studied by some of us in [112] and was shown to obey the same expressions as the bosonic negativity [70] for both even and odd values of $n$ .",
"In this appendix, we briefly report the results for various geometries.",
"A technical point is that the monodromy of the field around $ {\\widetilde{\\cal T}}_{k,n} $ for the susy trace is given by $\\psi _k\\mapsto e^{\\pm i(2\\pi k/n-\\varphi _n)} \\psi _k$ where $\\varphi _n=\\pi $ or $\\pi (n-1)/n$ for $n$ even or odd, respectively [110], [133].",
"1.",
"Disjoint intervals: in this case the moments (REF ) become ${\\cal N}_n^{({\\rm susy})}&=c^{(1)}_n \\ln (\\ell _1 \\ell _2)+ c^{(2)}_n \\ln (x)+ \\cdots $ where $x$ is defined in (REF ), $c^{(1)}_{n} = -(n^2-1)/6n$ , and $c^{(2)}_{n}=\\left\\lbrace \\begin{array}{ll}-\\frac{1}{12} \\left( n_o- \\frac{1}{n_o} \\right) & \\ \\ \\ \\ \\ n=n_o\\quad \\text{odd}, \\\\-\\frac{1}{6} \\left( \\frac{{n_e}}{2}+ \\frac{1}{{n_e}} \\right) &\\ \\ \\ \\ \\ n=n_e\\quad \\text{even}.",
"\\ \\end{array}\\right.$ 2.",
"Adjacent intervals: when the distance $d \\rightarrow 0$ , i.e., $x \\rightarrow 0$ in the above expression, the moments take the form ${\\cal N}_n^{({\\rm susy})}&= c^{(1)}_n \\ln (\\ell _1 \\ell _2)+ c^{(2)}_n \\ln (\\ell _1+\\ell _2) + \\cdots $ where $c^{(1)}_{n_o} &=c^{(2)}_{n_o} = -\\frac{1}{12} \\left( n_o- \\frac{1}{n_o} \\right),$ for odd $n=n_o$ , and $c^{(1)}_{n_e} & = -\\frac{1}{6} \\left( \\frac{{n_e}}{2}- \\frac{2}{{n_e}} \\right), \\\\c^{(2)}_{n_e} &= -\\frac{1}{6} \\left( \\frac{{n_e}}{2}+ \\frac{1}{{n_e}} \\right),$ for even $n=n_e$ .",
"3.",
"Bipartite geometry: finally, in this case one has ${\\cal N}_n^{({\\rm r})}&= c_n \\ln (\\ell _1) + \\cdots $ where $c_{n}=\\left\\lbrace \\begin{array}{ll}-\\frac{1}{6} \\left( n_o- \\frac{1}{n_o} \\right) & \\ \\ \\ \\ \\ n=n_o\\quad \\text{odd}, \\\\-\\frac{1}{3} \\left( \\frac{{n_e}}{2}- \\frac{2}{{n_e}} \\right) &\\ \\ \\ \\ \\ n=n_e\\quad \\text{even}.",
"\\ \\end{array}\\right.$"
],
[
"Negativity of bosonic scalar field theory",
"As we have seen in the main text, calculating negativity boils down to computing correlators of twist fields.",
"In this appendix, we briefly review the conformal weights of the twist fields in the complex scalar field theory, $ {\\cal L}_\\phi =\\frac{1}{4\\pi } \\int |\\nabla \\phi |^2,$ from which we can compute the correlators of twist fields and derive expressions for the entanglement of free bosons.",
"Similar to fermions, the moments of density matrix in the coherent basis read as $Z_{{\\cal R}_n}=\\text{Tr}[\\rho ^n] =& \\int \\prod _{i=1}^n d\\phi _i d \\phi ^\\ast _i \\ \\prod _{i=1}^n \\left[ \\rho (\\phi ^\\ast _i,\\phi _i)\\right] e^{\\sum _{i,j} \\phi ^\\ast _i T_{ij} \\phi _j },$ where $ T=\\left(\\begin{array}{cccc}0 & 1 & 0 & \\dots \\\\0 & 0 & 1 & 0 \\\\\\vdots & \\vdots & \\ddots & 1 \\\\1 & 0 & \\cdots & 0 \\\\\\end{array} \\right).$ For the moments of the partial transpose, we have ${Z}_{{\\cal N}_n}=\\text{Tr}[({\\rho ^{T_A}} )^n] =& \\int \\prod _{i=1}^n d\\phi _i d \\phi ^\\ast _i \\ \\prod _{i=1}^n \\left[ \\rho (\\phi ^\\ast _i,\\phi _i)\\right]e^{\\sum _{i,j} {\\phi ^A_i}^{\\ast } [ T^{-1}]_{ij} \\phi _j^{A} }e^{\\sum _{i,j} {\\phi ^B_i}^{\\ast } T_{ij} \\phi _j^{B} }.$ In the case of free bosons, the moments can be written as a product of partition functions of decoupled modes, $Z_{{\\cal R}_n}&= \\prod _{k=0}^{n-1} \\langle {\\cal T}_{k,n}^{-1} (0) {\\cal T}_{k,n}(\\ell ) \\rangle , \\\\{Z}_{{\\cal N}_n}&= \\prod _{k=0}^{n-1} \\langle {\\cal T}_{k,n}^{-1} (-\\ell _1) {\\cal T}_{k,n}^2(0) {\\cal T}_{k,n}^{-1}(\\ell _2) \\rangle .$ Hence, our objective here is to find the conformal weight of ${\\cal T}_{k,n}$ , ${\\cal T}_{k,n}^2$ , and their adjoints.",
"It is worth noting that in the case of bosons $k$ takes positive integer values, $k=0,1,\\cdots , n-1$ .",
"This is because the global boundary condition is periodic, i.e.",
"the twist matrix obeys $T^n=1$ .",
"As usual in the conformal field theory, the computation goes by placing a twist field ${\\cal T}_{k,n}$ at the origin which leads to a ground state ${\\cal T}_{k,n}(0) \\mathinner {|{0}\\rangle }$ where $\\phi (z)$ and $ \\phi ^\\ast (z)$ are multivalued fields with the boundary conditions $\\phi (e^{i2\\pi } z)=e^{i2\\pi k/n} \\phi (z)$ and $\\phi ^\\ast (e^{i2\\pi } z)=e^{-i2\\pi k/n} \\phi ^\\ast (z)$ .",
"Next, we compute the correlator $\\mathinner {\\langle {\\partial _z \\phi \\partial _w \\phi ^\\ast }\\rangle }_{k/n} := \\mathinner {\\langle {{\\cal T}_{k,n}^{-1}(\\infty )|\\partial _z \\phi \\partial _w \\phi ^\\ast |{\\cal T}_{k,n}(0)}\\rangle }, $ to find the expectation value of the energy-momentum tensor via $\\mathinner {\\langle { T(z) }\\rangle }_{k/n} &= - \\lim _{z\\rightarrow w} \\left\\langle \\frac{1}{2} \\partial _z \\phi \\partial _w \\phi ^\\ast + \\frac{1}{(z-w)^2} \\right\\rangle _{k/n}.$ Using the fact that $T(z) {\\cal T}_{k,n}(0)\\mathinner {|{0}\\rangle } \\sim \\frac{\\Delta _{{\\cal T}_{k,n}}}{z^2} {\\cal T}_{k,n}(0) \\mathinner {|{0}\\rangle } +\\cdots $ we can read off the conformal weight $\\Delta _{{\\cal T}_{k,n}}$ .",
"Let us start with ${\\cal T}_{k,n}$ and ${\\cal T}_{k,n}^{-1}$ .",
"The correlation function (REF ) can be directly computed by the mode expansion of $\\phi $ field or can be simply derived from the asymptotic behavior $z\\rightarrow w$ and $z\\rightarrow 0$ or $w\\rightarrow \\infty $ .",
"The result is found to be [134], [135], $-\\frac{1}{2} \\mathinner {\\langle {\\partial _z \\phi \\partial _w \\phi ^\\ast }\\rangle }_{k/n} &= z^{k/n-1} w^{-k/n} \\left[\\frac{z(1-k/n)+wk/n}{(z-w)^2}\\right],$ which leads to $ \\Delta _{{\\cal T}_{k,n}}=\\Delta _{{\\cal T}_{k,n}^{-1}}= \\frac{k}{2n}\\left(1-\\frac{k}{n}\\right).$ We should note that doing this calculation for complex Dirac fermions, instead, leads to $\\Delta _{{\\cal T}_{k,n}}= k^2/2n^2$ .",
"So, the Rényi entropies are given by ${\\cal R}_n = \\frac{2}{1-n} \\sum _{k=0}^{n-1} \\Delta _{{\\cal T}_{k,n}} \\cdot \\ln \\ell &=\\left(\\frac{n+1}{6n}\\right) \\ln \\ell .$ One can do a similar calculation for ${\\cal T}_{k,n}^2$ .",
"In this case, the boundary condition is $\\phi (e^{i2\\pi } z)=e^{i4\\pi k/n} \\phi (z)$ .",
"For $k/n<1/2$ , the result is identical to (REF ) up to replacing $k/n$ by $2k/n$ .",
"For $1/2<k/n<1$ however, the effective phase shift is $2\\pi (2k/n-1)$ and we need to substitute $k/n$ in (REF ) by $2k/n-1$ .",
"This result can also be understood from the mode expansion.",
"Consequently, we arrive at $\\Delta _{{\\cal T}_{k,n}^2}=\\Delta _{{\\cal T}_{k,n}^{-2}}=\\left\\lbrace \\begin{array}{ll}\\frac{k}{n}\\left(1-\\frac{2k}{n}\\right) &\\ \\ \\ \\frac{k}{n} \\le \\frac{1}{2},\\\\\\left(\\frac{2k}{n}-1\\right)\\left(1-\\frac{k}{n}\\right) &\\ \\ \\ \\frac{1}{2}\\le \\frac{k}{n} <1.\\end{array}\\right.$ Using the following expression for the moments of partial transpose, ${\\cal N}_n &= c^{(1)}_n \\ln (\\ell _1 \\ell _2)+ c^{(2)}_n \\ln (\\ell _1+\\ell _2) + \\cdots $ we find $-c_n^{(1)}=\\sum _{k=0}^{n-1} \\Delta _{{\\cal T}_{k,n}^2} =\\left\\lbrace \\begin{array}{ll}\\frac{n^2-4}{12 n} &\\ \\ \\ n\\ \\ \\text{even} \\\\\\frac{n^2-1}{12 n} &\\ \\ \\ n\\ \\ \\text{odd}\\end{array}\\right.$ and $-c_n^{(2)}=\\sum _{k=0}^{n-1}( 2\\Delta _{{\\cal T}_{k,n}}-\\Delta _{{\\cal T}_{k,n}^2}) =\\left\\lbrace \\begin{array}{ll}\\frac{n^2+2}{12 n} &\\ \\ \\ n\\ \\ \\text{even} \\\\\\frac{n^2-1}{12 n} &\\ \\ \\ n\\ \\ \\text{odd}\\end{array}\\right.$ which are the familiar results [70]."
]
] | 1906.04211 |
[
[
"Ultra light Thomas-Fermi Dark Matter"
],
[
"Abstract We investigate the viability of a simple dark matter (DM) model consisting of a single fermion in the context of galactic dynamics.",
"We use a consistent approach that does not presume a particular DM density profile but instead requires that the DM+baryon system is in hydrostatic equilibrium.",
"Using a phenomenological baryon density profile, the model then predicts the DM distribution with a core like behavior close to the galactic center.",
"The presence of supermassive black holes (SMBHs) in the center of large galaxies arises naturally in this framework.",
"Using data from a set of large elliptical and spiral galaxies, and from a small set of dwarf galaxies, we find that the model can explain most of the bulk galactic properties, as well as some of the features observed in the rotation curves, provided the DM mass is in the $\\mathcal{O}$(50 eV) range.",
"More precise tests of the model require better modeling of the baryon profile and better control on the uncertainties in the data."
],
[
"Introduction",
"The nature of dark matter (DM) remains one of the most pressing questions in modern cosmology and astrophysics; despite enormous theoretical and observational/experimental efforts, no definite DM candidate, or even paradigm for the dark sector, has been generally accepted.",
"Direct probes of the dark sector, such as the direct detection experiments [1], [2], [3], [4] and collider searches [5], [6], have placed only limits on some of the interactions of dark particles.",
"Cosmological and astrophysical observations have placed complementary constraints, such as those derived form the relic abundance requirement [7], and the need to address the core-cusp problem [8] in the DM galactic distribution.",
"For this last problem a popular approach has been to assume that the dark sector has appropriately strong, velocity-dependent self-interactions [9].",
"An alternative idea The possibility that DM consists of ultra-light bosons that form a Bose-Einstein condensate on galactic scales has also been studied [10], [11] as a way of addressing the cusp problem.",
"is to assume that the DM is composed of fermions [12], [13], [14], and to ascribe the absence of a cusp to the exclusion principle; in this paper we investigate in some detail the viability of this last possibility.",
"Qualitatively speaking the possibility that the Pauli principle is responsible for the smooth DM profile at the galactic cores can be realized only for sufficiently light fermionic DM: only if the wavelength of such fermions is large enough can we expect the exclusion principle to be effective over distances typical of galaxies.",
"This type of DM would be light; in fact, we will show below that the model provides reasonable results for masses $ \\sim 50 \\,\\hbox{eV}$ , consistent with qualitative arguments [12].",
"Such light DM candidates could not have been in thermal equilibrium during the big-bang nucleosynthesis and large scale structure formation epochs [15], [16], [17], [18], [19], [20], [21].",
"This can be achieved by assuming the DM fermions carry a conserved charge, under which all standard model (SM) particles are neutral, in which case there are no renormalizable couplings between the DM fermions and the SM There are, of course, non-renormalizable couplings, but these are proportional to inverse powers of some scale – the scale of the (heavy) physics that mediates such interactions.",
"We assume that such scale is sufficiently large to ensure absence of SM-DM equilibrium..",
"In this situation most constraints are easily met, with the exception of the relic abundance, for which existing approaches [21] can be adapted.",
"Alternatively (though this is less attractive), the relic abundance can be ascribed to some primordial abundance generated in the very early universe by a yet-unknown mechanism.",
"In this paper, however, we concentrate on galactic dynamics – cosmological considerations lie outside the scope of our investigation.",
"In the calculations below we obtain the DM distribution assuming only (i) hydrostatic equilibrium, (ii) non-interacting and isothermal DM, (iii) asymptotically flat rotation cuves, and (iv) a given baryon density.",
"More specifically, we do not make any assumptions about the shape of the DM distribution or its degree of degeneracy, which differs from the approach used in several related calculations that have appeared in in the literature [22], [23], [24], [12].",
"One additional salient trait of this model is that it generally requires the presence of a super-massive black hole (SMBH) at the galactic center, though in special cases it can also accommodate galactic configurations without a SMBH.",
"An interesting argument found in the literature [22], [23], [24], [12], based on the requirements that the assumed DM profile is consistent with the observational features (core size, velocity dispersion etc.)",
"or merely from the DM phase space distribution [25] Other lower bounds can be derived form the relic density constraint [13], which we do not consider here., leads to a lower-bound constraint on the mass of the DM candidate.",
"Our calculations do not generate this type of constraint because we make no a-priori assumptions about the DM distribution; in fact, we obtain consistent values as low as $ \\sim 20 $ eV (cf.",
"Sect.",
"REF ).",
"In contrast, we do obtain an upper bound for the DM mass that depends on the asymptotic value of the rotation velocity and the mass of the SMBH (if no black hole is present the bound is trivial).",
"The rest of the paper is organized as follows.",
"The equilibrium of the DM+baryon system is discussed in the next section; we then apply the results to spherically-symmetric configurations (Sec.",
").",
"In Sec.",
"we compare the model predictions with observational data for specific galaxies and obtain the DM mass values consistent with these observations.",
"Conclusions are presented in Sec.",
", while some details of the data we used are provided in the Appendix."
],
[
"Equilibrium equations",
"As indicated above, we will investigate the viability of a Fermi-Dirac gas as a galactic DM candidate; we will assume that the gas is in local equilibrium, and that its self-interactions can be neglected.",
"Additionally, we also assume the gas is non-relativistic, which we will justify a posteriori.",
"In this case the hydrostatic stability of a small volume of the DM gas requires $m n \\nabla \\Phi + \\nabla P =0 \\,,$ where $m $ is the DM mass, $n$ the density of the gas, $P$ its pressure, and $ \\Phi $ the gravitational potential.",
"Using the standard thermodynamic relation $ n \\, d\\mu = dP - s \\, dT $ , where $ \\mu $ is the chemical potential, $T$ the temperature and $s$ the entropy (volume) density of the gas, it follows that $\\nabla (m \\Phi + \\mu ) + \\frac{s}{n} \\nabla T=0 \\,.$ We will assume that $T$ is constant throughout the gas, in which case $m \\Phi + \\mu = E_0= \\mbox{constant}.$ The value of $E_0 $ will be discussed below.",
"Using eq:energy in the Poisson equation for $ \\Phi $ gives $\\nabla ^2 \\mu = - \\frac{4\\pi m}{M_{\\tt pl}^2} \\left( \\rho _B + m n \\right)\\,,$ where $ M_{\\tt pl}$ denotes the Planck mass We work in units where $k_B= \\hbar = c = 1 $ , where $ k_B$ is Boltzmann's constant, $ \\rho _B $ is the baryon mass density, and $n$ the DM number density (as noted previously); explicitly $n = - \\frac{2}{\\lambda ^3} \\mbox{Li}_{3/2}\\left( -e^{\\mu /T} \\right)\\, ;\\quad \\lambda = \\sqrt{\\frac{2\\pi }{m T}}\\,,$ where Li denotes the standard polylogarithm function and $\\lambda $ is the thermal wavelength; the factor of 2 is due to the spin degrees of freedom.",
"Using standard expressions for the ideal Fermi gas the average DM velocity dispersion is given by $\\sigma ^2_{\\text{\\tiny DM}} = \\frac{1}{3}\\left\\langle v^2 \\right\\rangle = \\frac{P}{m\\, n} \\,; \\quad P = - \\frac{2T}{\\lambda ^3} \\mbox{Li}_{5/2}\\left( -e^{\\mu /T} \\right)\\,.$ Within our model introduced above, the structure of the galaxy is determined by the solution to equation eq:eom with appropriate boundary conditions.",
"To do this our strategy will be to choose an analytic parameterization for $ \\rho _B $ consistent with observations, and impose boundary conditions at large distances from the galactic center which lead to the flat rotation curves; from this $ \\mu ({\\bf r})$ can be obtained.",
"The solution will depend on the parameters in $ \\rho _B$ , the DM mass $m$ and the asymptotic rotation velocity $\\overline{v_{\\tt rot}}$ .",
"The idea of constraining the DM mass using the phase space density evolution was first suggested by Tremaine and Gunn (TG) [25].",
"In their seminal approach the DM halo is assumed to be an isothermal classical ideal gas in hydrostatic equilibrium, with a phase space distribution of the form $ f({\\bf p},{\\bf r})= n(r) \\exp [ - p^2/(2 m^2 \\sigma ^2) ]$ , where $ n(r) = n_0/r^2 $ .",
"The exclusion principle then requires $ f(0,{\\bf r}) < 1 $ , which leads to the lower bound $m^4 \\gtrsim 0.004 M_{\\tt pl}^2/(\\sigma r^2)$ .",
"This bound then follows from a consistency requirement associated with the adopted form of $f$ .",
"The Milky Way dwarf spheroidal satellites, due to their high DM density, allow a simple and robust application of the TG bound (see for eg.",
"[26], [13]) obtaining, for example, $ m \\gtrsim 70$ eV using Fornax [13], though uncertainty in the DM core radius limits somewhat the reliability of this bound A large core size cannot be ruled out [24], while relaxing the dependence of the DM halo core radius on the stellar component and marginalizing the unknown stellar velocity dispersion anisotropy lead to mass bounds as low as 20 eV [24] (though such large haloes are unrealistic and would be at odds with their lifetime due to dynamical friction effects within the Milky Way)..",
"In contrast to these assumptions, we use the Fermi-Dirac distribution $ f({\\bf p},{\\bf r}) = \\lbrace \\exp [ p^2/(2m T) - \\mu (r)/T ] + 1 \\rbrace ^{-1}$ that, $i)$ automatically satisfies the exclusion principle constraint, $ii)$ does not factorize into a product of space and momentum functions and $iii)$ leads to a singular $n$ only when a central SMBH is present.",
"In our approach the DM density profile is determined by the baryon distribution by solving eq:eom; we make no assumptions about the the DM core radius or the DM distribution in general.",
"In particular, the degree of degeneracy of the fermion distribution function follows from the behavior of $ \\mu ({\\bf r}) $ ; we will see below that the DM approximates a classical Maxwell-Boltzmann gas far from the bulge and that its quantum nature only becomes important near the galactic center, leading to a core-like profile.",
"Despite these differences, we observe that the values we obtain for $m$ (see below) are roughly consistent with the bounds based on the extended TG approach, especially for smaller dwarf galaxies with higher DM density.",
"The value of $ \\mu $ at the origin will be of interest in interpreting the solutions to eq:eom.",
"If $ \\mu ({\\bf r}\\rightarrow 0) \\rightarrow + \\infty $ then $ \\phi \\rightarrow - \\infty $ , which, as we will show, corresponds to a point-like mass at origin, a black hole This scenario was recently considered in [27] with completely different boundary conditions, without baryons and DM mass in the keV range..",
"In these cases, the DM density exhibits a cusp at the origin, but for realistic parameters this cusp appears only in the immediate vicinity of the black hole.",
"Outside this region the DM density has a core-like profile.",
"Solutions for which $ \\mu (0) $ is finite corresponds to galaxies where no central black hole is present and exhibit `pure' core-like DM densities.",
"The remaining possibility, $ \\phi ({\\bf r}\\rightarrow 0)\\rightarrow + \\infty $ describe the unphysical situation of a repulsive point-like object."
],
[
"Spherically symmetric solutions",
"In the following section we will adopt the simplifying assumption that all quantities depend only on $r = |{\\bf r}|$ ; this is a reasonable assumption for ellipticals, but is problematic for spiral galaxies.",
"We will comment on this when we apply our formalism to specific cases.",
"It proves convenient to define $ \\overline{u}$ and $x$ by $x = \\frac{r}{A}\\,, \\quad \\frac{\\overline{u}(x)}{x} = \\frac{\\mu }{T}\\,; \\quad A = \\sqrt{\\frac{T M_{\\tt pl}^2 \\lambda ^3}{8\\pi m^2}}\\,,$ while the baryon density can be written in the form $\\rho _B = \\frac{M_{\\tt B}}{\\left(\\frac{4}{3}\\pi a^3 \\right)} F(r/ a)\\,,$ where $M_{\\tt B}$ is the total bulge mass and $ a$ denotes the scale radius which can be obtained from the effective radius using the explicit form of the baryonic profile function $F$ ; $ \\rho _B $ will be negligible for $ r \\gg a$ .",
"The normalization for $F$ is taken to be $\\int _0^\\infty dy\\,y^2 F(y) = \\frac{1}{3}\\,.$ With these definitions eq:eom becomes (a prime denotes an $x$ derivative) $\\overline{u}^{\\prime \\prime } = x \\mbox{Li}_{3/2}\\left( - e^{\\overline{u}/x} \\right) -{\\tt q}\\, x F(x/X_B)\\,, \\qquad X_B= a/A\\,,\\quad {\\tt q}=\\frac{3M_{\\tt B}\\lambda ^3}{8\\pi m a^3 } \\,.$ For most of the examples we consider $ X_B\\lesssim 1 $ .",
"Far from the galactic center $ \\rho _B $ can be neglected and the gas density will be small enough so that $ P = n T $ and Li$_{3/2}(-z) \\simeq -z $ .",
"In this region a `test' object in a circular orbit of radius $r$ will have velocity $ v_{\\tt rot}(r)$ determined by $v_{\\tt rot}^2(r) = \\frac{ M_{\\tt tot}(r)}{M_{\\tt pl}^2 r }\\,,$ where $ M_{\\tt tot}$ is the total mass ($M_{\\tt BH}$ + $M_{\\tt B}$ + $M_{\\tt DM}$ ) inside radius $r$ .",
"At large distances $v_{\\tt rot}(r)$ will approach an $r$ -independent value $ \\overline{v_{\\tt rot}}$ provided $ M_{\\tt tot}(r) \\propto r$ , which requires $ n \\sim 1/r^2$ (since the dark component dominates in the asymptotic region).",
"This then implies $ \\overline{u}= x \\ln (b/x^2) $ for some constant $b$ ; substituting in $ \\overline{u}^{\\prime \\prime } \\simeq - x \\exp (\\overline{u}/x) $ gives $ b=2$ : $\\overline{u}\\rightarrow x \\ln \\left( \\frac{2}{x^2}\\right)\\,, \\quad x \\gg X_B.$ The numerical solutions approach the asymptotic expression in eq:uas for $ x \\gtrsim 1$ .",
"Using the asymptotic expressions it follows that $ M_{\\tt tot}(r) \\simeq (16\\pi A^2/\\lambda ^3)m r $ , whence eq:vrot gives $T = \\frac{1}{2}m \\overline{v_{\\tt rot}}^2 \\,; \\qquad \\mbox{where}~~v_{\\tt rot}(r)\\, \\stackrel{r \\gg a}{\\longrightarrow }\\,\\overline{v_{\\tt rot}}\\,.$ Comparing this with the expression eq:sigma we find $\\sigma ^{}_{\\text{\\tiny DM}} = \\frac{\\overline{v_{\\tt rot}}}{\\sqrt{2}} \\,, \\quad (r \\gg a) \\,;$ it also follows that $ \\lambda = \\sqrt{4 \\pi }/( m \\overline{v_{\\tt rot}})$ We solve eq:ueom using eq:uas and its $x$ derivative as boundary conditions.",
"The solution For later convenience we explicitly display the dependence on the parameters $ X_B$ and ${\\tt q}$ .",
"$ \\overline{u}(x;X_B,{\\tt q})$ will then ensure that rotation curves are flat and is consistent with the chosen baryon profile.",
"Note that in general $ \\overline{u}$ will not vanish at the origin, which implies the behavior $\\Phi \\, \\stackrel{r \\rightarrow 0 }{\\longrightarrow }\\, - \\frac{A T}{m} \\frac{u_0 }{r}\\,, \\quad u_0 = \\overline{u}(0;X_B,{\\tt q}) \\,.$ For $u_0>0 $ this corresponds to the field generated by a point mass $M_{\\tt BH}= \\frac{AT M_{\\tt pl}^2}{m}\\, u_0 = \\left( \\frac{\\sqrt{\\pi } \\, \\overline{v_{\\tt rot}}^3}{8} \\right)^{1/2} \\frac{M_{\\tt pl}^3}{m^2} u_0$ that we interpret as a black hole at the galactic center: in these cases the boundary conditions are consistent only if a black hole with this particular mass is present.",
"For $u_0<0 $ the solution in eq:uo is unphysical, at least as far as classical non-relativistic configurations are concerned.",
"These two regimes are separated by the curve $ u_0 =0 $ in the $ X_B- {\\tt q}$ plane; solutions of this type correspond to galaxies without a central black hole.",
"For the examples that follow, we consider that the expression $ \\overline{u}(0;X_B,{\\tt q})= u_0 $ is equivalent (to a good approximation) to the simple relation This was obtained numerically, not derived rigorously using the properties of the solutions to eq:eom.",
"$\\ln X_B= \\nu (u_0) \\ln {\\tt q}+ c(u_0)\\,,$ where the functions $ \\nu $ and $c$ depend on the form of $F$ in eq:B-density, but are generally $\\mathcal {O}(1) $ .",
"For the choices of $F$ , and for $ u_0 $ not too close to zero, below they can be approximated by algebraic functions: $c(u_0) \\sim \\bar{c}_1 \\sqrt{\\bar{c}_2 - u_0}\\,, \\qquad \\nu (u_0 ) \\sim -\\bar{\\nu }_1 - \\bar{\\nu }_2 u_0^2 + \\bar{\\nu }_3 u_0^3\\,,$ where $ \\bar{c}_{1,2},\\,\\bar{\\nu }_{1,2,3} $ are positive and $ O(1) $ ; values for several choices of $F$ are provided in the next section, see table REF .",
"The errors in using these expressions are below $ 10\\% $ , so they are useful for $ u_0 \\gg 0.1$ .",
"Unfortunately, many cases of interest correspond to $ u_0 \\lesssim 0.1 $ , so in most results below we will not use eq:scal.rel,eq:c.nu, opting instead for a high-precision numerical calculation.",
"It is worth pointing out that once the boundary conditions at large $r$ are imposed, $u_0 $ is determined by $ X_B$ and ${\\tt q}$ , it is not a free parameter.",
"Equivalently, $ M_{\\tt BH}$ is determined by $m$ and $ \\rho _B$ , in particular, the presence (or absence) of a black-hole and its mass are not an additional assumption, but instead follow naturally from the choice of DM mass and baryon density profile.",
"The relation eq:scal.rel can be used to estimate the DM mass $m$ in terms of the galactic quantities $M_{\\tt B},\\, a$ and $M_{\\tt BH}$ .",
"Since $ c(u_0) $ in eq:c.nu should be real, a necessary condition for $m$ to be real as well is $ u_0 < \\bar{c}_2 $ .",
"This leads to the requirement: $m^2 < \\frac{\\bar{c}_2}{(64/\\pi )^{1/4}} \\frac{\\left( M_{\\tt pl}^2 \\overline{v_{\\tt rot}}\\right)^{3/2}}{M_{\\tt BH}}\\stackrel{\\bar{c}_2 =1.3}{\\sim }\\left( 180\\, \\hbox{eV}\\right)^2 \\frac{ \\left( 10^3 \\overline{v_{\\tt rot}}\\right)^{3/2}}{M_{\\tt BH}/\\left(10^9 M_\\odot \\right)}\\,;$ for most of the the specific examples studied below we find $ m \\lesssim 100\\,\\hbox{eV}$ (see Sec.",
").",
"To get an estimate of the values of the quantities involved, for $ m\\sim 10\\, \\hbox{eV}$ and $ \\overline{v_{\\tt rot}}\\sim 300 $ km/s, $A\\sim 20 $ kpc and $ M_{\\tt BH}\\sim 10^{11} u_0^2 M_\\odot $ , so that realistic situations will correspond to small values of $u_0 $ that will satisfy eq:con.con.",
"Since $ \\overline{v_{\\tt rot}}\\ll 1 $ for all cases of interest, the gas temperature will be much smaller than its mass.",
"In addition, $ \\mu /m = \\overline{v_{\\tt rot}}^2 \\overline{u}/(2x) $ (cf.",
"eq:ubdef), where we expect $ \\overline{u}\\sim \\mathcal {O}(1) $ (see Sec. )",
"and $ \\mu \\ll m$ ; except perhaps in the immediate vicinity of the galactic center and even then only when $ \\overline{u}(0)\\ne 0$ (corresponding to $M_{\\tt BH}\\ne 0$ ).",
"From this it follows that in general the Fermi gas will be non-relativistic, as we assumed above.",
"We will define the halo (or virial) radius $ R_{\\tt hal}$ by the condition $ m n(R_{\\tt hal}) = 200 \\times \\rho _c $ , where $ \\rho _c \\simeq 4.21 \\times 10^{-47} \\hbox{GeV}^4 $ is the critical density of the Universe.",
"For all cases considered here the density will take its asymptotic expression (corresponding to eq:uas) at $ r = R_{\\tt hal}$ , then we find $R_{\\tt hal}= \\left(10^3 \\overline{v_{\\tt rot}}\\right)\\times 240 \\, \\mbox{kpc} \\,,$ and depends only on $ \\overline{v_{\\tt rot}}$ ; the galactic radius is then $\\mathcal {O}(100\\,\\mbox{kpc})$ .",
"Taking the zero of energy at infinity imposes the boundary condition $ \\Phi (R_{\\tt hal}) = - M_{\\tt hal}/( M_{\\tt pl}^2 R_{\\tt hal}) $ so that, using eq:energy and eq:uas, $E_0 = T \\ln \\left( \\frac{2A^2}{R_{\\tt hal}^2} \\right) - \\frac{M_{\\tt hal}}{R_{\\tt hal}M_{\\tt pl}^2}\\,; \\quad M_{\\tt hal}= M_{\\tt B}+ 4\\pi m \\int _0^{R_{\\tt hal}} dr\\, r^2 n(r) \\,,$ so that $E_0 $ is then determined by the other parameters in the model."
],
[
"Sample calculation",
"To illustrate the model presented above we consider a set of 3 hypothetical galaxies (cf.",
"table REF ) for which we display some of the results derived from the calculations described above, where the black-hole mass $ M_{\\tt BH}$ is calculated using eq:mbh-u0.",
"In this section we will assume $ m = 50 $ eV and use the Plummer profile $ F(y) = (1 + y^{2})^{-5/2}$ (again, for illustration purposes); note that the solution is independent of $a$ when $ M_{\\tt B}=0$ .",
"Table: Sample galaxiesAll these galaxies have a halo radius (cf.",
"eq:rhal) $ \\sim 300 $ kpc.",
"The total mass density and circular velocity eq:vrot are plotted in Fig.",
"REF .",
"Galaxy $A$ shows a density profile with no evidence for a core while a clear constant-density core develops in galaxies $B$ and $C$ .",
"Note that for the latter, the density increases again for $ r \\lesssim 200 $ pc due to the relatively large central SMBH.",
"Similarly, galaxy $B$ has a density increase only at very small radii, $ r \\ll 100 $ pc, because of a smaller black hole at the galactic center.",
"The circular velocity profile is generally steepest for $A$ , decreasing for $B$ and even more for $C$ .",
"Figure: Density (left) and circular velocity (right) for the sample galaxies in table , the black, dark-gray and light-gray curves correspond, respectively, to galaxies AA, BB and CC (ρ 0 =2m/λ 3 \\rho _0 = 2 m /\\lambda ^3).As shown by this exercise, the solution is very sensitive to the particular combination of size and galaxy mass ($a$ and $M_{\\tt B}$ ).",
"For example, a change by a factor of 50 is predicted in $ M_{\\tt BH}$ due to a relatively small ($\\sim 40$ %) change in $a$ , leading ultimately to quite different density profiles.",
"While this can be considered a feature of the model, which is anticipated to have large predictive power, given the uncertainties that plague current astronomical measurements one may refrain from over-interpreting the results at such level of detail.",
"It is interesting to note that the case $ M_{\\tt B}=0 $ is universal, in the sense that the solution to eq:eom with boundary conditions eq:uas is unique and, in particular, has $ u_0 \\simeq 1.49 $ : within this model configurations without a smooth baryon density are consistent with flat rotation curves only if they contain a central SMBH with mass $\\sim (6 \\times 10^6/m_{\\tt eV})^2 M_\\odot $ (see eq:mbh-u0), where $ m_{\\tt eV} $ is the DM mass in eV units."
],
[
"The TFDM model in specific galaxies.",
"Given a spherically-symmetric galaxy with a known baryon density profile and a given black hole mass, the results of Sec.",
"predict a DM mass $m$ .",
"It is then important to determine whether the same value of $m$ is obtained for different galaxies, as required for consistency.",
"In this section we discuss this issue for a set of large galaxies (Sec.",
"REF ) and then for a set of dwarf galaxies (Sec.",
"REF ).",
"We note that we cannot expect a perfect agreement (that is, precisely the same $m$ in all cases), as we have ignored many of the details of the structure of the galaxies being considered (assuming, for example, spherical symmetry).",
"We will be satisfied instead to see if the values of $m$ derived for each galaxy cluster around a specific range."
],
[
"Large galaxies with SMBH",
"We will adopt the following three commonly used stellar density profiles (cf.",
"eq:B-density) [28], [29], [30] into our model.",
"$F(y) &=\\frac{1}{(1 + y^{2})^{\\frac{5}{2}}} \\qquad \\qquad \\text{\\text{(Plummer)}} \\,, \\nonumber \\\\[6pt]F(y) &=\\frac{2}{3y(1 + y)^{3}} \\qquad \\qquad \\textrm {\\text{(Hernquist)}} \\,, \\nonumber \\\\[6pt]F(y) &=\\frac{1}{3y^2(1 + y)^{2}} \\qquad \\qquad \\textrm {\\text{(Jaffe)}} \\,,$ for which the parameters in the scaling relations eq:scal.rel,eq:c.nu are provided in table REF .",
"We use different profiles in order to gauge the effect of baryon distribution on the DM mass in the set of galaxies that we study.",
"Table: Fit ParametersWe collected a dataset from several sources [31], [32], [33], [34], [35], [36] for a total of 60 galaxies, spanning a large range of Hubble types, and each of them containing a SMBH at their galactic center; details on data selection are provided in appendix .",
"Using the central values of $M_{\\tt B},\\, M_{\\tt BH},\\,\\overline{v_{\\tt rot}}$ , and $a$ provided in the above references, we calculate the DM mass $m$ for all the galaxies in this set To minimize inaccuracies, we do not use eq:scal.rel, but find $m$ by solving $ \\overline{u}(0;X_B,{\\tt q})= u_0$ numerically..",
"The results are shown in Table REF and Table REF for elliptical and spiral galaxies, respectively.",
"Table: Elliptical GalaxiesTable: Spiral GalaxiesFor spiral galaxies, we find that the DM mass lies in the range $30-100$ eV with a few outliers in the range $ \\sim 100-150$ eV.",
"For elliptical galaxies, $m$ has a tighter range, $10-60$ eV for all the three baryon profiles (excluding the one outlier, NGC 221).",
"The average and standard deviation for the calculated DM mass for the two different galaxy types and three baryonic profiles are listed in table REF .",
"Table: Statistics of the DM MassIt is important to note that the average value of $m$ for elliptical galaxies is lower than that for spiral galaxies.",
"This is due, to a great extent, to having ignored the spiral mass in the above calculations: if we add the spiral mass to the bulge and increase the effective radius (while keeping all other parameters fixed), the value of $m$ decreases considerably.",
"For example, in the Milky Way (spiral mass $5.17 \\times 10^{10} M_{\\odot } $ , bulge mass $0.91 \\times 10^{10} M_\\odot $ [37]), this shifts the DM mass from $51.8$ eV to $22.38$ eV for a change in effective radius from 0.7 kpc to 3 kpc (for Hernquist profile).",
"Even though adding the entire baryonic mass of the spiral to the bulge stellar mass by just increasing the bulge effective radius is probably a poor assumption, it can be expected that considering the disc structure would lead to a decrease in the mean value of $m$ , closer to the result for elliptical galaxies.",
"On the other hand $\\overline{v_{\\tt rot}}$ is not known for most bulge-dominated elliptical galaxies, so uncertainties in this parameter may shift the DM mass for ellipticals, but the change in that would be less significant.",
"Overall, it is remarkable that despite all its simplifying assumptions the model provides values of $m$ that lie within a relatively narrow range The case of fermionic DM for the Milky way considering most of the structural features of the galaxy has been studied [38].",
"However, they assume complete degeneracy at zero temperature and the mass range is obtained strictly from the constraints on the rotation curve..",
"Figure: DM chemical potential (left) and P/(nT)P/(n T) (middle) as functions of rr for the Milky Way (see eq: n,eq:sigma,eq:T-vrot) for three baryon density profiles; the classical Maxwell-Boltzmann equation of state is shown in red.",
"Right: comparison of the DM density profile for the model discussed here using the Hernquist profile with the (unnormalized) NFW profile to illustrate the presence of a core in the former.",
"All graphs are for the Milky Way.The histograms next to tables REF and REF exhibit a few “outliers”, for which the DM mass is in the $ \\gtrsim 100 $ eV range, though this is dependent on the baryon profile used.",
"For example, $m$ associated with NGC 2778 is $ \\sim 75$ eV for the Plummer and Hernquist profiles, but $ \\sim 100$ eV for the Jaffe profile, while $m$ for NGC 6068 and NGC 5576 exhibit the opposite behavior.",
"The case of NGC 221 is unique in that it requires $ m \\sim 200 $ eV, but it is also special in that it is the smallest galaxy in this set (with an effective radius of 40 pc), and categorized as a dwarf galaxy with a central black hole.",
"By comparing the bulge mass from two different sources (log($M_{\\tt B}$ ) of 9.05 in [32] and 8.53 in [31]) hint at larger uncertainties in the measurement of stellar mass and lead to a comparatively large value for the DM mass.",
"To further understand the spread of $m$ values we present in Fig.",
"REF a plot of $m$ against $M_{\\tt B}$ for the galaxies in our dataset, where we find that larger values of $m$ are associated with smaller, less massive galaxies.",
"This correlation may indicate a defect in the DM model (which should produce similar values of $m$ for all galaxies, without the correlation show in the figure), or it may indicate that the data we use underestimates $M_{\\tt B}$ for smaller galaxies, and over-estimates it for larger ones.",
"To examine this last possibility we took from our dataset the values of $ \\overline{v_{\\tt rot}}$ and $ a$ for each galaxy and then obtained the baryon mass that corresponds to a fixed choice of $m=50 $ eV.",
"We denote this `derived' baryon mass by $ M_{\\tt B}^{\\prime }$ , In Fig.",
"REF we also present a plot of $ M^{\\prime }_B / M_{\\tt B}$ vs $M_{\\tt B}$ , which shows that $ | M^{\\prime }_B| \\lesssim 3 M_{\\tt B}$ for the spiral galaxies in our set, and $ | M^{\\prime }_B| \\lesssim 1.5 M_{\\tt B}$ for the ellipticals, so that an $\\mathcal {O}(1)$ shift in $\\log M_{\\tt B}$ can explain the fact that we do not obtain the same value of $m$ for these galaxies.",
"Although we believe this argument is compelling, factors of order $\\sim 2$ -3 can easily be accommodated given the current systematic errors in the estimation of $M_{\\tt B}$ associated to stellar evolution, reddening and the past star formation history of each galaxy (see for instance [40]).",
"Therefore the viability of the dark matter model in this context then cannot be absolutely decided.",
"Figure: Left: scatter plot and linear fit illustrating the correlation between the obtained values of mm and M 𝙱 M_{\\tt B} for elliptical (top) and spiral (bottom) galaxies.",
"Right: relative shift in M 𝙱 M_{\\tt B} needed to obtain a fixed value of mm, chosen here as 50 eV, for elliptical (top) and spiral (bottom) galaxies.",
"NGC 221 is not included in the plots.",
"All the results are for the Hernquist profile.We now consider various aspects of the solutions to eq:eom, using the Milky Way as an example.",
"In Fig.",
"REF , we show the chemical potential for three different baryon profiles.",
"As expected, $\\mu (r)$ diverges as $r$ approaches the galactic center, indicating a the presence of a SMBH.",
"We also examine the degree to which the gas is degenerate by plotting $ P/(n T) $ .",
"Far from the galactic center, the gas obeys the classic (dilute) Maxwell-Boltzmann distribution $ P \\simeq n T$ (red line in the figure), while close to the galactic center, a significant deviation due to Fermi-Dirac statistics is observed, indicating strong degeneracy.",
"In the bottom panel of Fig.",
"REF we compare the obtained density profile in the inner regions to the empirical solution found for collisional cold dark matter model, or NFW profile [41].",
"At the centers of halos, the cold dark matter solution is characterized by a cuspy mass distribution while our model favors shallower inner dark matter cores, with the exception of the region surrounding the central black hole.",
"Figure: DM density (left column) and mass fraction (right column) for two spiral galaxies (Milky Way and N224) and two elliptical galaxies (N3379 and N4621) (we use ρ 0 =2m/λ 3 \\rho _0 = 2 m/\\lambda ^3).Figure: Circular velocity as a function of distance for four spiral galaxies : Milky Way, NGC 224, NGC 3079 and NGC 4258.",
"Dataset 1 (with no error bars) for all the four galaxies is from whereas dataset 2 for the Milky Way is taken from .",
"Dataset 3 for NGC 224 is obtained from The mass densities for DM, and the fraction of the DM mass inside a given radius are shown in Fig.",
"REF .",
"By construction, the DM mass density exhibits the $ 1/r^2 $ behavior at large $r$ required for the observed flat rotation curves.",
"It is also relatively flat inside the bulge except for the immediate vicinity of the origin where it spikes due to the accumulation of DM around the central black-hole ($\\mu $ diverges as $ r \\rightarrow 0 $ , which allows for a higher density of DM particles to be accommodated in a smaller volume, leading to the observed increase in $ \\rho $ ); though not obvious from the figure, this spike is significant only for $ r \\lesssim 1 $ pc.",
"Outside of the region immediately surrounding the black-hole the exclusion principle obeyed by our DM candidate does lead to a core-like behavior.",
"The plot of the DM mass fraction shows that, except for a few kiloparsecs from the galactic center, galaxies are DM dominated.",
"In fig REF we plot the circular velocity as a function of distance from the galactic center for four spiral galaxies, the Milky Way, NGC 224 (M31 or Andromeda), NGC 3079 and NGC 4258, using the three different baryonic profiles.",
"We also compare the model predictions with data obtained using CO, HI and H-alpha observations (elliptical galaxies are not included in the sample due to the lack of rotational curve data).",
"The outer region of the rotation curves are in good agreement with the data, as expected from our boundary conditions.",
"The inner dynamics is best reproduced for NGC 3079 followed by the Milky Way, but no so effectively for NGC 224 and NGC 4258.",
"This again, can be attributed to the fact that our model does not include the disc structure, which has a significant contribution to the dynamics of circular velocities, and also assumes complete spherical symmetry for these galaxies.",
"It is then remarkable that the overall qualitative features of the rotation curves for our model are a good fit to the available data.",
"The statistical errors in the above values for $m$ can be estimated using the scaling relation in eq:scal.rel.",
"Using the fact that $ u_0 $ is small for the examples being considered, and taking $ \\nu (0)\\sim -0.4,\\, c(0) \\sim 0.9$ (cf.",
"table REF ), we find (at 3 standard deviations) $\\frac{\\delta m}{m} \\sim 3 \\times \\left[\\frac{1}{2}\\frac{\\delta a}{a}- \\frac{\\delta M_{\\tt B}}{M_{\\tt B}} +2 \\frac{\\delta \\overline{v_{\\tt rot}}}{\\overline{v_{\\tt rot}}}\\right]$ assuming that $ M_{\\tt B}\\propto a \\sigma ^2 $ [32], using eq:sig-v-spiral,eq:sig-v-ell, and taking $ \\delta M_{\\tt B}/M_{\\tt B}\\sim \\delta \\overline{v_{\\tt rot}}/\\overline{v_{\\tt rot}}\\sim 0.1 $ we find $ \\delta m/m \\sim 0.4$ .",
"This, however, does not include the systematic errors associated with our applying the spherically symmetric model to spiral galaxies, or systematic errors with the data itself; as noted earlier, we expect these errors to be considerably larger."
],
[
"Galaxies without SMBH",
"Strong observational evidence suggests that almost all massive galaxies contain a supermassive black hole at their galactic center; most galaxies with no SMBH are small, dwarf galaxies.",
"The best studied members of the latter category are the Milky Way dwarf spheroidal galaxies (dSphs) and because of this, they are the best suited candidates to test our model in the special case where $M_{\\tt BH}=0 $ .",
"However, it is widely accepted that these dSphs are mostly dominated by dark matter, with mass-to-light ratios of $M/L_{V} \\sim 10^{1-2}$ [43].",
"Detailed studies of light fermionic DM in nearby dwarf spheroidal galaxies have already appeared in the literature [12], [13], [14], though the implementation of the Thomas-Fermi paradigm is different form the one being discussed here (cf.",
"the discussion in Sect.",
"and at the end of Sec.",
").",
"The DM profile in our model is determined based on the baryon distribution and hence we do not consider these galaxies due to their negligible baryonic content.",
"There is also the generally accepted picture that a majority of the dwarf galaxies have slowly rising rotation curves [44], [45], so our assumption of flattened out circular velocities for the boundary conditions no longer holds It is possible to adapt tour approach to these situations, but we will not pursue this here..",
"Therefore we will here restrict ourselves to somewhat larger dwarf galaxies without central black holes, but with flat asymptotic rotation curves and also with an estimate of the baryonic mass.",
"We choose a total of eight such dwarf galaxies (from the SPARC database [46]) based on their small bulge mass ($M_{\\tt B}\\lesssim 10^9 M_\\odot $ ) and small asymptotic rotational velocity ($\\overline{v_{\\tt rot}}\\lesssim 100$ km/s) There were a few other galaxies in the data set that satisfied these two constraints, but for which we found no real solutions for the DM mass..",
"Since we do not find a strong dependence with the baryonic profile function $F$ , in this section we restrict ourselves to the case of the Plummer profile.",
"Table: Dwarf galaxiesThe values of $m$ for the eight dwarf galaxies are listed in table REF ; the masses turn out to be on the higher end of the spectrum as compared to the galaxies with SMBHs in the previous section.",
"This can be understood using the scaling relations eq:scal.rel,eq:c.nu, which in this ($u_0 =0 $ ) case reduces to $0.412 \\ln \\left( \\frac{M_{\\tt B}}{10^9 M_\\odot } \\right)+ 0.352 \\ln \\left( \\frac{m}{30 \\hbox{eV}}\\right) = 0.236 \\ln \\left( \\frac{a}{2.5\\text{kpc}}\\right) + 0.736 \\ln \\left( \\frac{\\overline{v_{\\tt rot}}}{200 \\text{km/s}} \\right) + 1.493\\,,$ where we used the fit parameters for the Plummer model listed in table REF .",
"For the eight galaxies considered here, if we take the average value of $\\overline{v_{\\tt rot}}\\sim $ 70 km/s and a $\\sim $ 0.5 kpc, we get log ($M_{\\tt B}/M_{\\odot }$ ) as 9.17 for the DM mass of 50 eV which is not far off from the data available for $M_{\\tt B}$ (cf.",
"[46] ).",
"Also, the farthest outlier in our data, UGC 8550 requires log ($M_{\\tt B}/M_{\\odot }$ ) to be 9.16 as compared to the given value of 8.72.",
"The difference is far less compared to the case of galaxies with SMBH as dwarf galaxy NGC 221 with similar DM mass for the same Plummer profile requires much larger shift in baryonic mass (log ($M_{\\tt B}/M_{\\odot }$ ) of 9.61 as compared to 8.53 provided in the data).",
"This might hint that the large systematic errors in the measurement of $M_{\\tt B}$ are more impactful in the case of galaxies without SMBH causing considerable shift in the DM mass.",
"We again denote by $M^{\\prime }_B$ the total baryon mass when $m$ has the specific value of 50 eV, then we find that $ M^{\\prime }_{B}/M_{\\tt B}$ in the range $1.5-3$ for all the eight dwarfs we studied.",
"As for the case of large galaxies, it is currently impossible to exclude this possibility because of the large systematic errors in $ M_{\\tt B}$ .",
"It should be noted that some of these dwarf galaxies provide two real solutions for the DM mass.",
"In such cases, only the smaller of the two values are included in table REF because the larger mass solution (in the $\\mathcal {O}(500\\,\\hbox{eV})$ range) does not lead to a core-like profile or match with other observations (e.g.",
"rotation curves).",
"Figure: Properties of the solution to the TFDM equations for dwarf galaxies.",
"Top row: chemical potential (left) and P/(nT)P/(nT) (right) for the DM as a function of rr for 3 dwarf galaxies; middle row: DM density (left) and mass fraction (right) for the same galaxies; bottom row: rotation curve for NGC 2915 (left) and DDO 154 (right) with rotation data taken from ; also, ρ 0 =2m/λ 3 \\rho _0 = 2 m/\\lambda ^3.In Fig REF we illustrate the properties of the solutions by plotting various properties of model predictions for three dwarf galaxies, whose behavior away from the center is qualitatively similar to that of large galaxies with SMBHs.",
"We note that the predicted dark matter profiles show a central constant density core with core radii $r \\sim 100$ -400 pc (which is also the case for the other galaxies in our set).",
"Of special interest are the rotation curves (bottom line in the figure): for DDO 154 and NGC 2915, the predicted behavior of $v_{\\tt rot}(r)$ qualitatively matches quite well with the observations, but the rise in the curve is somewhat steeper compared to the data.",
"It is unclear whether these discrepancies are due to a shortcoming in the model itself or in the simplifying assumptions we adopted, or due to the specific baryonic profile (Plummer's) we use The match with observations does not improve if we use the Hernquist or Jaffe profiles.."
],
[
"Conclusions",
"In this paper we investigated the extent to which a DM model consisting of single, light fermion, is consistent with the observed bulk properties of galaxies (effective radius, baryon mass and profile, etc.).",
"To simplify the calculations we neglected possible fermion (non-gravitational) interactions, and assumed that the galaxies are well described by a spherically-symmetric configuration.",
"We also assumed a fixed baryon distribution that affects the mechanical equilibrium of the system, but we neglected any thermal or dynamical effects of the baryons.",
"The baryon profile, which is directly observable, together with the boundary conditions leading to flat rotation curves, completely determine the DM distribution in the system.",
"This is in contrast with other publications which assume a DM profile ab initio.",
"For the set of galaxies we considered (that includes spiral, ellipticals and several dwarf galaxies) the model is consistent with the observational data, in the sense that the values of $m$ we obtain lie in a relatively narrow range.",
"Admittedly, for the model to be convincing, the same value of $m$ should be obtained for all galaxies; but to test this would require a careful modeling of each galaxy, and solving the stability equation eq:eom without the assumption of spherical symmetry – which lies beyond the scope of this paper.",
"A stringent test of the model would also require more accurate data with reduced systematic errors; it is unclear whether any of these effects leads to the $ m - M_{\\tt B}$ correlation observed in Fig.",
"REF .",
"Given these uncertainties we limit ourselves to stating that the model is promising, but additional calculations and observations are necessary to fully determine its viability.",
"For galaxies with a SMBH we find that the preferred DM mass is $ \\sim 40 $ eV, and that the DM distribution has a central core region where the fermions are strongly degenerate, with the degeneracy increasing as the central black-hole is approached.",
"For galaxies without SMBHs the DM mass values we find are generally larger ($ \\gtrsim 70 $ eV).",
"Possible reasons for this discrepancy, as well as for the spread in the preferred values of $m$ within each galaxy class are discussed in sections REF and REF .",
"It is interesting to note that the lower bounds for $m$ obtained in [13], [12], [24] for the Milky Way dwarf spheroidal galaxies are in the range $20-100$ eV, which is consistent with our results for galaxies without a SMBH.",
"Interestingly, this model makes clear testable predictions that may be worth exploring in more detail.",
"For instance, at fixed $m$ and asymptotic outer velocity, the profile is fully determined by the equilibrium reached between dark matter and baryons.",
"This means that any detected difference in the shapes of the rotation curves measured in galaxies at fixed terminal rotation velocity [48], in particular for dark matter-dominated objects like dwarfs, should be accompanied by a significant difference in the baryonic mass distribution.",
"Such correlation has already been shown to help alleviate the problem of rotation velocity diversity in the case of self-interactive dark matter [49].",
"Exploring the correlation between observed baryonic properties (mass, gas fractions, size) and the shape of the velocity profiles in single fermion dark matter case would also help assess the viability of this model.",
"Small deviations from spherical symmetry can be implemented using perturbation theory, which would be applicable to elliptical galaxies or for studying the effects of rotation.",
"In contrast, a more accurate comparison of the model to spiral galaxies will require solving eq:eom assuming cylindrical symmetry, and including in $ \\rho _B$ bulge and spiral components.",
"Also of interest would be a study of the dynamic stability of the system, that can be approached using standard techniques [50]; in this case eq:eom is replaced by the Euler equation and complemented by the DM and baryon current conservation constraints.",
"Finally, we wish to comment on the possible effects of exchange interactions.",
"Inside an atom these effects are significant [51], but in the present situation they can be neglected since we assume the fermions experience only gravitational interactions.",
"This, however, will change dramatically should fermion self-interactions are included, and can lead to a further reduction of the DM pile-up at the core.",
"The authors would like to thank Hai-bo Yu for interesting and useful comments.",
"LVS acknowledges support from NASA through the HST Program AR-14582 and from the Hellman Foundation."
],
[
"Comments on the data used.",
"In this appendix we give some details on the data we used to obtain the results presented in the main text.",
"For galaxies with SMBHs, we consider in total a sample of 60 galaxies, 29 elliptical and 31 spiral galaxies.",
"For each of these galaxies, we needed the mass of the black hole $M_{\\tt BH}$ , bulge mass $M_{\\tt B}$ , scale radius $a$ and the asymptotic velocity $\\overline{v_{\\tt rot}}$ .",
"We got most of the entries in our dataset from [31] (we used $M_{\\tt B}$ calculated by K band M/L derived from B-V color, and excluded galaxies where this value of $M_{\\tt B}$ was unavailable).",
"In addition, we obtained $M_{\\tt BH},\\,M_{\\tt B}$ and $ \\overline{v_{\\tt rot}}$ from [32] for 3 elliptical (NGC 1332, NGC 1407 and NGC 7052 ) and 2 spiral galaxies (NGC 1277 and NGC 3945); for these 5 galaxies we obtained $ a$ from 3 sources: [33] for NGC 1332, NGC 1407 and NGC 3945; [31] for NGC 7052 and [34] for NGC 1277.",
"Other galaxies from [32] were not included due to the lack of data on the effective/half-light radius.",
"The asymptotic circular $\\overline{v_{\\tt rot}}$ for some of the spiral galaxies (Circinus, Milky Way, NGC 224, NGC 1023, NGC 1068, NGC 2787, NGC 3031, NGC 3115, NGC 3227, NGC 3384, NGC 3585, NGC 4026, NGC 4258, NGC 4596, NGC 7457 and IC2560) are listed in [32].",
"For all other spiral galaxies we use the empirical relation [52], $\\log \\overline{v_{\\tt rot}}= (0.8 \\pm 0.029) \\log \\sigma + (0.62 \\pm 0.062)\\,,$ where $\\sigma $ is the bulge velocity dispersion.",
"For elliptical galaxies, we assume a very similar relation from the same reference: $\\log \\overline{v_{\\tt rot}}= (0.82 \\pm 0.027) \\log \\sigma + (0.57 \\pm 0.058),$ that was obtained using a larger sample of galaxies including ellipticals.",
"The data for rotation curves of spiral galaxies is taken from [35] .",
"For galaxies with no central black hole, we obtained $ M_{\\tt B},\\,a$ and $ \\overline{v_{\\tt rot}}$ from the SPARC database [47], [46].",
"We note that this dataset has no information on the presence or absence of SMBHs, so we include only eight of the smallest dwarf galaxies."
]
] | 1906.04212 |
[
[
"Estimation of 2D Velocity Model using Acoustic Signals and Convolutional\n Neural Networks"
],
[
"Abstract The parameters estimation of a system using indirect measurements over the same system is a problem that occurs in many fields of engineering, known as the inverse problem.",
"It also happens in the field of underwater acoustic, especially in mediums that are not transparent enough.",
"In those cases, shape identification of objects using only acoustic signals is a challenge because it is carried out with information of echoes that are produced by objects with different densities from that of the medium.",
"In general, these echoes are difficult to understand since their information is usually noisy and redundant.",
"In this paper, we propose a model of convolutional neural network with an Encoder-Decoder configuration to estimate both localization and shape of objects, which produce reflected signals.",
"This model allows us to obtain a 2D velocity model.",
"The model was trained with data generated by the finite-difference method, and it achieved a value of 98.58% in the intersection over union metric 75.88% in precision and 64.69% in sensibility."
],
[
"Introduction",
"The inverse problem consists of posing an approximate model of a system in which its parameters could be estimated from measurements, which are usually indirect.",
"This kind of problem is present in multiple fields of science and engineering, moreover it is especially important in areas like geology and oceanography, where estimating soil properties or characteristics of underwater structures could only be possible through indirect measurements [1], [2].",
"Specifically, in the area of underwater acoustics, the estimation of position and shape of underwater objects is important for activities such as exploration and navigation, for that reason multiple approaches have been proposed, with analytical methods being the main ones [1], [2], [3].",
"However, with the rise of high-performance computational methods such as deep neural networks, a great variety convolutional networks models have been proposed to solve inverse problems to different applications [4], [5], [6].",
"For the above, in this work we propose a convolutional encoder-decoder architecture to estimate a 2D velocity model of an underwater environment, determining approximately the localization, shape and size of objects in the environment.",
"The estimation of the velocity model is made from the reception of the echoes in 11 points of the study medium.",
"To be able to train the convolutional neuronal network (CNN); a model, based on finite-differences method (FDM), is posed with which synthetic data are generated.",
"The remainder of this paper is as follows.",
"In the Section , we describe in detail the methodology used in this work, explaining the procedure to generate synthetic data and the architecture of the proposed CNN.",
"Then, in the Section , we present the results obtained in the training of the CNN and the analysis of CNN's performance.",
"Finally, in the Section , we point out the conclusions obtained."
],
[
"Materials and Methods",
"In this section, we describe the inverse problem studied here and, then, we present the methods used to generate data and estimate the solution of the inverse problem using convolutional neural networks."
],
[
"Inverse Problem",
"In the area of underwater acoustics, there is a variety of inverse problems, which, according to [2], are classified into two groups: remote sensing and source localization problems.",
"In this paper we study a remote sensing problem, which seeks to estimate the location and shape of objects found in an aquatic environment; therefore, a CNN model is proposed that can perform this estimation using signals received from eleven points and produced by the propagation of an acoustic signal through the medium and from a point source."
],
[
"Synthetic Data Generation",
"We decided to use synthetic data to train the proposed CNN due to the fact that no public dataset with the required characteristics could be found, and also that obtaining real data would be very expensive in resources.",
"For these reasons, we use the finite difference method to simulate the acoustic wave propagation in an underwater medium in order to generate 20,000 samples under different scenarios."
],
[
"Finite-difference method",
"In order to generate synthetic data, we model the forward problem as the propagation of a plane acoustic wave in an 2D heterogeneous medium [7], [8].",
"To this purpose, we solve the partial differential equation (REF ) using the finite-difference method under an extrapolation approach [9]; where $p$ is the pressure, $c$ is the wave propagation velocity in the medium, $s$ is the acoustic source, $x$ and $z$ are the spatial coordinates, and $t$ is the temporal coordinate.",
"$\\partial _{tt} p(x,z,t) & {}={} & c(x,z)^2 (\\partial _{xx} p(x,z,t)+ \\partial _{zz} p(x,z,t)) \\nonumber \\\\&&{+}\\:s(x, z, t)$ The FDM requires to discretize the variables that are shown in (REF ) and approximate partial derivatives as a combination of these variables.",
"To do this, the spatial coordinates $x$ and $z$ are divided into length spacings $dx$ and $dz$ , while the temporal coordinate is segmented in time intervals $dt$ .",
"In (REF ), it can be seen the relation between continuous and discrete variables, where the superscript $n$ is related with timesteps, and subscripts $i$ and $k$ are related with $x$ and $z$ , respectively .",
"$p_{i, k}^n = p(i dx,k dz,n dt)$ $\\partial _{tt} p \\approx (p_{i, k}^{n+1} - 2 p_{i, k}^n + p_{i, k}^{n-1})/dt^2$ $\\partial _{xx} p & {}\\approx {} & ((p_{i+3, k}^n + p_{i-3, k}^n)/90-3(p_{i+2, k}^n + p_{i-2, k}^{n})/20\\nonumber \\\\&&{+}\\:3(p_{i+1, k}^n + p_{i-1, k}^n)/20-49p_{i, k}^n/18)/dx^2$ $\\partial _{zz} p & {}\\approx {} & ((p_{i, k+3}^n + p_{i, k-3}^n)/90-3(p_{i, k+2}^n + p_{i, k-2}^n)/20\\nonumber \\\\&&{+}\\:3(p_{i, k+1}^n + p_{i, k-1}^n)/20-49p_{i, k}^n/18)/dz^2$ In the approximation of the partial derivatives, we use three-point stencil for the temporal derivative and seven-point stencil for the spatial derivatives; their expressions are (REF ), (REF ) and (REF ).",
"From these equations, the variable $p$ is isolated at the timestep $n+1$ as a function of the previous timesteps, which are used to obtain the temporal evolution of the pressure.",
"Then, we model the medium of propagation as a grid of points, where each point has a specific velocity, which is called the velocity model ($ c_ {i, k} $ ).",
"By doing so, we can emulate the heterogeneity of the medium and build structures of different shapes within it so that we can also measure echoes produced by the structures when the wave impacts with them, as observed in Fig.",
"REF .",
"The velocity values in the model are in the range of 0 m/s to 3000 m/s; in addition to this, we established the homogeneous medium as water with a velocity equal to 1500 m/s.",
"Moreover, the dimensions of the velocity model and variables $dx$ and $dz$ determine the physical dimensions of the simulated environment.",
"Given that $dx$ and $dz$ are equal to one-fifth of the minimum wavelength, that is 15 mm, and the model has a dimension equal to $256\\times 256$ , the simulated space has a surface of 14.74 m2.",
"Additionally, we considered a total of 1800 timesteps, each one with a duration of 2.5 $\\mu s$ , which was determined with the Courant-–Friedrichs-–Lewy condition [9].",
"Finally, an acoustic source is fixed in a point of the simulated space, as it is shown in Fig.",
"REF .",
"This source produces a waveform equal to the first derivative of the Gaussian function (REF ).",
"We choose this function because it has a limited bandwidth.",
"In (REF ), the parameter $f_0$ is the maximum signal frequency, and it has a value of 40 kHz.",
"$s^n = -2 (n-100) dt^2 f_0^2 e^{((n-100) dt f_0)^2}$ Figure: (a) Velocity model used in the simulation.",
"(b) Propagation of the acoustic wave in the medium.",
"Black inverted triangles represent measuring points while the green star represent the acoustic source."
],
[
"Structure of samples",
"Each individual sample in the dataset is an input-target pair, which is generated by the method described in Section REF with different velocity models.",
"We randomly generate velocity models (Fig.",
"REF ) for each simulation, where each one has a different number of objects, in the range of 0 to 10.",
"These objects are disk-or-square-shaped and randomly distributed over the medium; they have a propagation speed of 3000 m/s.",
"Each of the models represents a target that will be normalized and stored in a $256\\times 256$ -array; its respective input consists of the pressure measurements made in 11 fixed positions of the simulated medium, as indicated in Fig.",
"REF .",
"Each measurement starts in the 400th timestep and is stored in a $1400\\times 11$ -array re-scaling them in the range of -50 to 50 units."
],
[
"Convolutional Neural Network Architecture",
"The proposed CNN has an encoder-decoder architecture, this type of structure consists of two differentiated stages.",
"The first stage is the encoder, it extracts the most relevant features of the input signals and then it encodes them in order to reduce their dimensions.",
"The second stage is called decoder, it interprets the encoded information to produce an output with the desired characteristics.",
"Each of these can be treated as an independent neural network, so in the following sections, we detail each of them.",
"Figure: Expanded structure of the encoder and decoder implemented with convolutional layers."
],
[
"Encoder Structure",
"There is a wide range of encoder structures.",
"Selecting one depends of the shape of the input data and the nature of the phenomenon that produces them [5], [6], [10].",
"Since our input data are sequences, a traditional approach is using recurrent layers, such as LSTM or GRU, to encode the information; however, the recent use of 1D and 2D convolutional layers have shown great potential to manipulate sequences.",
"In addition to this, they require a lower computational cost, that is why we decided to implement an encoder architecture based on 1D convolutional layers (Conv1D).",
"The basic structure in the encoder is a sequence of four-layer blocks.",
"Each block is composed by a convolutional layer; followed by a Batch Normalization (BN) layer; an activation layer (ReLU or Sigmoid); and, finally, a max pooling layer.",
"Each layer has its own hyperparameters: kernel size (k) and the padding method (same or valid), for the convolutional layers; window size, for the max-pooling layers; and stride (s), for both.",
"All the hyperparameters of the encoder are shown in Fig.",
"REF .",
"Regarding the data dimension, the encoder receives an array of $1400\\times 11$ elements and generates, as an output, an array of $16\\times 16$ elements.",
"This output contains essential information of the input signals."
],
[
"Decoder Structure",
"The decoder structure, in a similar way to the encoder, could be built in many ways, so that its design has to be done considering the nature of both the input and the desired output.",
"Since both input and output of the decoder are two-dimensional arrays, they could be processed as images and, for that reason, 2D convolutional layers (Conv2D) are the main layers in the decoder implementation.",
"In addition to the Conv2D layers, we also use UpSampling 2D and Batch Normalization (BN) layers.",
"The Conv2D layers have almost the same hyperparameters as the Conv1D layers with the only difference that their kernels are two-dimensional.",
"Moreover, the Up Sampling layers have two hyperparameters: window size and stride, which have values equal to 2 in the decoder implementation.",
"The hyperparameters values for each layer are shown in Fig.",
"REF ."
],
[
"Results",
"In this section, we will show the training results of the CNN models, as well as some metrics used in the process to evaluate the their performance."
],
[
"CNN Training",
"In this stage, we propose two additional models in order to compare their performance with that of the model proposed in Section REF .",
"These models have residual layers, similar to that shown in Fig.",
"REF .",
"From now on, the CNN described in Section REF will be called InvNet, while the additional models will be called InvNet+1Res and InvNet+2Res; where, notation +1Res and +2Res are references to the number of residual layers added after each max pooling layer in the InvNet's encoder.",
"Figure: Residual LayerThe proposed CNNs were implemented with Python 3.6 in a server with an Intel Xeon E5-2620 CPU at 2.1 GHz, 128GB RAM and two Nvidia Tesla K40 GPU.",
"The dataset described in Section REF was divided in training, validation and test sets with a ratio of 70 %, 15 % and 15 %, respectively.",
"Then, the selected cost function was binary cross-entropy, and the optimizer was Adam with a learning rate equal to 0.0002, a first moment ($\\beta _1$ ) equal to 0.5 and a second moment ($\\beta _2$ ) equal to 0.99.",
"Finally, all CNNs were trained with a batch size of 20 during 30 epochs; accuracy and loss curves are shown in Fig.",
"REF , where it could be seen that they tend to over-fit around the 10th epoch.",
"In the validation set, the CNNs get accuracy and loss values around 97.6 % and 0.064, respectively.",
"Figure: Curves obtained during the training stage of the three CNNs."
],
[
"Evaluation of the CNN",
"After training, we proceed to evaluate and compare the performance achieved by each model.",
"Since the outputs of the CNNs are binary masks, the following metrics will be used in the analysis: accuracy, precision, sensitivity, specificity and intersection over union (IoU).",
"The accuracy, precision, sensitivity and specificity are calculated with (REF ), (REF ), (REF ) and (REF ); where $tp$ is the number of true positive pixels; $tn$ , true negatives; $fp$ , false positives; and $fn$ , false negatives.",
"Each one of this are measured in relation with pixels of the output image.",
"While accuracy is a global metric that indicates the percentage of pixels correctly classified as solid objects or water, the precision indicates the percentage of pixels properly detected over objects.",
"In a similar way, the sensitivity points out the percentage of pixels of the objects that have been omitted and the specificity is related with the percentage of pixels which are correctly classified as water.",
"Additionally, IoU measures the percentage of overlap between the ground truth and the estimated velocity model, calculated according to (REF ).",
"It gives us an intuition of how well located and sized the objects are.",
"Figure: Images (a), (b) y (c) show the ground truth velocity model, while images in right side (d), (e) y (f) show the model estimated by InvNet.$accuracy = (tp + tn) / (tp + fp + tn + fn)$ $precision = tp / (tp + fp)$ $sensitivity = tp / (tp + fn)$ $specificity = tn / (tn + fp)$ $IoU = (target \\cap prediction) / (target \\cup prediction)$ These metrics were used over the test set, which has 3000 samples, for each CNN and the results are shown in Table REF .",
"There, it could be seen that all the CNNs have high values in the accuracy and specificity metrics, but lower values in the precision and sensitivity metrics.",
"These results point out that they can detect the presence of objects but they still have difficulty interpreting interference and echoes.",
"In the case of the IoU metric, all the CNNs obtain high values which indicates that all of them can properly estimate the localization and size of the objects.",
"Despite the InvNet+1Res have a slightly higher performance than the others, it has a higher computational cost than that of InvNet, based on the required number of parameters shown in Table REF .",
"Since, InvNet has a good performance and low computational cost, it is the best of the three models.",
"Table: Performance obtained from the CNNs trainedFinally, we test InvNet under different situations and the results are shown in Fig.",
"REF .",
"There, it could be seen what was mentioned above; that is, the proposed CNN correctly locates most of the objects, it also estimates their shapes and sizes, but presents some false positives as observed in Fig.",
"REF and Fig.",
"REF , as well as some omissions of objects as seen in Fig.",
"REF and Fig.",
"REF ."
],
[
"Conclusions",
"In this work, a convolutional encoder-decoder architecture was proposed to estimate the velocity model of an underwater environment, managing to locate objects and approximate their shapes with a high value in the IoU metric equals to 98.58%.",
"Despite the fact that the CNN presents a good performance, it can still be improved for in some cases it makes mistakes in detecting objects due to multiple echoes and shadows.",
"These behaviors reflect their effects in the precision and sensitivity metrics where the CNN obtains values of 75.88% and 64.69%, respectively.",
"Additionally, it is important to point out that the proposed CNN shows characteristics such as quick calculation of a velocity model based on acoustic signals, and precision in objects localization and size estimation.",
"Since the proposed model was trained in a great variety of synthetic scenarios, we can infer that it may achieve similar results with signals from real scenarios.",
"This is part of a future work."
],
[
"Acknowledgment",
"The authors would like to thank the National Institute for Research and Training in Telecommunications (INICTEL-UNI) for the technical and financial support to carry out this work."
]
] | 1906.04310 |
[
[
"A Novel Cost Function for Despeckling using Convolutional Neural\n Networks"
],
[
"Abstract Removing speckle noise from SAR images is still an open issue.",
"It is well know that the interpretation of SAR images is very challenging and despeckling algorithms are necessary to improve the ability of extracting information.",
"An urban environment makes this task more heavy due to different structures and to different objects scale.",
"Following the recent spread of deep learning methods related to several remote sensing applications, in this work a convolutional neural networks based algorithm for despeckling is proposed.",
"The network is trained on simulated SAR data.",
"The paper is mainly focused on the implementation of a cost function that takes account of both spatial consistency of image and statistical properties of noise."
],
[
"Introduction",
"In the last decades, remote sensing has continuously grown providing more and more images of the planet.",
"The way to extract useful informations is still an open issue, even more when we are dealing with SAR sensors.",
"SAR images are affected by multiplicative noise called speckle, that impairs performances of different tasks such as classification, object detection and segmentation.",
"In fact, in these years a very big area of research has grown to tackle this problem and a lot of despeckling algorithms have been proposed.",
"As said before, speckle is a multiplicative noise given by the interaction of electromagnetic fields scattered in different directions from a rough surface.",
"Let's consider $Y$ a SAR image, it can be expressed as [1]: $Y = f(X,N) = X\\cdot N$ where $X$ is the noise-free image and $N$ is the multiplicative speckle.",
"In the hypothesis of fully developped speckle, its distribution is known and, for an intensity image, it is a Gamma distribution [2]: $p(N) = \\frac{1}{\\Gamma (L)} N^L e^{-NL}$ where $L$ is the number of looks of SAR image, (Fig.",
"REF ).",
"An ideal despeckling filter will remove the noise without introducing artefacts and preserving the spatial informations.",
"The despeckling filters are usually divided in two categories: local and non local filters.",
"The formers as Lee[3], Enhanced Lee[4] and Kuan filter[5] rely on similarity between the target and its adjacent pixels.",
"The latter as Patch Probabilist Based (PPB)[6], SAR-BM3D[7], NL-SAR [8] look for similarity in a wider window search.",
"Nowadays, with the increasing of deep learning solutions in a lot of fields related to image processing, another branch of filters has born.",
"Indeed, in the last years also convolutional neural networks (CNN) based solutions have been proposed such as [9], [10].",
"Using CNNs for despeckling is quite challenging because the lack of a clean reference: once a real SAR image is acquired, there is no possibility to have a speckle free image to use as reference.",
"The trends to overcome this problem are mainly two: training a network to perform one of despeckling filter as in [10], in which a CNN is proposed to perform multilook when there is no chance to have several acquisitions of same data; training on simulated data as in [9].",
"As in [9], in this work SAR simulated data are used.",
"Clean images are taken from three datasets: UCID, BSD[11] and scraped Google Maps[12].",
"The Google Maps dataset is composed by images in urban environment, instead in UCID and BSD there are generic images.",
"Figure: Simulated SAR image in hypothesis of multiplicative speckleFigure: Top-level workflow of the despeckling CNN."
],
[
"Proposed Approach",
"In this work a deep learning solution for despeckling is proposed.",
"It is focused on the use of deep convolutional neural networks and on their ability to predict the noise and provide a filtered image in which spatial and statistical details are preserved."
],
[
"Convolutional Neural Networks",
"A CNN is composed by a combination of several layers, connected in different ways (cascade, parallel, loop).",
"Each layer can perform different function: convolution, pooling, non-linearities.",
"A generic layer provides a set of $ M $ so-called feature maps.",
"Higher is the level of the layer, more abstract is its output and more representative of overall interaction between layers.",
"So the $l$ -th generic convolutional layer, for $N$ -bands input $\\mathbf {x}^{(l)}$ , yields an $M$ -band output $\\mathbf {z}^{(l)}$ $\\mathbf {z}^{(l)} = \\mathbf {w}^{(l)} \\ast \\mathbf {x}^{(l)} + \\mathbf {b}^{(l)},$ whose $m$ -th component is a combination of 2D convolutions: $\\mathbf {z}^{(l)}(m,\\cdot ,\\cdot ) = \\sum _{n=1}^N \\mathbf {w}^{(l)}(m,n,\\cdot ,\\cdot ) \\ast \\mathbf {y}^{(l)}(n,\\cdot ,\\cdot )+ \\mathbf {b}^{(l)}(m).$ The tensor $\\mathbf {w}$ is a set of $M$ convolutional $N\\times (K\\times K)$ kernels, with a $K\\times K$ spatial support (receptive field), while $\\mathbf {b}$ is a $M$ -vector bias.",
"These parameters, $\\Phi _l\\triangleq \\left(\\mathbf {w}^{(l)},\\mathbf {b}^{(l)}\\right)$ , are learnt during the training phase.",
"In this work we use a pointwise ReLU activation function $g_l(\\cdot )\\triangleq \\max (0,\\cdot )$ yielding the intermediate layer outputs $\\mathbf {y}^{(l)}\\triangleq f_l(\\mathbf {x}^{(l)},\\Phi _l) ={\\left\\lbrace \\begin{array}{ll}\\max (0,\\mathbf {w}^{(l)} \\ast \\mathbf {x}^{(l)} + \\mathbf {b}^{(l)}), & l<L\\\\\\mathbf {w}^{(l)} \\ast \\mathbf {x}^{(l)} + \\mathbf {b}^{(l)}, & l=L\\end{array}\\right.",
"}$ whose concatenation gives the overall CNN function $f(\\mathbf {x},\\Phi ) = f_L(f_{L-1}(\\ldots f_1(\\mathbf {x},\\Phi _1),\\ldots ,\\Phi _{L-1}),\\Phi _L)\\nonumber $ where $\\Phi \\triangleq (\\Phi _1,\\ldots ,\\Phi _L)$ is the whole set of parameters to learn.",
"In the proposed solution, the network (Fig.",
"REF ) is composed by 10 convolutional layers each, except the first and the last, followed by a Rectified Linear Unit (ReLu) activations to ensure fast convergence.",
"The network has a single band image affected by speckle noise $Y$ , the overall output is its filtered version $ \\hat{X}=f(\\mathbf {x},\\Phi )$"
],
[
"Training",
"The goal of the work is to provide a network for despeckling urban areas.",
"For this aim the CNN is trained on the Google Maps dataset that supply a set of urban images on which speckle is simulated according to (REF ) and (REF ).",
"Moreover, in order to give robustness to the network, also a set of generic grayscale images from the UCID and BSD dataset are taking in count for the training.",
"The training process is performed by the Stochastic Gradient Descent with momentum, with learning rate $ \\eta = 2 \\cdot 10^{-6}$ on $30000 \\times (65 \\times 65)$ training patches and $12000 \\times (65 \\times 65)$ for the validation.",
"The cost function $C(\\cdot )$ computes the distance between output and reference and according to its value, the parameters $\\Phi $ of the network are updated via the SGD optimization process $C = \\lambda C_1+ C_2$ $C_1 = \\textit {SID}( \\frac{Y}{\\hat{X}},\\frac{Y}{X} ) = \\textit {SID}(\\hat{N},N)$ $C_2 =||\\hat{X}-X||^2$ In this work $C(\\cdot )$ is a linear combination of two terms: $C_2$ is the mean squared error between filtered image and the noise-free reference; $C_1$ computes a single band adaptation of Spectral Information Divergence (SID) [13] between the estimated ratio image $\\hat{N}$ and the reference one $N$ .",
"Using $C_2$ ensures to minimize the spatial distance between $\\hat{X} $ and $X$ .",
"Minimizing $C_1$ makes the network able to predict the speckle noise and preserve its statistical properties.",
"The aim of using this cost function is two fold: first the network has to predict directly the clean image, second has to take care about the statistical properties of the noise and to do not remove spatial details from the noisy image, but just the speckle.",
"Table: Hyper-parameters of the proposed network"
],
[
"Experimental results",
"In order to assess the performance in an urban environment, the proposed solution is tested on Google Maps images.",
"The networks has never seen these images during the training process.",
"In Fig.",
"REF -REF is shown a comparison with PPB, one of the most well known solution in the state of art for despeckling.",
"Although the PPB filtered images seems to be very clean, the proposed solution preserves better the spatial details and give a closer result to the reference.",
"The network seems to remove the noise and to preserve spatial details that in PPB tend to disappear.",
"PPB works well on big scale object like large buildings and roads, but the overall result tends to be over smoothed and so the most of lower scale objects are filtered.",
"The proposed solution is able to generalize the object scale: it can remove the noise saving spatial details at different scales.",
"In fact, cars and trees are still visible in Fig.",
"REF , as well as the reconstruction of the roofs in Fig.",
"REF .",
"Given that a despeckling solution can be used as pre-processing for other tasks like classification and object detection, preserving objects at different scale plays a very important role in the assessment of performances.",
"Moreover, in Tab.",
"REF numerical results are shown.",
"For numerical assessment M-index [14] has been computed: this index takes into account the filtering accuracy in both regularizing homogeneous areas, computing the Equivalent Number of Looks (ENL), and preserving structures and details, computing homogeneity of ratio images.",
"An ideal filter would produce an M-index equal to zero.",
"The values of this index confirm what we say in the visual comparison.",
"Table: Numerical Results: M-index evaluated on clip1 and clip2Figure: Result on simulated data: clip1Figure: Result on simulated data: clip2Figure: Results on real dataTable: Numerical Results: M-index evaluated on real SAR imageSame considerations can be done for real data: in Fig.",
"REF results on a real SAR images are shown.",
"Without a reference it is difficult to state the quality of a filter, so together with filtered images (top row) we show also the ratio between noisy and filtered image (bottom row).",
"Even if Tab REF shows a better M-index for PPB, also in this case the proposed solution better preserves details than PPB that tends to present an over-smoothed filtered image as well.",
"Considering the ratio images, it is clear that PPB suppresses a lot of details, meanwhile the proposed solution faces some difficulties filtering strong scatterers."
],
[
"Conclusion and Future Works",
"In this work a deep convolutional neural network for despeckling in urban areas is proposed.",
"The network is trained and tested on simulated data.",
"Moreover, the CNN is trained to predict both the clean image and the noise, in order to ensure spatial and statistical consistency in the filtered image.",
"The results are encouraging, the estimated clean images show good details preservation and don't seem to create spatial artefacts on homogeneous areas.",
"In future works, the potential of CNN for despeckling in unsupervised learning will be explored in order to avoid the use of a clean reference."
]
] | 1906.04441 |
[
[
"Secondary atomization of liquid columns in compressible crossflows"
],
[
"Abstract The secondary atomization of liquid droplets is a common physical phenomenon in many industrial and engineering applications.",
"Atomization in high speed compressible flows is less well understood than its more frequently studied low Mach number counterpart.",
"The key to understanding the mechanisms of secondary atomization is examination of the breakup characteristics and droplet trajectories across a range of physical conditions.",
"In this study, a planar shock wave impacting a cylindrical water column ($\\rho_l=1000 \\rm~kg/m^3$) is simulated for a range of Weber numbers ranging three orders of magnitude ($\\sim 10^0-10^3$).",
"Four different incident shock speeds are simulated ($M_s=1.47, 2, 2.5, 3$) which induce subsonic, transonic, and supersonic crossflow across the column.",
"The flowfield is solved using a compressible multicomponent Navier-Stokes solver with capillary forces.",
"Fluid immiscibility is maintained with an interface sharpening scheme.",
"Overall, a diverse range of complex interface dynamics are captured across the range of physical conditions studied.",
"Additionally, while the unsteady drag coefficient of the liquid column shows a dependence on the Weber number using the undeformed diameter, calculations using the deformed diameter significantly reduce the dependence, particularly for the supersonic cases, with implications for subgrid droplet modeling in atomization simulations.",
"A preliminary under-resolved three-dimensional simulation of droplet breakup shows reasonable agreement with experimental data, indicating the potential of the numerical approach for future investigations."
],
[
"Introduction",
"Liquid atomization is an important physical process in a wide variety of applications ranging from manufacturing (including 3D printing) to drug delivery and fuel sprays.",
"The process of liquid breakup has a strong dependence on the Weber number which relates the inertial force to the surface tension.",
"As a large quantity of atomization applications occur in low Mach number flow regimes, significant numerical modeling effort has focused on incompressible schemes [1].",
"State of the art secondary atomisation modeling in the compressible flow regime has largely focused on the early stages of the breakup process and/or higher Weber numbers where the effects of surface tension are assumed to be negligible and are not considered [2], [3], [4].",
"Meanwhile, technical challenges involving supersonic combustion ramjets (scramjets) has identified a need for greater understanding of the penetration, mixing, and atomization of liquid jets injected into high-speed compressible crossflows [5].",
"Liquid jet atomization consists of primary and secondary breakup.",
"The former consists of the bulk liquid transforming into smaller jets, sheets, and droplets.",
"Secondary breakup consists of liquid droplets or ligaments undergoing further deformation and breakup and has generally been classified into vibrational, bag, multi-mode (or bag-and-stamen), sheet-thinning, and catastrophic regimes according to the Weber number [6], [7], [8], [9].",
"However, Theofanous et al.",
"[10] examined droplet breakup in highly rarefied supersonic flow conditions and instead proposed classification of the breakup into two primary criticalities, Rayleigh-Taylor piercing (RTP) and shear-induced entrainment (SIE).",
"The defining feature of RTP is the penetration of the droplet by the gas while SIE is demarcated by a breakup process involving a peeling of the outer surface of the droplet [11].",
"As noted by Guildenbecher et al.",
"[6], this departure from the traditional breakup morphology suggests more investigation of the topic is needed.",
"Moreover, several researchers have pointed out a dependence of the breakup behavior on the density ratio [12], [13] which is important in the context of high speed flows with varying post-shock gas densities and significant compressibility effects.",
"Simulating the entire atomization process requires extremely high resolution due to the multiscale nature of the features involved.",
"This is especially problematic at high Reynolds and Weber numbers where resolving the boundary layer on the droplet surface and becomes difficult and large numbers of small droplets can be generated.",
"Subgrid droplet models can relax the computational complexity and have been used to simulate liquid jet injection in supersonic crossflows [14], [15].",
"However they generally utilize steady-state empirical relations for the drag coefficient of solid spherical particles as a function of the particle Reynolds number to calculate drop trajectories [16].",
"To better understand the behavior of deforming droplets in crossflows and the secondary atomization process in general, various experimental and numerical studies have been performed and were recently reviewed by Guildenbecher et al [6].",
"With respect to the drag coefficient, Kim et al.",
"[17] found that the effects of the initial relative velocity and large relative acceleration or deceleration are significant when predicting rectilinear motion of spherical particles in crossflows.",
"Experiments by Temkin and Mehta [18] showed that the unsteady drag is always larger in decelerating or smaller in accelerating flows than the steady state value.",
"Wadhwa et al.",
"[19] coupled a compressible gas phase solver with an incompressible liquid phase solver and found for axisymmetric conditions the droplet Weber number affects the drag coefficient of a drop traveling at high speeds and placed in quiescent air.",
"Finally, the unsteady nature of the flow as well as the scales (both temporal and spatial) involved in droplet breakup means experimentally measuring the local drop and ambient flow fields during secondary atomization is incredibly challenging [6].",
"Therefore, numerical simulations are a valuable tool for providing important physical insight in such conditions.",
"While some experimental [10] and numerical [20] investigations exist on the interface dynamics and breakup behavior of liquid droplets at a handful of supersonic flow conditions and Weber numbers, the secondary atomization process across a diverse range of physical conditions has not yet been investigated thoroughly.",
"Experimental investigation of liquid columns (as opposed to spherical droplets) allows for easier visualization of the wave structures [21], [22], although difficulties remain in visualizing the later stages of the breakup process.",
"The deformation behavior of the two-dimensional liquid columns have also been found to follow similar trends as that of three-dimensional spherical droplets [23], [24].",
"Numerous researchers have simulated the two-dimensional shock-column interaction, commonly as a test case for compressible multicomponent flow solvers [24], [25], [26], [27], [28], [29], [30], [31].",
"Notable examples include the work of Terashima and Tryggvason [28] who simulated the entire evolution of the column breakup, while Meng and Colonius [25] and Chen [30] examined the sheet-thinning process and evaluated column trajectories and drag coefficients.",
"However, such studies focused on the early stages of breakup and neglected the effects of both surface tension and molecular viscosity.",
"As a result, questions remain as to the breakup process of a liquid column when accounting for molecular viscosity and surface tension effects and especially in the context of supersonic flows.",
"Fortunately, the cylindrical geometry of the water column can be efficiently modeled using a two-dimensional domain providing faster turnaround times compared to full three-dimensional simulations.",
"This allows a wider range of physical conditions to be efficiently examined where for similar reasons axisymmetric domains and/or lower gas-liquid density ratios have been employed in incompressible studies [32], [33], [13].",
"Garrick et al.",
"[34] performed a preliminary study of secondary atomization without molecular viscosity effects and while using a non-conservative interface sharpening scheme.",
"Several simulations of water column-shock interactions were performed including an $M_s=1.47$ shock with comparisons to experiment and an $M_s=3$ shock with and without surface tension.",
"These simulations considered the early stages of breakup and successfully highlighted the effects of surface tension on the dynamics of the gas-liquid interface.",
"The dependence of the breakup behavior on the Weber number for $\\mathrm {We}=5-100$ was also examined with an array of $M_s=1.39$ ($M=0.5$ crossflow) shock-column simulations.",
"The liquid-gas density ratio was set to $\\rho _l/\\rho _g=10$ to reduce computational effort.",
"Garrick et al.",
"[35] extended the numerical method to account for molecular viscosity and non-uniform grids and replaced the non-conservative interface sharpening scheme with a conservative reconstruction based interface sharpening scheme.",
"That approach was then applied to simulate primary and secondary atomization in high speed crossflow.",
"The present work applies the same approach to a wider range of secondary atomization conditions for a two-dimensional liquid column with a high density ($\\rho _l=1000 \\mbox{ kg}/\\mbox{m}^3$ ).",
"This should provide a first order estimate of the three-dimensional behavior but with the benefit of a significantly reduced computational cost.",
"To gain a better understanding of the secondary atomization process in high speed flows, the present work simulates shock-column interactions at various Weber and incident shock Mach numbers to examine the combined effects of surface tension and compressibility on the breakup process across a broad range of physical conditions.",
"This involves detailed two-dimensional simulations of column breakup in high speed compressible flows while accounting for capillary and viscous forces and utilizing an interface sharpening scheme to maintain the fluid immiscibility condition and prevent unphysical numerical smearing of the interface.",
"Particular focus is made on the breakup process and drag coefficient of the droplets over time.",
"The two-dimensional nature of the study is motivated by the focus on a broad range of physical conditions which would be otherwise cost prohibitive to simulate in three dimensions.",
"This follows prior studies which utilized two-dimensional or axisymmetric domains (see [25], [33], [13], [30], [36], [37]) and is also motivated by experimental observations of qualitatively similar breakup characteristics for two-dimensional liquid columns and three-dimensional spherical droplets [21], [23].",
"The paper is organized as follows.",
"Section describes the mathematical model and non-dimensionalization.",
"Section describes the numerical approach while the problem statement is reviewed in Section .",
"Section presents a two-dimensional investigation of the breakup process and drag coefficient of a liquid column across a range of Weber and incident shock Mach numbers.",
"This is followed with a three-dimensional droplet breakup simulation in Section and conclusions in Section ."
],
[
"Mathematical model",
"The present work utilizes the approach of Garrick et al.",
"[34], [35] for solving the flowfield.",
"A non-dimensional form of the quasi-conservative five equation model of Allaire [38] is employed with capillary and molecular viscosity terms.",
"As such, the compressible multicomponent Navier-Stokes equations govern the flowfield [39]: 1 1t + (1 1 u) = 0, 2 2t + (2 2 u) = 0, ut + (u u + p I) = 1Rea + 1Wea 1, Et + ( ( E + p) u ) = 1Rea ( u) + 1Wea 1 u, 1t + u 1 = 0, where $\\rho _1 \\phi _1$ , $\\rho _2 \\phi _2$ , and $\\rho $ are the liquid, gas, and total densities, $\\mathbf {u}=(u,v)^T$ is the velocity, $\\phi _1$ is the liquid volume fraction, $p$ is the pressure, $\\mathrm {We_a}$ and $\\mathrm {Re_a}$ are the acoustic Weber and Reynolds numbers, respectively, $\\kappa $ is the interface curvature, and $E$ is the total energy $E = \\rho e + \\frac{1}{2}\\rho \\mathbf {u} \\cdot \\mathbf {u}$ where $e$ is the specific internal energy.",
"The model is non-dimensionalized using the rules in Table REF where primes indicate dimensional quantities and the subscript `0' refers to a chosen reference state.",
"The dimensional distance $l^\\prime _0$ is chosen as the droplet diameter.",
"This results in the viscous and capillary forces being scaled by acoustic Reynolds and Weber numbers: Rea =0 a0 l00 Wea =0 a20 l00 where $\\mu ^\\prime _0$ and $\\sigma ^\\prime _0$ are the reference dimensional viscosity and surface tension coefficients, respectively.",
"The viscous stress tensor ${\\tau }$ is given with the non-dimensional mixture viscosity $\\mu $ : ${\\tau } = 2 \\mu \\left( \\mathbf {D} - \\frac{1}{3} (\\nabla \\cdot \\mathbf {u}) \\mathbf {I} \\right)$ where $\\mathbf {D}$ is the deformation rate tensor $\\mathbf {D} = \\frac{1}{2} \\left( \\nabla \\mathbf {u} + ( \\nabla \\mathbf {u} )^T \\right).$ The fluid components are considered immiscible and the liquid and gas volume fraction functions ($\\phi _1$ and $\\phi _2$ respectively) are used to capture the fluid interface.",
"Mass is discretely conserved for each phase via individual mass conservation equations.",
"Surface tension is implemented as a volume force as in the CSF model [40] with terms in both the momentum and energy equations [39].",
"While a conservative form of the surface tension term exists [41], the present model utilizes the non-conservative form which enables flexible treatment of the curvature term $\\kappa $ and its accuracy.",
"Table: Non-dimensional rules used in the model."
],
[
"Equation of state and mixture rules",
"To close the model, the stiffened gas equation of state (EOS) [42] is employed to model both the gas and liquid phases.",
"The stiffened gas equation of state utilises fitting parameters $\\gamma $ and $\\pi _\\infty $ to recreate the sonic speed in various materials based on experimental measurements.",
"In the case of air, $\\gamma =1.4$ becomes the specific heat ratio with $\\pi _\\infty =0$ and the stiffened gas equation of state simplifies to the ideal gas law.",
"For a given simulation containing a liquid (1) and gas (2), the stiffened gas equation of state fitting parameters are computed at every point within the domain as a function of the volume fraction: $\\Gamma = \\frac{1}{\\gamma -1} = \\frac{\\phi _2}{\\gamma _2 - 1} + \\frac{\\phi _1}{\\gamma _1-1}$ and $\\Pi = \\frac{\\gamma \\pi _\\infty }{\\gamma -1} = \\frac{\\phi _2 \\gamma _2 \\pi _{\\infty ,2}}{\\gamma _2 - 1} + \\frac{\\phi _1 \\gamma _1 \\pi _{\\infty ,1}}{\\gamma _1-1}.$ where $\\gamma _1$ , $\\gamma _2$ , $\\pi _{\\infty ,1}$ , and $\\pi _{\\infty ,2}$ are the specific stiffened gas EOS fitting parameters for the liquid (1) and gas (2).",
"Using the mixture quantities $\\Gamma $ and $\\Pi $ the total energy becomes $E = \\Gamma p + \\Pi + \\frac{1}{2}\\rho \\mathbf {u} \\cdot \\mathbf {u}.$ The speed of sound is given by $c = \\sqrt{ \\frac{ \\gamma (p + \\pi _\\infty ) }{\\rho } }$ where the stiffened gas EOS fitting parameters $\\gamma $ and $\\pi _\\infty $ are computed using the mixture quantities in Eqs.",
"REF and REF .",
"Similar to Coralic and Colonius [43], the mixture viscosity is determined following Perigaud and Saurel [39] but written in non-dimensional form for use in Eq.",
"REF : = 1 01 + 2 0 2 = N1 + 2 where the liquid (1) and gas (2) viscosities are assumed to remain constant with the gas viscosity used as the reference state $\\mu ^\\prime _0$ .",
"As a result, $\\mu ^\\prime _2/ \\mu ^\\prime _0=1$ and $N=\\mu ^\\prime _1/ \\mu ^\\prime _0$ becomes the liquid to gas viscosity ratio."
],
[
"Numerical method",
"The model (Eqs.",
"-) is discretized using a finite volume method on a non-uniform two-dimensional Cartesian grid.",
"The convective fluxes are upwinded using the Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver originally developed by Toro et al.",
"[44], [45] with modifications for surface tension by Garrick et al. [34].",
"Following the approach of Johnsen and Colonius [46], oscillation free advection of material interfaces is ensured with adaptations to the HLLC for a quasi-conservative form of the volume fraction transport equation.",
"Viscous terms are implemented following Coralic and Colonius [43].",
"Spatial reconstruction to cell faces is performed on the primitive variables using the second order MUSCL scheme with the minmod limiter.",
"The fluid immiscibility condition is maintained using the $\\rho $ -THINC interface sharpening procedure [35] for reconstructing the phasic densities and volume fraction within the interface.",
"The conserved variables are integrated in time using an explicit third order TVD Runge-Kutta scheme [47].",
"Interface curvature is calculated via the interface normals ($\\kappa =-\\nabla \\cdot \\mathbf {n}$ ) which are determined using the smoothed interface function of Shukla et al.",
"[26] and second order central differences.",
"A full description of the numerical method employed and the results of standard validation cases can be found in the work of Garrick et al.",
"[34], [35]."
],
[
"Water column attached domain",
"Additional computational efficiency is gained by translating the static domain with the $x$ component of the column center of mass.",
"This requires appropriate modifications to the fluxes via a simplified arbitrary Lagrangian Eulerian (ALE) formulation [48].",
"The liquid center of mass (and thus the moving grid) velocity $u_c$ is determined via [25]: $u_c = \\frac{\\int \\rho _1\\phi _1 u dV}{\\int \\rho _1 \\phi _1 dV}.$ The individual control volumes remain static, however, the overall computational domain translates downstream such that the liquid center of mass remains approximately centered throughout the simulation."
],
[
"Drag coefficient",
"In the present study the drag coefficient of the liquid is computed following the approach of Meng and Colonius [25]: $C_d = \\frac{m a_c}{\\frac{1}{2} \\rho _g ( u_g - u_c)^2 d_0} $ where $d_0$ is the undeformed diameter of the column, $\\rho _g$ and $u_g$ are the initial post-shock gas conditions and $u_c$ is the center of mass velocity given by equation REF .",
"The acceleration is then computed using finite differences in time [25]: $a_c = \\frac{d}{dt} \\frac{\\int \\rho _1\\phi _1 u dV}{\\int \\rho _1 \\phi _1 dV}.$"
],
[
"Problem statement",
"Standard benchmark cases to verify and validate the shock and interface capturing scheme and the implementation of surface tension were performed by Garrick et al.",
"[34], [35].",
"For the present simulations, the initial conditions are depicted in Figure REF and correspond to a liquid column ($\\rho _l=1000 \\mbox{ kg}/\\mbox{m}^3$ ) in air ($\\rho _g=1.2 \\mbox{ kg}/\\mbox{m}^3$ ) at ambient pressure ($p=101325\\mbox{ Pa}$ ).",
"The column has unity non-dimensional diameter and is centered at the origin.",
"Dirichlet and extrapolation conditions are enforced on the upstream and remaining boundaries respectively.",
"The domain consists of a block of uniform cells in the vicinity of the column corresponding to a resolution of 120 points across the initial column diameter.",
"Grid stretching to the boundary results in an overall domain of $1579\\times 1589$ cells.",
"Figure: Initial layout of the two-dimensional computational domain.The liquid column has a unity non-dimensional diameter and is centered at the origin.Simulations are performed for incident shock Mach numbers of $M_s=1.47$ , $M_s=2$ , $M_s=2.5$ , and $M_s=3$ .",
"The incident shock wave is traveling at a speed defined by the incident shock Mach number toward the liquid column which is stationary in ambient air conditions.",
"The Mach number of the induced crossflow for each simulation is determined by first employing the normal shock relations to compute the Mach number and local speed of sound in the gas behind the incident shock.",
"The crossflow Mach number is the ratio of the post-shock (crossflow) gas velocity in the shock moving reference frame to the post-shock speed of sound.",
"Passage of these incident shocks over the liquid column induces a crossflow with corresponding Mach numbers of $M=0.58$ , $M=0.96$ , $M=1.2$ , and $M=1.36$ , respectively, which range from subsonic to supersonic speeds.",
"These initial conditions are analagous to experimental shock tube setups whereby pressurized gas is released from a driver section into a driven section such that a shock wave develops and travels down the tube to produce a uniform step change in velocity over droplets inserted into the driven section [6].",
"The surface tension term in the momentum and energy conservation equations is scaled by the acoustic Weber number.",
"To examine the breakup behavior for a range of physical conditions, simulations with $\\mathrm {We_a}=1, 5, 10, 20, 50, 100, \\mbox{ and } 1000$ were performed for each incident shock speed.",
"In addition, the breakup behaviors for $\\mathrm {We_a}=0.05 \\mbox{ and } 0.2$ are considered for the $M_s=3$ incident shock speed.",
"The acoustic Reynolds number was held constant with a value of $\\mathrm {Re_a}=1000$ and a liquid to gas viscosity ratio of $N=\\mu _l/\\mu _g=45$ .",
"In the dimensional sense and for a given surface tension coefficient, each acoustic Weber number represents a different column diameter.",
"Of particular interest is the difference in breakup behavior in subsonic versus supersonic crossflow across the range of Weber numbers.",
"To quantify the strength of the surface tension for each simulation, several Weber numbers are described.",
"These are the acoustic, crossflow, and effective Weber numbers.",
"The acoustic Weber number is given in terms of the reference quantities used to non-dimensionalize the system: $\\mathrm {We_a} =\\frac{\\rho ^\\prime _0 a^{\\prime 2}_0 d_0^\\prime }{\\sigma ^\\prime _0}.$ Meanwhile the crossflow Weber number $\\mathrm {We_c}$ is computed using the post-shock crossflow conditions: $\\mathrm {We_c} =\\mathrm {We_a} \\rho u^2 $ where $u$ is the non-dimensional streamwise flow speed and $\\rho $ is the non-dimensional density behind the incident shockwave.",
"The crossflow Reynolds number is similarly estimated by scaling the acoustic Reynolds numbers by the initial post-shock conditions to give $\\mathrm {Re}_{1.47}=1430$ , $\\mathrm {Re}_{2}=4000$ , $\\mathrm {Re}_{2.5}=7000$ , and $\\mathrm {Re}_{3}=10290$ for the $M_s=1.47, 2, 2.5$ and $M_s=3$ cases respectively.",
"Based on the crossflow Reynolds and Weber numbers, these simulations correspond to Ohnesorge numbers ranging from 0.001 to 0.045.",
"Finally, all simulation times are scaled into their respective non-dimensional characteristic times given by [49]: $t^*= \\frac{t u}{D \\sqrt{\\epsilon }}$ where $u$ is the crossflow velocity and $\\epsilon $ is the liquid to gas density ratio using the post-shock conditions.",
"The presence of the density ratio in this equation indicates some dependence of the breakup behavior on the local density ratio which varies for each incident shock Mach number as the post-shock gas density varies depending on the strength of the incident shock.",
"In addition, for the simulations with supersonic crossflow a bow shock is generated in front of the liquid column, further compressing the gas.",
"As a result the local gas-liquid density ratio varies considerably for each incident shock Mach number.",
"One approach to quantify the compressibility effects is the computation of an effective Weber number which considers the local flow conditions that occur behind the bow shock for the simulations with a supersonic crossflow.",
"This effective Weber number can be computed using the crossflow Mach and Weber numbers and the velocity and density normal shock relations [50]: $\\mathrm {We_{eff}} = \\frac{2+(\\gamma -1)M^2}{(\\gamma +1)M^2}\\mathrm {We_c}.$"
],
[
"Grid resolution study and drag uncertainty estimation",
"Grid convergence studies on the shock and interface capturing behavior of the scheme were performed by Garrick et al.",
"[34], [35].",
"In the present work, the effect of grid resolution on the breakup behavior and drag coefficient is examined with several simulations of the $M_s=3$ , $\\mathrm {We_a}=100$ (crossflow $\\mathrm {We}\\approx 2300$ ) shock-column interaction with grid resolutions in the vicinity of the column of $D/60$ , $D/120$ , $D/240$ , and $D/480$ .",
"In lieu of performing an exhaustive grid resolution study at each shock speed and Weber number to be tested, it is assumed that relatively similar behavior trends will apply for the range of conditions in the production runs to follow.",
"First it is important to highlight the limitations of the present simulations.",
"As noted by Jain et al.",
"[51], liquid breakup is ultimately a molecular process and without multiscale modeling the breakup will be initiated by the grid resolution.",
"As noted by Meng and Colonius [2] in their recent paper, this means grid convergence of the breakup behavior is impossible to achieve in a traditional sense.",
"With regards to the viscous effects, direct numerical simulations that resolve the boundary layer on the liquid surface are impractical without highly specialized solvers capable of both significant adaptive mesh refinement and additional body fitted structured conformal meshes which can achieve effective grid resolutions of up to $D/4000$ [20].",
"For these reasons, recent studies of secondary atomisation in this flow regime have tended to consider flow conditions where viscous and surface tension effects can be safely neglected [2], [3], [4].",
"Therefore while both viscous and surface tension effects are included in the simulations presented here, it should be acknowledged that these effects will be under-resolved to some degree.",
"However, the goal is partly to determine to what degree the physics involved in secondary atomisation can be captured despite this limitation.",
"First, the drag coefficient is examined in Figure REF .",
"Note that the drag (Eq.",
"REF ) is determined by integrating the acceleration of the total liquid mass in the domain (Eq.",
"REF ), so as liquid mass is separated and swept downstream it will have a corresponding effect on the drag coefficient.",
"This is particularly noticable in Figure REF where the drag coefficients separate around $t^*=1$ , however, they remain reasonably correlated until approximately $t^*=2$ at which point they diverge.",
"The deformation and breakup behavior of the different simulations is depicted in Figure REF which depicts a time history of the gas-liquid interface (i.e.",
"$\\phi _1=0.5$ iso-line) throughout the simulations where each row depicts a different solution time and each column a different grid resolution.",
"Like with the drag coefficient, the early stages ($t^*<1$ ) of the deformation process does not vary significantly across the grid resolutions tested.",
"For $1< t^*<2$ more fine scale ligament and droplet features are observed in the finer grid resolutions but the general behavior remains similar in the three simulations.",
"The minor differences in the location and trajectory of the smaller droplet particles impact the computed drag coefficient and explains the previously discussed separation of the coefficients in Figure REF for $t^*>1$ .",
"For $t^*>2$ the general behavior consists of the flow “piercing\" through the center of the droplet.",
"This piercing is initiated sooner at the finer grid resolutions (or delayed on coarser grids) but the general breakup behavior is qualitatively similar in all three simulations, albeit with significantly more small droplets captured on the finest grid.",
"Finally, an additional $D/120$ simulation was performed with a domain twice as large and produced nearly identical results to the original $D/120$ simulation, verifying the domain size was not impacting the results.",
"Figure: Mean and standard deviation (SD) of drag coefficient from all grid resolutions at each time point as an estimate of drag coefficient uncertainty over time.Figure: M s =3M_s=3, We a =100\\mathrm {We_a}=100 breakup behavior at D/60D/60 (left), D/120D/120 (center), and D/240D/240 (right) grid resolutions.These results can be broken into several useful groups based on the observed behavior of the drag coefficient and breakup characteristics.",
"For $t^*<1$ the results converge and should provide a reasonable estimate of the drag coefficient and droplet deformation.",
"From $1\\le t^*< 2$ there is some uncertainty in the breakup behavior in terms of the presence and trajectory of smaller droplet clouds, however the general behavior remains the same and as the drag coefficients reasonably correlate across the grid resolutions they should provide at least a first order estimate.",
"For $t^*\\ge 2$ there is significantly more uncertainty in the drag coefficients which begin to diverge across the grid resolutions, however, the general breakup behavior is still observed at all three resolutions."
],
[
"Deformation and breakup behavior",
"The effect of Weber number on the deformation and breakup characteristics of the liquid column is investigated for each shock speed using a grid resolution of D/120.",
"Time histories of the gas-liquid interface (i.e.",
"$\\phi _1=0.5$ iso-line) are shown in corresponding figures where the Weber numbers are depicted at the bottom of each figure.",
"Each row depicts a different solution time and each column a different Weber number.",
"The characteristic time $t^*$ for each row of images is depicted on the left side of each figure.",
"In all cases the crossflow is traveling from left to right."
],
[
"$M_s=1.47$",
"Figure REF depicts the results for the $M_s=1.47$ simulations.",
"For this Mach number, the crossflow Weber numbers correspond closely to the acoustic Weber numbers.",
"For this shock strength the local gas-liquid density ratio using the initial post-shock gas conditions is $\\rho _l/\\rho _g\\approx 460$ .",
"The observed breakup characteristics exhibit reasonable qualitative agreement with the different regimes observed in subsonic experiments for $\\mathrm {Oh<0.1}$ .",
"The regimes are listed in Table REF where the transition Weber numbers are approximate partly due to the continuous nature of the breakup process and the arbitrary choice for specific transition points [6].",
"As a result, different researchers have reported slight variations on the transition between different regimes [7], however, the order in which they appear remains the same [51].",
"At lower Weber numbers (Figure REF (a) and (b)) a vibrational type mode is observed where the surface tension is large enough for the column to remain intact and oscillate as an ellipse.",
"Table: Breakup regimes and transition Weber number as given by .Figure REF (c) depicts various stages of what appears to be a bag breakup process.",
"Generally this regime is characterized by the growth of a bag structure where the center of the drop is blown downstream and attached to an outer rim.",
"Figure: M s =1.47M_s=1.47 deformation and breakup behavior.In the bag-and-stamen/multi-mode regime, the center of the droplet is driven downstream more slowly than the rim leading to the creation of a bag/plume structure [52].",
"Similar features are observed in the present liquid column simulations as depicted in Figure REF (d) and (e).",
"Figure REF (d) depicts the formation of this bag-and-stamen type structure at a slightly lower Weber number (20) compared to the breakup regimes observed for incompressible flow characterized in Table REF .",
"However in the present compressible flow simulations, a small standing shock is observed downstream of the liquid column.",
"A similar standing shock feature has been observed in prior numerical results without surface tension at this flow speed [25], [28].",
"The pressure disturbance caused by the presence of the standing shocks could contribute to the growth of the bag-and-stamen structure.",
"Figure REF (e) is characterized by a substantial plume/bag-and-stamen structure forming around $t^*=2.3$ before its subsequent rupture into numerous small droplets.",
"Finally, the breakup characteristics in Figure REF (g) correlate well with the so-called catastrophic regime where the drop surface is corrugated by large amplitude waves resulting in a large number of smaller droplets and ligaments [6]."
],
[
"$M_s=2$",
"Figure REF depicts the breakup behavior for the $M_s=2$ simulations.",
"The post-shock conditions are in the transonic regime with a crossflow Mach number of $M=0.96$ .",
"For this shock strength the local gas-liquid density ratio using the initial post-shock gas conditions is $\\rho _l/\\rho _g\\approx 312$ .",
"Across the range of Weber numbers the breakup behavior is very similar to the slower $M_s=1.47$ case even as late as $t^*=2$ .",
"However, at later times the general breakup behavior begins to noticably deviate from the lower Mach number case, especially with respect to the overall size of the ligament structures which were observed to stretch considerably further in the $M_s=1.47$ simulations.",
"As the $M_s=2$ shock induces a faster crossflow than the $M_s=1.47$ case, the crossflow Weber number corresponding to each acoustic Weber number is slightly higher.",
"Figure REF (b) depicts a bag-and-stamen type breakup structure with the outer rim of the column being swept downstream faster than the center of the column, resulting in the formation of several ligament structures.",
"Figures REF (c)-(e) depict a unique multimode type of asymmetric breakup culminating in the collapse of the droplet into a largely coherent ligament structure although an increasing number of smaller droplets are generated during this process at the higher Weber numbers.",
"This noticably asymmetric behavior appears to originate from small asymmetries which appear earlier during the deformation process, i.e.",
"in Figures REF (c)-(e) at $t^*=1.98, 2.47$ .",
"Finally, a catastrophic type breakup is observed at the highest Weber numbers in Figures REF (f) and (g).",
"Figure: M s =2M_s=2 deformation and breakup behavior."
],
[
"$M_s=2.5$",
"Figure REF depicts the breakup behavior for the $M_s=2.5$ simulations.",
"The higher incident shock speed means the post-shock conditions consist of a supersonic flow.",
"For this shock strength the local gas-liquid density ratio using the initial post-shock gas conditions is $\\rho _l/\\rho _g\\approx 250$ .",
"As a result, the estimated crossflow Weber number is much higher for each acoustic Weber compared to the corresponding $M_s=1.47$ and $M_s=2$ simulations.",
"With the presence of supersonic flow and an associated bow shock appearing in front of the droplet, the effective post-shock Weber number is computed using Eq.",
"REF to provide a comparable metric for subsonic simulations.",
"Using the same approach to compute an effective gas-liquid density ratio accounting for the bow-shock gives $\\rho _l/\\rho _g\\approx 187$ .",
"The breakup behavior is generally similar to the $M_s=2$ simulations with a vibrational type mode observed in Figure REF (a), multimode type behavior in Figures REF (b)-(d) and catastrophic type breakup in Figures REF (e)-(g).",
"Similarly to the $M_s=2$ simulations, a feature of this catastrophic breakup behavior is the generation of a “channel\" whereby the liquid column is pierced in the center into two separate chunks.",
"Figure: M s =2.5M_s=2.5 deformation and breakup behavior."
],
[
"$M_s=3$",
"Theofanous et al.",
"[10] performed experiments of aerobreakup of spherical liquid droplets in $M=3$ crossflows.",
"They observed “piercing\" ($44<\\mathrm {We}<10^3$ ) and “stripping\" ($\\sim 10^3<\\mathrm {We}$ ) breakup regimes.",
"Figure REF depicts the breakup behavior for the present simulations which considers an $M_s=3$ shock speed that results in a considerably slower $M=1.36$ crossflow compared to the experiments of Theofanous et al.",
"Despite this difference, the range of breakup features depicted in Figure REF with the estimated effective Weber numbers varying from approximately 0.7 in Figure REF (a) to 1400 in Figure REF (g) appear to qualitatively match descriptions of the experimentally observed breakup regimes despite the disparity in crossflow speeds and flow dimensionality.",
"As with the previous simulations, the higher crossflow speed in the $M_s=3$ case results in significantly higher crossflow Weber numbers for each acoustic Weber number.",
"As a result, a significant number of small droplets are generated even at relatively low acoustic Weber numbers such as Figure REF (e) and in the early stages of Figures REF (f)-(g).",
"Catastrophic breakup is observed in the later stages of Figures REF (f)-(g).",
"As in the $M_s=2$ and $M_s=2.5$ simulations, this catastrophic breakup is characterized by a channel which forms in the liquid column, splitting it into two.",
"This general behavior is similar to that experimentally observed for a waterdrop in a shocktube by Waldman et al [53].",
"They described the breakup process as an initially continuous stripping of liquid from the droplet surface followed by a growth in the amplitude of surface waves which lead to the final disintegration of the droplet.",
"This description appears qualitatively similar to the time history of breakup depicted in Figures REF (f)-(g).",
"For this shock strength the local gas-liquid density ratio using the initial post-shock gas conditions is $\\rho _l/\\rho _g\\approx 216$ .",
"The effective gas-liquid density ratio accounting for the bow-shock gives $\\rho _l/\\rho _g\\approx 134$ .",
"Figure: M s =3M_s=3 deformation and breakup behavior."
],
[
"Drag coefficient",
"Figure REF depicts comparisons of the early stages of the drag coefficient with prior numerical results of Meng and Colonius [25], Chen [30], and Terashima and Tryggvason [28].",
"The drag coefficient was computed following the approach of Meng and Colonius [25] as discussed in section REF .",
"Good agreement is obtained with the data of [25], disparities in the other results can likely be attributed to the use of a different approach to calculate the drag coefficient, where drift data (and not averaged fluid velocity) is used to estimate the column acceleration.",
"Further discussion of different approaches for computing the drag coefficient can be found in [23] and [25].",
"Figure: M s =2.5M_s=2.5Figure REF depicts the drag coefficient at the later stages of the simulations with comparisons to Meng and Colonius [25].",
"An extra simulation was also performed to provide a reference point to a stationary and rigid cylinder in crossflow where the drag coefficient is known.",
"This was approximated with a high liquid density ($\\rho _l=10,000\\mbox{ kg}/\\mbox{m}^3$ ) case with $\\mathrm {We_a}=1$ .",
"Note that even under these conditions, some deformation of the high density liquid does occur.",
"Generally for $1000 < \\mathrm {Re} < 3\\times 10^5$ , the drag coefficient of a cylinder is known to be approximately unity [54].",
"This value is plotted as a solid blue line in Figure REF and agrees well with the present subsonic simulation with a crossflow Reynolds number of 1430.",
"Meanwhile from Gowen and Perkins [55] the drag coefficient of a stationary cylinder in a $M=1.2$ crossflow (i.e.",
"the crossflow for $M_s=2.5$ ) is approximately 1.64 and is plotted as a solid blue line in Figure REF for reference.",
"This value reasonably predicts the minimum drag coefficient value for the $\\rho _l=10,000\\mbox{ kg}/\\mbox{m}^3$ simulation and which occurs around $t^*=0.3-0.4$ in Figure REF .",
"Figure: M s =3M_s=3While the general trend is similar, overall the drag coefficient exhibits less unsteady variation compared to the results of [25].",
"The inclusion of surface tension and especially interface sharpening employed in the current simulations reduces the amount of liquid material stripped from the interface where it would otherwise enter the highly chaotic wake region and contribute to unsteady liquid acceleration measurements.",
"Generally, lower drag coefficients are observed with lower Weber numbers for each shock Mach number except $M_s=3$ which shows less relative variation between the drag coefficients at Weber numbers in the range of 1 to 100 in Figure REF .",
"Significant differences in the drag as a function of the Weber number are observed in the $M_s=1.47$ and $M_s=2$ cases in Figures REF and REF .",
"Less variation is observed between the higher Weber numbers for the $M_s=2.5$ and $M_s=3$ cases depicted in Figure REF and REF .",
"Gowen and Perkins also noted there was almost no observed variation in the drag coefficient as a function of Reynolds number in the supersonic flow regime for a solid circular cylinder [55].",
"They stated that the suction pressures on the downstream side of the cylinder contribute a large part of the total drag in subsonic flows but as a percentage of the total drag this contribution rapidly decreases as the Mach number increases.",
"Supporting the experimental observations of Temkin and Mehta [18], the unsteady drag is found to be larger in the decelerating relative flows of the liquid columns compared to that of the rigid stationary column.",
"The coefficients are observed to be twice as large or more compared to the rigid case for all shock Mach numbers.",
"Interestingly, comparing the present supersonic cases to the subsonic cases shows that at higher Mach numbers there is significantly less variation in the drag coefficient as a function of the Weber number for the liquid columns.",
"Upon first inspection, this is perhaps surprising as section REF demonstrated a broad range of breakup behaviors at each Mach number as a function of the Weber number and the drag is computed as an integration over the acceleration of the total liquid volume as it undergoes breakup.",
"However, an examination of the breakup behaviors for the supersonic cases in Figures REF and REF appears to show a similar deformed diameter progression for the Weber number 1-100 cases within the respective Mach numbers.",
"To explore this, an effective diameter of the deformed drop was computed and the results are presented in Figure REF .",
"This value is computed as the total projected length of the liquid on an x-normal plane, where the liquid is defined as $\\phi >0.5$ .",
"Comparing the calculated effective diameters, a similar trend is observed for the effective diameter as the drag.",
"Significant differences are seen in the effective diameter of the subsonic cases while less variation is observed in the supersonic cases at higher Weber numbers.",
"This suggests the similarities in drag are a product of a similar effective diameter throughout the breakup process, even if the breakup itself differs.",
"Figure: M s =3M_s=3Figure: M s =3M_s=3Figure REF depicts the drag coefficient computed again using Eq.",
"REF but with the time dependent effective diameter used in place of the undeformed diameter term $d_0$ .",
"As noted by Meng and Colonius [25], the computed drag coefficients can largely be assumed as constant regardless of shock speed during the early stages of breakup when accounting for the effective deforming diameter of the droplets.",
"Interestingly, the present simulations show that this assumption is still relatively reasonable during the mid and later stages of breakup and even when accounting for the effects of surface tension across a wide range of Weber numbers.",
"This is a notable result given the wide range of breakup behaviours observed in the present simulations.",
"These results are especially relevant at supersonic speeds where less variation of the drag coefficient is observed as a function of Weber number."
],
[
"Three-dimensional simulation of droplet breakup",
"A three-dimensional simulation of droplet breakup was performed.",
"The objective being to further validate the ability of the numerical method to predict three-dimensional droplet breakup behaviour and to provide a point of comparison to the two-dimensional liquid column breakup simulations.",
"The flow conditions were set to match the experimental conditions in Figure 33 of Theofanous et al [11].",
"Specifically, the simulation consists of a water droplet impacted by a shockwave with post-shock crossflow conditions of $M=0.32$ , $\\mathrm {Re}_{g}=2.2 \\times 10^4$ , $\\mathrm {We}= 7.8 \\times 10^2$ , and $\\mathrm {Oh}=2.4 \\times 10^{-3}$ .",
"Given the computational complexity of such a three-dimensional simulation, the grid resolution in the vicinity of the droplet was set to a relatively coarse $D/80$ and symmetry boundary conditions were employed at the centerline such that the computational domain consisted of only a quarter of the overall droplet.",
"Figure REF shows the progression of the droplet deformation and breakup.",
"The present resolution is inadequate to capture the fine scale features of the breakup process however the overall droplet shape evolution over time reasonably agrees with the experimental behavior shown in the video supplementing Figure 33 of Theofanous et al [11] (see supplementary multimedia material of [11] for video).",
"Figure: Snapshots from three-dimensional droplet breakup simulation corresponding to experiment from Figure 33 of Theofanous et al .",
"Simulation consists of a water droplet impacted by a shock-wave with freestream flow conditions M=0.32M=0.32, Re g =2.2×10 4 \\mathrm {Re}_{g}=2.2 \\times 10^4, We =7.8×10 2 \\mathrm {We}= 7.8 \\times 10^2, and Oh =2.4×10 -3 \\mathrm {Oh}=2.4 \\times 10^{-3}.",
"Note the crossflow is moving from right to left."
],
[
"Conclusion",
"Numerical experiments are performed of $M_s=1.47, 2, 2.5$ and $M_s=3$ shockwaves interacting with liquid columns at various Weber numbers.",
"The simulations account for the effects of compressibility, molecular viscosity, and surface tension.",
"The shockwaves induce a crossflow leading to aerobreakup of the liquid column.",
"A diverse range of complex interface dynamics and breakup modes are observed with good correlation to experimentally observed behavior across the range of Weber numbers tested.",
"During the early stages of the breakup process (i.e deformation), similar behavior is observed across the range of Mach numbers tested.",
"However, at later times the breakup behavior varies significantly depending on both the Mach and Weber numbers.",
"Additionally, lower Weber numbers result in lower observed drag coefficients for the liquid columns.",
"Depending on the Weber number, the drag coefficients are still approximately two to three times those observed for a rigid liquid column.",
"As a function of the Weber number, significantly less variation in the drag coefficient and qualitative flow features is observed as the Mach number increases.",
"In addition, when utilizing a deformed diameter in the drag coefficient calculation the results show significantly reduced variation between Weber numbers across all Mach numbers.",
"This has implications for subgrid atomization models which determine droplet trajectories based on estimated particle drag coefficients.",
"A three-dimensional simulation, while under-resolved, displays reasonable agreement with the corresponding experimental breakup behavior, highlighting the potential of the numerical approach for future investigations."
],
[
"Acknowledgments",
"This work is supported by Taitech, Inc. under sub-contracts TS15-16-02-004 and TS16-16-61-004 (primary contract FA8650-14-D-2316).",
"The computational resources in this paper are partially supported by the HPC@ISU equipment at Iowa State University, some of which has been purchased through funding provided by NSF under MRI grant number CNS 1229081 and CRI grant number 1205413.",
"This work has been approved for unlimited release: LA-UR-19-25304."
],
[
"Appendix",
"The additional cases for the $M_s=3.0$ incident shock are presented in Fig REF for completeness.",
"Similar breakup characteristics are seen to those observed in Fig REF in the comparable Weber number ranges.",
"Figure: Additional M s =3M_s=3 deformation and breakup behavior."
]
] | 1906.04307 |
[
[
"ALMA observations of A0620-00: fresh clues on the nature of quiescent\n black hole X-ray binary jets"
],
[
"Abstract We report on ALMA continuum observations of the black hole X-ray binary A0620-00, at an X-ray luminosity nine orders of magnitude sub-Eddington.",
"The system was significantly detected at 98 GHz (at $44 \\pm 7~\\mu{\\rm Jy}$) and only marginally at 233 GHz ($20 \\pm 8~\\mu{\\rm Jy}$), about 40 days later.",
"These results suggest either an optically thin sub-mm synchrotron spectrum, or highly variable sub-mm jet emission on month timescales.",
"Although the latter appears more likely, we note that, at the time of the ALMA observations, A0620-00 was in a somewhat less active optical-IR state than during all published multi-wavelength campaigns when a flat-spectrum, partially self-absorbed jet has been suggested to extend from the radio to the mid-IR regime.",
"Either interpretation is viable in the context of an internal shock model, where the jet's spectral shape and variability are set by the power density spectrum of the shells' Lorentz factor fluctuations.",
"While strictly simultaneous radio-mm-IR observations are necessary to draw definitive conclusions for A0620-00, the data presented here, in combination with recent radio and sub-mm results from higher luminosity systems, demonstrate that jets from black hole X-ray binaries exhibit a high level of variability - either in flux density or intrinsic spectral shape, or both - across a wide spectrum of Eddington ratios.",
"This is not in contrast with expectations from an internal shock model, where lower jet power systems can be expected to exhibit larger fractional variability owing to an overall decrease in synchrotron absorption."
],
[
"Introduction",
"The Atacama Large Millimeter Array (ALMA) offers the opportunity for the first time to characterize the properties of low-luminosity Galactic black hole X-ray binaries in the mm and sub-mm band.",
"This window remains largely unexplored, and can prove to be crucial for our understanding of the interplay between accretion and the production of outflows at very low accretion rates.",
"Especially when it comes to winds and relativistic jets, our phenomenological knowledge (, and references therein) is largely based on observations taken during the bright phases of the hard X-ray state, i.e., at X-ray luminosity levels between about one thousandth to ten per cent of the Eddington luminosity ($L_{\\rm Edd}$ ; ).",
"In this regime, the X-ray emission is ascribed to non-thermal radiation from a hot electron/positron population that enshrouds the inner accretion flow, either via Comptonization or synchrotron self-Compton, or a combination of the two.",
"This is ubiquitously associated with persistent radio emission with a flat or slightly inverted spectrum extending up to IR/optical frequencies, where the outer disc and/or the donor star dominate the emission.",
"The combination of the observed radio spectral indices, polarization levels and high brightness temperatures point to synchrotron radiation from a relativistic, collimated outflow, whereby the radio-emitting plasma becomes progressively more transparent at lower frequencies as it propagates toward larger distances from the base , .",
"Arguably the best example of such a persistent, hard state, “steady\" jet is the high-mass black hole X-ray binary Cygnus X-1, whose persistent, flat (radio-mm) spectrum counterpart has been resolved into a compact core plus a variable, highly collimated jet with very long baseline interferometry .",
"In order to attain a steady, flat spectrum, the relativistic plasma within the jet must be constantly re-energized as it propagates downstream, so as to compensate for adiabatic energy losses.",
"In analogy to both gamma-ray bursts and blazars , repeated collisions between multiple plasma shells of different velocities/Lorentz factors have been proposed as a viable and efficient mechanism for providing the plasma with a constant energy replenishment , .",
"Within the framework of this internal shock model, so long as the energy dissipation occurs at a (nearly) constant rate downstream in the jet, the net, time-averaged result is a (nearly) constant flux density over a large frequency domain (i.e., up to the break frequency where the emission becomes optically thin), even though the physical nature of the so-called steady jet is intrinsically variable.",
"Indeed, the Cygnus X-1 radio flux density displays 20–30 per cent level variability on time-scales of hours to months (, and references therein).",
"Moving towards lower luminosities, it is generally believed that the jet properties are fundamentally unaltered, in that the flat radio spectra seem to be retained at Eddington ratios below $$ < $$ 10-6-10-5$, i.e.",
"in the so-called quiescent regime.",
"In a recent, comprehensive investigation of the radio variability properties of the black hole X-ray binary V404~Cygni in quiescence (at $ LX $\\simeq 3-4\\times 10^{-5}$ LEdd$~\\cite {bernardini14}), spanning over a quarter of a century, \\cite {plotkin19} find evidence for relatively large deviations from a flat spectrum (with an average value of $ =0.020.65$, where $ F$).",
"Moreover, 0.3--0.4 dex variability is found to be common on every observable timescale, from minutes up to decades (it should be noted that, owing to the sub-mJy flux densities involved, it is extremely challenging to spatially resolve V404 Cygni^{\\prime }s quiescent radio counterpart into multiple, milliarcsec-scale beams with current instrumentation \\cite {millerjones08}).Within the internal shock model scenario, this behaviour can be interpreted as arising from a shock with higher-than-average amplitude propagating through an otherwise ``steady\" (i.e., nearly constant dissipation throughout the) jet.",
"Multi-wavelength investigations suggest that V404 Cygni^{\\prime }s $$flat synchrotron spectrum extends into the near-IR regime, where it becomes optically thin \\cite {gallo07,hynes09,russellbreaks,plotkin15,plotkin17a}.$ Our knowledge of jet behaviour is comparatively limited at extremely sub-Eddington ratios; in terms of radio properties, most of it hinges on radio detections of three nearby ($<3$ kpc) systems at $L_{\\rm X}$$$ < $$ 10-8$ $ LEdd$:A0620--00 \\cite {gallo06,gallo07,dincer18}, XTE J1118+480 \\cite {gallo14,plotkin15}, and MWC~656 (\\cite {dzib}; \\cite {ribo17}; see \\cite {millerjones11} for a compilation of upper limits).",
"For all three systems, the measured radio flux densities are again indicative of synchrotron emission from a relativistic jet, as gyro-synchrotron emission from either donor star is estimated to be negligible.",
"Moreover, the luminosities are consistent (within the large errors) with the extrapolation of the empirical, non-linear radio:X-ray luminosity correlation established for higher Eddington ratio systems (\\cite {corbel03}; \\cite {corbel08}; \\cite {corbel13}; \\cite {gfp03,gallo12}; \\cite {gallo14}).",
"This has contributed significantly to the notion of quiescence as a dialed-down version of the more luminous hard state, whereby the jet power decreases along with the accretion flow^{\\prime }s radiative efficiency.",
"A critical, open question, however, is whether, at such low Eddington ratios, the jet remains optically thick all the way to IR wavelengths.",
"If so, and owing to the non-linear radio:X-ray luminosity scaling, then the jet -- rather than the accretion flow -- may dominate the power output from quiescent systems (\\cite {fender03}; \\cite {gallo05a}; \\cite {kording06a}; \\cite {russell10,coriat11,polko13,markoff15}).$ Determining the full spectral extent of the radio-emitting jet in hard state and quiescent black hole X-ray binaries essentially translates into a careful modelling of the system's optical and IR emission.",
"This aims to ascertain the presence of any excess IR emission with respect to the tail of the donor star's blackbody spectrum, and to determine its physical nature.",
"In addition to synchrotron radiation from a jet, excess IR emission can be ascribed to a variety of mechanisms, including synchrotron radiation from a hot, radiatively inefficient inflow, thermal emission from the accretion stream-outer flow impact point, thermal emission from the outer accretion disc and/or thermal emission from a cold, dusty disc of circumbinary material (see § for details and references).",
"At $L_{\\rm X}$$\\simeq 10^{-9}$ $L_{\\rm Edd}$ , and $1.06\\pm 0.12$ kpc distance , the quiescent black hole X-ray binary A0620–00 provides us with a testbed for competing models through a direct measurement of its mm spectrum.",
"In this Letter, we report on dual band ALMA observations of A0620–00, obtained in November and December 2016.",
"In concert with closely-spaced radio and IR-optical observations, these data enable us to place new constraints on the jet properties in this prototypical system.",
"Continuum observations in Band 3 at 98 GHz and Band 6 at 233 GHz were taken as part of project 2016.1.00773.S (P.I.",
"Gallo).",
"The Band 3 data were taken on the 12th and 14th November 2016 with 46 antennas, with baselines spanning the range 15 m to 1039 m, and for a total of 1.1 hr of integration.",
"The Band 6 data were taken on the 20th and 21st December 2016 with 48 antennas, baselines ranging between 15 m and 491 m, for 2.15 hr.",
"All observations were performed with good weather conditions with median precipitable water vapor of 0.36 mm for the Band 3 observations and 1.29 mm for Band 6.",
"The data were calibrated using $\\texttt {CASA}$ using the pipeline script provided by ALMA staff.",
"For both sets of observations the quasars J0750+1231 and J0641-0320 were used for flux and phase calibration, respectively.",
"After the initial calibration, there was insufficient signal to perform self-calibration.",
"The data were imaged using natural weighting to maximize sensitivity, achieving a beam size of $1.25\\times 1.10$ with a position angle of $89.0$ for Band 3, and a root mean square (rms) noise of $6.9~\\mu {\\rm Jy~beam}^{-1}$ .",
"For the Band 6 data the synthesized beam is $0.86\\times 0.77$ at a position angle of $-71.3$ , with an rms noise of $8.2~\\mu {\\rm Jy~beam}^{-1}$ .",
"The resulting images are shown in Fig.",
"REF .",
"A0620–00 is clearly detected in the Band 3 continuum, while only marginally significant emission is found for Band 6.",
"Figure: ALMA continuum images at 98 GHz (band 3), left, and 233 GHz (band 6), right.",
"The contours are in steps of 2σ2\\sigma where σ=7.9μ Jy beam -1 \\sigma = 7.9~\\mu {\\rm Jy~beam}^{-1} for Band 3 and 8.2μ Jy beam -1 8.2~\\mu {\\rm Jy~beam}^{-1} for Band 6, with dotted contours being negative.",
"The source center is shown by the black cross, and the synthesized beams are shown in the lower left corner of each panel.In order to check whether phase decorrelation could have reduced the detection significance in Band 6, we reran the calibration pipeline, flagging three scans on the phase calibrator (scans 8, 14 and 20) prior to the gain calibration, and treating them as a second target source, interpolating the complex gains from neighbouring calibrator scans.",
"Imaging the data from these three scans gave a peak flux density within 5 per cent of that measured from the self-calibrated data on the same source.",
"This confirms that phase decorrelation as a function of time was $<$ 5 per cent during the Band 6 observations.",
"While we cannot explicitly test for decorrelation as a function of position on the sky, the time stability and Band 6 weather conditions give us a degree of confidence that the (5.6 degree) positional shift between the phase calibrator and A0620–00 was unlikely to be responsible for decorrelation as a function of position.",
"The integrated flux densities were measured by fitting a 2D Gaussian with the same properties as the synthesized beam to the continuum images, allowing the total integrated flux density and source position to vary.",
"Posterior distributions were sampled using an affine invariant Markov chain Monte Carlo ensemble sampler, emcee .",
"In Band 3 the best-fitting integrated flux was $43.6 \\pm 0.8~\\mu {\\rm Jy}$ and in Band 6 it was $20 \\pm 1~\\mu {\\rm Jy}$ , where the 16th to 84th percentiles of the posterior distribution (which was symmetric about the median) were used to quantify the uncertainty in the Gaussian's amplitude.",
"Absolute flux calibration adds a systematic uncertainties of 5 per cent for Band 3 and 10 per cent for Band 6; overall, the measurement errors are dominated by rms noise uncertainties, yielding a robust detection in Band 3, at $44 \\pm 7~\\mu {\\rm Jy}$ , and a marginally significant detection in band 6, at $20 \\pm 8~\\mu {\\rm Jy}$ ."
],
[
"VLA",
"We obtained 13 observations with the Karl G. Jansky Very Large Array (VLA) between the 11th September 2017 and 22nd January 2018 (program 17B-233; PI: Plotkin).",
"Each observation lasted between 30 and 90 min, for a total of 666 min integration time on source across all 13 observations.",
"The data were taken in continuum mode at X-band (8–12 GHz) using 4 GHz total bandwidth (over 2$\\times $ 2 GHz basebands centred at 9.0 and 10.65 GHz).",
"Data were processed using CASA v5.1.1 .",
"We used calibrated data products provided by NRAO from the VLA CASA Calibration pipeline v5.1.2.",
"For these calibrations, for 11/13 observations, 3C 48 was used as the primary calibrator to perform delay calibration, to find complex bandpass solutions, and to set the flux density scale (3C 286 was used for the other two observations).",
"All 13 observations used J0643+0857 as a nearby secondary calibrator to derive the complex gain solutions.",
"Imaging was performed using clean in CASA.",
"We stacked all 13 images, but imaged each 2 GHz base-band separately to obtain spectral information.",
"We used natural weighting to maximise the sensitivity, and we accounted for the frequency dependence over the large fractional bandwidth using two Taylor terms (nterms=2).",
"We measured flux densities in the stacked images of each 2 GHz baseband using imfit, forcing a point source (fixed to the size of the synthesized beam) at the known location of A0620–00.",
"We measured flux densities of $12.9\\pm 1.5$ and $14.2 \\pm 1.8$ $\\mu $ Jy bm$^{-1}$ at 8.9 and 10.8 GHz, respectively.",
"The quoted error bars correspond to rms errors for each image, conservatively inflated by 10 per cent to account for potential systematic uncertainties in the flux density scale derived from 3C 48, which started a large flare around 2018 January (towards the end of our observations).",
"Radio variability is estimated to be well within a factor of 2 during this monitoring campaign (Plotkin et al., in prep.",
")."
],
[
"SMARTS",
"A0620–00 is routinely monitored at optical and near-IR (NIR) frequencies with the dual-channel imager ANDICAM on the SMARTS 1.3 m telescope, at the Cerro Tololo Inter-American Observatory.",
"Data presented here were reduced following standard procedures described in ; point-spread function photometry was first performed to measure instrumental magnitudes for A0620–00 and several nearby field stars, and then converted to the standard photometric system through differential photometry, with absolute calibration via optical primary standards on clear nights and the Two Micron All-Sky Survey catalog.",
"For calculating the intrinsic source fluxes, we used the zero points given in bessell98, and the color excess $E(B-V)=0.30\\pm 0.05$ (Cantrell et al.",
"2010), which was converted to total extinction values ${A}_{{\\rm {B}}}=1.23$ , ${A}_{{\\rm {V}}}\\,=0.93$ , ${A}_{{\\rm {I}}}=0.47$ , ${A}_{{\\rm {J}}}=0.26$ , ${A}_{{\\rm {H}}}=0.16$ , and ${A}_{{\\rm {K}}}=0.11$ .",
"Shown in Figure REF are A0620–00's, phase-folded, SMARTS V-band magnitudes as measured over a variety of epochs close to the multi-wavelength campaigns discussed below.",
"This is for the purpose of illustrating whether, at the time of a specific campaign, the source was in a passive vs. active quiescent state, as identified by Cantrell et al.",
"(2008; see § 3.1).",
"Figure shows instead the average H-I-V and B-magnitudes as measured on December 20 and 21, 2016, i.e.",
"simultaneous with the ALMA Band 6 data.",
"Figure: A0620–00's orbital-phased SMARTS V-band magnitudes, calculated using the updated ephemeris by .",
"Nearly simultaneous epochs with other multi-wavelength measurements quoted in the body of this paper are highlighted with different colors, and compared against the passive state “envelope\" defined by Cantrell et al.",
"(2008), in black, vs. active state data from Cantrell et al., as well as , in sand and light pink.",
"Blue and orange points, respectively, are simultaneous within ±\\pm 2 days with the 2016 Band 6 and Band 3 ALMA data presented here.",
"While they are formally inconsistent with a pure passive state, the Band 3 data (which, unlike Band 6, resulted in a significant detection) may correspond to a slightly higher activity level.To interpret the ALMA observations presented here, it is useful to start by critically reviewing the relative complexity of what we refer to as quiescence, and the many ways the black hole X-ray binary A0620–00 has played a key role in our understanding of it."
],
[
"A0620–00's jet: spectral extent and variability",
"In a seminal work, analysed optical and NIR orbital-phase resolved data of A0620–00 from 1981 to 2007 (see also ).",
"They first distinguished between two main quiescent states – passive and active, plus loops state, in between the two – each representing different levels of activity.",
"Passive state light curves display the lowest level variability; in this regime, the ellipsoidal modulation of the donor star fully accounts for the observed optical-NIR variability.",
"In the active state, the optical–NIR flux is 0.1–0.4 magnitudes brighter and variable, with prominent variations even on short timescales (down to seconds; e.g., ; ; ).",
"A0620–00, which has been in quiescence since the decline from the 1975 outburst that led to its discovery , can spend months or years in these different quiescent states, being in the passive state in 1997–2003, the active state in 2004–2007 and 2013, the passive state in 2015–2016, changing to the active state by 2016 November , , , .",
"Figure REF illustrates the phase-folded, V-band variability of A0620–00 during (or close to) the different campaigns and observations discussed below, vis-a-vis the well-identified passive and active states, shown by the black and sand/pink circles, respectively.",
"Several lines of evidence suggest that, at least during the active state, the spectral properties of the radio jet in A0620–00 resemble those of higher Eddington ratios systems, such as V404 Cygni, where a partially self-absorbed synchrotron spectrum extends all the way from radio up to mid-IR wavelengths , , , .",
"Arguably the most compelling piece of evidence for this comes from broadband SED modelling.",
"With an 8.5 GHz counterpart at 51$\\pm 7 \\mu $ Jy, A0620–00 was the first truly quiescent black hole X-ray binary to be detected in the radio band, in 2005 .",
"Earlier observations, performed in 2003 with Spitzer MIPS had shown evidence for excess mid-IR emission with respect to the tail of the donor star at 8 $\\mu $ m .",
"These authors ascribed the mid-IR excess to thermal emission from a cold, circumbinary dust disc, illuminated by the low-mass donor star (such discs may be formed as a result of mass outflow from the outer accretion disc; ).",
"Subsequently, combined optical and X-ray observations performed in 2005 (simultaneously with the VLA) with the (non-simultaneous) Spitzer MIPS data (all in active state – see Figure 2), and reinterpreted the mid-IR excess as arising from the extrapolation of a $\\sim $ flat-spectrum radio jet extending all the way to the Spitzer band (if true, this would imply that the radiative output of the jet in A0620–00 would be comparable to – if not greater than, depending on the cooling break frequency – that of the accretion flow).",
"Along the same lines, fitted the broadband SED of A0620–00, taken in 2010 March (likely during the active state, according to the same authors; no SMARTS data are available for this campaign), and including high-resolution optical and UV spectra from Keck and the Hubble Space Telescope.",
"They found that the non-stellar light had a peak at 0.3 $\\mu $ m that could be fitted by a black-body, likely from the hot spot/stream impact point onto the accretion disc from the donor star.",
"They also reported on a UV upturn in flux and a red excess that can both be fitted well by a model in which a partially self-absorbed jet extends all the way to the mid-IR, with the pre-acceleration inner jet component dominating the UV upturn, and the post-acceleration synchrotron dominating the red excess towards the IR (simultaneous radio observations with the Australia Telescope Compact Array did not detect A0620–00 down to 42 $\\mu $ Jy at 5.5 GHz).",
"inferred from their modelling that the mass accretion rate from the donor through the hot spot was five orders of magnitude higher than the accretion rate at the black hole inferred from the X-ray luminosity, implying either highly radiatively inefficient accretion, and/or that outflows expel almost all of the accreted mass, and/or the mass transfer rate from the outer disc into the inner hot region is very low as matter builds up in the disc.",
"Optical–IR variability also gives important clues to the nature of the IR emission in A0620–00, and other quiescent systems.",
"The power density spectrum of the highly variable optical-IR component detected in the active quiescent state, when measured, is similar to that of an X-ray power density spectrum of a black hole X-ray binary in the hard state, in which the X-ray variability originates in the inner regions of the accretion flow.",
"This “flickering\" component is stronger at longer wavelengths, with a fractional rms amplitude of $\\sim 15$ –24 per cent at optical wavelengths, rising to $\\sim 42$ per cent at ($K$ -band) NIR wavelengths .",
"The spectral index of the variable component has been inferred by various authors to be steeply red at optical–NIR wavelengths ($\\alpha = -1.4$ to $-0.7$ ; , , ), and flat/slightly inverted at NIR ($\\alpha \\sim 0.3$ to 0.4; ) and mid-IR ($\\alpha = 0.2$ to 0.3 at 3.6–8.0 $\\mu $ m; ) wavelengths.",
"Noting that the extrapolation of this non-stellar mid-IR spectrum of A0620–00 down to GHz frequencies was consistent with the radio flux measured by the VLA in 2005, concluded that the optically thick-to-thin jet break lies in the optical frequency range: $1.3\\pm 0.5\\times 10^{14}$ Hz.",
"Finally, reported an excess of linear polarization of $\\sim 1.25 \\pm 0.28$ per cent in the NIR for A0620–00 (from observations taken in February 2013, again during the active state), with a position angle of the magnetic field vector that is consistent with being parallel with the inferred axis for the transient radio ejection associated with the source 1975 outburst , providing further circumstantial evidence for the jet interpretation of the IR excess – albeit alternative mechanisms, such as coronal synchrotron emission or a circumbinary disc whose dust grains align with the global magnetic field of the accretion disc may be able to yield similar polarization signatures.",
"In spite of multiple, indirect lines of evidence for a $\\sim $ flat radio-mid-IR jet spectrum in the active state of A0620–00 , , there are reasons to remain skeptical.",
"It should be noted that, to first order, the jet break frequency is expected to scale inversely with the black hole mass , ; as jet breaks are often observed in the GHz/sub-mm regime in active galactic nuclei, they are expected to occur in the IR-optical band for $10^{5-7}$ times lighter objects.",
"At the same time, additional parameters are thought to affect the exact jet break frequency for a given system: e.g., the exact value is known to vary with the overall luminosity level as well as X-ray photon index (; , , ), likely reflecting changes in the magnetic field energy density, particle density and/or mass loading at the jet base , , .",
"While an optical jet break for a system at nine orders of magnitude sub-Eddington remains somewhat challenging in the context of basic scale-invariant jet models, recent developments allow for more nuanced solutions.",
"As an example, the 3 dex excursion in jet break frequency observed during a single state transition in the black hole X-ray binary MAXI J1836–194 has been successfully modelled through a semi-analytical treatment of the relativistic-magneto-hydrodynamic jet equations (; ), where the jet break frequency is assumed to correspond to the location of the jet's “modified fast point\".",
"Alternatively, it may be that an additional spectral component contributes some of the optical emission.",
"The last broadband SED investigation of A0620–00 prior to our ALMA observations was presented by , who collected simultaneous radio, IR, optical and X-ray data in December 2013 (see pink points “Dincer+18\" in Figure 2; also in the active state).",
"While they did not attempt an overall spectral fit, were the first to report a highly inverted radio spectrum, between 5–22 GHz, with $\\alpha ={+0.70\\pm 0.13}$ .",
"Taken at face value, this spectrum is inconsistent with an extrapolation of the known mid-IR excess to radio frequencies; in the absence of mm data, however, it could either signal a highly peaked radio-mm spectrum, or simply trace high-level variability, or, the low frequency curvature of an otherwise $\\sim $ flat 50 $\\mu $ Jy spectrum above 20 GHz, (as seen, e.g., in V404 Cygni; , ).",
"Figure shows the broadband SED of A0620–00 for three of the multi-wavelength campaigns discussed above: red open triangles correspond to the 2003-2005 campaign reported by Gallo et al.",
"(2007; VLA, SMARTS and Chandra), including the Spitzer data from ; green open squares refer to the 2013 campaign (VLA and SMARTS) reported by ; blue filled circles represent the most recent data set, consisting of the new VLA, ALMA and SMARTS observations presented in this work.",
"The ALMA data alone could easily be interpreted as indicative of optically thin synchrotron emission, with $\\alpha ={-0.56\\pm 0.47}$ (3$\\sigma $ C.L.)",
"over the 98–233 GHz frequency interval.",
"However, as the Band 3 and 6 observations were taken on 12-14th November and 20th-21st December 2016, respectively, it is also possible – and arguably more likely – that the measured values indicate a factor $$ > $$ 2$ variability in flux density over $$\\; > \\over \\sim \\;$ 1$-month timescale.",
"This would be broadly consistent both with the measured variability at 5--8 GHz over a time-scale of several years (see again Figure \\ref {fig:sed}), as well as with the much shorter term variability inferred for the jet of V404 Cygni at radio frequencies (albeit at 3 dex higher $ LEdd$; \\cite {plotkin19}).\\begin{figure}\\includegraphics [width=0.5]{sed.pdf}\\caption {Broadband SED of A0620--00 in quiescence, over different epochs.",
"Red open triangles are based on data taken over 2003-2005 (VLA, \\textit {Spizter}, SMARTS and \\textit {Chandra}) and reported by Gallo et al.",
"(2007); green open squares refer to the 2013 campaign (VLA and SMARTS) reported by (Din{ç}er et al.",
"2018); blue filled circles represent the most recent data set, consisting of the 2017 VLA and 2016 ALMA observations presented in this work, combined with SMARTS data.",
"}\\end{figure}$ That intrinsic variability may be at the origin of what we are observing is indirectly supported by the SMARTS monitoring, as illustrated in Figure REF , where data from within $\\pm 2$ days of the 98 GHz (Band 3) and 233 GHz (Band 6) ALMA observations, respectively, are shown as orange and blue points.",
"While both ALMA epochs appear inconsistent with a pure passive state (i.e.",
"with the black points), the 98 GHz data (which resulted in a significant detection) broadly overlap with active state data from previous campaigns (light pink and sand points), and, seem to correspond to a slightly higher activity level than the 233 GHz data.",
"At the same time, it should be noted that, during both ALMA observations, A0620–00 was somewhat less active than during other published multi-wavelength campaigns discussed above, when a flat-spectrum, partially self-absorbed jet been suggested to extend from the radio to at least the mid-IR regime."
],
[
"On the variable nature of black hole X-ray binary jets",
"While strictly simultaneous radio-mm-IR observations are necessary to draw definitive conclusions, the data presented here, when discussed in the context of previous multi-wavelength campaigns, suggest that A0620–00's jet has a highly variable nature.",
"Whether they are best explained by a variable jet break location (with a $$ < $$ 100$ GHz break frequency, compared to previous claims in the mid-IR/optical), or significant intrinsic flux variability within an otherwise partially self-absorbed jet, the 2016 ALMA observations of A0620--00, in combination with recent radio results from V404~Cygni \\cite {plotkin19} and Cygnus X-1 \\cite {teta19cyg}, and optical--IR results from Swift J1357.2--0933 \\cite {shahbaz13,russell18}, demonstrate that \\textit {jets from black hole X-ray binaries exhibit a high level of variability across a wide spectrum of Eddington ratios.",
"}$ In the context of the internal shock model, A0620–00's behaviour may be linked to a systematic change in the amplitude and/or injection time-scale of the shells' Lorentz factor fluctuations.",
"With respect to the latter, find that, whereas a $\\sim $ flat radio-mid-IR jet SED naturally arises from a flicker noise process, i.e., where the power spectral density of the Lorentz factor fluctuations is inversely proportional to the Fourier frequency ($P(f)\\propto 1/f$ ), a peaked spectrum at GHz frequencies can result from a steep power spectral density, where $P(f) \\propto 1/f^{\\beta }$ , with $\\beta >1$ .",
"For steeper $\\beta $ values, the Lorentz factor fluctuations have, on average, longer time-scales, meaning that most of the shell collisions, and hence energy dissipation, occur farther away along the jet (see figures 5e and 5e in ).",
"This would correspond to a scenario where the jet break frequency indeed occurred below $\\sim $ 100 GHz at the time of the ALMA observations.",
"Alternatively – and arguably more likely – the ALMA data could signal $$ > $$ 100$ per cent flux variability over a month timescale.",
"Going back to the internal shock model, a broad range of variability timescales can be induced if/when higher-than-average amplitude fluctuations propagate through an otherwise fainter, flat-spectrum jet which may or may not extend to the mid-IR (this has also been suggested to explain the optically thin radio spectral index that is sometimes observed in V404 Cygni; \\cite {plotkin19}).",
"Incidentally, for a conical jet geometry, the fractional variability is bound to be progressively higher at higher frequencies, where the size of the emitting region becomes progressively smaller (in spite of the very limited diagnostics available in the mm and sub-mm windows, observations of V404 Cygni during the decay from its 2015 outbursts appear to confirm enhanced variability at sub-mm wavelengths, compared to cm; \\cite {teta19v404}).",
"\\\\$ In the specific case of A0620–00, it is important to stress again that the ALMA observations were taken at a time when the system's optical-IR variability had not fully reached the active quiescent state, which previous investigations have claimed to be associated with a mid-IR jet.",
"It is interesting to recall that, in their seminal work, ascribe the transition from passive to loops and active states to the onset of clumpy accretion, where the loops are “qualitatively consistent with expanding shells of gas, initially heated to be bluer than A0620–00's secondary, then cooling adiabatically as they redden significantly while remaining bright, then finally fading from view\".",
"The active state is described as more erratic, with large color as well as magnitude fluctuations that are seldom as faint as the minimum passive state data.",
"We speculate that the passive quiescent state may correspond to a regime where the jet emission is negligible at mid-IR frequencies – possibly even suppressed entirely as a result of extremely low accretion rates.",
"The sudden onset of (clumpy) accretion events would initiate the building up of a jet.",
"Initially, the power density spectrum of the Lorentz factor fluctuations could be fairly steep ($\\beta $ > $$ 1 $), possibly corresponding to a peaked sub-mm/mm synchrotron spectrum, and would progressively flatten as the system settles into the active quiescent state.",
"If so, the jet itself may ultimately be responsible for \\textit {defining} the active and passive quiescent states of A0620--00 as identified by Cantrell et al.",
"More importantly, this behaviour should be common at extremely low accretion rates.",
"Higher Eddington-ratios systems, where accretion takes place in a more sustained fashion, would instead be characterized (in A0620--00^{\\prime }s jargon) by a \\textit {persistent} active state, where the jet is likely responsible for the observed mid-IR excess at all times.",
"\\\\$ In closing, no firm conclusions can be drawn based upon on the dual band ALMA observations of the quiescent black hole X-ray binary A0620–00 presented here.",
"The source was significantly detected at 98 GHz (Band 3) in mid November 2016, and only marginally so at 233 GHz (Band 6) some 40 days later.",
"This can either be interpreted as the signature of a partially self-absorbed jet whose break frequency occurred somewhere below 98 GHz, or, more simply, in terms of intrinsic flux variability on $\\sim $ month timescales in the ALMA bandpass.",
"Strictly simultaneous radio and mm observations of A0620–00, in concert with IR and optical, are necessary to discriminate between the two explanations, and whether indeed the peculiar active/passive quiescent state phenomenology of this system is directly linked to the jet itself.",
"Regardless of the correct interpretation, the data presented here, in combination with recent radio and sub-mm results from higher luminosity systems, demonstrate that jets from black hole X-ray binaries exhibit a high level of variability – either in flux density or intrinsic spectral shape, or both – across a wide spectrum of Eddington ratios.",
"At first glance, this may be surprising, in the sense that one might expect that, as the accretion rate decreases, so does the total jet power and, as a consequence, its degree of variability.",
"However, precisely as a result of a reduced jet power, the jet synchrotron emission becomes less absorbed, and the inner parts of the jet emit lower frequency radiation that they would at higher accretion rates/jet powers.",
"In other words, at any given frequency, the SED of a quiescent/low jet power source is going to be dominated by emission occurring at smaller (as in, closer to the jet base) physical scales than for brighter X-ray states.",
"Thus, in spite or the reduced overall flux levels, fractional variability can be safely expected to remain strong in the less powerful sources."
],
[
"Acknowledgements",
"This paper makes use of the following ALMA data: ADS/JAO.ALMA#2016.1.00773.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.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation (NSF) operated under cooperative agreement by Associated Universities, Inc. JCAM-J is the recipient of an Australian Research Council Future Fellowship (FT140101082) funded by the Australian government.",
"EG is grateful to Julien Malzac for his insightful comments on an earlier version of this manuscript."
]
] | 1906.04299 |
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