Add files using upload-large-folder tool

.gitattributes

@@ -4342,3 +4342,59 @@ _dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf filter=lfs diff=lfs merge=lfs -tex
 L9E0T4oBgHgl3EQfjAEs/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 SdAyT4oBgHgl3EQf8PoD/content/2301.00851v1.pdf filter=lfs diff=lfs merge=lfs -text
 lNE0T4oBgHgl3EQf7wKH/content/2301.02780v1.pdf filter=lfs diff=lfs merge=lfs -text
+iNE0T4oBgHgl3EQfpgG_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf filter=lfs diff=lfs merge=lfs -text
+dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf filter=lfs diff=lfs merge=lfs -text
+UNE3T4oBgHgl3EQfEQnM/content/2301.04295v1.pdf filter=lfs diff=lfs merge=lfs -text
+tNAzT4oBgHgl3EQfPft0/content/2301.01184v1.pdf filter=lfs diff=lfs merge=lfs -text
+wdE2T4oBgHgl3EQfgwdJ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+39AyT4oBgHgl3EQf1_mj/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+KtE4T4oBgHgl3EQf7w7r/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+XdFRT4oBgHgl3EQf-jgZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+SdAyT4oBgHgl3EQf8PoD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf filter=lfs diff=lfs merge=lfs -text
+h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf filter=lfs diff=lfs merge=lfs -text
+KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf filter=lfs diff=lfs merge=lfs -text
+4dE1T4oBgHgl3EQf6QUq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+jtAzT4oBgHgl3EQfbfyR/content/2301.01387v1.pdf filter=lfs diff=lfs merge=lfs -text
+xtE2T4oBgHgl3EQfhQc5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+A9AyT4oBgHgl3EQf3_rL/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+WNE5T4oBgHgl3EQfcQ-J/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+eNE4T4oBgHgl3EQfQgzc/content/2301.04983v1.pdf filter=lfs diff=lfs merge=lfs -text
+WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf filter=lfs diff=lfs merge=lfs -text
+c9E_T4oBgHgl3EQf0hwq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+jtAzT4oBgHgl3EQfbfyR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+K9E0T4oBgHgl3EQfSgAu/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+mtAzT4oBgHgl3EQfqP1R/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ptAzT4oBgHgl3EQfb_y2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+3NFAT4oBgHgl3EQflB25/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf filter=lfs diff=lfs merge=lfs -text
+v9FPT4oBgHgl3EQf-jXG/content/2301.13216v1.pdf filter=lfs diff=lfs merge=lfs -text
+O9E0T4oBgHgl3EQf0wLF/content/2301.02691v1.pdf filter=lfs diff=lfs merge=lfs -text
+39E2T4oBgHgl3EQfjgft/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+QdFRT4oBgHgl3EQf7Dgl/content/2301.13678v1.pdf filter=lfs diff=lfs merge=lfs -text
+c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf filter=lfs diff=lfs merge=lfs -text
+ctE5T4oBgHgl3EQffg_5/content/2301.05628v1.pdf filter=lfs diff=lfs merge=lfs -text
+FdE1T4oBgHgl3EQfqwVD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf filter=lfs diff=lfs merge=lfs -text
+4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf filter=lfs diff=lfs merge=lfs -text
+A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf filter=lfs diff=lfs merge=lfs -text
+39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf filter=lfs diff=lfs merge=lfs -text
+KNFIT4oBgHgl3EQfaCs0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DdE1T4oBgHgl3EQfEAP6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+_9FJT4oBgHgl3EQfqyzS/content/2301.11606v1.pdf filter=lfs diff=lfs merge=lfs -text
+DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf filter=lfs diff=lfs merge=lfs -text
+ddFJT4oBgHgl3EQfSCyU/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+lNE0T4oBgHgl3EQf7wKH/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+39E2T4oBgHgl3EQfOAbJ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+wNE5T4oBgHgl3EQfMQ59/content/2301.05480v1.pdf filter=lfs diff=lfs merge=lfs -text
+K9E1T4oBgHgl3EQfGgPl/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+UNE3T4oBgHgl3EQfEQnM/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DdE3T4oBgHgl3EQfUwqw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+KtE5T4oBgHgl3EQfYA_J/content/2301.05571v1.pdf filter=lfs diff=lfs merge=lfs -text
+utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf filter=lfs diff=lfs merge=lfs -text
+7tE1T4oBgHgl3EQfTwM_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf filter=lfs diff=lfs merge=lfs -text
+btAzT4oBgHgl3EQfZvwc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf filter=lfs diff=lfs merge=lfs -text
+ctE5T4oBgHgl3EQffg_5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text

0dE1T4oBgHgl3EQf4wVz/content/tmp_files/2301.03504v1.pdf.txt

@@ -0,0 +1,1940 @@
+Radial pulsations, moment of inertia and tidal deformability of dark energy stars
+Juan M. Z. Pretel1, ∗
+1Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud,
+150 URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil
+(Dated: January 10, 2023)
+We construct dark energy stars with Chaplygin-type equation of state (EoS) in the presence of
+anisotropic pressure within the framework of Einstein gravity. From the classification established by
+Iyer et al. [Class. Quantum Grav. 2, 219 (1985)], we discuss the possible existence of isotropic dark
+energy stars as compact objects. However, there is the possibility of constructing ultra-compact stars
+for sufficiently large anisotropies. We investigate the stellar stability against radial oscillations, and
+we also determine the moment of inertia and tidal deformability of these stars. We find that the usual
+static criterion for radial stability dM/dρc > 0 still holds for dark energy stars since the squared
+frequency of the fundamental pulsation mode vanishes at the critical central density corresponding
+to the maximum-mass configuration. The dependence of the tidal Love number on the anisotropy
+parameter α is also examined. We show that the surface gravitational redshift, moment of inertia and
+dimensionless tidal deformability undergo significant changes due to anisotropic pressure, primarily
+in the high-mass region. Furthermore, in light of the detection of gravitational waves GW190814,
+we explore the possibility of describing the secondary component of such event as a stable dark
+energy star in the presence of anisotropy.
+I.
+INTRODUCTION
+Different types of observations (such as Type Ia super-
+novae, structure formation and CMB anisotropies) indi-
+cate that our Universe is not only expanding, it is acceler-
+ating. Within the standard ΛCDM model (which is based
+on cold dark matter and cosmological constant in Ein-
+stein gravity), this cosmic acceleration is due to a smooth
+component with large negative pressure and repulsive
+gravity, the so-called dark energy. Such a model gives
+a good agreement with the recent observational data [1],
+but suffers from the well-known coincidence problem and
+the fine-tuning problem [2, 3]. The exact physical nature
+of dark energy is still a mystery and, consequently, the
+possibility that dark matter and dark energy could be
+different manifestations of a single substance has been
+considered [4–7]. In that regard, it was shown that the in-
+homogeneous Chaplygin gas offers a simple unified model
+of dark matter and dark energy [8]. It was also argued
+that if the Universe is dominated by the Chaplygin gas
+a cosmological constant would be ruled out with high
+confidence [9].
+Using the Planck 2015 CMB anisotropy, type-Ia su-
+pernovae and observed Hubble parameter data sets, the
+full parameter space of the modified Chaplygin gas was
+measured by Li et al. [10]. Based on recent observations
+of high-redshift quasars, Zheng and colleagues [11] inves-
+tigated a series of Chaplygin gas models as candidates
+for dark matter-energy unification. The application of
+the Hamilton-Jacobi formalism for generalized Chaplygin
+gas models was carried out in Ref. [12]. Additionally, it is
+worth mentioning that Odintsov et al. [13] considered two
+different equations of state for dark energy (i.e., power-
+law and logarithmic effective corrections to the pressure).
+∗ [email protected]
+They showed that the power-law model only yielded some
+modest results, achieved under negative values of bulk
+viscosity, while the logarithmic scenario provide good fits
+in comparison to the ΛCDM model.
+Another way to give rise to an accelerated expansion of
+the Universe is by modifying the geometry itself [14, 15],
+namely, considering higher curvature corrections to the
+standard Einstein-Hilbert action.
+Under this outlook,
+the cosmic acceleration can be modeled in the scope of
+a scalar-tensor gravity theory [16, 17]. Moreover, within
+the context of the so-called f(R) theories [18, 19], the
+quadratic term in the Ricci scalar R leads to an infla-
+tionary solution in the early Universe [20], although such
+a model does not provide a late-time accelerated expan-
+sion. Nevertheless, the late-time acceleration era can be
+realized by terms containing inverse powers of R [21],
+though it was shown that this is not compatible with
+the solar system experiments [22]. For a comprehensive
+study on the evolution of the early and present Universe
+in f(R) modified gravity, we refer the reader to the re-
+view articles [23–25] and references contained therein.
+On the other hand, the astrophysical implications due
+to the f(R) modified gravitational Lagrangian on com-
+pact stars have been intensively investigated in the past
+few years [26–33].
+According to the aforementioned works, different dark
+energy models have been proposed in order to explain the
+mechanisms that lead to the cosmic acceleration. Only
+about 4% of the Universe is made of familiar atomic mat-
+ter, 20% dark matter, and it turns out that roughly 76%
+of the Universe is dark energy [34]. Within the context
+of General Relativity, dark energy is an exotic negative
+pressure contribution that can lead to the observed accel-
+erated expansion. In the absence of consensus regarding a
+theoretical description for the current accelerated expan-
+sion of the Universe, theorists have proposed using the
+Chaplygin gas as a useful phenomenological description
+arXiv:2301.03504v1  [gr-qc]  9 Jan 2023
+
+2
+[4]. If dark energy is distributed anywhere permeating or-
+dinary matter, then it could be present in the interior of a
+compact star. Therefore, the purpose of this manuscript
+is to investigate the possible existence of compact stars
+with dark energy by assuming a Chaplygin-type EoS. For
+such stars to exist in nature, they need to be stable under
+small radial perturbations.
+Adopting a description of dark energy by means of a
+phantom (ghost) scalar field, Yazadjiev [35] constructed a
+general class of exact interior solutions describing mixed
+relativistic stars containing both ordinary matter and
+dark energy.
+The energy conditions and gravitational
+wave echoes of such stars were recently analyzed in
+Ref. [36]. Furthermore, the effect of the dynamical scalar
+field quintessence dark energy on neutron stars was inves-
+tigated in [37]. Panotopoulos and collaborators [38] stud-
+ied slowly rotating dark energy stars made of isotropic
+matter using the Chaplygin EoS. Bhar [39] proposed a
+model for a dark energy star made of dark and ordinary
+matter in the Tolman–Kuchowicz spacetime geometry.
+For further stellar models with dark energy we also refer
+the reader to Refs. [40–48].
+In addition, anisotropy in compact stars may arise due
+to strong magnetic fields, pion condensation, phase tran-
+sitions, mixture of two fluids, bosonic composition, rota-
+tion, etc. Thus, regardless of the specific source of the
+anisotropy, it is more natural to think of anisotropic fluids
+when studying compact stars at densities above nuclear
+saturation density. In that regard, the literature offers
+some physically motivated functional relations for the
+anisotropy, see for example Refs. [49–55]. However, we
+must point out that these anisotropic models are based
+on general assumptions (or ansatzes) that do not directly
+relate to exotic modifications of matter or gravity. In-
+deed, it has been argued that the deformation near the
+maximum neutron-star mass comes from the anisotropic
+pressure within these stars, which is caused by the distor-
+tion of Fermi surface predicted by the equation of state
+of the models [56]. Becerra-Vergara et al. [57] showed
+that the contribution of the fourth order corrections pa-
+rameter (a4) of the QCD perturbation on the radial
+and tangential pressure generate significant effects on the
+mass-radius relation and the stability of quark stars. It
+has also been shown that the stellar structure equations
+in Eddington-inspired Born-Infeld theory with isotropic
+matter can be recast into GR with a modified (apparent)
+anisotropic matter [58].
+Motivated by the several works already mentioned, we
+aim to discuss the impact of anisotropy on the macro-
+scopic properties of dark energy stars with Chaplygin-like
+EoS. We will address the following questions: Do these
+stars belong to families of compact or ultra-compact
+stars? How does anisotropy affect the compactness and
+radial stability of dark energy stars satisfying the causal-
+ity condition? In particular, by adopting the phenomeno-
+logical ansatz proposed by Horvat et al. [51], we deter-
+mine the radius, mass, gravitational redshift, frequency
+of the fundamental oscillation mode, moment of inertia
+and the dimensionless tidal deformability of anisotropic
+dark energy stars. The isotropic solutions are recovered
+when the anisotropy parameter vanishes, i.e. when α = 0.
+The organization of this paper is as follows: In Sec. II
+we start with a brief overview of relativistic stellar struc-
+ture, describing the basic equations for radial pulsations,
+moment of inertia and tidal deformability. We then in-
+troduce the Chaplygin-like EoS and discuss its relation to
+the cosmological context in Sec. III, as well as we present
+the anisotropy profile. Section IV provides a discussion
+of the numerical results for the different physical prop-
+erties of dark energy stars. Finally, our conclusions are
+summarized in Sec. V.
+II.
+STELLAR STRUCTURE EQUATIONS
+In order to study the basic features of compact stars
+with dark energy, in this section we briefly summarize the
+stellar structure equations in Einstein gravity. In particu-
+lar, we focus on hydrostatic equilibrium structure, radial
+pulsations, moment of inertia, and tidal deformability.
+The theory of gravity to be used in this work is general
+relativity, where the Einstein field equations are given by
+Gµν ≡ Rµν − 1
+2gµνR = 8πTµν,
+(1)
+with Gµν being the Einstein tensor, Rµν the Ricci tensor,
+R denotes the scalar curvature, and Tµν is the energy-
+momentum tensor.
+Since we are interested in isolated
+compact stars, we consider that the spacetime can be
+described by the spherically symmetric four-dimensional
+line element
+ds2 = −e2ψdt2 + e2λdr2 + r2(dθ2 + sin2 θdφ2).
+(2)
+In addition, we model the compact-star matter by an
+anisotropic perfect fluid, whose energy-momentum tensor
+is given by
+Tµν = (ρ + pt)uµuν + ptgµν − σkµkν,
+(3)
+where ρ is the energy density, σ ≡ pt − pr the anisotropy
+factor, pr the radial pressure, pt the tangential pressure,
+uµ the four-velocity of the fluid, and kµ is a unit four-
+vector.
+These four-vectors must satisfy uµuµ = −1,
+kµkµ = 1 and uµkµ = 0. Notice that the stellar fluid
+becomes isotropic when σ = 0.
+A.
+TOV equations
+When the stellar fluid remains in hydrostatic equilib-
+rium, neither metric nor thermodynamic quantities de-
+pend on the time coordinate.
+This allows us to write
+uµ = e−ψδµ
+0 and kµ = e−λδµ
+1 . Accordingly, the hydro-
+static equilibrium of an anisotropic compact star is gov-
+
+3
+erned by the TOV equations:
+dm
+dr = 4πr2ρ,
+(4)
+dpr
+dr = −(ρ + pr)
+�m
+r2 + 4πrpr
+� �
+1 − 2m
+r
+�−1
++ 2σ
+r ,
+(5)
+dψ
+dr = −
+1
+ρ + pr
+dpr
+dr +
+2σ
+r(ρ + pr),
+(6)
+which are obtained from Eqs. (1)-(3) together with the
+conservation law ∇µT µ
+1
+= 0. The metric function λ(r)
+is determined from the relation e−2λ = 1 − 2m/r, where
+m(r) is the gravitational mass within a sphere of radius
+r.
+By supplying an EoS for the radial pressure in the form
+pr = pr(ρ) and a defined anisotropy relation for σ, the
+system of differential equations (4)-(6) is then numeri-
+cally integrated from the center at r = 0 to the surface
+of the star r = R which correspond to a vanishing pres-
+sure. Therefore, the above equations will be solved under
+the requirement of the following boundary conditions
+ρ(0) = ρc,
+m(0) = 0,
+ψ(R) = 1
+2 ln
+�
+1 − 2M
+R
+�
+, (7)
+where ρc is the central energy density, and M ≡ m(R)
+is the total mass of the star calculated at its surface.
+The numerical solution of the TOV equations describes
+the equilibrium background and allow us to obtain the
+metric components and fluid variables.
+B.
+Radial oscillations
+A rigorous analysis of the radial stability of compact
+stars requires the calculation of the frequencies of nor-
+mal vibration modes. Such frequencies can be found by
+considering small deviations from the hydrostatic equi-
+librium state but maintaining the spherical symmetry of
+the star. In the linear treatment, where all quadratic (or
+higher-order) or mixed terms in the perturbations are
+discarded, one assumes that all perturbations in physical
+quantities are arbitrarily small.
+The fluid element lo-
+cated at r in the unperturbed configuration is displaced
+to radial coordinate r + ξ(t, r) in the perturbed config-
+uration, where ξ is the Lagrangian displacement.
+All
+perturbations have a harmonic time dependence of the
+form ∼ eiνt, where ν is the oscillation frequency to be
+determined. Consequently, defining ζ ≡ ξ/r, the adia-
+batic1 radial pulsations of anisotropic compact stars are
+1 In the adiabatic theory, it is assumed that the fluid elements of
+the star neither gain nor lose heat during the oscillation.
+governed by the following differential equations [55]
+dζ
+dr = − 1
+r
+�
+3ζ + ∆pr
+γpr
++
+2σζ
+ρ + pr
+�
++ dψ
+dr ζ,
+(8)
+d(∆pr)
+dr
+= ζ
+�
+ν2e2(λ−ψ)(ρ + pr)r − 4dpr
+dr
+−8π(ρ + pr)e2λrpr + r(ρ + pr)
+�dψ
+dr
+�2
++2σ
+�4
+r + dψ
+dr
+�
++ 2dσ
+dr
+�
++ 2σ dζ
+dr
+− ∆pr
+�dψ
+dr + 4π(ρ + pr)re2λ
+�
++ 2
+r δσ,
+(9)
+where ∆pr is the Lagrangian perturbation of the radial
+pressure and γ = (1+ρ/pr)dpr/dρ is the adiabatic index
+at constant specific entropy.
+The above first-order time-independent equations (8)
+and (9) require boundary conditions set at the center and
+surface of the star, similar to a vibrating string fixed at
+its ends. Since Eq. (8) has a singularity at the origin, the
+following condition must be required
+∆pr = − 2σζ
+ρ + pr
+γpr − 3γζpr
+as
+r → 0,
+(10)
+while the Lagrangian perturbation of the radial pressure
+at the surface must satisfy
+∆pr = 0
+as
+r → R.
+(11)
+C.
+Moment of inertia
+Suppose a particle is dropped from rest at a great dis-
+tance from a rotating star, then it would experience an
+ever increasing drag in the direction of rotation as it ap-
+proaches the star. Based on this description, we intro-
+duce the angular velocity acquired by an observer falling
+freely from infinity, denoted by ω(r, θ). Here we will cal-
+culate the moment of inertia of an anisotropic dark en-
+ergy star under the slowly rotating approximation [59].
+This means that when we consider rotational corrections
+only to first order in the angular velocity of the star Ω,
+the line element (2) is replaced by its slowly rotating
+counterpart, namely
+ds2 = − e2ψ(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2 θdφ2)
+− 2ω(r, θ)r2 sin2 θdtdφ,
+(12)
+and following Ref. [59], it is pertinent to define the differ-
+ence ϖ ≡ Ω−ω as the coordinate angular velocity of the
+fluid element at (r, θ) seen by the freely falling observer.
+Keep in mind that Ω is the angular velocity of the stel-
+lar fluid as seen by an observer at rest at some spacetime
+point (t, r, θ, φ), and hence the four-velocity up to linear
+terms in Ω can be written as uµ = (e−ψ, 0, 0, Ωe−ψ). To
+this order, the spherical symmetry is still preserved and
+
+4
+it is possible to extend the validity of the TOV equations
+(4)-(6). Nonetheless, the 03-component of the field equa-
+tions contributes an additional differential equation for
+angular velocity. By retaining only first-order terms in
+Ω, such component becomes
+eψ−λ
+r4
+∂
+∂r
+�
+e−(ψ+λ)r4 ∂ϖ
+∂r
+�
++
+1
+r2 sin3 θ
+∂
+∂θ
+�
+sin3 θ∂ϖ
+∂θ
+�
+= 16π(ρ + pt)ϖ.
+(13)
+As in the case of isotropic fluids, we follow the same
+treatment carried out by Hartle [59, 60] and we assume
+that ϖ can be written as
+ϖ(r, θ) =
+∞
+�
+l=1
+ϖl(r)
+� −1
+sin θ
+dPl
+dθ
+�
+,
+(14)
+where Pl are Legendre polynomials. Taking this expan-
+sion into account, Eq. (13) becomes
+eψ−λ
+r4
+d
+dr
+�
+e−(ψ+λ)r4 dϖl
+dr
+�
+− l(l + 1) − 2
+r2
+ϖl
+= 16π(ρ + pt)ϖl.
+(15)
+At a distance far away from the star, where e−(ψ+λ)
+becomes unity, the asymptotic solution of Eq. (15) takes
+the form ϖl(r) → a1r−l−2 + a2rl−1. If spacetime is to
+be flat at large r, then ω → 2J/r3 (or equivalently, ϖ →
+Ω − 2J/r3) for r → ∞, where J is the total angular
+momentum of the star [59, 61].
+Therefore, comparing
+this with the asymptotic behavior of ϖl(r), we find that
+l = 1. As a result, ϖ is a function only of the radial
+coordinate, and Eq. (15) reduces to
+eψ−λ
+r4
+d
+dr
+�
+e−(ψ+λ)r4 dϖ
+dr
+�
+= 16π(ρ + pt)ϖ,
+(16)
+which can be integrated to give
+�
+r4 dϖ
+dr
+�
+R
+= 16π
+� R
+0
+(ρ + pt)r4eλ−ψϖdr.
+(17)
+In view of Eq. (17), we can obtain the angular mo-
+mentum J and hence the moment of inertia I = J/Ω of
+a slowly rotating anisotropic star:
+I = 8π
+3
+� R
+0
+(ρ + pr + σ)eλ−ψr4 �ϖ
+Ω
+�
+dr,
+(18)
+which reduces to the expression given in Ref. [61] for
+isotropic compact stars when σ = 0. For an arbitrary
+choice of the central value ϖ(0), the appropriate bound-
+ary conditions for the differential equation (16) come
+from the requirements of regularity at the center of the
+star and asymptotic flatness at infinity, namely
+dϖ
+dr
+����
+r=0
+= 0,
+lim
+r→∞ ϖ = Ω.
+(19)
+Once the solution for ϖ(r) is found, we can then deter-
+mine the moment of inertia through the integral (18). It
+is remarkable that the above expression for I is referred
+to as the “slowly rotating” approximation because it was
+obtained to lowest order in the angular velocity Ω [61].
+This means that the stellar structure equations are still
+given by the TOV equations (4)-(6).
+D.
+Tidal deformability
+It is well known that the tidal properties of neutron
+stars are measurable in gravitational waves emitted from
+the inspiral of a binary neutron-star coalescence [62, 63].
+In that regard, here we also study the dimensionless tidal
+deformability of individual dark energy stars. To do so,
+we follow the procedure carried out by Hinderer et al. [64]
+(see also Refs. [65–70] for additional results). The basic
+idea is as follows: In a binary system, the deformation
+of a compact star due to the tidal effect created by the
+companion star is characterized by the tidal deformabil-
+ity parameter ¯λ = −Qij/Eij, where Qij is the induced
+quadrupole moment tensor and Eij is the tidal field ten-
+sor [68]. Namely, the latter describes the tidal field from
+the spacetime curvature sourced by the distant compan-
+ion.
+The tidal parameter is related to the tidal Love number
+k2 through the relation2
+¯λ = 2
+3k2R5,
+(20)
+but it is common in the literature to define the dimen-
+sionless tidal deformability Λ = ¯λ/M 5, so in our results
+we will focus on Λ. The calculation of ¯λ requires consider-
+ing linear quadrupolar perturbations (due to the external
+tidal field) to the equilibrium configuration. Thus, the
+spacetime metric is given by gµν = g0
+µν + hµν, where g0
+µν
+describes the equilibrium configuration and hµν is a lin-
+earized metric perturbation. For static and even-parity
+perturbations in the Regge-Wheeler gauge [71], the per-
+turbed metric can be written as [64]
+hµν =
+diag
+�
+−e2ψ(r)H0, e2λ(r)H2, r2K, r2 sin2 θK
+�
+Y2m(θ, φ),
+(21)
+where H0, H2 and K are functions of the radial coordi-
+nate, and Ylm are the spherical harmonics for l = 2.
+Since the perturbed energy-momentum tensor is given
+by δT ν
+µ = diag(−δρ, δpr, δpt, δpt), the linearized field
+2 It should be noted that the tidal deformability parameter is be-
+ing denoted by ¯λ in order not to be confused with the metric
+component λ.
+
+5
+equations imply that:
+�
+�
+�
+�
+�
+H0 = −H2 ≡ H
+from
+δG2
+2 − δG3
+3 = 0,
+K′ = 2Hψ′ + H′
+from
+δG2
+1 = 0,
+δpt =
+H
+8πre−2λ(λ′ + ψ′)Y2m
+from
+δG2
+2 = 8πδpt.
+In addition, from δG0
+0 − δG1
+1 = −8π(δρ + δpt), we can
+obtain the following differential equation [72]
+H′′ + PH′ + QH = 0,
+(22)
+or alternatively,
+ry′ = −y2 + (1 − rP)y − r2Q,
+(23)
+where we have defined
+y ≡ rH′
+H ,
+(24)
+P ≡ 2
+r + e2λ
+�2m
+r2 + 4πr(pr − ρ)
+�
+,
+(25)
+Q ≡ 4πe2λ
+�
+4ρ + 8pr + ρ + pr
+Av2sr
+(1 + v2
+sr)
+�
+− 6e2λ
+r2
+− 4ψ′2,
+(26)
+with A ≡ dpt/dpr and vsr being the radial speed of
+sound.
+By matching the internal solution with the external
+solution of the perturbed variable H at the surface of the
+star r = R, we obtain the tidal Love number [72]
+k2 = 8
+5(1 − 2C)2C5 [2C(yR − 1) − yR + 2]
+×
+�
+2C[4(yR + 1)C4 + (6yR − 4)C3
++ (26 − 22yR)C2 + 3(5yR − 8)C − 3yR + 6
+�
++ 3(1 − 2C)2 [2C(yR − 1) − yR + 2] log(1 − 2C)
+�−1 ,
+(27)
+where C ≡ M/R is the compactness of the star, and
+yR ≡ y(R) is obtained by integrating equation (23) from
+the origin up to the stellar surface.
+III.
+EQUATION OF STATE AND ANISOTROPY
+MODEL
+As it is well known, a possible alternative to the Phan-
+tom and Quintessence fields is the Chaplygin gas, where
+the EoS assumes the form pr = −B/ρ, with B being a
+positive constant (given in m−4 units). In fact, it was ar-
+gued that such gas could provide a solution to unify the
+effects of dark matter in the early times and dark energy
+in late times [4, 11]. Although the literature provides a
+more generalized version for such EoS in the context of
+the Friedmann-Lemaˆıtre-Robertson-Walker Universe [5–
+7, 73–77], here we will use the simplest form plus a linear
+term corresponding to a barotropic fluid, namely
+pr = Aρ − B
+ρ ,
+(28)
+where A is a positive dimensionless constant. Our model
+is characterized by two free parameters A and B. Never-
+theless, we must emphasize here that Li et al. [10] consid-
+ered an equation of state with three degrees of freedom,
+specifically p = Aρ − B/ρα, where α is an extra param-
+eter. They carried out a statistical treatment of astro-
+nomical data in order to constrain the parameter space.
+In the light of the Markov chain Monte Carlo method,
+they found that at 2σ level, α = −0.0156+0.0982+0.2346
+−0.1380−0.2180
+and A = 0.0009+0.0018+0.0030
+−0.0017−0.0030 from CMB+JLA+CC data
+sets.
+In other words, the constants α and A are very
+close to zero and hence the nature of unified dark matter-
+energy model is very similar to the cosmological standard
+ΛCDM model.
+On the other hand, at astrophysics level, compact stars
+obeying the EoS (28) have been investigated by several
+authors, see for example Refs. [38, 41, 43–45]. In this
+work we will adopt values of A and B for which appre-
+ciable changes in the mass-radius diagram can be visu-
+alized in order to compare our theoretical results with
+observational measurements of massive pulsars.
+In order to describe physically realistic compact stars,
+the causality condition must be respected throughout the
+interior region of the star. In other words, the speed of
+sound (defined by vs ≡
+�
+dp/dρ) cannot be greater than
+the speed of light. Thus, in view of Eq. (28), we have
+v2
+sr ≡ dpr
+dρ = A + B
+ρ2 ,
+(29)
+and since the radial pressure vanishes at the surface of
+the star, then B = Aρ2. Thereby, the causality condition
+v2
+sr(R) = 2A < 1 implies that A < 0.5.
+Besides, it is more realistic to consider stellar models
+where there exists a tangential pressure as well as a radial
+one, since anisotropies arise at high densities, i.e. above
+the nuclear saturation density as considered in this work.
+Although the literature offers different functional rela-
+tions to model anisotropic pressures at very high densities
+inside compact stars [49–54], here we adopt the simplest
+model, which was proposed by Horvat and collaborators
+[51]
+σ = α
+�2m
+r
+�
+pr = α
+�
+1 − e−2λ�
+pr,
+(30)
+where α is a dimensionless parameter that controls the
+amount of anisotropy within the stellar fluid. This pa-
+rameter can assume positive or negative values of the
+order of unity, see Refs. [26, 32, 51, 52, 55, 78–82]. No-
+tice that the isotropic solutions are recovered when the
+value of α vanishes. Specifically, the anisotropy ansatz
+(30) has two important characteristics: (i) the fluid be-
+comes isotropic at the center generating regular solutions
+and (ii) the effect of anisotropy vanishes in the hydro-
+static equilibrium equation in the Newtonian limit. Un-
+like this profile, the effect of anisotropy does not van-
+ish in the hydrostatic equilibrium equation in the non-
+relativistic regime for the Bowers-Liang model [49], which
+
+6
+could be an unphysical trait as argued in Ref. [79]. For
+a broader discussion on the different ways of generating
+static spherically symmetric anisotropic fluid solutions,
+we refer the reader to the recent review article [83].
+Since the Eulerian perturbation for the metric poten-
+tial λ can be written as δλ = −4πr(ρ+pr)e2λξ [55], then
+δσ takes the form
+δσ = α
+�
+(1 − e−2λ)δpr − 8πpr(ρ + pr)r2ζ
+�
+,
+(31)
+where it should be noted that the relation between the
+Eulerian and Lagrangian perturbations for radial pres-
+sure is given by ∆pr = δpr + rζp′
+r. The above expression
+will be substituted in Eq. (9) when we discuss later the
+radial pulsations in the stellar interior for at least some
+values of α.
+IV.
+NUMERICAL RESULTS
+A.
+Equilibrium configurations
+So far we do not know exactly whether the millisecond
+pulsars (observed in compact binaries from optical spec-
+troscopic and photometric measurements) are hadronic,
+quark or hybrid stars.
+In fact, it has been theorized
+that cold quark matter might exist at the core of heavy
+neutron stars [84]. Despite the precise measurements of
+masses [85–87] and radii [88–90], such constraints are still
+unable to distinguish the theoretical predictions coming
+from the different models for strange stars and (hybrid)
+neutron stars. This means that the dense matter EoS
+within compact stars still remains poorly understood.
+Furthermore, a realistic compact star possesses high mag-
+netic fields and rotation properties, which significantly
+alter its internal structure. For comparison reasons, it is
+therefore common to use the observational mass-radius
+measurements (in view of the detection of gravitational
+waves and electromagnetic signals) on the mass-radius
+diagrams for any type of EoS even being of different mi-
+croscopic compositions. In that perspective, our theoret-
+ical results will be compared with observational measure-
+ments.
+We begin our discussion of dark energy stars by con-
+sidering the isotropic case (i.e., when σ = 0 in the TOV
+equations). We numerically integrate Eqs. (4)-(6) from
+the center up to the surface of the star through the
+boundary conditions (7). As usual, the radius R is de-
+termined when the pressure vanishes, and the total mass
+M is calculated at the surface. The felt panel of Fig. 1
+exhibits the mass-radius relations of dark energy stars for
+different values of parameters A and B in the EoS (28).
+Remark that we have adopted values of A less than 0.5 in
+order to respect the causality condition. One can observe
+that small values of A (see black curve) do not provide
+compact stars that fit current observational data. How-
+ever, higher values of maximum mass can be obtained for
+larger values of A, see for example red and green curves.
+For a fixed value of A, the maximum mass decreases as
+the parameter B increases.
+We perceive that the sec-
+ondary component resulting from the gravitational-wave
+signal GW190814 [91] can be consistently described as a
+compact star with Chaplygin EoS (28) for A = 0.4 and
+B ∈ [4, 5]µ. Furthermore, the magenta curve fits very
+well with all observational data, but its maximum-mass
+value is above 3M⊙.
+Another interesting feature of these stars is their com-
+pactness, defined by C ≡ M/R. According to the clas-
+sification adopted by Iyer et al. [92], the configurations
+shown in the mass-radius diagram correspond to compact
+stars, see the right plot of Fig. 1. Besides, we can appre-
+ciate that the compactness of dark energy stars is of the
+order of the compactness of hadronic-matter stars, as is
+the case of the SLy EoS [93], despite the fact that the
+maximum mass in the magenta configuration sequence
+can exceed 3M⊙. Nonetheless, as we will see later, the
+introduction of anisotropy can turn such stars into ultra-
+compact objects.
+Of course, this will depend on the
+amount of anisotropy in the stellar interior.
+In order to include anisotropic pressures and investi-
+gate their effects on the internal structure of dark energy
+stars, we will adopt two specific models with the following
+parameters
+⋆ Model I: A = 0.3, B = 6.0µ ,
+⋆ Model II: A = 0.4, B = 5.2µ ,
+which are models favored by observational measurements
+according to the left panel of Fig. 1. Moreover, model II
+precisely corresponds to the first model considered by
+Panotopoulos et al. [38].
+Similar to the isotropic case, we numerically solve the
+hydrostatic background equations (4)-(6) with boundary
+conditions (7), but taking into account the anisotropy
+profile (30). For instance, for the model I and a central
+density ρc = 2.0×1018 kg/m3, Fig. 2 illustrates the mass
+density, pressure and squared speed of sound as functions
+of the radial coordinate for different values of the free pa-
+rameter α. We can see that the internal structure of a
+dark energy star is affected by the presence of anisotropy.
+In effect, the radius of the star increases (decreases) for
+more positive (negative) values of α. In addition, we re-
+mark that the speed of sound, both radial and tangential,
+respect the causality condition. This has also been veri-
+fied for other values of central density considered in the
+construction of Fig. 1.
+Varying the central density, we obtain the mass-radius
+diagrams and mass-central density relations for models I
+and II, as shown in Fig. 3.
+We observe that the sub-
+stantial changes introduced by anisotropy in dark en-
+ergy stars occur in the high-mass branch (close to the
+maximum-mass point), while the effects are irrelevant at
+low central densities. The maximum-mass values increase
+as the parameter α increases (see also the data in Table
+I). Note that model I without anisotropic pressures is
+not capable of generating maximum masses above 2M⊙.
+
+7
+SLy
+A = 0.2, B = 6μ
+A = 0.3, B = 3μ
+A = 0.3, B = 4μ
+A = 0.3, B = 5μ
+A = 0.3, B = 6μ
+A = 0.4, B = 3μ
+A = 0.4, B = 4μ
+A = 0.4, B = 5μ
+A = 0.4, B = 6μ
+A = 0.48, B = 3μ
+4
+6
+8
+10
+12
+14
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+R [km]
+M [M⊙]
+C > 1/3
+(ultra-compact objects)
+1/6 < C < 1/3
+(compact objects)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+M [M⊙]
+C
+FIG. 1. Left panel: Mass-radius diagrams for dark energy stars with Chaplygin-like EoS (28) and isotropic pressure (σ = 0)
+for several values of the positive parameters A and B.
+Here the constant B is given in µ = 10−20 m−4 units.
+The gray
+horizontal stripe at 2.0M⊙ stands for the two massive NS pulsars J1614-2230 [85] and J0348+0432 [86]. Yellow and blue regions
+represent the observational measurements of the masses of the highly massive NS pulsars J0740+6620 [87] and J2215+5135
+[94], respectively. The filled pink band stands for the lower mass of the compact object detected by the GW190814 event
+[91], and the cyan area is the mass-radius constraint from the GW170817 event. Moreover, the NICER measurements for PSR
+J0030+0451 are displayed by black dots with their respective error bars [95, 96]. Right panel: Variation of the compactness
+with total gravitational mass, where the gray and orange stripes represent compact and ultra-compact objects, respectively,
+according to the classification given in Ref. [92]. For comparison reasons, we have included the results corresponding to the
+SLy EoS [93] by blue curves in both plots.
+Nevertheless, the inclusion of anisotropies (see the blue
+curve for α = 0.4) allows a significant increase in the
+maximum mass and hence a more favorable description
+of the compact objects observed in nature. On the other
+hand, model II with anisotropies (see orange curves) fits
+better with the observational measurements. In particu-
+lar, in view of the lower mass of the compact object from
+the coalescence GW190814 [91], two curves are partic-
+ularly outstanding. In other words, such object can be
+well described as an anisotropic dark energy star when
+α = 0.2 and α = 0.4. Moreover, model II with negative
+anisotropies (such as α = −0.4) favors the description of
+the massive pulsar J2215+5135 [94].
+The left panel of Fig. 4 describes the behavior of
+compactness as a function of central density.
+Positive
+anisotropies lead to an increase in compactness, mainly
+in the high-central-density branch. Remarkably, for suf-
+ficiently large values of α (see purple curve), it is possible
+to obtain anisotropic dark energy stars as ultra-compact
+objects.
+The gravitational redshift, conventionally defined as
+the fractional change between observed and emitted
+wavelengths compared to emitted wavelength, in the case
+of a Schwarzschild star is given by [61]
+zsur = eλ(R) − 1 =
+�
+1 − 2M
+R
+�−1/2
+− 1.
+(32)
+In the right plot of Fig. 4, the surface gravitational red-
+TABLE I.
+Maximum-mass configurations with Chaplygin-
+like EoS (28) for model I and II. The energy density values
+correspond to the critical central density where the function
+M(ρc) is a maximum on the right plot of Fig. 3.
+Model
+α
+ρc [1018 kg/m3]
+R [km]
+M [M⊙]
+−0.4
+2.424
+9.812
+1.786
+−0.2
+2.364
+9.902
+1.852
+I
+0
+2.295
+9.994
+1.919
+0.2
+2.219
+10.086
+1.988
+0.4
+2.135
+10.180
+2.059
+−0.4
+1.777
+11.630
+2.320
+−0.2
+1.721
+11.738
+2.402
+II
+0
+1.661
+11.845
+2.486
+0.2
+1.594
+11.955
+2.570
+0.4
+1.523
+12.065
+2.565
+shift is plotted as a function of the total mass for both
+models I and II. This plot indicates that the gravita-
+tional redshift of light emitted at the surface of a dark
+energy star is substantially affected by the anisotropy in
+the high-mass region, while the changes are negligible for
+sufficiently low masses. For a fixed value of central den-
+sity, Table II shows that positive (negative) anisotropy
+increases (decreases) the value of the redshift.
+
+8
+TABLE II.
+Radius, mass, redshift, fundamental mode frequency (f0 = ν0/2π), moment of inertia and dimensionless tidal
+deformability of dark energy stars with central energy density ρc = 1.5 × 1018 kg/m3 as predicted by models I and II for
+several values of the anisotropy parameter α. Remarkably, with the exception of the fundamental mode frequency and tidal
+deformability, these properties undergo a significant increase as α increases.
+Model
+α
+R [km]
+M [M⊙]
+zsur
+f0 [kHz]
+I [1038 kg · m2]
+Λ
+−0.4
+10.062
+1.713
+0.418
+2.414
+1.695
+13.278
+−0.2
+10.163
+1.781
+0.440
+2.312
+1.820
+10.709
+I
+0
+10.263
+1.852
+0.463
+2.201
+1.957
+8.598
+0.2
+10.361
+1.926
+0.489
+2.081
+2.105
+6.868
+0.4
+10.456
+2.003
+0.518
+1.950
+2.265
+5.454
+−0.4
+11.767
+2.310
+0.543
+1.131
+3.298
+4.889
+−0.2
+11.859
+2.395
+0.574
+0.998
+3.531
+3.823
+II
+0
+11.944
+2.481
+0.609
+0.840
+3.778
+2.978
+0.2
+12.019
+2.569
+0.647
+0.637
+4.037
+2.309
+0.4
+12.083
+2.656
+0.688
+0.315
+4.303
+1.782
+α = -0.6
+α = -0.3
+α = 0
+α = 0.3
+α = 0.6
+0
+2
+4
+6
+8
+10
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+r [km]
+ρ [kg/m3]
+Solid lines: pr
+Dashed lines: pt
+α = -0.6
+α = -0.3
+α = 0
+α = 0.3
+α = 0.6
+0
+2
+4
+6
+8
+10
+0
+1
+2
+3
+4
+5
+r [km]
+Pressure [1034 Pa]
+Solid lines: vsr
+2
+Dashed lines: vst
+2
+α = -0.6
+α = -0.3
+α = 0
+α = 0.3
+α = 0.6
+0
+2
+4
+6
+8
+10
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+r [km]
+(Speed of sound)2/c2
+FIG. 2. Radial behavior of the mass density (left panel), pressures (middle panel) and squared speed of sound (right panel)
+inside an anisotropic dark energy star with central density ρc = 2.0 × 1018 kg/m3 and several values of the parameter α. All
+plots correspond to model I and the black curves represent the isotropic solutions. Note that both the radial and tangential
+speed of sound obey the causality condition. Furthermore, one can observe that the increase in α leads to larger radii, and the
+anisotropy is more pronounced in the intermediate regions.
+B.
+Oscillation spectrum
+A necessary condition (the well-known M(ρc) method)
+for stellar stability is that stable stars must lie in the re-
+gion where dM/dρc > 0.
+According to the right plot
+of Fig. 3, the full blue and orange circles on each curve
+indicate the onset of instability for each family of equi-
+librium solutions. However, a sufficient condition is to
+calculate the frequencies of the radial vibration modes
+for each central density [61]. Here we will analyze if both
+methods are compatible in the case of dark energy stars
+including anisotropic pressure.
+Once the equilibrium equations (4)-(6) are integrated
+from the center to the surface of the star, we then pro-
+ceed to solve the radial pulsation equations (8) and (9)
+with the corresponding boundary conditions (10) and
+(11) using the shooting method. Namely, we integrate
+from the origin (where we consider the normalized eigen-
+functions ζ(0) = 1) up to the stellar surface for a set
+of trial values ν2 satisfying the condition (10). In this
+way, the appropriate eigenfrequencies correspond to the
+values for which the boundary condition (11) is fulfilled.
+For instance, for a central density ρc = 1.5×1018 kg/m3,
+α = 0.4 and parameters given by model I, Fig. 5 dis-
+plays the radial behavior of the perturbation variables
+for the first five squared eigenfrequencies ν2
+n, where n
+indicates the number of nodes inside the star. This fre-
+quency spectrum forms an infinite discrete sequence, i.e.
+ν2
+0 < ν2
+1 < ν2
+2 < · · · , where the eigenvalue corresponding
+to n = 0 is the lowest one (or equivalently, the longest
+period of all the allowed vibration modes) and it is known
+as the fundamental mode.
+Such mode has no nodes,
+
+9
+Model I
+Model II
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+7
+8
+9
+10
+11
+12
+0.5
+1.0
+1.5
+2.0
+2.5
+R [km]
+M [M⊙]
+Model I
+Model II
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+17.8
+18.0
+18.2
+18.4
+18.6
+0.5
+1.0
+1.5
+2.0
+2.5
+Log ρc [kg/m3]
+M [M⊙]
+FIG. 3. Mass-radius diagram (left panel) and mass-central density relation (right panel) for anisotropic dark energy stars as
+predicted by model I (blue curves) and II (orange curves) with anisotropy profile (30) for several values of α. The colored
+bands in the left plot represent the same as in Fig. 1. Moreover, the full blue and orange circles on the right plot indicate the
+maximum-mass points for model I and II, respectively. Note that the maximum-mass values for model II correspond to lower
+central densities than those for model I, however, model II allows larger masses (see also Table I). The critical central density
+corresponding to the maximum point on the M(ρc) curve is modified by the presence of anisotropy for both models.
+Model I
+Model II
+C > 1/3
+(ultra-compact objects)
+1/6 < C < 1/3
+(compact objects)
+α = 0.7
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+17.8
+18.0
+18.2
+18.4
+18.6
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Log ρc [kg/m3]
+C
+Model I
+Model II
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0.5
+1.0
+1.5
+2.0
+2.5
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+M [M⊙]
+zsur
+FIG. 4.
+Left panel: Variation of the compactness with central density for several anisotropic dark energy star sequences.
+The gray and light-green stripes represent compact and ultra-compact objects, respectively, according to the classification
+established by Iyer et al. [92]. Positive anisotropy results in increased compactness for sufficiently high central densities, while
+the opposite occurs for negative anisotropy. Note also that dark energy stars would correspond to ultra-compact objects if
+α > 0.4 for model II, see for instance the purple curve for α = 0.7. Right panel: Surface gravitational redshift as a function
+of the total mass. In the high-redshift region it can be observed that positive (negative) anisotropy increases (decreases) the
+value of zsur. Meanwhile, the effect of anisotropy is irrelevant for sufficiently low redshifts.
+whereas the first overtone (n = 1) has one node, the
+second overtone (n = 2) has two, and so on. Stable stars
+are described by their oscillatory behavior so that ν2
+n > 0
+(i.e., νn is purely real). On the other hand, if any of these
+is negative for a particular star, the frequency is purely
+imaginary and hence the star is unstable.
+Since each higher-order mode has a squared eigenfre-
+quency that is larger than in the case of the preceding
+mode, it is enough to calculate the frequency of the fun-
+damental pulsation mode for the equilibrium sequences
+presented in Fig. 3.
+With this in mind, in Fig. 6 we
+plot the squared frequency of the fundamental oscilla-
+tion mode as a function of the central density (left panel)
+and gravitational mass (right panel). According to the
+left plot, the squared frequency of the fundamental mode
+is exactly zero at the critical-central-density value corre-
+sponding to the maximum-mass configuration as shown
+in the right plot of Fig. 3, see the full blue and orange cir-
+
+10
+cles for both models. Furthermore, according to the right
+plot of Fig. 6, the maximum-mass values (that is, when
+dM/dρc = 0) can be used as turning points from stability
+to dynamical instability. Therefore, we can conclude that
+the usual criterion to guarantee stability dM/dρc > 0 is
+still valid for the case of anisotropic dark energy stars.
+In other words, the conventional M(ρc) method is com-
+patible with the calculation of the eigenfrequencies of the
+normal vibration modes.
+If the anisotropic dark energy star has a central den-
+sity higher than one corresponding to the maximum-mass
+configuration (indicated by full blue and orange circles
+in Figs. 3 and 6), the star will become unstable against
+radial perturbations and collapse to form a black hole.
+For further details on the dissipative gravitational col-
+lapse of compact stellar objects we also refer the reader
+to Refs. [55, 97–99].
+Nonetheless, we must point out
+that there are EoS models that allow a compact star to
+migrate to another branch of stable solutions instead of
+forming a black hole when it is subjected to a perturba-
+tion. As a matter of fact, the first-order phase transition
+between nuclear and quark matter can generate multiple
+stable branches in the mass-radius diagram for hybrid
+stars [100].
+C.
+Moment of inertia
+To calculate the moment of inertia of anisotropic dark
+energy stars, we first need to solve the differential equa-
+tion for the rotational drag (16) with boundary condi-
+tions (19). In particular, for model I and central density
+ρc = 1.5 × 1018 kg/m3, figure 7 illustrates the angular
+velocity everywhere for several values of α. As can be
+observed in the right plot, the dragging angular velocity
+outside the star has the behavior ω(r) ∼ r−3, so that at
+infinity (where spacetime is flat) the distant local inertial
+frames do not rotate around the star, namely, ω(r) → 0
+for r → ∞. Moreover, anisotropy significantly affects the
+angular velocity of the local inertial frames in the inte-
+rior region of the star. More specifically, the dragging
+angular velocity increases (decreases) for positive (nega-
+tive) values of the anisotropy parameter α. We can then
+determine the moment of inertia using the integral given
+in Eq. (18). For the above central density, we present the
+moment of inertia of some dark energy configurations for
+both models in Table II, where it can be noticed that I
+increases as the value of α increases.
+We can now calculate the moment of inertia for a whole
+sequence of dark energy stars by varying the central den-
+sity ρc. The left panel of Fig. 8 displays the moment of
+inertia as a function of the gravitational mass for both
+models. Remarkably, model II provides larger values for
+the moment of inertia than model I. Indeed, the maxi-
+mum value Imax depends quite sensitively on the free pa-
+rameters A and B in the EoS (28). In addition, the main
+effect of anisotropy on the moment of inertia for slow ro-
+tation occurs in the high-mass region, while its influence
+is irrelevant for sufficiently low masses. In order to bet-
+ter quantify the changes in the maximum values of the
+moment of inertia induced by the anisotropic pressure,
+we can define the following relative difference
+∆I = Imax,ani − Imax,iso
+Imax,iso
+,
+(33)
+where Imax,iso and Imax,ani are the maximum values of the
+moment of inertia for isotropic and anisotropic configura-
+tions, respectively. In the right plot of Fig. 8 we present
+the dependence ∆I against the anisotropy parameter α.
+The impact of anisotropy is getting stronger as |α| grows,
+reaching variations (with respect to the isotropic case) of
+up to ∼ 20% for α = 0.5. We can also note that such
+relative variations are almost independent of the model
+adopted.
+D.
+Tidal properties
+We will now investigate how the anisotropy parameter
+α affects the tidal properties of dark energy stars. Given
+a specific value of α, this requires solving the differential
+equation (23) for a range of central densities. The left
+panel of Fig. 9 is the result of calculating the tidal Love
+number (27) for a sequence of stellar configurations by
+considering different values of α, where the isotropic case
+corresponds to α = 0. Similar to the trends in strange
+quark stars, as reported in Ref. [70], the Love number of
+dark energy stars grows until it reaches a maximum value
+and then decreases as compactness increases. Note also
+that the maximum value of k2 is sensitive to the value
+of α, indicating that the Love number decreases as the
+parameter α increases for both models. Although model
+II provides larger maximum masses (as well as redshift
+and moment of inertia) than model I, we see that the
+behavior is different for the maximum values in the tidal
+Love number.
+Ultimately, in the right plot of Fig. 9, the dimensionless
+tidal deformability Λ = ¯λ/M 5 is plotted as a function of
+mass, where it can be observed that smaller masses yield
+higher deformabilities.
+In each model, the presence of
+anisotropy has a negligible effect on Λ for small masses,
+while slightly more significant changes take place only in
+the high-mass region.
+V.
+CONCLUSIONS AND OUTLOOK
+In this work, we have focused on the equilibrium struc-
+ture of dark energy stars by using a Chaplygin-like equa-
+tion of state under the presence of both isotropic and
+anisotropic pressures within the context of standard GR.
+Our goal was to construct stable compact stars whose
+characteristics could be compared with the observational
+data on the mass-radius diagram.
+In this perspective,
+the global properties of a compact star such as radius,
+mass, redshift, moment of inertia, oscillation spectrum
+
+11
+n=0 mode
+n=1 mode
+n=2 mode
+n=3 mode
+n=4 mode
+n=5 mode
+0
+2
+4
+6
+8
+10
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+r [km]
+ζn (r)
+n=0 mode
+n=1 mode
+n=2 mode
+n=3 mode
+n=4 mode
+n=5 mode
+0
+2
+4
+6
+8
+10
+-1.5
+-1.0
+-0.5
+0.0
+r [km]
+Δpr,n (r) [1035 Pa]
+FIG. 5. Numerical solution of the radial pulsation equations (8) and (9) in the case of an anisotropic dark energy star with
+central density ρc = 1.5 × 1018 kg/m3, α = 0.4 and EoS parameters given by model I. The radius, mass and the fundamental
+mode frequency for such configuration are found in Table II. The lines with different colors and styles indicate different overtones
+so that the solution corresponding to the nth vibration mode contains n nodes in the internal structure of the star. Note that
+the eigenfunctions ζn(r) have been normalized assuming ζ = 1 at r = 0, and the Lagrangian perturbation of the radial pressure
+∆pr,n(r) obeys the boundary condition (11) at the stellar surface. Since f0 is real, this configuration corresponds to a stable
+anisotropic dark energy star.
+Model I
+Model II
+17.8
+18.0
+18.2
+18.4
+18.6
+0.0
+0.5
+1.0
+1.5
+Log ρc [kg/m3]
+ν0
+2 [109 s-2]
+-0.05
+0.
+0.05
+18.12
+18.22
+18.32
+18.42
+Model I
+Model II
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+1.0
+1.5
+2.0
+2.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+M [M⊙]
+ν0
+2 [109 s-2]
+FIG. 6. Left panel: Squared frequency of the fundamental pulsation mode as a function of central mass density for anisotropic
+dark energy stars predicted by Einstein gravity. The full blue and orange circles indicate the central density values where
+ν2
+0 = 0, whose values precisely correspond to the maximum-mass points on the M(ρc) curves on the right plot of Fig. 3. Right
+plot: Squared frequency of the fundamental mode versus gravitational mass, where it can be observed that the maximum-mass
+values determine the boundary between stable and unstable stars.
+and tidal deformability have been calculated. To describe
+the anisotropic pressure within the dark energy fluid we
+have adopted the anisotropy profile proposed by Horvat
+et al. [51], where a free parameter α measures the degree
+of anisotropy.
+We have discussed the possibility of observing sta-
+ble dark energy stars made of a negative pressure fluid
+“−B/ρ” plus a barotropic component “Aρ”. By way of
+comparison, the EoS parameters A and B have been cho-
+sen in such a way that they agree sufficiently with the
+observational data, e.g. the mass-radius constraint from
+the GW170817 event. For isotropic configurations, we
+have shown that various sets of values {A, B} can be
+chosen since they obey the causality condition and con-
+sistently describe compact stars observed in the Universe.
+Furthermore, we saw that the secondary component re-
+sulting from the gravitational-wave signal GW190814 [91]
+can be described as a dark energy star using A = 0.4 and
+
+12
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0
+10
+20
+30
+40
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+r [km]
+ϖ/Ω
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0
+10
+20
+30
+40
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+r [km]
+ω/Ω
+FIG. 7. Left panel: Numerical solution of the differential equation (16) for a dark energy star described by model I and central
+density ρc = 1.5 × 1018 kg/m3 in the presence of anisotropy for several values of the free parameter α. The solid and dashed
+lines represent the interior and exterior solutions, respectively. Right panel: Ratio of frame-dragging angular velocity to the
+angular velocity of the star, namely ω(r)/Ω = 1 − ϖ(r)/Ω. It can be observed that the outer solution behaves asymptotically
+at large distances from the surface of the star (this is, ω → 0 for r → ∞). Furthermore, appreciable changes in the angular
+velocity due to anisotropy can be noticeable, mainly in the interior region of the star.
+Model I
+Model II
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0.5
+1.0
+1.5
+2.0
+2.5
+0
+1
+2
+3
+4
+M [M⊙]
+I [1038 kg.m2]
+Model I
+Model II
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-15
+-10
+-5
+0
+5
+10
+15
+20
+α
+ΔI [%]
+FIG. 8. Left panel: Moment of inertia versus mass for anisotropic dark energy stars, where a higher mass results in larger
+moment on inertia for both models. It is observed that the substantial impact of anisotropy on the moment of inertia occurs
+predominantly in the high-mass branch. Right panel: Relative deviation (33) as a function of the anisotropy parameter. The
+maximum value of the moment of inertia can undergo variations with respect to its isotropic counterpart of up to ∼ 20% for
+α = 0.5.
+B ∈ [4, 5]µ.
+Based on these results, we have established two mod-
+els with different values A and B in order to explore
+the effects of anisotropy in the interior region of a dark
+energy star. In particular, the maximum-mass values in-
+crease as the parameter α increases.
+We noticed that
+model I without anisotropic pressures is not capable of
+generating maximum masses above 2M⊙. However, the
+inclusion of anisotropies (α = 0.4) allows a significant in-
+crease in the maximum mass and thus a more favorable
+description of the compact objects observed in nature.
+On the other hand, model II with anisotropies fits bet-
+ter with the observational measurements, although such
+a model can lead to the formation of ultra-compact ob-
+jects for sufficiently large values of α. We also calculated
+the surface gravitational redshift for such stars, and our
+results indicated that zsur is substantially affected by the
+anisotropy in the high-mass branch, while the changes
+are irrelevant for sufficiently low masses.
+A star exists in the Universe only if it is dynamically
+stable, so our second task was to investigate whether the
+dark energy stars are stable or unstable with respect to an
+adiabatic radial perturbation. Our results showed that
+the standard criterion for radial stability dM/dρc > 0
+still holds for dark energy stars since the squared fre-
+quency of the fundamental pulsation mode (ν2
+0) van-
+
+13
+Model I
+Model II
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.010
+0.015
+0.020
+C
+k2
+Model I
+Model II
+α = -0.4
+α = -0.2
+α = 0
+α = 0.2
+α = 0.4
+0.5
+1.0
+1.5
+2.0
+2.5
+1
+5
+10
+50
+100
+500
+1000
+M [M⊙]
+Λ
+FIG. 9. Left panel: Tidal Love number plotted as a function of the compactness C ≡ M/R. Right panel: Dimensionless tidal
+deformability versus gravitational mass predicted by each model, where larger masses yield smaller deformabilities. Note also
+that the Love number is substantially modified by the anisotropy parameter α for both models, while its greatest effect on tidal
+deformability Λ occurs only in the high-mass region.
+ishes at the critical central density corresponding to the
+maximum-mass configuration. This has been examined
+in detail for both isotropic (α = 0) and anisotropic
+(α ̸= 0) stellar configurations.
+In the slowly rotating approximation, where only first-
+order terms in the angular velocity are kept, we have also
+determined the moment of inertia of anisotropic dark en-
+ergy stars. For this purpose, we first had to calculate the
+frame-dragging angular velocity for each central density.
+The presence of anisotropic pressure results in a substan-
+tial increase (decrease) of the angular velocity ω for more
+positive (negative) values of α. We found that the signif-
+icant impact of the anisotropy on the moment of inertia
+occurs mainly in the high-mass branch for both models.
+Furthermore, the maximum value of the moment of in-
+ertia can undergo variations of up to ∼ 20% for α = 0.5
+as compared with the isotropic case.
+We have analyzed the effect of anisotropic pressure on
+the tidal properties of such stars. In particular, our out-
+comes revealed that the tidal Love number is sensitive to
+moderate variations of the parameter α, indicating that
+the maximum value of k2 can increase as α decreases.
+In addition, the greatest effect of anisotropy on the di-
+mensionless tidal deformability takes place only in the
+high-mass region.
+Based on the foregoing results, the
+present work thereby serves to develop a comprehensive
+perspective on the relativistic structure of dark energy
+stars in the presence of anisotropy.
+Summarizing, we have explored the possible existence
+of stable dark energy stars whose masses and radii are
+not in disagreement with the current observational data.
+The Chaplygin-like EoS predicts maximum-mass values
+consistent with observational measurements of highly
+massive pulsars. Future research includes the adoption
+of widespread versions of Chaplygin gas that best fit
+key cosmological parameters. In future studies we will
+thereby take further steps in that direction, focusing on
+the different types of generalized Chaplygin gas models
+as discussed in Ref. [11]. In addition, as carried out in the
+case of boson stars [101], it would be interesting to em-
+ploy a Fisher matrix analysis in order to distinguish dark
+energy stars from black holes and neutron stars from tidal
+interactions in inspiraling binary systems. It is also worth
+mentioning that Romano [102] has recently discussed the
+effects of dark energy on the propagation of gravitational
+waves. In that regard, we expect that future electromag-
+netic observations of compact binaries and gravitational-
+wave astronomy will provide a better understanding of
+compact stars in the presence of dark energy, and even
+help us answer the most basic question: How did dark
+energy form in the Universe? Anyway, our results sug-
+gest that dark energy stars deserve further investigation
+by taking into account the cosmological aspects as well
+as the gravitational-wave signals from binary mergers.
+ACKNOWLEDGMENTS
+The author would like to acknowledge the anonymous
+reviewer for useful constructive feedback and valuable
+suggestions. The author would also like to thank Maria
+F. A. da Silva for giving helpful comments. This research
+work was financially supported by the PCI program of
+the Brazilian agency “Conselho Nacional de Desenvolvi-
+mento Cient´ıfico e Tecnol´ogico”–CNPq.
+
+14
+[1] N. Aghanim et al., A&A 641, A6 (2020).
+[2] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
+[3] T. Padmanabhan, Physics Reports 380, 235 (2003).
+[4] A. Kamenshchik, U. Moschella,
+and V. Pasquier,
+Physics Letters B 511, 265 (2001).
+[5] M. C. Bento, O. Bertolami, and A. A. Sen, Phys. Rev.
+D 66, 043507 (2002).
+[6] R. R. R. Reis, I. Waga, M. O. Calv˜ao, and S. E. Jor´as,
+Phys. Rev. D 68, 061302 (2003).
+[7] L. Xu, J. Lu, and Y. Wang, Eur. Phys. J. C 72, 1883
+(2012).
+[8] N. Bili´c, G. Tupper, and R. Viollier, Physics Letters B
+535, 17 (2002).
+[9] M. Makler, S. Q. de Oliveira,
+and I. Waga, Physics
+Letters B 555, 1 (2003).
+[10] H. Li, W. Yang, and L. Gai, A&A 623, A28 (2019).
+[11] J. Zheng et al., Eur. Phys. J. C 82, 582 (2022).
+[12] Y. Ignatov and M. Pieroni, arXiv:2110.10085 [astro-
+ph.CO] (2021).
+[13] S. D. Odintsov, D. S.-C. G´omez,
+and G. S. Sharov,
+Phys. Rev. D 101, 044010 (2020).
+[14] E. J. Copeland, M. Sami,
+and S. Tsujikawa, Int. J.
+Mod. Phys. D 15, 1753 (2006).
+[15] K. Koyama, Rep. Prog. Phys. 79, 046902 (2016).
+[16] B. Boisseau, G. Esposito-Far`ese, D. Polarski, and A. A.
+Starobinsky, Phys. Rev. Lett. 85, 2236 (2000).
+[17] G. Esposito-Far`ese and D. Polarski, Phys. Rev. D 63,
+063504 (2001).
+[18] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451
+(2010).
+[19] A. De Felice and S. Tsujikawa, Living Rev. Relativ. 13,
+3 (2010).
+[20] A. Starobinsky, Physics Letters B 91, 99 (1980).
+[21] S. M. Carroll et al., Phys. Rev. D 70, 043528 (2004).
+[22] T. Chiba, Physics Letters B 575, 1 (2003).
+[23] S. Nojiri and S. D. Odintsov, Physics Reports 505, 59
+(2011).
+[24] T. Clifton et al., Physics Reports 513, 1 (2012).
+[25] S. Nojiri, S. Odintsov, and V. Oikonomou, Physics Re-
+ports 692, 1 (2017).
+[26] V. Folomeev, Phys. Rev. D 97, 124009 (2018).
+[27] G. J. Olmo, D. Rubiera-Garcia, and A. Wojnar, Physics
+Reports 876, 1 (2020).
+[28] A. Astashenok et al., Physics Letters B 811, 135910
+(2020).
+[29] A. Astashenok et al., Physics Letters B 816, 136222
+(2021).
+[30] K. Numajiri, T. Katsuragawa,
+and S. Nojiri, Physics
+Letters B 826, 136929 (2022).
+[31] K. Nobleson, A. Ali, and S. Banik, Eur. Phys. J. C 82,
+32 (2022).
+[32] J. M. Z. Pretel and S. B. Duarte, Class. Quantum Grav.
+39, 155003 (2022).
+[33] J. M. Z. Pretel et al., JCAP 2022, 058 (2022).
+[34] J. A. Frieman, M. S. Turner,
+and D. Huterer, Annu.
+Rev. Astron. Astrophys. 46, 385 (2008).
+[35] S. S. Yazadjiev, Phys. Rev. D 83, 127501 (2011).
+[36] M. F. A. R. Sakti and A. Sulaksono, Phys. Rev. D 103,
+084042 (2021).
+[37] S. Smerechynskyi, M. Tsizh, and B. Novosyadlyj, JCAP
+2021, 045 (2021).
+[38] G. Panotopoulos, ´Angel Rinc´on, and I. Lopes, Physics
+of the Dark Universe 34, 100885 (2021).
+[39] P. Bhar, Physics of the Dark Universe 34, 100879
+(2021).
+[40] R. Chan, M. F. A. da Silva,
+and J. F. V. da Rocha,
+Gen. Relativ. Gravit. 41, 1835 (2009).
+[41] F. Rahaman, S. Ray, A. K. Jafry, and K. Chakraborty,
+Phys. Rev. D 82, 104055 (2010).
+[42] C. R. Ghezzi, Astrophys. Space Sci. 333, 437 (2011).
+[43] P. Bhar, M. Govender, and R. Sharma, Pramana 90, 5
+(2018).
+[44] F. Tello-Ortiz et al., Eur. Phys. J. C 80, 371 (2020).
+[45] J. Estevez-Delgado et al., Mod. Phys. Lett. A 36,
+2150213 (2021).
+[46] L. S. M. Veneroni, A. Braz, and M. F. A. da Silva, Int.
+J. Mod. Phys. D 30, 2150039 (2021).
+[47] T. Grammenos et al., Advances in High Energy Physics
+2021, 6966689 (2021).
+[48] Z. Haghani and T. Harko, Phys. Rev. D 105, 064059
+(2022).
+[49] R. L. Bowers and E. P. T. Liang, Astrophys. J. 188, 657
+(1974).
+[50] M. Cosenza, L. Herrera, M. Esculpi, and L. Witten, J.
+Math. Phys. 22, 118 (1981).
+[51] D. Horvat, S. Iliji´c, and A. Marunovi´c, Class. Quantum
+Grav. 28, 025009 (2010).
+[52] D. D. Doneva and S. S. Yazadjiev, Phys. Rev. D 85,
+124023 (2012).
+[53] L. Herrera and W. Barreto, Phys. Rev. D 88, 084022
+(2013).
+[54] G. Raposo et al., Phys. Rev. D 99, 104072 (2019).
+[55] J. M. Z. Pretel, Eur. Phys. J. C 80, 726 (2020).
+[56] R. Rizaldy,
+A. R. Alfarasyi,
+A. Sulaksono,
+and
+T. Sumaryada, Phys. Rev. C 100, 055804 (2019).
+[57] E. A. Becerra-Vergara, S. Mojica, F. D. Lora-Clavijo,
+and A. Cruz-Osorio, Phys. Rev. D 100, 103006 (2019).
+[58] M. D. Danarianto and A. Sulaksono, Phys. Rev. D 100,
+064042 (2019).
+[59] J. B. Hartle, Astrophys. J. 150, 1005 (1967).
+[60] J. B. Hartle, Astrophys. Space Sci. 24, 385 (1973).
+[61] N. K. Glendenning, Compact Stars: Nuclear Physics,
+Particle Physics, and General Relativity, 2nd ed. (As-
+tron. Astrophys. Library, Springer, New York, 2000).
+[62] E. R. Most, L. R. Weih, L. Rezzolla, and J. Schaffner-
+Bielich, Phys. Rev. Lett. 120, 261103 (2018).
+[63] K. Chatziioannou, Gen. Relativ. Gravit. 52, 109 (2020).
+[64] T. Hinderer, Astrophys. J. 677, 1216 (2008).
+[65] T. Damour and A. Nagar, Phys. Rev. D 80, 084035
+(2009).
+[66] T. Binnington and E. Poisson, Phys. Rev. D 80, 084018
+(2009).
+[67] S. Postnikov, M. Prakash,
+and J. M. Lattimer, Phys.
+Rev. D 82, 024016 (2010).
+[68] A. G. Chaves and T. Hinderer, J. Phys. G: Nucl. Part.
+Phys. 46, 123002 (2019).
+[69] T. Dietrich, T. Hinderer, and A. Samajdar, Gen. Rel-
+ativ. Gravit. 53, 27 (2021).
+[70] M. Kumari and A. Kumar, Eur. Phys. J. C 81, 791
+(2021).
+[71] T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063
+(1957).
+
+15
+[72] B. Biswas and S. Bose, Phys. Rev. D 99, 104002 (2019).
+[73] J. V. Cunha, J. S. Alcaniz,
+and J. A. S. Lima, Phys.
+Rev. D 69, 083501 (2004).
+[74] V. Gorini et al., JCAP 2008, 016 (2008).
+[75] O. F. Piattella, JCAP 2010, 012 (2010).
+[76] S. F. Salahedin et al., J. Astrophys. Astron. 43, 14
+(2022).
+[77] R.
+von
+Marttens,
+D.
+Barbosa,
+and
+J.
+Alcaniz,
+arXiv:2208.06302 [astro-ph.CO] (2022).
+[78] H. O. Silva et al., Class. Quantum Grav. 32, 145008
+(2015).
+[79] K. Yagi and N. Yunes, Phys. Rev. D 91, 123008 (2015).
+[80] A. Rahmansyah et al., Eur. Phys. J. C 80, 769 (2020).
+[81] A. Rahmansyah and A. Sulaksono, Phys. Rev. C 104,
+065805 (2021).
+[82] J. M. Z. Pretel, Mod. Phys. Lett. A 37, 2250188 (2022).
+[83] J. Kumar and P. Bharti, New Astronomy Reviews 95,
+101662 (2022).
+[84] E. Annala et al., Nature Phys. 16, 907 (2020).
+[85] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and
+J. Hessels, Nature 467, 1081 (2010).
+[86] J. Antoniadis et al., Science 340, 6131 (2013).
+[87] H. T. Cromartie et al., Nature Astronomy 4, 72 (2019).
+[88] M. C. Miller et al., Astrophys. J. Lett. 887, L24 (2019).
+[89] T. E. Riley et al., Astrophys. J. Lett. 887, L21 (2019).
+[90] G. Raaijmakers et al., Astrophys. J. Lett. 887, L22
+(2019).
+[91] R. Abbott et al., Astrophys. J. Lett. 896, L44 (2020).
+[92] B. R. Iyer, C. V. Vishveshwara, and S. V. Dhurandhar,
+Class. Quantum Grav. 2, 219 (1985).
+[93] F. Douchin and P. Haensel, A&A 380, 151 (2001).
+[94] M. Linares, T. Shahbaz, and J. Casares, Astrophys. J.
+859, 54 (2018).
+[95] M. C. Miller et al., Astrophys. J. Lett. 887, L24 (2019).
+[96] T. E. Riley et al., Astrophys. J. Lett. 887, L21 (2019).
+[97] J. M. Z. Pretel and M. F. A. da Silva, MNRAS 495,
+5027 (2020).
+[98] R. S. Bogadi, M. Govender,
+and S. Moyo, Eur. Phys.
+J. C 81, 922 (2021).
+[99] R. S. Bogadi and M. Govender, Eur. Phys. J. C 82, 475
+(2022).
+[100] M. G. Alford, S. Han, and M. Prakash, Phys. Rev. D
+88, 083013 (2013).
+[101] N. Sennett et al., Phys. Rev. D 96, 024002 (2017).
+[102] A. E. Romano, arXiv:2211.05760 [gr-qc] (2022).
+

39AyT4oBgHgl3EQf1_mj/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c479ba589b0f0a41e8896c9f5d9321fe57ff76085e0e24dcc095d7bbdd6c2582
+size 3735597

39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d7e6506c4c57c14422803fa575e15cd67efa2441b7fe6a7862a8e3d1986d6a3a
+size 4513237

39E2T4oBgHgl3EQfOAbJ/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2ead374b1abbe53cb6aef85b039d7b452433a08682fa237e0753efee3add6fdb
+size 3735597

39E2T4oBgHgl3EQfOAbJ/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c77f7eb4f1f096593fae880df245bb5b5908be119ad461766c1fa88472cc02e9
+size 130029

39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:34da1a629eda3f5af485ea0122f90ff4eeee1ded08654b65a8255b675d45ce2c
+size 365394

39E2T4oBgHgl3EQfjgft/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e7ef2aed3b2076af3f9d575be8e760318027de794adeb010a14f574d61da3372
+size 5505069

3NFAT4oBgHgl3EQflB25/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b4cc9f1204c8fa7d0935449b65c67ebce70352db358ee5764eff198115f673c4
+size 2883629

3dFKT4oBgHgl3EQfQy0a/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:fc581c23c6976646d7740ffdf7d61fc0661e1ec40941b14abd3378d15d998385
+size 359057

4NE1T4oBgHgl3EQf6AXQ/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1402d7bdaf4e6b67794300c0927351e07820fb69030daa45b4c294e6b481a447
+size 167174

4dE1T4oBgHgl3EQf6QUq/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:25509116b51cf3c3293fa38cd37edd670f65e4842d3e2b4c0377b3863ce72564
+size 1966125

4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b128b796fc7d155961fe0081b5499ab1e1e774170403d47dfe43444f82b8f8c4
+size 538501

4tAzT4oBgHgl3EQffvxD/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9930099a7b7a4ac122859ddc8d500bfd3454ef42ab2ee8723afe2ffa1c981e99
+size 130607

79E1T4oBgHgl3EQfBwKM/content/tmp_files/2301.02856v1.pdf.txt

@@ -0,0 +1,1946 @@
+1
+Neural Network-Based DOA Estimation in the
+Presence of Non-Gaussian Interference
+S. Feintuch, J. Tabrikian, Fellow, IEEE, I. Bilik, Senior Member, IEEE, and H. Permuter, Senior Member, IEEE
+Abstract—This work addresses the problem of direction- of-
+arrival (DOA) estimation in the presence of non-Gaussian,
+heavy-tailed, and spatially-colored interference. Conventionally,
+the interference is considered to be Gaussian-distributed and
+spatially white. However, in practice, this assumption is not guar-
+anteed, which results in degraded DOA estimation performance.
+Maximum likelihood DOA estimation in the presence of non-
+Gaussian and spatially colored interference is computationally
+complex and not practical. Therefore, this work proposes a
+neural network (NN) based DOA estimation approach for spa-
+tial spectrum estimation in multi-source scenarios with a-priori
+unknown number of sources in the presence of non-Gaussian
+spatially-colored interference. The proposed approach utilizes
+a single NN instance for simultaneous source enumeration and
+DOA estimation. It is shown via simulations that the proposed
+approach significantly outperforms conventional and NN-based
+approaches in terms of probability of resolution, estimation
+accuracy, and source enumeration accuracy in conditions of low
+SIR, small sample support, and when the angular separation
+between the source DOAs and the spatially-colored interference
+is small.
+Index Terms—Array Processing, DOA Estimation, Source
+Enumeration, Spatially-Colored Interference, Non-Gaussian In-
+terference, Neural Networks, Deep Learning, Machine Learning,
+MVDR, MDL, AIC, Radar.
+I. INTRODUCTION
+Direction-of-arrival (DOA) estimation using a sensor array
+is required in multiple applications, such as radar, sonar,
+ultrasonic, wireless communications, and medical imaging [1].
+In real-world applications, the signal received at the sensor
+array is a superposition of signals from the sources of interest,
+interference, and receiver thermal noise. In radars, the received
+signal consists of a target echo, clutter, and thermal noise. In
+multiple scenarios, the radar clutter has a spatially-colored,
+heavy-tailed non-Gaussian distribution [2], which can signifi-
+cantly degrade the performance of conventional estimators.
+Minimum-variance-distortionless-response (MVDR) [3], is
+a conventional adaptive beamforming approach for DOA es-
+timation. MVDR estimates the spatial spectrum and obtains
+the source DOAs via a one-dimensional peak search on a
+predefined grid. The estimation of signal parameters using
+rotational invariance techniques (ESPIRIT) [4], multiple signal
+classification (MUSIC) [5], and root-MUSIC (R-MUSIC) [6]
+are additional widely used DOA estimation approaches. These
+approaches involve received signal autocorrelation matrix
+processing, which conventionally is performed via the sam-
+ple autocorrelation matrix estimation [3]–[6]. However, the
+Stefan Feintuch, Joseph Tabrikian, Igal Bilik, and Haim H. Permuter
+are with the School of Electrical and Computer Engineering, Ben Gurion
+University of the Negev, Beer Sheva, Israel. (e-mails: [email protected],
+[email protected], [email protected], [email protected])
+performance of the sample autocorrelation matrix estimator
+degrades in small sample support or non-Gaussian scenarios.
+Furthermore, these methods use the second-order statistics
+only and omit the higher-order statistics on non-Gaussian-
+distributed interference. In addition, ESPRIT, MUSIC, and R-
+MUSIC approaches require a-priori knowledge of the number
+of sources (or targets), which limits their practical use.
+The problem of DOA estimation in the presence of non-
+Gaussian interference is of great practical interest. The max-
+imum likelihood estimator (MLE) for DOA estimation in the
+presence of non-Gaussian interference does not have a closed-
+form analytical solution [7], [8]. Multiple model-based DOA
+estimation approaches have been intensively studied in the
+literature [7]–[18].
+Robust covariance matrix-based DOA estimation and source
+enumeration methods have been studied in the literature. For
+complex elliptically symmetric (CES) distributed data, the
+authors in [9] showed that a scatter matrix-based beamformer
+is consistent, and the semiparametric lower bound and Slepian-
+Bangs formula for DOA estimation were derived in [10].
+In [11], a generalized covariance-based (GC) approach for
+the covariance matrix estimation in scenarios with impulsive
+alpha-stable noise was proposed for MUSIC DOA estimation.
+However, these methods consider a specific family of distri-
+butions, such as the CES or alpha-stable, and are therefore,
+limited in the case of model mismatch. In [12], a probability
+measure transform (MT) based covariance matrix estimator
+was proposed for MUSIC-based DOA estimation and mini-
+mum descriptive length (MDL) based source enumeration. The
+MT-based covariance estimator was also adopted for robust
+MVDR beamformer [13]. These methods are usually based
+on setting a parameter that determines the tradeoff between
+the level of robustness and performance.
+The problem of DOA estimation in the presence of a mix-
+ture of spatially-white K-distributed and Gaussian-distributed
+noise under a deterministic and unknown (conditional) source
+model was studied in [7]. An iterative MLE-based approach
+for the conditional and joint likelihood of interference distri-
+bution’s parameters was derived in [14], [15]. This approach
+was further extended in [16] to marginal likelihood function.
+However, this approach is computationally complex due to
+numerical integral evaluation that involves a 2M dimensional
+grid search for M targets [8]. Therefore, [8] proposed a kernel
+minimum error entropy-based adaptive estimator and a novel
+criterion to reduce the estimator’s computational complexity.
+The expectation-maximization (EM) with a partial relaxation-
+based DOA estimation algorithm under the conditional model
+assumption was proposed in [17]. In [18] a sparse Bayesian
+learning (SBL) approach for outlier rejection of impulsive
+arXiv:2301.02856v1  [eess.SP]  7 Jan 2023
+
+2
+and spatially-white interference was proposed. This EM-based
+approach does not require a-priori knowledge of the number
+of sources and was shown to resolve highly-correlated and co-
+herent sources. However, none of these model-based DOA es-
+timation approaches considered an a-priori unknown number
+of sources and spatially-colored interference and therefore are
+limited for real-world applications. Although source enumera-
+tion methods, such as MDL and Akaike information criterion
+(AIC) [19] can be used, they assume signal Gaussianity, and
+can therefore be inaccurate in non-Gaussian scenarios.
+Deep learning and machine learning approaches were re-
+cently adopted for radar signal processing. Three types of
+NN-based DOA estimation approaches have been introduced
+in literature [20]. The first approach assumes a-priori known
+number of sources, and uses a NN, which is optimized to
+output a vector of the estimated DOAs [21]–[27]. The second
+approach does not assume a-priori known number of sources
+and uses a NN for source enumeration [25]–[31]. The third
+approach uses a NN to estimate source presence probability
+at each DOA on a predefined angular grid and obtains the
+source DOAs via a peak search [32]–[41]. However, all these
+approaches have not addressed non-Gaussian and spatially-
+colored interference [20]–[41].
+The cases of non-Gaussian and/or spatially-colored inter-
+ference have been addressed using machine learning-based
+approaches. For massive MIMO cognitive radar, a reinforce-
+ment learning-based approach for multi-target detection un-
+der heavy-tailed spatially-colored interference was proposed
+in [42]. In [43], authors addressed the MIMO radar target
+detection under non-Gaussian spatially-colored interference
+by using a CNN architecture that is optimized according to
+a novel loss. A radial-basis-function (RBF) NN [44] and a
+convolutional neural network (CNN) [45] architectures were
+proposed for DOA estimation in the presence of non-Gaussian
+impulsive noise. In [46], a CNN-based architecture that in-
+cludes denoising NN, source enumeration NN, and DOA esti-
+mation sub-NNs, was introduced. However, [44]–[46] consider
+spatially-white noise and are suboptimal for scenarios with
+spatially-colored interference.
+This work addresses the problem of DOA estimation of a-
+priori unknown number of sources in the presence of non-
+Gaussian, heavy-tailed, spatially-colored interference at a low
+signal-to-interference ratio (SIR) and small sample size. The
+contribution of this work include:
+1) A novel NN-based processing mechanism is used for
+array processing within non-Gaussian spatially-colored
+interference. The proposed NN architecture utilizes the
+structure of information within the set of received com-
+plex snapshots.
+2) The proposed NN is optimized to output an interference-
+mitigated spatial spectrum, and is used for simultaneous
+source enumeration and DOA estimation of sources
+within non-Gaussian spatially-colored interference.
+The proposed approach outperforms conventional adaptive
+beamforming and competing straightforward NN-based meth-
+ods in terms of probability of resolution and estimation
+accuracy in scenarios with non-Gaussian spatially-colored
+interference. In addition, the proposed approach outperforms
+conventional source enumeration techniques in scenarios char-
+acterized by non-Gaussian spatially-colored interference.
+The following notations are used throughout the paper.
+Roman boldface lower-case and upper-case letters represent
+vectors and matrices, respectively while Italic letters stand for
+scalars. IN is the identity matrix of size N × N and 1N
+is a column vector of length N whose entries are equal to
+one. E(·), (·)T , and (·)H are the expectation, transpose, and
+Hermitian transpose operators, respectively. Vec(·), diag(·),
+and | · | stand for the vectorization, diagonalization, and
+absolute value operators, respectively. [a]n and [A]n,m are the
+n-th and n, m-th elements of the vector a and the matrix A,
+respectively.
+The remainder of this paper is organized as follows. The
+addressed problem is stated in Section II. Section III intro-
+duces the proposed NN-based DOA estimation approach. The
+proposed approach is evaluated via simulations in Section IV.
+Our conclusions are summarized in Section V.
+II. PROBLEM DEFINITION
+This work considers the problem of DOA estimation using
+an array of L receiving elements and M distinct and unknown
+sources with DOAs, Θ = {θ1, . . . , θM}. The measurements
+contain K spatial snapshots, {xk}K
+k=1:
+xk = A (Θ) sk + σcck + nk ,
+(1)
+=
+M
+�
+m=1
+a (θm) sk,m + σcck + nk , k = 1, . . . , K ,
+where A (Θ) =
+�a (θ1)
+· · ·
+a (θM)�
+, with a (θm) ∈ CL
+denoting the steering vector for source at direction θm,
+and sk ≜
+�sk,1
+· · ·
+sk,M
+�T is the source signal vector.
+We assume an unconditional model [47], where {sk}
+i.i.d.
+∼
+CN
+�
+0M, diag
+�
+σ2
+1, . . . , σ2
+M
+��
+, is temporally uncorrelated be-
+tween pulses. The targets are assumed to be spatially distinct.
+The receiver thermal noise, denoted by nk, is considered to be
+complex Gaussian-distributed {nk}
+i.i.d.
+∼ CN
+�
+0L, σ2
+nIL
+�
+. The
+heavy-tailed non-Gaussian and spatially-colored interference is
+modeled by the interference amplitude σc, and the interference
+component ck ∈ CL. The considered compound-Gaussian
+distributed interference, {ck}
+i.i.d.
+∼ K (ν, θc) represents a non-
+Gaussian interference with angular spread around an unknown
+direction θc, such that c ∼ K (ν, θc) implies
+c = √τz ,
+(2)
+τ
+|=
+z, τ ∼ Γ (ν, ν) , z ∼ CN (0L, Mθc) .
+The compound-Gaussian statistical model is conventionally
+used in the literature to model heavy-tailed non-Gaussian
+interference [7], [8], [14], [16], [43], [48]. The texture com-
+ponent, τ ∈ R+, determines the heavy-tailed behavior and
+is characterized by, ν. The speckle component, z ∈ CL,
+determines the spatial distribution of the interference and
+is characterized by the covariance matrix, Mθc. The spatial
+covariance matrix of the interference upholds:
+E
+�
+σ2
+cccH�
+=σ2
+cE [τ] E
+�
+zzH�
+= σ2
+cMθc ,
+(3)
+
+3
+where Mθc can be modeled as [14]–[16], [43], [48]:
+[Mθc]m,l = ρ|m−l|ej(m−l)π sin θc .
+(4)
+The model in (3) and (4), represents the spatial interference,
+characterized by ρ, with a spread around the interference DOA,
+θc.
+III. THE PROPOSED DAFC-BASED NEURAL NETWORK
+The proposed approach generalizes the NN architecture that
+was introduced for linear-frequency-modulated (LFM) radar
+target detection in the range-Doppler domain [49]. In the
+following, the data pre-processing and the proposed NN-based
+processing mechanism are introduced in Subsections III-A and
+III-B. The proposed NN architecture and loss function are
+detailed in Subsections III-C and III-D, respectively.
+A. Pre-Processing
+The input matrix, X ∈ CL×K is constructed from the set
+of K snapshots in (1), {xk}:
+X =
+�
+x1
+x2
+· · ·
+xK
+�
+,
+(5)
+where the k-th column of X contains the k-th snapshot.
+The variation between the columns of X is induced by the
+statistical characteristics of the source signal sk, interference
+signal ck, and thermal noise nk. Therefore, each column in
+X can be interpreted as a complex “feature” vector containing
+essential information for DOA estimation. The set of columns
+in X can be interpreted as “realizations” of that feature.
+The complex-valued matrix, X, is converted into real-valued
+representation needed for the NN-based processing. To keep
+consistency with [49], we apply a transpose operator to the
+input matrix, such that the snapshots are stacked in rows. The
+output of the pre-processing denoted by Z0 ∈ CK×2L, is:
+Z0 =
+�
+Re
+�
+XT �
+, Im
+�
+XT ��
+.
+(6)
+B. Dimensional Alternating Fully-Connected
+The dimensional alternating fully-connected (DAFC) block
+was introduced to process measurements in a form similar to
+the model in Section II [49]. Fig. 1 schematically shows the
+DAFC mechanism.
+For arbitrary dimensions D1, D2, D3, the formulation of a
+general fully-connected (FC) layer applied to each row in a
+given matrix Z ∈ RD1×D2 can be represented by the transform
+F (·):
+F : RD1×D2 → RD1×D3 ,
+(7)
+F (Z) ≜ h
+�
+ZW + 1D1bT �
+.
+This matrix-to-matrix transformation is characterized by the
+“learnable” weight matrix, W ∈ RD2×D3, the bias vector,
+b ∈ RD3, and a scalar element-wise activation function, h(·).
+Let Fr (·) and Fc (·) be two separate, and not necessarily
+identical instances of F (·) from (7), and Zin be an arbitrary
+input matrix. The DAFC mechanism is formulated by the
+following operations:
+Dimensional Alternating Fully Connected
+• Input: Zin ∈ RH×W
+Fr : RH×W → RH×W ′
+Fc : RW ′×H → RW ′×H′
+1) Apply a single FC layer to each row in Zin:
+Zr = Fr (Zin)
+2) Apply a single FC layer to each column in Zr:
+Zc = Fc
+�
+ZT
+r
+�
+3) Transpose to keep orientation:
+Zout = ZT
+c
+• Output: Zout ≜ S (Z) ∈ RH′×W ′
+In the following, three DAFC design principles are detailed.
+1) Structured transformation
+The input to the first DAFC block is the pre-processed, Z0,
+given in (6). Therefore, the first FC layer, Fr, of the first DAFC
+block extracts spatial-related features from each row in Z0.
+The second FC layer, Fc, of the first DAFC block, introduces
+an interaction between transformed rows. This implies that
+a) Fr performs “spatial-feature” extraction by transforming
+the pre-processed i.i.d. snapshots (the rows of Z0) to a
+high-dimensional feature space, and b) the Fc performs a
+nonlinear transformation of the extracted features (the columns
+of Fr (Z0)) from each snapshot. In this way, the DAFC utilizes
+both spatial and statistical information. In addition, it can
+exploit high-order statistics-related features. Thus, the DAFC
+mechanism can contribute to estimating the source DOAs and
+mitigating the interference when incorporated into a NN.
+2) Sparsity
+Conventional DOA estimation considers the input data as
+the collection of measurement vectors (the snapshots {xk}) in
+a matrix form. One straightforward approach to processing
+the input data using a NN is to reshape it and process it
+via an FC-based architecture. In this way, each neuron in the
+layer’s output interacts with every neuron in the input. On
+the other hand, the DAFC block transforms the data using
+a structured transformation, which is significantly sparser in
+terms of learnable parameters compared to the straightforward
+FC-based approach.
+This parameter reduction can be observed in the following
+typical case. Consider an input matrix Z1 ∈ RD1×D1, which
+is transformed to an output matrix Z2 ∈ RD2×D2. The
+number of learnable parameters in the FC- and the proposed
+DAFC-based approaches is of the order of O
+�
+D2
+1D2
+2
+�
+, and
+O (D1D2), respectively. Notice that the DAFC-based transfor-
+mation complexity grows linearly with the number of learnable
+parameters compared to the quadratic complexity growth of
+the straightforward, FC-based approach.
+The contribution of learnable parameters dimension reduc-
+tion is twofold. First, the conventional NN optimization is
+gradient-based [50]. Therefore, a significant reduction in the
+learnable parameter dimension reduces the degrees of freedom
+in the optimizable parameter space and improves the gradient-
+based learning algorithm convergence rate. Second, reduction
+
+4
+Figure 1: The DAFC mechanism concept. Each row of dimen-
+sion W in Zin, represented by the red color, is transformed by
+Fr to a row of dimension W ′ in the middle matrix, represented
+by the transparent red color. Next, each column of dimension
+H in the middle matrix, represented by the blue color, is
+transformed by Fc to a column of dimension H′ in Zout,
+represented by the transparent blue color.
+in the learnable parameter dimension can be interpreted as
+increasing the “inductive bias” of the NN model [51], which
+conventionally contributes to the NN statistical efficiency and
+generalization ability, thus, reducing the NNs tendency to
+overfit the training data.
+3) Nonlinearity
+The proposed DAFC considers an additional degree of
+nonlinearity compared to the straightforward FC-based ap-
+proach. A straightforward matrix-to-matrix approach includes
+an interaction of every neuron in the output matrix with
+every neuron in the input matrix, followed by an element-wise
+nonlinear activation function. On the other hand, the proposed
+DAFC consists of two degrees of nonlinearity, in Fr and Fc.
+Although the weight matrices applied as part of Fr and Fc
+are of lower dimension than the weight matrix used in the
+straightforward approach, the extra degree of nonlinearity can
+increase the NN’s capacity [50]. Therefore, a NN architecture
+with the proposed DAFC is capable of learning a more abstract
+and rich transformation of the input data.
+C. NN Architecture
+The continuous DOA space is discretized into a d-
+dimensional grid: φ =
+�φ1
+φ2
+· · ·
+φd
+�T . This implies
+that the entire field-of-view (FOV) is partitioned into d DOAs,
+{φi}d
+i=1, determined by the selected grid resolution, ∆φ ≜
+φi+1 − φi. The proposed NN is designed to represent a
+mapping from the input set of snapshots, {xk} given in (1),
+into the probability of source present in the DOAs {φi}d
+i=1.
+The proposed NN architecture is formulated as follows:
+Z0 = P (X) ,
+(8)
+zvec = Vec (S6 (. . . S1 (Z0))) ,
+ˆy = G3 (G2 (G1 (zvec))) ,
+Operator
+Output
+Dimension
+Activation
+# Parameters
+P
+K × 2L
+-
+-
+S1
+64 × 256
+tanh-
+ReLu
+9,536
+S2
+128 × 512
+tanh-
+ReLu
+139,904
+S3
+256 × 1024
+tanh-
+ReLu
+558,336
+S4
+64 × 512
+tanh-
+ReLu
+541,248
+S5
+16 × 256
+tanh-
+ReLu
+132,368
+S6
+4 × 128
+tanh-
+ReLu
+32,964
+vec
+512
+-
+-
+G1
+1024
+tanh
+525,312
+G2
+256
+tanh
+262,400
+G3
+d
+sigmoid
+31,097
+Table I:
+Specification of the proposed NN architecture for
+K = 16, L = 16, d = 121. “tanh-ReLu” activation stands
+for tanh in Fr and ReLU in Fc of each DAFC block. The
+number of total learnable parameters is 2, 233, 165.
+where Z0 is the output of the pre-processing procedure,
+denoted as P (·) and detailed in Section III-A, and X is the
+input matrix in (5).
+In the next stage, six DAFC instances, represented by
+S1 (·) , . . . , S6 (·), of different dimensions with tanh activa-
+tion for the row transform (Fr in Section III-B) and ReLu
+activation for the column transform (Fc in Section III-B), are
+used to generate the vectorized signal zvec. Our experiments
+showed that this configuration of row and column activation
+functions provides the best performance. At the last stage, the
+signal, zvec, is processed by three FC layers, where the first
+two use tanh activation, and the final (output) layer of equal
+size to the DOA grid dimension, d, uses sigmoid activation
+function to output ˆy ∈ [0, 1]d. Thus, {[ˆy]i}d
+i=1 represent the
+estimated probabilities of a source presence at {φi}d
+i=1. Table I
+and Fig. 2 summarize the parameters and architecutre of the
+proposed NN-based approach.
+The estimated source DOAs are extracted from the spatial
+spectrum via peak search and applying 0.5 threshold:
+{i1, . . . , i ˆ
+N} = peak search
+�
+{[ˆy]i}d
+i=1
+�
+(9)
+ˆΘ =
+�
+φin : [ˆy]in > 0.5
+� ˆ
+N
+n=1 .
+Namely, the set of estimated DOAs, ˆΘ, consists of the grid
+points corresponding to the peaks of ˆy that exceed the 0.5
+threshold. The number of peaks that exceed this threshold is
+used for source enumeration, and therefore the proposed NN
+can be utilized as a source enumeration method as well.
+The dimensionality of the hidden layers in the proposed
+
+5
+Figure 2: Proposed NN architecture. The pre-processing P is described in Section III-A and appears in yellow. The purple
+matrices denote the concatenation of DAFC blocks, which is detailed in Section III-B. The blue vector represents a vectorization
+of the last DAFC output, and the orange vectors stands for FC layers with tanh activation function. The last green vector is
+the output of the last FC layer, which consists of sigmoid activation function and yields the estimated spatial spectrum ˆy.
+NN architecture expands in the first layers and then reduces.
+This trend resembles the NN architecture presented in [49] and
+characterizes both the DAFC-based and FC-based processing
+stages. This expansion-reduction structure can be explained
+by a) the early NN stages need to learn an expressive and
+meaningful transformation of the input data by mapping it to
+a higher dimensional representation and b) the late stages need
+to extract significant features from the early mappings, and are
+therefore limited in dimensionality. In addition, the late stages
+are adjacent to the output vector and therefore need to be of
+similar dimension.
+D. Loss Function
+The label used for the supervised learning process, y ∈
+{0, 1}d, is defined as a sparse binary vector with the value 1,
+at the grid points that correspond to the source DOAs, and
+0, otherwise. In practice, the DOAs in Θ do not precisely
+correspond to the grid points. Therefore, for each DOA in
+Θ, the nearest grid point in {φi}d
+i=1 is selected as the
+representative grid point in the label. Each training example
+is determined by the input-label pair, (X, y). Using the NN
+feed-forward in (8), X is used to generate the output spatial
+spectrum, ˆy, which is considered as the estimated label.
+The loss function, L, is a weighted mean of the binary cross
+entropy (BCE) loss computed at each grid point:
+L (y, ˆy, t) = 1
+d
+d
+�
+i=1
+w(t)
+i BCE ([y]i , [ˆy]i) ,
+(10)
+BCE (y, ˆy) = −y log (ˆy) − (1 − y) log (1 − ˆy) ,
+where w(t)
+i
+represents the loss weight of the i-th grid point at
+the t-th epoch. The loss value for equally-weighted BCEs eval-
+uated per grid point (w(t)
+i
+= 1 in (10)) does not significantly
+increase in the case of a large error in source/interference esti-
+mated probability, due to the sparsity of the label y. This forces
+the NN convergence into a sub-optimal solution that is prone
+to “miss” the sources. Therefore, the loss weights, {w(t)
+i }d
+i=1,
+are introduced to “focus” the penalty on source/interference
+grid points.
+The loss weight of the i-th grid point, w(t)
+i , is determined
+by the presence of source or interference in the corresponding
+label entry [y]i. This relation is defined using the epoch and
+label dependent factors e(t)
+0 , e(t)
+1 , according to:
+w(t)
+i
+=
+�
+1/e(t)
+1 ,
+if φi contains source or interference
+1/e(t)
+0 ,
+else
+.
+(11)
+For t = 0, the factor e(0)
+1
+is determined by the fraction of
+label grid points that contain source or interference out of
+the total label grid points in the training set, and e(0)
+0
+is the
+corresponding complement. For subsequent epochs, the factors
+are updated according to a predefined schedule, similarly to a
+predefined learning rate schedule. The loss weights are updated
+Nw times with spacing of ∆t epochs during training. The
+update values are determined by updating e(t)
+0 , e(t)
+1 , according
+to the following decaying rule:
+e(t)
+q
+= (1 − β(l))e(l∆t)
+q
++ β(l), l∆t ≤ t < (l + 1)∆t
+(12)
+q = 0, 1, l = 1, . . . , Nw,
+
+6
+where l is the loss weight update iteration, and {β(l)}Nw
+l=1
+represent the loss weight update factors which uphold, 0 ≤
+β(l) ≤ 1. Note that for Nw∆t ≤ t, the weight factor
+remains e(Nw∆t)
+i
+during the rest of the training stage. No-
+tice that as β(l) → 1, the corresponding loss weights will
+tend to be equally distributed across the grid points, i.e.,
+e(t)
+1
+≈ e(t)
+0 . In this case, an erroneously estimated proba-
+bility for source/interference containing grid point is equally
+weighted to a neither-containing grid point. On the other
+hand, as β(l) → 0, the corresponding factors will uphold
+e(t)
+1
+≪ e(t)
+0 , yielding a significantly larger contribution of
+source/interference containing grid points to the loss value.
+The rule in (12) enables a “transition of focus” throughout
+the training. That is, during the early epochs β(l) → 0, which
+contributes more weight to the source/interference containing
+areas in the estimated label ˆy (i.e., the estimated spatial spec-
+trum) to focus the NN to being correct for source/interference.
+During the later epochs, β(l) is incrementally increased, which
+relaxes the focus on source/interference from early epochs.
+Thus, reducing erroneously estimated sources in areas that do
+not contain source/interference (i.e. “false-alarms”).
+IV. PERFORMANCE EVALUATION
+This section evaluates the performance of the proposed
+DAFC-based NN approach and compares it to the conventional
+approaches, summarized in Subsection IV-A1. The data for
+all considered scenarios is simulated using the measurement
+model from Section II.
+A. Setup & Training
+This work considers a uniform linear array (ULA) with
+half-wavelength-spaced L elements. Each simulated example
+consists of the input-label pair, (X, y), where the input X is
+defined in (5), and the label y is defined in Section III-D.
+The simulation configurations are detailed in Table II. The
+performance of the proposed approach is evaluated using
+a single NN instance. Therefore, a single NN model is
+used for various signal-to-interference ratios (SIRs), signal-
+to-noise ratios (SNRs), interference-to-noise ratios (INRs),
+DOAs, interference distribution, and the number of sources for
+joint DOA estimation and source enumeration. The following
+definitions for the m-th source are used in all experiments:
+INR
+=
+E[∥c∥2]
+E[∥n∥2] = σ2
+c/σ2
+n ,
+(13)
+SNRm
+=
+E[∥a(θm)sm∥2]
+E[∥n∥2]
+= σ2
+m/σ2
+n ,
+(14)
+SIRm
+=
+E[∥a(θm)sm∥2]
+E[∥c∥2]
+= σ2
+m/σ2
+c .
+(15)
+The NN optimization for all evaluated architectures is
+performed using the loss function in (10) and Adam opti-
+mizer [52] with a learning rate of 10−3, and a plateau learning
+rate scheduler with a decay of 0.905. The set of loss weight up-
+date factors, {β(l)}Nw
+l=1, in (12) is chosen as the evenly-spaced
+logarithmic scale between 10−5 and 10−2 with Nw = 6, that
+is {10−5, 7.25 · 10−5, 5.25 · 10−4, 3.8 · 10−3, 2.78 · 10−2, 0.2}.
+The chosen batch size is 512, the number of epochs is 500,
+and early stopping is applied according to the last 200 epochs.
+Notation
+Description
+Value
+Mmax
+Maximal
+number
+of
+sources
+4
+L
+Number of sensors
+16
+K
+Number of snapshots
+16
+d
+Angular grid dimension
+121
+∆φ
+Angular grid resolution
+1◦
+FOV
+Field of view
+[−60◦, 60◦]
+σ2
+n
+Thermal noise power
+1
+Table II: Simulation Configurations.
+1) DOA Estimation Approaches: This subsection briefly
+summarizes the conventional DOA estimation approaches. The
+performance of the proposed approach is compared to the
+conventional MVDR, CNN, and FC-based NN. All the NN-
+based approaches were implemented using similar number of
+layers and learnable parameters. In addition, the FC-based NN
+and CNN were optimized using the same learning algorithm
+and configurations.
+(a) Conventional Adaptive Beamforming
+The MVDR [3] estimator is based on adaptive beamforming,
+and it is the maximum likelihood estimator in the presence
+of unknown Gaussian interference [53]. The MVDR estimates
+DOAs by a peak search on the MVDR spectrum:
+PMV DR (φ) =
+1
+aH (φ) ˆR−1
+x a (φ)
+,
+(16)
+where ˆRx =
+1
+K
+�K
+k=1 xkxH
+k is the sample covariance ma-
+trix estimator. Notice that the MVDR spectrum utilizes only
+second-order statistics of the received signal xk. For Gaussian-
+only interference (i.e. ck = 0 in (1)), the second-order statistics
+contains the entire statistical information. However, for non-
+Gaussian interference, information from higher-order statistics
+is needed.
+(b) CNN Architecture
+We consider a CNN-based DOA estimation approach using a
+CNN architecture that is similar to the architecture provided
+in [38]. The input to the CNN of dimension L × L × 3
+consists of the real, imaginary, and angle parts of ˆRx. The
+CNN architecture consists of 4 consecutive CNN blocks,
+such that each block contains a convolutional layer, a batch
+normalization layer, and a ReLu activation. The convolutional
+layers consist of [128, 256, 256, 128] filters. Kernel sizes of
+3 × 3 for the first block and 2 × 2 for the following three
+blocks are used. Similarly to [38], 2 × 2 strides are used
+for the first block and 1 × 1 for the following three blocks.
+Next, a flatten layer is used to vectorize the hidden tensor,
+and 3 FC layers of dimensions 1024, 512, 256 are used
+with a ReLu activation and Dropout of 30%. Finally, the
+output layer is identical to the proposed DAFC-based NN
+as detailed in Subsection III-C. The considered loss function
+is identical to the proposed DAFC-based approach in (10).
+The number of trainable parameters in the considered CNN
+
+7
+architecture accounts for 3, 315, 449. Notice that the CNN-
+based architecture utilizes the information within the sample
+covariance matrix and therefore, is limited to second-order
+statistics only.
+(c) FC Architecture
+A straightforward implementation of an FC-based architecture,
+as mentioned in Subsection III-B, was implemented. The
+data matrix, X, is vectorized, and the real and imaginary
+parts of the values were concatenated to obtain a 2KL-
+dimension input vector. The selected hidden layers are of
+sizes: [512, 512, 1024, 1024, 512, 256] where each hidden layer
+is followed by a tanh activation function. The output layer
+is identical to the proposed DAFC-based NN approach as
+detailed in Subsection III-C. The considered loss function
+is (10), and the number of trainable parameters in the FC-
+based NN accounts for 2, 787, 449. Notice that the FC-based
+NN architecture utilizes all the measurements by interacting
+with all samples in the input data. However, this processing is
+not specifically tailored to the structure of information within
+the measurements. On the other hand, the proposed DAFC-
+based NN utilizes the information structure to process the input
+data. Therefore, for the considered DOA estimation problem,
+the “inductive bias” [51] for this approach is improper and can
+result in under-fitted NN architecture.
+2) Performance Evaluation Metrics: This subsection dis-
+cusses the criteria for the performance evaluation of the
+proposed DOA estimation approach. In this work, similarly
+to [38], the DOA estimation accuracy of a set of sources
+is evaluated by the Hausdorff distance between sets. The
+Hausdorff distance, dH between the sets, A, and B, is defined
+as:
+dH (A, B) = max {d (A, B) , d (B, A)} ,
+(17)
+d (A, B) = sup {inf {|α − β| : β ∈ B} : α ∈ A} .
+Notice that d (A, B) ̸= d (B, A). Let Θ = {θm}M
+m=1 and ˆΘ =
+{ˆθm} ˆ
+M
+m=1 be the sets of true and estimated DOAs, respectively.
+The estimation error is obtained by evaluating the Hausdorff
+distance, dH(Θ, ˆΘ). We define the root mean squared distance
+(RMSD) for an arbitrary set of N examples (e.g., test set),
+�
+X(n), y(n)�N
+n=1, with the corresponding true and estimated
+DOAs,
+�
+Θ(n), ˆΘ(n)�N
+n=1 as:
+RMSD ≜
+�
+�
+�
+� 1
+N
+N
+�
+n=1
+d2
+H
+�
+Θ(n), ˆΘ(n)
+�
+.
+(18)
+Angular resolution is one of the key criteria for DOA
+estimation performance. The probability of resolution is com-
+monly used as a performance evaluation metric for angular
+resolution. In the considered problem, resolution between two
+sources and between source and interference are used for
+performance evaluation. For an arbitrary example with M
+sources, the resolution event Ares is defined as:
+Ares
+�
+Θ, ˆΘ
+�
+≜
+�
+1,
+�M
+m=1 ξm ≤ 2◦ and | ˆΘ| ≥ M
+0,
+else
+,
+(19)
+ξm ≜ min
+ˆθ∈ ˆΘ
+|θm − ˆθ|, m = 1, . . . , M .
+For example, a scene with M sources is considered success-
+fully resolved if for each true DOA a) there exists a close-
+enough estimated DOA, ˆθ ∈ ˆΘ, that is at most 2◦ apart, and
+b) there exists at least M DOA estimations. According to (18),
+the probability of resolution, can be defined as:
+Pres = 1
+N
+N
+�
+n=1
+Ares
+�
+Θ(n), ˆΘ(n)�
+.
+(20)
+3) Data Sets: This subsection describes the structure and
+formation of Training & Test sets.
+(a) Training Set
+The considered training set contains Ntrain
+=
+10, 000
+examples re-generated at each epoch. For each exam-
+ple, i.e. an input-label pair (X, y), the number of DOA
+sources, M, is generated from uniform and i.i.d. distribution,
+{1, . . . , Mmax}. The training set contains 10% of interference-
+free examples and 90% of interference-containing. Out of the
+interference-containing examples, 90% generated such that the
+source DOAs, {θm}M
+m=1, and the interference’s DOA, θc, are
+distributed uniformly over the simulated FOV. The remaining
+10% are generated such that θc is distributed uniformly over
+the FOV, and the source DOAs, {θm}M
+m=1, are distributed
+uniformly over the interval [θc − 8◦, θc + 8◦]. This data set
+formation enables to “focus” the NN training on the chal-
+lenging scenarios where the source and interference DOAs are
+closely spaced. The generalization capabilities of the proposed
+NN to variations in interference statistics are achieved via the
+interference angular spread parameter, ρ, from the uniform dis-
+tribution, U ([0.7, 0.95]), and the interference spikiness param-
+eter, ν, from the uniform distribution, U ([0.1, 1.5]). The INR
+for each interference-containing example and {SIRm}M
+m=1 or
+{SNRm}M
+m=1 are drawn independently according to Table III.
+(b) Test Set
+The test set consists of Ntest = 20, 000 examples. The results
+are obtained by averaging the evaluated performance over 50
+independent test set realizations. Considering the low-snapshot
+support regime, the number of snapshots is set to K = 16,
+except for experiment (c) in IV-B2. Considering heavy-tailed
+interference, the spikiness parameter is set to ν = 0.2. The
+INR is set to INR = 5 dB, and the interference angular spread
+parameter is set to ρ = 0.9. The signal amplitude was set to be
+identical for all sources, σ1 = · · · = σm, except for experiment
+(b) in IV-B2.
+B. Experiments
+1) Single Source Within Interference: In this scenario, the
+ability to resolve a single source from interference is evaluated.
+Let M = 1 with θ1 = 0.55◦, and θc = θ1 + ∆θc such
+that ∆θc is the angular separation between the single source
+and interference. The 0.55◦ offset is considered to impose
+a realistic off-grid condition. Fig. 3 shows the RMSD and
+probability of resolution for all evaluated approaches.
+Fig. 3a shows that the FC-based NN approach does not
+manage to resolve the single source from the interference
+for all evaluated angular separations. This result supports the
+
+8
+(a)
+(b)
+Figure 3: Scenario with a single source at θ1 = 0.55◦ and interference located at θc = θ1 + ∆θc. (a) probability of resolution
+and (b) RMSD.
+Notation Description
+Value
+ρ
+Interference angular
+spread parameter
+∼ U ([0.7, 0.95])
+ν
+Interference
+spikiness parameter
+∼ U ([0.1, 1.5])
+INR
+INR
+∼ U ([0, 10]) [dB]
+SIRm
+SIR of m-th source
+∼ U ([−10, 10]) [dB]
+SNRm
+SNR of m-th source
+∼ U ([−10, 10]) [dB]
+Table III: Training set parameters. SNRm distribution applies
+to interference-free examples.
+under-fitting limitation of the FC-based NN approach for the
+DOA estimation, which can be explained by the architecture
+that processes the input data as-is, without any structured
+transformation or model-based pre-processing.
+The MVDR and CNN performance in terms of the resolu-
+tion are similar since both rely only on second-order statistics,
+which is sufficient in scenarios with widely separated sources
+and interference. Fig. 3a shows that the proposed DAFC-based
+NN approach outperforms all other considered approaches in
+low angular separation scenarios. This can be explained by the
+fact that the DAFC uses the high-order statistics needed for
+the resolution of closely spaced sources and interference.
+Fig. 3b shows the RMSD of all considered DOA estimation
+approaches. The proposed DAFC-based NN approach outper-
+forms the other tested approaches in low SIR. At high SIR
+and small angular separation, ∆θc = 5◦, the interference
+is negligible with respect to the strong source signal, and
+therefore, the DAFC-based, CNN, and MVDR approaches
+obtain similar performance. For large angular separation,
+∆θc = 30◦, the source and the interference are sufficiently
+separated, and therefore, DOA estimation errors are mainly
+induced by the interference DOA, θc. The MVDR spectrum
+contains a peak at θc = 30.55◦, and therefore, MVDR’s
+RMSD = 30◦ is approximately constant. The NNs are trained
+to output a 0-probability for the interference, therefore, the
+NN-based approaches: FC, CNN, and DAFC achieve a smaller
+DOA estimation error. The DAFC-based NN and CNN utilize
+structured transformations, which better fit the input data, and
+therefore, they outperform the FC-based NN approach in terms
+of RMSD.
+2) Resolving Two Sources from Interference: This subsec-
+tion evaluates the performance of the tested DOA estimation
+approaches in scenarios with two sources within AWGN and
+interference.
+(a) Resolution of Equal-Strength Sources
+In the following experiment, the resolution between two equal-
+power sources, M = 2, with θ1 = − ∆θ
+2 + 0.55◦, and θ2 =
+∆θ
+2 +0.55◦, is evaluated. The off-grid additional 0.55◦ offset to
+the ∆θ angular separation between the sources represents the
+practical scenario. The interference at θc = 0.55◦ influences
+the two sources similarly. Fig. 4 shows the probability of
+resolution of the tested approaches in scenarios with (a) the
+AWGN only and (b) spatially-colored interference.
+The FC-based NN approach does not resolve the two targets
+in both evaluated scenarios. Subplot (a) in Fig. 4 shows
+that the proposed DAFC-based NN approach outperforms the
+MVDR and the CNN at low-SNR and small angular separation
+scenarios due to its generalization ability to spatially-white
+interference. Subplot (b) in Fig. 4 shows that at low SIR
+of SIR = −5 dB, the performances of MVDR and CNN
+significantly degrade compared to the proposed DAFC-based
+NN approach. Comparing subplots in Fig. 4, notice that at
+SIR = −5 dB, the MVDR fails to resolve the sources with
+angular separation ∆θ < 20◦ due to the presence of the heavy-
+tailed spatially-colored interference in the proximity of the
+sources. However, the proposed DAFC-based NN approach
+mitigates this interference and resolves the sources, and hence,
+outperforms other tested approaches at both SIR = 0 dB and
+SIR = −5 dB.
+Subplot (b) in Fig. 4 shows the non-monotonic trend of
+CNN and MVDR performance at 4◦ < ∆θ < 18◦ and
+
+1.0
+0.8
+MVDR, SIR=0
+DAFC, SIR=0
+FC, SIR=0
+S
+0.6
+res
+CNN. SIR=O
+P
+MVDR, SIR=-5
+DAFC, SIR=-5
+0.4
+FC, SIR=-5
+CNN, SIR=-5
+0.2
+10
+15
+20
+25
+30
+△0c [Deg]RMSD [Deg]
+101
+MVDR, △0c=5
+DAFC, A0c=5
+FC, △Qc=5
+CNN, △0c=5
+MVDR, △0c=30
+DAFC, △0c=30
+100
+FC, △0c=30
+CNN, △0.=30
+-10
+-5
+0
+5
+10
+15
+20
+SIR[dB]9
+(a)
+(b)
+Figure 4: Probability of resolution for two sources located at θ1,2 = θc ± ∆θ/2, and interference located at θc = 0.55◦. (a)
+AWGN-only scenario and (b) interference-containing scenario.
+(a) FC
+(b) MVDR
+(c) CNN
+(d) DAFC
+Figure 5: Spatial spectrum, two sources with SIR = −5 dB
+located at θ1,2 = θc ± ∆θ/2 with ∆θ = 12◦ and θc = 0.55◦.
+The dashed blue lines represent the mean spatial spectrum,
+and the color fill represents the standard deviation around the
+mean obtained from 2, 000 i.i.d. examples. The solid vertical
+orange lines represent the true source DOAs, and the dashed
+vertical green line represents the interference DOA.
+SIR = −5 dB. For 4◦ < ∆θ < 8◦ the sources are closer
+to the peak of the interference’s lobe and are therefore less
+mitigated by it. As ∆θ initially increases, 8◦ < ∆θ < 12◦,
+the sources reach DOAs which are in the proximity of the
+interference lobe’s “nulls” which explains the reduction in
+resolution, and as ∆θ further increases, 16◦ < ∆θ, the
+sources are sufficiently separated from the interference such
+that the resolution increases. As a result, MVDR and CNN-
+based approaches that use second-order statistics only, can not
+resolve the sources in the vicinity of a stronger interference.
+Fig. 5 shows the average spatial spectrum of all tested
+approaches for ∆θ = 12◦ and SIR = −5 dB. The average
+spatial spectrum of the FC-based NN approach does not show
+two prominent peaks, which results in its poor probability of
+resolution in Fig. 4. The MVDR “bell-shaped” spatial spec-
+trum does not contain the two prominent peaks at θ1,2 since the
+interference “masks” the two sources. The CNN and proposed
+DAFC-based NN approaches show two peaks at the average
+spatial spectrum. The peaks at the CNN’s average spatial
+spectrum are lower, resulting in a low-resolution probability.
+The average spatial spectrum of the proposed DAFC-based
+NN approach contains two high peaks, resulting in a superior
+probability of resolution in Fig. 4.
+(b) Resolution of Unequal-Power Sources
+Fig. 6 shows the probability of resolution in a scenario
+with two sources, M = 2, at θ1 = −∆θ/2 + 0.55◦, and
+θ2 = +∆θ/2 + 0.55◦ with interference located between the
+sources at θc = 0.55◦. The signal strength of the second
+source is set to SIR1 = SIR2 + 10 dB. Comparing Fig. 6 to
+Fig. 4b, the competing methods show similar trends, except
+the degradation of the CNN’s probability of resolution for the
+SIR = 0 dB case. On the other hand, the proposed DAFC-
+based NN approach outperforms other tested approaches in
+terms of the probability of resolution. Therefore, Fig. 6 demon-
+strates the generalization ability of the proposed DAFC-based
+NN approach to a variance between source strengths.
+(c) Effect of the Number of Snapshots on the Resolution
+This experiment investigates the influence of the number
+of snapshots, K, on the ability to resolve two proximate
+sources from heavy-tailed spatially-colored interference. The
+equal-strength resolution scenario is repeated using K =
+4, 8, 16, 32, 64 with different instances of NN training for
+each K value. Fig. 7 shows the probability of resolution for
+two equal-strength sources at θ1,2 = θc ±∆θ/2 for ∆θ = 12◦
+and θc = 0.55◦.
+The FC-based NN approach fails to resolve the two sources.
+For SIR = 0 dB, the MVDR, CNN, and DAFC-based NN
+
+DAFC
+0.8
+SIR=-5.00 dB
+INR=5.00 dB
+0.6
+0.4
+0.2
+0.0
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]1.0
+0.8
++*+
+MVDR, SNR=O
+DAFC, SNR=O
+0.6
+FC, SNR=0
+res
+CNN, SNR=0
+MVDR, SNR=-5
+P
+0.4
+DAFC, SNR=-5
+FC, SNR=-5
+CNN, SNR=-5
+0.2
+0.0
+10
+20
+30
+40
+50
+60
+Ae [Deg]1.0
+0.8
+MVDR, SIR=O
+DAFC, SIR=O
+0.6
+FC, SIR=0
+res
+CNN. SIR=0
+MVDR, SIR=-5
+P
+0.4
+DAFC, SIR=-5
+FC, SIR=-5
+CNN, SIR=-5
+0.2
+0.0
+10
+20
+30
+40
+50
+60
+Ae [Deg]0.35
+FC
+SIR=-5.00 dB
+0.30
+NR=5.00 dB
+0.25
+0.20
+0.15
+0.10
+0.05
+0.00
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]MVDR
+1.2
+SIR=-5.00 dB
+1.0
+INR=5.00 dB
+0.8
+0.6
+0.4
+0.2 
+0.0
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]0.6
+CNN
+SIR=-5.00 dB
+0.5
+INR=5.00 dB
+0.4
+0.3
+0.2
+0.1
+0.0
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]10
+Figure 6: Probability of resolution for two sources located at
+θ1,2 = θc ±∆θ/2, and interference located at θc = 0.55◦. The
+SIR in the legend represents the SIR of the first source, SIR1.
+The SIR of the second source is set to SIR2 = SIR1 + 10 dB.
+approaches achieve a monotonic increasing probability of
+resolution with increasing K. The proposed DAFC-based NN
+approach slightly outperforms other tested approaches. At low
+SIR of SIR = −5 dB, the proposed DAFC-based NN ap-
+proach significantly outperforms the other tested approaches.
+This can be explained by the fact that increasing K increases
+the probability for outliers to be present in the input data
+matrix, X. Therefore, the estimated autocorrelation matrix,
+ˆRx, is more likely to be biased by the interference-related
+outliers, which results in interference “masking” the sources.
+The proposed DAFC-based NN approach is immune to these
+outliers and successfully exploits the information from the
+additional snapshots to improve the probability of resolution.
+Figure 7: Probability of resolution for two sources located at
+θ1,2 = θc ± ∆θ/2 with ∆θ = 12◦, and interference located at
+θc = 0.55◦, as a function of the number of snapshots, K.
+Figs. 4, 5, 6, and 7 show the ability of the proposed
+DAFC-based NN approach to utilize the information structure
+of the input data by exploiting the higher-order statistics
+and performing the domain-fitted transformation in order to
+provide superior resolution ability in the case of proximate
+heavy-tailed spatially-colored interference, low SIR and small
+sample size.
+3) Multiple Source Localization: The performances of the
+tested DOA estimation approaches are evaluated and compared
+in a multi-source scenario. Four sources, (M = 4) were
+simulated with angular separation, ∆θ: {θ1, θ2, θ3, θ4} =
+θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, where θc = 0.51◦ represents
+a realistic off-grid condition. The RMSD of evaluated meth-
+ods is depicted in Fig. 8. The proposed DAFC-based NN
+approach outperforms the other tested approaches at low SIR
+(SIR < 0 dB) for large and small angular separations. For
+high SIR and low angular separation, ∆θ = 5◦, the MVDR
+achieves the lowest RMSD. The reason is that for this case, the
+interference is negligible with respect to the lobe of the strong
+source in the MVDR’s spectrum. However, at high angular
+separation, ∆θ = 20◦, the proposed DAFC-based NN ap-
+proach significantly outperforms the other tested approaches.
+This is explained by Fig. 9, that shows the spectrum of the
+tested DOA estimation approaches. Notice that the proposed
+DAFC-based NN mitigates interference, while the spectra of
+other tested approaches contain high peaks at the interference
+DOA, θc. These peaks increase the Hausdorff distance in (17),
+increasing the RMSD of other tested approaches in Fig. 8.
+Figure 8: RMSD in scenarios with M = 4 sources located at
+{θ1, θ2, θ3, θ4} = θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, where θc =
+0.51◦.
+4) Multiple Source Enumeration: The source enumeration
+performance is evaluated in this experiment. The DOAs of
+the sources are selected from the set of following values:
+{10.51◦, −9.49◦, −19.49◦, 10.51◦} such that for M sources,
+the DOAs are selected to be the first M DOAs. The interfer-
+ence is located at θc = 0.51◦. The proposed DAFC-based NN
+approach is compared to the MDL and AIC [19]. Fig. 10 shows
+the source enumeration confusion matrices for the MDL, AIC,
+and the proposed DAFC-based NN with SIR = 0 dB.
+Figs. 10a, 10b show that in both the MDL and the AIC, the
+predicted number of sources has a constant bias for each true
+M due to the spatially-colored interference. Fig. 10c shows the
+source enumeration performance of the proposed DAFC-based
+NN approach in the presence of spatially colored interference.
+The DAFC-based NN identifies the interference and does not
+count it as one of the sources by outputting a low probability
+for angular grid points near θc, resulting in a better source
+enumeration performance.
+
+1.0
+0.8
+MVDR, SIR=O
+DAFC, SIR=O
+0.6
+FC, SIR=0
+res
+CNN. SIR=0
+DP
+MVDR, SIR=-5
+0.4
+DAFC, SIR=-5
+FC, SIR=-5
+CNN, SIR=-5
+0.2
+0.0
+10
+20
+30
+40
+50
+60
+Ae [Deg]1.0
+0.8
+0.6
+res
+MVDR. SIR=O
+0.4
+DAFC, SIR=0
+FC, SIR=0
+CNN, SIR=0
+0.2
+MVDR,SIR=-5
+DAFC, SIR=-5
+FC, SIR=-5
+0.0
+CNN, SIR=-5
+4
+8
+16
+32
+64
+KRMSD [Deg]
+101
+MVDR. A0=5
+DAFC, △0=5
+FC, △0=5
+CNN, △0=5
+MVDR,A0=20
+100
+DAFC, △0=20
+FC, △0=20
+CNN, △0=20
+-10
+-5
+0
+5
+10
+15
+20
+SIR[dB]11
+(a) FC
+(b) MVDR
+(c) CNN
+(d) DAFC
+Figure 9: Spatial spectrum, four sources with SIR = 0 dB
+located at {θ1, θ2, θ3, θ4} = θc + {−2∆θ, −∆θ, ∆θ, 2∆θ},
+where θc = 0.51◦ and ∆θ = 20◦. The dashed blue lines rep-
+resent the mean spatial spectrum, and the color fill represents
+the standard deviation around the mean obtained from 2, 000
+i.i.d. examples. The solid vertical orange lines represent the
+true source DOAs and the dashed vertical green line represents
+the interference DOA.
+5) Loss Weights: This experiment evaluates the effect of the
+loss weight update factors, {β(l)}Nw
+l=1, introduced in (12), on
+the confidence level in the spatial spectrum. Let �B denote the
+set of {β(l)}Nw
+l=1 values used in the proposed approach. The
+loss weights, {w(t)
+i }d
+i=1, are defined by the factors e(t)
+0 , e(t)
+1
+according to (11), and are introduced to provide a trade-off
+between the penalty obtained on source/interference and the
+penalty obtained for the rest of the output spatial spectrum.
+For comparison, we set B0 = {10−6, 3.98 · 10−6, 1.58 ·
+10−5, 6.31 · 10−5, 2.51 · 10−4, 10−3}, and B1 = {10−3, 3.98 ·
+10−3, 0.0158, 0.063, 0.25, 0.1} as two sets of loss weight up-
+date factors. For B0, the loss weight update factors are closer
+to 0, hence the loss weights emphasize the source/interference,
+since e(t)
+1
+≪ e(t)
+0
+which, according to (11), translates to larger
+w(t)
+i
+for source/interference grid points. For B1 the values are
+closer to 1, hence the loss weights are more equally distributed
+among grid points, since e(t)
+1
+≈ e(t)
+0 . The experiment in IV-B1
+is repeated here for the DAFC-based NN approach with the
+two additional B0, B1 values mentioned above.
+Let ˆp1 represent the probability assigned for the source-
+containing grid point in the estimated label ˆy. Let ˆp0 represent
+the maximum over probabilities assigned for non-source grid
+points in ˆy, excluding a 5-grid point guard interval around the
+source. Fig. 11 shows ˆp1 and ˆp0 for various angular separations
+between the source and interference for SIR = −5 dB. For
+B0, the source’s contribution to the loss value is substan-
+tially higher, which results in a higher probability for the
+(a) MDL
+(b) AIC
+(c) DAFC
+Figure 10: Confusion matrix for source enumeration, SIR =
+0 dB, sources located at {10.51◦, −9.49◦, −19.49◦, 10.51◦}.
+(a) MDL, (b) AIC, (c) proposed DAFC-based NN.
+source-containing grid point. However, this results in a higher
+probability obtained for non-source grid points, since their
+contribution to the loss value is negligible compared to the
+source-containing grid point, increasing “false-alarm” peaks
+in the spatial spectrum, subsequently increasing the estimation
+error. Correspondingly, for B1 the source’s contribution to the
+loss value is less significant, which results in low probability
+assigned for the source-containing grid points, as well as low
+probability for non-source grid points.
+V. CONCLUSION
+This work addresses the problem of DOA estimation and
+source enumeration of an unknown number of sources within
+heavy-tailed, non-Gaussian, and spatially colored interference.
+A novel DAFC-based NN approach is proposed for this
+
+FC
+0.4
+SIR=0.00 dB
+INR=5.00 dB
+0.3
+0.2
+0.1
+0.0
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]1.6
+1.4
+1.2
+1.0
+MVDR
+SIR=0.00 dB
+0.8
+INR=5.00 dB
+0.6
+0.4
+0.2
+0.0
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]0.8
+0.7
+0.6
+0.5
+CNN
+0.4
+SIR=0.00 dB
+NR三5.00 dB
+0.3
+0.2
+0.1
+0.0
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]0.8
+0.6
+DAFC
+SIR0.00 dB
+0.4
+INR=5.00dB
+0.2
+0.0
+-60
+-40
+-20
+0
+20
+40
+60
+Φ[Deg]0
+0
+0
+0
+0
+0
+0
+0
+0.6
+0.5
+0
+0.13
+0.66
+0.2
+0
+0
+0
+True M
+- 0.4
+0
+0
+0.16
+0.68
+0.16
+0
+0
+0.3
+0
+0
+0
+0.18
+0.68
+0.14
+0
+3
+0.2
+- 0.1
+0
+0
+0
+0
+0.21
+0.66
+0.12
+4
+- 0.0
+1
+1
+1
+1
+0
+1
+2
+3
+4
+5
+6
+Predicted M0.6
+0
+0
+0
+0
+0
+0
+0
+0
+0.5
+0
+0.06
+0.57
+0.36
+0.01
+0
+0
+- 0.4
+True M
+0
+0
+0.07
+0.61
+0.32
+0
+0
+0.3
+0.2
+0
+0
+0
+0.08
+0.63
+0.29
+0
+3
+- 0.1
+0
+0
+0
+0
+0.09
+0.65
+0.26
+4
+- 0.0
+1
+1
+1
+1
+0
+1
+2
+3
+4
+5
+6
+Predicted M0
+0
+0
+0
+0
+0
+0.8
+0
+0.7
+0
+0.83
+0.17
+0.01
+0
+0
+-0.6
+True M
+-0.5
+0
+0
+0.85
+0.15
+0
+0
+2
+- 0.4
+0.3
+0
+0
+0.01
+0.89
+0.1
+0
+- 0.2
+0
+0
+0
+0.01
+0.89
+0.1
+- 0.1
+4
+- 0.0
+1
+1
+1
+1
+1
+0
+1
+2
+3
+4
+5
+Predicted M12
+Figure 11: Loss weight update factor impact on probability
+levels obtained in the DAFC-based NN’s spatial spectrum,
+single target at θ1 = 0.55◦ with interference at θc = θ1 +∆θc,
+SIR = −5 dB. ˆp1 represents the probability obtained for
+source-containing grid points. ˆp0 represents the probability
+obtained for non-source grid points.
+problem. The DAFC mechanism applies a structured transfor-
+mation capable of exploiting the interference non-Gaussianity
+for its mitigation while retaining a low complexity of learnable
+parameters. The proposed DAFC-based NN approach is opti-
+mized to provide an interference-mitigated spatial spectrum
+using a loss weight scheduling routine, performing DOA
+estimation and source enumeration using a unified NN.
+The performance of the proposed approach is compared to
+MVDR, CNN-based, and FC-based approaches. Simulations
+showed the superiority of the proposed DAFC-based NN ap-
+proach in terms of probability of resolution and estimation ac-
+curacy, evaluated by RMSD, especially in weak signal power,
+small number of snapshots, and near-interference scenarios.
+The source enumeration performance of the proposed DAFC-
+based NN approach was compared to the MDL and AIC. It was
+shown that in the considered scenarios, the proposed approach
+outperforms the MDL and the AIC in the source enumeration
+accuracy.
+REFERENCES
+[1] H. L. Van Trees, Optimum Array Processing: Part IV of Detection,
+Estimation, and Modulation Theory.
+John Wiley & Sons, 2004.
+[2] E. Ollila, D. E. Tyler, V. Koivunen, and H. V. Poor, “Complex elliptically
+symmetric distributions: Survey, new results and applications,” IEEE
+Transactions on signal processing, vol. 60, no. 11, pp. 5597–5625, 2012.
+[3] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,”
+Proceedings of the IEEE, vol. 57, no. 8, pp. 1408–1418, 1969.
+[4] R. Roy and T. Kailath, “ESPRIT-estimation of signal parameters via ro-
+tational invariance techniques,” IEEE Transactions on Acoustics, Speech,
+and Signal Processing, vol. 37, no. 7, pp. 984–995, 1989.
+[5] R. Schmidt, “Multiple emitter location and signal parameter estimation,”
+IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp.
+276–280, 1986.
+[6] A. Barabell, “Improving the resolution performance of eigenstructure-
+based direction-finding algorithms,” in ICASSP’83. IEEE International
+Conference on Acoustics, Speech, and Signal Processing, vol. 8. IEEE,
+1983, pp. 336–339.
+[7] O. Besson, Y. Abramovich, and B. Johnson, “Direction-of-Arrival es-
+timation in a mixture of K-distributed and Gaussian noise,” Signal
+Processing, vol. 128, pp. 512–520, 2016.
+[8] U. K. Singh, R. Mitra, V. Bhatia, and A. K. Mishra, “Kernel minimum
+error entropy based estimator for mimo radar in non-Gaussian clutter,”
+IEEE Access, vol. 9, pp. 125 320–125 330, 2021.
+[9] E. Ollila and V. Koivunen, “Influence function and asymptotic efficiency
+of scatter matrix based array processors: Case MVDR beamformer,”
+IEEE Transactions on Signal Processing, vol. 57, no. 1, pp. 247–259,
+2008.
+[10] S. Fortunati, F. Gini, M. S. Greco, A. M. Zoubir, and M. Rangaswamy,
+“Semiparametric CRB and Slepian-Bangs formulas for complex ellipti-
+cally symmetric distributions,” IEEE Transactions on Signal Processing,
+vol. 67, no. 20, pp. 5352–5364, 2019.
+[11] S. Luan, M. Zhao, Y. Gao, Z. Zhang, and T. Qiu, “Generalized
+covariance for non-Gaussian signal processing and GC-MUSIC under
+Alpha-stable distributed noise,” Digital Signal Processing, vol. 110, p.
+102923, 2021.
+[12] K. Todros and A. O. Hero, “Robust multiple signal classification
+via probability measure transformation,” IEEE Transactions on Signal
+Processing, vol. 63, no. 5, pp. 1156–1170, 2015.
+[13] N. Yazdi and K. Todros, “Measure-transformed MVDR beamforming,”
+IEEE Signal Processing Letters, vol. 27, pp. 1959–1963, 2020.
+[14] X. Zhang, M. N. El Korso, and M. Pesavento, “Maximum likelihood and
+maximum a posteriori Direction-of-Arrival estimation in the presence
+of SIRP noise,” in 2016 IEEE International Conference on Acoustics,
+Speech and Signal Processing (ICASSP).
+IEEE, 2016, pp. 3081–3085.
+[15] ——, “MIMO radar target localization and performance evaluation
+under SIRP clutter,” Signal Processing, vol. 130, pp. 217–232, 2017.
+[16] B. Meriaux, X. Zhang, M. N. El Korso, and M. Pesavento, “Iterative
+marginal maximum likelihood DOD and DOA estimation for MIMO
+radar in the presence of SIRP clutter,” Signal Processing, vol. 155, pp.
+384–390, 2019.
+[17] M. Trinh-Hoang, M. N. El Korso, and M. Pesavento, “A partially-
+relaxed robust DOA estimator under non-Gaussian low-rank interference
+and noise,” in ICASSP 2021-2021 IEEE International Conference on
+Acoustics, Speech and Signal Processing (ICASSP).
+IEEE, 2021, pp.
+4365–4369.
+[18] J. Dai and H. C. So, “Sparse Bayesian learning approach for outlier-
+resistant direction-of-arrival estimation,” IEEE Transactions on Signal
+Processing, vol. 66, no. 3, pp. 744–756, 2017.
+[19] M. Wax and T. Kailath, “Detection of signals by information theoretic
+criteria,” IEEE Transactions on Acoustics, Speech, and Signal Process-
+ing, vol. 33, no. 2, pp. 387–392, 1985.
+[20] J. Fuchs, M. Gardill, M. L¨ubke, A. Dubey, and F. Lurz, “A machine
+learning perspective on automotive radar direction of arrival estimation,”
+IEEE Access, 2022.
+[21] J. Fuchs, R. Weigel, and M. Gardill, “Single-snapshot direction-of-arrival
+estimation of multiple targets using a multi-layer perceptron,” in 2019
+IEEE MTT-S International Conference on Microwaves for Intelligent
+Mobility (ICMIM).
+IEEE, 2019, pp. 1–4.
+[22] A. H. El Zooghby, C. G. Christodoulou, and M. Georgiopoulos,
+“Performance of radial-basis function networks for direction of arrival
+estimation with antenna arrays,” IEEE Transactions on Antennas and
+Propagation, vol. 45, no. 11, pp. 1611–1617, 1997.
+[23] B. Milovanovic, M. Agatonovic, Z. Stankovic, N. Doncov, and
+M. Sarevska, “Application of neural networks in spatial signal process-
+ing,” in 11th Symposium on Neural Network Applications in Electrical
+Engineering.
+IEEE, 2012, pp. 5–14.
+[24] G. Ofek, J. Tabrikian, and M. Aladjem, “A modular neural network for
+direction-of-arrival estimation of two sources,” Neurocomputing, vol. 74,
+no. 17, pp. 3092–3102, 2011.
+[25] A. Barthelme and W. Utschick, “A machine learning approach to DoA
+estimation and model order selection for antenna arrays with subarray
+sampling,” IEEE Transactions on Signal Processing, vol. 69, pp. 3075–
+3087, 2021.
+[26] O. Bialer, N. Garnett, and T. Tirer, “Performance advantages of deep
+neural networks for angle of arrival estimation,” in ICASSP 2019-
+2019 IEEE International Conference on Acoustics, Speech and Signal
+Processing (ICASSP).
+IEEE, 2019, pp. 3907–3911.
+[27] J. Cong, X. Wang, M. Huang, and L. Wan, “Robust DOA estimation
+method for MIMO radar via deep neural networks,” IEEE Sensors
+Journal, vol. 21, no. 6, pp. 7498–7507, 2020.
+[28] M. Gardill, J. Fuchs, C. Frank, and R. Weigel, “A multi-layer perceptron
+applied to number of target indication for direction-of-arrival estimation
+in automotive radar sensors,” in 2018 IEEE 28th International Workshop
+on Machine Learning for Signal Processing (MLSP).
+IEEE, 2018, pp.
+1–6.
+[29] J. Fuchs, R. Weigel, and M. Gardill, “Model order estimation using a
+multi-layer perceptron for direction-of-arrival estimation in automotive
+
+0.8
+Bo, p1
+B,P1
+0.6
+B1,P1
+Bo, Po
+B, po
+0.4
+B1, po
+0.2
+5
+10
+15
+20
+25
+30
+△c [Deg]13
+radar sensors,” in 2019 IEEE Topical Conference on Wireless Sensors
+and Sensor Networks (WiSNet).
+IEEE, 2019, pp. 1–3.
+[30] J. Rogers, J. E. Ball, and A. C. Gurbuz, “Estimating the number of
+sources via deep learning,” in 2019 IEEE Radar Conference (Radar-
+Conf).
+IEEE, 2019, pp. 1–5.
+[31] ——, “Robust estimation of the number of coherent radar signal sources
+using deep learning,” IET Radar, Sonar & Navigation, vol. 15, no. 5,
+pp. 431–440, 2021.
+[32] Z.-M. Liu, C. Zhang, and S. Y. Philip, “Direction-of-arrival estimation
+based on deep neural networks with robustness to array imperfections,”
+IEEE Transactions on Antennas and Propagation, vol. 66, no. 12, pp.
+7315–7327, 2018.
+[33] E. Ozanich, P. Gerstoft, and H. Niu, “A deep network for single-snapshot
+direction of arrival estimation,” in 2019 IEEE 29th International Work-
+shop on Machine Learning for Signal Processing (MLSP). IEEE, 2019,
+pp. 1–6.
+[34] M. Gall, M. Gardill, T. Horn, and J. Fuchs, “Spectrum-based single-
+snapshot super-resolution direction-of-arrival estimation using deep
+learning,” in 2020 German Microwave Conference (GeMiC).
+IEEE,
+2020, pp. 184–187.
+[35] M. Gall, M. Gardill, J. Fuchs, and T. Horn, “Learning representa-
+tions for neural networks applied to spectrum-based direction-of-arrival
+estimation for automotive radar,” in 2020 IEEE/MTT-S International
+Microwave Symposium (IMS).
+IEEE, 2020, pp. 1031–1034.
+[36] A. M. Ahmed, O. Eissa, and A. Sezgin, “Deep autoencoders for DOA
+estimation of coherent sources using imperfect antenna array,” in 2020
+Third International Workshop on Mobile Terahertz Systems (IWMTS).
+IEEE, 2020, pp. 1–5.
+[37] G. K. Papageorgiou and M. Sellathurai, “Direction-of-arrival estimation
+in the low-SNR regime via a denoising autoencoder,” in 2020 IEEE
+21st International Workshop on Signal Processing Advances in Wireless
+Communications (SPAWC).
+IEEE, 2020, pp. 1–5.
+[38] G. K. Papageorgiou, M. Sellathurai, and Y. C. Eldar, “Deep networks
+for direction-of-arrival estimation in low SNR,” IEEE Transactions on
+Signal Processing, vol. 69, pp. 3714–3729, 2021.
+[39] E. Ozanich, P. Gerstoft, and H. Niu, “A feedforward neural network for
+direction-of-arrival estimation,” The journal of the acoustical society of
+America, vol. 147, no. 3, pp. 2035–2048, 2020.
+[40] Y. Yao, H. Lei, and W. He, “A-CRNN-based method for coherent DOA
+estimation with unknown source number,” Sensors, vol. 20, no. 8, p.
+2296, 2020.
+[41] A. Barthelme and W. Utschick, “DoA estimation using neural network-
+based covariance matrix reconstruction,” IEEE Signal Processing Let-
+ters, vol. 28, pp. 783–787, 2021.
+[42] A. M. Ahmed, A. A. Ahmad, S. Fortunati, A. Sezgin, M. S. Greco,
+and F. Gini, “A reinforcement learning based approach for multitarget
+detection in massive MIMO radar,” IEEE Transactions on Aerospace
+and Electronic Systems, vol. 57, no. 5, pp. 2622–2636, 2021.
+[43] D. Luo, Z. Ye, B. Si, and J. Zhu, “Deep MIMO radar target detector in
+Gaussian clutter,” IET Radar, Sonar & Navigation, 2022.
+[44] W. Guo, T. Qiu, H. Tang, and W. Zhang, “Performance of RBF neural
+networks for array processing in impulsive noise environment,” Digital
+Signal Processing, vol. 18, no. 2, pp. 168–178, 2008.
+[45] D. Chen and Y. H. Joo, “A novel approach to 3D-DOA estimation of
+stationary EM signals using convolutional neural networks,” Sensors,
+vol. 20, no. 10, p. 2761, 2020.
+[46] ——, “Multisource DOA estimation in impulsive noise environments
+using convolutional neural networks,” International Journal of Antennas
+and Propagation, vol. 2022, 2022.
+[47] P. Stoica and A. Nehorai, “Performance study of conditional and
+unconditional direction-of-arrival estimation,” IEEE Transactions on
+Acoustics, Speech, and Signal Processing, vol. 38, no. 10, pp. 1783–
+1795, 1990.
+[48] M. Viberg, P. Stoica, and B. Ottersten, “Maximum likelihood array pro-
+cessing in spatially correlated noise fields using parameterized signals,”
+IEEE Transactions on Signal Processing, vol. 45, no. 4, pp. 996–1004,
+1997.
+[49] S. Feintuch, H. Permuter, I. Bilik, and J. Tabrikian, “Neural network-
+based multi-target detection within correlated heavy-tailed clutter,” Sub-
+mitted to IEEE Transactions on Aerospace and Electronic Systems, 2022.
+[50] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning. MIT Press,
+2016, http://www.deeplearningbook.org.
+[51] S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning:
+From theory to algorithms.
+Cambridge university press, 2014.
+[52] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”
+arXiv preprint arXiv:1412.6980, 2014. [Online]. Available: https:
+//doi.org/10.48550/arXiv.1412.6980
+[53] K. Harmanci, J. Tabrikian, and J. L. Krolik, “Relationships between
+adaptive minimum variance beamforming and optimal source localiza-
+tion,” IEEE Transactions on Signal Processing, vol. 48, no. 1, pp. 1–12,
+2000.
+

7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:da21566038279c42bc5fc2fc8e971fb2cfc236e570f67072d17d0a2823356813
+size 1319299

7tE1T4oBgHgl3EQfTwM_/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5fd418cf89e2447be5f7c2bc821ed25b12c9f4ecba15aaf22baea471d79339af
+size 3538989

7tE1T4oBgHgl3EQfTwM_/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7037249fb25803f93575f4db98dac81dac606cf1ec4debe20836856da82412f8
+size 133538

8NE2T4oBgHgl3EQfPgby/content/tmp_files/2301.03761v1.pdf.txt

@@ -0,0 +1,1718 @@
+1
+Tensor Denoising via Amplification and Stable
+Rank Methods
+Jonathan Gryak1, Kayvan Najarian2,3,4,5,6, and Harm Derksen7
+Abstract—Tensors in the form of multilinear arrays are ubiq-
+uitous in data science applications. Captured real-world data,
+including video, hyperspectral images, and discretized physical
+systems, naturally occur as tensors and often come with attendant
+noise. Under the additive noise model and with the assumption
+that the underlying clean tensor has low rank, many denoising
+methods have been created that utilize tensor decomposition
+to effect denoising through low rank tensor approximation.
+However, all such decomposition methods require estimating the
+tensor rank, or related measures such as the tensor spectral and
+nuclear norms, all of which are NP-hard problems.
+In this work we adapt the previously developed framework of
+tensor amplification, which provides good approximations of the
+spectral and nuclear tensor norms, to denoising synthetic tensors
+of various sizes, ranks, and noise levels, along with real-world
+tensors derived from physiological signals. We also introduce de-
+noising methods based on two variations of rank estimates called
+stable X-rank and stable slice rank. The experimental results
+show that in the low rank context, tensor-based amplification
+provides comparable denoising performance in high signal-to-
+noise ratio (SNR) settings and superior performance in noisy
+(i.e., low SNR) settings, while the stable X-rank method achieves
+superior denoising performance on the physiological signal data.
+Index Terms—Tensors, Denoising, Tensor Amplification, Stable
+Rank Methods
+I. INTRODUCTION
+T
+ENSORS in the form of multilinear arrays are ubiquitous
+in data science applications. Captured real-world data,
+including color and hyperspectral images (HSIs), video, and
+discretized physical systems, naturally occur as tensors and
+often come with attendant noise. As is common in other
+signal processing applications, the captured tensor T
+∈
+Rp1×p2×···×pd is modeled as T = D + N, where D is a pure
+or “clean” tensor D that has been corrupted by additive noise
+D, which is typically assumed to be Gaussian. Additionally,
+the clean tensor D is assumed to be low rank.
+Under this framework, tensor denoising can be achieved by
+utilizing tensor decompositions methods, such as the canonical
+1Department of Computer Science, Queens College, City University of New
+York, New York, NY, USA
+2Department of Computational Medicine and Bioinformatics, University of
+Michigan, Ann Arbor, MI, USA
+3Department of Emergency Medicine, University of Michigan, Ann Arbor,
+MI, USA
+4Electrical and Computer Engineering, College of Engineering, University
+of Michigan, Ann Arbor, MI, USA
+5Michigan Institute for Data Science, University of Michigan, Ann Arbor,
+MI, USA
+6Max Harry Weil Institute for Critical Care Research and Innovation,
+University of Michigan, Ann Arbor, MI, USA
+7Department of Mathematics, Northeastern University, Boston, MA, USA
+polyadic (CP) [1], [2] and Tucker [3], [4] decompositions, to
+determine a low-rank approximation of the observed tensor.
+These decomposition algorithms require a pre-specified rank
+to compute an approximation, however, determining the rank
+of a tensor is NP-hard [5]. Thus, tensor decomposition-based
+methods utilize some estimate of the tensor rank to effect
+tensor denoising.
+CP decomposition has been frequently used for HSI de-
+noising, such as in Liu et al., [6], which estimated the tensor
+rank using covariance matrices of the n-model flattenings; in
+Veganzones et al. [7], which used a non-negative variant of
+CP decomposition; and in [8], in which a CP decomposition
+regularized by the nuclear norm of clustered 3D patches of
+the HSI was employed. Tucker decomposition based denoising
+include two works by Rajwade et al. that utilized higher order
+singular value decomposition [9], a tensor analog of matrix
+SVD, to denoise video [10] and images [11]; as well Lee
+et al. [12], which focused on denoising tensors with ordinal
+values. More recently, a tensor train (matrix product state)
+decomposition [13] was used for denoising of HSIs [14].
+A general framework for understanding tensor denoising in
+the additive model was developed in [15], that relates the
+problem of denoising in the low rank context to the mini-
+mization of dual norms ∥D∥X and ∥N∥Y , such as the nuclear
+∥·∥⋆ and spectral ∥·∥σ norms, respectively. The calculation
+of these norms for tensors is also NP-hard [16],[5], thus in
+order make use of the denoising framework in [15] the co-
+authors developed the method of tensor amplification [17],
+which provides good approximations of the tensor spectral
+norm and its dual the nuclear norm.
+In this work, we utilize the general framework of [15] and
+the approximations of the spectral norm [17] to devise three
+novel tensor denoising methods - amplification-based, stable
+slice rank, and stable X-rank denoising, the latter two methods
+based on their eponymous rank estimates. The performance of
+these methods is compared to several standard decomposition-
+based denoising methods on synthetic tensors of various sizes,
+ranks, and noise levels, along with real-world tensors derived
+from physiological signals. The experimental results show that
+in the low rank context, tensor-based amplification provides
+comparable denoising performance in high signal-to-noise
+ratio (SNR) settings and superior performance in noisy (i.e.,
+low SNR) settings, while the stable X-rank method achieves
+superior denoising performance on the physiological signal
+data.
+arXiv:2301.03761v1  [cs.LG]  10 Jan 2023
+
+2
+II. PRELIMINARIES AND RELATED WORK
+A. Basic Notation
+Let T ∈ Rp1×p2×···×pd denote a real-valued tensor of order
+d. In the denoising experiments that are performed in this study
+we will assume that the tensor T is the noisy version of a
+pure tensor D ∈ Rp1×p2×···×pd corrupted by additive noise
+N ∈ Rp1×p2×···×pd, that is,
+T = D + N.
+(1)
+The Frobenius norm of T is denoted ∥T ∥ and defined as
+∥T ∥ =
+�
+�
+�
+�
+p1
+�
+i1
+p2
+�
+i2
+· · ·
+pd
+�
+id
+t2
+i1i2···id,
+(2)
+while the tensor inner product of two tensors T , S of matching
+order and dimension is defined as
+⟨T , S⟩ =
+p1
+�
+i1
+p2
+�
+i2
+· · ·
+pd
+�
+id
+ti1i2···idsi1i2···id.
+(3)
+The induced norm of the tensor inner product is the Frobenius
+norm defined above, with the typical relation ⟨T , T ⟩ = ∥T ∥2.
+Given a tensor T and a permutation q = ⟨q1, . . . , qd⟩ of
+the indices 1 : d, the q-transpose of T is the tensor T ⟨q⟩ ∈
+Rpq1×pq2×···×pqd with entries
+(T ⟨q⟩)i1i2...id = tiq1iq2...iqd .
+(4)
+At times we will need to matricize the tensors under con-
+sideration by rearranging their entries in specific ways, as well
+as employ various tensor-tensor, tensor-matrix, and matrix-
+matrix products. In the definitions below and throughout the
+manuscript we will primarily follow the notational conventions
+introduced by Kolda and Bader in [18].
+The mode-n flattening or unfolding of the tensor T is the
+matrix T(n) ∈ Rpn×N/pn, where N = �
+i pi, whose columns
+are the mode-n fibers of T .
+The n-mode product of a tensor T and a matrix A ∈ RJ×pn
+is the tensor T ×n A of size p1 ×p2 ×· · ·×pn−1 ×J ×pn+1 ×
+· · · × pd with entries
+(T ×n A)i1i2...in−1jin+1...id =
+pn
+�
+in=1
+ti1i2...idujin.
+(5)
+If S = T ×n A, then the n-mode product as defined above is
+equivalent to S(n) = AT(n).
+The Kronecker product of two matrices A ∈ RI×J and
+B ∈ RK×L is the matrix A ⊗ B ∈ RIK×JL defined by
+A ⊗ B =
+�
+����
+a11B
+a12B
+· · ·
+a1JB
+a21B
+a22B
+· · ·
+a2JB
+...
+...
+...
+...
+aI1B
+aI2B
+· · ·
+aIJB
+�
+���� .
+(6)
+Finally, given two tensors T
+∈ Rp1×···×pd and S
+∈
+Rq1×···×qe, their outer product is the tensor T ◦ S of size
+p1 × · · · × pd × q1 × · · · × qe with entries
+(T ◦ S)i1i2...idj1j2...je = ti1ti2 . . . tidsj1sj2 . . . sje.
+(7)
+B. Decomposition-based Denoising
+Tensor decomposition methods seek to represent a given
+tensor by decomposing it into factors such as simple tensors or
+matrices and whose combination results in a “good” approx-
+imation of the original tensor. In the context of denoising,
+it is typical to assume that a noisy signal is sparse, in the
+sense that its ℓ1 norm is small. In the case of matrices and
+tensors, this assumption corresponds to the original tensor
+having low rank, with the high rank components corresponding
+to additive noise. Thus, computing a low rank approximation
+of the original tensor via tensor decomposition is a means to
+effect tensor denoising.
+In the case of matrices (order two tensors), singular value
+decomposition yields the exact rank r of the matrix and its
+decomposition into r factor, with the best low rank approxi-
+mation for a given rank l < r provided by choosing the factors
+corresponding to the l largest singular values. For higher order
+tensors, calculating the exact rank is NP hard [5]. Moreover,
+unlike the matrix case, the factors used to create the best rank
+r − 1 approximation need not be those used to produce the
+best rank r approximation [19], and for degenerate tensors,
+the best rank r approximation may not even exist [20].
+Despite these theoretical limitations, in practice one can
+utilize tensor decomposition methods to effect denoising by
+creating decompositions for a range of rank values, then choos-
+ing the best rank r decomposition D that best approximates
+the original tensor T , e.g., min ∥T − D∥. This strategy for
+tensor denoising was evaluated using three common tensor
+decomposition methods: canonical polyadic decomposition,
+higher-order orthogonal iteration, and multiway Wiener filters.
+1) CP Decomposition via Alternating Least Squares (CP-
+ALS): Let U (j) = [uj,1uj,2 . . . uj,r] ∈ Rpj×r, 1 ≤ j ≤ d. CP
+decomposition factorizes a d-way tensor into d factor matrices
+and a vector Λ = [λ1, λ2, . . . , λr] ∈ Rr:
+S =
+r
+�
+i=1
+λiu1,i ◦ u2,i ◦ . . . ◦ ud,i.
+(8)
+The best rank r approximation problem for a tensor T
+∈
+Rp1×p2×...×pd can be given as:
+min
+Λ,U (1),...,U (d) ∥T − S∥ where S = [Λ ; U (1), U (2), . . . , U (d)].
+(9)
+This can be found by employing alternating least squares
+(ALS), wherein each iteration of the algorithm an approxima-
+tion of the flattening for one mode is found by fixing all other
+modes of the tensors and solving a least squares problem.
+This process is repeated, cycling through all modes, until
+convergence or a maximum number of iterations is reached. In
+this work, the implementation of CP-ALS from TensorToolbox
+[21] was utilized with the default level of tolerance (10−4)
+and maximum number of iterations (50). CP-ALS was run
+for specified rank values r ∈ [1, min(pi)], with the rank r∗
+approximation
+Dr∗ = min
+r
+∥T − Dr∥
+(10)
+chosen as the best denoised tensor.
+
+3
+2) Higher-Order Orthogonal Iteration (HOOI): For matri-
+ces, orthogonal iteration produces a sequence orthonormal
+bases for each subspace of the vector space. De Lathauwer
+et al. [22] extended this to tensors, developing the technique
+known as higher-order orthogonal iteration (HOOI). This
+method uses ALS to estimate the best rank-[r1, . . . , rd] ap-
+proximation for a tensor, and is achieved by iteratively solving
+the optimization problem
+argminU(i)|ri ∥T − G ×1 U(1)|r1 ×2 U(2)|r2 × . . . ×N U(d)|rd∥,
+(11)
+where G is a core tensor of size r1 × . . . × rd and each U(i)|ri
+is a matrix comprised of the ri leftmost singular vectors of
+the singular value decomposition of the modal flattening U(i).
+HOOI-based
+denoising
+was
+implemented
+using
+the
+tucker_als method in TensorToolbox [21] to determine
+the best rank [r∗
+1, . . . , r∗
+d] approximation, where each ri was
+chosen equally and uniformly from r ∈ [1, min(pi)], with the
+rank r∗ approximation (Eq. 10) chosen as the best denoised
+tensor.
+3) Multiway Wiener Filter: For a discrete signal y[n] and
+filter output ˆy[n], the Wiener filter h[n] is the filter that
+minimizes the mean squared error between ˆy[n] and y[n]:
+argminh[·]E[(ˆy[n] − y[n])2)].
+(12)
+Wiener filters have been used in a variety of denoising appli-
+cations, such as for images [23], [24], physiological signals
+[25], [26] and speech [27], [28].
+Muti et al. [29] created a multiway Wiener filter that can be
+used to denoise tensors of arbitrary size. Given a noisy tensor
+T , their method uses an ALS approach to learn Wiener filters
+{Hn} for each mode n so that the mean squared error between
+T and the denoised tensor D is minimized, where
+D = T ×1 H1 ×2 H2 ×3 · · · ×d Hd.
+(13)
+The implementation of the multiway Wiener filter utilized in
+this study and the exposition below follows [30]. The filters
+Hn in each mode n are initialized to the identity matrix
+of Rpn. At each stage k of the algorithm, the filter Hk
+n is
+computed for each mode as
+Hk
+n = VnΛnV ⊺
+n ,
+(14)
+where Vn is a matrix containing the Kn orthonormal basis
+vectors of the signal subspace in the column space of T(n),
+the mode-n flattening of T , and
+Λn = diag
+�
+λγ
+1 − ˆσγ2
+n
+λΓ
+1
+, . . . , λγ
+Kn − ˆσγ2
+n
+λΓ
+Kn
+�
+,
+(15)
+where {λγ
+i , i = 1, . . . , Kn} and {λΓ
+i , i = 1, . . . , Kn} are
+respectively the Kn largest eigenvalues of the matrices γn and
+Γn, defined as
+γn =
+E
+�
+T(n)qnT(n)
+⊺�
+(16)
+Γn =
+E
+�
+T(n)QnT(n)
+⊺�
+(17)
+with
+qn
+=
+d
+�
+i̸=n
+Hi
+(18)
+Qn
+=
+d
+�
+i̸=n
+H⊺
+i Hi.
+(19)
+The values ˆσγ2
+n in Equation 15 are estimates of the pn − Kn
+smallest eigenvalues of γn, calculated as
+ˆσγ2
+n =
+1
+pn − Kn
+pn
+�
+i=Kn+1
+λγ
+i .
+(20)
+Following [31], the optimal Kn for mode n is estimated using
+the Akaike Information Criterion (AIC). Please refer to [30]
+for further details.
+III. AMPLIFICATION AND STABLE RANK DENOISING
+In this section we introduce three different denoising meth-
+ods – Amplification-based, Stable Slice Rank, and Stable X-
+Rank denoising – that leverage the general framework for
+denoising based on dual norms as introduced in Derksen [15],
+to effect denoising on tensors.
+A. A Framework for Denoising Using Dual Norms
+The model T = D +N utilized in this work can be viewed
+as an instance of the additive noise model c = a + b, where
+a, b, c ∈ V are elements of a vector space V . In Derksen
+[15], a general framework for understanding the denoising of
+vectors under the additive model was developed that relates the
+problem of denoising the vector c to the minimization of ∥a∥X
+and ∥b∥Y , where ∥ · ∥X and ∥ · ∥Y are dual norms. Moreover,
+the framework makes the assumptions that the original vector
+(or tensor) a is sparse, e.g., that it has few non-zero values or
+is of low rank, while the additive noise b is dense or of high
+rank. Thus, the norms ∥ · ∥X and ∥ · ∥Y can be interpreted as
+respectively measuring the sparsity and noise of the vector (or
+tensor) under consideration. The prototypical ∥ · ∥X norm is
+the nuclear norm, which for a matrix is the sum of its singular
+values, while for a tensor the tensor nuclear norm ∥T ∥⋆, is
+defined as
+∥T ∥⋆ = min
+r
+�
+i=1
+∥vi∥2,
+where v1, . . . , vr are rank-1 tensors and T = �r
+i=1 vi.
+The prototypical ∥·∥Y norm and dual to ∥·∥X is the spectral
+norm, which for a matrix is the absolute value of its largest
+singular value, while for a tensor the tensor spectral norm
+∥T ∥σ is defined as
+∥T ∥σ = sup |T · u1 ⊗ u2 ⊗ . . . ⊗ ud|,
+where uj ∈ Rpj and ∥uj∥ = 1 for 1 ≤ j ≤ d.
+If V is also an inner product space we also have the induced
+norm
+�
+⟨c, c⟩ that corresponds to the standard Euclidean norm
+∥c∥2 for vectors or the Frobenius norm ∥ · ∥, introduced in
+Section II-A, for matrices and tensors. As shown in [15],
+the denoising of a vector c via a decomposition c = a + b
+
+4
+that simultaneously minimizes the values ∥a∥X and ∥b∥Y is
+governed by the Pareto frontier, which models the competing
+objectives of minimizing the two norms in terms of Pareto
+efficiency, and the above XY -decomposition that achieves this
+is deemed Pareto efficient. Moreover, [15] defines the related
+notion of the Pareto subfrontier, which relates the three norms
+∥ · ∥X, ∥ · ∥Y , ∥ · ∥2 and their induced decompositions XY ,
+X2, and 2Y , describing the conditions under which these
+decompositions can achieve Pareto efficiency.
+B. Amplification-based Denoising
+To make use of the denoising framework introduced in
+[15] requires the calculations of various norms for the vec-
+tors of interest. While the Frobenius norm of a tensor is
+easily obtained, computing either the nuclear norm [16] or
+the spectral norm [5] for tensors is NP-hard. In order to
+obtain an approximation to the tensor spectral norm, the co-
+authors developed the methodology of tensor amplification
+[17]. For a matrix A with singular values λ1, . . . , λr, the
+function φ : A → AA⊺A produces the matrix AA⊺A whose
+singular values are λ3
+1, . . . , λ3
+r. Repeated applications of φ(·)
+will amplify the larger singular values, which correspond to
+the sparse or low rank components of the matrix, while
+minimizing smaller singular values that likely correspond to
+noise.
+Analogously, tensor amplification utilizes degree d polyno-
+mial functions on tensors to amplify the low rank structure.
+Moreover, for each amplification map Φσ′ there exists a cor-
+responding norm ∥·∥σ′,d that approximates the tensor spectral
+norm, in the sense that limd→∞ ∥T ∥σ′,d = ∥T ∥σ. Two such
+amplification maps – Φσ,4 and Φ# – were introduced for
+order 3 tensors in [17], with Φ# being show to be a better
+approximation to the tensor spectral norm than Φσ,4.
+The method presented in Algorithm 1 utilizes the 2Y -
+decomposition framework of [15] and the tensor spectral norm
+approximations Φ to denoise a given tensor T . The algorithm
+allows for the choice of amplification map as well as the
+number of amplifications per round. For third order tensors
+the amplification map Φ# was used, while for fourth order
+tensors a compatible version of Φσ,4 was employed as there
+is no currently known analogue of the Φ# map for fourth
+order tensors. Multiple experiments were performed with m,
+the number of amplifications per round, ranging from 1 to
+10, with m = 5 being found to produce the best denoising
+performance.
+C. Stable Slice Rank Denoising
+Slice rank was introduced in [32] in relation to the cap set
+problem. Following Tao [33], the slice rank of a tensor T
+is the least non-negative integer srk such that T is a sum
+of tensors with slice rank 1, i.e., T = �r
+i=1 Ti, where Ti is
+contained in the tensor product space
+V1 ◦ · · · Vi−1 ◦ s ◦ Vi−1 ◦ · · · Vd,
+(21)
+where Vj are vector spaces and s is a vector in some Vi. In
+[34], the notion of a stable rank for matrices was introduced,
+in which the matrix rank function rank(A), is replaced by the
+Algorithm 1 Amplification-based tensor denoising.
+D ← DENOISE AMPLIFICATION(T , Φ, m)
+ϵ ← ∥T ∥
+N ← T
+while true do
+A ← Φm(N)
+A ←
+A
+∥A∥
+N ← N − ⟨A, N⟩A
+Break if ∥N∥ < ϵ
+end while
+D ← T − N
+numerical rank function,
+∥A∥2
+∥A∥2σ , or the related stable nuclear
+rank
+∥A∥2
+⋆
+∥A∥2 . These ranks are stable in the sense that small
+perturbations of the values of the matrix A will not change
+their value. Extending this methodology to tensors, the stable
+slice rank is defined as
+�d
+i=1 ∥T(i)∥2
+⋆
+∥T ∥2
+,
+(22)
+where T(i) are the mode-i flattenings of T .
+For a given value of the parameter λ, the stable slice rank
+(SliceRank) method denoises a tensor by finding a decompo-
+sition T = D + N that minimizes the Frobenius norm of
+D = �
+i Si under the constraints that the nuclear norms of
+the flattenings of N are all ≤ λ. The method also produces
+a decomposition D = �
+i Si that minimizes the sum of the
+nuclear norms of S(i), the mode-i flattenings of Si. Typically,
+the S(i) have low rank.
+Algorithm 2 Stable SliceRank denoising.
+(D, {Si}, ssrk) ← DENOISE SLICERANK(T , λ, acc)
+Si ← 0 ∈ Rp1×p2×···×pd
+curr acc ← 0
+while curr acc < acc do
+for i ← 1 : d do
+A ← T − �
+j̸=i Sj
+q ← CIRCSHIFT([1, . . . , d], −(i − 1))
+A ← A⟨q⟩
+(U, D, V ) ← SVD(A(i))
+Ei ← MAX(D − λ, 0)
+ei ← DIAG(Ei)
+F ← U · Ei · V T
+F ← RESHAPE(F, pi, . . . , pd, p1, . . . , pi−1)
+q ← CIRCSHIFT([1, . . . , d], (i − 1))
+Si ← F⟨q⟩
+end for
+curr acc ←
+⟨T − �d
+j=1 Sj, �d
+j=1 Sj⟩
+λ �d
+j=1 ∥A(j)∥⋆
+end while
+D ← �d
+i=1 Si
+ssrk ←
+�d
+i=1 ∥A(i)∥2
+⋆
+∥D∥2
+Algorithm 2 depicts the implementation of SliceRank de-
+noising, which utilizes a number of auxiliary functions from
+
+5
+MATLAB [35]: circshift performs a cyclic permutation of
+an index set [1, . . . , d], with the second parameter determining
+the number of forward or backwards shifts; reshape is used
+to flatten a tensor into a matrix with the specified dimensions;
+and diag returns a vector comprising the entries on the
+main diagonal of the specified matrix. The algorithm takes
+as hyperparameters λ as described above and acc ∈ (0, 1],
+the specified accuracy level that once achieved the algorithm
+terminates. The algorithm returns the denoised tensor D, the
+decomposition factors Si, and ssrk, the stable slice rank of
+D. The hyperparameters were optimized via grid search over
+the ranges λ ∈ {10−2, 0.1, 1, 10} and acc ∈ {0.90, 0.95}.
+D. Stable X-Rank Denoising
+As noted in Section II-B, a degenerate tensor T of rank r
+may not have a best rank k < r approximation for a given
+rank k. In such cases, a tensor may be approximated to any
+desired precision by rank j < k tensors. This is due to the set
+of all tensors for a given rank r not being Zariski closed [20].
+In Derksen [36], the G-stable rank of a tensor was introduced
+that, among its other advantages, is Zariski closed. Thus, every
+tensor T has a best G-stable rkG < r approximation. The G-
+stable α rank of a tensor can be defined as
+rkG
+α (T ) = sup
+g∈G
+min
+i
+αi∥g · T ∥2
+∥ (g · T )(i) ∥2σ
+,
+(23)
+where α = (α1, . . . , αd) and g is an element of a reductive
+group G, i.e., g ∈ SL(Rp1) × · · · × SL(Rpd).
+Using the above definition we can define the related concept
+of stable X-rank, which is
+sXrkG(T ) = max
+α
+rkG
+α (T ),
+(24)
+where α is subject to the restriction that �
+i αi = d. Algorithm
+4 depicts the implementation of the stable X-Rank (XRank)
+denoising method. Like SliceRank, the method imposes a con-
+straint on the nuclear norm of the flattenings of N. However,
+in the XRank method, this cutoff is determined automatically
+using Algorithm 3. The hyperparameters were optimized via
+grid search over the ranges λ ∈ {10−2, 0.1, 1, 10} and acc ∈
+{0.90, 0.95}.
+IV. EXPERIMENTAL RESULTS AND DISCUSSION
+In order to evaluate the various denoising methods under
+consideration, two sets of synthetic tensors were generated
+with varying orders, ranks, and dimensions, resulting in 512
+parameter combinations. For each combination, one hundred
+(100) tensors were generated. For all synthetic tensors, varying
+amounts of noise were added from a standard Gaussian
+distribution N(0, 1), with the resulting noisy tensors having
+signal-to-noise ratios (SNR) ranging from 20 dB to −20 dB.
+The full range of parameters is provided in Table I.
+Additionally, two sets of tensors were extracted from elec-
+trocardiogram (ECG) signals to which Gaussian noise was
+added prior to tensor extraction, using the same range of
+resultant SNRs as those employed in the generation of the
+synthetic tensors.
+Algorithm 3 Determine the nuclear norm cutoff for XRank
+denoising.
+c ← FIND CUTOFF(f = [λ1, . . . , λr]⊺, λ)
+r ← |f|
+t ← 0 ∈ Rr
+for i ← 1 : r do
+ti ← λ
+�i
+j=1 λj
+1 + λ · i
+end for
+S ← 0 ∈ Rr×r
+for i ← 1 : r do
+for j ← 1 : r do
+sij ← MAX(fi − tj, 0)
+end for
+end for
+v ← 0 ∈ Rr
+for j ← 1 : r do
+vj ← �r
+i=1(fij − sij)2 + λ �r
+i=1(sij)2
+end for
+k ← ARGMIN(v)
+c ← tk
+Algorithm 4 Stable XRank denoising.
+(D, {Si}, sxrk) ← DENOISE XRANK(T , λ, acc)
+Si ← 0 ∈ Rp1×p2×···×pd
+curr acc ← 0
+while curr acc < acc do
+for i ← 1 : d do
+A ← T − �
+j̸=i Sj
+q ← CIRCSHIFT([1, . . . , d], −(i − 1))
+A ← A⟨q⟩
+(U, D, V ) ← SVD(A(i))
+c ← FIND CUTOFF(DIAG(D),λ)
+Ei ← MAX(D − c, 0)
+ei ← DIAG(Ei)
+F ← U · Ei · V T
+F ← RESHAPE(F, pi, . . . , pd, p1, . . . , pi−1)
+q ← CIRCSHIFT([1, . . . , d], (i − 1))
+Si ← F⟨q⟩
+end for
+Scurr ← �d
+i=1 Si
+T S = T − Scurr
+y ← 0
+for i ← 1 : d do
+q ← CIRCSHIFT([1, . . . , d], −(i − 1))
+B ← T S⟨q⟩
+(U, D, V ) ← SVD(B(i))
+y ← y + d2
+1
+end for
+y ← √y
+curr acc ←
+⟨T , Scurr⟩
+y ·
+��d
+j=1 ∥A(j)∥2⋆
+end while
+D �� �d
+i=1 Si
+sxrk ←
+�d
+j=1 ∥A(j)∥2
+⋆
+∥D∥2
+
+6
+TABLE I: Parameters and their respective values used to
+generate the synthetic tensor datasets.
+Parameter
+Range/Values
+Distribution
+Normal N(0, 1)
+Order
+3, 4
+Rank
+[1, 5], 10, 20, 25
+Size
+5, 10, 25, 50
+SNR
+20, 10, 5, 1, −1, −5, −10, −20
+1) Uniform Synthetic Tensors: In this dataset, the dimen-
+sions of a given tensor are chosen uniformly across each mode.
+To generate synthetic tensors from a distribution D of a given
+rank r, size s, and order d, scalar values λ1, . . . , λr are chosen
+from D, then for each mode j, r random vectors xj,i ∈ Rs
+are chosen from D. The synthetic tensor is then
+r
+�
+i=1
+λix1,i ◦ x2,i ◦ · · · ◦ xd,i.
+2) Non-Uniform Synthetic Tensors: In this dataset, one
+mode mk of a given tensor is “stretched” to a different
+dimension dk by choosing a number uniformly in the range
+dk = [s, min(500, sd)], i.e., the lower bound is the dimension
+of the other models while the upper bound is the product of
+the dimensions of each mode or 500, whichever is lower. After
+choosing the stretch mode and its dimension the tensors are
+generated in the same manner as for the uniform tensors above.
+3) ECG Waveform Tensors: The PTB Diagnostic ECG
+Database [37] is comprised of high resolution (1 kHz) digitized
+recordings of electrocardiograms (ECGs) from patients with
+various cardiovascular diseases, including myocardial infarc-
+tion, heart failure, and arrhythmia, as well as healthy controls.
+The database is publicly available via Physionet [38].
+Tensor-based methods have been shown to be effective for
+a number of ECG analytical tasks, a survey of such methods
+can be found in [39]. Given the utility of tensor-based methods
+in this context and that such that recordings of physiological
+signals may be corrupted by noise yields a natural application
+of the proposed denoising methods. In order to evaluate these
+methods, we first must construct tensors from the ECGs. In
+forming these tensors, one has to consider the amount of
+signal over which to perform subsequent signal processing and
+feature extraction: two methods were employed. In the first,
+ninety (90) seconds of a patient’s ECG recording was sampled
+across all twelve ECG leads, while in the second method three
+windowed samples of thirty (30) seconds each were extracted.
+Using these two sampling strategies we adapted the tensor
+formation method introduced in [40] that has been shown
+to be effective for subsequent applications of machine learn-
+ing for prognosticating severe cardiovascular conditions [41],
+[42]. In this method, each ECG signal is preprocessed using
+the taut string method, which produces a piecewise linear
+approximation of a given signal, parametrized by ϵ, which
+controls the coarseness of the approximation. Given a discrete
+signal x = (x1, . . . , xn) one can define the finite difference
+D(x) = (x2 − x1, . . . , xn − xn−1). For a fixed ϵ > 0, the
+taut string estimate of x is the unique function y such that
+∥x − y∥∞ ≤ ϵ and ∥D(y)∥2 is minimal. The taut string
+approximation can be found efficiently using the method in
+[43].
+After the taut string approximation for a given signal is
+found, six morphological and statistical features are extracted
+following [40]. This process is repeated for five values of
+epsilon: (0.0100, 0.1575, 0.3050, 0.4525, 0.6000). As each pa-
+tient’s ECG recording is comprised of the standard 12 leads,
+the approximation of each 90 second ECG sample via taut
+string and the extraction of taut string features yields third
+order tensors of size 5 × 6 × 12 for each patient. For the
+windowed samples, fourth order tensors were formed of size
+5 × 6 × 12 × 3, with the fourth mode corresponding to the
+features extracted in each window.
+4) Adding Noise: For every generated synthetic tensor, a
+set of noisy tensors was created by adding Gaussian noise
+(N(0, 1)) so that the resultant tensors had SNRs in the
+range [20, 10, 5, 1, −1, −5, −10, −20]. For the ECG waveform
+tensors, Gaussian noise was added to each ECG signal to
+produce a set of noisy signals with the same SNR range as
+for the synthetic tensors. However, this was performed prior
+to tensor formation given that in practical applications ECG
+signals themselves may come with some intrinsic amount of
+noise, rather than noise being introduced to the tensors directly.
+A. Results: Synthetic Data
+The overall denoising performance for third order tensors
+across various ranks and tensors sizes are presented in Table II.
+The performance statistics for only one order are presented due
+to the incomparable sizes of the non-uniform tensors across
+orders, please see Table IV in Appendix A for the fourth order
+4. The best performing denoising algorithms for uniformly
+sized tensors, as depicted in Table II (a), varied by noise level.
+For cleaner tensors (20 and 10 dB), the multiway Wiener
+filter performed best overall, achieving mean and standard
+deviations in denoised SNRs of 20.95 (13.19) and 20.22 (10.1)
+dBs, respectively. For moderately noisy tensors (5, 1, and
+−1 dB), ALS was the best performing denoising method,
+achieving denoised SNRs of 15.11 (8.26), 11.12 (7.98), and
+9.1 (7.88) dBs. For nosier tensors with starting SNRs of −5
+and −10, tensor amplification produced the best denoised
+SNRs of 3.37 (5.69) and −2.25 (4.57) dBs. Finally for tensors
+with starting SNRs of −20 dB, the noisiest tensors evaluated,
+XRank produced on average the highest denoised SNR of
+−9.22 (4.79) dB. The results for non-uniformly sized tensors,
+as depicted in Table II (b), are much clearer, with ALS
+achieving the best denoised SNRs across all starting SNRs,
+ranging from 30.81 (12.7) dBs for tenors with starting SNRs
+of 20 dB to −6.6 (8.57) dB for the noisiest tensors (starting
+SNRs of −20 dB).
+The relationship between tensor size (dimension of each
+mode) and achieved denoised SNRs is depicted in Figure 1.
+With the exception of HOOI, all other denoising algorithms
+see improvements in achieved SNR as the size of the tensor
+increases. The multiway Wiener filter maintains the best de-
+noising performance as size increases, followed by ALS. Both
+amplification and XRank have similar denoising performances,
+while slice rank and HOOI having the lowest performance
+overall.
+
+7
+TABLE II: Mean (SD) SNR, in decibels, after tensor denoising across all parameters.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+20
+19.57 (3.32)
+28.67 (11.1)
+29.05 (13.19)
+10.0 (14.13)
+18.79 (1.59)
+10.89 (5.64)
+10
+10.59 (1.12)
+19.91 (8.88)
+20.22 (10.1)
+8.65 (11.09)
+13.21 (2.21)
+9.84 (4.15)
+5
+5.81 (0.75)
+15.11 (8.26)
+14.94 (8.59)
+7.85 (9.64)
+8.18 (2.52)
+8.56 (3.13)
+1
+1.87 (0.7)
+11.12 (7.98)
+10.34 (7.64)
+7.01 (8.56)
+3.54 (2.25)
+6.91 (2.49)
+-1
+-0.12 (0.7)
+9.1 (7.88)
+7.96 (7.01)
+6.48 (8.1)
+1.18 (2.03)
+5.86 (2.36)
+-5
+-4.12 (0.69)
+4.99 (7.77)
+3.37 (5.69)
+5.17 (7.45)
+-3.46 (1.57)
+3.24 (2.6)
+-10
+-9.12 (0.69)
+-0.19 (7.65)
+-2.25 (4.57)
+2.85 (7.35)
+-9.02 (1.12)
+-0.58 (3.43)
+-20
+-19.11 (0.7)
+-10.38 (7.39)
+-11.17 (7.38)
+-11.79 (6.98)
+-19.61 (0.6)
+-9.22 (4.79)
+(a) Uniformly sized tensors.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+20
+23.91 (7.33)
+30.81 (12.7)
+29.46 (13.3)
+11.68 (14.81)
+18.7 (1.53)
+11.85 (5.83)
+10
+15.45 (4.71)
+22.75 (10.29)
+21.39 (10.79)
+10.33 (11.77)
+13.28 (2.11)
+10.75 (4.34)
+5
+10.99 (3.94)
+18.25 (9.55)
+16.28 (9.86)
+9.45 (10.34)
+8.26 (2.44)
+9.4 (3.3)
+1
+7.27 (3.67)
+14.49 (9.23)
+11.84 (9.03)
+8.25 (9.36)
+3.63 (2.19)
+7.68 (2.6)
+-1
+5.38 (3.63)
+12.57 (9.12)
+9.54 (8.41)
+7.32 (9.03)
+1.28 (2.01)
+6.6 (2.42)
+-5
+1.5 (3.64)
+8.66 (8.99)
+5.38 (6.92)
+4.89 (9.02)
+-3.37 (1.61)
+3.95 (2.56)
+-10
+-3.51 (3.61)
+3.65 (8.86)
+0.08 (5.37)
+1.65 (9.72)
+-8.92 (1.29)
+-0.13 (3.39)
+-20
+-13.59 (3.51)
+-6.6 (8.57)
+-7.94 (7.44)
+-11.74 (8.56)
+-19.56 (0.64)
+-9.01 (4.66)
+(b) Non-uniformly sized tensors.
+(a)
+(b)
+Fig. 1: Denoising performance with respect to tensor size.
+The relationship between tensor rank and achieved de-
+noised SNRs is depicted in Figure 2. Tensor amplification
+achieves the best rank-1 performance for both uniformly
+and non-uniformly sized tensors, followed by the multiway
+Wiener filter. The denoising performance of both methods
+decreases as tensor rank increases, with the multiway Wiener
+filter maintaining denoising performance for higher ranks
+than amplification. ALS has lower performance at low ranks
+but generally maintains its denoising performance as rank
+increases, ultimately achieving the best results by rank 20.
+XRank has greater performance than SliceRank for uniformly
+sized tensors, but their denoising performances converge prior
+to rank 20 for non-uniformly sized ones. HOOI has the
+lowest denoising performance for uniformly sized tensors, but
+performs slightly better than the amplification and SliceRank
+methods for non-uniformly sized tensors with ranks greater
+than 5.
+The results depicted in Table III provide a further investiga-
+tion into the denoising performance for low rank (ranks 1 and
+2) and high noise (SNRs ≤ 1) tensors. From these results one
+can observe that amplification achieves the best performance
+for uniformly sized tensors, with the multiway Wiener filter
+achieving the second-best denoising performance. In the case
+of non-uniformly sized tensors, the Wiener filter and amplifi-
+cation achieve comparable results for starting SNRs of 1 and
+−1 dB, while amplification achieving better performance for
+starting SNRs of −5,−10, and −20 dB.
+B. Results: Real Data - ECG Waveform Tensors
+Figure 3 shows the denoising performance on the tensors
+derived from the PTB dataset. For the tensors derived from
+90 second samples of ECG signal (Figure 3 a), only the
+stable rank methods (XRank and SliceRank) were able to
+achieve any effective denoising, with XRank achieving the
+best denoising performance with a modest ≈ 4 dB denoised
+SNR for tensors whose signals had SNR ratios of 20 dB
+prior to tensor formation. This performance was maintained
+for tensors from signals with starting SNRs down to 5 dB,
+after which the denoising performance of XRank declines. The
+SliceRank method does not yield any tensor denoising until the
+starting signal SNR dropped below −5 dB, after which it too
+experiences a continued decline in denoising performance. All
+
+Uniform Synthetic Tensors
+1OOH
+14 -
+ALS
+Wiener
+12 -
+Amp
+Denoised SNR (dB)
+10
+SliceRank
+XRank
+8
+6 ·
+4
+2 -
+5
+10
+25
+50
+SizeNon-uniform Synthetic Tensors
+16
+IOOH
+ALS
+14 -
+Wiener
+12
+Amp
+Denoised SNR (dB)
+SliceRank
+10
+XRank
+8
+6 -
+4 -
+2 
+01
+5
+10
+25
+50
+Size8
+(a)
+(b)
+Fig. 2: Denoising performance with respect to tensor rank.
+TABLE III: Mean (SD) denoised SNR, in decibels, for low rank and noisy tensors.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+1
+1.59 (0.36)
+5.97 (2.93)
+12.21 (4.05)
+14.98 (7.72)
+5.64 (2.29)
+8.89 (1.91)
+-1
+-0.42 (0.35)
+3.92 (2.92)
+10.31 (3.85)
+13.66 (7.28)
+3.33 (2.11)
+7.34 (2.01)
+-5
+-4.43 (0.32)
+-0.17 (2.91)
+6.42 (3.78)
+10.68 (6.90)
+-1.52 (1.70)
+3.77 (2.34)
+-10
+-9.45 (0.30)
+-5.24 (2.93)
+1.90 (3.84)
+6.16 (7.47)
+-7.49 (1.39)
+-0.79 (2.88)
+-20
+-19.45 (0.30)
+-15.29 (2.91)
+-8.86 (5.90)
+-4.36 (7.84)
+-18.79 (1.03)
+-10.31 (3.48)
+(a) Uniformly sized tensors.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+1
+4.59 (1.82)
+8.32 (4.32)
+16.96 (8.56)
+16.15 (8.84)
+5.85 (2.30)
+9.68 (1.96)
+-1
+2.58 (1.82)
+6.25 (4.30)
+14.99 (8.41)
+14.88 (8.34)
+3.51 (2.24)
+8.25 (2.03)
+-5
+-1.46 (1.84)
+2.12 (4.29)
+10.77 (8.21)
+12.06 (7.72)
+-1.25 (2.06)
+4.89 (2.39)
+-10
+-6.50 (1.87)
+-2.98 (4.31)
+4.07 (6.49)
+7.84 (7.93)
+-7.19 (1.87)
+0.21 (3.09)
+-20
+-16.52 (1.87)
+-13.05 (4.31)
+-7.18 (6.61)
+-1.79 (9.00)
+-18.86 (0.84)
+-9.05 (3.87)
+(b) Non-uniformly sized tensors.
+other methods - HOOI, ALS, and amplification - introduced
+noise into the tensors across all starting signal SNRs. No
+appreciable difference was observed in tensors derived from
+the two patient cohorts (healthy and unhealthy). The denoising
+results for the tensors derived from windowed samples (Figure
+3 b) are essentially the same as those derived from the
+90 second samples, with the only exception being a slight
+increase in SliceRank’s denoising performance for tensors
+corresponding to unhealthy patients with a starting signal SNR
+of −5 dB.
+
+Uniform Synthetic Tensors
+1OOH
+17.5
+ALS
+Wiener
+15.0
+Amp
+Denoised SNR (dB)
+SliceRank
+12.5
+XRank
+10.0
+7.5
+5.0
+2.5
+0.0
+10
+20
+25
+RankNon-uniform Synthetic Tensors
+IOOH
+20
+ALS
+Wiener
+Amp
+Denoised SNR (dB)
+15
+SliceRank
+XRank
+10
+5
+0
+10
+20
+25
+4.
+5
+Rank9
+(a)
+(b)
+Fig. 3: Denoising performance on the PTB tensors formed
+from a) 90 second samples, and b) windowed samples.
+C. Discussion
+Overall alternating least squares (ALS) was the best per-
+forming method for denoising synthetic tensors across all
+tensor orders, sizes, ranks, and starting noise levels, with the
+multiway Wiener filter (MWF) also performing well across
+all parameters. Amplification-based denoising performed well
+for low ranked tensors as well as very noisy (< 0 dB)
+tensors. The performance of amplification at low ranks and
+its decreased performance at higher ranks is to be expected,
+as the amplification maps correspond to approximations of the
+spectral norm, which only measures the highest singular value
+for a given tensor. Amplification-based denoising for higher
+rank tensors may be improved through the development of
+a decomposition method that can find successively smaller
+singular values and their corresponding rank 1 components,
+such as through a gradient-based descent optimization method.
+For the tensors derived from physiological signals, only the
+XRank method had any appreciable denoising performance.
+Such a method may find applications as a preprocessing step
+in a machine learning pipeline that utilizes tensorial data, such
+as that used for the prediction of hemodynamic decomposition
+in [40]. One limitation of the amplification-based denois-
+ing method is that amplification requires determining order-
+specific amplification maps; currently only those for orders
+three and four have been computed. Other tested methods have
+no such restriction. However, many real-world data modalities,
+such as images (order 3) and video (order 4) can potentially
+be denoised using current amplification maps.
+V. CONCLUSION
+In this work, we utilize the general framework of tensor
+denoising introduced [15] and previously developed approxi-
+mations of the spectral norm [17] to devise three novel tensor
+denoising methods based on tensor amplification and two
+notions of tensor rank related to the G-stable rank [36] - stable
+slice rank and stable X-rank. The performance of these meth-
+ods was compared to several standard decomposition-based
+denoising methods on synthetic tensors of various sizes, ranks,
+and noise levels, along with real-world tensors derived from
+electrocardiogram (ECG) signals. The experimental results
+show that in the low rank context, tensor-based amplification
+provides comparable denoising performance in high signal-to-
+noise ratio (SNR) settings (> 0 dB) and superior performance
+in noisy (< 1 dB) settings, while the stable X-rank method
+achieves superior denoising performance on the ECG signal
+data. Future work will seek to improve the performance of
+amplification-based methods for higher rank tensors.
+ACKNOWLEDGMENT
+This work was partially supported by the National Science
+Foundation under Grant No. 1837985 and by the Department
+of Defense under Grant No. BA150235.
+APPENDIX A
+ORDER 4 DENOISING RESULTS
+
+Healthy Patients
+Unhealthy Patients
+5
+5
+0 
+-0
+-5 -
+(dB)
+Denoised SNR
+-10
+-10
+-15
+-15
+1O0H
+20
+-20
+ALS
+Wiener
+-25 
+Amp
+-25 
+SliceRank
+XRank
+-30 
+-30 
+20
+10
+5
+1-1 -5
+-10
+-20
+20
+10
+5
+1 -1
+-5
+-10
+-20
+Starting SNR (dB)
+Starting SNR (dB)Healthy Patients
+Unhealthy Patients
+5
+IOOH 
+ALS
+0 
+Wiener
+0
+Amp
+SliceRank
+-5 
+ XRank
+-5 
+(dB)
+Denoised SNR (
+-10
+-10
+-15
+-15 
+-20 
+-20
+-25 -
+-25 
+-30 -
+-30
+20
+10
+5
+1-1 -5
+-10
+-20
+20
+10
+5
+1 -1
+-5
+-10
+-20
+Starting SNR (dB)
+Starting SNR (dB)10
+TABLE IV: Mean (SD) SNR, in decibels, after tensor denoising across all parameters.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+20
+19.68 (3.70)
+33.04 (12.81)
+32.12 (15.50)
+10.93 (15.73)
+18.96 (1.43)
+9.82 (4.10)
+10
+10.82 (1.35)
+24.54 (9.92)
+22.99 (11.98)
+9.60 (12.65)
+13.31 (2.18)
+9.20 (3.60)
+5
+6.11 (0.88)
+20.04 (8.73)
+17.08 (10.24)
+8.81 (11.16)
+8.29 (2.63)
+8.31 (3.10)
+1
+2.20 (0.82)
+16.21 (8.10)
+11.73 (9.21)
+7.92 (10.06)
+3.62 (2.38)
+7.05 (2.67)
+-1
+0.22 (0.81)
+14.24 (7.85)
+8.90 (8.47)
+7.36 (9.60)
+1.23 (2.12)
+6.22 (2.58)
+-5
+-3.78 (0.80)
+10.14 (7.64)
+3.68 (6.62)
+6.06 (8.98)
+-3.44 (1.61)
+4.10 (2.77)
+-10
+-8.79 (0.79)
+4.89 (7.52)
+-2.18 (4.03)
+3.93 (8.78)
+-9.03 (1.09)
+0.69 (3.64)
+-20
+-18.78 (0.81)
+-5.44 (7.21)
+-9.70 (6.52)
+-16.67 (3.64)
+-19.64 (0.49)
+-7.44 (5.15)
+(a) Uniformly sized tensors of order 4.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+20
+25.83 (8.53)
+35.74 (14.42)
+32.02 (15.29)
+14.20 (15.98)
+18.82 (1.44)
+11.16 (4.57)
+10
+17.69 (5.21)
+28.12 (11.06)
+24.55 (12.18)
+12.83 (12.86)
+13.34 (2.14)
+10.41 (3.85)
+5
+13.47 (3.77)
+24.00 (9.59)
+18.98 (11.13)
+11.83 (11.43)
+8.32 (2.49)
+9.31 (3.17)
+1
+10.00 (2.87)
+20.60 (8.58)
+13.77 (10.31)
+10.14 (10.68)
+3.67 (2.21)
+7.90 (2.67)
+-1
+8.21 (2.55)
+18.86 (8.14)
+10.99 (9.52)
+8.70 (10.62)
+1.29 (1.99)
+6.97 (2.49)
+-5
+4.49 (2.20)
+15.20 (7.47)
+6.25 (7.26)
+4.99 (11.29)
+-3.41 (1.49)
+4.63 (2.69)
+-10
+-0.49 (2.06)
+10.30 (7.00)
+0.78 (4.80)
+0.76 (12.49)
+-8.95 (1.16)
+0.91 (3.77)
+-20
+-10.64 (1.95)
+-0.10 (6.65)
+-6.69 (6.44)
+-18.46 (2.99)
+-19.53 (0.67)
+-7.47 (5.21)
+(b) Non-uniformly sized tensors of order 4.
+(a)
+(b)
+Fig. 4: Denoising performance with respect to tensor rank for fourth-order tensors.
+TABLE V: Mean (SD) denoised SNR, in decibels, for low rank and noisy tensors.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+1
+1.59 (0.36)
+5.97 (2.93)
+12.21 (4.05)
+14.98 (7.72)
+5.64 (2.29)
+8.89 (1.91)
+-1
+-0.42 (0.35)
+3.92 (2.92)
+10.31 (3.85)
+13.66 (7.28)
+3.33 (2.11)
+7.34 (2.01)
+-5
+-4.43 (0.32)
+-0.17 (2.91)
+6.42 (3.78)
+10.68 (6.90)
+-1.52 (1.70)
+3.77 (2.34)
+-10
+-9.45 (0.30)
+-5.24 (2.93)
+1.90 (3.84)
+6.16 (7.47)
+-7.49 (1.39)
+-0.79 (2.88)
+-20
+-19.45 (0.30)
+-15.29 (2.91)
+-8.86 (5.90)
+-4.36 (7.84)
+-18.79 (1.03)
+-10.31 (3.48)
+(a) Uniformly sized fourth-order tensors.
+Starting SNR
+HOOI
+ALS
+Wiener
+Amp
+SliceRank
+XRank
+1
+10.68 (2.08)
+22.45 (7.05)
+24.99 (11.39)
+23.47 (14.61)
+5.82 (2.20)
+10.97 (2.55)
+-1
+8.67 (2.08)
+20.36 (7.02)
+21.36 (11.26)
+22.36 (13.82)
+3.43 (2.08)
+9.71 (2.59)
+-5
+4.65 (2.07)
+16.19 (6.95)
+14.20 (9.11)
+20.10 (12.42)
+-1.59 (1.63)
+6.94 (2.98)
+-10
+-0.38 (2.05)
+11.03 (6.88)
+4.95 (5.79)
+17.07 (10.93)
+-7.48 (1.42)
+3.01 (4.21)
+-20
+-10.56 (1.98)
+0.71 (6.97)
+-6.40 (5.90)
+-15.23 (4.45)
+-18.66 (0.87)
+-5.51 (5.53)
+(b) Non-uniformly sized fourth-order tensors.
+
+Uniform Synthetic Tensors, Order 4
+IOOH
+ALS
+25
+Wiener
+Amp
+SliceRank
+20 
+XRank
+Denoised SNR (dB)
+15
+10
+5
+0
+3 
+4
+5
+10
+20
+25
+RankNon-uniform Synthetic Tensors, Order 4
+IOOH
+30 
+ALS
+Wiener
+Amp
+25 
+SliceRank
+XRank
+Denoised SNR (dB)
+20
+15
+10
+5
+0
+2
+3 
+4
+5
+10
+20
+25
+Rank11
+REFERENCES
+[1] J. D. Carroll and J.-J. Chang, “Analysis of individual differences in
+multidimensional scaling via an n-way generalization of “eckart-young”
+decomposition,” Psychometrika, vol. 35, no. 3, pp. 283–319, 1970.
+[2] R. A. Harshman et al., “Foundations of the parafac procedure: Models
+and conditions for an” explanatory” multimodal factor analysis,” 1970.
+[3] L. R. Tucker, “Implications of factor analysis of three-way matrices for
+measurement of change,” Problems in measuring change, vol. 15, no.
+122-137, p. 3, 1963.
+[4] L. R. Tucker et al., “The extension of factor analysis to three-
+dimensional matrices,” Contributions to mathematical psychology, vol.
+110119, 1964.
+[5] C. J. Hillar and L.-H. Lim, “Most tensor problems are NP-hard,” Journal
+of the ACM (JACM), vol. 60, no. 6, pp. 1–39, 2013.
+[6] X. Liu, S. Bourennane, and C. Fossati, “Denoising of hyperspectral
+images using the parafac model and statistical performance analysis,”
+IEEE Transactions on Geoscience and Remote Sensing, vol. 50, no. 10,
+pp. 3717–3724, 2012.
+[7] M. A. Veganzones, J. E. Cohen, R. C. Farias, J. Chanussot, and
+P. Comon, “Nonnegative tensor cp decomposition of hyperspectral data,”
+IEEE Transactions on Geoscience and Remote Sensing, vol. 54, no. 5,
+pp. 2577–2588, 2015.
+[8] J. Xue, Y. Zhao, W. Liao, and J. C.-W. Chan, “Nonlocal low-rank
+regularized tensor decomposition for hyperspectral image denoising,”
+IEEE Transactions on Geoscience and Remote Sensing, vol. 57, no. 7,
+pp. 5174–5189, 2019.
+[9] L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear
+singular value decomposition,” SIAM journal on Matrix Analysis and
+Applications, vol. 21, no. 4, pp. 1253–1278, 2000.
+[10] A. Rajwade, A. Rangarajan, and A. Banerjee, “Using the higher order
+singular value decomposition for video denoising,” in International
+Workshop on Energy Minimization Methods in Computer Vision and
+Pattern Recognition.
+Springer, 2011, pp. 344–354.
+[11] ——, “Image denoising using the higher order singular value de-
+composition,” IEEE Transactions on Pattern Analysis and Machine
+Intelligence, vol. 35, no. 4, pp. 849–862, 2012.
+[12] C. Lee and M. Wang, “Tensor denoising and completion based on
+ordinal observations,” in International Conference on Machine Learning.
+PMLR, 2020, pp. 5778–5788.
+[13] I. V. Oseledets, “Tensor-train decomposition,” SIAM Journal on Scien-
+tific Computing, vol. 33, no. 5, pp. 2295–2317, 2011.
+[14] X. Gong, W. Chen, J. Chen, and B. Ai, “Tensor denoising using low-rank
+tensor train decomposition,” IEEE Signal Processing Letters, vol. 27, pp.
+1685–1689, 2020.
+[15] H. Derksen, “A general theory of singular values with applications to
+signal denoising,” SIAM Journal on Applied Algebra and Geometry,
+vol. 2, no. 4, pp. 535–596, 2018.
+[16] S. Friedland and L.-H. Lim, “Nuclear norm of higher-order tensors,”
+Mathematics of Computation, vol. 87, no. 311, pp. 1255–1281, 2018.
+[17] N. Tokcan, J. Gryak, K. Najarian, and H. Derksen, “Algebraic methods
+for tensor data,” SIAM Journal on Applied Algebra and Geometry, vol. 5,
+no. 1, pp. 1–27, 2021.
+[18] T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,”
+SIAM review, vol. 51, no. 3, pp. 455–500, 2009.
+[19] T. G. Kolda, “A counterexample to the possibility of an extension of
+the eckart–young low-rank approximation theorem for the orthogonal
+rank tensor decomposition,” SIAM Journal on Matrix Analysis and
+Applications, vol. 24, no. 3, pp. 762–767, 2003.
+[20] V. De Silva and L.-H. Lim, “Tensor rank and the ill-posedness of the best
+low-rank approximation problem,” SIAM Journal on Matrix Analysis and
+Applications, vol. 30, no. 3, pp. 1084–1127, 2008.
+[21] B. W. Bader, T. G. Kolda et al. (2022) Tensor Toolbox for MATLAB,
+Version 3.4. [Online]. Available: www.tensortoolbox.org
+[22] L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1
+and rank-(r1, r2, . . . , rn) approximation of higher-order tensors,” SIAM
+journal on Matrix Analysis and Applications, vol. 21, no. 4, pp. 1324–
+1342, 2000.
+[23] F. Jin, P. Fieguth, L. Winger, and E. Jernigan, “Adaptive wiener
+filtering of noisy images and image sequences,” in Proceedings 2003
+International Conference on Image Processing (Cat. No. 03CH37429),
+vol. 3.
+IEEE, 2003, pp. III–349.
+[24] X. Zhang, “Image denoising using local wiener filter and its method
+noise,” Optik, vol. 127, no. 17, pp. 6821–6828, 2016.
+[25] L. Smital, M. Vitek, J. Kozumpl´ık, and I. Provaznik, “Adaptive wavelet
+wiener filtering of ECG signals,” IEEE transactions on biomedical
+engineering, vol. 60, no. 2, pp. 437–445, 2012.
+[26] B. Somers, T. Francart, and A. Bertrand, “A generic EEG artifact
+removal algorithm based on the multi-channel wiener filter,” Journal
+of neural engineering, vol. 15, no. 3, p. 036007, 2018.
+[27] A. Spriet, M. Moonen, and J. Wouters, “Spatially pre-processed speech
+distortion weighted multi-channel wiener filtering for noise reduction,”
+Signal Processing, vol. 84, no. 12, pp. 2367–2387, 2004.
+[28] J. Chen, J. Benesty, Y. Huang, and S. Doclo, “New insights into the
+noise reduction wiener filter,” IEEE Transactions on audio, speech, and
+language processing, vol. 14, no. 4, pp. 1218–1234, 2006.
+[29] D. Muti and S. Bourennane, “Multidimensional filtering based on a
+tensor approach,” Signal Processing, vol. 85, no. 12, pp. 2338–2353,
+2005.
+[30] T. Lin and S. Bourennane, “Survey of hyperspectral image denoising
+methods based on tensor decompositions,” EURASIP journal on Ad-
+vances in Signal Processing, vol. 2013, no. 1, pp. 1–11, 2013.
+[31] N. Renard, S. Bourennane, and J. Blanc-Talon, “Denoising and dimen-
+sionality reduction using multilinear tools for hyperspectral images,”
+IEEE Geoscience and Remote Sensing Letters, vol. 5, no. 2, pp. 138–
+142, 2008.
+[32] J. Blasiak, T. Church, H. Cohn, J. A. Grochow, E. Naslund, W. F. Sawin,
+and C. Umans, “On cap sets and the group-theoretic approach to matrix
+multiplication,” arXiv preprint arXiv:1605.06702, 2016.
+[33] “Notes on the “slice rank” of tensors,” https://terrytao.wordpress.com/
+2016/08/24/.
+[34] M. Rudelson and R. Vershynin, “Sampling from large matrices: An
+approach through geometric functional analysis,” Journal of the ACM
+(JACM), vol. 54, no. 4, pp. 21–es, 2007.
+[35] MATLAB, “Statistics and machine learning toolbox,” R2022a, the
+MathWorks Inc., Natick, MA, USA.
+[36] H. Derksen, “The g-stable rank for tensors and the cap set problem,”
+Algebra & Number Theory, vol. 16, no. 5, pp. 1071–1097, 2022.
+[37] R. Bousseljot, D. Kreiseler, and A. Schnabel, “Nutzung der ekg-
+signaldatenbank cardiodat der ptb ¨uber das internet,” 1995.
+[38] A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. C.
+Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C.-K. Peng, and H. E.
+Stanley, “PhysioBank, PhysioToolkit, and PhysioNet: Components of a
+new research resource for complex physiologic signals,” Circulation,
+vol. 101, no. 23, pp. e215–e220, 2000.
+[39] S. Padhy, G. Goovaerts, M. Bouss´e, L. D. Lathauwer, and S. V. Huffel,
+“The power of tensor-based approaches in cardiac applications,” in
+Biomedical Signal Processing.
+Springer, 2020, pp. 291–323.
+[40] L. Hernandez, R. Kim, N. Tokcan, H. Derksen, B. E. Biesterveld,
+A. Croteau, A. M. Williams, M. Mathis, K. Najarian, and J. Gryak,
+“Multimodal tensor-based method for integrative and continuous patient
+monitoring during postoperative cardiac care,” Artificial Intelligence in
+Medicine, p. 102032, 2021.
+[41] R. B. Kim, O. P. Alge, G. Liu, B. E. Biesterveld, G. Wakam, A. M.
+Williams, M. R. Mathis, K. Najarian, and J. Gryak, “Prediction of post-
+operative cardiac events in multiple surgical cohorts using a multimodal
+
+12
+and integrative decision support system,” Scientific reports, vol. 12,
+no. 1, pp. 1–11, 2022.
+[42] M. R. Mathis, M. C. Engoren, A. M. Williams, B. E. Biesterveld,
+A. J. Croteau, L. Cai, R. B. Kim, G. Liu, K. R. Ward, K. Najarian
+et al., “Prediction of postoperative deterioration in cardiac surgery
+patients using electronic health record and physiologic waveform data,”
+Anesthesiology, 2022.
+[43] P. L. Davies and A. Kovac, “Local extremes, runs, strings and multires-
+olution,” The Annals of Statistics, vol. 29, no. 1, pp. 1–65, 2001.
+

8dFLT4oBgHgl3EQfBS4r/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8f57cba4ab03adcbbede5f87aaa2d6ada1897cdcab2d2d594e31a75e551e5e6c
+size 85657

A9AyT4oBgHgl3EQf3_rL/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6862893d256697beab1db79564dd2ae417ff3c69b825c9f86e87de1920d7ebf0
+size 7798829

A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ee310b65655c75a6074f37fcdd0bb5b4e42d720c313e5fee437900a11276858a
+size 1341835

A9AzT4oBgHgl3EQf__9t/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4f586d69e2574c51995d8d2a950d1673ad20d2b7e0bddea79844bd734dd7ab81
+size 197933

C9FQT4oBgHgl3EQfPDYj/content/tmp_files/2301.13277v1.pdf.txt

@@ -0,0 +1,1080 @@
+arXiv:2301.13277v1  [cond-mat.soft]  30 Jan 2023
+Transport properties in liquids from first principles: the case of
+liquid water and liquid Argon
+Pier Luigi Silvestrelli
+Dipartimento di Fisica e Astronomia “G. Galilei”,
+Universit`a di Padova, via Marzolo 8, I-35131 Padova, Italy
+(Dated: February 1, 2023)
+Abstract
+Shear and bulk viscosity of liquid water and Argon are evaluated from first principles in the Den-
+sity Functional Theory (DFT) framework, by performing Molecular Dynamics simulations in the
+NVE ensemble and using the Kubo-Greenwood equilibrium approach. Standard DFT functional is
+corrected in such a way to allow for a reasonable description of van der Waals (vdW) effects. For
+liquid Argon the thermal conductivity has been also calculated. Concerning liquid water, to our
+knowledge this is the first estimate of the bulk viscosity and of the shear-viscosity/bulk-viscosity
+ratio from first principles. By analyzing our results we can conclude that our first-principles sim-
+ulations, performed at a nominal average temperature of 366 K to guarantee that the systems is
+liquid-like, actually describe the basic dynamical properties of liquid water at about 330 K. In
+comparison with liquid water, the normal, monatomic liquid Ar is characterized by a much smaller
+bulk-viscosity/shear-viscosity ratio (close to unity) and this feature is well reproduced by our first-
+principles approach which predicts a value of the ratio in better agreement with experimental
+reference data than that obtained using the empirical Lennard-Jones potential. The computed
+thermal conductivity of liquid Argon is also in good agreement with the experimental value.
+1
+
+I.
+INTRODUCTION
+Transport properties are among the most important and useful features of condensed-
+matter systems, particularly for characterizing the dynamical behavior of liquids, since they
+play an important role in many technical and natural processes. Therefore their estimate
+represents one of the most relevant goal of Molecular Dynamics (MD) simulation techniques
+which become particularly useful in cases where experimental data are not available or
+difficult to obtain. Different theoretical approaches can be adopted with a varying degree of
+accuracy (see, for instance, refs. 1–27, and further references therein).
+Basically, in MD simulation transport properties can be evaluated either through a gen-
+uine nonequilibrium approach by applying an explicit external perturbation (such as a shear
+flow or a temperature gradient), which is clearly direct and intuitive but is affected by non-
+trivial technical issues (in particular the need to generate nonequilibrium steady states in
+typical systems characterized by finite-size supercells with periodic boundary conditions and
+to extrapolate to the limit of zero driving force). Alternatively, the transport coefficients
+can be more easily estimated from equilibrium MD simulations by using the Green-Kubo
+relations28–30 of statistical mechanics (dissipation-fluctuation theorem) which allow the cal-
+culation of transport coefficients by integration of suitable autocorrelation functions. This
+latter approach is simpler because standard equilibrium MD simulations can be easily car-
+ried out and estimated transport coefficients exhibit a weaker system-size dependence.26 An
+equivalent17 equilibrium method exploits the Einstein–Helfand expressions2 to get transport
+coefficients directly from the particle displacements and velocities;18 for instance, the shear
+viscosity can be computed in terms of the mean-square x displacement of the center of y mo-
+mentum, while the thermal conductivity is proportional to the mean square x displacement
+of the center of energy.
+The shear viscosity describes the resistance of a fluid to shear forces and is a measure
+of the shear stress induced by an applied velocity gradient,1 while the bulk viscosity refers
+to the resistance to dilatation of an infinitesimal volume element at constant shape and
+measures the resistance of a fluid to compression. It is closely connected with absorption
+and dispersion of ultrasonic waves in a fluid, so it can provide valuable information about
+intermolecular forces. Moreover, the role of the bulk viscosity is acquiring more and more
+importance, for instance in the area of surface and interface-related phenomena and for
+2
+
+the interpretation of acoustic sensor data.31 In spite of its relevance, bulk viscosity has
+received less experimental and theoretical attention, partly due to the greater difficulties in
+obtaining accurate measurements and estimates. In principle it should be evaluated in the
+microcanonical (NVE) ensemble where there is no need to evaluate an additional term which
+would be required if, for instance, the canonical NVT ensemble were used.13,31 Moreover,
+bulk viscosity is subject to much larger statistical error caused by the fact that it must
+be calculated by the regression of fluctuations about a nonzero mean.3 While the shear
+viscosity is associated with changes in water Hydrogen-bond network connectivity and is
+mostly related to translational molecular motion, the bulk viscosity is associated with local
+density fluctuations and reflects the relaxation of both rotational and vibrational modes.32,33
+The thermal conductivity describes instead the capability of a substance to allow molecular
+transport of energy driven by temperature gradients.
+In general dynamical properties such as the transport coefficients are much more depen-
+dent on the simulation size and timescale than structural properties.23 One must also point
+out that shear and bulk viscosities, and thermal conductivity are even more difficult to be
+evaluated accurately than, for instance, the diffusion coefficient (a single-particle property)
+since they are collective transport properties involving all the particles.14 In fact, for esti-
+mating the diffusion coefficient one can perform a statistical average over the particles in
+addition to the average over time because every particle diffuses individually but any stress
+or energy fluctuation is an event involving the system as a whole. As a consequence, in
+order to obtain the same statistical accuracy, collective properties need much longer runs
+than single particle properties by a factor proportional to the size of the system.12
+We here estimate from first principles simulations, in the framework of the Density Func-
+tional Theory (DFT), the shear and bulk viscosity of liquid water and Argon. For liquid
+Argon the thermal conductivity is also calculated. By analyzing our results we can con-
+clude that our first-principles simulations, performed at a nominal average temperature of
+366 K to guarantee that the systems is liquid-like, actually describe the basic dynamical
+properties of liquid water at about 330 K. Our approach is also able to reproduce well the
+bulk-viscosity/shear-viscosity ratio of liquid Ar which is much smaller than that of liquid
+water.
+3
+
+II.
+METHOD
+We have performed first principles MD simulations of liquid water using the CPMD
+package,34 at constant volume, considering the experimental density of water at room tem-
+perature. The computations were performed at the Γ-point only of the Brillouin zone, using
+norm-conserving pseudopotentials35 and a basis set of plane waves to expand the wavefunc-
+tions with an energy cutoff of 250 Ry; we have explicitly tested that this energy cutoff, much
+higher than that used in standard DFT simulations of liquid water, is required to have a
+good convergence also for the stress tensor components.
+We have adopted the gradient-corrected BLYP36 density functional augmented by van
+der Waals (vdW) corrections, hereafter referred to as DFT-D2(BLYP).37 This choice is
+motivated both by the fact that BLYP has been shown38–42 to give an acceptable description
+of Hydrogen bonding in water, and because it represents a good reference DFT functional to
+add vdW corrections.43–46 A good description of Hydrogen bonding is essential here since, in
+liquid water, the shear viscosity mostly originates from covalent interactions in the Hydrogen-
+bond dynamics of water molecules.19 Moreover, vdW corrections to BLYP are important
+because it was shown that BLYP significantly underestimates (by 25%) the equilibrium
+density of liquid water; the experimental density can be recovered by adding the vdW
+corrections proposed by Grimme,37 which have the further effect of making the oxygen-
+oxygen radial distribution function in better agreement with experiment.47,48 Our system
+consists of 64 water molecules contained in a supercell with simple-cubic symmetry and
+periodic boundary conditions. Hydrogen nuclei have been treated as classical particles with
+the mass of the deuterium isotope which allows us to use larger time steps. The effective
+mass determining the time scale of the fictitious dynamics of the electrons was 700 a.u. and
+the equations of motion were integrated with a time step of 3 a.u. (=0.073 fs).
+Our simulation consisted of an initial equilibration phase, lasting about 0.15 ps, in which
+the ionic temperature was simply controlled by velocity rescaling, followed by a much longer
+(about 22 ps) canonical (NVT) MD simulation (using suitable thermostats for a Nos´e-Hoover
+dynamics), followed by a final 22 ps microcanonical (NVE) production MD run. A common
+drawback of most standard DFT functionals applied to liquid water at room temperature
+is their tendency to ”freeze” the system which therefore exhibits an ice-like behavior. By
+applying vdW corrections the problem is reduced but it still present. In particular, since the
+4
+
+melting temperature of water estimated by DFT-D2(BLYP) is 360 K49 (while it is 411 K with
+BLYP), following a common strategy, we performed NVT simulations with an average ionic
+temperature of 380 K to be sure that the system is indeed liquid-like. This use of artificially
+increased temperature also serves to mimic Nuclear Quantum Effects in simulations of liquid
+water.23 The average ionic temperature of the subsequent NVE MD simulation was 366
+K. Several data (atomic coordinates, velocities, stress-tensor components,...) relevant for
+characterizing structural and dynamical properties of the system were recorded every 20
+steps in the production stage.
+As far as liquid Ar is concerned, before starting MD simulations, we have performed exten-
+sive preliminary calculations to choose optimal parameters and a suitable DFT functional.
+Clearly in this case even an empirical Lennard-Jones potential reference could probably give
+reasonable results but here we are interested in studying transport properties using DFT
+functionals in a first-principle framework, which has the advantage of explicitly accounting
+for the electronic structure of matter. Application to the face-centered cubic (fcc) Ar crystal
+(considering a fcc supercell with 32 Ar atoms) and comparison with experimental reference
+values for the equilibrium Ar-Ar distance and the cohesive energy, suggests that, among
+many tested, vdW-corrected DFT functionals, DFT-D2(PBE)37,50 is the most adequate to
+describe extended systems made by Ar atoms, hence we mainly use it for the MD simula-
+tions of liquid Ar. In this case we have checked that a suitable energy cutoff to get a good
+convergence for the stress tensor components is 110 Ry.
+The liquid Ar sample was prepared starting from an initial (unfavorable) simple cubic
+lattice configuration with 64 Ar atoms and considering the experimental Ar density (1.4
+g/cm3) at melting point (84 K). Then the systems was heated by gradually increasing the
+ionic temperature (by velocity rescaling) to 500 K (in a time of 1.3 ps) to be sure that the
+system was truly melted; then the temperature was gradually decreased (in 1.0 ps) to 150
+K, which is a temperature sufficiently higher than the experimental melting point that it
+can be assumed that the system is indeed in a liquid phase; this has been explicitly checked
+looking at the translational order parameter.1
+Then a 60 ps canonical (NVT) MD simulation (with a ionic temperature of 150 K) was
+performed, followed by a 60 ps microcanonical (NVE) MD production runs with an average
+ionic temperature of 129 K. In this case the electronic effective mass was 700 a.u. and the
+equations of motion were integrated with a time step of 5 a.u. (=0.121 fs). Data (atomic
+5
+
+coordinates, velocities, stress-tensor components,...) relevant for structural and dynamical
+properties of the system were recorded every 10 steps in the production stage.
+As mentioned above, different approaches exist for the calculation of shear, ηS, and
+bulk, ηB, viscosity from MD simulations.1,8–11,13 The most used technique is based on the
+evaluation of the autocorrelation functions of stress-tensor components; in particular,1
+ηS =
+V
+kBT
+� ∞
+0
+dt⟨Pαβ(0)Pαβ(t)⟩ ,
+(1)
+ηB =
+V
+9kBT
+� ∞
+0
+dt⟨δPαα(0)δPββ(t)⟩ =
+V
+kBT
+� ∞
+0
+dt⟨δP(0)δP(t)⟩ ,
+(2)
+where, in practice the upper limit of integration (∞) is replaced by a reasonably-long
+simulation time, tmax, ⟨...⟩ denotes average over different time origins, V is the system
+volume, T the ionic temperature, kB the Boltzmann constant, Pαβ quantities denote the
+components of the stress tensor, the instantaneous pressure is given by P(t) = 1/3 �
+α Pαα
+(that is the average of the diagonal elements of the stress tensor), and the fluctuations are
+defined as:
+δPαα(t) = Pαα(t) − ⟨Pαα⟩ = Pαα(t) − P , δP(t) = P(t) − ⟨P⟩ = P(t) − P ,
+(3)
+where P is the system pressure obtained as the ensemble average of P(t). In isotropic
+fluids (with rotational invariance) there are only 5 independent (and equivalent) components
+of the traceless stress tensor: Pxy, Pyz, Pzx, (Pxx − Pyy)/2, and (Pyy − Pzz)/2, so that it is
+convenient to compute the shear viscosity ηS by averaging over these 5 components to get
+better statistics.
+Instead, the bulk viscosity ηB has only one component, moreover the diagonal stresses
+must be evaluated carefully since a non-vanishing equilibrium average must be subtracted.
+In oder to get more accurate evaluations of transport properties and also reliable estimates
+of the associated statistical errors, we adopt the block-average technique,51 which consists
+of dividing the whole simulation into a sequence of several shorter intervals (“blocks”),
+each with an equal number of samples; then block averages are calculated which allow to
+estimate means and variances.15 In the case of the bulk-viscosity calculation, to reduce the
+error, it is convenient to take for the system pressure the average value of the pressures
+over all blocks.13 Clearly the choice of the block size must be made with care; in fact,
+6
+
+samples become uncorrelated as the block size increases so for small block sizes, the error is
+underestimated while for large block sizes the error estimate is inaccurate due to insufficient
+sampling (see detailed discussion below).
+Typically transport coefficients are estimated from classical MD simulations based on
+empirical interatomic potentials. The practical feasibility of calculating transport coefficients
+in liquids using instead first principles MD simulations, was demonstrated by D. Alf´e and
+M. J. Gillan,12 who used the Green-Kubo relations to compute the shear viscosity of liquid
+iron and aluminum, with a statistical error of about 5%. However, the simulations of D. Alf´e
+and M. J. Gillan12 were performed in the NVT ensemble, while our simulations have been
+carried out using the NVE ensemble, since the NVE simulations also allow the evaluation
+of the bulk viscosity without any correction term (see above).
+A simpler alternative method exists (valid for temperatures that are not too low52) to
+obtain an approximate estimate of the shear viscosity, by exploiting its connection with the
+self-diffusion coefficient D via the Stokes-Einstein relation:4,12
+ηS = kBT
+2πaD ,
+(4)
+where a is an effective atomic diameter. Such relation is exact for the Brownian motion
+of a macroscopic particle of diameter a in a liquid of shear viscosity ηS, but it is only
+approximate when applied to atoms; however if a is chosen to be the radius r1 of the first
+peak in the radial distribution function, the relation usually predicts ηS to within 40%.12
+Here we take for r1 the position of the first peak in the O-O and Ar-Ar radial distribution
+function for liquid water and liquid Ar, respectively, while the diffusion coefficient D can
+be computed1 from the mean square displacement of the oxygen atoms (for liquid water)
+or Ar atoms (for liquid Ar). The validity of the Stokes–Einstein relation has been recently
+discussed in detail by Herrero et al.52 who also explored the connection between structural
+properties and transport coefficients.
+For liquid Argon the thermal conductivity has been also calculated, using the formula:1
+λT =
+V
+kBT 2
+� ∞
+0
+dt⟨jE
+α (0)jE
+α (t)⟩ ,
+(5)
+where jE
+α is the α component of the energy current defined as the time derivative of
+7
+
+δEα = 1
+V
+�
+i
+riα(Ei − ⟨Ei⟩) ,
+(6)
+and Ei is the energy of the i−th Ar atom (located at coordinates rix, riy, riz), which can
+be evaluated as
+Ei = p2
+i /2mi + 1/2
+�
+j̸=i
+v(rij) ,
+(7)
+by assuming a pairwise interatomic potential. In order to obtain a pair potential for
+evaluating the thermal conductivity of liquid Ar using configurational data from our first-
+principles DFT simulations, we have adopted a strategy similar to that proposed in ref. 26:
+we assume for the pair potential a Lennard-Jones analytical form:
+v(r) = a(b2/r12 − b/r6) ,
+(8)
+where a and b are parameters optimized by fitting the potential-energy curve of the Ar
+dimer (at different interatomic distances) obtained by using our DFT approach.
+III.
+RESULTS AND DISCUSSION
+In Fig. 1 and 2 we plot the behavior of the temperature and pressure as a function of time
+in the NVE simulation for liquid water and Ar, respectively. As can be seen, these quantities
+turn out to be stable and exhibit only moderate oscillations around the average values, which
+are, for liquid water, 0.132 GPa and 366 K for the pressure and the temperature, respectively,
+while for liquid Ar the values are 0.173 GPa and 129 K.
+In Fig. 3 and 4 we instead plot the auto-correlation functions (ACFs), corresponding to
+the integrands (considering the average over the components for the shear viscosity) of eqs.
+(1) and (2). Differently from what observed in monatomic systems (such as liquid Ar) or
+in classical MD simulations where waters are modeled by rigid molecules, in first-principles
+simulations of liquid water, high-frequency intermolecular vibrations lead to corresponding
+high-frequency oscillations in the pressure and in related ACFs. In order to better appreciate
+the global decay behavior of ACFs, in the case of liquid water, high-frequency components
+have been cut by Fourier-transforming the ACFs.
+A quantitative estimate of the ACFs
+relaxation times can be obtained assuming a global exponential decay (≃ e−t/τ) of the
+
+0
+5
+10
+15
+20
+time (ps)
+0
+0
+100
+100
+200
+200
+300
+300
+400
+400
+T(K)
+P (10 MPa)
+FIG. 1: Temperature and pressure of liquid water plotted as a function of the simulation time.
+integrands and computing:
+τS =
+� ∞
+0
+dt ⟨Pαβ(0)Pαβ(t)⟩
+⟨Pαβ(0)Pαβ(0)⟩ ,
+(9)
+and
+τB =
+� ∞
+0
+dt ⟨δPαα(0)δPββ(t)⟩
+⟨δPαα(0)δPββ(0)⟩
+(10)
+For liquid water we find τS ≃ 6 fs and τB ≃ 4 fs, while for liquid Ar τS ≃ 340 fs and
+τB ≃ 410 fs.
+
+0
+10
+20
+30
+40
+50
+60
+time (ps)
+0
+0
+50
+50
+100
+100
+150
+150
+T(K)
+P (10 MPa)
+FIG. 2: Temperature and pressure of liquid Ar plotted as a function of the simulation time.
+The shear and bulk viscosity, computed using eqs. (1) and (2), are plotted as a function
+of the upper limit of the integrals in Fig. 5 and 6, while the thermal conductivity of liquid
+Ar is reported in Fig. 7. From these curves an approximate estimate of the shear and
+bulk viscosity can be obtained considering the values of the quantities corresponding to the
+position of the first pronounced maximum-plateau; in fact this indicates that the running
+integral starts becoming nearly independent of time implying that the corresponding ACF
+has decayed to zero and is fluctuating along the horizontal time axis. Clearly, considering
+
+0
+2
+4
+6
+8
+10
+time (ps)
+-0,005
+0
+0,005
+0,01
+0,015
+0,02
+ACF (Pa  )
+for shear viscosity
+for bulk viscosity
+2
+FIG. 3: Auto-correlation functions (ACFs) used for the evaluation of the shear and bulk viscosities
+of liquid water (see text) plotted as a function of the simulation time.
+longer times only introduces additional noise to the signal and the beginning of a plateau
+represents the desired value of the viscosity with the smallest uncertainty. As can be seen,
+the maximum-plateau is reached at about t = 0.8 ps for both the shear and bulk viscosity of
+liquid water, while the corresponding values for liquid Ar are 3.0, 5.0 ps, and 0.5 ps for the
+shear viscosity, the bulk viscosity, and the thermal conductivity, respectively. As expected,
+these times are much larger than the corresponding relaxation times τS and τB estimated
+
+0
+2
+4
+6
+8
+10
+time (ps)
+0
+0,0002
+0,0004
+0,0006
+0,0008
+ACF (Pa  )
+for shear viscosity
+for bulk viscosity
+2
+FIG. 4: Auto-correlation functions (ACFs) used for the evaluation of the shear and bulk viscosities
+of liquid Ar (see text) plotted as a function of the simulation time.
+above.
+As already discussed, a more accurate evaluation, with also a reliable estimate of the
+associated statistical error, can be obtained by adopting a block-average technique.
+In
+this case a proper choice of the block size is crucial: with many, small-size blocks, the
+statistical error is small but the blocks are probably correlated and the viscosity is typically
+underestimated (not yet converged); on the contrary, with just a few, large-size blocks, these
+
+0
+0,5
+1
+1,5
+2
+time (ps)
+0
+5
+10
+(        Pa s)
+bulk viscosity
+shear viscosity
+10-4
+FIG. 5: Shear and bulk viscosity of liquid water plotted as a function of the upper limit of the
+integrals of the ACFs.
+are probably uncorrelated and the viscosity is converged but the statistical error is large.
+In Figs. 8, 9, 10, and 11 we plot the values of the shear and bulk viscosity of liquid
+water and Ar evaluated by using different numbers of blocks (keeping constant the total
+number of configurations) with the relative statistical errors. The dashed horizontal lines
+indicate the corresponding values inferred by considering the maximas-plateaus of the curves
+in Figs. 5 and 6. As can be seen, in the case of liquid water, the maxima of the shear and
+
+0
+1
+2
+3
+4
+5
+6
+time (ps)
+0
+1
+2
+3
+4
+ (       Pa s)
+bulk viscosity
+shear viscosity
+10-4
+FIG. 6: Shear and bulk viscosity of liquid Ar plotted as a function of the upper limit of the integrals
+of the ACFs.
+bulk viscosities are obtained considering 16 blocks, each equivalent to a simulation time
+of about 1.4 ps. Interestingly, taking statistical uncertainties into account, these maxima
+are compatible with the rough estimates obtained before and, for the shear viscosity, also
+with the values obtained using the Stokes-Einstein formula (Eq. (4)). As already described
+above, in the Stokes-Einstein estimate the shear viscosity is obtained in terms of the diffusion
+coefficient D and the radius of the first peak in the radial distribution function (see Eq. (4),
+
+0
+1
+2
+3
+4
+time (ps)
+0
+0,05
+0,1
+thermal conductivity (W/m K)
+FIG. 7: Thermal conductivity of liquid Ar plotted as a function of the upper limit of the integral
+of the ACF.
+for liquid water we have considered the first peak in the oxygen-oxygen radial distribution
+function, see below). Actually our reported Stokes-Einstein estimated values are corrected
+by finite-size effects: in fact D can be extrapolated to infinite size of the simulation box (see,
+for instance, ref. 52) by just considering the shear-viscosity value:
+
+D∞ = D + 2.837 kBT
+6πηSL ,
+(11)
+where L is the size of the cubic simulation box. Therefore, by simultaneously taking
+into account Eqs. (4) and (11), one can get a “self-consistent”, finite-size corrected Stokes-
+Einstein estimate for ηS :
+η∗
+S = kBT
+2πaD − 2.837 kBT
+6πLD .
+(12)
+Quantitative data are collected in Table I where they are also compared with some the-
+oretical and experimental reference values.
+As far as the shear viscosity is concerned, for liquid water our estimated value, obtained
+from the NVE simulation at an average temperature of 366 K, agrees with the experimental
+reference data at a lower temperature of about 330 K. This is in line with the performances
+of other DFT functionals; for instance (see Table I), in recent simulations52 of liquid water
+based on the SCAN functional,59 the shear viscosity estimate is close to that obtained from
+a force-field approach (that, for this quantity, well reproduces the experimental behavior)
+only between 330 and 360 K, while it is severely overestimated at 300 K. With the OPTB88-
+vdW functional60 reasonable agreement with experimental data at room temperature is only
+found52 at 360 K.
+For liquid Ar the behavior is qualitatively similar (see Figs. 10, 11, and 12 for the shear
+viscosity, the bulk viscosity, and the thermal conductivity, respectively). In this case both the
+maxima of the shear and bulk viscosities are obtained considering 5 blocks, each equivalent
+to a simulation time of about 12.0 ps. Even in this case, taking statistical uncertainties into
+account, these maxima are compatible with the plateau positions and, for the shear viscosity,
+also with the estimate from the Stokes-Einstein formula. The maximum of the thermal
+conductivity is instead reached with 25 blocks, each equivalent to a simulation time of about
+2.4 ps, and its value (0.11±0.02 W/m K) is again compatible with that estimated considering
+the maximum-plateau position and in good agreement with the literature reference value
+(0.12 W/m K) at 90 K61 and that obtained by classical MD simulations based on the
+Lennard-Jones potential (0.119 W/m K).62
+An interesting physical quantity is represented by the ratio between bulk and shear
+viscosity, which can be related to the ratio of observed to classical absorption coefficients in
+
+0
+5
+10
+15
+20
+25
+30
+35
+# of blocks
+0
+2
+4
+6
+8
+shear viscosity (       Pa s) 
+*
+10-4
+expt. 303 K
+expt. 323 K
+expt. 333 K
+FIG. 8: Shear viscosity of liquid water evaluated by using different numbers of blocks (the smaller
+is the block number the larger is the number of configurations of each block) with the relative sta-
+tistical errors. The dashed horizontal line indicates the position of the first-pronounced maximum-
+plateau of the corresponding curve of Fig.
+5.
+The asterisk denotes the value obtained by the
+Stokes-Einstein formula (Eq.12), while the triangles indicate experimental estimates at different
+temperatures.
+
+TABLE I: Shear and bulk viscosity of liquid water and Ar, in 10−4 Pa s, compared with theoretical
+and experimental reference data.
+Statistical errors are in parenthesis.
+η∗
+S indicates the shear
+viscosity estimate obtained by the Stokes-Einstein relation (see text).
+system
+ηS
+η∗
+S
+ηB
+ηB/ηS 3/4ηB/ηS + 1
+water (366 K)
+4.8(0.7) 5.7 11.3(2.9) 2.4(0.8)
+2.8(0.6)
+water DFT SCANa (300K)
+23
+—
+—
+—
+—
+water DFT SCANa (330K)
+6
+—
+—
+—
+—
+water DFT SCANa (360K)
+5
+—
+—
+—
+—
+water DFT OPTB88-vdWa (300K)
+30
+—
+—
+—
+—
+water DFT OPTB88-vdWa (330K)
+15
+—
+—
+—
+—
+water DFT OPTB88-vdWa (360K)
+8
+—
+—
+—
+—
+water force fielda (300K)
+8
+—
+—
+—
+—
+water force fielda (330K)
+5
+—
+—
+—
+—
+water force fielda (360K)
+3.5
+—
+—
+—
+—
+water force fieldb (303K)
+6.5(0.4) — 15.5(1.6) 2.4(0.3)
+2.8(0.2)
+water expt.c (298 K)
+8.90
+—
+—
+—
+—
+water expt.b,d,e (303 K)
+7.97
+—
+21.5
+2.7
+3.0
+water expt.f (323 K)
+5.47
+—
+14.8
+2.7
+3.0
+water expt.c (333 K)
+4.67
+—
+—
+—
+—
+Ar (129 K)
+3.7(1.6) 2.0 4.0(2.2) 1.1(0.8)
+1.8(0.6)
+Ar expt.g (90 K)
+2.33
+—
+1.82
+0.8
+1.6
+Ar expt.h (90 K)
+2.57
+—
+—
+—
+aref.52.
+bref.13.
+cref.55.
+dref.53.
+eref.54.
+fref.56.
+gref.57.
+href.58.
+
+0
+5
+10
+15
+20
+25
+30
+35
+# of blocks
+0
+5
+10
+15
+bulk viscosity (       Pa s)
+10-4
+FIG. 9: Bulk viscosity of liquid water evaluated by using different numbers of blocks (the smaller
+is the block number the larger is the number of configurations of each block) with the relative sta-
+tistical errors. The dashed horizontal line indicates the position of the first-pronounced maximum-
+plateau of the corresponding curve of Fig. 5.
+ultrasonic absorption experiments.13 In fact, under the condition that the heat conductivity
+contribution to the ultrasonic absorption may be neglected,
+
+0
+10
+20
+30
+40
+50
+# of blocks
+0
+1
+2
+3
+4
+5
+6
+shear viscosity (       Pa s) 
+*
+10-4
+expt. 90 K
+FIG. 10: Shear viscosity of liquid Ar evaluated by using different numbers of blocks (the smaller is
+the block number the larger is the number of configurations of each block) with the relative statis-
+tical errors. The dashed horizontal line indicates the position of the first-pronounced maximum-
+plateau of the corresponding curve of Fig.
+6.
+The asterisk denotes the value obtained by the
+Stokes-Einstein formula (Eq.4), while the triangle indicates the experimental estimate at 90 K.
+α
+αclass
+= 3/4ηB
+ηS
++ 1 ,
+(13)
+and water belongs to the group of the so-called ”associated liquids”, characterized by a
+ratio from 1 to 3, where structural relaxation is dominant.
+Classical MD simulations based on the SPC/E semiempirical potential predict13 a ηB
+ηS
+ratio of 2.4, leading to a
+α
+αclass ratio of 2.79, in reasonable agreement with the experimental
+
+0
+10
+20
+30
+40
+50
+# of blocks
+0
+1
+2
+3
+4
+5
+6
+7
+bulk viscosity (       Pa s)
+10-4
+FIG. 11: Bulk viscosity of liquid Ar evaluated by using different numbers of blocks (the smaller is
+the block number the larger is the number of configurations of each block) with the relative statis-
+tical errors. The dashed horizontal line indicates the position of the first-pronounced maximum-
+plateau of the corresponding curve of Fig. 6.
+value of 3.0.53 Instead normal liquids, such as monatomic liquids (for instance liquid Ar)
+are characterized by a ratio no greater than 1.2. Although in general the ratio varies with
+temperature and pressure, in liquid water it is found to remain constant within 20% in
+the temperature range 0-90 C (273-363 K).63 By taking statistical errors into account, our
+estimated value of the
+α
+αclass ratio (2.8 ± 0.6) is compatible with the available experimental
+
+0
+10
+20
+30
+40
+50
+# of blocks
+0
+0,05
+0,1
+0,15
+0,2
+0,25
+thermal conductivity (W/m K)
+FIG. 12: Thermal conductivity of liquid Ar evaluated by using different numbers of blocks (the
+smaller is the block number the larger is the number of configurations of each block) with the relative
+statistical errors. The dashed horizontal line indicates the position of the maximum-plateau of the
+corresponding curve in Fig. 7.
+data at ambient temperature (3.0). This is a remarkable result, considering that most of the
+reported classical MD simulations13 predict a bulk viscosity lower than the the experimental
+one, leading to an underestimated value of the
+α
+αclass ratio.
+One should also point out that a proper comparison with experimental data requires a
+careful analysis taking into account the pronounced temperature dependence of shear and
+
+bulk viscosity. In fact, according to a common empirical model,56,64 the viscosity strongly de-
+creases with increasing temperature following an exponential decay. By fitting experimental
+data56 with an exponential function and taking statistical errors into account, our estimated
+values of the shear and bulk viscosity of liquid water are compatible with experimental
+data in the temperature range of 323-344 K. One should also consider that also the bulk-
+viscosity/shear-viscosity ratio for liquid water tends to decrease slightly with temperature,56
+suggesting an even better agreement between our estimated value and the experimental
+data.56 We remind that our simulations have been carried out at temperatures higher than
+ambient temperature to guarantee that the systems is liquid-like. By considering that our
+estimate (after finite-size correction) for the diffusion coefficient, D = 5.02 × 10−5 cm2/s,
+corresponds to the experimental value measured at about 336 K,65 we can conclude that,
+our DFT simulations based on the DFT-D2(BLYP) functional and performed at a nominal
+average temperature of 366 K, actually describe the basic dynamical properties of liquid
+water at about 330 K. One should also mention that bulk-viscosity measurements are in-
+direct and affected by considerable errors.13,27,33,56,66,67 In summary, we can conclude that
+our adopted BLYP-D2 functional is able to describe reasonably well the density fluctuations
+of liquid water; the discrepancy with respect to experimental data at ambient conditions
+can be to a large extend explained in terms of the pronounced temperature dependence of
+both shear and bulk viscosity and the need to perform first-principles MD simulations at
+temperatures higher than ambient temperature.
+As far as liquid Ar is concerned, our shear and bulk viscosities, computed by first-
+principles at a nominal average simulation temperature of 129 K, turn out to be some-
+how overestimated with respect to the reference experimental values at 90 K, although
+they are compatible with them if statistical errors are taken into account. Moreover our
+bulk-viscosity/shear-viscosity ratio (close to unity) agrees well with the reference estimate,
+while interestingly this is not the case if a standard Lennard-Jones empirical potential is
+adopted using classical MD simulations that predict instead a very low value13,62 of the ratio
+(0.17-0.35 at high densities), thus showing that this popular potential cannot properly re-
+produce all the dynamical properties of liquid Ar and underlining once again the superiority
+of first-principles approaches.
+We conclude our study by reporting some basic structural properties of our investigated
+systems. In particular, in Fig. 13, for liquid water we plot our computed O-O pair correlation
+
+2
+3
+4
+5
+6
+7
+r (A)
+0
+0,5
+1
+1,5
+2
+2,5
+3
+g(r)
+FIG. 13: O-O pair correlation function, gOO(r), compared with that obtained experimentally from
+X-ray diffraction measurements at ambient conditions.68–70
+function, gOO(r), compared with that obtained experimental from X-ray diffraction measure-
+ments at ambient conditions.68–70 The main features of the gOO(r) curves are reported in
+Table II. As can be seen, there is a good agreement between the two curves; the fact the
+oscillations of our computed curve are slightly reduced with respect to the experimental one
+can again be related to the higher effective temperature of our simulation.
+
+2
+3
+4
+5
+6
+7
+r (A)
+0
+0,5
+1
+1,5
+2
+2,5
+3
+3,5
+g(r)
+FIG. 14: Ar-Ar pair correlation function, g(r), compared with that obtained experimentally from
+neutron-scattering measurements at 85K.73
+In Fig. 14, for liquid Ar our computed Ar-Ar pair correlation function, g(r), is compared
+with that obtained experimentally from neutron-scattering measurements at 85 K,73 while
+again the main features of the g(r) curves are reported in Table II. Even in this case there
+is a reasonable agreement between the simulation and experimental curve, by considering
+that simulations for liquid Ar have been performed at significantly higher temperature (129
+K) than experiments (85 K) (note that the experimental melting and boiling points of Ar
+are at 84 and 87 K, respectively). After applying the same finite-size correction adopted
+
+TABLE II: Main features of the O-O pair correlation function, gOO(r), of liquid water and of
+the Ar-Ar pair correlation function, g(r) of liquid Ar compared with experimental reference data,
+obtained from X-ray diffraction measurements at ambient conditions for liquid water and neutron-
+scattering measurements for liquid Ar. rmax and rmin indicate the position of the first maximum
+(the main peak) and the first minimum of gOO(r) and g(r), respectively, and gmax and gmin the
+corresponding values of the gOO(r) and g(r) functions.
+system
+rmax(˚A)
+gmax
+rmin(˚A)
+gmin
+water (366 K)
+2.79
+2.42
+3.66
+0.88
+water expt.a (293 K) 2.80(1) 2.55(5) 3.41(4) 0.85(2)
+Ar (129 K)
+3.70
+2.80
+5.29
+0.64
+Ar expt.b (85 K)
+3.68
+3.05
+5.18
+0.56
+aref.68–70.
+bref.73.
+above for liquid water, our estimated diffusion coefficient for liquid Ar, D = 3.82 × 10−5
+cm2/s, evaluated at a nominal simulation temperature of 129 K is significantly higher than
+the reference value (1.6 × 10−5 cm2/s) reported at 84 K.71 Again this discrepancy can be
+explained in terms of the higher temperature of the liquid Ar simulation.
+IV.
+CONCLUSIONS
+Shear and bulk viscosity of liquid water and Argon have been evaluated, together with
+other structural and dynamical properties, from first principles by adopting a vdW-corrected
+DFT approach, by performing Molecular Dynamics simulations in the NVE ensemble and
+using the Kubo-Greenwood equilibrium approach. For liquid Argon the thermal conductivity
+has been also calculated. Concerning liquid water, to our knowledge this is the first estimate
+of the bulk viscosity and of the shear-viscosity/bulk-viscosity ratio from first principles. By
+analyzing our results and comparing then with reference experimental data, we can conclude
+that our first-principles simulations, performed at a nominal average temperature of 366
+K to guarantee that the systems is liquid-like, actually describe well the basic dynamical
+
+properties of liquid water at about 330 K. In comparison with liquid water, the normal,
+monatomic liquid Ar is characterized by a much smaller bulk-viscosity/shear-viscosity ratio
+(close to unity) and this feature is well reproduced by our first-principles approach which
+predicts a value of the ratio in better agreement with experimental reference data than that
+obtained using the empirical Lennard-Jones potential. The computed thermal conductivity
+of liquid Argon is also in good agreement with the experimental value.
+V.
+ACKNOWLEDGEMENTS
+We acknowledge funding from Fondazione Cariparo, Progetti di Eccellenza 2017, relative
+to the project: ”Engineering van der Waals Interactions: Innovative paradigm for the control
+of Nanoscale Phenomena”.
+VI.
+DATA AVAILABILITY
+The data that support the findings of this study are available from the corresponding
+author upon reasonable request.
+1 M. P. Allen and D. J. Tildesley, Computer Simulations of Liquids (Oxford Science Publications,
+Clarendon Press, Oxford 1987).
+2 E. Helfand, ”Transport Coefficients from Dissipation in a Canonical Ensemble”, Phys. Rev.
+119, 1 (1960).
+3 B. J. Alder, D. M. Gass, T. E. Wainwright, ”Studies in Molecular Dynamics. VIII. The Trans-
+port Coefficients for a Hard-Sphere Fluid”, J. Chem. Phys. 53, 3813 (1970).
+4 E. M. Gosling, I. R. McDonald, K. Singer, ”On the calculation by molecular dynamics of the
+shear viscosity of a simple fluid”, Mol. Phys. 26, 1475 (1973).
+5 G. Ciccotti, G. Jacucci, I. R. McDonald, ”Transport properties of molten alkali halides”, Phys.
+Rev. A 13, 426 (1976).
+6 G. Ciccotti, G. Jacucci, K. R. McDonald, ”Thermal response to a weak external field”, J. Phys.
+C: Solid State Phys. 11, L509 (1978).
+
+7 G. Ciccotti, G. Jacucci, I. R. McDonald, ”Thought-experiments by molecular dynamics”, J.
+Stat. Phys. 21, 1 (1979).
+8 M. Schoen, C. Hoheisel, ”The shear viscosity of a Lennard-Jones fluid calculated by equilibrium
+molecular dynamics”, Mol. Phys. 56, 653 (1985).
+9 C. Hoheisel, ”Bulk viscosity of model fluids. A comparison of equilibrium and nonequilibrium
+molecular dynamics results”, J. Chem. Phys. 86, 2328 (1987).
+10 J. J. Erpenbeck, ”Einstein-Kubo-Helfand and McQuarrie relations for transport coefficients”,
+Phys. Rev. E 51, 4296 (1995).
+11 S. Balasubramanian, C. J. Mundy, M. L. Klein, ”Shear viscosity of polar fluids: Molecular
+dynamics calculations of water”, J. Chem. Phys. 105, 11190 (1996).
+12 D. Alf´e, M. J. Gillan, ”First-principles calculation of transport coefficients”, Phys. Rev. Lett.
+81, 5161 (1998).
+13 G.-J. Guo, Y. Zhang, ”Equilibrium molecular dynamics calculation of the bulk viscosity of
+liquid water”, Mol. Phys. 99, 283 (2001).
+14 S. Viscardy, J. Servantie and P. Gaspard, ”Transport and Helfand moments in the Lennard-
+Jones fluid. I. Shear viscosity”, J. Chem. Phys. 126, 184512 (2007).
+15 R. E. Jones, K. K. Mandadapu, ”Adaptive Green-Kubo estimates of transport coefficients from
+molecular dynamics based on robust error analysis”, J. Chem. Phys. 136, 154102 (2012).
+16 S. V. Lishchuk, ”Role of three-body interactions in formation of bulk viscosity in liquid argon”,
+preprint (2012): arXiv:1204.1235 [cond-mat.soft]
+17 C. Kim, O. Borodin, G. Emkarniadakis, ”Quantification of sampling uncertainty for molecular
+dynamics simulation: Time-dependent diffusion coefficient in simple fluids”, J. Comput. Phys.
+302, 485 (2015).
+18 E. M. Kirova, G. E. Norman, ”Viscosity calculations at molecular dynamics simulations”, Jour-
+nal of Physics: Conference Series 653, 012106 (2015).
+19 Y. Shi, H. Scheiber, R. Z. Khaliullin, ”Contribution of the Covalent Component of the Hydrogen-
+Bond Network to the Properties of Liquid Water”, J. Phys. Chem. A 122, 7482 (2018).
+20 F. Z. Chen, N. A. Mauro, S. M. Bertrand, P. McGrath, L. Zimmer, K. F. Kelton, ”Breakdown of
+the Stokes-Einstein relationship and rapid structural ordering in CuZrAl metallic glass-forming
+liquids”, J. Chem. Phys. 155, 104501 (2021).
+21 R. Rabani, M. H. Saidi, L. Joly, S. Merabia, A. Rajabpour, ”Enhanced local viscosity around
+
+colloidal nanoparticles probed by equilibrium molecular dynamics simulations”, J. Chem. Phys.
+155, 174701 (2021).
+22 H. Kusudo, T. Omori, Y. Yamaguchi, ”Local stress tensor calculation by the method-of-plane
+in microscopic systems with macroscopic flow: A formulation based on the velocity distribution
+function”, J. Chem. Phys. 155, 184103 (2021).
+23 A. Torres, L. S. Pedroza, M. Fernandez-Serra, A. R. Rocha, ”Using Neural Network Force
+Fields to Ascertain the Quality of Ab Initio Simulations of Liquid Water”, J. Phys. Chem. B
+125, 10772 (2021).
+24 R. Vogelsang, C. Hoheisel, G. Ciccotti, ”Thermal conductivity of the Lennard-Jones liquid by
+molecular dynamics calculations”, J. Chem. Phys. 86, 6371 (1987).
+25 Z. Fan, L. F. C. Pereira, H.-Q. Wang, J.-C. Zheng, D. Donadio, A. Harju, ”Force and heat
+current formulas for many-body potentials in molecular dynamics simulations with applications
+to thermal conductivity calculations”, Phys. Rev. B 92, 094301 (2015).
+26 J. Kang, L.-W. Wang, ”First-principles Green-Kubo method for thermal conductivity calcula-
+tions”, Phys. Rev. B 96, 020302(R) (2017).
+27 G. A. Fernandez, J. Vrabec, and H. Hasse, ”A molecular simulation study of shear and bulk
+viscosity and thermal conductivity of simple real fluids”, Fluid Phase Equilibria 221, 157 (2004).
+28 M. S. Green, ”Markoff Random Processes and the Statistical Mechanics of Time-Dependent
+Phenomena. II. Irreversible Processes in Fluids”, J. Chem. Phys. 22, 398 (1954).
+29 R. Kubo, M. Yokota, and S. Nakajima, ”Statistical-Mechanical Theory of Irreversible Processes.
+II. Response to Thermal Disturbance”, J. Phys. Soc. Jpn 12, 1203 (1957).
+30 D. McQuarrie, ”Electronic Transport in Mesoscopic Systems”, (University Science Books,
+Sausalito, 2000).
+31 R. Hafner, G. Guevara-Carrion, J. Vrabec, P. Klein, ”Sampling the Bulk Viscosity of Water
+with Molecular Dynamics Simulation in the Canonical Ensemble”, J. Phys. Chem. B 126, 10172
+(2022).
+32 A. Yahya, L. Tan, S. Perticaroli, E. Mamontov, D. Pajerowski, J. Neuefeind, G. Ehlersd, J. D.
+Nickels, ”Molecular origins of bulk viscosity in liquid water”, Phys. Chem. Chem. Phys. 22,
+9494 (2020).
+33 A. S. Dukhin and P. J. Goetz, ”Bulk viscosity and compressibility measurement using acoustic
+spectroscopy”, J. Chem. Phys. 130, 124519 (2009).
+
+34 R. Car and M. Parrinello, ”Unified Approach for MD and DFT”, Phys. Rev. Lett. 55, 2471
+(1985). We have used the code CPMD: http://www.cpmd.org/, Copyright IBM Corp 1990-2022,
+Copyright MPI f¨ur Festk¨orperforschung Stuttgart 1997-2001.
+35 N. Troullier and J. L. Martins, ”Efficient pseudopotentials for plane-wave calculations”, Phys.
+Rev. B 43, 1993 (1991).
+36 A. D. Becke, ”Density-functional exchange energy approximation with correct asymptotic be-
+havior”, Phys. Rev. A 38, 3098 (1988); C. Lee, W. Yang, and R. C. Parr, ”Development of the
+Colle-Salvetti correlation energy formula into a functional of the electron density”, Phys. Rev.
+B 37, 785 (1988).
+37 S. Grimme, ”Semiempirical GGA-type density functional constructed with a long-range disper-
+sion correction”, J. Comp. Chem. 27, 1787 (2006).
+38 M. Sprik, J. Hutter, and M. Parrinello, ”Ab initio MD simulation of liquid water: Comparison
+of 3 gradient-corrected density functionals”, J. Chem. Phys. 105, 1142 (1996).
+39 P. L. Silvestrelli and M. Parrinello, ”Water Molecule Dipole in the Gas and in the Liquid Phase”,
+Phys. Rev. Lett. 82, 3308 (1999).
+40 P. L. Silvestrelli and M. Parrinello, ”Structural, electronic, and bonding properties of liquid
+water from first principles”, J. Chem. Phys. 111, 3572 (1999).
+41 M. Boero, K. Terakura, T. Ikeshoji, C. C. Liew, and M. Parrinello, ”Hydrogen Bonding and
+Dipole Moment of Water at Supercritical Conditions: A First-Principles Molecular Dynamics
+Study”, Phys. Rev. Lett. 85, 3245 (2000).
+42 M. Boero, K. Terakura, T. Ikeshoji, C. C. Liew, and M. Parrinello, ”Water at Supercritical
+Conditions: A First-Principles Study”, J. Chem. Phys. 115, 2219 (2001).
+43 P. L. Silvestrelli, ”Van der Waals Interactions in DFT Made Easy by Wannier Functions”, Phys.
+Rev. Lett. 100, 053002 (2008).
+44 P. L. Silvestrelli, K. Benyahia, S. Grubisic, F. Ancilotto, and F. Toigo, ”Van der Waals interac-
+tions at surfaces by density functional theory using Wannier functions”, J. Chem. Phys. 130,
+074702 (2009).
+45 P. L. Silvestrelli, ”Van der Waals Interactions in DFT using Wannier Functions”, J. Phys. Chem.
+A 113, 5224 (2009).
+46 F. O. Kannemann and A. D. Becke, ”Van der Waals Interactions in Density-Functional Theory:
+Rare-Gas Diatomics”, J. Chem. Theory Comput. 5, 719 (2009).
+
+47 J. Schmidt, J. VandeVondele, I.-F. W. Kuo, D. Sebastiani, J. I. Siepmann, J. Hutter, C.
+J. Mundy, ”Isobaric-Isothermal Molecular Dynamics Simulations Utilizing Density Functional
+Theory: An Assessment of the Structure and Density of Water at Near-Ambient Conditions”,
+J. Phys. Chem. B 113, 11959 (2009).
+48 J. Wang, G. Rom´an-P´erez, J. M. Soler, E. Artacho, M.-V. Fern´andez-Serra, ”Density, structure,
+and dynamics of water: The effect of van der Waals interactions”, J. Chem. Phys. 134, 024516
+(2011).
+49 S. Yoo and S. S. Xantheas, ”Communication: The effect of dispersion corrections on the melting
+temperature of liquid water”, J. Chem. Phys. 134, 121105 (2011).
+50 J. P. Perdew, K. Burke, and M. Ernzerhof, ”Generalized Gradient approximation made simple”,
+Phys. Rev. Lett. 77, 3865 (1996).
+51 D. Frenkel and R. Smit, Understanding Molecular Simulation (Academic Press, San Diego,
+1996).
+52 C. Herrero, M. Pauletti, G. Tocci, M. Iannuzzi, L. Joly, ”Connection between water’s dynamical
+and structural properties: Insights from ab initio simulations”, PNAS 119, e2121641119 (2022).
+53 T. A. Litovitz, E. H. Carnevale, ”Effect of Pressure on Sound Propagation in Water”, J. Appl.
+Phys. 26, 816 (1955).
+54 J. V. Sengers, J. T. R. Watson, ”Improved International Formulations for the Viscosity and
+Thermal Conductivity of Water Substance”, J. Phys. Chem. Ref. Data 15, 1291 (1986).
+55 K. R. Harris and L. A. Woolf, ”Temperature and Volume Dependence of the Viscosity of Water
+and Heavy Water at Low Temperatures”, J. Chem. Eng. Data 49, 1064 (2004).
+56 M. J. Holmes, N. G. Parker, M. J. W. Povey, ”Temperature dependence of bulk viscosity in
+water using acoustic spectroscopy”, J. Phys.: Conf. Ser. 269, 012011 (2011).
+57 J. A. Cowan, R. N. Ball, ”Temperature dependence of bulk viscosity in liquid argon”, Can. J.
+Phys. 50, 1881 (1972).
+58 P. J. Linstrom, G. W. Mallard eds., NIST Chemistry WebBook, NIST Standard Reference
+Database Number 69
+59 J. Sun, A. Ruzsinszky, J. P. Perdew, ”Strongly constrained and appropriately normed semilocal
+density functional”, Phys. Rev. Lett. 115, 036402 (2015).
+60 J. Klimeˆs, D. R. Bowler, A. Michaelides, ”Chemical accuracy for the van der Waals density
+functional”, J. Phys. Condens.Matter 22, 022201 (2010); ”Van der Waals density functionals
+
+applied to solids”, Phys. Rev. B 83, 195131 (2011).
+61 B. A. Younglove and H. J. M. Hanley, ”The Viscosity and Thermal Conductivity Coefficients of
+Gaseous and Liquid Argon”, Journal of Physical and Chemical Reference Data 15, 1323 (1986).
+62 C. Hoheisel, R. Vogelsang and M. Schoen, ”Bulk viscosity of the Lennard-Jones fluid for a wide
+range of states computed by equilibrium molecular dynamics”, J. Chem. Phys. 87, 7195 (1987).
+63 C. M. Davis, J. Jarzynski, “Water: A Comprehensive Treatise, Vol. 1, edited by F. Franks
+(1972) p. 443.
+64 O. Reynolds, ”IV. On the theory of lubrication and its application to Mr. Beauchamp tower’s
+experiments, including an experimental determination of the viscosity of olive oil”, Phil. Trans.
+R. Soc. Lond. 177, 157 (1886).
+65 See, for instance, values tabulated in https://dtrx.de/od/diff/
+66 T. A. Litovitz and C. M. Davis, ”Physical Acoustics”, Vol. 2, edited by W. P. Mason, New
+York: Academic, Chap. 5.
+67 J. Xu, X. Ren, W. Gong, R. Dai, D. Liu, ”Measurement of the bulk viscosity of liquid by
+Brillouin scattering”, Appl. Opt. 42, 6704 (2003).
+68 L. B. Skinner, C. Huang, D. Schlesinger, L. G. M. Pettersson, A. Nilsson, C. J. Benmore,
+”Benchmark oxygen-oxygen pair-distribution function of ambient water from x-ray diffraction
+measurements with a wide Q-range”, J. Chem. Phys. 138, 074506 (2013).
+69 L. B. Skinner, C. J. Benmore, J. C. Neuefeind, J. B. Parise, ”The structure of water around
+the compressibility minimum”, J. Chem. Phys. 141, 214507 (2014).
+70 J. Daru, H. Forbert, J. Behler, D. Marx, ”Coupled Cluster Molecular Dynamics of Condensed
+Phase Systems Enabled by Machine Learning Potentials: Liquid Water Benchmark”, Phys.
+Rev. Lett. 129, 226001 (2022).
+71 J.-P. Hansen, I. R. McDonald, ”Theory of simple liquids”, Academic Press (1990).
+72 E. H. Hardy, A. Zygar, M. D. Zeidler, M. Holz, F. D. Sacher, ”Isotope effect on the translational
+and rotational motion in liquid water and ammonia”, J. Chem. Phys. 114, 3174 (2001).
+73 J. L. Yarnell, M. J. Katz, and R. G. Wenzel, ”Structure Factor and Radial Distribution Function
+for Liquid Argon at 85 K”, Phys. Rev. A 7, 2130 (1973).
+

CNA0T4oBgHgl3EQfAP_i/content/tmp_files/2301.01961v1.pdf.txt

@@ -0,0 +1,1182 @@
+arXiv:2301.01961v1  [math.AG]  5 Jan 2023
+SOME MORE FANO THREEFOLDS WITH A MULTIPLICATIVE
+CHOW–K ¨UNNETH DECOMPOSITION
+ROBERT LATERVEER
+ABSTRACT. We exhibit several families of Fano threefolds with a multiplicative Chow–K¨unneth
+decomposition, in the sense of Shen–Vial. As a consequence, a certain tautological subring of the
+Chow ring of powers of these threefolds injects into cohomology. As a by-product of the argu-
+ment, we observe that double covers of projective spaces admit a multiplicative Chow–K¨unneth
+decomposition.
+1. INTRODUCTION
+Given a smooth projective variety Y over C, let Ai(Y ) := CHi(Y )Q denote the Chow groups
+of Y (i.e. the groups of codimension i algebraic cycles on Y with Q-coefficients, modulo rational
+equivalence). The intersection product defines a ring structure on A∗(Y ) = �
+i Ai(Y ), the Chow
+ring of Y [14].
+In the special case of K3 surfaces, this ring structure has remarkable properties:
+Theorem 1.1 (Beauville–Voisin [3]). Let S be a projective K3 surface. The Q-subalgebra
+�
+A1(S), cj(S)
+�
+⊂ A∗(S)
+injects into cohomology under the cycle class map.
+Theorem 1.2 (Voisin [54], Yin [57]). Let S be a projective K3 surface, and m ∈ N. The Q-
+subalgebra
+R∗(Sm) :=
+�
+A1(S), ∆S
+�
+⊂ A∗(Sm)
+(generated by pullbacks of divisors and pullbacks of the diagonal ∆S ⊂ S × S) injects into
+cohomology under the cycle class map for all m ≤ 2 dim H2
+tr(S, Q)+1 (where H2
+tr(S, Q) denotes
+the transcendental part of cohomology). Moreover, R∗(Sm) injects into cohomology for all
+m ∈ N if and only if S is Kimura finite-dimensional.
+The Chow ring of abelian varieties also has an interesting property: there is a multiplicative
+splitting, defined by the Fourier transform [1].
+Motivated by the particular behaviour of K3 surfaces and abelian varieties, Beauville [2] has
+conjectured that for certain special varieties, the Chow ring should admit a multiplicative split-
+ting. In the wake of Beauville’s “splitting property conjecture”, Shen–Vial [47] have introduced
+the concept of multiplicative Chow–K¨unneth decomposition (we will abbreviate this to “MCK
+Key words and phrases. Algebraic cycles, Chow group, motive, Beauville’s “splitting property” conjecture, mul-
+tiplicative Chow–K¨unneth decomposition, Fano threefolds, tautological ring.
+2020 Mathematics Subject Classification: 14C15, 14C25, 14C30.
+Supported by ANR grant ANR-20-CE40-0023.
+1
+
+2
+ROBERT LATERVEER
+decomposition”). With the concept of MCK decomposition, it is possible to make concrete sense
+of this elusive “splitting property conjecture” of Beauville.
+It is hard to understand precisely which varieties admit an MCK decomposition. To give an
+idea of what is known: hyperelliptic curves have an MCK decomposition [47, Example 8.16],
+but the very general curve of genus ≥ 3 does not have an MCK decomposition [12, Example
+2.3]; K3 surfaces have an MCK decomposition, but certain high degree surfaces in P3 do not
+have an MCK decomposition (cf. the examples given in [43], cf. also section 2 below).
+In this note, we will focus on Fano threefolds and ask the following question:
+Question 1.3. Let X be a Fano threefold with Picard number 1. Does X admit an MCK decom-
+position ?
+The restriction on the Picard number is necessary to rule out a counterexample of Beauville
+[2, Examples 9.1.5]. The answer to Question 1.3 is affirmative for cubic threefolds [8], [12], for
+intersections of 2 quadrics [32], for intersections of a quadric and a cubic [34], and for prime
+Fano threefolds of genus 8 [37] and of genus 10 [38].
+The main result of this paper answers Question 1.3 for several more families of Fano three-
+folds:
+Theorem (=Theorem 4.1). The following smooth Fano threefolds have a multiplicative Chow–
+K¨unneth decomposition:
+• hypersurfaces of weighted degree 6 in weighted projective space P(13, 2, 3);
+• quartic double solids;
+• sextic double solids;
+• double covers of a quadric in P4 branched along the intersection with a quartic;
+• special Gushel–Mukai threefolds.
+In Table 1 (at the end of this paper), we have listed all Fano threefolds of Picard number 1 and
+what is known about MCK for them.
+To prove Theorem 4.1, we provide a general criterion (Proposition 3.3), that may be useful in
+other situations. For example, using this criterion we also prove the following:
+Proposition (=Proposition 3.6). Let X be a smooth projective variety such that X → Pn is a
+double cover ramified along a smooth divisor D ⊂ Pn of degree d > n. Then X admits an MCK
+decomposition.
+As a consequence of Theorem 4.1, we obtain an injectivity result similar to Theorem 1.2:
+Corollary (cf. Theorem 5.1). Let Y be a Fano threefold as in Theorem 4.1, and m ∈ N. Let
+R∗(Y m) :=
+�
+h, ∆Y
+�
+⊂
+A∗(Y m)
+be the Q-subalgebra generated by pullbacks of the polarization h ∈ A1(Y ) and pullbacks of the
+diagonal ∆Y ∈ A3(Y × Y ). The cycle class map induces injections
+R∗(Y m) ֒→ H∗(Y m, Q) for all m ��� N .
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+3
+Conventions. In this paper, the word variety will refer to a reduced irreducible scheme of finite
+type over C. A subvariety is a (possibly reducible) reduced subscheme which is equidimensional.
+All Chow groups will be with rational coefficients: we will denote by Aj(Y ) the Chow group
+of j-dimensional cycles on Y with Q-coefficients; for Y smooth of dimension n the notations
+Aj(Y ) and An−j(Y ) are used interchangeably. The notation Aj
+hom(Y ) will be used to indicate
+the subgroup of homologically trivial cycles. For a morphism f : X → Y , we will write Γf ∈
+A∗(X × Y ) for the graph of f.
+The contravariant category of Chow motives (i.e., pure motives with respect to rational equiv-
+alence as in [46], [41]) will be denoted Mrat.
+2. MCK DECOMPOSITION
+Definition 2.1 (Murre [40]). Let X be a smooth projective variety of dimension n. We say that
+X has a CK decomposition if there exists a decomposition of the diagonal
+∆X = π0
+X + π1
+X + · · · + π2n
+X
+in An(X × X) ,
+such that the πi
+X are mutually orthogonal idempotents and (πi
+X)∗H∗(X, Q) = Hi(X, Q).
+(NB: “CK decomposition” is shorthand for “Chow–K¨unneth decomposition”.)
+Remark 2.2. Murre has conjectured that any smooth projective variety should have a CK de-
+composition [40], [20].
+Definition 2.3 (Shen–Vial [47]). Let X be a smooth projective variety of dimension n, and let
+∆sm
+X ∈ A2n(X × X × X) denote the class of the small diagonal
+∆sm
+X :=
+�
+(x, x, x) | x ∈ X
+�
+⊂ X × X × X .
+An MCK decomposition is defined as a CK decomposition {πi
+X} of X that is multiplicative, i.e.
+it satisfies
+πk
+X ◦ ∆sm
+X ◦ (πi
+X × πj
+X) = 0 in A2n(X × X × X) for all i + j ̸= k .
+(NB: “MCK decomposition” is shorthand for “multiplicative Chow–K¨unneth decomposition”.)
+Remark 2.4. The small diagonal (when considered as a correspondence from X × X to X)
+induces the multiplication morphism
+∆sm
+X :
+h(X) ⊗ h(X) → h(X) in Mrat .
+Let us assume X has a CK decomposition
+h(X) =
+2n
+�
+i=0
+hi(X) in Mrat .
+By definition, this decomposition is multiplicative if for any i, j the composition
+hi(X) ⊗ hj(X) → h(X) ⊗ h(X)
+∆sm
+X
+−−→ h(X) in Mrat
+factors through hi+j(X).
+
+4
+ROBERT LATERVEER
+If X has an MCK decomposition, then setting
+Ai
+(j)(X) := (π2i−j
+X
+)∗Ai(X) ,
+one obtains a bigraded ring structure on the Chow ring: that is, the intersection product sends
+Ai
+(j)(X) ⊗ Ai′
+(j′)(X) to Ai+i′
+(j+j′)(X).
+It is conjectured that for any X with an MCK decomposition, one has
+Ai
+(j)(X)
+??= 0 for j < 0 ,
+Ai
+(0)(X) ∩ Ai
+hom(X)
+??= 0 ;
+this is related to Murre’s conjectures B and D, that have been formulated for any CK decompo-
+sition [40].
+For more background on the concept of MCK, and for examples of varieties with an MCK
+decomposition, we refer to [47, Section 8], as well as [53], [48], [13], [28], [39], [29], [30], [31],
+[12], [33], [34], [36], [42].
+3. A GENERAL CRITERION
+We develop a general criterion for having an MCK. The criterion hinges on the Franchetta
+property for families of varieties, which is defined as follows:
+Definition 3.1. Let X → B be a smooth projective morphism, where X , B are smooth quasi-
+projective varieties, and let us write Xb for the fiber over b ∈ B. We say that X → B has the
+Franchetta property in codimension j if the following holds: for every Γ ∈ Aj(X ) such that the
+restriction Γ|Xb is homologically trivial for the very general b ∈ B, the restriction Γ|b is zero in
+Aj(Xb) for all b ∈ B.
+We say that X → B has the Franchetta property if X → B has the Franchetta property in
+codimension j for all j.
+This property is studied in [45], [5], [10], [11].
+Definition 3.2. Given a family X → B as in Definition 3.1, we use the shorthand
+GDAj
+B(Xb) := Im
+�
+Aj(X ) → Aj(Xb)
+�
+⊂ Aj(Xb)
+(GDA∗() stands for the “generically defined cycles”).
+The Franchetta property for X → B means that the generically defined cycles inject into
+cohomology.
+Proposition 3.3. Let X → B be a family of smooth projective varieties of relative dimension n,
+with fiber Xb. Assume the following:
+(i) the family X ×B X → B has the Franchetta property;
+(ii) there exists a projective quotient variety P (i.e. P = P ′/G where P ′ is smooth projective
+and G ⊂ Aut(P ′) is a finite cyclic group) with trivial Chow groups (i.e. A∗
+hom(P) = 0), such
+that Xb → P is a double cover with branch locus a smooth ample divisor, for all b ∈ B.
+Then Xb admits an MCK decomposition, for all b ∈ B.
+Proof. We have the following Lefschetz-type result in cohomology:
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+5
+Lemma 3.4. Let Xb → P be as in the proposition. Then pullback
+Hi(P, Q) → Hi(Xb, Q)
+is an isomorphism for i < n, and injective for i = n.
+Proof. In case P is smooth, this is a result of Cornalba [6]. The general case is readily deduced
+from this: assume P = P ′/G where P ′ is smooth projective and G ⊂ Aut(P ′) is a finite cyclic
+group, and consider the fiber square
+X′
+b
+→
+Xb
+↓
+↓
+P ′
+→
+P .
+Cornalba’s result applies to the double cover of the left-hand vertical arrow, and so pullback
+Hi(P ′, Q) → Hi(X′
+b, Q)
+is an isomorphism for i < n, and injective for i = n. The G-action on P ′ lifts to X′
+b, and taking
+G-invariants we find that
+Hi(P, Q) = Hi(P ′, Q)G → Hi(X′
+b, Q)G = Hi(Xb, Q)
+is an isomorphism for i < n, and injective for i = n.
+□
+Since H∗(P, Q) is algebraic (this is a general fact for any variety with trivial Chow groups, cf.
+[23]), this implies that also Hi(Xb, Q) is algebraic, for all i ̸= n. More precisely, for i ̸= n odd,
+one has Hi(Xb, Q) = 0 while for i < n even, one has isomorphisms
+Ai/2(P) ∼= Hi(Xb, Q) ,
+induced by pullback. This implies that for i < n the K¨unneth components πi
+Xb are algebraic, and
+generically defined. To define the K¨unneth components πi
+Xb explicitly, let p: Xb → P denote
+the projection morphism, and let πi
+P denote the (unique) CK decomposition of P. One can then
+define
+πi
+Xb := 1/2 tΓp ◦ πi
+P ◦ Γp if i < n ,
+πi
+Xb := π2n−i
+Xb
+if i > n ,
+πn,fix
+Xb
+:= 1/2 tΓp ◦ πn
+P ◦ Γp ,
+πn,var
+Xb
+:= ∆Xb −
+�
+j̸=n
+πj
+Xb − πn,fix
+Xb
+,
+πn
+Xb := πn,fix
+Xb
++ πn,var
+Xb
+∈ An(Xb × Xb) .
+(Note that πn
+Xb = 0 in case n is odd.) The notation is meant to remind the reader that πn,fix
+Xb
+and
+πn,var
+Xb
+are projectors on the fixed part resp. the variable part of cohomology in degree n.
+
+6
+ROBERT LATERVEER
+These projectors define a generically defined CK decomposition for each Xb, i.e. all projectors
+are in GDAn
+B(Xb × Xb). This CK decomposition has the property that
+hj(Xb) := (Xb, πj
+Xb, 0) = ⊕1(∗) ∀j ̸= n ,
+hn,fix(Xb) := (Xb, πn,fix
+Xb
+, 0) = ⊕1(∗) in Mrat .
+(1)
+Let us now proceed to verify that this CK decomposition is MCK. What we need to check is
+the vanishing
+πk
+Xb ◦ ∆sm
+Xb ◦ (πi
+Xb × πj
+Xb) = 0 in A2n(Xb × Xb × Xb) for all i + j ̸= k .
+First, let us assume that at least one of the 3 integers (i, j, k) is different from n, and i+j ̸= k.
+In this case, we have that
+πk
+Xb ◦ ∆sm
+Xb ◦ (πi
+Xb × πj
+Xb) = (tπi
+Xb × tπj
+Xb × πk
+Xb)∗∆sm
+Xb
+= (π2n−i
+Xb
+× π2n−j
+Xb
+× πk
+Xb)∗∆sm
+Xb
+֒→
+�
+A∗(Xb × Xb) .
+Here the first equality is an application of Lieberman’s lemma [41, Lemma 2.1.3], and the in-
+clusion follows from property (1). The resulting cycle in � A∗(Xb × Xb) is generically defined
+(since the π∗
+Xb and ∆sm
+Xb are) and homologically trivial (since i + j ̸= k). By assumption (i), the
+resulting cycle in � A∗(Xb × Xb) is rationally trivial, and so
+πk
+Xb ◦ ∆sm
+Xb ◦ (πi
+Xb × πj
+Xb) = 0 in A2n(Xb × Xb × Xb) ,
+as desired.
+It remains to treat the case i = j = k = n. The decomposition πn
+Xb := πn,fix
+Xb
++ πn,var
+Xb
+induces
+a decomposition
+πn
+Xb ◦ ∆sm
+Xb ◦ (πn
+Xb × πn
+Xb) =πn,fix
+Xb
+◦ ∆sm
+Xb ◦ (πn,fix
+Xb
+× πn,fix
+Xb
+)
++ πn,fix
+Xb
+◦ ∆sm
+Xb ◦ (πn,fix
+Xb
+× πn,var
+Xb
+)
++ · · · · · ·
++ πn,var
+Xb
+◦ ∆sm
+Xb ◦ (πn,var
+Xb
+× πn,var
+Xb
+) in A2n(Xb × Xb × Xb) .
+Using property (1) and the Franchetta property for Xb × Xb, all summands containing πn,fix
+Xb
+vanish. One is left with the last term. To deal with the last term, we observe that the covering
+involution ι ∈ Aut(Xb) of the double cover p: Xb → P induces a splitting of the motive
+h(Xb) =h(Xb)+ ⊕ h(Xb)−
+:=(Xb, 1/2 (∆Xb + Γι), 0) ⊕ (Xb, 1/2 (∆Xb − Γι), 0) in Mrat ,
+where Γι denotes the graph of the involution ι. Moreover, there is equality
+hn,var(Xb) = h(Xb)− in Mrat .
+But the intersection product map
+h(Xb)− ⊗ h(Xb)−
+∆sm
+Xb
+−−→ h(Xb)
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+7
+factors over h(Xb)+, as is readily seen (cf. Lemma 3.5 below), which is saying exactly that
+πn,var
+Xb
+◦ ∆sm
+Xb ◦ (πn,var
+Xb
+× πn,var
+Xb
+) = 0 in A2n(Xb × Xb × Xb) .
+This closes the proof, modulo the following lemma (which is probably well-known, but we
+include a proof for completeness):
+Lemma 3.5. Let X → P be a double cover, where X and P are quotient varieties, and let
+ι ∈ Aut(X) be the covering involution. Let
+h(X)+ := (X, 1/2 (∆X + Γι, 0) ,
+h(X)− := (X, 1/2 (∆X − Γι), 0) in Mrat .
+The map of motives
+h(X)− ⊗ h(X)−
+∆sm
+X
+−−→ h(X)
+factors over h(X)+.
+To prove the lemma, let ι ∈ Aut(X) denote the covering involution. The motive h(X)− is
+defined by the projector
+∆−
+X := 1/2 (∆X − Γι)
+∈ An(X × X) .
+Plugging this in and developing, it follows that
+∆−
+X ◦ ∆sm
+X ◦ (∆−
+X × ∆−
+X) = 1/8 (∆X − Γι) ◦ ∆sm
+X ◦ (∆X×X − ∆X × Γι − Γι × ∆X + Γι × Γι)
+= 1/8
+�
+∆X ◦ ∆sm
+X ◦ (∆X × ∆X) + · · · − Γι ◦ ∆sm
+X ◦ (Γι × Γι)
+�
+= 1/8
+�
+∆sm
+X
+− (id × id ×ι)∗(���sm
+X ) − (id ×ι × id)∗(∆sm
+X ) − (ι × id × id)∗(∆sm
+X )
++ (id ×ι × ι)∗(∆sm
+X ) + (ι × id ×ι)∗(∆sm
+X ) + (ι × ι × id)∗(∆sm
+X )
+− (ι × ι × ι)∗(∆sm
+X )
+�
+in A2n(X × X × X) .
+Here the last equality is by virtue of Lieberman’s lemma [41, Lemma 2.1.3]. However, we have
+equality
+∆sm
+X = {(x, x, x) | x ∈ X} = (ι × ι × ι)∗(∆sm
+X ) in A2n(X × X × X) ,
+and so the sum of the first and last summand vanish. Likewise, we have equality
+(id ×ι×ι)∗(∆sm
+X ) = (id ×ι×ι)∗(ι×ι×ι)∗(∆sm
+X ) = (ι×id × id)∗(∆sm
+X ) in A2n(X ×X ×X) ,
+and so the other summands cancel each other pairwise. This proves the lemma.
+□
+As a first application of our general criterion, we now proceed to show the following:
+Proposition 3.6. Let X be a smooth projective variety such that X → Pn is a double cover
+ramified along a smooth divisor D ⊂ Pn, and assume either dim Hn(X, Q) > 1, or D has
+degree d > n. Then X admits an MCK decomposition.
+
+8
+ROBERT LATERVEER
+Proof. Double covers X as in the proposition are exactly the smooth hypersurfaces of degree 2d
+in the weighted projective space P := P(1n+1, d), where 2d := deg D. Let
+B ⊂ ¯B := PH0(P, OP(2d))
+denote the Zariski open parametrizing smooth hypersurfaces, and let
+B × P ⊃ X → B
+denote the universal family. In view of Proposition 3.3, it suffices to check that the family
+X ×B X → B has the Franchetta property.
+To this end, we remark that the line bundle OP(2d) is very ample (cf. Lemma 3.7 below),
+which means that the set-up verifies condition (∗2) of [12, Definition 2.5]. An application of the
+stratified projective bundle argument [12, Proposition 2.6] then implies that
+(2)
+GDA∗
+B(Xb × Xb) =
+�
+(pi)∗(h), ∆Xb
+�
+,
+where we write h ∈ A1(Xb) for the hyperplane class. The excess intersection formula [14,
+Theorem 6.3] gives an equality
+∆Xb · (pi)∗(h) = 2d
+�
+j
+(p1)∗(hj) · (p2)∗(hn+1−j) in An+1(Xb × Xb) ,
+and so equality (2) reduces to the equality
+GDA∗
+B(Xb × Xb) =
+�
+(p1)∗(h), (p2)∗(h)
+�
+⊕ Q[∆Xb] .
+The “decomposable part” ⟨(p1)∗(h), (p2)∗(h)⟩ injects into cohomology, because of the K¨unneth
+formula for H∗(Xb × Xb, Q). The class of the diagonal in cohomology is linearly independent
+from the decomposable part: indeed, if the diagonal were decomposable it would act as zero on
+the primitive cohomology
+Hn
+prim(Xb, Q) := Coker
+�
+Hn(Pn, Q) → Hn(Xb, Q)
+�
+.
+But the assumption dim Hn(Xb, Q) > 1 is equivalent to having Hn
+prim(Xb, Q) ̸= 0. This proves
+the Franchetta property for X ×B X → B, and closes the proof.
+The case d > n is a special case where Hn
+prim(Xb, Q) ̸= 0, because it is known that the
+geometric genus of Xb is
+pg(Xb) =
+�d − 1
+n
+�
+[9, Section 3.5.4].
+It remains to prove the following, which we have used above:
+Lemma 3.7. Let P := P(1n+1, d). The sheaf OP(d) is locally free and very ample.
+The assertion about the sheaf being locally free is just because d is a multiple of the weights
+of P (cf. [7, Remarques 1.8]). As for the very ampleness, we apply Delorme’s criterion [7,
+Proposition 2.3(iii)] (cf. also [4, Theorem 4.B.7]). To prove very ampleness of OP(d), we need
+to prove that the integer E as defined in [7] and [4] is equal to 0.
+Let us write x0, . . . , xn, y for the weighted homogeneous coefficients of P, where xj and y
+have weight 1 resp. d. It is readily seen that every monomial in xj, y of (weighted) degree
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+9
+m + dk (where m is a positive multiple of d, and k is any positive integer) is divisible by a
+monomial of (weighted) degree dk. This means that the integer E defined in loc. cit. is 0, and so
+[7, Proposition 2.3(iii)] implies the very ampleness of OP(d).
+This proves the lemma, and ends the proof of the proposition.
+□
+Here is another sample application of our general criterion:
+Proposition 3.8. Let X ⊂ P(1n, 2, 3) be a smooth hypersurface of (weighted) degree 6. Assume
+dim Hn(X, Q) > 1. Then X has an MCK decomposition.
+Proof. The varieties X as in the proposition are exactly the smooth double covers of P :=
+P(1n, 2) branched along a (weighted) degree 6 divisor (cf. [26, Remark 2.3] and for n = 3
+also [17, Theorem 4.2]). Let X → B denote the family of such double covers. We are going to
+check that the family X ×B X → B has the Franchetta property. Proposition 3.8 is then a special
+case of our general criterion Proposition 3.3.
+Let ¯
+X → ¯B ∼= Pr denote the universal family of all (possibly singular) hypersurfaces of
+weighted degree 6 in P. The line bundle OP(6) is very ample (cf. Lemma 3.9 below), and so the
+projection
+¯
+X × ¯B ¯
+X → P × P
+has the structure of a stratified projective bundle (with strata the diagonal ∆P and its comple-
+ment). One can thus use the stratified projective bundle argument [12, Proposition 2.6] to deduce
+the identity
+GDA∗
+B(X × X) =
+�
+(pi)∗GDA∗
+B(X), ∆X
+�
+=
+�
+(pi)∗(h), ∆X
+�
+(here, h ∈ A1(X) denotes the restriction to X of an ample generator of A1(P) ∼= Q).
+Since X ⊂ P is a hypersurface, the excess intersection formula gives
+∆X · (pi)∗(h) = ∆P|X
+∈
+�
+(pi)∗(h)
+�
+.
+The above identification thus simplifies to
+GDA∗
+B(X × X) =
+�
+(pi)∗(h)
+�
+⊕ Q[∆X] .
+The assumption that dim Hn(X, Q) > 1 implies that the diagonal ∆X is linearly independent
+in cohomology from the decomposable classes
+�
+(pi)∗(h)
+�
+(indeed, the decomposable classes act
+as zero on the primitive cohomology of X, while the diagonal acts as the identity). This shows
+that GDA∗
+B(X × X) injects into cohomology, as requested.
+Lemma 3.9. Let P := P(1n, 2, 3). The sheaf OP(6) is (locally free and) very ample.
+The assertion about the sheaf being locally free is just because 6 is a multiple of all the weights
+(cf. [7, Remarques 1.8]). As for the very ampleness, we apply Delorme’s criterion [7, Proposition
+2.3(iii)] (cf. also [4, Theorem 4.B.7]). To prove very ampleness of OP(6), we need to prove that
+the integer E defined in [7] and [4] is equal to 0.
+Let us write x1, . . . , y, z for the weighted homogeneous coefficients of P, where y and z have
+weight 2 resp. 3. We need to check that every monomial in xj, y, z of (weighted) degree 6 + 6k
+is divisible by a monomial of (weighted) degree 6k (if this is the case, then E = 0 and [7,
+
+10
+ROBERT LATERVEER
+Proposition 2.3(iii)] implies the very ampleness of OP(6)). In case the monomial contains z2, it
+is divisible by z2 and so the condition is satisfied. Assume now the monomial contains only one
+z. In case the monomial contains y3 it is divisible by y3. Next, if the monomial contains y (or
+y2) it is divisible by zyxj (for some j) and so the condition is satisfied. A monomial in z and xj
+obviously satisfies the condition. Finally, monomials in xj satisfy the condition.
+This proves the lemma, and ends the proof of the proposition.
+□
+4. MAIN RESULT
+Theorem 4.1. The following Fano threefolds admit an MCK decomposition:
+(i) hypersurfaces of weighted degree 6 in weighted projective space P(13, 2, 3);
+(ii) quartic double solids;
+(iii) sextic double solids;
+(iv) double covers of a quadric in P4 branched along the intersection with a quartic;
+(v) special Gushel–Mukai threefolds.
+Proof. The cases (ii) and (iii) are immediate applications of Proposition 3.6. The case (i) is a
+special case of Proposition 3.8.
+Before proving case (iv), let us first state a preparatory lemma:
+Lemma 4.2. Let Z ⊂ P := P(15, 2) be a smooth weighted hypersurface of degree 2. Then
+∆Z = 1
+2
+4
+�
+j=0
+hj × h4−j
+in A4(Z × Z) .
+Proof. Z is a quotient of a non-singular quadric in P5 and so Z has trivial Chow groups (i.e.
+A∗
+hom(Z) = 0). Using [9, 4.4.2], one can compute the Betti numbers of Z and one finds that
+they are the same as those of projective space P4. This means that there is a cohomological
+decomposition of the diagonal
+∆Z = 1
+2
+4
+�
+j=0
+hj × h4−j
+in H8(Z × Z, Q) .
+Since Z (and hence also Z × Z) has trivial Chow groups, the same decomposition holds modulo
+rational equivalence, proving the lemma.
+□
+Now, to prove case (iv) of Theorem 4.1, we apply our general criterion Proposition 3.3. Let
+P := P(15, 2), and let Y → B be the universal family of smooth dimensionally transverse
+complete intersections of OP(2) ⊕ OP(4), where the base B is a Zariski open
+B ⊂ ¯B := PH0(P, OP(2) ⊕ OP(4)) .
+It follows from Lemma 3.7 that OP(2) and OP(4) are very ample line bundles on P, and so
+¯Y × ¯B ¯Y → P × P is a stratified projective bundle with strata ∆P and its complement. The usual
+stratified projective bundle argument [12, Proposition 2.6] applies, and we find that
+GDA∗
+B(Y × Y ) =
+�
+(pi)∗GDA∗
+B(Y ), ∆Y
+�
+=
+�
+(pi)∗(h), ∆Y
+�
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+11
+(here, h ∈ A1(Y ) denotes the restriction to Y of an ample generator of A1(P) ∼= Q). Let
+Y = Z ∩ Z′, where Z and Z′ ⊂ P are hypersurfaces of (weighted) degree 2 and 4. Up to
+shrinking B, we may assume the hypersurface Z is smooth. Since Y ⊂ Z is a divisor, the excess
+intersection formula gives
+∆Y · (pi)∗(h) = ∆Z|Y
+in A4(Y × Y ) .
+Using Lemma 4.2, it follows that
+∆Y · (pi)∗(h) ∈
+�
+(pi)∗(h)
+�
+.
+The above identification thus simplifies to
+GDA∗
+B(Y × Y ) =
+�
+(pi)∗(h)
+�
+⊕ Q[∆Y ] .
+As before, the fact that the diagonal ∆Y is linearly independent from the decomposable corre-
+spondences in cohomology now shows that
+GDA∗
+B(Y × Y ) → H∗(Y × Y, Q)
+is injective, and so Y verifies the hypotheses of Proposition 3.3.
+The argument for case (v) is similar to that of (iv). First, in view of the spread argument
+[55, Lemma 3.2], it suffices to establish an MCK decomposition for the generic special Gushel–
+Mukai threefold Y . Thus we may assume that there exists P ⊂ Gr(2, 5), a smooth complete
+intersection of Pl¨ucker hyperplanes, and a double cover p: Y → P branched along a smooth
+Gushel–Mukai surface. We now consider the family Y → B of all double covers of P branched
+along smooth Gushel–Mukai surfaces (so B ⊂ ¯B is a Zariski open in the projectivized space of
+quadratic sections of the cone over P), and we apply our general criterion Proposition 3.3 to this
+family.
+Lemma 4.3. Let Y → B be the family of double covers of P branched along smooth Gushel–
+Mukai surfaces. The family Y → B has the Franchetta property.
+Proof. We consider the family ¯Y → ¯B with the projection to the cone C over P. This is a
+projective bundle, and so for any fiber Y = Yb with b ∈ B we have
+GDA∗
+B(Y ) = Im
+�
+A∗(C) → A∗(Y )
+�
+.
+The condition b ∈ B means exactly that Y avoids the summit of the cone C, and so (writing
+C◦ ⊂ C for the complement of the summit of the cone) we have
+(3)
+GDA∗
+B(Y ) = Im
+�
+A∗(C◦) → A∗(Y )
+�
+.
+But C◦ → P is an affine bundle, and
+A∗(P) = Im
+�
+A∗(Gr(2, 5)) → A∗(P)
+�
+=
+�
+h
+�
+,
+where h denotes the restriction to P of a Pl¨ucker hyperplane (this follows from [35, Theorem
+3.17], or alternatively from the fact that the derived category of P has a full exceptional collection
+of length 4 [44]). Thus, (3) reduces to
+GDA∗
+B(Y ) =
+�
+h
+�
+.
+This proves the Franchetta property for Y .
+□
+
+12
+ROBERT LATERVEER
+Lemma 4.4. Let Y → B be as in Lemma 4.3. The family Y ×B Y → B has the Franchetta
+property.
+Proof. Let us consider the family ¯Y × ¯B ¯Y → ¯B with the projection to C × C. This is a stratified
+projective bundle, with strata ∆C and its complement. Thus, the stratified projective bundle
+argument [12, Proposition 2.6] implies that
+GDA∗
+B(Y × Y ) =
+�
+Im
+�
+A∗(C◦ × C◦) → A∗(Y × Y )
+�
+, ∆Y
+�
+.
+Since A∗(C◦) = Im
+�
+A∗(Gr(2, 5)) → A∗(C◦), we find that
+GDA∗
+B(Y × Y ) =
+�
+Im
+�
+A∗(Gr(2, 5) × Gr(2, 5)) → A∗(Y × Y )
+�
+, ∆Y
+�
+.
+But A∗(Gr(2, 5) × Gr(2, 5)) = A∗(Gr(2, 5)) ⊗ A∗(Gr(2, 5)) since the Grassmannian has trivial
+Chow groups, and so
+GDA∗
+B(Y × Y ) =
+�
+GDB(Y ), ∆Y
+�
+=
+�
+h, ∆Y
+�
+(where the last equality follows from Lemma 4.3).
+To finish the proof of the lemma, we now claim that for any (ordinary or special) Gushel–
+Mukai threefold Y we have
+(4)
+∆Y · h ∈
+�
+Im
+�
+A∗(Gr(2, 5)) → A∗(Y )
+��
+.
+Combined with Lemma 4.3, this means that for a special Gushel–Mukai threefold Y (and Y → B
+as above) there is equality
+GDA∗
+B(Y × Y ) =
+�
+h
+�
+⊕ Q[∆Y ] .
+Then, since the diagonal is linearly independent in cohomology of
+�
+h
+�
+(since h1,2(Y ) ̸= 0), this
+proves the lemma.
+It remains to prove the claim (4). Using the spread argument [55, Lemma 3.2], it suffices to
+prove equality (4) for the very general Gushel–Mukai threefold. Thus, we may assume that Y is
+ordinary, and moreover that
+Y = Y ′ ∩ Q ,
+where Q is a quadric and Y ′ = Gr(2, 5) ∩ H1 ∩ H2 is a smooth fourfold (where H1, H2 are
+Pl¨ucker hyperplanes) and Y ′ is such that
+A∗(Y ′) = Im
+�
+A∗(Gr(2, 5)) → A∗(Y ′)
+�
+.
+(Indeed, the smooth fourfold Y ′ has trivial Chow groups [35, Corollary 4.6], and the very general
+Y ′ has no primitive cohomology, as follows from [35, Lemma 3.15]). The excess intersection
+formula then implies that
+∆Y · h = 1
+2 ∆Y ′|Y ×Y ,
+and the claim (4) follows.
+□
+Lemma 4.4 being proven, all conditions of Proposition 3.3 are met with, and so fibers Y of the
+family Y → B have an MCK decomposition; this settles (v).
+□
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+13
+5. THE TAUTOLOGICAL RING
+Theorem 5.1. Let Y be a Fano threefold of Picard number 1. Assume that Y has an MCK
+decomposition, and Y is member of a family Y → B such that Y ×B Y → B has the Franchetta
+property. For m ∈ N, let
+R∗(Y m) :=
+�
+(pi)∗(h), (pij)∗(∆Y )
+�
+⊂
+A∗(Y m)
+be the Q-subalgebra generated by pullbacks of the polarization h ∈ A1(Y ) and pullbacks of the
+diagonal ∆Y ∈ A3(Y × Y ). (Here pi and pij denote the various projections from Y m to Y resp.
+to Y × Y ). The cycle class map induces injections
+R∗(Y m) ֒→ H∗(Y m, Q) for all m ∈ N .
+Proof. This is inspired by the analogous result for cubic hypersurfaces [11, Section 2.3]. In its
+turn, the result of [11] was inspired by analogous results for hyperelliptic curves [49], [50] (cf.
+Remark 5.2 below) and for K3 surfaces [54], [57].
+Let d denote the degree of Y , and let 2b := dim H3(Y, Q). As in [11, Section 2.3], let us write
+o := 1
+dh3 ∈ A3(Y ) (the “distinguished zero-cycle”) and
+τ := ∆Y − 1
+d
+3
+�
+j=0
+hj × h3−j
+∈ A3(Y × Y )
+(this cycle τ is nothing but the projector on the motive h3(Y ) considered above). Moreover, let
+us write
+hi := (pi)∗(h) ∈ A1(Y m) ,
+oi := (pi)∗(o) ∈ A3(Y m) ,
+τi,j := (pij)∗(τ) ∈ A3(Y m) .
+We define the Q-subalgebra
+¯R∗(Y m) := ⟨oi, hi, τi,j⟩
+⊂ H∗(Y m, Q)
+(where i ranges over 1 ≤ i ≤ m, and 1 ≤ i < j ≤ m). One can prove (just as [11, Lemma 2.11]
+and [57, Lemma 2.3]) that the Q-algebra ¯R∗(Y m) is isomorphic to the free graded Q-algebra
+generated by oi, hi, τij, modulo the following relations:
+(5)
+oi · oi = 0,
+hi · oi = 0,
+h3
+i = d oi ;
+(6)
+τi,j · oi = 0,
+τi,j · hi = 0,
+τi,j · τi,j = 2b oi · oj ;
+(7)
+τi,j · τi,k = τj,k · oi ;
+(8)
+�
+σ∈S2b+2
+b+1
+�
+i=1
+τσ(2i−1),σ(2i) = 0 .
+To prove Theorem 5.1, we need to check that these relations are also verified modulo ratio-
+nal equivalence. The relations (5) take place in R∗(Y ) and so they follow from the Franchetta
+
+14
+ROBERT LATERVEER
+property for Y . The relations (6) take place in R∗(Y 2). The first and the last relations are triv-
+ially verified, because Y being Fano one has A6(Y 2) = Q. As for the second relation of (6),
+this follows from the Franchetta property for Y × Y . (Alternatively, it is possible to deduce the
+second relation from the MCK decomposition: indeed, the product τ · hi lies in A4
+(0)(Y 2), and it
+is readily checked that A4
+(0)(Y 2) injects into H8(Y 2, Q).)
+Relation (7) takes place in R∗(Y 3) and follows from the MCK relation. Indeed, we have
+∆sm
+Y
+◦ (π3
+Y × π3
+Y ) = π6
+Y ◦ ∆sm
+Y
+◦ (π3
+Y × π3
+Y ) in A6(Y 3) ,
+which (using Lieberman’s lemma) translates into
+(π3
+Y × π3
+Y × ∆Y )∗∆sm
+Y
+= (π3
+Y × π3
+Y × π6
+Y )∗∆sm
+Y
+in A6(Y 3) ,
+which means that
+τ1,3 · τ2,3 = τ1,2 · o3 in A6(Y 3) .
+It is left to consider relation (8), which takes place in R∗(Y 2b+2). To check that this relation is
+also verified modulo rational equivalence, we observe that relation (8) involves a cycle contained
+in
+A∗�
+Sym2b+2(h3(Y )
+�
+.
+But we have vanishing of the Chow motive
+Sym2b+2 h3(Y ) = 0 in Mrat ,
+because dim H3(Y, Q) = 2b and h3(Y ) is oddly finite-dimensional in the sense of Kimura [22]
+(all Fano threefolds are known to have Kimura finite-dimensional motive [51, Theorem 4]). This
+establishes relation (8), modulo rational equivalence, and ends the proof.
+□
+Remark 5.2. Given a curve C and an integer m ∈ N, one can define the tautological ring
+R∗(Cm) :=
+�
+(pi)∗(KC), (pij)∗(∆C)
+�
+⊂ A∗(Cm)
+(where pi, pij denote the various projections from Cm to C resp. C × C). Tavakol has proven
+[50, Corollary 6.4] that if C is a hyperelliptic curve, the cycle class map induces injections
+R∗(Cm) ֒→ H∗(Cm, Q) for all m ∈ N .
+On the other hand, there are many (non hyperelliptic) curves for which the tautological ring
+R∗(C3) does not inject into cohomology (this is related to the non-vanishing of the Ceresa cycle,
+cf. [50, Remark 4.2] and also [12, Example 2.3 and Remark 2.4]).
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+15
+6. A TABLE
+Table 1 below lists all Fano threefolds with Picard number 1 (the classification of Fano three-
+folds is contained in [18]). The last column indicates the existence of an MCK decomposition.
+Note that a Fano threefold X with h1,2(X) = 0 has trivial Chow groups (i.e. A∗
+hom(X) = 0), and
+so these Fano threefolds have an MCK decomposition for trivial reasons. The asterisks indicate
+new cases settled in this paper. Question marks indicate cases I am not able to settle.
+Label
+Index
+Degree
+h1,2
+Description
+MCK
+4
+4
+1
+0
+P3
+trivial
+3
+3
+2
+0
+X2 ⊂ P4
+trivial
+2.1
+2
+1
+21
+X6 ⊂ P(13, 2, 3)
+∗
+2.2
+2
+2
+10
+X4 ⊂ P(14, 2)
+∗
+2.3
+2
+3
+5
+X3 ⊂ P4
+[8], [12]
+2.4
+2
+4
+2
+X(2,2) ⊂ P5
+[32]
+2.5
+2
+5
+0
+Gr(2, 5) ∩ L ⊂ P9
+trivial
+1.2
+1
+2
+52
+X6 ⊂ P(14, 3)
+∗
+1.4.a
+1
+4
+30
+X4 ⊂ P4
+?
+1.4.b
+1
+4
+30
+X
+2:1
+−→ Q with quartic branch locus
+∗
+1.6
+1
+6
+20
+X(2,3) ⊂ P5
+[34]
+1.8
+1
+8
+14
+X(2,2,2) ⊂ P6
+?
+1.10.a
+1
+10
+10
+ordinary Gushel–Mukai 3fold
+?
+1.10.b
+1
+10
+10
+special Gushel–Mukai 3fold
+∗
+1.12
+1
+12
+7
+OGr+(5, 10) ∩ L ⊂ P15
+?
+1.14
+1
+14
+5
+Gr(2, 6) ∩ L ⊂ P14
+[37]
+1.16
+1
+16
+3
+LGr(3, 6) ∩ L ⊂ P13
+?
+1.18
+1
+18
+2
+G2/P ∩ L ⊂ P13
+[38]
+1.22
+1
+22
+0
+V (s) ⊂ Gr(3, 7)
+trivial
+TABLE 1. All Fano threefolds with Picard number 1. Here, X(d1,...,dr) denotes
+a complete intersection of multidegree (d1, . . . , dr), Q is a quadric, and L ⊂ Pr
+is a linear subspace of the appropriate dimension. The notations LGr(3, 6) and
+OGr+(5, 10) indicate the Lagrangian Grassmannian, resp. a connected compo-
+nent of the orthogonal Grassmannian. In 1.22, V (s) denotes the zero locus of a
+section of some vector bundle.
+Acknowledgements. Thanks to Mr. Kai Laterveer of the Lego University of Schiltigheim who
+provided inspiration for this work.
+
+16
+ROBERT LATERVEER
+REFERENCES
+[1] A. Beauville, Sur l’anneau de Chow d’une vari´et´e ab´elienne. Math. Ann. 273 (1986), 647—651,
+[2] A. Beauville, On the splitting of the Bloch–Beilinson filtration, in: Algebraic cycles and motives (J. Nagel
+and C. Peters, editors), London Math. Soc. Lecture Notes 344, Cambridge University Press 2007,
+[3] A. Beauville and C. Voisin, On the Chow ring of a K3 surface, J. Alg. Geom. 13 (2004), 417—426,
+[4] M. Beltrametti and L. Robbiano, Introduction to the theory of weighted projective spaces, Exposition.
+Math. 4 (1986), no. 2, 111—162,
+[5] N. Bergeron and Z. Li, Tautological classes on moduli space of hyperk¨ahler manifolds, Duke Math. J.,
+arXiv:1703.04733,
+[6] M. Cornalba, Una osservazione sulla topologia dei rivestimenti ciclici di variet`a algebriche, Bolletino
+U.M.I. (5) 18-A (1981), 323—328,
+[7] C. Delorme, Espaces projectifs anisotropes, Bull. Soc. Math. France 103 (1975), 203—223,
+[8] H. Diaz, The Chow ring of a cubic hypersurface, International Math. Research Notices 2021 no. 22
+(2021), 17071—17090,
+[9] I. Dolgachev, Weighted projective varieties, in: Group Actions and Vector Fields. Springer Berlin Hei-
+delberg, 1982, pp. 34—71,
+[10] L. Fu, R. Laterveer and Ch. Vial, The generalized Franchetta conjecture for some hyper-K¨ahler varieties
+(with an appendix joint with M. Shen), Journal Math. Pures et Appliqu´ees (9) 130 (2019), 1—35,
+[11] L. Fu, R. Laterveer and Ch. Vial, The generalized Franchetta conjecture for some hyper-K¨ahler varieties,
+II, Journal de l’Ecole Polytechnique–Math´ematiques 8 (2021), 1065—1097,
+[12] L. Fu, R. Laterveer and Ch. Vial, Multiplicative Chow–K¨unneth decompositions and varieties of coho-
+mological K3 type, Annali Mat. Pura ed Applicata 200 no. 5 (2021), 2085—2126,
+[13] L. Fu, Z. Tian and Ch. Vial, Motivic hyperk¨ahler resolution conjecture: I. Generalized Kummer varieties,
+Geometry & Topology 23-1 (2019), 427—492,
+[14] W. Fulton, Intersection theory, Springer–Verlag Ergebnisse der Mathematik, Berlin Heidelberg New York
+Tokyo 1984,
+[15] M. Green and P. Griffiths, An interesting 0-cycle, Duke Math. J. 119 no. 2 (2003), 261—313,
+[16] A. Iliev and L. Manivel, Prime Fano threefolds and integrable systems, Math. Ann. 339 no. 4 (2007),
+937—955,
+[17] V. Iskovskikh, Fano 3-folds. I, Izv. Akad. Nauk SSSR Ser. Mat. Tom. 41 no. 3 (1977), 485—527,
+[18] V. Iskovskih and Yu. Prokhorov, Algebraic Geometry V: Fano varieties, Encyclopaedia of Math. Sciences
+47, Springer-Verlag, Berlin 1999,
+[19] U. Jannsen, Motivic sheaves and filtrations on Chow groups, in: Motives (U. Jannsen et alii, eds.), Pro-
+ceedings of Symposia in Pure Mathematics Vol. 55 (1994), Part 1,
+[20] U. Jannsen, On finite-dimensional motives and Murre’s conjecture, in: Algebraic cycles and motives (J.
+Nagel and C. Peters, editors), Cambridge University Press, Cambridge 2007,
+[21] S.-I. Kimura, On the characterization of Alexander schemes, Comp. Math. 92 no. 3 (1994), 273—284,
+[22] S.-I. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 no. 1 (2005), 173—201,
+[23] S.-I. Kimura, Surjectivity of the cycle map for Chow motives, in: Motives and algebraic cycles, Fields
+Inst. Commun., vol. 56, Amer. Math. Soc., Providence, RI, 2009, pp. 157—165,
+[24] A. Kuznetsov, Hyperplane sections and derived categories, Izv. Math. 70 (2006), 447—547,
+[25] A. Kuznetsov, Derived categories of Fano threefolds, Proc. V. A. Steklov Inst. Math 264 (2009), 110—
+122,
+[26] A. Kuznetsov and Y. Prokhorov, On higher-dimensional del Pezzo varieties, arXiv:2206.01549,
+[27] R. Laterveer, A family of cubic fourfolds with finite-dimensional motive, Journal of the Mathematical
+Society of Japan 70 no. 4 (2018), 1453—1473,
+[28] R. Laterveer, A remark on the Chow ring of K¨uchle fourfolds of type d3, Bulletin Australian Math. Soc.
+100 no. 3 (2019), 410—418,
+[29] R. Laterveer, Algebraic cycles and Verra fourfolds, Tohoku Math. J. 72 no. 3 (2020), 451—485,
+
+SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION
+17
+[30] R. Laterveer, On the Chow ring of certain Fano fourfolds, Ann. Univ. Paedagog. Crac. Stud. Math. 19
+(2020), 39—52,
+[31] R. Laterveer, On the Chow ring of Fano varieties of type S2, Abh. Math. Semin. Univ. Hambg. 90 (2020),
+17—28,
+[32] R. Laterveer, Algebraic cycles and intersections of 2 quadrics, Mediterranean Journal of Mathematics 18
+no. 4 (2021),
+[33] R. Laterveer, Algebraic cycles and intersections of three quadrics, Mathematical Proceedings of the Cam-
+bridge Philosophical Society 173 no. 2 (2022), 349—367,
+[34] R. Laterveer, Algebraic cycles and intersections of a quadric and a cubic, Forum Mathematicum 33 no. 3
+(2021), 845—855,
+[35] R. Laterveer, Motives and the Pfaffian–Grassmannian equivalence, Journal of the London Math. Soc. 104
+no. 4 (2021), 1738—1764,
+[36] R. Laterveer, On the Chow ring of Fano varieties on the Fatighenti-Mongardi list, Communications in
+Algebra 50 no. 1 (2022), 131—145,
+[37] R. Laterveer, Algebraic cycles and Fano threefolds of genus 8, Portugal. Math. (N.S.) Vol. 78, Fasc. 3-4
+(2021), 255—280,
+[38] R. Laterveer, Algebraic cycles and Fano threefolds of genus 10, preprint,
+[39] R. Laterveer and Ch. Vial, On the Chow ring of Cynk–Hulek Calabi–Yau varieties and Schreieder vari-
+eties, Canadian Journal of Math. 72 no. 2 (2020), 505—536,
+[40] J. Murre, On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II, Indag.
+Math. 4 (1993), 177—201,
+[41] J. Murre, J. Nagel and C. Peters, Lectures on the theory of pure motives, Amer. Math. Soc. University
+Lecture Series 61, Providence 2013,
+[42] A. Negut, G. Oberdieck and Q. Yin, Motivic decompositions for the Hilbert scheme of points of a K3
+surface, arXiv:1912.09320v1,
+[43] K. O’Grady, Decomposable cycles and Noether-Lefschetz loci, Doc. Math. 21 (2016), 661—687,
+[44] D. Orlov, An Exceptional Collection of Vector Bundles on the Threefold V5, Vestn. Mosk. Univ., Ser. 1:
+Mat., Mekh., No. 5, 69–71 (1991) [Moscow Univ. Math. Bull. 46 (5), 48–50 (1991)],
+[45] N. Pavic, J. Shen and Q. Yin, On O’Grady’s generalized Franchetta conjecture, Int. Math. Res. Notices
+(2016), 1—13,
+[46] T. Scholl, Classical motives, in: Motives (U. Jannsen et alii, eds.), Proceedings of Symposia in Pure
+Mathematics Vol. 55 (1994), Part 1,
+[47] M. Shen and Ch. Vial, The Fourier transform for certain hyperK¨ahler fourfolds, Memoirs of the AMS
+240 (2016), no. 1139,
+[48] M. Shen and Ch. Vial, The motive of the Hilbert cube X[3], Forum Math. Sigma 4 (2016), 55 pp.,
+[49] M. Tavakol, The tautological ring of the moduli space M rt
+2 , International Math. Research Notices 2014
+no. 24 (2014), 6661—6683,
+[50] M. Tavakol, Tautological classes on the moduli space of hyperelliptic curves with rational tails, J. Pure
+Applied Algebra 222 no. 8 (2018), 2040—2062,
+[51] Ch. Vial, Projectors on the intermediate algebraic Jacobians, New York J. Math. 19 (2013), 793—822,
+[52] Ch. Vial, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proceedings of the
+LMS 106 no. 2 (2013), 410—444,
+[53] Ch. Vial, On the motive of some hyperk¨ahler varieties, J. Reine Angew. Math. 725 (2017), 235—247,
+[54] C. Voisin, On the Chow ring of certain algebraic hyperk¨ahler manifolds, Pure Appl. Math. Q. 4 no. 3 part
+2 (2008), 613—649,
+[55] C. Voisin, Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Princeton Univer-
+sity Press, Princeton and Oxford, 2014,
+[56] Q. Yin, The generic nontriviality of the Faber–Pandharipande cycle, International Math. Res. Notices
+2015 no. 5 (2015), 1263—1277,
+
+18
+ROBERT LATERVEER
+[57] Q. Yin, Finite-dimensionality and cycles on powers of K3 surfaces, Comment. Math. Helv. 90 (2015),
+503–511.
+INSTITUT DE RECHERCHE MATH´EMATIQUE AVANC´EE, CNRS – UNIVERSIT´E DE STRASBOURG, 7 RUE
+REN´E DESCARTES, 67084 STRASBOURG CEDEX, FRANCE.
+Email address: [email protected]
+

CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:fc2afdd2090143536748eeb9b7768b7a4db132711abb4f702191ae6b42a0882c
+size 8730594

CdE4T4oBgHgl3EQfFgzX/content/tmp_files/2301.04887v1.pdf.txt

@@ -0,0 +1,1466 @@
+Learning Partial Differential Equations by Spectral Approximates of General
+Sobolev Spaces
+Juan Esteban Suarez Cardona 1 Michael Hecht 1
+Abstract
+We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev
+polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving
+a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the
+physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as
+PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast
+class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the
+exponential convergence rate of the underlying variational gradient descent.
+As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance
+Computing (HPC) on a local machine. This gain in efficiency is complemented by an increase of approximation
+power, outperforming PINN alternatives in both accuracy and runtime.
+Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev
+space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse
+PDE problems.
+1. Introduction
+Partial differential equations (PDEs) are omnipresent mathematical models governing the dynamics and (physical)
+laws of complex systems (Jost, 2002; Brezis, 2011). However, analytic PDE solutions are rarely known for most of the
+systems being the centre of current research. Therefore, there is a strong demand on efficient and accurate numerical solvers
+and simulations.
+Main classic numerical solvers divide into: Finite Elements (Ern & Guermond, 2004); Finite Differences (LeVeque, 2007);
+Finite Volumes(Eymard et al., 2000); Spectral Methods (Bernardi & Maday, 1997; Canuto et al., 2007) and Particle Methods
+(Li & Liu, 2007).
+Machine learning methods such as: Physics-Informed GAN (Arjovsky et al., 2017), Deep Galerkin Method (Sirignano
+& Spiliopoulos, 2018), and Physics Informed Neural Networks (PINNs) (Raissi et al., 2019), gain big traction in the
+scientific computing community. In contrast to classic solvers, PINNs provide a neural net (NN) surrogate model e.g.,
+ˆu : (−1, 1)m −→ R, m ∈ N, parametrising the solution space of the PDEs and enabling to solve inverse problems like
+inference of PDE parameters or initial condition detection. PINN-learning is given by minimising a variational problem,
+which is typically formulated in L2-loss terms
+�
+Ω
+��ˆu(x) − u(x)
+��2dΩ ≈
+1
+|P|
+�
+p∈P
+��ˆu(p) − u(p)
+��2
+(1)
+being approximated by the mean square error (MSE) in random (data) nodes P, (Yang et al., 2020),(Long et al., 2018).
+The applications of PINNs range from fluid mechanics (Jin et al., 2020) to biology (Lagergren et al., 2020) or medicine
+(Sahli Costabal et al., 2020), physics (Ellis et al., 2021) and beyond.
+1CASUS - Center for Advanced System Understanding, Helmholtz-Zentrum Dresden-Rossendorf e.V. (HZDR), G¨orlitz, Germany.
+Correspondence to: Juan Esteban Suarez Cardona <[email protected]>, Michael Hecht <[email protected]>.
+This work was partially funded by the Center of Advanced Systems Understanding (CASUS), financed by Germany’s Federal Ministry of
+Education and Research (BMBF) and by the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis of the
+budget approved by the Saxon State Parliament.
+arXiv:2301.04887v1  [math.NA]  12 Jan 2023
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+1.1. Related work – Physics Informed Neural Nets (PINNs)
+We identify the essential approaches addressing stability and accuracy of PINNs below.
+1.1.1. VARIATIONAL PINNS (VPINNS)
+VPINNs were introduced in (Kharazmi et al., 2019; 2020) resting on variational Sobolev losses for PINN-training.
+The approach exploits analytic integration and differentiation formulas of shallow neural networks with specified activation
+functions. The method is extended by using quadrature rules and automatic differentiation for computing the losses and is
+complemented by a domain decomposition approach. The drawback of VPINNs, we identify and demonstrate here, is their
+highly consuming runtime performance, preventing the approach to be applicable for multi-dimensional PDE problems.
+1.1.2. INVERSE DIRICHLET LOSS BALANCING
+The Inverse Dirichlet method (Maddu et al., 2021) was shown to increase the numerical stability of PINNS by
+dynamically balancing the occurring variational gradient amplitudes, which if unbalanced cause numerical stiffness
+phenomena (Wang et al., 2021). However, the PINN formulation rests on classic MSE losses, limiting the approach to
+consider only strong PDE problem formulations.
+1.1.3. SOBOLEV CUBATURES PINNS (SC-PINN)
+In our prior work (Cardona & Hecht, 2022) we gave a PINN formulation, by replacing the MSE loss by Sobolev
+Cubatures. In contrast to ID-PINNs approximating Sobolev losses enables the approach to consider PDE problems in the
+weak and strong sense. As a consequence, the automatic differentiation (A.D.) is replaced by polynomial differentiation
+implicitly realised in the Sobolev cubatures. As we demonstrated this results in an increase of accuracy and runtime
+efficiency by several orders of magnitude compared to PINNs relying on A.D.
+1.2. Related Work - Classic spectral methods
+Spectral methods are well established techniques solving PDEs and ODEs. Hereby, one aims to approximate the PDE
+solution by an expansion u = �
+α∈A cαϕα, A ⊆ Nm with respect to a specific finite dimensional space Π = span{ϕα}α∈A
+generated by a chosen basis, e.g., Fourier basis for periodic PDEs or Jacobi-Chebyshev polynomials for general, non-periodic
+problems. The coefficients of the expansion are constrained by the PDE and its corresponding boundary conditions. For
+example: Consider a (non-linear) differential operator L and the equation
+Lu = f
+in Ω,
+with homogeneous Dirichlet boundary conditions. By sampling the function f = f(pα)α∈A ∈ R|A|, A ⊆ Nm in some
+node set P = {pα}α∈A determination of the coefficients C := (cα)α∈A ⊆ R|A| demands solving the truncated (non-linear)
+system:
+L[C] − f
+!= 0 ,
+where L = L|Π denotes the truncated operator. This system of equations is typically formulated as the solution of the
+weighted residual:
+⟨ϕi, L[C] − f⟩
+!= 0 ,
+∀α ∈ A.
+Depending on the choice of the test functions ϕi we obtain pseudo-spectral methods or Galerkin spectral methods (Kang &
+Suh, 2008; Canuto et al., 2007; Bernardi & Maday, 1997). If the operator L is linear, the problem is reduced to solving a
+linear system. In the non-linear case, least square methods with Newton-Raphson minimiser are commonly used (Hessari
+& Shin, 2013; Kim & Shin, 2006). Extending this formulation to inverse problems (inferring parameters) with general
+boundary conditions and/or additional constraints without causing ill-conditioned problems is a unresolved challenge for
+classic spectral methods. Our contribution relies on providing the demanded extensions, enabling to addresses general
+forward and inverse PDE problems in a numerically stable, efficient and accurate fashion.
+1.3. Contribution
+We present a generalised soft-constrained spectral method that results in a λ-convex variational optimisation problem
+for linear and a class of non-linear PDEs. We theoretically guarantee exponentially fast convergence of the resulting
+variational gradient descent. While established PINN alternatives result in non-convex variational problems, already for
+linear PDEs, the spectral polynomial surrogate models (PSMs) provide approximates of the PDE solutions outperforming
+PINNs in runtime and accuracy, as demonstrated in Section 4.
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+Our approach rests on using Chebyshev Polynomial Surrogate Models (PSMs):
+ˆu(x, Θ) =
+�
+α∈Am,n
+θαTα(x) ,
+Θ = (θα)α∈Am,n ∈ R|Am,n|, x ∈ Rm ,
+(2)
+where Am,n denotes a multi-index set, see Section 2.1, and Tα denotes the Cheybshev polynomial basis of first kind given
+by the relation:
+Tα(cos(x)) = Tα(cos(x1), . . . , cos(xm)) =
+m
+�
+i=1
+cos(αixi) = cos(αx)
+(3)
+for all α ∈ Am,n. The Chebyshev polynomials are widely used due to their excellent approximation properties extensively
+discussed in (Trefethen, 2019). In our recent work (Cardona & Hecht, 2022), we already formulated (weak) PDE losses by
+generalising classic Gauss-Legendre cubature rules, we termed Sobolev cubatures. As aforementioned, for linear and a
+class of non-linear PDEs the induced variational λ-convex gradient flows possess an exponential rate of convergence. The
+resulting PSMs deliver an increase of accuracy up to 10 orders of magnitude, by reducing the runtime costs up to 3 orders of
+magnitude compared to PINN alternatives. Moreover, we demonstrate the PSMs to be as flexible as PINNs for addressing
+inverse PDE problems, such as PDE-parameter inference.
+In contrast to PINNs, the prominent PDE problems considered in Section 4 were solved by our PSM-method without
+High Performance Computing (HPC) on a local machine. We consequently expect the approach to deeply impact current
+methodology addressing computational challenges arising across all scientific disciplines and believe that even currently
+non-reachable (high-dimensional, strongly varying) PDE problems can be successfully resolved due to our contribution.
+2. PDE theory
+In this section we introduce the mathematical concepts on which our approach rest. This includes the formulation of
+Sobolev cubatures (Cardona & Hecht, 2022), approximating general Sobolev norms. To start with we fix the notation used
+throughout this article.
+2.1. Notation and basic concepts
+We denote with Ω = (−1, 1)m the open m-dimensional standard hypercube, with ��Ω = [−1, 1]m its closure, and with
+∂Ω its boundary. ∥x∥p = (�m
+i=1 |xi|p)1/p, x = (x1, . . . , xm) ∈ Rm, 1 ≤ p < ∞, ∥x∥∞ = max1≤i≤m |xi| denotes the
+lp-norm, and ⟨x, y⟩, ∥x∥, x, y ∈ Rm the standard Euclidean inner product and norm on Rm.
+Moreover, Πm,n = span{xα}∥α∥∞≤n denotes the R-vector space of all real polynomials in m variables spanned by
+all monomials xα = �m
+i=1 xαi
+i
+of maximum degree n ∈ N, whereas Πm,n(∂Ω) = {Q|Ω : Q ∈ Πm,n} denotes the space of
+restricted polynomials with support Ω.
+We consider the multi-index set Am,n = {α ∈ Nm : ∥α∥∞ ≤ n} with |Am,n| = (n + 1)m and order Am,n with
+respect to the lexicographic order ⪯ on Nm starting from last entry to the 1st, e.g., (5, 3, 1) ⪯ (1, 0, 3) ⪯ (1, 1, 3). Let
+D ∈ R|Am,n|×|Am,n| be a matrix we slightly abuse notation by writing
+D = (dα,β)α,β∈Am,n ,
+(4)
+where dα,β ∈ R is the α-th, β-th entry of D.
+2.2. Sobolev space theory
+We recommend (Adams & Fournier, 2003; Neuberger, 2008; Brezis, 2011) for an excellent overview on functional
+analysis and Sobolev space theory including the concepts we shortly summarise: We denote with Ck(Ω, R), k ∈ N∪{∞} the
+Banach spaces of all k-times continuously differentiable functions with norm ∥f∥Ck(Ω) = �k
+i=0 supx∈Ω,∥α∥1=i |Dαf(x)|.
+The Sobolev spaces
+Hk(Ω, R) =
+�
+f ∈ L2(Ω, R) : Dαf ∈ L2(Ω, R)
+�
+,
+∥α∥1 = �m
+i=1 αi ≤ k, k ∈ N are given by all L2-integrable functions f : Ω −→ R with existing L2-integrable weak
+derivatives Dαf = ∂α1
+x1 . . . ∂αm
+xm f up to order k. In fact, Hk(Ω, R) is a Hilbert space with inner product
+⟨f, g⟩Hk(Ω) =
+�
+0≤∥α∥1≤k
+⟨Dαf, Dαg⟩L2(Ω)
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+and norm ∥f∥2
+Hk(Ω) = ⟨f, f⟩Hk(Ω). Thus, the embeddings j : Hk(Ω, R) �→ Hk′(Ω) are well defined and continuous for
+all k′ ≤ k due to ∥ · ∥Hk′(Ω ≤ ∥ · ∥Hk(Ω,R), whereas H0(Ω, R) = L2(Ω, R), with ⟨f, g⟩L2(Ω) =
+�
+Ω
+f · g dΩ.
+For k ≥ 1 the trace operator
+tr : Hk(Ω, R) −→ L2(∂Ω, R)
+(5)
+is defined as usual as the Hk-extension of the classic continuous trace tr(u) = u|∂Ω with domain dom(tr) = C0(¯Ω, R).
+The Sobolev spaces with zero trace are denoted as usual with Hk
+0 (Ω, R) = {u ∈ Hk(Ω, R) : tr(u) = 0}, k ≥ 1 and can be
+alternatively defined as completion of the space of smooth functions that vanish on the boundary ∂Ω of Ω, i.e.,
+Hk
+0 (Ω, R) = C∞
+0 (Ω, R)
+∥·∥Hk(Ω) ,
+C∞
+0 (Ω, R) = {f ∈ C∞(Ω, R) : f|∂Ω = 0} .
+We further consider the space of all distributions D′(Ω) = {F : C∞
+0 (¯Ω) −→ R} also known as generalised functions
+(being the dual space of all test functions C∞
+0 (¯Ω) = {f ∈ C∞(Ω) : f|∂Ω = 0} with respect to the canonical LF topology).
+We associate the negative order Sobolev space as the completion of D′(Ω) with respect to the following norm
+H−k(Ω, R) := D′(Ω)
+∥·∥H−k(Ω) ,
+∥F∥H−k(Ω,R) =
+sup
+u∈Hk(Ω,R)
+|Fu|
+∥u∥Hk(Ω,R)
+,
+(6)
+yielding a separable, reflexive Hilbert space (Lax, 1955).
+The weak PDE formulations and their underlying Hilbert space choice we will propose later on require the notion of
+adjoint (differential) operators. We recall the definition.
+Definition 1 (Adjoint operators). Let (K, ∥ · ∥K), (H, ∥ · ∥H) be Hilbert spaces and T : dom(T) ⊆ K −→ H, T ∗ :
+dom(T ∗) ⊆ H −→ K be linear operators with dense domains. Then T ∗ is called an adjoint operator of T if and only if
+⟨Tx, y⟩H = ⟨x, T ∗y⟩K
+for all x ∈ dom(T) and y ∈ dom(T ∗).
+Example 2. Consider ∂xi : L2(Ω, R) −→ L2(Ω, R) as the differential operator in the weak sense. Then its domain is given
+by dom(∂xi) = H1(Ω, R) ⊆ L2(Ω, R), which is a dense subset. Following Definition 1, and applying integration by parts,
+an adjoint operator ∂∗
+xi : L2(Ω, R) −→ L2(Ω, R), with domain dom(∂∗
+xi) = H1
+0(Ω, R) is given by ∂∗
+xi = −∂xi.
+We link the spaces H−k(Ω, R) and Hk(Ω, R) due to the following fact.
+Proposition 3. Let j : Hk(Ω, R) �→ L2(Ω, R), k ∈ N be the embedding with adjoint operator j∗ : L2(Ω, R) −→
+Hk(Ω, R). Let f, g ∈ L2(Ω, R) and the distributions F = ⟨f, ·⟩L2(Ω,R), G = ⟨g, ·⟩L2(Ω,R) ∈ H−k(Ω, R), with f ∈
+L2(Ω, R). Then
+∥F∥H−k(Ω,R) = ∥j∗f∥Hk(Ω) ,
+⟨F, G⟩H−k(Ω) = ⟨j∗f, j∗g⟩Hk(Ω) .
+Proof. The proof is derived directly from the definition of the H−k(Ω, R)-norm in Eq. (6):
+∥j∗f∥Hk(Ω) =
+∥j∗f∥2
+Hk(Ω)
+∥j∗f∥Hk(Ω)
+= |⟨jf, j∗f⟩L2(Ω)|
+∥j∗f∥Hk(Ω)
+= |⟨f, j∗f⟩L2(Ω)|
+∥j∗f∥Hk(Ω)
+≤
+sup
+u∈Hk(Ω,R)
+|⟨f, u⟩L2(Ω)|
+∥u∥Hk(Ω)
+= ∥F∥H−k(Ω) .
+Vice versa, applying the Cauchy-Schwarz inequality yields
+∥F∥H−k(Ω,R) =
+sup
+u∈Hk(Ω,R)
+|⟨f, ju⟩L2(Ω)|
+∥u∥Hk(Ω)
+=
+sup
+u∈Hk(Ω,R)
+|⟨j∗f, u⟩Hk(Ω)|
+∥u∥Hk(Ω)
+≤
+sup
+u∈Hk(Ω,R)
+∥j∗f∥Hk(Ω)∥u∥Hk(Ω)
+∥u∥Hk(Ω)
+= ∥j∗f∥Hk(Ω) ,
+implying the claimed equality. The statement for the inner product follows analogously.
+A main ingredient of all further considerations are the truncated L2- or Hk-inner products that rest on adaptions of
+classic Gauss-Legendre cubatures, which we provide next.
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+2.3. Orthogonal polynomials and Gauss-Legendre cubatures
+Here, we recapture the underlying concept of orthogonal polynomials: Let m, n ∈ N and Pm,n = ⊕m
+i=1Legn ⊆ Ω be
+the we the m-dimensional Legendre grids, where Legn = {p0, . . . , pn} are the n + 1 Legendre nodes given by the roots of
+the Legendre polynomials of degree n + 2 We denote pα = (pα1, . . . , pαm) ∈ Pm,n, α ∈ Am,n. It is a classic fact (Stroud,
+1971; 2011; Trefethen, 2017; 2019), that the Lagrange polynomials Lα ∈ Πm,n, α ∈ Am,n given by
+Lα =
+m
+�
+i=1
+lαi,i ,
+lj,i =
+m
+�
+j̸=i,j=0
+xi − pj
+pi − pj
+,
+(7)
+satisfy Lα(pβ) = δα,β, ∀ α, β ∈ Am,n and form an orthogonal L2-basis of Πm,n, i.e.,
+⟨Lα, Lβ⟩L2(Ω) =
+�
+Ω
+Lα(x)Lβ(x)dΩ = wαδα,β ,
+∀ α, β ∈ Am,n, where δ·,· denotes the Kronecker delta and
+wα = ∥Lα∥2
+L2(Ω)
+(8)
+the efficiently computable Gauss-Legendre cubature weight (Stroud, 1971; 2011; Trefethen, 2017; 2019). Consequently, for
+any polynomial Q ∈ Πm,2n+1 of degree 2n + 1 the following cubature rule applies:
+�
+Ω
+Q(x)dΩ =
+�
+α∈Am,n
+wαQ(pα) .
+(9)
+Summarising: Polynomials of degree 2n + 1 can be (numerically) integrated exactly when sampled on the Legendre grid
+Pm,n of order n + 1. Thanks to |Pm,n| = (n + 1)m ≪ (2n + 1)m this makes Gauss-Legendre integration a very powerful
+scheme yielding
+⟨Q1, Q2⟩L2(Ω) =
+�
+Ωm
+Q1(x)Q2(x)dΩm =
+�
+α∈Am,n
+Q1(pα)Q2(pα)wα ,
+(10)
+for all Q1, Q2 ∈ Πm,n. In light of this fact, we propose the following definition.
+Definition 4 (Legendre interpolation and L2-projection ). Let m, n ∈ N, Pm,n be the Legendre grid and Lα, α ∈ Am,n be
+the corresponding Lagrange polynomials from Eq.(7). For continuous functions f : ¯Ω −→ R we denote with
+Im,n : C0(Ω, R) −→ Πm,n ,
+Im,n(f) =
+�
+α∈Am,n
+f(pα)Lα ∈ Πm,n
+(11)
+the interpolation operator. Moreover, we denote with
+πm,n : L2(Ω, R) −→ Πm,n ,
+πm,n(f) =
+�
+α∈Am,n
+1
+wα
+⟨f, Lα⟩L2(Ω)Lα ∈ Πm,n
+(12)
+the L2-projection.
+Remark 5. It is important to note that Im,n(f) ̸= πm,n(f) in general. However, both operators are projections that due to
+Eq. (10) satisfy
+πm,n(πm,n(f)) = πm,n(f) ,
+Im,n(Im,n(f)) = Im,n(f) ,
+Im,n(πm,n(f)) = Im,n(f) ,
+πm,n(Im,n(f)) = Im,n(f) .
+In fact, both concepts can deliver exponential fast approximation rates (truncation errors) in case the considered function f
+is analytic (Trefethen, 2019).
+How differential operators acting on polynomial spaces can be understood due to these concepts is proposed in the
+next section.
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+2.4. Truncated differential and adjoint operators
+Based on Eq. (7) we derive exact matrix representations of differential operators acting on the polynomial spaces
+Πm,n. This allows to extend Eq. (10) and deliver approximates of the Sobolev norms for general functions f ∈ Hk(Ω, R),
+k ∈ N.
+For Lα ∈ Πm,n from Eq. (7) and 1 ≤ i ≤ m the computation of the values dα,β = ∂xiLα(pβ), pβ ∈ Pm,n,
+∀ β ∈ Am,n yield the Lagrange expansion
+∂xiLα(x) =
+�
+β∈Am,n
+dα,βLβ(x) .
+(13)
+Consequently, the matrix
+Di = (dα,β)α,β∈Am,n ∈ R|Am,n|×|Am,n| ,
+(14)
+represents the finite dimensional truncation of the differential operator ∂xi : C1(Ω, R) −→ C0(Ω, R) to the polynomial
+space Πm,n and for β ∈ Nm we set
+Dβ =
+m
+�
+j=1
+Dβi ,
+with D0 = I ,
+(15)
+to be the approximation of the differential operator ∂β := ∂β1
+x1 . . . ∂βm
+xm .
+For representing the truncation of general adjoint operators we we consider the Legendre grid Pm,n = {pα : α ∈
+Am,n}, m, n, ∈ N the positive, symmetric Gauss-Legendre cubature weight matrix Wm,n = diag(wα)α∈Am,n, and the
+evaluation vector f = (f(Pα))α∈Am,n ∈ R|Am,n| for a given function f : Ω −→ R. With these ingredients we state:
+Proposition 6. Let Dβ : L2(Ω, R) −→ L2(Ω, R), β ∈ Nm be a differential operator and Dβ : Πm,n(Ω) −→ Πm,n(Ω) be
+its truncation to the polynomial space. Then the matrix representation of the truncated adjoint operator D∗
+β : Πm,n(Ω) −→
+Πm,n(Ω) is given by:
+D∗
+β = W−1
+m,nD⊤
+β Wm,n .
+(16)
+Proof. We derive Eq. (16) due to the Gauss-cubature in terms of Eq. (10). Let Q1, Q2 ∈ Πm,n, and denote with q1 =
+(Q1(pα))α∈Am,n, q2 = (Q2(pα))α∈Am,n ∈ R|Am,n| the corresponding evaluation vectors. Then we compute
+⟨DβQ1, Q2⟩L2(Ω,R) = ⟨Dq1, Wm,nq2⟩ = q⊤
+1 D⊤
+β Wm,nq2 = q⊤
+1 Wm,nW−1
+m,nD⊤
+β Wm,nq2
+= ⟨W⊤
+m,nq1, D∗
+βq2⟩ = ⟨q1, Wm,nD∗
+βq2⟩ = ⟨Q1, D∗
+βQ2⟩L2(Ω,R) ,
+proving the statement.
+We provide a matrix representation of the truncation of the adjoint operator j∗ : Hk(Ω, R) −→ L2(Ω, R) of the
+embedding j : Hk(Ω, R) −→ L2(Ω, R).
+Theorem 7. Let j∗ : L2(Ω, R) −→ Hk(Ω, R) be the adjoint operator of the embedding j : Hk(Ω, R) −→ L2(Ω, R).
+Denote with Dβ the representations of the derivatives from Eq. (15) then its truncation J∗ : Πm,n(Ω) ⊆ L2(Ω, R) −→
+Πm,n(Ω) ⊆ Hk(Ω, R) can be represented by the matrix J∗ ∈ R|Am,n|×|Am,n| given by
+J∗ =
+� �
+|β|≤k
+D∗
+βDβ
+�−1
+.
+(17)
+Proof. Let Q1, Q2 ∈ Πm,n, Pm,n the Legendre grid and q1 = (Q1(pα))α∈Am,n, q2 = (Q2(pα))α∈Am,n ∈ R|Am,n| the
+evaluation vectors,respectively. Then we compute
+⟨Q1, Q2⟩Hk(Ω) =
+�
+|β|≤k
+⟨DβQ1, DβQ2⟩L2(Ω,R) =
+�
+|β|≤k
+⟨D∗
+βDβQ1, Q2⟩L2(Ω,R)
+= ⟨
+� �
+|β|≤k
+D∗
+βDβ
+�
+Q1, Q2⟩L2(Ω,R) .
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+Thus, setting J∗−1 := �
+|β|≤k D∗
+βDβ yields that due to the identity above J∗−1 is a symmetric and positive definite linear
+operator on a finite dimensional space implying its invertibility. Due to
+⟨
+� �
+|β|≤k
+D∗
+βDβ
+�
+Q1, Q2⟩L2(Ω,R) = ⟨
+� �
+|β|≤k
+D∗
+βDβ
+�
+q1, q2⟩
+we realise that J∗−1 := �
+|β|≤k D∗
+βDβ represents J∗−1.
+As introduced, the PSMs rely on the Chebyshev polynomials {Tα}α∈Am,n, m, n ∈ N, Eq. (3). For later purpose we
+provide the basis transformation between the Tα and the Lagrange basis Lα in the Legendre grid Pm,n. That is to consider
+the matrix
+T = (Tβ(pα))α,β∈Am,n ∈ R|Am,n|×|Am,n|
+and its inverse
+T−1 ∈ R|Am,n|×|Am,n| .
+(18)
+Given Lagrange coefficients C = (cα)α∈Am,n of a polynomial Q = �
+α∈Am,n cαLα, Θ = (θα)α∈Am,n = T−1C yields the
+coefficients of its Chebyshev representation Q = �
+α∈Am,n θαTα. Vice versa D = (dα)α∈Am,n = TΘ yields the Lagrange
+coefficients of its Chebyshev expansion. We close this section, by deriving a matrix representation of the trace operator,
+Eq. (5):
+Definition 8 (Truncated trace operator). Let tr : Hk(Ω, R) −→ L2(∂Ω, R) be the trace operator, Eq. (5). Denote with
+P ±
+m−1,n,j ⊆ ∂Ω±
+j the m-1-dimensional Legendre grids for each of the faces ∂Ω±
+j = {x ∈ Ω : xj = ±1} of the hypercube
+Ω. Then the matrix S±
+m,n,j ∈∈ R|Am−1,n|×|Am,n| with
+S±
+m,n,j = (Tα(pγ))(γ,α)∈Am−1,n×Am,n ,
+pγ ∈ P ±
+m−1,n,j , j = 1, . . . , m .
+(19)
+represents the truncated trace operator tr : Πm,n −→ Πm−1,n(∂Ω±
+j ) for each of the faces ∂Ω±
+j .
+The derived representations of the truncated differential and adjoint operators enable to derive cubature rules for the
+truncated Sobolev spaces.
+2.5. Sobolev cubatures
+Based on the classic Gauss-Legendre cubature Eq. (10) we, here, derive general Sobolev cubatures. We start by
+defining:
+Definition 9 (Truncated (dual) inner product and norm). For β ∈ Nm, ∥β∥1 ≤ k, m, n ∈ N we consider the truncated
+differential operator Dβ and its adjoint Dβ : Πm,n(Ω) −→ Πm,n(Ω), D∗
+β : Πm,n(Ω) −→ Πm,n(Ω) satisfying
+⟨DβQ1, Q2⟩L2(Ω) = ⟨Q1, D∗
+βQ2⟩L2(Ω) ,
+∀Q1, Q2 ∈ Πm,n
+Given the matrix representations Dβ, D∗
+β = W −1
+m,nDT
+β Wm,n from Proposition 6, J∗ from Eq. (17) and its formal dual
+J∗ =
+� �
+|β|≤k
+D∗
+βDβ
+�−1
+,
+J∗ =
+� �
+|β|≤k
+DβD∗
+β
+�−1
+,
+we introduce
+Wm,n,k = Wm,nJ∗−1 , Wm,n,−k = Wm,nJ∗ ,
+Wm,n,k = Wm,nJ∗−1 , Wm,n,−k = Wm,nJ∗ ,
+and for f, g ∈ Πm,n and their dual distributions F = ⟨f, ·⟩L2(Ω), G = ⟨g, ·⟩L2(Ω) we set
+⟨f, g⟩Hk(Ω)
+=
+�
+β∈Nm,∥β∥1≤k
+⟨Dβf, Dβg⟩L2(Ω)
+=⟨f, Wm,n,kg⟩
+⟨f, g⟩Hk(Ω),∗
+=
+�
+β∈Nm,∥β∥1≤k
+⟨D∗
+βf, D∗
+βg⟩L2(Ω)
+=⟨f, Wm,n,kg⟩
+⟨F, G⟩H−k(Ω) =
+�
+β∈Nm,∥β∥1≤k
+⟨DβJ∗f, DβJ∗g⟩L2(Ω)=⟨f, Wm,n,−kg⟩
+⟨F, G⟩H−k(Ω),∗=
+�
+β∈Nm,∥β∥1≤k
+⟨D∗
+βJ∗f, D∗
+βJ∗g⟩L2(Ω)=⟨f, Wm,n,−kg⟩ ,
+(20)
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+where f = (f(pα))α∈Am,n ∈ R|Am,n|, g = (g(pα))α∈Am,n ∈ R|Am,n| are the evaluation vectors of f, g in the Legendre
+nodes pα ∈ Pm,n, respectively. The corresponding norms are given by
+∥f∥Hk(Ω) = ⟨f, f⟩1/2
+Hk(Ω) ,
+∥f∥Hk(Ω),∗ = ⟨f, f⟩1/2
+Hk(Ω),∗
+∥F∥H−k(Ω) = ⟨F, F⟩1/2
+H−k(Ω) ,
+∥F∥H−k(Ω),∗ = ⟨F, F⟩1/2
+H−k(Ω),∗ .
+(21)
+In fact, while including the L2-inner product for β = 0, the expressions above define inner products and norms. We
+deduce the exactness of the equations.
+Theorem 10 (Sobolev cubatures). Let f, g ∈ Hk(Ω, R) and F = ⟨f, ·⟩, G = ⟨g, ·⟩ ∈ H−k(Ω, R). Then the approximations
+given by Definition 9, Eq. (20), are exact for all f, g ∈ Πm,n.
+Proof. By combining Proposition 3, Theorem 7 and Im,n(πm,n(f)) = πm,n(f) the proof follows.
+The following observation is helpful for computing the Sobolev cubatures.
+Corollary 11. Let f ∈ Πm,n and the assumptions of Definition 9 be fulfilled. Then the following identities hold:
+⟨Dβf, Dβf⟩L2(Ω,R) =
+�
+α∈Am,n
+1
+wα
+⟨Dβf, Lβ⟩2
+L2(Ω,R)
+⟨D∗
+βf, D∗
+βf⟩L2(Ω,R) =
+�
+α∈Am,n
+1
+wα
+⟨f, DβLα⟩2
+L2(Ω,R)
+(22)
+Proof. We use Proposition 6 in terms of D∗
+β = W−1
+m,nDT
+β Wm,n and due to Theorem 10 compute
+⟨D∗
+βf, D∗
+βf⟩L2(Ω,R) = ⟨D∗
+βf, Wm,nD∗
+βf⟩ = ⟨W−1
+m,nD⊤
+β Wm,nf, D⊤
+β Wm,nf⟩
+=
+�
+α∈Am,n
+1
+wα
+⟨f, D⊤
+β Wm,neα⟩2 =
+�
+α∈Am,n
+1
+wα
+⟨f, DβLα⟩2
+L2(Ω,R) ,
+where eα is the α-th standard basis vector of R|Am,n|. The analog computation applies for Dβ.
+In fact, when considering the truncated (dual) norms (∥·∥H−k(Ω),∗, ∥·∥Hk(Ω),∗), ∥·∥H−k(Ω), ∥·∥Hk(Ω), computations
+based on Eq. (22) are straightforwardly achieved and documented in (ABC, 2021). We provide the formal setup next.
+3. PDE formulations
+In light of the provided perspectives, we follow (Jost, 2002; Brezis, 2011) to propose the following formalization of
+classic PDE problems. For the sake of simplicity, we focus on classic Poisson type equations. Extensions to more general
+PDE problems can be derived once the notion is given, see Section 4.
+3.1. Poisson equation
+Let us consider the Poisson equation, for f ∈ C0(Ω, R). The strong Poisson problem with Dirichlet boundary
+condition g ∈ C0(∂Ω, R) seeks for solutions u ∈ C2(Ω, R) fulfilling:
+� −∆u(x) − f(x)
+= 0
+, ∀x ∈ Ω
+u(x) − g(x)
+= 0
+, ∀x ∈ ∂Ω .
+(23)
+By using the notion of weak derivatives we can formulate a weaker version of the Poisson equation. That is, finding
+u ∈ H2(Ω, R) ⊆ C0(Ω, R) fulfilling
+�
+Ω
+(−∆u − f)φ dx, ∀φ ∈ C∞(Ω, R),
+(24)
+subjected to the same Dirichlet boundary conditions as in equation (23). The notions give rise to the following optimisation
+problems.
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+3.2. PDE loss
+We use the Sobolev space setting Hk(Ω, R), Hl(∂Ω, R), k, l ∈ Z for introducing soft-constrained PDE-losses that
+impose the Poisson-PDE-solution with general boundary condition as one global variational optimisation problem.
+Definition 12. Given the setup of Eq. (23) the strong PDE-loss Lstrong : Hk+2(Ω, R) ∩ Hl(∂Ω, R) −→ R, k, l ∈ N is
+defined by
+Lstrong(u) = rstrong(u) + sstrong(u) = ∥ − ∆u − f∥2
+Hk(Ω) + ∥u|∂Ω − g∥2
+L2(Ω) .
+(25)
+The weak PDE-loss Lweak : Hk+2(Ω, R) ∩ Hl(∂Ω, R) −→ R, reflecting the weak formulation in Eq. (24), is given by
+Lweak(u) = rweak(u) + sweak(u)
+=
+sup
+φ∈C∞(Ω,R)
+⟨−∆u − f, φ⟩2
+Hk(Ω) +
+sup
+φ∈C∞(∂Ω,R)
+⟨u − g, φ⟩2
+L2(Ω) .
+(26)
+Truncations of the the strong loss Lstrong : Πm,n −→ R+ can be derived by applying the Sobolev cubatures from
+Definition 9. A truncation Lweak : Πm,n −→ R+ of the weak PDE-loss, Eq. (26) is given by requiring Eq. (24) to be
+fulfilled only for all polynomial test functions ϕ ∈ Πm,n = span(Lα)α∈Am,n spanned by the Lagrange polynomials. Hence,
+we consider
+rweak(u) ≈
+�
+α∈Am,n
+⟨−∆u − f, Lα⟩2
+Hk(Ω) ,
+sweak(u) ≈
+�
+α∈Am,n
+⟨u − g, Lα⟩2
+Hl(Ω) .
+(27)
+While Definition 12 includes the case k, l < 0 the corresponding losses occur when replacing ∥ · ∥Hk(Ω), ∥ · ∥H−k(Ω)
+with ∥·∥Hk(Ω), ∥·∥H−k(Ω),∗, yielding well-defined notions due to Proposition 3. Next, we derive the corresponding gradient
+flows of the given losses.
+3.3. Variational gradient flows
+Given a polynomial QC0
+=
+�
+α∈Am,n cαLα in Lagrange expansion with respect to the Legendre
+grid Pm,n
+⊆
+Ω with coefficients C0
+=
+(cα)α∈Am,n
+∈
+R|Am,n|.
+We consider the truncated loss
+L : R|Am,n| −→ R+, L = L[C] acting on the coefficients and the gradient flow ODE
+∂tC(t) = −∇L(QC(t))
+, C(0) = C0 .
+(28)
+Combining the identity QC(pα) = cα, with Definition 9 for the evaluation vector f = (f(pα))α∈Am,n we derive the
+following expression for the L2-gradient in case for the strong loss L = Lstrong from Eq. (25),i.e,
+∇C(rstrong) = ∇C⟨
+�
+(D2
+x1 + · · · + D2
+xm)C + f
+�
+, Wm,n
+�
+(D2
+x1 + · · · + D2
+xm)C + f
+�
+⟩ ,
+where according to Eq. (15), D2
+xi = D2ei with ei ∈ Rm being the standard basis, i = 1, . . . , m. Thus,
+∇C(rstrong) = −2(D2
+x1 + · · · + D2
+xm)T Wm,n
+�
+(D2
+x1 + · · · + D2
+xm)C + f
+�
+,
+(29)
+∇C(sstrong)±
+j = 2Wm��1,n(S±
+m,n,jC − g±
+j ) ,
+j = 1, . . . , m ,
+where g±
+j is the evaluation vector of g in the m-1-dimensional Legendre grid P ±
+m−1,n,j ⊆ ∂Ω±
+j contained in each face ∂Ω±
+j
+of Ω, and S±
+m,n,j denotes the truncated trace operator, Definition 8.
+Analogously, in case of the weak loss L = Lweak from Eq. (26) we derive
+∇C(rweak) = −2(D2
+x1 + · · · D2
+xm)T W2
+m,n
+�
+(D2
+x1 + · · · + D2
+xm)C + f
+�
+(30)
+∇C(sweak)±
+j = 2W2
+m−1,n(S±
+m,n,jC − g±
+j ) .
+Formulas for choosing truncated dual norms ∥ · ∥Hk(Ω), ∥ · ∥Hk(Ω),∗, 0 < k < ∞ as in Definition 9 result when replacing
+Wm,n with the corresponding cubature matrix, e.g. Wm,nJ∗−1, from Definition 9 in Eq. (29), while in Eq. (30) W2
+m,nJ∗−1
+occurs.
+For all cases, Corollary 11 provides the baseline for numerical stable implementations, which are realised and
+documented in (ABC, 2021).
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+3.3.1. ANALYTIC VARIATION OF LINEAR PDES
+Given the analytic expressions of the variational gradients in Eq. (29),(30) we derive the analytic solution of the
+gradient descent, Eq. (28): To do so, we shorten D := (D2
+x1 + · · · + D2
+xm), D∗ := DT Wm,n, S := �m
+j=1 S±
+m,n,j,
+S∗g := Wm−1,n
+�m
+j=1 g±
+j and realise that Eq. (28) becomes:
+d
+dtC(t) = −2(D∗D + S∗S)C(t) + 2(S∗g − D∗f) .
+By applying the variation of parameters we derive the solution of the ODE as:
+C(t) = exp(−t · K∗K)C0 + 2(I − exp(−t · K∗K))(K∗K)+(S∗g − D∗f) ,
+where K∗K := 2(D∗D+S∗S), and (K∗K)+ denotes the Moore–Penrose pseudo-left-inverse, see e.g., (Ben-Israel & Greville,
+2003; Trefethen & Bau III, 1997). In case, where K∗K is a positive definite matrix that imples
+C∞ := lim
+t→∞ C(t) = (K∗K)−1(S∗g − D∗f) .
+(31)
+While we expect that K∗K is positive definite, and thus invertible, whenever the underlying PDE problem is well posed and
+posses a unique solution a formal proof of this implication requires a deeper theoretical study that is out of scope of this
+article. Empirical demonstrations in Section 4, however, suggest this expectation to be genuine.
+Whatsoever, non-linear PDEs or inverse PDE problems can not be solved due to Eq. (31) and require gradient descent
+methods, realising Eq. (28). A deeper investigation of such approaches is given in the next section.
+3.4. Exponential convergence of λ-convex gradient flows
+In practice more general problems than linear (forward) PDE problems occur. We motivate this section by considering
+an inverse problem for the Poisson equation (23). That is to consider a function f : Ω −→ R and an unknown parameter
+µ ∈ R and pose the PDE problem
+� −∆u(x) − µf(x)
+= 0
+, ∀x ∈ Ω
+u(x) − g(x)
+= 0
+, ∀x ∈ Ω
+(32)
+where g is one specific Poisson solution, i.e., ∆g = µf on Ω. For inferring the parameter µ ∈ R and the PDE solutions
+simultaneously we assume that g can be sampled at the Legendre grid Pm,n and formulate the truncated (polynomial) loss
+by:
+L[C, µ] = ∥ − ∆QC − µf∥2
+Hk(Ω) + ∥QC − g∥2
+Hl(Ω) ,
+k, l ∈ N .
+(33)
+While the PDE solution depends on µ itself, we cannot compute the analytic solution directly. Instead, we apply an iterative
+gradient descent for deriving the solution based on Eq. (33). We prove that the proposed approach converges exponentially
+fast for even more general problems.
+Definition 13. A differentiable functional F : R|Am,n| → R is called λ-convex if there is a λ > 0 such that:
+F[x] ≥ F[y] + ∇F[y]T (x − y) + λ
+2 ∥x − y∥2, ∀x, y ∈ R|A|
+(34)
+Theorem 14. Given a truncated loss L : R|Am,n| −→ R+, m, n ∈ N, as in Section 3.2, that is λ-convex and differentiable
+and assume that the optimal solution C∞ := argminC∈R|Am,n|L[C] minimizing the variational problem exists and is unique.
+Then both the loss and the gradient descent
+∂tC(t) = −∇L(QC(t))
+, C(0) = C0 .
+converge exponentially fast as t → ∞:
+λ
+2 ∥C(t) − C∞∥2 ≤ L[C(t)] − L[C∞] ≤ e−2λt(L[C0] − L[C∞]).
+(35)
+Proof. The proof of the statement is given in the appendix.
+We give some insights to assert in which situations Theorem 14 applies:
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+Proposition 15. Let A ∈ Rr×s, r ≥ s ∈ N be a positive definite matrix, λ > 0 be the smallest eigenvalue of A then the
+affine loss
+L(C) = ∥AC + b∥2 ,
+b ∈ Rr
+(36)
+is λ-convex.
+Proof. We start by observing that any norm is 1−convex, in particular it holds:
+∥x∥2 = ∥y∥2 + (∇∥y∥2)T (x − y) + ∥x − y∥2 ,
+(37)
+where (∇∥y∥2)T (x − y) = 2⟨y, x − y⟩.
+By replacing the roles of x, y with Ax + b, Ay + b, respectively, we compute:
+∥Ax + b∥2 = ∥Ay + b∥2 + 2⟨Ay + b, A(x − y)⟩ + ∥A(x − y)∥2
+= ∥Ay + b∥2 + 2⟨AT (Ay + b), x − y⟩ + ∥A(x − y)∥2
+≥ ∥Ay + b∥2 + 2(∇(∥Ay + b∥2), x − y) + λ∥x − y∥2 ,
+where ∇(∥Ay + b∥2) = 2(AT (Ay + b)).
+We want to note that the assumption on A in Proposition 15 can be relaxed:
+Remark 16 (Exponential convergence of non-unique solutions). Given that ker A ̸= 0, but b ∈ Rr in Eq. (36) satisfies
+b ∈ cokerAT = {x ∈ Rs : AT x ̸= 0} we observe that solving AC = b is equivalent to minimising
+L(C) = ∥AT AC + AT b∥2 = ∥A′C + b′∥2 ,
+(38)
+with b′ = AT b, A′ = AT A. Let λ > 0 be the smallest non-vanishing eigenvalue of A′ = AT A. While cokerAT ∼= imA,
+L is λ-convex on (ker A)⊥. Due to Theorem 14 and Proposition 15 this implies that the gradient descent of well-posed
+problems, Eq. (38), converges exponentially fast to a solution as long as the initial coefficients C0 = C(0) ̸∈ ker A were
+proper chosen.
+The practical relevance of the observation above is part of the empirical demonstrations of our proposed concepts
+given in the next section.
+4. Numerical experiments
+We designed several numerical experiments for validating our theoretical results. The computations of the PSMs were
+executed on a standard Linux laptop (Intel(R) Core(TM) i7-1065G7 CPU @ 1.30GHz, 32 GB RAM). Precomputation of the
+Sobolev cubature matrices is realised as a feature of the open source package (Hernandez Acosta et al., 2021). The PSMs are
+realised by Chebyshev polynomials, Eq. (3), constrained on Legendre grids as asserted in Eq. (18). All PINN experiments
+were executed on the NVIDIA V100 cluster at HZDR. Complete code and benchmark sets is available at (ABC, 2021). We
+intensively compared several PINN approaches in our previous work (Cardona & Hecht, 2022). That is why, apart from
+classic PINNs, here, we focus on comparing our approach with the PINN-methods that turned out to be most reliable:
+i) Classic PINNs with the strong L2-MSE loss based on (Raissi et al., 2019), as described in the introduction.
+ii) Inverse Dirichlet Balancing (ID-PINNs) with the L2-MSE loss (Maddu et al., 2021), as described in the introduction.
+iii) Sobolev Cubature PINNs (SC-PINNs) (Cardona & Hecht, 2022), with the weak L2-loss for all the experiments unless
+specified otherwise.
+iv) Gradient flow optimised PSMs (GF-PSM), using the LBFGS-optimiser (Byrd et al., 1995) for the forward problem
+with the H−1
+⋆ -norm for the PDE loss and the strong L2−loss for the other terms (unless further specified). Poisson and
+QHO Inverse problems are solved by an Implicit-Euler time integration (Butcher, 2001) with the strong L2 loss and
+Newton-Raphson (Chong & Zak, 1996) for the Navier Stokes inverse problem, with the H−1
+⋆
+loss.
+iv) Analytic Descent (AD-PSM), deriving the PSM by the analytic descent given in Eq. (31) by choosing the dual H−1
+⋆ -loss,
+Eq. (20), for the PDE-loss and the strong L2-loss for the remaining terms.
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+For measuring the approximation errors of a ground truth function g : Ω −→ R by a surrogate model u we evaluate
+both on equidistant grids g = (g(pi))i=1,...,N ∈ RN u(u(pi))i=1,...,N ∈ RN of size N and compute the l1, l∞-errors
+ϵ1 := ∥g − u∥1/N, ϵ∞ := ∥g − u∥∞. We used N = 1002 points for the 2D problems and N = 204 points for the 4D
+problem. The parameter inference error is denoted with ϵµ := |µ − µgt|.
+All models are trained with the same number of training points T. For the PINN and ID-PINN methods, the training
+points are given by randomly sampling from an equidistant grid G of size |G| ≫ N. For the SC-PINN and the PSM methods
+the training points are given by the Legendre grids. CPU-training-runtimes are reported in seconds.
+4.1. 2D and 4D Poisson equations
+We start by considering the Poisson problem in dimension m = 2 in the strong formulation with Dirichlet boundary
+conditions, Eq. (23).
+Figure 1. Solution for 2D Poisson problem
+Approximation error
+Runtime (s)
+dim = 2
+ϵ1
+ϵ∞
+PINN
+4.43 · 10−3
+5.2 · 10−2
+t = 886
+ID-PINN
+5.23 · 10−3
+1.9 · 10−2
+t = 1356
+SC-PINN
+2.52 · 10−3
+3.33 · 10−2
+t = 79.2
+GF-PSM
+5.37 · 10−5
+2.94 · 10−3
+t = 12.84
+AD-PSM
+8.79 · 10−10
+1.25 · 10−8
+t = 1.21
+Approximation error
+Runtime (s)
+dim = 4
+ϵ1
+ϵ∞
+GF-PSM
+1.33 · 10−6
+1.0 · 10−3
+t = 173.59s
+AD-PSM
+5.42 · 10−8
+6.37 · 10−7
+t = 7.66s
+Table 1. Errors for 2D and 4D Poisson forward problem
+Experiment 4.1 (Non-periodic 2D-Poisson forward problem with hard transitions). We consider the Poisson equation with
+right hand side function f given by
+f(x, y) =C(A sin(ωy) + tanh(βy))(−Aω2 sin(ωx) − 2β2 tanh(βx)sech2(βx))
++ C(A sin(ωx) + tanh(βx))(−Aω2 sin(ωy) − 2β2 tanh(βy)sech2(βy)),
+with C = 0.1, A = 0.1, β = 5, ω = 10π. All the experiments where conducted with the same number of training points, as
+required for the Sobolev cubatures of degree n = 50 in the domain and n = 100 for the boundary. For the SC-PINN the
+weak L2 -loss was used for the PDE loss and for the boundary.
+Table 1 (top) reports the results and shows that the PSM methods outperform all PINN approaches, both, in accuracy
+and runtime. AD-PSM reaches seven orders of magnitude smaller ϵ1-error and requires up to three orders of magnitude
+less runtime. The GF-PSM performance is non-compatible to AD-PSM, but still far better than the PINN alternatives. The
+results clearly demonstrate the PSM method to be capable of finding solutions to non-trivial linear PDEs with general
+non-periodic boundary conditions.
+The following experiment indicates that this observation maintains true even for higher dimensional problems.
+Experiment 4.2 (4D Poisson equation forward problem). We seek for a solution of a Poisson problem in dimension m = 4.
+We choose
+f(x) := −4ω2g(x),
+with ω = 1 and periodic boundary condition g(x) := sin(ωx1) cos(ωx2) sin(ωx3) cos(ωx4) yielding u(x) = g(x) to be
+the analytic solution. We choose Sobolev cubatures of degree n = 8 for both, the domain and the boundary loss.
+In Table 1 (bottom) the approximation errors are reported. While all PINN approaches failed to provide any reasonable
+solution, the PINN-results were skipped. In contrast, the PSMs can recover the solution accurately. We want to stress that
+the PSM runtimes are still smaller than the training runtimes of ID-PINN or the standard PINNs occuring for the analogue
+2D Poisson problem, validating again its superior efficiency.
+
+1.0
+0.10
+0.5
+0.05
+> 0.0
+0.00
+-0.05
+-0.5
+-0.10
+.0
+-0.5
+0.0
+0.5
+1.0
+XLearning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+Figure 2. Solution for 2D inverse Poisson
+problem with ωgt = π.
+Approximation error
+Runtime (s)
+ϵµ
+ϵ1
+ϵ∞
+PINN
+4.63 · 10−1
+1.13 · 10−2
+1.24 · 10−1
+t ≈ 1592
+ID-PINN
+2.14 · 10−2
+8.09 · 10−4
+1.52 · 10−2
+t ≈ 2184
+SC-PINN
+3.0 · 10−4
+5.49 · 10−4
+1.01 · 10−2
+t ≈ 103
+GF-PSM
+5.8 · 10−8
+6.0 · 10−10
+3.47 · 10−9
+t ≈ 0.49
+Table 2. Errors for 2D Poisson inverse problem
+Figure 4. Solution for 2D QHO with µ = 31 on Ω′ = 5.3Ω due to AD-PSM.
+Experiment 4.3 (2D Poisson inverse problem). We consider the inverse 2D-Poisson problem, as introduced in Section 3.4,
+Eq. (32): We are seeking for inferring the parameter µ in the right hand side f(x) = µ cos(ωx) sin(ωy), for the unknown
+ground truth µgt = 2ω2
+gt, ωgt = π and the corresponding PDE solution simultaneously, with the L2-loss (k = l = 0) given
+in equation (33). The GF-PSM is applied for a Sobolev cubature with degree n = 100 for the boundary and n = 30 for the
+PDE loss. Benchmarks for the standard PINN and the ID-PINN are executed with the same number of training points.
+Table 2 reports the reached accuracy and the required runtimes. The GF-PSM outperforms all other methods by
+several orders of magnitude in accuracy for both the solution of the PDE, as well as the inferred parameter µ. As discussed in
+Section 3.4 the analytic variation, Eq. (31), does not directly apply for this task and is, thus, omitted here. The exponentially
+fast convergence of the GF-PSM, Section 3.4, is reflected in the required runtime being 4 orders of magnitude less than the
+PINN alternatives.
+4.2. Quantum Harmonic Oscillator in 2D
+We consider eigenvalue problem for the time-independent Quantum Harmonic Oscillator in dimension m = 2, which
+is a special case of the Schr¨odinger equation with linear potential V (u(x)) := (x2
+1 + x2
+2)u(x), u ∈ C2(Ω, R), see e.g.,
+(Liboff, 1980; Griffiths & Schroeter, 2018):
+�
+−∆u(x) + V (u(x))
+= µu(x)
+, ∀x ∈ Ω
+u(x) − g(x)
+= 0
+, ∀x ∈ ∂Ω ,
+It is a classic fact, that the the eigenvalues are given by µ = n1 + n2 + 1, n1, n2 ∈ N with corresponding eigenfunctions
+g(x1, x2) =
+π−1/4
+√2n1+n2n1!n2!e−
+(x2
+1+x2
+2)
+2
+Hn1(x1)Hn2(x2) ,
+whereas Hn denotes the n-th Hermite polynomial.
+Experiment 4.4 (QHO forward problem). For solving the QHO forward problem with eigenvalue µ = 21 and extended
+domain Ω′ = [−5.3, 5.3], GF-PSM and the AD-PSM use Sobolev cubatures of degree n = 100 for the boundary and
+
+1.0
+1.00
+0.75
+0.5
+0.50
+0.25
+V0.0
+0.00
+0.25
+-0.5
+0.50
+0.75
+-1.0 -
+1.00
+-1.0
+-0.5
+0.0
+0.5
+1.0
+XGround Truth
+Prediction
+Point-wise Error le-8
+5.3
+0.4
+5.3
+0.4
+5.3
+2.5
+2.6
+2.6
+2.6
+2.0
+0.2
+0.2
+1.5
+0.0
+y
+0.0
+y 0.0
+0.0
+0.0
+1.0
+-2.6
+-2.6
+-2.6
+0.5
+-0.2
+-0.2
+-5.3
+-5.3
+-5.3
+5.3
+-2.6
+0.0
+2.6
+5.3
+-5.3
+-2.6
+0.0
+2.6
+5.3
+-5.3
+-2.6
+0.0
+2.6
+5.3
+x
+xLearning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+Figure 3. Solution of 2D QHO
+forward problem with µ = 21.
+Approximation error
+Runtime (s)
+µ = 21
+ϵ1
+ϵ∞
+PINN
+6.97 · 10−2
+1. · 10−3
+t ≈ 776
+ID-PINN
+4.29 · 10−2
+1.30 · 10−1
+t ≈ 948
+SC-PINN
+8.16 · 10−4
+7.27 · 10−3
+t ≈ 167
+GF-PSM
+1.6 · 10−8
+5.4 · 10−8
+t ≈ 0.16
+AD-PSM
+7.61 · 10−13
+2.37 · 10−12
+t ≈ 0.07
+µ = 31
+ϵ1
+ϵ∞
+GF-PSM
+1.09 · 10−9
+1.45 · 10−8
+t ≈ 2.39
+AD-PSM
+2.25 · 10−9
+9.82 · 10−9
+t ≈ 1.07
+Table 3. Errors for 2D QHO forward problem with µ = 21, 31.
+Figure 5. Solution for 2D QHO with
+µgt = 9 on Ω′ = 5.3Ω.
+Approximation error
+Runtime (s)
+ϵµ
+ϵ1
+ϵ∞
+PINN
+6.01
+7.32 · 10−2
+4.37 · 10−1
+t ≈ 1414
+ID-PINN
+6.21 · 10−2
+7.51 · 10−3
+9.40 · 10−2
+t ≈ 1346
+SC-PINN
+2.18 · 10−4
+5.68 · 10−4
+1.39 · 10−2
+t ≈ 192
+GF-PSM
+9.50 · 10−11
+1.49 · 10−12
+5.13 · 10−10
+t ≈ 5
+Table 4. Errors for 2D QHO inverse problem with µgt = 9
+n = 30 for the PDE loss, whereas we choose n = 200 and n = 50 for eigenvalue µ = 31 on the standard hypercube Ω,
+respectively. The AD-PSM uses the by default chosen H−1(Ω), ∗ norm, while the GF-PSM was applied with weak L2-loss,
+as in Eq. (26).
+Results are reported in Table 3. SC-PINN was the only PINN method that gains reasonable results for µ = 31
+and Ω = [−1, 1]2. However, as in Section 4.1 the PSMs-methods outperform SC-PINN in both runtime and accuracy
+performance. In the second scenario, µ = 21, Ω′ = 5.3Ω, none of PINN approaches was able to reach close approximations,
+while AD-PSM and GF-PSM do. AD-PSM performs best and its solution is visualised in Fig. 4.
+Experiment 4.5 (QHO inverse problem). Similar to Exp. 4.3 we seek for inferring the unknown eigenvalue µ, set to µgt = 9,
+and the corresponding continuous approximation of the PDE solution simultaneously, with given data u ∈ R|Am,n| sampled
+on the Legendre grid by optimising the loss:
+L[C, µ] = ∥∆Qc + V (Qc) − µQC∥2
+L2 + ∥QC − u∥2
+L2
+(39)
+We choose a n = 50 degree Sobolev cubature for the domain and n = 200 on the boundary and compare it with the PINN
+and the ID-PINN for the same number of training points.
+As shown in Table 4 the GF-PSM outperforms the ID-PINN by several orders of magnitude in both accuracy and
+runtime. This reflects the strength and flexibility of the method when addressing linear inverse problems. While na¨ıve,
+unconditioned Implicit-Euler implementations are inherently unstable the insights of Section 3.4 enable us to exploit the
+structure of the gradient flow to realize stable numerical integrators. Applying the PSM method to non-linear forward
+problems is our next demonstration task.
+
+1.0
+0.15
+0.10
+0.5
+0.05
+y 0.0
+0.00
+-0.05
+-0.5
+0.10
+-1.0
+-0.15
+-1.0
+-0.5
+0.0
+0.5
+1.0
+X5.3
+0.4
+0.3
+2.6
+0.2
+0.1
+> 0.0
+0.0
+-0.1
+-2.6
+-0.2
+-0.3
+-5.35.3
+-0.4
+-2.6
+0.0
+2.6
+5.3
+XLearning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+4.3. 2D Incompressible Navier Stokes equation
+We consider the incompressible 2D Navier Stokes equation as an example of a non-linear PDE problem: Let
+u = (u1, u2), u ∈ C2(Ω, R2) be the vector velocity field and p ∈ C1(Ω; R) the scalar pressure field the equation becomes:
+�
+�
+�
+−ν∆u(x, y) + (u(x, y) · ∇)u(x, y) + ∇p(x, y)
+= f(x, y)
+, ∀(x, y) ∈ Ω
+∇ · u(x, y)
+= 0
+, ∀(x, y) ∈ Ω
+u(x, y) − g(x, y)
+= 0
+, ∀(x, y) ∈ ∂Ω ,
+where
+f(x, y) = 2νπ2(u1(x, y), u2(x, y)) + π cos(πx) cos(πy)(−u1(x, y), u2(x, y))
++ π sin(πx) sin(πy)(u2, −u1) + exp(πy)(1, πx) ,
+g(x, y) = [− sin(πx) cos(πy), cos(πx) sin(πy)]T
+Experiment 4.6 (Navier-Stokes Forward and Inverse Problem). We solve the Navier-Stokes forward problem by applying
+GF-PSM with n = 100 and n = 30 degree Sobolev cubature for the boundary and the domain respectively. We set the
+viscosity to ν = 0.05 and use the analytic pressure field p = x exp(πy) with Dirichlet boundary conditions.
+The inverse problem seeks for inferring ν and the scalar pressure field p for the ground truth viscosity νgt = 0.05
+and u1 = − sin(πx) cos(πy), u2 = cos(πx) sin(πy). The errors ϵ1 and ϵ∞ reported for this experiment, correspond to the
+predicted pressure against the ground truth one.
+Figure 6. Solution u1.
+Approximation error
+Runtime (s)
+Forward Problem
+ϵ1
+ϵ∞
+GF-PSM
+u1
+3.31 · 10−10
+2.35 · 10−9
+t ≈ 405.22
+GF-PSM
+u2
+3.28 · 10−10
+2.35 · 10−9
+t ≈ 405.22
+Table 5. Approximation errors of the forward problem.
+Approximation error
+Runtime (s)
+Inverse Problem
+ϵν
+ϵ1
+ϵ∞
+GF-PSM
+2.91 · 10−16
+2.63 · 10−14
+1.21 · 10−11
+t ≈ 0.79
+Table 6. Approximation errors of the inverse problem.
+While none of the PINN approaches was able to address the problem reasonably the PSM methods reach similar
+accuracy as in the prior (linear) experiments, as reported in Tables 5,6.
+We summarise the experimental and theoretical findings in the concluding thoughts below.
+5. Conclusion
+We introduced a novel variational spectral method solving linear, non-linear, forward and inverse PDE problems.
+In contrast to neural network - PINN approaches Chebyshev polynomials surve as a polynomial surrogate model - PSM,
+maintainig the same flexibility as PINNs.
+Based on our prior work (Cardona & Hecht, 2022), we gave weak PDE formulations, resting on the novel Sobolev
+cubatures approximating general Sobolev norms. Allowing us to formulate and compute the resulting finite-dimensional
+gradient flow for finding the optimal coefficients for the PSMs, in the case of linear PDEs, we could even derive the analytical
+solution of the gradient flow. In particular, the resulting efficient computation of the negative order dual Sobolev norm
+
+1.0
+1.0
+0.5
+0.5
+0.0
+0.0
+-0.5
+-0.5
+-1.0
+-1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+∥ · ∥H−k(Ω),∗ was demonstrated to perform best compared to the alternative formulations. While we meanwhile deepened
+the theoretical insights, presented here, to deliver the optimal choice of the Sobolev norm beforehand these subjects are
+part of a follow-up study. This includes a relaxation of the Sobolev cubatures, resisting the curse of dimensionality when
+addressing higher dimensional problems.
+In summary, the PSMs methods outperformed all other benchmark methods by far, showing the superiority in runtime
+and accuracy performance of the PSMs formulation on the whole spectrum of the considered problems. Since the PSMs
+offer the same flexibility and capabilities of PINNs, we propose to extend the presented approach in order to learn PDE
+solutions for ranges of boundary conditions, parameters (like diffusion constants) or dynamic time ranges. Because the
+gain in efficiency allowed to compute the presented benchmarks without High Performance Computing (HPC) on a local
+machine, we expect so far non-reachable high-dimensional dim ≥ 3, strongly varying PDE problems, appearing for instance
+for dynamic phase space simulations, to become solvable when being addressed by a parallelised HPC version of the current
+implementation (ABC, 2021).
+References
+ABC. Repository with documentation and implementations under construction. https://github.com/XYZ, 2021.
+Adams, R. A. and Fournier, J. J. Sobolev spaces, volume 140. Academic press, 2003.
+Arjovsky, M., Chintala, S., and Bottou, L. Wasserstein generative adversarial networks. In Precup, D. and Teh, Y. W.
+(eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine
+Learning Research, pp. 214–223. PMLR, 06–11 Aug 2017. URL https://proceedings.mlr.press/v70/
+arjovsky17a.html.
+Ben-Israel, A. and Greville, T. N. Generalized inverses: theory and applications, volume 15. Springer Science & Business
+Media, 2003.
+Bernardi, C. and Maday, Y. Spectral methods. Handbook of numerical analysis, 5:209–485, 1997.
+Brezis, H. Functional analysis, Sobolev spaces and partial differential equations, volume 2. Springer, 2011.
+Butcher, J. Numerical methods for ordinary differential equations in the 20th century. 12 2001. ISBN 9780444506177. doi:
+10.1016/B978-0-444-50617-7.50018-5.
+Byrd, R. H., Lu, P., Nocedal, J., and Zhu, C. A limited memory algorithm for bound constrained optimization. SIAM
+Journal on Scientific Computing, 16(5):1190–1208, 1995. doi: 10.1137/0916069. URL https://doi.org/10.
+1137/0916069.
+Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. Spectral methods: fundamentals in single domains. Springer
+Science & Business Media, 2007.
+Cardona, J. E. S. and Hecht, M. Replacing automatic differentiation by sobolev cubatures fastens physics informed neural
+nets and strengthens their approximation power. arXiv preprint arXiv:2211.15443, 2022.
+Chong, E. and Zak, S. An introduction to optimization. Antennas and Propagation Magazine, IEEE, 38:60, 05 1996. doi:
+10.1109/MAP.1996.500234.
+Ellis, J. A., Fiedler, L., Popoola, G. A., Modine, N. A., Stephens, J. A., Thompson, A. P., Cangi, A., and Rajamanickam, S.
+Accelerating finite-temperature kohn-sham density functional theory with deep neural networks. Physical Review B, 104
+(3):035120, 2021.
+Ern, A. and Guermond, J.-L. Theory and practice of finite elements, volume 159. Springer, 2004.
+Eymard, R., Gallou¨et, T., and Herbin, R. Finite volume methods. Handbook of numerical analysis, 7:713–1018, 2000.
+Griffiths, D. J. and Schroeter, D. F. Introduction to quantum mechanics. Cambridge University Press, 2018.
+Hernandez Acosta, U., Krishnan Thekke Veettil, S., Wicaksono, D., and Hecht, M. MINTERPY - multivariate interpolation
+in python. https://github.com/casus/minterpy/, 2021.
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+Hessari, P. and Shin, B.-C. The least-squares pseudo-spectral method for navier–stokes equations. Computers & Mathematics
+with Applications, 66(3):318–329, 2013. ISSN 0898-1221. doi: https://doi.org/10.1016/j.camwa.2013.05.009. URL
+https://www.sciencedirect.com/science/article/pii/S0898122113003118.
+Jin, X., Cai, S., Li, H., and Karniadakis, G. E. NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for
+the incompressible Navier-Stokes equations. arXiv:2003.06496 [physics], March 2020. URL http://arxiv.org/
+abs/2003.06496. arXiv: 2003.06496.
+Jost, J. Partial Differential Equations. New York: Springer-Verlag, 2002.
+Kang, S. and Suh, Y. K. Spectral Methods, pp. 1875–1881. Springer US, Boston, MA, 2008. ISBN 978-0-387-48998-8.
+URL https://doi.org/10.1007/978-0-387-48998-8_1442.
+Karimi, H., Nutini, J., and Schmidt, M. Linear convergence of gradient and proximal-gradient methods under the polyak-
+łojasiewicz condition. In Frasconi, P., Landwehr, N., Manco, G., and Vreeken, J. (eds.), Machine Learning and Knowledge
+Discovery in Databases, pp. 795–811, Cham, 2016. Springer International Publishing. ISBN 978-3-319-46128-1.
+Kharazmi, E., Zhang, Z., and Karniadakis, G. E. Variational physics-informed neural networks for solving partial differential
+equations. arXiv preprint arXiv:1912.00873, 2019.
+Kharazmi, E., Zhang, Z., and Karniadakis, G. E. hp-vpinns: Variational physics-informed neural networks with domain
+decomposition. ArXiv, abs/2003.05385, 2020.
+Kim, S. D. and Shin, B. C. Chebyshev weighted norm least-squares spectral methods for the elliptic problem. Journal of
+Computational Mathematics, pp. 451–462, 2006.
+Lagergren, J. H., Nardini, J. T., Baker, R. E., Simpson, M. J., and Flores, K. B. Biologically-informed neural networks
+guide mechanistic modeling from sparse experimental data. arXiv:2005.13073 [math, q-bio], May 2020. URL http:
+//arxiv.org/abs/2005.13073. arXiv: 2005.13073.
+Lax, P. D. On cauchys problem for hyperbolic equations and the differentiability of solutions of elliptic equations. Comm.
+Pure Appl. Math. 8, 615-633, 1955.
+LeVeque, R. J. Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent
+problems. SIAM, 2007.
+Li, S. and Liu, W. K. Meshfree particle methods. Springer Science & Business Media, 2007.
+Liboff, R. L. Introductory Quantum Mechanics. Addison-Wesley Publishing Company. Canad´a, 1980.
+Long, Z., Lu, Y., Ma, X., and Dong, B. Pde-net: Learning pdes from data. ArXiv, abs/1710.09668, 2018.
+Maddu, S., Sturm, D., M¨uller, C. L., and Sbalzarini, I. F. Inverse dirichlet weighting enables reliable training of physics
+informed neural networks. Machine Learning: Science and Technology, 2021. URL http://iopscience.iop.
+org/article/10.1088/2632-2153/ac3712.
+Neuberger, P. K. J. Potential theory and applications in a constructive method for finding critical points of ginzburg–landau
+type equations. Nonlinear Analysis: Theory, Methods & Applications vol. 69 iss. 3, 69, aug 2008. doi: 10.1016/j.na.2008.
+02.074. URL libgen.li/file.php?md5=871f710130ca8f46f6cc6df7e25eb611.
+Raissi, M., Perdikaris, P., and Karniadakis, G. Physics-informed neural networks: A deep learning framework for
+solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational
+Physics, 378:686–707, 2019. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2018.10.045. URL https://www.
+sciencedirect.com/science/article/pii/S0021999118307125.
+Sahli Costabal, F., Yang, Y., Perdikaris, P., Hurtado, D. E., and Kuhl, E. Physics-Informed Neural Networks for Cardiac
+Activation Mapping. Frontiers in Physics, 8:42, February 2020. ISSN 2296-424X. doi: 10.3389/fphy.2020.00042. URL
+https://www.frontiersin.org/article/10.3389/fphy.2020.00042/full.
+Sirignano, J. A. and Spiliopoulos, K. Dgm: A deep learning algorithm for solving partial differential equations. Journal of
+Computational Physics, 2018.
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+Stroud, A. Approximate calculation of multiple integrals: Prentice-Hall series in automatic computation. Prentice-Hall
+(Englewood Cliffs, NJ), 1971.
+Stroud, A. Secrest. d.(1966). Gaussian quadrature formulas, 2011.
+Trefethen, L. N. Cubature, approximation, and isotropy in the hypercube. SIAM Review, 59(3):469–491, 2017.
+Trefethen, L. N. Approximation theory and approximation practice, volume 164. SIAM, 2019.
+Trefethen, L. N. and Bau III, D. Numerical linear algebra, volume 50. SIAM, 1997.
+Wang, S., Teng, Y., and Perdikaris, P. Understanding and mitigating gradient flow pathologies in physics-informed neural
+networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021.
+Yang, L., Zhang, D., and Karniadakis, G. E. Physics-informed generative adversarial networks for stochastic differential
+equations. ArXiv, abs/1811.02033, 2020.
+Appendix
+The result provided in Theorem 14 is a known fact and could be also found for example in (Karimi et al., 2016) in
+a more general setting. We prove it by combining the following lemmas. Given a differentiable λ-convex truncated loss
+L : R|Am,n| −→ R+, m, n ∈ N, as in Theorem 14, inducing the gradient descent ODE
+∂tC(t) = −∇L(QC(t))
+, C(0) = C0 ,
+where C0 ∈ R|Am,n| is some initial guess of the coefficients. The Implicit Euler discretisation of the ODE is given by
+Cn+1 = Cn − τ∇L[Cn+1] ,
+(40)
+where τ ∈ R is the learning rate. We will use the following two definitions:
+Definition 17. A functional F : R|Am,n| → R is convex if:
+F[tx + (1 − t)y] ≤ tF[x] + (1 − t)F[y],
+(41)
+it is called strictly convex, if the inequality is strict.
+Definition 18. A functional F : R|Am,n| → R is coercive if:
+lim
+||u||→∞ F[u] = ∞
+(42)
+Lemma 19. Let the assumptions of Theorem 14 be fulfilled then the following estimate applies:
+λ
+2 ∥Cn − C∞∥2 ≤ L[Cn] − L[C∞] ≤ 1
+2λ∥∇L[Cn]∥2 .
+Proof. We prove the first inequality by rephrasing the λ- convexity property,Eq. (34). Let γt := tx + (1 − t)y, then
+L = L(x) is λ-convex if
+L[γt] ≤ tL[x] + (1 − t)L[y] − λ
+2 t(1 − t)∥x − y∥2 .
+By replacing x and y with Cn and C∞, respectively, and re-arranging, we obtain:
+λ
+2 t(1 − t)∥Cn − C∞∥2 ≤ t(L[Cn] − L[C∞]) + L[C∞] − L[γt] ≤ t(L[Cn] − L[C∞]) ,
+where we used the minimality of C∞ for the last inequality. Dividing by t and taking the limit for t → 0 yields the first
+inequality of Lemma 19. The second inequality follows directly from the λ-convexity, Eq. (34), implying
+L[Cn] − L[C∞] ≤ −∇L[Cn]T (C∞ − Cn) − λ
+2 ∥C∞ − Cn∥2,
+
+Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
+We set F[C∞] := ∇L[Cn]T (C∞ − Cn) + λ
+2 ∥C∞ − Cn∥2
+2 and realise that F is a coercive, strictly convex functional with
+respect to C∞. Hence, the uniquely determined minimum C∗
+∞ is given by:
+∇F
+!= 0 ⇐⇒ (C∗
+∞ − Cn) = − 1
+λ∇L[Cn].
+In light of this fact, we can bound −∇F by
+L[Cn] − L[C∞] ≤ ( 1
+λ − 1
+2λ)∥∇L[Cn]∥2 ,
+yielding the desired result.
+The following lemma provides the monotonicity property of the gradient flow, being a necessary ingredient for proving
+the exponential convergence.
+Lemma 20. Let the assumptions of Theorem 14 be fulfilled the the following estimate holds:
+L[Cn−1] − L[C∞] ≥ (1 + λτ)2(L[Cn] − L[C∞])
+Proof. Due to the λ-convexity and the Implicit Euler update, Eq. (40), we realise that:
+L[Cn−1] ≥ L[Cn] + ∇L[Cn](Cn−1 − Cn) + λ
+2 ∥Cn−1 − Cn∥2
+= L[Cn] + τ(τλ
+2 + 1)∥∇L[Cn]∥2 .
+Due to Lemma 19 we further conclude
+L[Cn−1] ≥ L[Cn] + 2λτ(τλ
+2 + 1)(L[Cn] − L[C∞]) .
+(43)
+Adding −L[C∞] at both sides provides the claim.
+Lemma 21. Let the assumptions of Theorem 14 be fulfilled and define ˆλ := 1
+τ log(1 + λτ). Then the sequence:
+∆nL := L[Cn] − L[C∞],
+decreases monotonically with an exponential rate of e−2ˆλτn, i.e.
+∆nL ≤ e−2ˆλτn(L[C0] − L[C∞])
+(44)
+Proof. Due to Lemma (20) we compute
+e2ˆλτn(L[Cn] − L[C∞]) = (1 + λτ)2n(L[Cn] − L[C∞])
+≤ (1 + λτ)2(n−1)(L[Cn−1] − L[C∞])
+· · ·
+≤ L[C0] − L[C∞] .
+Proof of Theorem 14. Theorem (14) now follows by combing Lemma (19) and (21) yielding:
+1
+λ∥Cn − C∞∥2
+2 ≤ L[Cn] − L[C∞] ≤ e−2ˆλτn(L[C0] − L[C∞]) .
+(45)
+Thus, for τ → 0, it follows by the definition of ˆλ that ˆλ → λ and Cn → C(t), with t = nτ due to the continuity of
+C = C(t) inherited from the differentiability of F. Hence, the continuity of the norm implies the statement.
+Remark 22. Lemma 20 implies that also the Implicit Euler discretised gradient flow, converges exponentially fast.
+

DNAzT4oBgHgl3EQfwf6z/content/tmp_files/2301.01724v1.pdf.txt

@@ -0,0 +1,2838 @@
+1
+Super-resolution with Binary Priors: Theory and
+Algorithms
+Pulak Sarangi, Ryoma Hattori, Takaki Komiyama and Piya Pal
+Abstract—The problem of super-resolution is concerned with
+the reconstruction of temporally/spatially localized events (or
+spikes) from samples of their convolution with a low-pass filter.
+Distinct from prior works which exploit sparsity in appropriate
+domains in order to solve the resulting ill-posed problem, this
+paper explores the role of binary priors in super-resolution,
+where the spike (or source) amplitudes are assumed to be
+binary-valued. Our study is inspired by the problem of neural
+spike deconvolution, but also applies to other applications such
+as symbol detection in hybrid millimeter wave communication
+systems. This paper makes several theoretical and algorithmic
+contributions to enable binary super-resolution with very few
+measurements. Our results show that binary constraints offer
+much stronger identifiability guarantees than sparsity, allowing
+us to operate in “extreme compression" regimes, where the num-
+ber of measurements can be significantly smaller than the sparsity
+level of the spikes. To ensure exact recovery in this "extreme
+compression" regime, it becomes necessary to design algorithms
+that exactly enforce binary constraints without relaxation. In
+order to overcome the ensuing computational challenges, we
+consider a first order auto-regressive filter (which appears in
+neural spike deconvolution), and exploit its special structure. This
+results in a novel formulation of the super-resolution binary spike
+recovery in terms of binary search in one dimension. We perform
+numerical experiments that validate our theory and also show
+the benefits of binary constraints in neural spike deconvolution
+from real calcium imaging datasets.
+Index Terms—Binary compressed sensing, super-resolution,
+spike deconvolution, sparsity, binary search, beta-expansions
+I. INTRODUCTION
+The problem of recovering localized events (spikes) from
+their convolution with a blurring kernel, arises in a wide range
+of scientific and engineering applications such as fluorescence
+microscopy [1], neural spike deconvolution [2]–[4], hybrid
+millimeter wave (mmWave) communication [5], to name a few.
+Consider K temporal spikes, which can be represented as:
+xhi(t) =
+K
+�
+k=1
+ckδ(t − nkThi)
+Here, the high-rate spikes are supported on a fine temporal grid
+with spacing Thi, nk ∈ Z is an integer corresponding to the
+time index of the kth spike and ck denotes its amplitude. The
+convolution of spikes with a filter h(t) is typically uniformly
+(down)sampled at a (low) rate Tlo = DThi (D > 1), yielding
+measurements:
+y[n] = xhi(t) ⋆ h(t)|t=nT lo =
+K
+�
+k=1
+ckh(nTlo − nkThi)
+(1)
+The goal of super-resolution is to recover the spike locations nk
+and amplitudes ck, k = 1, 2, · · · , K from a limited number (M)
+of low-rate samples {y[n]}M−1
+n=0 . The problem is typically ill-
+posed due to systematic attenuation of high-frequency contents
+of the spikes by the low-pass filter h(t). In order to make the
+problem well-posed, it becomes necessary to exploit priors such
+as sparsity [6]–[9] and/or non-negativity [10], [11]. In recent
+times, there has been a substantial progress towards developing
+efficient algorithms for provably solving the super-resolution
+problem [7]–[19].
+In this paper, we investigate the problem of binary super-
+resolution, where the amplitudes of the spikes are known
+apriori to be ck = A, but their number (K) and locations
+(nk) are unknown. Motivated by the problem of neural spike
+deconvolution in two-photon calcium imaging [2], [20], we
+will focus on a blurring kernel that can be represented as a
+stable first order auto-regressive (AR(1)) filter. Each neural
+spike results in a sharp rise in Ca2+ concentration followed by
+a slow exponential decay (modeled as the impulse response of
+an AR(1) filter), which results in an overlap of the responses
+from nearby spiking events, leading to poor temporal resolution
+[2], [21].
+A. Related Works
+Early
+works
+on
+super-resolution
+date
+back
+to
+algebraic/subspace-based
+techniques
+such
+as
+Prony’s
+method, MUSIC [12], [22], ESPRIT [8], [23] and matrix
+pencil [9], [24]. Following the seminal work in [6], substantial
+progress has been made in understanding the role of sparsity
+as a prior for super-resolution [7], [25], [26]. In recent times,
+convex optimization-based techniques have been developed
+that employ Total Variational (TV) norm and atomic norm
+regularizers, in order to promote sparsity [7], [18], [19], [25],
+[26] and/or non-negativity [10], [11], [27]. These techniques
+primarily employ sampling in the Fourier/frequency domain by
+assuming the kernel h(t) to be (approximately) bandlimited.
+However, selecting the appropriate cut-off frequency is crucial
+for super-resolution and needs careful consideration [25],
+[28]. Unlike subspace-based methods, theoretical guarantees
+for these convex algorithms rely on a minimum separation
+between the spikes, which is also shown to be necessary even
+in absence of noise [29]. The finite rate of innovation (FRI)
+framework [30]–[34] also considers the recovery of spikes
+from measurements acquired using an exponentially decaying
+kernel, which includes the AR(1) filter considered in this
+paper. In the absence of noise, FRI enables the exact recovery
+of K spikes with arbitrary amplitudes from M = Ω(K)1
+measurements, without any separation condition [32]. It is
+to be noted that all of the above methods require M > K
+measurements for resolving K spikes. In contrast, we will
+show that it is possible to recover K spikes from M ≪ K
+1This notation essentially means that there exists a positive constant c such
+that M ≥ cK.
+arXiv:2301.01724v1  [eess.SP]  4 Jan 2023
+
+2
+measurements by exploiting the binary nature of the spiking
+signal. The above algorithms are designed to handle arbitrary
+real-valued amplitudes and as such, they are oblivious to
+binary priors. Therefore, they cannot successfully recover
+spikes in the regime M < K, which is henceforth referred to
+as the extreme compression regime.
+The problem of recovering binary signals from underde-
+termined linear measurements (with more unknowns than
+equations/measurements) has been recently studied under the
+parlance of Binary Compressed Sensing (BCS) [35]–[42].
+In BCS, the undersampling operation employs random (and
+typically dense) sampling matrices, whereas we consider a
+deterministic and structured measurement matrix derived from
+a filter, followed by uniform downsampling. Moreover, existing
+theoretical guarantees for BCS crucially rely on sparsity
+assumptions that will be shown to be inadequate for our
+problem (discussed in Section II-C). Most importantly, in order
+to achieve computational tractability, BCS relaxes the binary
+constraints and solves continuous-valued optimization problems.
+Consequently, their theoretical guarantees do not apply in the
+extreme compression regime M < K.
+As mentioned earlier, our study is motivated by the problem
+of neural spike deconvolution arising in calcium imaging [3],
+[4], [20], [32], [43]–[45]. A majority of the existing spike
+deconvolution techniques [4], [43], [44] infer the spiking
+activity at the same (low) rate at which the fluorescence signal
+is sampled, and a single estimate such as spike counts or
+rates are obtained over a temporal bin equal to the resolution
+of the imaging rate. Although sequential Monte-Carlo based
+techniques have been proposed that generate spikes at a rate
+higher than the calcium frame rate [3], no theoretical guarantees
+are available that prove that these methods can indeed uniquely
+identify the high-rate spiking activity. Algorithms that rely
+on sparsity and non-negativity [43], [44] alone are ineffective
+for inferring the neural spiking activity that occurs at a much
+higher rate than the calcium sampling rate. On the other hand,
+at the high-rate, the spiking activity is often assumed to be
+binary since the probability of two or more spikes occurring
+within two time instants on the fine temporal grid is negligible
+[2], [46]. Therefore, we propose to exploit the inherent binary
+nature of the neural spikes and provide the first theoretical
+guarantees that it is indeed possible to resolve the high-rate
+binary neural spikes from calcium fluorescence signal acquired
+at a much lower rate.
+B. Our Contributions
+We make both theoretical and algorithmic contributions to
+the problem of binary super-resolution in the setting when
+the spikes lie on a fine grid. We theoretically establish that
+at very low sampling rates, sparsity and non-negativity are
+inadequate for the exact reconstruction of binary spikes (Lemma
+2). However, by exploiting the binary nature of the spiking
+activity, much stronger identifiability results can be obtained
+compared to classical sparsity-based results (Theorem 1). In the
+absence of noise, we show that it is possible to uniquely recover
+K binary spikes from only M = Ω(1) low-rate measurements.
+The analysis also provides interesting insights into the interplay
+between binary priors and the “infinite memory" of the AR(1)
+filter.
+Although it is possible to uniquely identify binary spikes in
+the extreme compression regime (M ≪ K), the combinatorial
+nature of binary constraints introduce computational hurdles in
+exactly enforcing them. Our second contribution is to leverage
+the special structure of the AR(1) measurements to overcome
+this computational challenge in the extreme compression
+regime M < K (Section III-A). Our formulation reveals
+an interesting and novel connection between binary super-
+resolution, and finding the generalized radix representation of
+real numbers, known as β-expansion [47]–[49] (Section III). In
+order to circumvent the problem of exhaustive search, we pre-
+construct and store (in memory) a binary tree that is completely
+determined by the model parameters (filter and undersampling
+factor). When the low-rate measurements are acquired, we can
+efficiently perform a binary search to traverse the tree and find
+the desired binary solution. This ability to trade-off memory
+for computational efficiency is made possible by the unique
+structure of the measurement model governed by the AR(1)
+filter. The algorithm guarantees exact super-resolution even
+when the measurements are corrupted by a small bounded
+(adversarial) noise, the strength of which depends on the
+AR filter parameter and the undersampling factor. When the
+measurements are corrupted by additive Gaussian noise, we
+characterize the probability of erroneous decoding (Theorem
+3) in the extreme compression regime M < K and indicate
+the trade-off among the filter parameter, SNR and the extent
+of compression. Finally, we also demonstrate how binary
+priors can improve the performance of a popularly used spike
+deconvolution algorithm (OASIS [43]) on real calcium imaging
+datasets.
+II. FUNDAMENTAL SAMPLE COMPLEXITY OF BINARY
+SUPER-RESOLUTION
+Let yhi[n] be the output of a stable first-order Autoregressive
+AR(1) filter with parameter α, 0 < α < 1, driven by an
+unknown binary-valued input signal xhi[n] ∈ {0, A}, A > 0:
+yhi[n] = αyhi[n − 1] + xhi[n]
+(2)
+In this paper, we consider a super-resolution setting where
+we do not directly observe yhi[n], and instead acquire M
+measurements {ylo[n]}M−1
+n=0
+at a lower-rate by uniformly
+subsampling yhi[n] by a factor of D:
+ylo[n] = yhi[Dn],
+n = 0, 1, · · · , M − 1,
+(3)
+The signal ylo[n] corresponds to a filtered and downsampled
+version of the signal xhi[n] where the filter is an infinite impulse
+response (IIR) filter with a single pole at α. Let ylo ∈ RM
+be a vector obtained by stacking the low-rate measurements
+{ylo[n]}M−1
+n=0 :
+ylo = [ylo[0], ylo[1], · · · , ylo[M − 1]]⊤
+Since (2) represents a causal filtering operation, the low rate
+signal ylo only depends on the present and past high-rate
+binary signal. Denote L := (M − 1)D + 1. The M low-rate
+measurements in ylo are a function of L samples of the high
+
+3
+rate binary input signal {xhi[n]}L−1
+n=0. These L samples are
+given by the following vector xhi ∈ {0, A}L:
+xhi := [xhi[0], xhi[1], · · · , xhi[L − 1]]⊤.
+Assuming the system to be initially at rest, i.e., yhi[n] = 0, n <
+0, we can represent the M samples from (3) in a compact
+matrix-vector form as:
+ylo := SDyhi = SDGαxhi
+(4)
+where Gα ∈ RL×L is a Toeplitz matrix given by:
+Gα =
+�
+����
+1
+0
+· · ·
+0
+α
+1
+· · ·
+0
+...
+...
+...
+...
+αL−1
+αL−2
+· · ·
+1
+�
+����
+(5)
+and SD ∈ RM×L is defined as:
+[SD]i,j =
+�
+1,
+j = (i − 1)D + 1
+0, else
+.
+The matrix SD represents the D−fold downsampling operation.
+Our goal is to infer the unknown high-rate binary input signal
+xhi[n] from the low-rate measurements ylo[n]. This is essentially
+a “super-resolution" problem because the AR(1) filter first
+attenuates the high-frequency components of xhi[n], and
+the uniform downsampling operation systematically discards
+measurements. As a result, it may seem that the spiking activity
+{xhi[(n − 1)D + k]}D
+k=1 occurring “in-between" two low-rate
+measurements ylo[n − 1] and ylo[n] is apparently lost. One can
+potentially interpolate arbitrarily, making the problem hopeless.
+In the next section, we will show that surprisingly, xhi still
+remains identifiable from ylo in the absence of noise, due to
+the binary nature of xhi and “infinite memory" of the AR(1)
+filter.
+A. Identifiability Conditions for Binary super-resolution
+Consider the following partition of xhi into M disjoint blocks,
+where the first block is a scalar and the remaining M −1 blocks
+are of length D, xhi = [xhi(0), xhi(1)⊤, . . . , xhi(M−1)⊤]⊤. Here,
+xhi(0) = xhi[0] and xhi(n) ∈ {0, A}D is given by:
+[xhi
+(n)]k = xhi[(n − 1)D + k],
+1 ≤ n ≤ M − 1
+(6)
+The sub-vectors xhi(n), and xhi(n−1) (n ≥ 1) represent consec-
+utive and disjoint blocks (of length D) of the high-rate binary
+spike signal. In order to study the identifiability of xhi from ylo,
+we first introduce an alternative (but equivalent) representation
+for (4), by constructing a sequence c[n] as follows c[0] = ylo[0],
+c[n] = ylo[n] − αDylo[n − 1], 1 ≤ n ≤ M − 1
+(7)
+Given the high rate AR(1) model defined in (2), it is possible
+to recursively represent yhi[Dn] in terms of yhi[Dn − 1], which
+in turn, can be represented in terms of yhi[Dn − 2], and so
+on. By this recursive relation, we can represent yhi[Dn − 1] in
+terms of yhi[Dn−D] and {xhi[Dn−i]}D−1
+i=0 and re-write ylo[n]
+as
+ylo[n] = yhi[Dn] = αyhi[Dn − 1] + xhi[Dn]
+= αDyhi[Dn − D] + αD−1xhi[D(n − 1) + 1] + · · ·
++ αxhi[Dn − 1] + xhi[Dn],
+ylo[n] − αDylo[n − 1] = αD−1xhi[D(n − 1) + 1] + · · ·
++ αxhi[Dn − 1] + xhi[Dn]
+(8)
+The last equality holds due to the fact that ylo[n−1] = yhi[Dn−
+D]. Combining (7) and (8), the sequence c[n] can be re-written
+as c[0] = ylo[0] = xhi(0), and for 1 ≤ n ≤ M − 1
+c[n] =
+D
+�
+i=1
+αD−ixhi[(n − 1)D + i] = hT
+αxhi
+(n)
+(9)
+where hα = [αD−1, αD−2, . . . , α, 1]T ∈ RD. This implies
+that c[n] depends only on the block xhi(n). Denote c :=
+[c[0], c[1], . . . , c[M − 1]]⊤ ∈ RM. For any D, (9) can be
+compactly represented as:
+c = HD(α)xhi
+(10)
+where HD(α) ∈ RM×L is given by:
+HD(α) =
+�
+������
+1
+0⊤
+0⊤
+· · ·
+0⊤
+0
+h⊤
+α
+0⊤
+· · ·
+0⊤
+0
+0⊤
+h⊤
+α
+· · ·
+0⊤
+...
+...
+...
+...
+...
+0
+0⊤
+0⊤
+· · ·
+h⊤
+α
+�
+������
+The following Lemma establishes the equivalence between (4)
+and (10).
+Lemma 1. Given ylo, construct c following (7). Then, there
+is a unique binary xhi ∈ {0, A}L satisfying (4) if and only if
+xhi is a unique binary vector satisfying (10).
+Proof. First suppose that there is a unique binary xhi ∈ {0, A}L
+satisfying (4) but (10) has a non-unique binary solution, i.e.,
+there exists xhi′ ∈ {0, A}L, xhi′ ̸= xhi, such that
+c = HD(α)xhi = HD(α)xhi
+′
+(11)
+Define yhi′ := Gαxhi′ whose entries are given by:
+yhi
+′[n] =
+n
+�
+k=0
+αn−kxhi
+′[k],
+0 ≤ n ≤ L − 1
+(12)
+Notice that (7) can be re-written as
+ylo[0] = c[0] = xhi[0], ylo[1] = c[1] + αDylo[0] = c[1] + αDc[0]
+ylo[2] = c[2] + αDylo[1] = c[2] + αDc[1] + α2Dc[0]
+...
+Following this recursive relation, and using (9) and (11), we
+can further re-write ylo[n] as:
+ylo[n] =
+n
+�
+i=0
+α(n−i)Dc[i] = αnDx′
+hi
+(0) +
+n
+�
+i=1
+α(n−i)Dh⊤
+α xhi
+′(i)
+= αnDx′
+hi
+(0) +
+n
+�
+i=1
+D
+�
+j=1
+αnD−(i−1)D−jx′
+hi[(i − 1)D + j]
+(a)
+=
+nD
+�
+k=0
+αnD−kx′
+hi[k]
+(b)
+= y′
+hi[nD]
+(13)
+
+4
+The equality (a) follows by a re-indexing of the summation
+into a single sum, and (b) follows from (12). By arranging
+(13) in a matrix form we obtain the following relation:
+ylo = SDGαxhi
+′
+However from (4), we have ylo = SDGαxhi. This contradicts
+the supposition that (4) has a unique binary solution.
+Next, suppose that (10) has a unique binary solution but the
+binary solution to (4) is non-unique, i.e., there exists xhi′ ∈
+{0, A}L, xhi′ ̸= xhi such that
+ylo = SDGαxhi
+′ = SDGαxhi
+By following (7) and (10), we also have c = HD(α)xhi′ =
+HD(α)xhi which contradicts the assumption that solution of
+(10) is unique.
+Lemma 1 assures that a binary xhi is uniquely identifiable
+from measurements ylo if and only if there is a unique binary
+solution xhi ∈ {0, A}L to (10). From (9), it can be seen that
+c[n] and c[n − 1] have contributions from only disjoint blocks
+of high rate spikes xhi(n), and xhi(n−1). Hence effectively,
+we only have a single scalar measurement c[n] to decode an
+entire block xhi(n) of length D, regardless of how sparse it
+is. The task of decoding xhi(n) from a single measurement
+seems like a hopelessly “ill-posed" problem, caused by the
+uniform downsampling operation. But this is precisely where
+the binary nature of xhi can be used as a powerful prior to
+make the problem well-posed. Theorem 1 specifies conditions
+under which it is possible to do so.
+Theorem 1. (Identifiability) For any α ∈ (0, 1), with the
+possible exception of α belonging to a set of Lebesgue measure
+zero, there is a unique xhi ∈ {0, A}L that satisfies (10) for
+every D ≥ 1.
+Proof. In Appendix A.
+Using Lemma 1 and Theorem 1, we can conclude that xhi
+is uniquely identifiable from ylo for almost all α ∈ (0, 1). It
+can be verified that for α = 1 the mapping is non-injective.
+Theorem 1 establishes that it is fundamentally possible to
+decode each block xhi(n) of length D, from effectively a single
+measurement c[n]. Since xhi(n) can take 2D possible values, in
+principle, one can always perform an exhaustive search over
+these 2D possible binary sequences and by Theorem 1, only
+one of them will satisfy c[n] = h⊤
+α xhi(n). Since exhaustive
+search is computationally prohibitive, this leads to the natural
+question regarding alternative solutions. In Section III, we will
+develop an alternative algorithm that leverages the trade-off
+between memory and computation to achieve a significantly
+lower run-time decoding complexity.
+B. Comparison with Finite Rate of Innovation Approach
+In a related line of work [30]–[32], [34], the FRI framework
+has been developed to reconstruct spikes from the measurement
+model considered here. However, in the general FRI framework,
+there is no assumption on the amplitude of the spikes, and there
+are a total of 2D real valued unknowns corresponding to the
+locations and amplitudes of D spikes. In [32], it was shown that
+by leveraging the property of exponentially reproducing kernels,
+it is possible to recover arbitrary amplitudes and spike locations
+using Prony-type algorithms, provided at least 2D+1(> D) low-
+rate measurements are available. However, since we exploit
+the binary nature of spiking activity, we can operate at a
+much smaller sample complexity than FRI. In fact, Theorem
+1 shows that when we exploit the fact that the spikes occur
+on a high-resolution grid with binary amplitudes, M = Ω(1)
+measurements suffice to identify D spikes regardless of how
+large D is. A direct application of the FRI approach cannot
+succeed in this regime, since the number of spikes is larger than
+the number of measurements. That being said, with enough
+measurements, FRI techniques are powerful, and they can also
+identify off-grid spikes. In future, it would be interesting to
+combine the two approaches by incorporating binary priors to
+FRI based techniques and remove the grid assumptions.
+C. Curse of Uniform Downsampling: Inadequacy of sparsity
+and non-negativity
+By virtue of being a binary signal, xhi is naturally sparse and
+non-negative. Therefore, one may ask if sparsity and/or non-
+negativity are sufficient to uniquely identify xhi from c, without
+the need for imposing any binary constraints. In particular, we
+would like to understand if the solution to the following problem
+that seeks the sparsest non-negative vector in RL satisfying
+(10) indeed coincides with the true xhi ∈ {0, A}L
+min
+x∈RL
+∥x∥0
+subject to c = HD(α)x,
+x ≥ 0
+(P0)
+Lemma 2. For every xhi ∈ {0, A}L (except xhi = Ae1),
+and c ∈ RM satisfying (10), the following are true
+(i) There exists a solution x⋆ ̸= xhi to (P0) satisfying
+∥x⋆∥0 ≤ ∥xhi∥0
+(14)
+(ii) The inequality in (14) is strict as long as there exists an
+integer n0 ≥ 1 such that the block x(n0)
+hi
+of xhi (defined
+in (6)) satisfies ∥x(n0)
+hi
+∥0 ≥ 2.
+Proof. The proof is in Appendix B.
+Lemma 2 shows there exist other non-binary solution(s) to
+(10) (different from xhi) that have the same or smaller sparsity
+as the binary signal xhi ∈ {0, A}L. Furthermore, there exist
+problem instances where the sparsest solution to (P0) is strictly
+sparser than xhi. Hence, sparsity and/or non-negativity are
+inadequate to identify the ground truth xhi uniquely.
+Implicit Bias of Relaxation: The optimization problem (P0)
+is non-convex and the binary constraints are not enforced. In
+binary compressed sensing [35], [36], it is common to relax the
+binary constraints using box-constraint and l0 norm is relaxed
+to l1 norm in the following manner:
+min
+x∈RL ∥x∥1
+subject to c = HD(α)x, 0 ≤ x ≤ A1 (P1-B)
+In the following Lemma, we show that there is an implicit bias
+introduced to the solution of (P1-B).
+Lemma 3. For every xhi ∈ {0, A}L, and c ∈ RM satisfying
+(10). There exists a solution x⋆ to (P1-B) satisfying
+∥x⋆∥1 ≤ ∥xhi∥1.
+(15)
+
+5
+Moreover, for all n ≥ 1, the blocks x(n)⋆ ∈RD of x⋆ satisfy:
+supp(x(n)⋆) = {D, D − 1, · · · , D − jn}, if c[n] ̸= 0
+(16)
+for some 0 ≤ jn ≤ D − 1 and x(n)⋆ = 0 if c[n] = 0,
+irrespective of the support of xhi.
+Proof. The proof is in Appendix B.
+Lemma 3 shows that even in the noiseless setting, introducing
+the box-constraint as a means of relaxing the binary constraint
+introduces a bias in the support of the recovered spikes.
+The optimal solution always results in spikes with support
+clustered towards the end of each block of length D, irrespective
+of the ground truth spiking pattern xhi that generated the
+measurements. This bias is a consequence of the nature of
+relaxation, as well as the specific structure of the measurement
+matrix HD(α) arising in the problem.
+D. Role of Memory in Super-resolution: IIR vs. FIR filters
+The ability to identify the high-rate binary signal xhi ∈
+{0, A}L from D−fold undersampled measurements ylo (for
+arbitrarily large D) in the absence of noise, is in parts also due to
+the “infinite memory" or infinite impulse response of the AR(1)
+filter. Indeed, for an Finite Impulse Response (FIR) filter, there
+is a limit to downsampling without losing identifiability. This
+was recently studied in our earlier work [40] where we showed
+that the undersampling limit is determined by the length of
+the FIR filter. To see this, consider the convolution of a binary
+valued signal xhi with a FIR filter u = [u[0], u[1], · · · , u[r −
+1]]T ∈ Rr of length r: zf[n] = �r−1
+i=0 u[r − 1 − i]xhi[n + i].
+These samples are represented in the vector form as zf :=
+u⋆xhi ∈ RL (by suitable zero padding). Suppose, as before, we
+only observe a D−fold downsampling of the output zD[n] =
+zf[Dn]. Two consecutive samples zD[p], zD[p + 1] of the low-
+rate observation are given by:
+zD[p] =
+r−1
+�
+i=0
+u[r − 1 − i]xhi[Dp + i],
+zD[p + 1] =
+r−1
+�
+i=0
+u[r − 1 − i]xhi[D(p + 1) + i]
+If D > r, notice that none of the measurements is a function of
+the samples xhi[Dp+r], xhi[Dp+r +1], · · · , xhi[D(p+1)−1].
+Hence, it is possible to assign them arbitrary binary values and
+yet be consistent with the low-rate measurements zD[n]. This
+makes it impossible to exactly recover xhi (even if it is known
+to be binary valued) if the decimation is larger than the filter
+length (D > r). The following lemma summarizes this result.
+Lemma 4. For every FIR filter u ∈ Rr, if the undersampling
+factor exceeds the filter length, i.e. D > r, there exist x0, x1 ∈
+{0, A}L, x0 ̸= x1 such that SD(u ⋆ x0) = SD(u ⋆ x1).
+This shows that the identifiability result presented in Theorem
+1 is not merely a consequence of binary priors but the infinite
+memory of the autoregressive process is also critical in allowing
+arbitrary undersampling D > 1 in absence of noise. For such
+IIR filters, the memory of all past (binary) spiking activity
+is encoded (with suitable weighting) into every measurement
+captured after the spike, which would not be the case for a
+finite impulse response filter.
+III. EFFICIENT BINARY SUPER-RESOLUTION USING
+BINARY SEARCH WITH STRUCTURED MEASUREMENTS
+By Theorem 1, we already know that it is possible to uniquely
+identify xhi from c (or equivalently, each block xhi(n) from
+a single measurement c[n]) by exhaustive search. We now
+demonstrate how this exhaustive search can be avoided by
+formulating the decoding problem in terms of “binary search"
+over an appropriate set, and thereby attaining computational
+efficiency. We begin by introducing some notations and
+definitions. Given a non-negative integer k, 0 ≤ k ≤ 2D − 1,
+let (b1(k), b2(k), · · · , bD(k)) be the unique D-bit binary repre-
+sentation of k: k = �D
+d=1 2D−dbd(k),
+bd(k) ∈ {0, 1} ∀ 1 ≤
+d ≤ D. Here b1(k) is the most significant bit and bD(k) is
+the least significant bit. Using this notation, we define the
+following set:
+Sall := {v0, v1, v2, · · · , v2D−1},
+(17)
+where each vk ∈ {0, A}D is a binary vector given by
+[vk]d = Abd(k).
+1 ≤ d ≤ D
+(18)
+In other words, the binary vector
+1
+Avk is the D-bit binary
+representation of its index k. Using this convention, v0 = 0
+(i.e., a binary sequence of all 0′s) and v2D−1 = A1 (i.e., a
+binary sequence of all A′s). Recall the partition of xhi defined
+in (6), where each block xhi(n) (n ≥ 1) is a binary vector of
+length D and xhi(0) ∈ {0, A} is a scalar. It is easy to see that
+(17) comprises of all possible values that each block xhi(n) can
+assume. According to (9) each scalar measurement c[n] can be
+written as: c[0] = x(0),
+c[n] = hα⊤xhi(n), 1 ≤ n ≤ M − 1.
+For every α, we define the following set:
+Θα := {θ0, θ1, · · · , θ2D−1}, where θk := h⊤
+α vk
+(19)
+Observe that every measurement c[n] = �D
+i=1 αD−ixhi[(n −
+1)D+i] takes values from this set Θα, depending on the value
+taken by the underlying block of spiking pattern from Sall. Our
+goal is to recover the spikes {xhi[(n − 1)D + i]}D
+i=1 from c[n].
+In the following, we show that this problem is equivalent to
+finding the representation of a real number over an arbitrary
+radix, which is known as “β-expansion" [49]. Given a real
+(potentially non-integer) number β > 1, the representation of
+another real number p ≥ 0 of the form:
+p =
+∞
+�
+n=1
+anβ−n, where 0 ≤ an < ⌊β⌋
+(20)
+is referred to as a β-expansion of p. The coefficients 0 ≤ an <
+⌊β⌋ are integers. This is a generalization of the representation
+of numbers beyond integer-radix to a system where the radix
+can be chosen as an arbitrary real number. This notion of
+representation over arbitrary radix was first introduced by Renyi
+in [49], and since then has been extensively studied [47], [48],
+[50]. There is a direct connection between β-expansion and
+the binary super-resolution problem considered here. In the
+problem at hand, any element θk ∈ Θα can be written as:
+θk = h⊤
+α vk =
+D
+�
+i=1
+αD−i[vk]i
+When 1/2 < α < 1, by letting β = 1/α, we see that the
+coefficients in (20) must satisfy 0 ≤ an < ⌊1/α⌋ < 2, i.e.,
+
+6
+they are restricted to be binary valued an ∈ {0, 1}. Therefore,
+decoding the spikes vk from the observation θk is equivalent
+to finding a D−bit representation for the number θk/A over
+the non-integer radix β = 1/α. Questions regarding the
+existence of β-expansion, and finding the coefficients of a finite
+β−expansion (whenever it exists) has been an active topic of
+research [47], [48], [50], [51]. When β ≥ 2 (equivalently,
+0 < α ≤ 1/2), it is possible to find the coefficients using
+a greedy algorithm which proceeds in a fashion similar to
+finding the D-bit binary representation of an integer [47], [51].
+However, the regime β ∈ (1, 2) (equivalently 1/2 < α < 1),
+is significantly more complicated and is of continued research
+interest [47], [48], [50]. To the best of our knowledge, there
+are no known computationally efficient ways to find the finite
+β-expansion when 1/2 < α < 1 (if it exists) [N. Sidorov,
+personal communication, May 24, 2022]. In practice, we
+encounter filter values α (= 1/β) that are much closer to
+1, and hence, we need an alternative approach to find this
+finite β-radix representation for θk. In the next section, we
+show that by performing a suitable preprocessing, finite β-radix
+representation can be formulated as a binary search problem
+which is guaranteed to succeed for all values of β that permit
+unique finite β−expansions.
+A. Formulation as a Binary Search Problem
+Before describing the algorithm, we first introduce the notion
+of a collision-free set.
+Definition 1 (Collision Free set). Given an undersampling
+factor D, define a class of “collision free" AR(1) filters as:
+GD = {α ∈ (0, 1) s.t. h⊤
+α vi ̸= h⊤
+α vj ∀ i ̸= j, vi, vj ∈ Sall}
+The set GD denotes permissible values of the AR(1) filter
+parameter α such that each of the 2D binary sequences in
+Sall maps to a unique element in the set Θα. In other words,
+every θk ∈ Θα has a unique D−bit expansion for all α ∈ GD.
+This naturally raises the question “How large is the set GD?".
+Theorem 1 already provided the answer to this question, where
+the identifiability result implies that for every D, almost all
+α ∈ (0, 1) belong to this set GD (with the possible exception
+of a measure zero set). Hence, Theorem 1 ensures that there
+are infinite choices for collision-free filter parameters.
+Lemma 5. For every α ∈ GD, the mapping Φα(.) : Sall → Θα,
+Φα(v) = h⊤
+α v forms a bijection between Sall and Θα.
+Proof. Since α ∈ GD, from the definition of the set GD, it is
+clear that for any vi, vj ∈ Sall, vi ̸= vj we have hα⊤vi ̸=
+hα⊤vj. Therefore, the mapping is injective. Furthermore, from
+(19) we also have |Θα| ≤ |Sall| = 2D. Since Φα(·) is injective,
+we must also have |Θα| = 2D and hence the mapping Φα(.)
+forms a bijection between Sall and Θα.
+When α ∈ GD, Lemma 5 states that the finite beta expansion
+for every θk ∈ Θα is unique. Lemma 5 provides a way to avoid
+exhaustive search over Sall, and yet identify xhi(n) from c[n] in
+a computationally efficient way. From Lemma 5, we know that
+each of the 2D spiking patterns in Sall maps to a unique element
+in Θα, and each element in Θα has a corresponding spiking
+pattern. Hence instead of searching Sall, we can equivalently
+search the set Θα in order to determine the unknown spiking
+pattern. Since Θα permits “ordering", searching Θα has a
+distinct computational advantage over searching Sall. This
+ordering enables us to employ binary search over (an ordered)
+Θα and find the desired element in a computationally efficient
+manner. To do this, we first sort the set Θα (in ascending order)
+and arrange the corresponding elements of Sall in the same
+order. Given Θα as an input, the function SORT(·) returns
+a sorted list Θsort
+α , and an index set I = {i0, i1, · · · , i2D−1}
+containing the indices of the sorted elements in the list Θα.
+Θsort
+α , I ← SORT(Θα)
+Let us denote the elements of the sorted lists as Θsort
+α
+=
+{�θ0, · · · , �θ2D−1}, and Ssort
+all = {�v0, · · · , �v2D−1} where:
+�θ0 < �θ1 < · · · < �θ2D−1
+and �θj = θij,
+�vj = vij
+∀j.
+It is important to note that this sorting step does not depend
+on the measurements c, and can therefore be part of a pre-
+processing pipeline that can be performed offline. However,
+it does require memory to store the sorted lists. In the
+Algorithm 1 Noiseless Spike Recovery
+1: Input: Measurement c[n], Sorted list Θsort
+α
+and the corre-
+sponding (ordered) spike patterns Ssort
+all
+2: Output: Decoded spike block �xhi(n)
+3: i⋆ ← BINSEARCH(Θsort
+α , c[n])
+4: Return �xhi(n) ← �vi⋆
+noiseless setting, we know that every scalar measurement
+c[n] = h⊤
+α xhi(n) belongs to the set Θsort
+α . Therefore, if we
+identify its index, say i⋆, then we can successfully recover
+xhi(n) by returning the corresponding binary vector �vi⋆ from
+Ssort
+all . Therefore, we can formulate the decoding problem as
+searching for the input c[n] in the sorted list Θsort
+α . This can be
+efficiently done by using “Binary Search". The noiseless spike
+decoding procedure is summarized as Algorithm 1. Since the
+complexity of performing a binary search over an ordered list
+of N elements is O(log N), the complexity of Algorithm 1
+is logarithmic in the cardinality of Θsort
+α , which results in a
+complexity of O(log(2D)) = O(D). We summarize this result
+in the following Lemma.
+Lemma 6. Assume α ∈ GD. Given the ordered set Θsort
+α
+, and
+an input c[n] = h⊤
+α xhi(n), Algorithm 1 terminates in O(D)
+steps and its output �xhi(n) satisfies �xhi(n) = xhi(n).
+B. Noisy Measurements and 1 D Nearest Neighbor Search
+We demonstrate how binary search can still be useful in
+presence of noise by formulating noisy spike detection as a
+one dimensional nearest neighbor search problem. Suppose
+{zlo[n]}M−1
+n=0 denote noisy D-fold decimated filter output
+zlo[n] = ylo[n] + w[n],
+0 ≤ n ≤ M − 1
+(21)
+
+7
+Here w[n] represents the additive noise term that corrupts the
+(noiseless) low-rate measurements ylo[n]. Similar to (7), we
+compute ce[n] from zlo[n] as follows:
+ce[n] = zlo[n] − αDzlo[n − 1]
+(22)
+=
+D
+�
+i=1
+αD−ixhi[(n − 1)D + i] + e[n]= c[n] + e[n] (23)
+where c[n] = h⊤
+α xhi(n) ∈ Θsort
+α , and e[n] = w[n] − αDw[n −
+1]. We can interpret ce[n] as a noisy/perturbed version of an
+element c[n] ∈ Θsort
+�� , with e[n] representing the noise. This
+perturbed signal may no longer belong to Θsort
+α
+(i.e. ce[n] ̸∈
+Θsort
+α ) and hence, we cannot find an exact match in the set
+Θsort
+α . Instead, we aim to find the closest element in Θsort
+α
+(the
+nearest neighbor of ce[n]) by solving the following problem:
+�xhi
+(n) = arg min
+v∈Ssort
+all
+|ce[n] − h⊤
+α v|
+(24)
+Solving (24) is equivalent to finding the spike sequence
+�v ∈ Ssort
+all
+that maps to the nearest neighbor of ce[n] in the
+set Θsort
+α . By leveraging the sorted list Θsort
+α , it is no longer
+necessary to parse the list sequentially (which would incur
+O(2D) complexity), instead we can perform a modified binary
+search as summarized in Algorithm 2, that keeps track of
+additional indices compared to the vanilla binary search. Finally,
+we return the unique spiking pattern from Ssort
+α
+that gets
+mapped to the nearest neighbor of the noisy measurement
+ce[n]. It is well-known that the nearest neighbor for any query
+could be found in O(log(2D)) = O(D) steps, instead of the
+linear complexity of O(2D). This guarantees a computationally
+efficient decoding of spikes by solving (24).
+Next, we characterize the error events that lead to erroneous
+detection of a block of spikes. Recall that the set Θsort
+α
+is sorted,
+and its elements satisfy the ordering:
+0 = �θ0 < �θ1 < · · · < �θlD = 1 + α + · · · + αD−1
+where lD := 2D−1. We also have �θk = h⊤
+α �vk, where �vk ∈ Ssort
+all
+is a binary spiking sequence of length D.
+For each �vk and each n, we will determine the error event
+�xhi(n) ̸= xhi(n), when xhi(n) = �vk. First, consider the scenario
+when xhi(n) = �vk for some 0 < k < lD (excluding �v0, �vlD).
+The corresponding noiseless measurement is c[n] = �θk =
+h⊤
+α �vk which satisfies �θk−1 < c[n] = �θk < �θk+1. Since Θsort
+α
+is
+sorted, it can be easily verified that the nearest neighbor of
+ce[n] will be �θk, if and only if ce[n] satisfies the following
+condition:
+(�θk−1 + �θk)/2 ≤ ce[n] ≤ (�θk+1 + �θk)/2
+(25)
+Since �θk = h⊤
+α �vk, the solution to (24) is attained at �vk ∈ Ssort
+all ,
+and the decoding is successful. Therefore Algorithm 2 produces
+an erroneous estimate of �vk if and only if ce[n] violates (25).
+The event ce[n] ̸∈ [
+�θk−1+�θk
+2
+,
+�θk+1+�θk
+2
+] is equivalent to e[n] ∈
+Ek (e[n] is defined earlier in (23)), where
+Ek = {e[n] < −
+�θk − �θk−1
+2
+, or e[n] >
+�θk+1 − �θk
+2
+}
+(26)
+Finally, we characterize the error events for k = 0, lD. The
+error events for c[n] = θ0 = 0 or c[n] = θlD are given by:
+E0 = {e[n] ≥ �θ1/2}, ElD = {e[n] ≤ −(�θlD − �θlD−1)/2} (27)
+Define the “minimum distance" between points in Θsort
+α :
+∆θmin(α, D) =
+min
+1≤k≤lD |�θk − �θk−1|.
+This minimum distance depends on A, α and D. From (26),
+(27) it can be verified that if 2|w[n]| < ∆θmin(α, D)/2 (which
+would imply |e[n]| < ∆θmin(α, D)/2) for all n, then �xhi(n) =
+xhi(n). As summarized in Theorem 2, Algorithm 2 can exactly
+recover the ground truth spikes from measurements corrupted
+by bounded adversarial noise, the extent of the robustness is
+determined by the parameters A, α, D.
+Algorithm 2 Noisy Spike Recovery
+1: Input: Measurement ce[n], Sorted list Θsort
+α
+and the
+corresponding (ordered) spike patterns Ssort
+all
+2: Output: Decoded spike block �xhi(n)
+3: Set l ← 0, u ← 2D − 1
+4:
+while u − l > 1
+5:
+Set m ← l + ⌊(u − l)/2⌋
+6:
+if �θm > ce[n] then
+7:
+u ← m
+8:
+else
+9:
+l ← m
+10:
+end if
+11:
+end while
+12: Find the nearest neighbor i⋆ = arg mini∈{l,u}(ce[n]− �θi)2
+13: Return �xhi(n) ← �vi⋆
+Theorem 2. Assume α ∈ GD. Given the ordered set Θsort
+α , the
+output of Algorithm 2 with input ce[n] exactly coincides with
+the solution of the optimization problem (24) in at most O(D)
+steps. Furthermore, if for all n, |w[n]| < ∆θmin(α, D)/4, then
+the output of Algorithm 2 satisfies �xhi(n) = xhi(n).
+From Theorem 2, it is evident that ∆θmin(α, D) plays an
+important role in characterizing the upper bound on noise.
+We attempt to gain insight into how ∆θmin(α, D) varies as a
+function of α when D is held fixed.
+Lemma 7. Given D, ∆θmin(α, D) = αD−1 for α ∈ (0, 0.5].
+Proof. The proof is in Appendix C.
+When α ∈ (0, 0.5], ∆θmin(α, D) is monotonically increasing
+with α. However, for α > 0.5 the trend fluctuates with α
+differently for different D, and becomes quite challenging to
+predict. This is also confirmed by the empirical plot in Fig. 1.
+A refined analysis of ∆θmin(α, D) to gain insight into desirable
+filter parameters α is an interesting direction for future work.
+C. Trade-off between memory and computational complexity
+A crucial aspect of Algorithms 1 and 2 is that they
+achieve efficient run-time complexity by leveraging the off-
+line construction of the sorted list Θsort
+α
+and Ssort
+all . These lists,
+each with 2D elements, need to be stored in memory and
+made available during run-time. Since there is no free lunch,
+the resulting computational efficiency of O(D) at run-time
+is attained at the expense of the additional memory that is
+required to store the sorted lists Θsort
+α , Ssort
+all .
+
+8
+D. Parallelizable Implementation
+Algorithm 2 (also Algo. 1) only takes ce[n](c[n]) as input
+and returns �xhi(n), and is completely de-coupled from any
+other �xhi(n′), n′ ̸= n. Recall that in reality, we are provided
+with measurements zlo[n](ylo[n]), and ce[n](respectively c[n])
+needs to be computed. Due to this de-coupling, we can compute
+ce[n]′s in parallel using two consecutive low-rate samples
+zlo[n], zlo[n−1] and perform a nearest neighbor search without
+waiting for any previously decoded spikes. Therefore, the total
+decoding complexity can be further improved depending on
+the available parallel computing resources.
+IV. ERROR ANALYSIS FOR GAUSSIAN NOISE
+Algorithm 2 solves (24) without requiring any knowledge
+of the noise statistics. However, in order to analyze its per-
+formance, we will make the following (standard) assumptions
+on the statistics of the high-rate spiking signal xhi and the
+measurement noise w[n] as follows:
+• (A1) The entries of the binary vector xhi ∈ {0, A}L are
+i.i.d random variables distributed as xhi[n] ∼ ABern(p).
+• (A2) The additive noise w[n], 0 ≤ n ≤ M − 1 is
+independent of xhi[n], and distributed as w[n] ∼ N(0, σ2)
+A. Probability of Erroneous Decoding
+Under assumption (A2), the ML estimate of xhi is given by
+the solution to the following problem:
+�xML = arg
+min
+v∈{0,A}L ∥zlo − SDGαv∥2
+(PNN)
+The proposed Algorithm 2 does not attempt to solve
+(PNN), which is computationally intractable. Instead, it solves
+a set of M − 1 one dimensional nearest neighbor search
+problems, by finding the nearest neighbor of ce[n] for each
+n = 1, 2, · · · , M − 1. This scalar nearest neighbor search is
+implemented in a computationally efficient manner by using
+parallel binary search on a pre-sorted list. Notice that by the
+operation (22), the variance of the equivalent noise term e[n]
+gets amplified by a factor of at most (1+α2D) < 2. This can be
+thought of as a price paid to achieve computational efficiency
+and parallelizability. The following theorem characterizes the
+dependence of certain key quantities of interest, such as the
+signal-to-noise ratio (SNR), undersampling factor D, and filter’s
+frequency response (controlled by α) on the performance of
+Algorithm 2.
+Theorem 3. Suppose α ∈ GD and assumptions (A1-A2) hold.
+Given δ > 0, if the following condition is satisfied:
+∆θ2
+min(α, D)/σ2 ≥ 4 ln (2M/δ)
+(28)
+then Algorithm 2 can exactly recover the binary signal xhi
+with probability at least 1 − δ.
+Proof. The proof follows standard arguments for computing
+the probability of error for symbol detection in Gaussian noise,
+followed by certain simplifications and is included in Appendix
+D for completeness.
+In Fig. 1, we plot ∆θmin(α, D) as a function of D for
+different values of α. As expected, ∆θmin(α, D) decays as the
+D increases. Understandably, for a fixed α, as D increases,
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+Minimum distance
+For D=4
+For D=5
+Min. Dist (D=4)
+Min. Dist (D=5)
+Cluster Min. Dist (D=4)
+Cluster Min. Dist (D=5)
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Undersampling factor (D)
+0
+0.2
+0.4
+0.6
+0.8
+1
+Minimum distance
+=0.2
+=0.5
+=0.9
+Fig. 1: Variation of ∆θmin(α, D) as a function of undersampling factor
+D and α. The cluster-distance ∆c
+min(α, D) vs. α is also overlaid. Each
+dotted line denotes the start of the interval FD.
+it becomes harder to recover the spikes exactly, and higher
+SNR is needed to compensate for the lower sampling rate.
+This can be interpreted as the price paid for super-resolution
+in presence of noise. This phenomenon is also reminiscent of
+the noise amplification effect in super-resolution, where the
+ability to super-resolve point sources becomes more severely
+hindered by noise as the target resolution grid becomes finer
+[6]. In Fig. 1, we plot ∆θmin(α, D) as a function of α and as
+predicted by Lemma 7, it monotonically increases upto 0.5,
+but for α > 0.5, the behavior becomes much more erratic
+and a precise characterization becomes challenging. It is to
+be noted that in Theorem 3, we aim to exactly recover xhi.
+The SNR requirement can be relaxed if our goal is to recover
+only spike counts instead of the true spikes as discussed in the
+next subsection. One can define other notions of approximate
+recovery, the analysis of which will be a topic of future research.
+B. Relaxed Spike reconstruction: Count Estimation
+As shown in Theorem 2, exact recovery of spikes is possible
+under somewhat restrictive condition on the noise in terms
+of ∆θmin(α, D), which becomes quite small as D increases.
+This naturally calls for other relaxed notions of recovery
+which can handle larger noise levels. In neuroscience, it is
+believed that information is encoded as either the spike timing
+(temporal code) or the firing rates (rate coding) of individual
+neurons in the brain. Therefore, the spike counts over an
+interval can be informative to understand neural functions, even
+when it is impossible to temporally localize the neural spikes.
+For example, neurons in the visual cortex encode stimulus
+orientations as their firing rates [52]. We will therefore focus
+on spike count as an approximate recovery metric, which
+concerns estimating the number of spikes occurring between
+two consecutive low-rate measurements instead of resolving
+the individual spiking activity at a higher resolution.
+Let γ[n] denote the total number of spikes occurring between
+two consecutive low-rate samples zlo[n] and zlo[n − 1]. Since
+xhi and its estimate �xhi are both binary valued (amplitude A),
+the true spike count (γ[n]) and estimated count (�γ[n]) are given
+by: γ[n] = ∥xhi(n)∥0,
+�γ[n] = ∥�x(n)
+hi ∥0, n = 1, · · · , M − 1,
+
+9
+γ[0] = xhi[0]/A and �γ[0] = �xhi[0]/A since the first block is of
+size 1 as described in (6). Define a set CD
+k as:
+CD
+k := {v ∈ {0, A}D, ∥v∥0 = k},
+0 ≤ k ≤ D
+It is a collection of all binary vectors (of length D) with spike
+count k. The ground truth spike block belongs to CD
+γ[n]. Any
+element from CD
+γ[n] will give the true spike count. Hence, exact
+recovery of count can be possible even when spikes cannot be
+recovered.
+For a fixed D, we define a set of α denoted by FD:
+FD := {α ∈ (0, 1)|αD − αD−k0−1 − αk0 + 1 < 0}
+(29)
+where k0 = ⌊D/2⌋. We will obtain a sufficient condition for
+robust spike count estimation when α ∈ FD. It can be shown
+that for any D, FD will always be non-empty. Define
+θk
+min := min
+u∈CD
+k
+h⊤
+α u
+θk
+max := max
+u∈CD
+k
+h⊤
+α u
+(30)
+Observe that if
+θk+1
+min > θk
+max, k = 0, 1, · · · , D − 1
+(31)
+then all spike patterns ui ∈ CD
+k (with the same spike count k)
+are clustered together when mapped on to the real line by the
+transformation h⊤
+α u as shown in Figure 2. When (31) holds,
+we can define a “cluster-restricted minimum distance" as:
+∆c
+min(α, D) :=
+min
+0≤k≤D−1 θk+1
+min − θk
+max
+(32)
+Given a noisy observation ce[n] = h⊤
+α xhi(n)+e[n], the solution
+to the nearest neighbor problem (24) may return an incorrect
+neighbor θj ̸= h⊤
+α xhi(n). However, when (31) holds and if
+the noisy observation satisfies the following conditions:
+(θγ[n]
+min + θγ[n]−1
+max
+)/2 < ce[n] < (θγ[n]+1
+min
++ θγ[n]
+max)/2
+(33)
+then the nearest-neighbor decision rule in Algorithm 2 will still
+ensure that θj ∈ CD
+γ[n]. This has also been visualized in Fig. 2
+where each colored band represents the “safe-zone" for each
+count and the black dotted-line denotes the boundary. This will
+result in correct identification of the spike count but will incur
+error in terms of spiking pattern. We formally summarize this
+in the following Theorem that provides robustness guarantee
+for exact count recovery from measurements corrupted by
+adversarial noise (similar to Theorem 2 for spike recovery).
+Theorem 4. Assume α ∈ FD. Given the ordered set Θsort
+α , let
+�γ[n] be the estimated spike count obtained from Algorithm 2
+with input ce[n]. If for all n, |w[n]| < ∆c
+min(α, D)/4, then the
+count can be exactly recovered, i.e., �γ[n] = γ[n].
+Proof. Proof is in Appendix E.
+It is clear that when (31) holds, ∆c
+min(α, D) is no smaller
+than ∆θmin(α, D), since the former is computed over neigh-
+boring elements of the cluster whereas ∆θmin(D, α) computes
+the minimum distance over all consecutive elements (both
+inter-cluster as well as intra-cluster) in Θsort
+α . This essentially
+suggests that estimation of counts (for this range of α and
+D) can be more robust compared to inferring the individual
+spiking patterns. We also illustrate this numerically in Figure
+1 (top), where we plot both ∆c
+min and ∆θmin as a function of
+α and the start of the interval FD (computed numerically) is
+C0
+C1
+C2
+C3
+000
+100
+010
+001
+110
+101
+011
+111
+Fig. 2: Visualization of the sets CD
+k for D = 3. In this scenario, the
+spiking patterns corresponding to the same count are clustered together
+and hence, are favorable for robust count estimation.
+denoted using dotted lines. For both values of D, we can see
+that ∆c
+min > ∆θmin and the gap grows as α increases.
+V. NUMERICAL EXPERIMENTS
+We conduct numerical experiments to evaluate the per-
+formance of the proposed super-resolution spike decoding
+algorithm on both synthetic and real calcium imaging datasets.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Undersampling Factor (D)
+0
+0.2
+0.4
+0.6
+0.8
+1
+F-score
+p=0.35, s=350
+Algo 2 ( =0.5)
+l1 Box ( =0.5)
+Algo 2 ( =0.9)
+l1 Box ( =0.9)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Undersampling factor (D)
+0.6
+0.7
+0.8
+0.9
+1
+F1-score
+p=0.5, s=500
+D=3
+D=5
+D=7
+AR(1), =0.5
+FIR (r=3)
+FIR (r=5)
+FIR (r=7)
+Fig. 3: (Top) Quantitative comparison of Algorithm 2 against box-
+constrained l1 minimization method with noiseless measurements
+(with tolerance t0 = 0). (Bottom) (Role of Filter Memory): Average
+F-score vs. D for FIR and IIR (AR(1)) filters. Each dotted line indicates
+the corresponding theoretical transition point (D = r).
+A. Synthetic Data Generation and Evaluation Metrics
+We create a synthetic dataset by generating high-rate binary
+spike sequence xhi ∈ {0, 1}L (A = 1 and L = 1000) that
+satisfies assumption (A1). The spiking probability p controls
+the average sparsity level given by s := E[∥xhi∥0] = Lp. We
+aim to reconstruct xhi from M ≈ L/D low-rate measurements
+zlo[n] defined in (21). Notice that we operate in a regime where
+the expected sparsity is greater than the total number of low-
+rate measurements, i.e., s > M. We employ the widely-used
+F-score metric to evaluate the accuracy of spike detection [4],
+[10]. The F-score is computed by first matching the estimated
+and ground truth spikes. An estimated spike is considered a
+“match" to a ground truth spike if it is within a distance of t0
+of the ground truth (many-to-one matching is not allowed) [4],
+[10]. Let K and K′ be the total number of ground truth and
+estimated spikes, respectively. The number of spikes declared as
+true positives is denoted by Tp. After the matching procedure,
+we compute the recall (R =
+Tp
+K ) which is defined as the
+ratio of true positives (Tp) and the total number of ground
+truth spikes (K). Precision (P = Tp
+K′ ) measures the fraction
+of the total detected spikes which were correct. Finally, the
+F-score is given by the harmonic mean of recall and precision
+F-score = 2PR/(P + R).
+
+10
+0 
+2 
+4 
+6 
+8 
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+42
+44
+46
+48
+50
+0
+4.99
+yhi[n]
+D = 5
+(Top)
+(Bottom)
+0
+4.99
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+ylo[n]
+0
+1.02
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+xhi[n]
+0
+1.02
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+�xhi[n]
+0
+1.02
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+�xl1[n]
+0 
+2 
+4 
+6 
+8 
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+42
+44
+46
+48
+50
+0
+4.47
+yhi[n]
+D = 10
+0
+4.47
+0 
+10
+20
+30
+40
+50
+ylo[n]
+0
+1.02
+0 
+10
+20
+30
+40
+50
+xhi[n]
+0
+1.02
+0 
+10
+20
+30
+40
+50
+�xhi[n]
+0
+1.02
+0 
+10
+20
+30
+40
+50
+�xl1[n]
+0
+4.99
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+ylo[n]
+0
+1.02
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+xhi[n]
+0
+1.02
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+�xhi[n]
+0
+1.02
+0 
+5 
+10
+15
+20
+25
+30
+35
+40
+45
+50
+�xl1[n]
+0
+3.45
+0 
+10
+20
+30
+40
+50
+ylo[n]
+0
+1.02
+0 
+10
+20
+30
+40
+50
+xhi[n]
+0
+1.02
+0 
+10
+20
+30
+40
+50
+�xhi[n]
+0
+1.02
+0 
+10
+20
+30
+40
+50
+�xl1[n]
+xhi[n]: Ground Truth Spikes, �xhi[n]: Output of Algorithm 2, �xl1[n]: Output of l1 minimization,
+yhi[n]: High rate waveform, ylo[n]: Low rate samples
+Fig. 4: Qualitative comparison of Algorithm 2 and box-constrained l1 minimization on simulated data. For each simulation noisy measurements
+are generated with α = 0.9 such that the noise realization (Top) obeys the bound |w[n]| ≤ ∆θmin (from Theorem 2) and (Bottom) violates
+the bound. For larger noise (Bottom), the spike recovery is imperfect but the spike count can still be exactly recovered using Algorithm 2.
+B. Noiseless Recovery: Role of Binary priors and memory
+We first consider the noiseless setting (w[n] = 0 in (21)).
+We compare the performance of Algorithm 2 against box-
+constrained l1 minimization method [35], [36], where we solve:
+min
+x∈RL ∥x∥1 s.t. ∥ylo − SDGαx∥2 ≤ ϵ, 0 ≤ x ≤ A1
+(P1)
+For synthetic data, ϵ is chosen using the norm of the noise term
+∥w∥2. This oracle choice ensures most favorable parameter
+tuning for the (P1), although a more realistic choice would
+be to set ϵ =
+√
+Mσ according to the noise power (σ). In the
+noiseless setting, we choose ϵ = 0. The problem (P1) is a
+standard convex relaxation of (P0) which promotes sparsity
+as well as tries to impose the binary constraint via the box-
+relaxation (introduced in Section II-C). In Fig. 3 (Top), we plot
+the F-score (t0 = 0) as a function of D. As can be observed,
+Algorithm 2 consistently achieves an F-score of 1, whereas the
+F-score of l1 minimization shows a decay as D increases. This
+confirms Lemma 3 that for D > 1, using box-constraints with l1
+norm minimization is not enough to enable exact recovery from
+low rate measurements. In absence of noise, the performance
+of Algorithm 2 is not affected by the filter parameter α as
+shown in Fig. 3 (Top).
+Next, we compare the reconstruction from the decimated
+output of (i) an AR(1) filter and (ii) an FIR filter of length
+r driven by the same input xhi ∈ {0, 1}1000. We choose the
+FIR filter h = [1, α, · · · , αr−1]⊤ (truncation of the IIR filter)
+with α = 0.5. Algorithm 2 is applied to the low-rate AR(1)
+measurements, whereas the algorithm proposed in [40] is used
+for the FIR case. The algorithm applied for the FIR case can
+provably operate with the optimal number of measurements
+when α = 0.5 and hence, we chose this specific value for
+the filter parameter. In Figure 3 (Bottom), we again compare
+the average F-score as a function of D, averaged over 10000
+Monte Carlo runs, for p = 0.5. As predicted by Lemma 4,
+despite utilizing binary priors, the error for the FIR filter shows
+a phase transition when D > r. This demonstrates the critical
+role played by the infinite memory of the AR(1) filter in
+achieving exact recovery with arbitrary D.
+C. Performance of noisy spike decoding
+We generate noisy measurements of the form (21), where
+w[n] and xhi[n] satisfy assumptions (A1-A2). We illustrate
+some representative examples of recovered spikes on synthetic
+data. In Fig. (4), we display the recovered super-resolution
+estimates on synthetically generated measurements for two
+undersampling factors D = 5 (left), 10 (right). For each D, the
+top plots show the spikes recovered using Algorithm 2 and l1
+minimization with box-constraint where the noise realization
+obeys the bound in Theorem 2, while the bottom plots show
+the same for noise realization violating the bound. The output
+of l1 minimization with box-constraint is inaccurate, and the
+spikes are clustered towards the end of each block of length
+D. This bias is consistent with the prediction made by our
+theoretical results in Lemma 3. When the noise is small enough
+(top), Algorithm 2 exactly decodes the spikes, including the
+ones occurring between two consecutive low-rate samples as
+predicted by Theorem 2. In presence of larger noise (violating
+the bound), the spikes estimated using l1 minimization continue
+to be biased to be clustered towards the end of the block.
+Although the spikes recovered using Algorithm 2 are not exact,
+most of the detected spikes are within a tolerance window of
+ground truth spikes. In fact, the spike count estimation is perfect
+as predicted by Theorem 4. We next quantitatively evaluate
+the performance in presence of noise, where the metrics are
+computed with t0 = 2. In Fig. 5 (Top), we plot the F-score
+as a function of D for different values of α. For a fixed α,
+the F-score of both methods decays with increasing D, but
+Algorithm 2 consistently attains a higher F-score compared to
+
+11
+3
+4
+5
+6
+7
+8
+9
+10
+Undersampling Factor (D)
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+F-score
+p=0.35, s=350>M
+Algo 2 (alpha=0.9)
+l1 Box (alpha=0.9)
+Algo 2 (alpha=0.5)
+l1 Box (alpha=0.5)
+Fig. 5: Spike detection performance with noisy measurements. (Top)
+F-score vs. D for different filter parameters α (σ = 0.01). Here,
+L = 1000 and expected sparsity s = 350 where we operate in the
+regime s > M. The F-score is computed with a tolerance of t0 = 2.
+l1 minimization. We observe that α = 0.5 leads to a higher F-
+score potentially due to having a larger ∆θmin(α, D) compared
+to α = 0.9. Next, in Fig. 7, we study the behavior of spike
+detection as a function of the spiking probability p, while
+keeping D fixed at D = 5. When σ is fixed, the performance
+trend is not significantly affected by the spiking probability.
+At first, this may seem surprising as the expected sparsity
+is growing while the number of measurements is unchanged.
+However, since our algorithm exploits the binary nature of
+the spikes (and not just sparsity), it can handle larger sparsity
+levels. The spikes reconstructed using l1 minimization achieve
+a much lower F-score than Algorithm 2 since the former fails
+to succeed when the sparsity is large. As expected, smaller σ
+leads to higher F-scores.
+In Fig. 8, we study the probability of erroneous spike
+detection as a function of D and validate the upper bound
+derived in Theorem 3. Recall that the decoding is considered
+successful if “every" spike is detected correctly. Therefore, it
+becomes more challenging to “exactly super-resolve" all the
+spikes in presence of noise as the desired resolution becomes
+finer. We calculate the empirical probability of error and overlay
+the corresponding theoretical bound. As shown in Fig. 8, the
+empirical probability of error is indeed upper bounded by the
+bound computed by our analysis. The empirical probability of
+error increases as a function of undersampling factor D.
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Noise Level
+0.2
+0.4
+0.6
+0.8
+1
+F-score
+D=5, M=200, s/M>1
+Algo 2 ( =0.5)
+l1 Box ( =0.5)
+Algo 2 ( =0.9)
+l1 Box ( =0.9)
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Noise Level
+10-10
+10-5
+100
+105
+Count Estimation Error
+D=5, M=200, s/M>1
+Algo 2 ( =0.5)
+l1 Box ( =0.5)
+Algo 2 ( =0.9)
+l1 Box ( =0.9)
+Fig. 6: Spike detection performance with noisy measurements for
+different filter parameters α. (Top) F-score vs. noise level (σ) (Bottom)
+Count estimation error vs. noise level. Here, L = 1000 and expected
+sparsity is fixed at s = 350 where we operate in the regime s > M.
+The F-score is computed with a tolerance of t0 = 2.
+Finally, we evaluate the noise tolerance of the proposed
+methodology by comparing the average F-score as a function
+of the noise level σ, while keeping the spiking rate and
+undersampling factor fixed at p = 0.35 and D = 5, respectively.
+As seen in Fig. 6 (Top), the performance of both algorithms
+degrades with increasing noise level and this is also consistent
+with the intuition that it becomes harder to super-resolve spikes
+with more noise. However, for both filter parameters considered
+in this experiment Algorithm 2 has a higher F-score compared
+to box-constrained l1 minimization. For large noise levels
+(comparable to spike amplitude A = 1), the performance gap
+decreases for α = 0.9 but Algorithm 2 achieves a much higher
+F-score for α = 0.5 at all noise levels.
+As discussed in Section IV-B, we next study a relaxed
+notion of spike recovery which focuses on the spike counts
+occurring between two consecutive low-rate samples. Let Γ =
+[γ[0], · · · , γ[M − 1]]⊤ be the vector of counts and �Γ be its
+estimate. In Fig. 6 (Bottom) we plot the average l1 distance
+∥Γ − �Γ∥1 as a function of the noise level. We observe that for
+α = 0.9 (it can be verified from Fig. 1 (Top) that 0.9 ∈ F5), it
+is possible to exactly recover the spike counts at higher noise
+even though the F-score (for timing recovery) has dropped
+below 1. However, this is not the case for α = 0.5, since
+0.5 ̸∈ F5. This is consistent with the conclusion of Theorem 4
+which states that when α ∈ FD, the noise tolerance for exact
+count recovery can be much larger than exact spike recovery
+since ∆c
+min(α, D) > ∆θmin(α, D).
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+Spiking Probability (p)
+0.2
+0.4
+0.6
+0.8
+1
+F-score
+D=5, M=200, s/M>1
+Algo 2 (sigma=0.001)
+l1 Box (sigma=0.001)
+Algo 2 (sigma=0.01)
+l1 Box (sigma=0.01)
+Fig. 7: Spike detection performance with noisy measurements. F-score
+vs. spiking probability (p) for different noise levels σ (fix α = 0.9,
+D = 5, L = 1000) in the extreme compression regime s > M.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Undersampling factor (D)
+0
+0.2
+0.4
+0.6
+0.8
+1
+Probability of Error
+s=30, L=100
+Algo 2 ( =0.9)
+Theoretical Bound ( =0.9)
+Algo 2 ( =0.95)
+Theoretical Bound ( =0.95)
+Fig. 8: Probability of erroneous detection of high-rate spikes xhi ∈
+{0, 1}L as a function of the undersampling factor D. Theoretical
+upper bounds are overlaid using dotted lines. Here, L = 100.
+D. Spike Deconvolution from Real Calcium Imaging Datasets
+We now discuss how the mathematical framework developed
+in this paper can be used for super-resolution spike deconvo-
+lution in calcium imaging. Two-photon calcium imaging is a
+widely used imaging technique for large scale recording of
+neural activity with high spatial but poor temporal resolution. In
+calcium imaging, the signal xhi corresponds to the underlying
+neural spikes which is modeled to be binary valued on a finer
+temporal scale [2], [46]. Each neural spike results in a sharp
+rise in Ca2+ concentration followed by a slow exponential
+decay, leading to superposition of the responses from nearby
+
+12
+spiking events [2]–[4]. This calcium transient can be modeled
+by the first order autoregressive model introduced in Section
+II. The decay time constant depends on the calcium indicator
+and essentially determines the filter parameter α. The signal
+yhi[n] is an unobserved signal corresponding to sampling the
+calcium fluorescence at a high sampling rate (at the same rate
+as the underlying spikes). The observed calcium signal ylo[n]
+corresponds to downsampling yhi[n] at an interval determined
+by the frame rate of the microscope. The frame rate of a
+typical scanning microscopy system (that captures the changes
+in the calcium fluorescence) is determined by the amount of
+time required to spatially scan the desired field of view, which
+makes it significantly slower compared to the temporal scale
+of the neural spiking activity. We model this discrepancy by
+the downsampling operation (by a factor D). Therefore, the
+mathematical framework developed in this paper can be directly
+applied to reconstruct the underlying spiking activity at a
+temporal scale finer than the sampling rate of the calcium signal.
+Using real calcium imaging data, we demonstrate a way to fuse
+our algorithm with a popular spike deconvolution algorithm
+called OASIS [43]. OASIS solves an l1 minimization problem
+similar to (P1) with only the non-negativity constraint, in order
+to exploit the sparse nature of the spiking activity. Unlike our
+approach where we wish to obtain spikes representation on a
+finer temporal scale, OASIS returns the spike estimates on the
+low-resolution grid. This is typically used to infer the spiking
+rate over a temporal bin equal to the sampling interval. We
+demonstrate that our proposed framework can be integrated with
+OASIS and improve its performance. As we saw in the synthetic
+experiments, the noise level is an important consideration. By
+augmenting Algorithm 2 with OASIS, referred as “B-OASIS",
+the denoising power of l1 minimization can be leveraged.Let
+�xl1 ∈ RM be the estimate obtained on a low-resolution grid
+by solving the l1 minimization problem such as the one
+implemented in OASIS. We can obtain an estimate of the
+denoised calcium signal as �ylo[n] = αD�ylo[n] + �xl1[n], n ≥ 1
+and �ylo[0] = �xl1[0]. We can now utilize the denoised calcium
+signal �ylo[n] generated by OASIS to obtain the estimate ce[n]
+indirectly. Due to the non-linear processing done by OASIS, it
+is difficult to obtain the resulting noise statistics. An important
+advantage of Algorithm 2 is that it does not rely on the
+knowledge of the noise statistics. Hence, we can directly apply
+Algorithm 2 on �ce[n] = �ylo[n]−αD�ylo[n−1] (instead of ce[n])
+to obtain a binary “fused super-resolution spike estimate".
+B-OASIS
+OASIS
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Recall
+F-score
+B-OASIS
+OASIS
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Recall
+F-score
+Fig. 9: Spike detection performance of OASIS and B-OASIS on
+GCaMP6f dataset sampled at (Left) 60 Hz and (Right) 30 Hz. We
+compare the average F-score of data points where the F-score of
+OASIS is < 0.5. Standard deviation is depicted using the error bars.
+20
+20.2
+20.4
+20.6
+20.8
+21
+21.2
+21.4
+21.6
+21.8
+22
+ylo[n]
+20
+20.2
+20.4
+20.6
+20.8
+21
+21.2
+21.4
+21.6
+21.8
+22
+xhi[n]
+20
+20.2
+20.4
+20.6
+20.8
+21
+21.2
+21.4
+21.6
+21.8
+22
+�xhi
+B-OA[n]
+20
+20.2
+20.4
+20.6
+20.8
+21
+21.2
+21.4
+21.6
+21.8
+22
+�xhi
+OA[n]
+Fig. 10: Example of spike reconstruction on GENIE dataset (GCaMP6f
+indicator) using OASIS and B-OASIS (binary augmented) with
+calcium signal sampled at 30Hz.
+E. Results
+We evaluate the algorithms on the publicly available GENIE
+dataset [53], [54] which consists of simultaneous calcium imag-
+ing and in vivo cell-attached recording from the mouse visual
+cortex using genetically encoded GCaMP6f calcium indicator
+GCaMP6f [53], [54]. The calcium images were acquired at a
+frame rate of 60 Hz and the ground truth electrophysiology
+signal was digitized at 10 KHz and synchronized with the
+calcium frames. In addition to using the original data, we also
+synthetically downsample it to emulate the effect of a lower
+frame rate of 30 Hz, and evaluate how the performance changes
+by this downsampling operation.
+In Fig. 10, we extract an interval of ∼ 2 sec (from the neuron
+1 of the GCaMP6f indicator dataset) and qualitatively compare
+the detected spikes with the ground truth. We downsample
+the data by a factor of 2 to emulate frame rate of 30 Hz,
+the low-rate grid becomes coarser. As a result of which, we
+observe an offset between ground truth spikes and estimate
+produced by OASIS. However, with the help of binary priors
+(B-OASIS), we can output spikes that are not restricted to be
+on the coarser scale, and this mitigates the offset observed in
+the raw estimates obtained by OASIS.
+We quantify the improvement in the performance by com-
+paring the F-scores of OASIS and B-OASIS at both sampling
+rates (60 and 30 Hz). Since the output of OASIS is non-
+binary, the estimated spikes are binarized by thresholding.
+To ensure a fair comparison, we select the threshold by a
+80 − 20 cross-validation scheme that maximizes the average
+F-score on a held-out validation set (averaged over 3-random
+selections of the validation set). The tolerance for the F-score
+was set at 100 ms. The dataset consisted of 34 traces of
+length ∼ 234 s. The OASIS algorithm has an automated
+routine to estimate the parameter α, which we utilize for
+our experiments. The amplitude A is estimated using the
+procedure described in Appendix F. We use D = 12 to obtain
+the spike representation for B-OASIS. In order to quantify
+the performance boost achieved by augmentation, we isolate
+the traces where the F−score of OASIS drops below 0.5
+and compare the average F-score and recall for these data
+points. As shown in Fig. 9, at both sampling rates, we see a
+significant improvement in the average F-score of B-OASIS
+over OASIS, attributed to an increase in recall while keeping the
+precision unchanged. Additionally, despite downsampling, the
+spike detection performance is not significantly degraded with
+binary priors, although the detection criteria were unchanged.
+
+13
+VI. CONCLUSION
+We theoretically established the benefits of binary priors in
+super-resolution, and showed that it is possible to achieve
+significant reduction in sample complexity over sparsity-
+based techniques. Using an AR(1) model, we developed
+and analyzed an efficient algorithm that can operate in the
+extreme compression regime ( M ≪ K) by exploiting the
+special structure of measurements and trading memory for
+computational efficiency at run-time. We also demonstrated that
+binary priors can be used to boost the performance of existing
+neural spike deconvolution algorithms. In the future, we will
+develop algorithmic frameworks for incorporating binary priors
+into different neural spike deconvolution pipelines and evaluate
+the performance gain on diverse datasets. The extension of
+this binary framework for higher-order AR filters is another
+exciting future direction.
+APPENDIX
+APPENDIX A: PROOF OF THEOREM 1
+Proof. We show that for any α in 0 < α < 1, except possibly
+for a set consisting of only a finite number of points, (10)
+always has a unique binary solution. Consider all possible
+D−dimensional ternary vectors with their entries chosen from
+{−1, 0, 1}, and denote them as v(i) = [v(i)
+1 , v(i)
+2 , · · · , v(i)
+D ]T ∈
+{−1, 0, 1}D, 0 ≤ i ≤ 3D − 1. We use the convention that
+v(0) = 0. For every i > 0, we define a set Zv(i) determined
+by v(i) as Zv(i) :=
+�
+x ∈ (0, 1)
+�� �D
+k=1 v(i)
+k xD−k = 0
+�
+. Notice
+that pi(x) := �D
+k=1 v(i)
+k xD−k denotes a polynomial (in x) of
+degree at most D−1, whose coefficients are given by the ternary
+vector v(i). The set Zv(i) denotes the set of zeros of pi(x) that
+are contained in (0, 1). Since the degree of pi(x) is at most
+D−1, Zv(i) is a finite set with cardinality at most D−1.
+Now suppose that the binary solution of (10) is non-unique,
+i.e., there exist u, w ∈ {0, A}L, u ̸= w, such that
+HD(α)u = HD(α)w ⇒ HD(α)u − HD(α)w = 0
+(34)
+By partitioning u, w into blocks u(n), w(n) in the same way
+as in (6), we can re-write (34) as u(0) = w(0) and
+D
+�
+i=1
+1
+A([u(j)]i − [w(j)]i)αD−i = 0,
+1 ≤ j ≤ M − 1
+(35)
+Since u ̸= w, they differ at least at one block, i.e., there exists
+some j0, 1 ≤ j0 ≤ M − 1 such that u(j0) ̸= w(j0). Define
+b := 1
+A(u(j0) − w(j0)). Then, b is a non-zero ternary vector,
+i.e., b ∈ {−1, 0, 1}D. Now from (35), we have
+D
+�
+i=1
+[b]iαD−i = 0,
+(36)
+which implies that α ∈ Zb. Since b can be any one of the 3D−1
+ternary vectors {v(i)}3D−1
+i=1 , (36) holds if and only if α ∈ S :=
+�3D−1
+i=1 Zv(i), i.e., α is a root of at least one of the polynomials
+pi(x) defined by the vectors v(i) as their coefficients. For each
+v(i), since the cardinality of Zv(i) is at most D−1, S is a finite
+set (of cardinality at most (D − 1)(3D − 1)), and therefore its
+Lebesgue measure is 0. This implies that (10) has a non-unique
+binary solution only if α belongs to the measure zero set S,
+thereby proving the theorem.
+APPENDIX B: PROOF OF LEMMA 2 AND LEMMA 3
+Proof. (i) Let sn denote the sparsity (number of non-zero
+elements) of the nth block xhi(n) of xhi. Then, the total
+sparsity is ∥xhi∥0 = �M−1
+n=0 sn. We will construct a vec-
+tor v ∈ RL, v ̸= xhi that satisfies c = HD(α)v and
+∥xhi∥0 ≥ ∥v∥0. Following (6), consider the partition of v
+v = [v(0), v(1)⊤, · · · , v(M−1)⊤]⊤. Firstly, we assign v(0) =
+c[0] = xhi(0). We construct v(n) as follows. For each n ≥ 1,
+there are three cases:
+Case I: sn = 0. In this case, xhi(n) = 0 and hence c[n] = 0.
+Therefore, we assign v(n) = xhi(n) = 0.
+Case II: sn = 1. First suppose that [xhi(n)]D = 0. We
+construct v(n) as follows:
+[v(n)]k =
+�
+c[n],
+if k = D
+0,
+else
+.
+(37)
+Next suppose that [xhi(n)]D ̸= 0. Since sn = 1, this implies
+that [xhi(n)]k = 0, k = 1, · · · , D−1. In this case, we construct
+v(n) as follows:
+[v(n)]k =
+�
+c[n]/α,
+if k = D − 1
+0,
+else
+.
+(38)
+Notice that both (37) and (38) ensure that v(n) ̸= xhi(n) and
+c[n] = hT
+αv(n). Moreover, ∥v(n)∥0 = sn.
+Case III: sn ≥ 2. In this case, we follow the same
+construction as (37). As before v(n) satisfies c[n] = h⊤
+α v(n).
+Since ∥xhi(n)∥0 ≥ 2 and ∥v(n)∥0 = 1, we automatically have
+v(n) ̸= xhi(n), and ∥v(n)∥0 < sn. Therefore, combining the
+three cases, we can construct the desired vector v that satisfies
+v ̸= xhi, c = HD(α)v, and ∥v∥0 ≤ �M−1
+n=0 sn = ∥xhi(n)∥0.
+Therefore, the solution x⋆ to (P0) satisfies ∥x⋆∥0 ≤ ∥v∥0 ≤
+∥xhi(n)∥0.
+(ii) In this case, we construct v(n0) according to Case III.
+Since ∥v(n0)∥0 < sn0, and ∥v(n)∥0 ≤ sn, n ̸= n0, we have
+∥v∥0 < ∥xhi∥0, implying ∥x⋆∥0 ≤ ∥v∥0 < ∥xhi∥0.
+A. Proof of Lemma 3
+Proof. We will construct a vector v ∈ RL whose support is of
+the form (16), that is feasible for (P1-B), and we will prove
+that it has the smallest l1 norm. Using the block structure given
+by (6), we choose v(0) = c[0]. For each n ≥ 1, we construct
+v(n) based on the following two cases:
+Case I: c[n] ≥ A. Let kn be the largest integer such that the
+following holds: µ[n] := A(1 + α + · · · + αkn−1) ≤ c[n],
+where 1 ≤ kn ≤ D. Note that kn = 1 always produces a valid
+lower bound. However, we are interested in the largest lower
+bound on c[n] of the above form. We choose
+[v(n)]k =
+�
+�
+�
+�
+�
+A,
+if D − kn + 1 ≤ k ≤ D
+(c[n] − µ[n])/αkn, if k = D − kn
+0, else
+It is easy to verify that h⊤
+α v(n) = c[n]. From the definition
+of kn, it follows that µ[n] ≤ c[n] < µ[n] + Aαkn and hence,
+0 ≤ (c[n] − µ[n])/αkn < A, which ensures that v obeys the
+box-constraints in (P1-B). Now, let vf ∈ RL be any feasible
+point of (P1-B) which must be of the form v(0)
+f
+= c[0], v(n)
+f
+=
+v(n) + r(n), where r(n) ∈ N(h⊤
+α ) is a vector in the null-space
+
+14
+of h⊤
+α . It can be verified that the following vectors {wt}D−1
+t=1
+form a basis for N(h⊤
+α ):
+[wt]k =
+�
+�
+�
+�
+�
+1,
+k = t
+−α,
+k = t + 1
+0,
+else
+,
+Therefore, ∃ {β(n)
+t
+}D−1
+t=1 such that r(n) = �D−1
+t=1 β(n)
+t
+wt. We
+further consider two scenarios: (i) 1 ≤ kn ≤ D − 2. In this
+case [v(n)]1 = 0, and for k = 1, 2, · · · D, [v(n)
+f ]k satisfies 2
+[v(n)
+f ]k =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+β(n)
+k , if k = 1
+β(n)
+k
+− αβ(n)
+k−1, if 2 ≤ k ≤ D − kn − 1
+[v(n)]k + β(n)
+k
+− αβ(n)
+k−1, if k = D − kn
+A + β(n)
+k
+− αβ(n)
+k−1, if D − kn + 1 ≤ k ≤ D − 1
+A − αβ(n)
+k−1, if k = D
+To ensure v(n)
+f
+is a feasible point for (P1-B), the following must
+hold: 0 ≤ β(n)
+D−1 ≤ A/α and 0 ≤ β(n)
+1
+≤ A. For 2 ≤ k ≤ D −
+kn−1, the constraint [v(n)
+f ]k ≥ 0 implies β(n)
+k
+≥ αβ(n)
+k−1. Since
+β(n)
+1
+≥ 0, it follows that β(n)
+k
+≥ 0 for all 2 ≤ k ≤ D − kn − 1.
+For D−kn+1 ≤ k ≤ D−1, the constraint [v(n)
+f ]k ≤ A implies
+β(n)
+k−1 ≥ β(n)
+k /α. Since β(n)
+D−1 ≥ 0, it follows that β(n)
+k
+≥ 0 for
+all D − kn ≤ k ≤ D − 1. (ii) kn ∈ {D − 1, D}. In this case,
+for k = 1, 2, · · · , D, [v(n)
+f ]k satisfies
+[v(n)
+f ]k =
+�
+�
+�
+�
+�
+[v(n)]1 + β(n)
+1
+, if k = 1
+A + β(n)
+k
+− αβ(n)
+k−1, if 2 ≤ k ≤ D − 1
+A − αβ(n)
+k−1, if k = D
+For 2 ≤ k ≤ D − 1, the box-constraint [v(n)
+f ]k ≤ A implies
+β(n)
+k−1 ≥ β(n)
+k /α. Since β(n)
+D−1 ≥ 0, it follows that β(n)
+k
+≥ 0 for
+all 1 ≤ k ≤ D − 1. Summarizing, we have established that
+β(n)
+i
+≥ 0, ∀i.
+Case II: c[n] < A. In this case, v(n) is constructed following
+(37), and hence v(n)
+f
+has the following structure:
+[v(n)
+f ]k =
+�
+�
+�
+�
+�
+β(n)
+k , if k = 1
+−αβ(n)
+k−1 + β(n)
+k , if 2 ≤ k ≤ D − 1
+c[n] − αβ(n)
+k−1, if k = D
+To ensure v(n)
+f
+is a feasible point, it must hold that β(n)
+1
+≥
+0, β(n)
+k
+≥ αβ(n)
+k−1 ≥ 0 for 2 ≤ k ≤ D − 1. Hence, in both
+Cases I and II, we established that β(n)
+k
+≥ 0. For each case,
+since v(n)
+f
+is a non-negative vector ∀n, it can be verified that
+∥vf∥1 =
+M−1
+�
+n=0
+∥v(n)
+f ∥1 = v(0)
+f
++
+M−1
+�
+n=1
+D
+�
+k=1
+[v(n)
+f ]k
+= c[0] +
+M−1
+�
+n=1
+D
+�
+k=1
+[v(n)]k
+�
+��
+�
+∥v∥1
++
+M−1
+�
+n=1
+D−1
+�
+k=1
+(1 − α)β(n)
+k
+2In the definition of v(n)
+f
+, an assignment will be ignored if the specified
+interval for k is empty.
+We used the fact that �D
+k=1
+�D−1
+t=1 β(n)
+t
+[wt]k = �D−1
+t=1 (1 −
+α)β(n)
+t
+. If vf ̸= v, we must have β(n)
+k
+̸= 0 for some k and
+n > 0. This implies that ∥vf∥1 > ∥v∥1. It is easy to see
+that the support of the constructed vector is of the form (16).
+Moreover, based on the above argument, v is the only vector
+that has the minimum l1 norm among all possible feasible
+points of (P1-B).
+APPENDIX C: PROOF OF LEMMA 7
+Proof. For any 0 < α ≤ 0.5, we begin by showing that for an
+integer p ≥ 1 the following inequality holds:
+p
+�
+k=1
+αD−k = αD−p−1
+� 1 − αp
+1/α − 1
+�
+< αD−p−1
+(39)
+since 1/α − 1 ≥ 1 and 1 − αp < 1 in the regime 0 < α ≤ 0.5.
+Let S1 = {0, αD−1, αD−2, αD−1 + αD−2}. Notice that the
+elements of S1 are sorted in ascending order for any α and D.
+Now, we recursively define the sets Si as follows:
+Si := {Si−1, Si−1 + αD−1−i}, 2 ≤ i ≤ D − 1
+(40)
+Our hypothesis is that for every 2 ≤ i ≤ D − 1 α ∈ (0, 0.5]
+and D, the set Si as defined in (40), is automatically sorted in
+ascending order. We prove this via induction. For i = 2, the
+sets S1 and S1 + αD−3 are individually sorted. Moreover from
+(39), we can show that: maxa∈S1 a = αD−1+αD−2 < αD−3 =
+minb∈S1+αD−3 b. This shows that S2 is ordered, establishing the
+the base case of our induction. Now, assume Si is ordered for
+some 2 ≤ i ≤ D−2. We need to show that Si+1 is also ordered.
+As a result of the induction hypothesis, both Si and Si+αD−2−i
+are ordered. Using the ordering of Si, we have: maxa∈Si a =
+�i+1
+j=1 αD−j, minb∈Si+αD−2−i b = αD−(i+1)−1. From (39), we
+can conclude that maxa∈Si a < minb∈Si+αD−2−i b and hence,
+Si+1 is also ordered. This completes the induction proof. Also,
+note that for α ∈ (0, 0.5], we have Θsort
+α
+= SD−1.
+Let ∆min(Si) be the min. distance between the elements of the
+set Si. It is easy to see that ∆min(Si) = ∆min(Si + αD−2−i).
+Since Si is sorted for α ∈ (0, 0.5], ∆min(Si) is given by:
+∆min(Si) = min(∆min(Si−1),
+min
+x∈Si−1+αD−1−i x − max
+y∈Si−1 y)
+= min{∆min(Si−1), αD−i−1 −
+i
+�
+j=1
+αD−j}.
+(41)
+Now, we use induction to establish the following conjecture:
+���min(Si) = αD−1, 1 ≤ i ≤ D − 1
+(42)
+For the base case i = 1, ∆min(S1) = min(αD−1, αD−2 −
+αD−1) = αD−1, where the last equality holds since α ∈
+(0, 0.5] ⇒ αD−1(1/α − 1) ≥ αD−1. Suppose (42) holds for
+some 1 ≤ i ≤ D − 2. From the definition of ∆min(Si+1) and
+the induction hypothesis that ∆min(Si) = αD−1, it follows that
+∆min(Si+1) = min{αD−1, αD−(i+1)−1 −�i+1
+j=1 αD−j}. Again,
+from the definition of ∆min(Si) in (41), and the induction
+hypothesis we also have αD−i−1 −�i
+j=1 αD−j ≥ ∆min(Si) =
+αD−1. Using this and the fact that α ≤ 0.5, we can show:
+αD−i−2 −αD−i−1 − �i
+j=1 αD−j ≥ αD−i−2 − 2αD−i−1 + αD−1
+≥ αD−1 + αD−i−1(1/α − 2) ≥ αD−1
+
+15
+Therefore ∆min(Si+1)=min{αD−1, αD−i−2−�i+1
+j=1 αD−j} =
+αD−1.
+Thus,
+we
+can
+conclude
+that
+∆min(α, D)
+=
+∆min(SD−1)=αD−1.
+APPENDIX D: PROOF OF THEOREM 3
+Proof. The probability of incorrectly identifying xhi(n) from a
+single measurement ce[n] is given by
+pe := P(�xhi
+(n) ̸= xhi
+(n))
+=
+lD
+�
+k=0
+P(�xhi
+(n) ̸= xhi
+(n)|xhi
+(n) = �vk)P(xhi
+(n) = �vk)
+Given a binary vector z ∈ {0, 1}D, define the function ψ(z) :=
+�D
+k=1 zk, which denotes the count of ones in z. Since the
+noisy observations are given by ce[n] = c[n] + e[n], where
+e[n] = w[n] − αDw[n − 1], it follows from assumption (A2)
+that e[n] ∼ N(0, σ2
+1) where σ2
+1 = (1 + α2D)σ2. From (27),
+we obtain P(�xhi(n) ̸= xhi(n)|xhi(n) = �v0) = P(e[n] ∈ E0) =
+Q(αD−1/(2σ1)). Similarly, P(�xhi(n) ̸= xhi(n)|xhi(n) = �vlD) =
+P(e[n] ∈ ElD) = Q((�θlD − �θlD−1)/(2σ1)) = Q(αD−1/(2σ1)).
+The last equality follows from the fact that �θlD − �θlD−1 = αD−1.
+Finally, when conditioned on xhi(n) = �vk for 0 < k < lD,
+from (26), we obtain P(�x(n) ̸= xhi(n)|xhi(n) = �vk) = P(e[n] ∈
+Ek) = Q(
+�θk−�θk−1
+2σ1
+) + Q(
+�θk+1−�θk
+2σ1
+). Due to Assumption (A1)
+on xhi, we have P(xhi(n) = �vk) = pψ(�vk)(1 − p)D−ψ(�vk).
+Therefore, pe is given by
+pe = Q(αD−1/(2σ1))(1 − p)D + Q(αD−1/(2σ1))pD+
+lD−1
+�
+k=1
+�
+Q(
+�θk − �θk−1
+2σ1
+) + Q(
+�θk+1 − �θk
+2σ1
+)
+�
+pψ(vk)(1 − p)D−ψ(vk)
+(43)
+The spike train xhi is incorrectly decoded if at least one of the
+blocks are decoded incorrectly, hence, the total probability of
+error is given by:
+P(
+M−1
+�
+n=0
+�x(n) ̸= xhi
+(n)) ≤
+M−1
+�
+n=0
+P(�x(n) ̸= xhi
+(n)) = Mpe
+(a)
+≤ 2MQ(∆θmin(α, D)/(2σ1))
+D
+�
+j=0
+pj(1 − p)D−j
+�D
+j
+�
+(b)
+≤ 2M exp(−∆θ2
+min(α, D)/(4σ2
+1))
+(44)
+where the first inequality follows from union bound and second
+equality is a consequence of (43). The inequality (a) follows
+from the monotonically decreasing property of Q(.) function
+and the sum can be re-written by grouping all terms with the
+same count, i.e., ψ(vk) = j. The inequality (b) follows from
+the inequality Q(x) ≤ exp(−x2/2) for x > 0. If the SNR
+condition (28) holds then from (44) the total probability of
+error is bounded by δ.
+APPENDIX E: PROOF OF THEOREM 4
+Proof. We first begin by showing that α ∈ FD implies that (31)
+holds and hence the mapping of spikes with the same counts are
+clustered. Notice that for k = 0, θk
+max = θk
+min = 0. For k ≥ 1,
+it is easy to verify that θk
+max and θk
+min are attained by the spiking
+patterns 00...1111 (with k consecutive spikes at the indices
+D − k + 1 to D) and 111...000 (with consecutive spikes at the
+indices 1 to k), which allows us to simplify (31) as αD−1 > 0
+for k = 0 and �k+1
+i=1 αD−i > �k−1
+j=0 αj, k = 1, · · · , D − 1.
+The values of α that satisfy each of these relations can be
+described by the following sets:
+G0 = {α ∈ (0, 1)|αD−1 > 0}, Gk = {α ∈ (0, 1)|rk(α) < 0},
+where rk(α) = αD − αD−k−1 − αk + 1 for 1 ≤ k ≤ D − 1. It
+is easy to see that FD = Gk0. Observe that the relations are
+symmetric, i.e., Gk = GD−k−1. Furthermore, for 1 ≤ k ≤ D/2,
+we show that Gk ⊆ Gk−1 as follows. Trivially, G1 ⊂ G0.
+For 2 ≤ k ≤ D/2, observe that
+rk(α) − rk−1(α) =
+αD−k(1 − 1/α) − αk(1 − 1/α) = (1/α − 1)(αk − αD−k) ≥ 0.
+Therefore, α ∈ Gk ⇒ α ∈ Gk−1, k = 1, 2 · · · , k0. Moreover,
+since Gk = GD−k−1, it follows that FD = Gk0 = ∩D−1
+k=0Gk.
+Hence, α ∈ FD ⇒ α ∈ Gi for all 0 ≤ i ≤ D − 1, which
+implies that (31) holds. If the noise perturbation satisfies
+|w[n]| < ∆c
+min(α, D)/4, it implies |e[n]| < ∆c
+min(α, D)/2.
+For any block xhi(n) ∈ CD
+k , θk
+min ≤ h⊤
+α xhi(n) ≤ θk
+max. If
+|e[n]| < ∆c
+min(α, D)/2, we have
+h⊤
+α xhi
+(n) + e[n] < θk
+max + ∆c
+min(α, D)
+2
+< θk
+max + θk+1
+min − θk
+max
+2
+h⊤
+α xhi
+(n) + e[n] > θk
+min − ∆c
+min(α, D)
+2
+> θk
+min − θk
+min − θk−1
+max
+2
+This shows that
+whenever α ∈ FD, the condition |e[n]| <
+∆c
+min(α, D)/2 is sufficient for (33) to hold ∀ γ[n] and hence
+the spike count can be exactly recovered.
+APPENDIX F: AMPLITUDE ESTIMATION
+We suggest a procedure to estimate the binary amplitude A, if
+it is unknown. We first evaluate the signal c[n] from different
+time instants n = 1, 2, · · · , M − 1. For some 1 ≤ n0 ≤
+M − 1, we estimate a set A = {Ak} of candidate amplitudes:
+Ak := c[n0]/hT
+αvk where vk ∈ Sall. Only a certain amplitudes
+can generate c[n0] from a valid binary spiking pattern vk ∈ Sall.
+Our goal is to prune A by sequentially eliminating certain
+candidate amplitudes from the set based on a consistency
+test across the remaining measurements c[n]. At the tth stage
+(t = 2, 3, · · · ), for every remaining candidate amplitude Ak ∈
+A, we perform the following consistency test with c[n], to
+identify if a candidate amplitude can potentially generate the
+corresponding measurement c[n]. Suppose there exists a spiking
+pattern vl ∈ Sall such that
+c[n] = AkhT
+αvl
+(45)
+then Ak remains a valid candidate. If we cannot find a
+corresponding vl ∈ Sall for an amplitude Ak, we remove
+it, A = A \ Ak. In presence of noise, (45) can be modified
+to allow a tolerance γ as we may not find an exact match.
+The tolerance γ is chosen to be 0.5 in the experiments on
+the GENIE dataset. This procedure prunes out possible values
+for the amplitude by leveraging the shared amplitude across
+multiple measurements c[n].
+ACKNOWLEDGEMENT
+The authors would like to thank Prof. Nikita Sidorov,
+Department of Mathematics at the University of Manchester,
+for helpful discussions regarding computational challenges
+in finding finite β-expansion in the range β ∈ (1, 2). This
+work was supported by Grants ONR N00014-19-1-2256, DE-
+SC0022165, NSF 2124929, and NSF CAREER ECCS 1700506.
+
+16
+REFERENCES
+[1] A. Small and S. Stahlheber, “Fluorophore localization algorithms for
+super-resolution microscopy,” Nature methods, vol. 11, no. 3, pp. 267–
+279, 2014.
+[2] R. Brette and A. Destexhe, Handbook of neural activity measurement.
+Cambridge University Press, 2012.
+[3] J. T. Vogelstein, B. O. Watson, A. M. Packer, R. Yuste, B. Jedynak, and
+L. Paninski, “Spike inference from calcium imaging using sequential
+monte carlo methods,” Biophysical journal, vol. 97, no. 2, pp. 636–655,
+2009.
+[4] T. Deneux, A. Kaszas, G. Szalay, G. Katona, T. Lakner, A. Grinvald,
+B. Rózsa, and I. Vanzetta, “Accurate spike estimation from noisy
+calcium signals for ultrafast three-dimensional imaging of large neuronal
+populations in vivo,” Nature communications, vol. 7, p. 12190, 2016.
+[5] S. Yang and L. Hanzo, “Fifty years of mimo detection: The road to
+large-scale mimos,” IEEE Communications Surveys & Tutorials, vol. 17,
+no. 4, pp. 1941–1988, 2015.
+[6] D. L. Donoho, “Superresolution via sparsity constraints,” SIAM journal
+on mathematical analysis, vol. 23, no. 5, pp. 1309–1331, 1992.
+[7] E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory
+of super-resolution,” Communications on pure and applied Mathematics,
+vol. 67, no. 6, pp. 906–956, 2014.
+[8] W. Li, W. Liao, and A. Fannjiang, “Super-resolution limit of the esprit
+algorithm,” IEEE Transactions on Information Theory, vol. 66, no. 7,
+pp. 4593–4608, 2020.
+[9] D. Batenkov, G. Goldman, and Y. Yomdin, “Super-resolution of near-
+colliding point sources,” Information and Inference: A Journal of the
+IMA, vol. 10, no. 2, pp. 515–572, 2021.
+[10] G. Schiebinger, E. Robeva, and B. Recht, “Superresolution without
+separation,” Information and Inference: A Journal of the IMA, vol. 7,
+no. 1, pp. 1–30, 2017.
+[11] T. Bendory, “Robust recovery of positive stream of pulses,” IEEE
+Transactions on Signal Processing, vol. 65, no. 8, pp. 2114–2122, 2017.
+[12] W. Liao and A. Fannjiang, “Music for single-snapshot spectral estimation:
+Stability and super-resolution,” Applied and Computational Harmonic
+Analysis, vol. 40, no. 1, pp. 33–67, 2016.
+[13] H. Qiao and P. Pal, “Guaranteed localization of more sources than
+sensors with finite snapshots in multiple measurement vector models
+using difference co-arrays,” IEEE Transactions on Signal Processing,
+vol. 67, no. 22, pp. 5715–5729, 2019.
+[14] ——, “A non-convex approach to non-negative super-resolution: Theory
+and algorithm,” in ICASSP 2019-2019 IEEE International Conference
+on Acoustics, Speech and Signal Processing (ICASSP).
+IEEE, 2019,
+pp. 4220–4224.
+[15] H. Qiao, S. Shahsavari, and P. Pal, “Super-resolution with noisy
+measurements: Reconciling upper and lower bounds,” in ICASSP 2020-
+2020 IEEE International Conference on Acoustics, Speech and Signal
+Processing (ICASSP).
+IEEE, 2020, pp. 9304–9308.
+[16] S. Shahsavari, J. Millhiser, and P. Pal, “Fundamental trade-offs in noisy
+super-resolution with synthetic apertures,” in ICASSP 2021-2021 IEEE
+International Conference on Acoustics, Speech and Signal Processing
+(ICASSP).
+IEEE, 2021, pp. 4620–4624.
+[17] H. Qiao and P. Pal, “On the modulus of continuity for noisy positive
+super-resolution,” in 2018 IEEE International Conference on Acoustics,
+Speech and Signal Processing (ICASSP).
+IEEE, 2018, pp. 3454–3458.
+[18] Y. Chi and M. F. Da Costa, “Harnessing sparsity over the continuum:
+Atomic norm minimization for superresolution,” IEEE Signal Processing
+Magazine, vol. 37, no. 2, pp. 39–57, 2020.
+[19] B. N. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with
+applications to line spectral estimation,” IEEE Transactions on Signal
+Processing, vol. 61, no. 23, pp. 5987–5999, 2013.
+[20] B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen,
+“High-speed in vivo calcium imaging reveals neuronal network activity
+with near-millisecond precision,” Nature methods, vol. 7, no. 5, p. 399,
+2010.
+[21] E. A. Pnevmatikakis, D. Soudry, Y. Gao, T. A. Machado, J. Merel, D. Pfau,
+T. Reardon, Y. Mu, C. Lacefield, W. Yang et al., “Simultaneous denoising,
+deconvolution, and demixing of calcium imaging data,” Neuron, vol. 89,
+no. 2, pp. 285–299, 2016.
+[22] R. Schmidt, “Multiple emitter location and signal parameter estimation,”
+IEEE transactions on antennas and propagation, vol. 34, no. 3, pp.
+276–280, 1986.
+[23] R. Roy and T. Kailath, “Esprit-estimation of signal parameters via
+rotational invariance techniques,” IEEE Transactions on acoustics, speech,
+and signal processing, vol. 37, no. 7, pp. 984–995, 1989.
+[24] Y. Hua and T. K. Sarkar, “Matrix pencil method for estimating
+parameters of exponentially damped/undamped sinusoids in noise,” IEEE
+Transactions on Acoustics, Speech, and Signal Processing, vol. 38, no. 5,
+pp. 814–824, 1990.
+[25] B. Bernstein and C. Fernandez-Granda, “Deconvolution of point sources:
+a sampling theorem and robustness guarantees,” Communications on
+Pure and Applied Mathematics, vol. 72, no. 6, pp. 1152–1230, 2019.
+[26] A. Koulouri, P. Heins, and M. Burger, “Adaptive superresolution in
+deconvolution of sparse peaks,” IEEE Transactions on Signal Processing,
+vol. 69, pp. 165–178, 2020.
+[27] V. I. Morgenshtern and E. J. Candes, “Super-resolution of positive sources:
+The discrete setup,” SIAM Journal on Imaging Sciences, vol. 9, no. 1,
+pp. 412–444, 2016.
+[28] D. Batenkov, A. Bhandari, and T. Blu, “Rethinking super-resolution: the
+bandwidth selection problem,” in ICASSP 2019-2019 IEEE International
+Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE,
+2019, pp. 5087–5091.
+[29] M. F. Da Costa and W. Dai, “A tight converse to the spectral resolution
+limit via convex programming,” in 2018 IEEE International Symposium
+on Information Theory (ISIT).
+IEEE, 2018, pp. 901–905.
+[30] T. Blu, P.-L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, “Sparse
+sampling of signal innovations,” IEEE Signal Processing Magazine,
+vol. 25, no. 2, pp. 31–40, 2008.
+[31] J. A. Urigüen, T. Blu, and P. L. Dragotti, “Fri sampling with arbitrary
+kernels,” IEEE Transactions on Signal Processing, vol. 61, no. 21, pp.
+5310–5323, 2013.
+[32] J. Onativia, S. R. Schultz, and P. L. Dragotti, “A finite rate of innovation
+algorithm for fast and accurate spike detection from two-photon calcium
+imaging,” Journal of neural engineering, vol. 10, no. 4, p. 046017, 2013.
+[33] R. Tur, Y. C. Eldar, and Z. Friedman, “Innovation rate sampling of pulse
+streams with application to ultrasound imaging,” IEEE Transactions on
+Signal Processing, vol. 59, no. 4, pp. 1827–1842, 2011.
+[34] S. Rudresh and C. S. Seelamantula, “Finite-rate-of-innovation-sampling-
+based super-resolution radar imaging,” IEEE Transactions on Signal
+Processing, vol. 65, no. 19, pp. 5021–5033, 2017.
+[35] M. Stojnic, “Recovery thresholds for l1 optimization in binary com-
+pressed sensing,” in 2010 IEEE International Symposium on Information
+Theory.
+IEEE, 2010, pp. 1593–1597.
+[36] S. Keiper, G. Kutyniok, D. G. Lee, and G. E. Pfander, “Compressed
+sensing for finite-valued signals,” Linear Algebra and its Applications,
+vol. 532, pp. 570–613, 2017.
+[37] A. Flinth and S. Keiper, “Recovery of binary sparse signals with biased
+measurement matrices,” IEEE Transactions on Information Theory,
+vol. 65, no. 12, pp. 8084–8094, 2019.
+[38] S. M. Fosson and M. Abuabiah, “Recovery of binary sparse signals from
+compressed linear measurements via polynomial optimization,” IEEE
+Signal Processing Letters, vol. 26, no. 7, pp. 1070–1074, 2019.
+[39] Z. Tian, G. Leus, and V. Lottici, “Detection of sparse signals under
+finite-alphabet constraints,” in 2009 IEEE International Conference on
+Acoustics, Speech and Signal Processing.
+IEEE, 2009, pp. 2349–2352.
+[40] P. Sarangi and P. Pal, “No relaxation: Guaranteed recovery of finite-valued
+signals from undersampled measurements,” in ICASSP 2021-2021 IEEE
+International Conference on Acoustics, Speech and Signal Processing
+(ICASSP).
+IEEE, 2021, pp. 5440–5444.
+[41] ——, “Measurement matrix design for sample-efficient binary com-
+pressed sensing,” IEEE Signal Processing Letters, 2022.
+[42] S. Razavikia, A. Amini, and S. Daei, “Reconstruction of binary shapes
+from blurred images via hankel-structured low-rank matrix recovery,”
+IEEE Transactions on Image Processing, vol. 29, pp. 2452–2462, 2019.
+[43] J. Friedrich, P. Zhou, and L. Paninski, “Fast online deconvolution of
+calcium imaging data,” PLoS computational biology, vol. 13, no. 3, p.
+e1005423, 2017.
+[44] S. W. Jewell, T. D. Hocking, P. Fearnhead, and D. M. Witten, “Fast
+nonconvex deconvolution of calcium imaging data,” Biostatistics, vol. 21,
+no. 4, pp. 709–726, 2020.
+[45] P. Sarangi, M. C. Hücümeno˘glu, and P. Pal, “Effect of undersampling on
+non-negative blind deconvolution with autoregressive filters,” in ICASSP
+2020-2020 IEEE International Conference on Acoustics, Speech and
+Signal Processing (ICASSP).
+IEEE, 2020, pp. 5725–5729.
+[46] A. Rupasinghe and B. Babadi, “Robust inference of neuronal correlations
+from blurred and noisy spiking observations,” in 2020 54th Annual
+Conference on Information Sciences and Systems (CISS).
+IEEE, 2020,
+pp. 1–5.
+[47] N. Sidorov, “Almost every number has a continuum of β-expansions,”
+The American Mathematical Monthly, vol. 110, no. 9, pp. 838–842, 2003.
+
+17
+[48] P. Glendinning and N. Sidorov, “Unique representations of real numbers
+in non-integer bases,” Mathematical Research Letters, vol. 8, no. 4, pp.
+535–543, 2001.
+[49] A. Rényi, “Representations for real numbers and their ergodic properties,”
+Acta Mathematica Academiae Scientiarum Hungarica, vol. 8, no. 3-4,
+pp. 477–493, 1957.
+[50] C. Frougny and B. Solomyak, “Finite beta-expansions,” Ergodic Theory
+Dynam. Systems, vol. 12, no. 4, pp. 713–723, 1992.
+[51] V. Komornik and P. Loreti, “Expansions in noninteger bases.” Integers,
+vol. 11, no. A9, p. 30, 2011.
+[52] D. H. Hubel and T. N. Wiesel, “Receptive fields of single neurones in
+the cat’s striate cortex,” The Journal of physiology, vol. 148, no. 3, p.
+574, 1959.
+[53] T.-W. Chen, T. J. Wardill, Y. Sun, S. R. Pulver, S. L. Renninger,
+A. Baohan, E. R. Schreiter, R. A. Kerr, M. B. Orger, V. Jayaraman
+et al., “Ultrasensitive fluorescent proteins for imaging neuronal activity,”
+Nature, vol. 499, no. 7458, pp. 295–300, 2013.
+[54] H. K. S. c. GENIE Project, Janelia Farm Campus, “Simultaneous imaging
+and loose-seal cell-attached electrical recordings from neurons expressing
+a variety of genetically encoded calcium indicators,” CRCNS. org, 2015.
+

DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:fdc31ec3a0bada040aa1098e317565147bf8b9a8e83938767dbc9708caaf49d6
+size 408975

DdE1T4oBgHgl3EQfEAP6/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b358752aa3314e0496e862d49c925310ec05f0fdf0734c1adbb09bc1ddee3e20
+size 1245229

DdE1T4oBgHgl3EQfEAP6/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2383999abd4787a467c171d66463323162ac53301e34e9ebfa52e9fbe0e1501a
+size 56968

DdE3T4oBgHgl3EQfUwqw/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:47a32bae090c3a2c3c6d74bb096cb67fbe01bd7f7ba22faee055e4fcc1c1d6c1
+size 1966125

DdE3T4oBgHgl3EQfUwqw/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:94e6a0bae7e287da2603bc75f9f564d826e7f5425dc37d6d83005773596fc57c
+size 76558

DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:20c52a50e8a492d8a8bd35df915ec3435b10a8d1ace85d3703a54deb4fda22fa
+size 514705

DdFJT4oBgHgl3EQfBSxP/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6d5013ea6328bbec096d654e957e709e63ef415015ac6fe0a6688e751a808bc6
+size 279975

EtFJT4oBgHgl3EQfCSzh/content/tmp_files/2301.11429v1.pdf.txt

@@ -0,0 +1,982 @@
+Just Another Day on Twitter: A Complete 24 Hours of Twitter Data
+J¨urgen Pfeffer1, Daniel Matter1, Kokil Jaidka2, Onur Varol3, Afra Mashhadi4, Jana Lasser5, 15,
+Dennis Assenmacher6, Siqi Wu7, Diyi Yang8, Cornelia Brantner9, Daniel M. Romero7, Jahna
+Otterbacher10, Carsten Schwemmer11, Kenneth Joseph12, David Garcia13, Fred Morstatter14
+1School of Social Sciences and Technology, Technical University of Munich, 2Centre for Trusted Internet and Community,
+National University of Singapore, 3Sabanci University, 4University of Washington (Bothell), 5Graz University of Technology,
+6GESIS - Leibniz Institute for the Social Sciences, 7University of Michigan, 8Stanford University, 9Karlstad University,
+10Open University of Cyprus & CYENS CoE, 11Ludwig Maximilian University of Munich, 12University at Buffalo,
+13University of Konstanz, 14Information Sciences Institute, University of Southern California 15Complexity Science Hub
+Vienna
+Abstract
+At the end of October 2022, Elon Musk concluded his acqui-
+sition of Twitter. In the weeks and months before that, sev-
+eral questions were publicly discussed that were not only of
+interest to the platform’s future buyers, but also of high rele-
+vance to the Computational Social Science research commu-
+nity. For example, how many active users does the platform
+have? What percentage of accounts on the site are bots? And,
+what are the dominating topics and sub-topical spheres on the
+platform? In a globally coordinated effort of 80 scholars to
+shed light on these questions, and to offer a dataset that will
+equip other researchers to do the same, we have collected all
+375 million tweets published within a 24-hour time period
+starting on September 21, 2022. To the best of our knowl-
+edge, this is the first complete 24-hour Twitter dataset that
+is available for the research community. With it, the present
+work aims to accomplish two goals. First, we seek to an-
+swer the aforementioned questions and provide descriptive
+metrics about Twitter that can serve as references for other
+researchers. Second, we create a baseline dataset for future
+research that can be used to study the potential impact of the
+platform’s ownership change.
+Introduction
+On March 21, 2006, Twitter’s first CEO Jack Dorsey sent
+the first message on the platform. In the subsequent 16 years,
+close to 3 trillion tweets have been sent.1 Roughly two-thirds
+of these have been either removed from the platform be-
+cause the senders deleted them or because the accounts (and
+all their tweets) have been banned from the platform, have
+been made private by the users, or are otherwise inaccessi-
+ble via the historic search with the v2 API endpoints. We
+estimate that about 900 billion public tweets were on the
+platform when Elon Musk acquired Twitter in October 2022
+for $44B., i.e., he paid about 5 cents per tweet.
+Besides its possible economic value, Twitter has been
+instrumental in studying human behavior with social me-
+dia data and the entire field of Computational Social Sci-
+ence (CSS) has heavily relied on data from Twitter. At the
+Copyright © 2022, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+1While we do not have an official source for this number, it rep-
+resents an educated guess from a collaboration of dozens of schol-
+ars of Twitter.
+AAAI International Conference on Web and Social Media
+(ICWSM), in the past two years alone (2021-2022), over
+30 scientific papers analyzed a subset of Twitter for a wide
+range of topics ranging from public and mental health anal-
+yses to politics and partisanship. Indeed, since its emer-
+gence, Twitter has been described as a digital socioscope
+(i.e., social telescope) by researchers in fields of social sci-
+ence (Mejova, Weber, and Macy 2015), “a massive antenna
+for social science that makes visible both the very large (e.g.,
+global patterns of communications) and the very small (e.g.,
+hourly changes in emotions)”. Beyond CSS, there is increas-
+ing use of Twitter data for training large pre-trained language
+models in the field of natural language processing and ma-
+chine learning, such as Bernice (DeLucia et al. 2022), where
+2.5 billion tweets are used to develop representations for
+Twitter-specific languages, and TwHIN-BERT (Zhang et al.
+2022) that leverages 7 billion tweets covering over 100 dis-
+tinct languages to model short, noisy, and user-generated
+text.
+Although Twitter data has fostered interdisciplinary re-
+search across many fields and has become a “model organ-
+ism” of big data, scholarship using Twitter data has also
+been criticized for various forms of bias that can emerge
+during analyses (Tufekci 2014). One major challenge giv-
+ing rise to these biases is getting access to data and knowing
+about data quality and possible data biases (Ruths and Pfef-
+fer 2014; Gonz´alez-Bail´on et al. 2014; Olteanu et al. 2019).
+While Twitter has long served as one of the most collabora-
+tive big social media platforms in the context of data-sharing
+with academic researchers, there nonetheless exists a lack
+of transparency in sampling procedures and possible biases
+created from technical artifacts (Morstatter et al. 2013; Pf-
+effer, Mayer, and Morstatter 2018). These unknown biases
+may jeopardize research quality. At the same time, access to
+unfiltered/unsampled Twitter data is nearly impossible to ac-
+cess, and thus the above mentioned studies, as well as thou-
+sands of others, still retain unknown and potentially signifi-
+cant biases in their use of sampled data.
+Contributions.
+The data collection efforts presented in
+this paper were driven by a desire to address these concerns
+about sampling bias that exist because of the lack of a com-
+plete sample of Twitter data. Consequently, the main contri-
+arXiv:2301.11429v1  [cs.SI]  26 Jan 2023
+
+bution of this article is to create the first complete dataset of
+24 hours on Twitter and make these Tweets available via fu-
+ture collaborations with the authors and contributors of this
+article. The dataset collected and described here can be used
+by the research community to:
+• Promote a better understanding of the communication
+dynamics on the platform. For example, it can be used
+to answer questions like, how many active (posting) ac-
+counts are on the platform? And, what are the dominating
+languages and topics?
+• Create a set of descriptive metrics that can serve as refer-
+ences for the research community and provide context to
+past and present research papers on Twitter.
+• Provide a baseline for the situation before the recent
+sale of Twitter. With the new ownership of Twitter, plat-
+form policies as well as the company structures are un-
+der significant change, which will create questions about
+whether previous Twitter studies will be still valuable ref-
+erences for future studies.
+In the following sections, we describe the data collection
+process and provide some descriptive analyses of the dataset.
+We also discuss ethical considerations and data availability.
+Data
+Data Collection.
+We have collected 24 hours of Twitter
+data from September 20, 15:00:00 UTC to September 21
+14:59:59 UTC. The data collection was accomplished by
+utilizing the Academic API (Pfeffer et al. 2022) that is free
+and openly available for researchers. The technical setup of
+the data collection pipeline was dominated by two major
+challenges: First, how can we avoid—at least to a satisfying
+extent—a temporal bias in data collection? Second, how can
+we get a good representation of Twitter? In the following,
+these two aspects are discussed in more detail.
+What is a complete dataset?
+What does complete mean
+when we want to collect a day’s worth of Twitter data? It has
+been shown previously that the availability of tweets fluctu-
+ates, especially in the first couple of minutes (Pfeffer et al.
+2022)—people might delete their tweets because of typos,
+tweets might be removed because of violations of terms of
+service, etc. To reduce this initial uncertainty, we have de-
+cided to collect the data 10 minutes after the tweets were
+sent. Consequently, this dataset does not include all tweets
+that were sent on the collection day but instead tries to create
+a somewhat stable representation of Twitter.
+Avoiding temporal collection bias.
+We wanted to collect
+a set of tweets close to the time when they were created.
+However, collecting data takes time, which can introduce
+possible temporal bias, e.g., if we want to collect data from
+the previous hour and the data collection job takes three
+hours, then the data that is collected at the end of the col-
+lection job will be much older (with potentially more tweet
+removals) than the data that is collected at the beginning. To
+tackle this challenge, we have split the day into 86,400 col-
+lection tasks, each consisting of 1 second of Twitter activ-
+ity. The collection of every second of data started exactly 10
+Time
+Tweets per minute
+200,000
+300,000
+400,000
+15
+18
+21
+00
+03
+06
+09
+12
+Figure 1: Tweets per minute over the 24-hour collection pe-
+riod, time in UTC.
+minutes after the data creation time. Because the data collec-
+tion of a second took more than a minute during peak times,
+we have distributed the workload to 80 collection processes,
+i.e., Academic API tokens, in order to avoid backlogs.
+Number of tweets.
+With the above-described process, we
+have collected 374,937,971 tweets within the 24 hours time
+span. On average, this amounts to 4,340 [2,989 – 8,955]
+tweets per second. Fig. 1 plots the number of tweets per
+Minute (avg=260,374, min=192,322, max=435,721). The
+data collection started at 15:00 UTC, when almost the en-
+tire Twitter world is awake. Then, we can see from Japan to
+Europe time zone after time zone getting off the platform.
+While Europe and the Americas are sleeping, Asia keeps
+the number of tweets at around 200,000. Starting at 7:00
+UTC, Europe is getting active again, followed by the Amer-
+icas from East to West. Another astonishing observation of
+this time series is that the first minute of every hour has on
+average 15.5% more tweets than the minute before—most
+likely due to bot activities and other timed tweet releases,
+e.g., news.
+Descriptive Analyses
+Active Users
+The 375 million tweets in our dataset were sent by
+40,199,195 accounts. While the publicly communicated
+numbers of users of a platform are often based on the num-
+ber of active and passive visitors, we can state that Twitter
+has (or at least had on our observed day) 40 million active
+contributors who have sent at least one tweet. Less than 100
+accounts have created about 1% (=3.5M) tweets. ∼175, 000
+accounts (0.44%) created 50% of all tweets.
+These numbers are not surprising when we consider that
+> 95% of active accounts have sent one or two tweets. How-
+ever, these numbers lend more nuance to recent reports from
+the Pew Research Center, which reported that while the ma-
+jority of Americans use social media, approximately 97% of
+all tweets were posted by 25% of the users (McClain 2021).
+
+hi
+fr
+qme
+in
+zxx
+fa
+ko
+th
+pt
+und
+ar
+tr
+es
+ja
+en
+Proportion of Tweets
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.01
+0.015
+0.017
+0.022
+0.023
+0.024
+0.03
+0.04
+0.044
+0.049
+0.05
+0.053
+0.073
+0.165
+0.31
+Figure 2: All languages occurring in at least 1% of the
+tweets.
+In fact, our dataset suggests that worldwide, the numbers
+may be more skewed than previously suggested.
+User metrics
+Followers.
+The active accounts on our day of Twitter
+data have a mean of 2,123 followers (median=99). We can
+find six accounts with more than 100 million followers
+(max=133,301,854), and 427/8,635 accounts with more than
+10/1 million followers. Exactly 50% of accounts that were
+active on our collection day have less than 100 followers.
+Following.
+These accounts follow much fewer other ac-
+counts: mean=547, median=197, range: 0–4,103,801. Inter-
+estingly, there are 2,377 accounts that follow more than
+100,000 other accounts. One-third of accounts follow less
+than 100 accounts, but only 1.7% of accounts follow zero
+other accounts.
+Listed.
+Lists are a Twitter feature for users to organize
+accounts around topics and filter tweets. While there is lit-
+tle evidence that lists are used widely on the platform, this
+feature might be useful for getting an impression about the
+number of interesting content creators on the platform. The
+40 million active accounts in our dataset are listed (i.e.,
+number of lists that include a user) in 0 to 3,086,443 lists
+(mean=10.1, median=0). 1,692/46,139 accounts are in lists
+of at least 10,000/1,000 accounts.
+Tweets sent.
+The user information of the tweet metadata
+also includes the number of tweets that a user has sent—or
+at least how many of those tweets are still available on Twit-
+ter. The sum of the sent tweets variable of all 40 million ac-
+Table 1: Distribution of user activity
+% Total Tweets
+% Total Users
+Min. no. of Tweets
+1%
+0.00023%
+2,267
+10%
+0.01199%
+465
+25%
+0.07284%
+152
+50%
+0.43526%
+39
+75%
+1.70955%
+11
+90%
+4.18836%
+3
+counts is ∼404 billion (mean=9,704, median=1,522). If we
+assume that our initial estimate of having 900 billion tweets
+on the platform at the time of data collection is somewhat
+correct, the accounts active in our dataset have contributed
+∼45% of all of the available tweets over the entire lifetime
+of Twitter.
+Verified accounts.
+At the time of our data collection, we
+can identify 221,246 verified accounts among the 40 million
+active users.
+Tweets and retweets
+79.2% of all tweets refer to other tweets, i.e. they are
+retweets or quotes of or replies to other tweets. Conse-
+quently, 20.8% of the tweets in our dataset are original
+tweets. The tweets with references are of the following
+types: 50.7% retweets, 4.3% quotes, 24.2% replies, i.e. half
+of all tweets are retweets and a fourth are replies.
+Retweeted and liked.
+Studying the retweet and like num-
+bers from the tweets’ metadata has created little insight since
+the top retweeted tweets are very old tweets that have been
+retweeted by chance on our collection day. Furthermore, we
+can see the number of likes only for tweets that have been
+tweeted and retweeted. In any case, the retweeted number
+is interesting—the 374 million tweets have been retweeted
+401 billion times. In other words, significant parts of historic
+Twitter get retweeted on a daily basis.
+Languages
+Twitter annotates a language variable for every tweet. Fig. 2
+shows those languages that were annotated on at least 1% of
+our dataset. Together, these 15 languages make up 92.5% of
+all tweets. Besides the most common languages on Twitter,
+we can also find interesting language codes in this list: und
+stands for undefined and represents tweets for which Twitter
+was not able to identify a language; qme and zxx seem to
+be used by Twitter for tweets consisting of only media or a
+Twitter card.
+Media
+There are 112,779,266 media attachments in our data collec-
+tion (76.9% photos, 20.7% videos, 2.4% animated GIFs), of
+which 37,803,473 have unique media keys (83.8% photos,
+10.0% videos, 6.2% animated GIFs).
+Geo-tags
+We found only 0.5% of tweets to be geo-tagged. This is
+not surprising as previous works have shown that the per-
+centage of geo-tagging in Twitter has been declining (Ajao,
+Hong, and Liu 2015). Fig. 3 shows the distribution of the
+geo-tagged tweets across the world, with USA (20%), Brazil
+(11%), Japan (8%), Saudi Arabia (6%) and India (4%) being
+the top five countries.
+Estimating prevalence of bot accounts
+Twitter has a pivotal role in public discourse and entities
+that are after power and influence often utilize this platform
+
+Figure 3: Choropleth map of the geo-tagged tweets across
+the world.
+through social bots and other means of automated activi-
+ties. Since the early days of Twitter, researchers have been
+studying bot behavior, and it has become an active research
+area (Ferrara et al. 2016; Cresci 2020). The first estimation
+of bot prevalence on Twitter indicates that 9-15% of Twit-
+ter accounts exhibit automated behavior (Varol et al. 2017),
+while others have observed significantly higher percentages
+of tweets produced by bot-likely accounts on specific dis-
+courses (Uyheng and Carley 2021; Antenore, Camacho Ro-
+driguez, and Panizzi 2022). One major challenge in estimat-
+ing bot prevalence is the variety of definitions, datasets, and
+models used for detection (Varol 2022).
+In this study, we employed BotometerLite (Yang et al.
+2020), a scalable and light-weight version of the Botome-
+ter (Sayyadiharikandeh et al. 2020), for computing bot
+scores for unique accounts in our collection. In Fig. 4a, we
+present the distribution of bot scores and nearly 20% of the
+40 million active accounts have scores above 0.5 suggesting
+bot-likely behavior.
+While identification of bots is a complex and possi-
+bly controversial challenge, plotting the distributions of
+BotometerLite scores grouped by account age in Fig. 4b sug-
+gests the proportions of accounts that show bot-like behavior
+has increased dramatically in recent years. This result may
+also suggest that the longevity of simpler bot accounts is
+limited and they are no longer active on the platform. In Fig.
+4c, we also present the distribution of bot scores for differ-
+ent rates of activities in our dataset. Accounts that have over
+1,000 posts exhibit higher rates of bot-like behaviors.
+It is important to mention that accounts studied in this pa-
+per were identified due to their content creation activities.
+Our collection cannot capture passive accounts that are sim-
+ply used to boost follower counts without visible activity on
+tweet streams. Fair assessment of bot prevalence is only pos-
+sible with complete access to Twitter’s internal database;
+since activity streams, network data, and historical tweet
+archives can capture different sets of accounts (Varol 2022).
+Content on Twitter
+The top 500 hashtags occurred 81,468,508 times in the
+tweets. Via manual inspection, we were able to identify the
+meaning of 95% of these top hashtags. They can be aggre-
+gated into ten the categories.
+Table 2 suggests that a large proportion of tweets referred
+to entertainment, which together comprised about 30% of
+tweets. These included mentions of celebrities (25.5%) and
+other entertainment-related tweets (5.4%) such as mentions
+of South Korean boy band members, and other references
+to music, movies, and TV shows. Our data collection time
+window occurred during Fall/Winter 2022, when the world
+was discussing the protests in Iran after the death of Mahsa
+Amini. Therefore, the Iranian protests also comprised a large
+proportion of the hashtag volume at 16.6%.
+Finally, and perhaps surprisingly, the category sex com-
+prised over a quarter of all content covered by the top hash-
+tags, and was almost completely related to escorts. “Other”
+topics reflect that on “regular” Twitter days, sports, tech, and
+art may take up only about 3.3% of Twitter volume.
+Fig. 5 is a hashtag visualization that attempts to provide an
+overview of the entire content on Twitter. We first removed
+all tweets from accounts with more than 240 tweets to re-
+duce the noise from bots using random trending hashtags.
+From the remaining tweets, we extracted the 10,000 most
+often used hashtags in our dataset and created a hashtag sim-
+ilarity matrix with the number of accounts that have used a
+pair of two hashtags on the day of data collection. Every el-
+ement in Fig. 5 represents a hashtag. The position is the re-
+sult of Multidimensional Scaling (MDS) and the color shows
+the dominant language that was used in the tweets with the
+particular hashtag. In this figure, we can see how languages
+separate the Twitter universe but that there are also topical
+sub-communities within languages.
+Discussion and Potential Applications
+Twitter is a social media platform with a worldwide user-
+base. Open access to its data also makes it attractive to a
+large community of researchers, journalists, technologists,
+and policymakers who are interested in examining social
+and civic behavior online. Early studies of Twitter explored
+who says what to whom on Twitter (Wu et al. 2011), char-
+acterizing its primary use as a communication tool. Other
+early work mapped follower communities through ego net-
+works (Gruzd, Wellman, and Takhteyev 2011). However,
+Twitter has since expanded into its own universe, with a
+plethora of users, uses, modalities, communities, and real-
+life implications. Twitter is increasingly the source of break-
+Table 2: The categories of the top 500 hashtags in the dataset
+Category
+Hashtags
+Occurrence
+Celebrities
+159
+20,809,742
+25.5%
+Sex
+104
+20,529,196
+25.2%
+Iranian Protests
+15
+13,488,295
+16.6%
+Entertainment
+45
+4,392,227
+5.4%
+Advertisement
+32
+4,644,540
+5.7%
+Politics
+38
+3,858,550
+4.7%
+Finance
+30
+3,549,107
+4.4%
+Games
+21
+3,348,128
+4.1%
+Other
+31
+2,672,291
+3.3%
+Unknown
+25
+4,176,432
+5.1%
+Sum
+500
+81,468,508
+100.0%
+
+GeotaggedTweets
+500k
+400k
+300k
+200k
+100k0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BotometerLite score
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Histogram of botscores
+1e6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+CDF
+(a)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BotometerLite score
+0
+1
+2
+3
+4
+5
+6
+Density
+2007
+2008
+2009
+2010
+2011
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+2022
+0
+2
+4
+6
+8
+# of accounts
+1e6
+(b)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BotometerLite score
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Density
+Nt = Tweet count
+Nt < 101
+101
+Nt < 102
+102
+Nt < 103
+103
+Nt
+(c)
+Figure 4: BotometerLite scores distribution: (a) histogram and cumulative distribution, (b) by account age, (c) by tweet counts
+in our dataset.
+ing news, and many studies from the U.S. and Europe have
+reported that Twitter is one of the primary sources of news
+for their citizens. Twitter has been used for political engage-
+ment and citizen activism worldwide. During the COVID-
+19 pandemic, Twitter even assumed the role of the official
+mouthpiece and crisis communication tool for many gov-
+ernments to contact their citizens, and from which citizens
+could seek help and information.
+Fig. 3 confirms prior reports that geotagging practices are
+limited in many low- and middle-income countries (Malik
+et al. 2015); however, this should not deter scholars from ex-
+ploring alternative methods of triangulating the location of
+users (Schwartz et al. 2013), and creating post-stratified es-
+timates of regional language use (Jaidka et al. 2020; Giorgi
+et al. 2022). In prior studies, the difficulties in widespread
+data collection and analyses have so far implied that most
+answers are based on smaller samples (usually constrained
+by geography, for convenience) of a burgeoning Twitter pop-
+ulation. Fig. 5 and Table 2 also impressively illustrate that
+Twitter is about so much more than US politics.
+We hope that our dataset is the first step in creating al-
+ternatives for conducting a representative and truly inclusive
+analysis of the Twitterverse. Temporal snapshots are invalu-
+able to map the national and international migration patterns
+that increasingly blur geopolitical boundaries (Zagheni et al.
+2014).
+The increasing popularity of Twitter has led it into issues
+of scale, where its moderation can no longer check the large
+proportion of bots on the platform. Our findings in Fig. 4
+indicate that the infestation of bots may be more pernicious
+than previously imagined. We are especially concerned that
+the escalation of the war on Ukraine by Russia may reflect a
+spike (in our dataset) in the online activity of bots from Rus-
+sia operated either by the Russian government or its allied
+intelligence agencies (Badawy, Ferrara, and Lerman 2018).
+These and other bots serve to amplify trending topics and
+facilitate the spread of misinformation (though, perhaps, at
+a rate less than humans do (Vosoughi, Roy, and Aral 2018)).
+They may also misuse hashtags to divert attention away from
+social or political topics (Earl, Maher, and Pan 2022; Broni-
+atowski et al. 2018) or strategically target influential users
+(Shao et al. 2018; Varol and Uluturk 2020). We hope that
+our work will spur more studies on these topics, and we wel-
+come researchers to explore our data.
+By observing bursts of discussions around politically
+charged events and characterizing the temporal spikes in
+Twitter topics, we can better rationalize how our experience
+of Twitter as a political hotbed differs from the simplified
+understanding of the American Twitter landscape reported
+in Mukerjee, Jaidka, and Lelkes (2022), which suggested
+that politics is largely a sideshow on Twitter. It is worth con-
+sidering that these politically active users may not be rep-
+resentative of social media users at large (McClain 2021;
+Wojcieszak et al. 2022).
+Twitter is also under scrutiny for how its platform gover-
+nance may conflict with users’ interests and rights (Van Di-
+jck, Poell, and De Waal 2018). Concerns have been raised
+about alleged biases in the algorithmic amplification (and
+deamplification) of content, with evidence from France,
+Germany, Turkey, and the United States, among other coun-
+tries (Maj´o-V´azquez et al. 2021; Tanash et al. 2015; Jaidka,
+Mukerjee, and Lelkes 2023). Other scholars have also criti-
+cized Twitter’s use as a censorship weapon by governments
+and political propagandists worldwide (Varol 2016; Elmas,
+Overdorf, and Aberer 2021; Jakesch et al. 2021). They, and
+others, may be interested in examining the trends in the en-
+forcement of content moderation policies by Twitter.
+Besides answering questions of data, representativeness,
+access, and censorship, we anticipate that our dataset
+is suited to explore the temporal dynamics of online
+(mis)information in the following directions:
+• Content characteristics: We have provided a high-level
+exploration of the topics on Twitter. However, more can
+
+Figure 5: MDS of top 10,000 hashtags based on co-usage by same accounts; colors represent dominant language in tweets using
+a hashtag.
+be done with regard to understanding users’ concerns and
+priorities. While hashtags act as signposts for the broader
+Twitter community to find and engage in topics of mu-
+tual interest (Cunha et al. 2011), tweets without hashtags
+may offer a different understanding of Twitter discourse,
+where users may engage in more interpersonal discus-
+sions of news, politics, and sports than the numbers sug-
+gest (Rajadesingan, Budak, and Resnick 2021).
+• Patterns of information dissemination: Informational
+exchanges occurring on Twitter can overcome spatio-
+temporal limitations as they essentially reconfigure user
+connections to create newly emergent communities.
+However, these communities may vanish as quickly as
+they are created, as the lifecycle of a tweet determines
+how long it continues to circulate on Twitter timelines.
+To the best of our knowledge, no prior research has re-
+ported on the average “age” of a tweet, and we hope that
+a 24-hour snapshot will enable us to answer this question
+empirically.
+• Content moderation and fake news: Prior research
+suggests that 0.1% of Twitter users accounted for 80%
+of all fake news sources shared in the lead-up to a
+US election (Grinberg et al. 2019). However, we ex-
+pect there to be cross-lingual differences in this distri-
+bution, especially for low- or under-resourced languages
+with fewer open tools for fact-checking. Similarly, we
+expect that the quality of moderation and hate speech
+will vary by geography and language, and recommend
+the use of multilingual large language models to explore
+these trends (with attention to persisting representative-
+
+fa
+hi
+ko
+th
+tr
+un
+ar
+en
+it
+de
+es
+ja
+pt
+und
+zh
+frness caveats (Wu and Dredze 2020)).
+• Mass mobilization: Twitter is increasingly the hotbed of
+protest, which has led to some activists donning the role
+of “movement spilloverers” (Zhou and Yang 2021) or se-
+rial activists (Bastos and Mercea 2016) who broker infor-
+mation across different online movements, thereby acting
+as key coordinators, itinerants, or gatekeepers in the ex-
+change of information. Such users, as well as the constant
+communities in which they presumably reside (Chowd-
+hury et al. 2022), may be easier to study through tempo-
+ral snapshots, as facilitated by this dataset.
+• Echo chambers and filter bubbles: On Twitter, algo-
+rithms can affect the information diets of users in over
+200 countries, with an estimated 396.5 million monthly
+users (Kemp 2022). Recent surveys of the literature have
+considered the evidence on how platforms’ designs and
+affordances influence users behaviors, attitudes, and be-
+liefs (Gonz´alez-Bail´on and Lelkes 2022). Studies of the
+structural and informational networks based on snapshots
+of Twitter can offer clues to solving these puzzles with-
+out the constraints of data selection.
+Ethics Statement and Data Availability
+Ethics statement.
+We acknowledge that privacy and ethi-
+cal concerns are associated with collecting and using social
+media data for research. However, we took several steps to
+avoid risks to human subjects since participants no longer
+opt into being part of our study, in a traditional sense (Zim-
+mer 2020). In our analysis, we only studied and reported
+population level, and aggregated observations of our dataset.
+We share publicly only the tweet IDs with the research com-
+munity to account for privacy issues and Twitter’s TOS. For
+this purpose, we use a data sharing and long-term archiving
+service provided by GESIS - Leibniz Institute for the Social
+Sciences, a German infrastructure institute for the social sci-
+ences 2.
+With regards to data availability, this repository adheres
+to the FAIR principles (Wilkinson et al. 2016) as follows:
+• Findability: In compliance with Twitter’s terms of ser-
+vice, only tweet IDs are made publicly available at DOI:
+https://doi.org/10.7802/2516. A unique Document Ob-
+ject Identifier (DOI) is associated with the dataset. Its
+metadata and licenses are also readily available.
+• Accessibility: The dataset can be downloaded using stan-
+dard APIs and communications protocol (the REST API
+and OAI-PMH).
+• Interoperability: The data is provided in raw text for-
+mat.
+• Reusability: The CC BY 4.0 license implies that re-
+searchers are free to use the data with proper attribution.
+Furthermore, we want to invite the broader research com-
+munity to approach one or more of the authors and collab-
+orators (see Acknowledgments) of this paper with research
+ideas about what can be done with this dataset. We will be
+very happy to collaborate with you on your ideas!
+2https://www.gesis.org/en/data-services/share-data
+Acknowledgments
+The data collection effort described in this paper could
+not have been possible without the great collaboration of
+a large number of scholars, here are some of them (in
+random order): Chris Schoenherr, Leonard Husmann, Diyi
+Liu, Benedict Witzenberger, Joan Rodriguez-Amat, Flo-
+rian Angermeir, Stefanie Walter, Laura Mahrenbach, Isaac
+Bravo, Anahit Sargsyan, Luca Maria Aiello, Sophie Brandt,
+Wienke Strathern, Bilal C¸ akir, David Schoch, Yuliia Holu-
+bosh, Savvas Zannettou, Kyriaki Kalimeri.
+References
+Ajao, O.; Hong, J.; and Liu, W. 2015. A survey of loca-
+tion inference techniques on Twitter. Journal of Information
+Science, 41(6): 855–864.
+Antenore, M.; Camacho Rodriguez, J. M.; and Panizzi, E.
+2022. A Comparative Study of Bot Detection Techniques
+With an Application in Twitter Covid-19 Discourse. Social
+Science Computer Review, 08944393211073733.
+Badawy, A.; Ferrara, E.; and Lerman, K. 2018. Analyzing
+the digital traces of political manipulation: The 2016 Rus-
+sian interference Twitter campaign. In 2018 IEEE/ACM in-
+ternational conference on advances in social networks anal-
+ysis and mining (ASONAM), 258–265. IEEE.
+Bastos, M. T.; and Mercea, D. 2016. Serial activists: Polit-
+ical Twitter beyond influentials and the twittertariat. New
+Media & Society, 18(10): 2359–2378.
+Broniatowski, D. A.; Jamison, A. M.; Qi, S.; AlKulaib, L.;
+Chen, T.; Benton, A.; Quinn, S. C.; and Dredze, M. 2018.
+Weaponized health communication: Twitter bots and Rus-
+sian trolls amplify the vaccine debate. American journal of
+public health, 108(10): 1378–1384.
+Chowdhury, A.; Srinivasan, S.; Bhowmick, S.; Mukherjee,
+A.; and Ghosh, K. 2022. Constant community identifica-
+tion in million-scale networks. Social Network Analysis and
+Mining, 12(1): 1–17.
+Cresci, S. 2020. A decade of social bot detection. Commu-
+nications of the ACM, 63(10): 72–83.
+Cunha, E.; Magno, G.; Comarela, G.; Almeida, V.;
+Gonc¸alves, M. A.; and Benevenuto, F. 2011. Analyzing the
+dynamic evolution of hashtags on twitter: a language-based
+approach. In Proceedings of the workshop on language in
+social media (LSM 2011), 58–65.
+DeLucia, A.; Wu, S.; Mueller, A.; Aguirre, C.; Dredze, M.;
+and Resnik, P. 2022. Bernice: A Multilingual Pre-trained
+Encoder for Twitter.
+Earl, J.; Maher, T. V.; and Pan, J. 2022. The digital repres-
+sion of social movements, protest, and activism: A synthetic
+review. Science Advances, 8(10): eabl8198.
+Elmas, T.; Overdorf, R.; and Aberer, K. 2021. A Dataset of
+State-Censored Tweets. In ICWSM, 1009–1015.
+Ferrara, E.; Varol, O.; Davis, C.; Menczer, F.; and Flammini,
+A. 2016. The rise of social bots. Communications of the
+ACM, 59(7): 96–104.
+
+Giorgi, S.; Lynn, V. E.; Gupta, K.; Ahmed, F.; Matz, S.; Un-
+gar, L. H.; and Schwartz, H. A. 2022. Correcting Sociode-
+mographic Selection Biases for Population Prediction from
+Social Media.
+In Proceedings of the International AAAI
+Conference on Web and Social Media, volume 16, 228–240.
+Gonz´alez-Bail´on, S.; and Lelkes, Y. 2022. Do social media
+undermine social cohesion? A critical review. Social Issues
+and Policy Review.
+Gonz´alez-Bail´on,
+S.;
+Wang,
+N.;
+Rivero,
+A.;
+Borge-
+Holthoefer, J.; and Moreno, Y. 2014. Assessing the bias in
+samples of large online networks. Social Networks, 38: 16 –
+27.
+Grinberg, N.; Joseph, K.; Friedland, L.; Swire-Thompson,
+B.; and Lazer, D. 2019. Fake news on Twitter during the
+2016 US presidential election.
+Science, 363(6425): 374–
+378.
+Gruzd, A.; Wellman, B.; and Takhteyev, Y. 2011. Imagining
+Twitter as an imagined community. American Behavioral
+Scientist, 55(10): 1294–1318.
+Jaidka, K.; Giorgi, S.; Schwartz, H. A.; Kern, M. L.; Ungar,
+L. H.; and Eichstaedt, J. C. 2020. Estimating geographic
+subjective well-being from Twitter: A comparison of dictio-
+nary and data-driven language methods. Proceedings of the
+National Academy of Sciences, 117(19): 10165–10171.
+Jaidka, K.; Mukerjee, S.; and Lelkes, Y. 2023. Silenced on
+social media: the gatekeeping functions of shadowbans in
+the American Twitterverse. Journal of Communication.
+Jakesch, M.; Garimella, K.; Eckles, D.; and Naaman, M.
+2021.
+Trend alert: A cross-platform organization manip-
+ulated Twitter trends in the Indian general election. Pro-
+ceedings of the ACM on Human-Computer Interaction,
+5(CSCW2): 1–19.
+Kemp, S. 2022. Digital 2022: Global overview report. Tech-
+nical report, DataReportal.
+Maj´o-V´azquez, S.; Congosto, M.; Nicholls, T.; and Nielsen,
+R. K. 2021. The Role of Suspended Accounts in Political
+Discussion on Social Media: Analysis of the 2017 French,
+UK and German Elections. Social Media+ Society, 7(3):
+20563051211027202.
+Malik, M.; Lamba, H.; Nakos, C.; and Pfeffer, J. 2015. Pop-
+ulation bias in geotagged tweets. In proceedings of the in-
+ternational AAAI conference on web and social media, vol-
+ume 9, 18–27.
+McClain, C. 2021. 70% of U.S. social media users never or
+rarely post or share about political, social issues. Technical
+report, Pew Research Center.
+Mejova, Y.; Weber, I.; and Macy, M. W. 2015. Twitter: a
+digital socioscope. Cambridge University Press.
+Morstatter, F.; Pfeffer, J.; Liu, H.; and Carley, K. M. 2013. Is
+the Sample Good Enough? Comparing Data from Twitter’s
+Streaming API with Twitter’s Firehose. In Seventh Inter-
+national AAAI Conference on Weblogs and Social Media,
+400–408.
+Mukerjee, S.; Jaidka, K.; and Lelkes, Y. 2022. The Political
+Landscape of the US Twitterverse. Political Communica-
+tion, 1–31.
+Olteanu, A.; Castillo, C.; Diaz, F.; and Kıcıman, E. 2019.
+Social data: Biases, methodological pitfalls, and ethical
+boundaries. Frontiers in Big Data, 2: 13.
+Pfeffer, J.; Mayer, K.; and Morstatter, F. 2018. Tampering
+with Twitter’s Sample API. EPJ Data Science, 7(50).
+Pfeffer, J.; Mooseder, A.; Lasser, J.; Hammer, L.; Stritzel,
+O.; and Garcia, D. 2022. This Sample seems to be good
+enough! Assessing Coverage and Temporal Reliability of
+Twitter’s Academic API.
+Rajadesingan, A.; Budak, C.; and Resnick, P. 2021. Political
+discussion is abundant in non-political subreddits (and less
+toxic). In Proceedings of the Fifteenth International AAAI
+Conference on Web and Social Media, volume 15.
+Ruths, D.; and Pfeffer, J. 2014. Social Media for Large Stud-
+ies of Behavior. Science, 346(6213): 1063–1064.
+Sayyadiharikandeh, M.; Varol, O.; Yang, K.-C.; Flammini,
+A.; and Menczer, F. 2020. Detection of novel social bots
+by ensembles of specialized classifiers. In Proceedings of
+the 29th ACM international conference on information &
+knowledge management, 2725–2732.
+Schwartz, H.; Eichstaedt, J.; Kern, M.; Dziurzynski, L.; Lu-
+cas, R.; Agrawal, M.; Park, G.; Lakshmikanth, S.; Jha, S.;
+Seligman, M.; et al. 2013. Characterizing geographic varia-
+tion in well-being using tweets. In Proceedings of the Inter-
+national AAAI Conference on Web and Social Media, vol-
+ume 7, 583–591.
+Shao, C.; Ciampaglia, G. L.; Varol, O.; Yang, K.-C.; Flam-
+mini, A.; and Menczer, F. 2018.
+The spread of low-
+credibility content by social bots. Nature communications,
+9(1): 1–9.
+Tanash, R. S.; Chen, Z.; Thakur, T.; Wallach, D. S.; and Sub-
+ramanian, D. 2015. Known unknowns: An analysis of Twit-
+ter censorship in Turkey. In Proceedings of the 14th ACM
+Workshop on Privacy in the Electronic Society, 11–20.
+Tufekci, Z. 2014. Big questions for social media big data:
+Representativeness, validity and other methodological pit-
+falls. In Eighth international AAAI conference on weblogs
+and social media.
+Uyheng, J.; and Carley, K. M. 2021. Computational Analy-
+sis of Bot Activity in the Asia-Pacific: A Comparative Study
+of Four National Elections.
+In Proceedings of the Inter-
+national AAAI Conference on Web and Social Media, vol-
+ume 15, 727–738.
+Van Dijck, J.; Poell, T.; and De Waal, M. 2018. The plat-
+form society: Public values in a connective world. Oxford
+University Press.
+Varol, O. 2016. Spatiotemporal analysis of censored content
+on twitter. In Proceedings of the 8th ACM Conference on
+Web Science, 372–373.
+Varol, O. 2022. Should we agree to disagree about Twitter’s
+bot problem? arXiv preprint arXiv:2209.10006.
+Varol, O.; Ferrara, E.; Davis, C.; Menczer, F.; and Flam-
+mini, A. 2017. Online human-bot interactions: Detection,
+estimation, and characterization. In Proceedings of the in-
+ternational AAAI conference on web and social media, vol-
+ume 11, 280–289.
+
+Varol, O.; and Uluturk, I. 2020. Journalists on Twitter: self-
+branding, audiences, and involvement of bots. Journal of
+Computational Social Science, 3(1): 83–101.
+Vosoughi, S.; Roy, D.; and Aral, S. 2018. The spread of true
+and false news online. science, 359(6380): 1146–1151.
+Wilkinson, M. D.; Dumontier, M.; Aalbersberg, I. J.; Apple-
+ton, G.; Axton, M.; Baak, A.; Blomberg, N.; Boiten, J.-W.;
+da Silva Santos, L. B.; Bourne, P. E.; et al. 2016. The FAIR
+Guiding Principles for scientific data management and stew-
+ardship. Scientific data, 3(1): 1–9.
+Wojcieszak, M.; Casas, A.; Yu, X.; Nagler, J.; and Tucker,
+J. A. 2022. Most users do not follow political elites on Twit-
+ter; those who do show overwhelming preferences for ideo-
+logical congruity. Science advances, 8(39): eabn9418.
+Wu, S.; and Dredze, M. 2020. Are All Languages Created
+Equal in Multilingual BERT?
+In Proceedings of the 5th
+Workshop on Representation Learning for NLP, 120–130.
+Wu, S.; Hofman, J. M.; Mason, W. A.; and Watts, D. J. 2011.
+Who says what to whom on twitter. In Proceedings of the
+20th international conference on World wide web, 705–714.
+Yang, K.-C.; Varol, O.; Hui, P.-M.; and Menczer, F. 2020.
+Scalable and generalizable social bot detection through data
+selection. In Proceedings of the AAAI conference on artifi-
+cial intelligence, volume 34, 1096–1103.
+Zagheni, E.; Garimella, V. R. K.; Weber, I.; and State, B.
+2014. Inferring international and internal migration patterns
+from twitter data. In Proceedings of the 23rd international
+conference on world wide web, 439–444.
+Zhang, X.; Malkov, Y.; Florez, O.; Park, S.; McWilliams, B.;
+Han, J.; and El-Kishky, A. 2022. TwHIN-BERT: A Socially-
+Enriched Pre-trained Language Model for Multilingual
+Tweet Representations. arXiv preprint arXiv:2209.07562.
+Zhou, A.; and Yang, A. 2021. The Longitudinal Dimension
+of Social-Mediated Movements: Hidden Brokerage and the
+Unsung Tales of Movement Spilloverers.
+Social Media+
+Society, 7(3): 20563051211047545.
+Zimmer, M. 2020. “But the data is already public”: on the
+ethics of research in Facebook. In The Ethics of Information
+Technologies, 229–241. Routledge.
+

FdE1T4oBgHgl3EQfqwVD/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5d604dc22a48a403a0d4f47e9a3f58cf6ac38c97a628338f8c07334aa470c1dc
+size 6553645

INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:304f5c1c69a663779b55c81bafa9f84ec2033c041339f998169d050035f126b7
+size 1378689

INFLT4oBgHgl3EQfIy9R/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:3aacccb9a57f502227a34dc65859db9c43457853e6ece7493aa62e393b419a28
+size 177226

IdAyT4oBgHgl3EQfr_me/content/tmp_files/2301.00569v1.pdf.txt

@@ -0,0 +1,420 @@
+arXiv:2301.00569v1  [math.AC]  2 Jan 2023
+ELIAS IDEALS
+HAILONG DAO
+Abstract. Let (R, m) be a one dimensional local Cohen-Macaulay ring. An m-primary
+ideal I of R is Elias if the types of I and of R/I are equal. Canonical and principal ideals
+are Elias, and Elias ideals are closed under inclusion. We give multiple characterizations
+of Elias ideals and concrete criteria to identify them. We connect Elias ideals to other
+well-studied definitions: Ulrich, m-full, integrally closed, trace ideals, etc. Applications are
+given regarding canonical ideals, conductors and the Auslander index.
+Introduction
+Let (R, m) be a local Cohen-Macaulay ring of dimension one and I be an m-primary ideal
+of R. We say that I is Elias if the Cohen-Macaulay types of I and R/I coincide. From
+standard facts, principal ideals or canonical ideals are Elias, and we will soon see that this
+property begets a rather rich and interesting theory.
+Our work is heavily influenced by a nice result in [7], where Elias proves that any ideal
+ω that lies inside a high enough power of m and such that R/ω is Gorenstein must be a
+canonical ideal. Although not stated explicitly there, the proof showed that any ideal that
+lies in a high enough power of m is Elias, in our sense. Another inspiration for the present
+work is [2], where De Stefani studies, in our language, powers of m that are Elias in a
+Gorenstein local ring, and gives a counter-example to a conjecture by Ding (see Section 4
+for the precise connection).
+In this note, we study Elias ideals in depth. They admit many different characterizations,
+and enjoy rather useful properties. For instance, they are closed under inclusion, and
+principal or canonical ideals are Elias. On the other hand, conductor ideals or regular trace
+ideals are not Elias. When R is Gorenstein, they are precisely ideals such that the Auslander
+index δ(R/I) is 1.
+We are able to obtain many criteria to check whether an ideal is Elias, using very accessible
+information such as the minimal number or valuations of generators.
+Combining them
+immediately gives sharp bounds and information on conductor or canonical ideals, which
+can be tricky to obtain otherwise.
+There are several obvious ways to extend the present definitions and results to higher
+dimension rings or to modules. However, we choose to focus on the ideals in dimension one
+case here as they are already interesting enough, and also to keep the paper short. We hope
+to address the more general theory in future works.
+We now describe briefly the structure and key results of the paper.
+• In section 1 we give the formal definition of Elias ideals and prove several key results.
+Theorem 1.2 contains several equivalent characterizations of Elias ideals. Corollary
+1.3 collects important consequences, for instance that Elias ideals are closed under
+ideal containment. Also, criteria for Elias ideals using colon ideals are given. Next,
+2020 Mathematics Subject Classification. Primary: 13D02, 13H10. Secondary: 14B99.
+1
+
+Proposition 1.4 establishes the fundamental change of rings result that are used
+frequently in the sequence.
+• Section 2 connects Elias ideals to several well-studied class of ideals: Ulrich ideals,
+m-full ideals, full ideals, integrally closed ideals, etc. After some basic observations,
+(2.3, 2.4, 2.5), we give Theorems 2.7 and Proposition 2.14, which contain concrete
+ways to recognize Elias ideals using basic information such as number of generators
+or valuations. We also derive that conductor ideals or regular trace ideals are not
+Elias (Corollary 2.13).
+This indicates one of the useful application: if we know,
+for instance, that m2 is Elias, then the conductor or any regular trace ideal must
+contains an element of m-adic order 1.
+• Given the previous section, it is natural to study the Elias index eli(R), namely
+the first power of m that is Elias, and we do so in Section 3. The first main result
+here is Theorem 3.2, connecting this index to the generalized L¨oewy length and the
+regularity of the associated graded ring. Next, in Theorem 3.3, we characterize rings
+with small indexes: eli(R) = 1 if and only if R is regular, and eli(R) = 2 plus R is
+Gorenstein is equivalent to e(R) = 2. We give a large class of non-Gorenstein rings
+with Elias index 2 (3.4).
+• Lastly, in Section 4 we focus on the special case of Gorenstein rings. In such situation,
+we observe that Elias ideals are precisely ones whose quotient has Auslander δ-
+invariant one. This immediately allows us to apply what we have to recover old
+results about the Auslander invariant and Auslander index in 4.1 and 4.3. We give
+a counter-example to a Theorem by Ding and also revisit a counter-example to a
+conjecture by Ding given in [2] (Examples 4.4 and 4.5).
+Acknowledgements: It is a pleasure to thank Juan Elias and Alessandro Di Stefani for
+helpful comments and encouragements. The author is partially supported by the Simons
+Collaboration Grant FND0077558.
+1. Elias ideals: definitions and basic results
+Throughout the paper, let (R, m, k) be Cohen-Macaulay local ring of dimension one. For
+a module M, set typeR(M) = dimk Extdim M
+R
+(k, M). Set Q = Q(R) to be the total ring of
+fractions of R. Set e = e(R), the Hilbert-Samuel multiplicity of R. For an element x ∈ R,
+the m-adic order of x, denoted ord(x) is the smallest a such that x ∈ ma. The order of an
+ideal I, denoted ord(I), is the minimum order of its elements.
+Definition 1.1. We say that a m-primary ideal I is an Elias ideal if it satisfies type(I) =
+type(R/I).
+Theorem 1.2. We always have type(I) ≥ type(R/I). The following are equivalent.
+(1) type(I) = type(R/I).
+(2) For any NZD x ∈ m, xI : m ⊆ (x).
+(3) For any NZD x ∈ m, xI : m = x(I : m).
+(4) For some NZD x ∈ m, xI : m ⊆ (x).
+(5) For some NZD x ∈ m, xI : m = x(I : m).
+(6) I :Q m ⊆ R.
+(7) K ⊆ m(K :Q I) (assuming R admits a canonical ideal K).
+Proof. Let x be a NZD. Then
+type(I) = type(I/xI) = dimk
+xI : m
+xI
+≥ dimk
+x(I : m)
+xI
+= dimk
+I : m
+I
+= type(R/I)
+2
+
+Thus, type(I) = type(R/I) if and only if xI : m = x(I : m). Now, xI : m ⊆ (x) is equivalent
+to xI : m = xJ for some ideal J, as x is a NZD. Rewriting it as xJm ⊆ xI, which is
+equivalent to Jm ⊆ I, we get J ⊆ I : m. On the other hand x(I : m) ⊆ xI : m, thus
+J = I : m. That establishes the equivalence of first five items.
+Note that for any NZD x ∈ m, xI : m = x(I :Q m). Thus, (6) is equivalent to (3).
+Let K be a canonical ideal.
+Apply HomR(−, K) to the sequence 0 → I → R →
+R/I → 0, and indentifying HomR(I, K) with K :Q I, we get 0 → K → K :Q I →
+Ext1
+R(R/I, K) = ωR/I → 0.
+Since type(I) = µ(K :Q I) and type(R/I) = µ(ωR/I), the
+equivalence of (7) and (1) follows.
+□
+Corollary 1.3. We have:
+(1) If I is isomorphic to R or the canonical module of R (assuming its existence), then
+I is Elias.
+(2) If I is Elias, then so is J for any ideal J ⊆ I. (being Elias is closed under inclusion)
+(3) Let K be a canonical ideal of R and I be an ideal containing K. Then I is Elias if
+and only if K ⊆ m(K :R I).
+(4) Let K be a canonical ideal of R and I be an ideal such that K ⊆ I. Then K : I is
+Elias if and only if K ⊆ mI.
+(5) Suppose that I contains a canonical ideal K such that ord(K) = 1. Then I is Elias
+if and only if I = K.
+Proof. For the first claim, I :Q m ⊂ I :Q I = R. For the second claim, we have J :Q m ⊂
+I :Q m. For (3), first note that K :Q I ⊂ K :Q K = R, so K :Q I = K :R I, and we can use
+part (7) of Theorem 1.2.
+For part (4), note that K : (K : I) = I hence we can apply part (3).
+For part (5), we again apply part (3): if K ⊊ I, then m(K :R I) ⊆ m2, contradicting
+ord(K) = 1.
+□
+The following change of rings result would be used frequently in what follows.
+Proposition 1.4. Let (R, m) → (S, n) be a local, flat rings extension such that dim S = 1
+and S is Noetherian. Then I is an Elias ideal of R if and only if IS is an Elias ideal of S.
+Proof. Under the assumption we have typeR(M) typeS/mS(S/mS) = typeS(M ⊗R S) for any
+finitely generated R-module M (see for instance [11]), thus the result follows.
+□
+2. Elias ideals and other special ideals
+Definition 2.1. Let I be an m-primary ideal.
+• I is called Ulrich (as an R-module) if µ(I) = e(R). Assuming k is infinite, then I is
+Ulrich if and only if xI = mI for some x ∈ m (equivalently, for any x ∈ m such that
+ℓ(R/xR) = e(R)).
+• I is called m-full if Im : x = I for some x ∈ m.
+• I is called full (or basically full) if Im : m = I.
+Remark 2.2. When the definition of special ideals such as Ulrich or m-full ones involves
+an element x, we say that the property is witnessed by x. Note that being such x is a
+Zariski-open condition (for the image of x in the vector space m/m2). For more on these
+ideals, see [3, 10, 9, 12].
+3
+
+Proposition 2.3. Let I be an m-primary ideal. Let e be the Hilbert-Samuel multiplicity of
+R. The following are equivalent.
+(1) I is Ulrich.
+(2) type(I) = e.
+Proof. We can assume k is infinite by making the flat extension R → R[t](m,t). Let x ∈ m be
+such that ℓ(R/xR) = e. Then ℓ(I/xI) = e. Note that type(I) = ℓ(soc(I/xI)) ≤ ℓ(I/xI) =
+e, and equality happens precisely when m(I/xI) = 0, in other words, I is Ulrich.
+□
+Proposition 2.4. Let I be an m-primary ideal.
+(1) Suppose k is infinite. If I is Ulrich, then it is m-full.
+(2) Suppose k is infinite. If I is integrally closed, then it is m-full.
+(3) If I is m-full, then it is full.
+Proof. (1): We can find a NZD x such that Ix = Im, so Im : x = Ix : x = I.
+(2): see [8, Theorem 2.4].
+(3): We have I ⊆ Im : m ⊆ Im : x, from which the assertion is clear.
+□
+Proposition 2.5. If I is m-full, witnessed by a NZD x ∈ m. The following are equivalent:
+(1) I is Elias.
+(2) I = xJ for some Ulrich ideal J.
+Proof. Assume I is Elias, witnessed by a NZD x, so Im : x = I.
+We will show that
+I ⊆ (x). If not, then I contains an element s whose image in R/(x) is in the socle. Thus
+sm ⊂ Im ∩ (x) = x(Im : x) = xI, so s ∈ xI : m ⊂ (x), a contradiction.
+Since I ⊆ (x) we must have I = xJ for some J. We have Jx = I = Im : x = Jxm : x =
+Jm, so J is Ulrich.
+Assume (2). Then I is Ulrich and also full by 2.4, so xI : m = mI : m = I = xJ ⊂ (x),
+thus I is Elias.
+□
+Corollary 2.6. If e = 2 and k is infinite, then I is Elias if and only if I ⊆ (x) for some
+NZD x ∈ m.
+Proof. Since e = 2, any ideal is either principal or Ulrich, and 2.4 together with 2.5 give
+what we want.
+□
+Theorem 2.7. The following hold for an m primary ideal I.
+(1) If µ(I) < e and type(R/I) ≥ e − 1, then I is Elias.
+(2) Assume µ(mI) ≤ µ(I) = e − 1. Then Im is Elias and Im : m = I.
+(3) Furthermore, assume R = S/(f) is a hypersurface, here S is a regular local ring of
+dimension 2. Let J be an S ideal minimally generated by e elements, one of them is
+f. Then JR is Elias.
+Proof. By the inequality type(I) ≥ type(R/I), we must have type(I) is e or e − 1. But if
+type(I) = e, then µ(I) = e by 2.3, contradiction.
+Next, we have:
+type(R/Im) = dimk
+Im : m
+Im
+≥ dimk
+I
+Im = µ(I) ≥ e − 1
+and Im is not Ulrich by assumption. So Im is Elias and type(Im) = e − 1, which by the
+chain above implies that Im : m = I.
+4
+
+For the last part, let I = JR. Then µR(I) = e − 1 and type(R/I) = type(S/J) = e − 1,
+and we can apply the first part.
+□
+Example 2.8. Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 − ac, c2 − a3b2). Then m2 is
+Elias: one can check directly or note that µ(m) = µ(m2) = 3 = e(R) − 1 and use 2.7. But
+m2 is not contained in (x) for any (x).
+Example 2.9. Let R = k[[t6, t7, t15]] ∼= k[[a, b, c]]/(a5 − c2, b3 − ac).
+Then the Hilbert
+function is {1, 3, 4, 5, 5, 6, . . .}, thus m4 is Elias. In this case, m4 ⊆ (a), so m4 is trivially
+Elias.
+Let R ⊂ S be a finite birational extension. We recall that the conductor of S in R,
+denoted cR(S), is R :Q(R) S.
+Proposition 2.10. Let R ⊂ S be a finite birational extension. If IS = I (i.e, I is an
+S-module) and I is Elias, then I : m ⊆ cR(S).
+Proof. Let Q = Q(R). We have R ⊃ I :Q m = IS :Q mS ⊃ (I : m)S, so I : m ⊆ R :Q S =
+cR(S) as desired.
+□
+Note that if IS = I, then trace(I) ⊆ cR(S). So naturally, one can ask to extend 2.10 as
+follows:
+Question 2.11. If I is Elias, do we have I : m ⊆ trace(I)?
+The answer is no. In Example 2.8 above, Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 −
+ac, c2 − a3b2). One can check that trace(m2) = (a2, ab, b2, c) while m2 : m = m.
+Corollary 2.12. Suppose m2 is Elias (e.g., if R has minimal multiplicity) and is integrally
+closed. If m2 ⊆ cR(R) then m ⊆ cR(R).
+Proof. Apply 2.10 to I = m2.
+□
+Corollary 2.13. Assume that the integral closure R is finite. Then the conductor of R in
+R is not Elias. A regular trace ideal is not Elias.
+Proof. Let c = cR(R). Then c is a R-module, so if it is Elias we would have c : m ⊆ c,
+absurd! Any regular trace ideal must contain c, see for instance [3], so it can not be Elias
+either by 1.3.
+□
+The following is simple but quite useful for constructing Elias ideals from minimal gener-
+ators of Ulrich ideals. See the examples that follow.
+Proposition 2.14. Let I ⊂ J be regular ideals with J Ulrich. Let x ∈ m be a minimal
+reduction of m. Assume that my ̸⊆ xI for any minimal generator of J. Then I is Elias.
+Proof. The assumption implies that xI : m ⊆ mJ = xJ ⊂ (x).
+□
+Example 2.15. Let R = k[[a1, . . . , an]]/(aiaj)1≤i<j≤n. Apply 2.14 with J = m, x = a1 +
+a2 + · · · + an. Note that each element f ∈ m has the form f = � αiasi
+i where αis are units
+or 0. Then aif = αiasi+1
+i
+and xf = � αiasi+1
+i
+. It follows easily then that the condition
+my ̸⊆ xI for any minimal generator y of m is equivalent to a2
+i /∈ xI for each i, which is
+equivalent to ai /∈ I for each i.
+For instance, if R = Q[[a, b, c]]/(ab, bc, ca), I = (a − b, b − c) is Elias.
+Since R/I =
+Q[[a]]/(a2) is Gorenstein, I is a canonical ideal.
+5
+
+One can use valuations to construct Elias ideals from part of a minimal generating set of
+some Ulrich ideal.
+Example 2.16. Let R = k[[tn, tn+1, . . . , t2n−1]]. Let I = (tn, . . . , t2n−2). Apply 2.14 with
+J = m, x = tn. Let ν be the t-adic valuation on R. Note that for any minimal generator of
+y ∈ J = m, 3n − 2 ∈ ν(ym). On the other hand 3n − 2 /∈ ν(xI), so ym ̸⊆ xI. It follows
+that I, and any ideal contained in I, is Elias. Note that again, since R/I is Gorenstein, I is
+actually a canonical ideal.
+3. Elias index
+Definition 3.1. One defines the following:
+• Let the Elias index of R, denoted by eli(R) be the smallest s such that ms is Elias.
+• Let the generalized L¨oewy length of R, denoted by gll(R), be the infimum of s such
+that ms ⊆ (x) for some x ∈ m.
+• Let the Ulrich index of R, denoted by ulr(R) be the smallest s such that ms is Ulrich,
+that is µ(ms) = e.
+Theorem 3.2. We have:
+(1) eli(R) ≤ gll(R).
+(2) gll(R) ≤ ulr(R) + 1, if the residue field k is infinite.
+(3) Suppose that the associated graded ring grm(R) is Cohen-Macaulay and the residue
+field k is infinite. Then eli(R) = gll(R) = ulr(R) + 1.
+Proof. If ms ⊆ (x) then x must be a NZD. Thus ms is Elias by 1.3. The second statement
+follows from definition. The condition that grm(R) is Cohen-Macaulay implies that ms is
+m-full for all s > 0, so the last assertion follows from 2.5.
+□
+Theorem 3.3. We have:
+(1) eli(R) = 1 if and only if R is regular.
+(2) Assume R is Gorenstein, then eli(R) = 2 if and only if e(R) = 2.
+(3) Let (A, n) be a Gorenstein local ring of dimension one. Suppose that R = n :Q(A) n
+is local. Then eli(R) ≤ 2.
+Proof. (1): Assume m is Elias.
+To show that R is regular, we can make the extension
+R → R[t](m,t) and assume k is infinite. Choose a NZD x ∈ m − m2, we have m2 : x = m,
+that is m is m-full witnessed by x. Then 2.5 shows that m ⊂ (x), thus m is principal.
+(2): We can assume again by 1.4 that k is infinite. If e = 2, then m2 ⊂ (x) for a minimal
+reduction x of m, thus m2 is Elias. Now, suppose m2 is Elias and e ≥ 3, and we need a
+contradiction. We first claim that any Ulrich ideal I of R must lie in m2. Take any minimal
+reduction x of m. Then Im = xI ⊆ (x), so I ⊂ (x) : m ⊆ (x) + m2 (otherwise the socle of
+R′ = R/xR has order 1, impossible as R′ is Gorenstein of length at least 3). As x is general,
+working inside the vector space m/m2, we see that I ⊆ m2.
+The set of m-primary Ulrich ideals in R is not empty, as it contains high enough powers
+of m. Thus, we can pick an element I in this set maximal with respect to inclusion. By the
+last claim, I ⊆ m2, and hence I is also Elias by 1.3. Now 2.4 and 2.5 imply that I = xJ for
+some NZD x ∈ m, so J is an Ulrich ideal strictly containing I, and that’s the contradiction
+we need.
+(3): If R = A, then n is Elias by 1.2, hence A is regular by part (1). Thus R is also
+regular, and eli(R) = 1. If R strictly contains A, then cA(R) = A :Q(A) R = n, hence
+6
+
+n ∼= HomA(R, A) ∼= ωR. So n is a canonical ideal of R. On the other hand, as A is not
+regular, µA(R) = 2 (dualize the exact sequence 0 → n → A → A/n → 0 and identify R with
+n∗ = HomA(n, A)). Thus ℓA(R/n) = 2, so ℓR(R/n) ≤ 2, which forces m2 ⊂ n, and since n is
+Elias, so is m2 by 1.3.
+□
+Example 3.4. We give some examples of item (3) in the previous Theorem.
+First let
+A = R[[t, it]] with i2 = −1. Then R = C[[t]].
+Next, let H = ⟨a1, . . . , an⟩ be any symmetric semigroup and b be the Frobenius number
+of H. Let A = k[[H]] be the complete Gorenstein numerical semigroup ring of H. Then
+R = k[[⟨a1, . . . , an, b⟩]] has Elias index 2, unless if H = ⟨2, 3⟩, in which case eli(R) = 1.
+Examples are R = k[[te, te+1, te2−e−1]] for e ≥ 3. For such ring we have type(R) = 2,
+e(R) = e, gll(R) = e − 1, ulr(R) = e − 1, yet eli(R) = 2. These examples show that one can
+not hope to get upper bounds for gll(R) or ulr(R) just using eli(R).
+4. Elias ideals in Gorenstein rings and Auslander index
+In this section we focus on Gorenstein rings. Throughout this section, let (R, m, k) be
+a local Gorenstein ring of dimension one and I ⊂ R an m-primary ideal. Recall that for
+a finitely generated module M, the Auslander δ invariant of M, δ(M) is defined to be the
+smallese number s such that there is a surjection Rs ⊕ N → M. The first s such that
+δ(R/ms) = 1 is called the Auslander index of R, denoted index(R).
+It turns out that Elias ideals are precisely those who quotient has Auslander invariant
+one. We collect here this fact and a few others. They are mostly known or can be deduced
+easily from results in previous sections, or both.
+Proposition 4.1. Let (R, m, k) be a local Gorenstein ring of dimension one and I ⊂ R an
+m-primary ideal. We have:
+(1) δ(R/I) = 1 if and only if I is Elias.
+(2) Suppose R is Gorenstein.
+Then I is Elias if and only if for each NZD x ∈ I,
+x ∈ m(x : I).
+(3) Suppose R is Gorenstein. For a NZD x ∈ I, x : I is Elias if and only if x ∈ mI. In
+particular, if x ∈ m2, then x : m is Elias.
+(4) I is Elias if and only if 1 ∈ mI−1, where I−1 = R :Q I. If I is Elias, then I ⊆
+m trace(I).
+Proof. Part (1) is a special case of a result by Ding, [6, Proposition 1.2] and our definition
+of Elias ideal. Part (2) and (3) are special cases of (3) and (4) of 1.3, as in that case (x) is
+isomorphic to the canonical module.
+Part (4) is [6, 2.4, 2.5], and also follows easily from results above: the first assertion is
+just a rewriting of (2). For the second assertion, it follows from the first that I ⊆ mII−1 =
+m trace(I).
+□
+There have been considerable interest in the following question:
+Question 4.2. Given an ideal I with δ(R/I) = 1, when can one say that I ⊂ (x) for some
+NZD x ∈ m?
+For instance, a conjecture of Ding asks whether index(R) = gll(R) always. From our
+point of view, this is of course just a question about Elias ideals and Elias index. Thus, one
+immediately obtains the following.
+7
+
+Corollary 4.3. Let (R, m, k) be a local Gorenstein ring of dimension one and I ⊂ R an
+m-primary ideal.
+(1) If I contains a NZD x of order 1, then I is Elias if and only if I = (x).
+(2) index(R) = eli(R).
+(3) index(R) = gll(R) = ulr(R) + 1 if k is infinite and grm(R) is Cohen-Macaulay (this
+happens for instance if R is standard graded or if R is a hypersurface).
+Proof. For part (1), we apply (5) of Corollary 1.3. Part (2) is trivial from part (1) of 4.1.
+Part (3) is [5, Theorem 2.1], [2, Corollary 2.11], and is also a consequence of 3.2.
+□
+Example 4.4. (Counter-examples to a result by Ding) In this example, we construct ex-
+amples of homogenous Elias ideals that are not inside principal ideals.
+Let S = k[[x1 . . . , xn]], and J be a homogenous ideal such that R = S/J is Gorenstein.
+Let f ∈ S be an irreducible element of degree at least 2 but lower than the initial degree of
+J, and such that the image of f in R is a NZD. Then I = fR : m is Elias by 4.1 but I is
+not inside any principal ideal. For by the irreducibility of f, we must have fR : m = (f),
+absurd.
+This class of examples contradicts Theorem 3.1 in [6], which claims that for I homogenous
+in a graded Gorenstein R, δ(R/I) = 1 (equivalently, I is Elias) if and only if I ⊆ (x) for
+some x ∈ m.
+For concrete examples, one can take S = Q[[a, b]], J = (a3 − b3), and f = a2 + b2. If one
+wants algebraically closed field, one can take S = C[[a, b, c]], J is a complete intersection of
+two general cubics, and f = a2 + b2 + c2.
+The mistake in [6, Theorem 3.1] is as follows. First, one derives that 1 = � ziyi
+xi
+with
+zi ∈ m and yi
+xi ∈ I−1 and hence there is i such that deg(ziyi) = deg(xi), which is correct.
+Then Ding claimed that there is u ∈ k such that ziyi = uxi. But this is not true. In the
+first example above we have z1 = y1 = a, z2 = y2 = b, x1 = x2 = a2 + b2.
+Example 4.5. (De Stefani’s counter-example to a conjecture of Ding, revisited) As men-
+tioned above, Ding conjectured that index(R) = gll(R) always when R is Gorenstein. De Ste-
+fani gives a clever counter-example in [2]. Let S = k[x, y, z](x,y,z), I = (x2−y5, xy2+yz3−z5).
+Then index(R) = 5 but gll(R) = 6. We now show how some parts of the proof in [2], which
+is quite involved, can be shortened using our results.
+We note that since the Hilbert functions of R are (1, 3, 5, 6, 7, 7, 8, 8...) and e(R) = 8, we
+get that m5 is Elias by Theorem 2.7. To conclude we need to show that m5 is not contained
+in (y) for any NZD y ∈ m. Note that m6 is Ulrich by Hilbert functions. We first show one
+can assume ord(y) = 1. Assume m5 ⊂ (y), m5 = yI, then m5 ∼= I. If ord(y) ≥ 2, then
+ym3 ⊂ m5 = yI, so m3 ⊂ I. But as mI ∼= m6 is Ulrich, we get m2I ⊂ (x) for some minimal
+reduction of m, thus m5 ⊂ m2I ⊂ (x). For the rest, one can follow [2].
+References
+[1] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39,
+Cambridge, Cambridge University Press, 1993.
+[2] A. De Stefani, A counterexample to conjecture of Ding, J. Algebra, 452, pp. 324–337, 2016.
+[3] H. Dao, S. Maitra, P. Sridhar, On reflexive and I-Ulrich modules over curves, arXiv:2101.02641, Trans.
+of Amer. Math. Soc., to appear.
+[4] H. Dao, T. Kobayashi and R. Takahashi, Burch ideals and Burch rings, Algebra Number Theory,
+Algebra Number Theory 14 (2020), no. 8, 2121–2150.
+8
+
+[5] S. Ding, The associated graded ring and the index of a Gorenstein local ring, Proc. Amer. Math. Soc.,
+120 (4) (1994),1029–1033.
+[6] S. Ding, Auslander’s δ-invariants of Gorenstein local rings, Proc. Amer. Math. Soc., 122 (3) (1994),
+649–656.
+[7] J. Elias, On the canonical ideals of one-dimensional Cohen-Macaulay local rings, Proc. Edinb. Math.
+Soc. (2) 59 (2016), no. 1, 77–90.
+[8] S. Goto, Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), no. 1, 151–160.
+[9] W. Heinzer, L.J Ratliff and D.E. Rush, Basically full ideals in local rings, Journal of Algebra 250
+(2002), 371–396.
+[10] C. Huneke and I. Swanson, Integral closures of ideals, rings and modules, London Math. Society Lecture
+Note Series 336, Cambridge University Press, 2006.
+[11] H-B. Foxby and A. Thorup, Minimal injective resolutions under flat base change, Proc. Amer. Math.
+Soc., 67 (1): 27–31, 1977.
+[12] J. Watanabe, m-full ideals, Nagoya Math. J. 106 (1987), 101–111.
+Hailong Dao, Department of Mathematics, University of Kansas, 405 Snow Hall, 1460
+Jayhawk Blvd., Lawrence, KS 66045
+Email address: [email protected]
+9
+

IdAyT4oBgHgl3EQfr_me/content/tmp_files/load_file.txt

@@ -0,0 +1,519 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf,len=518
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='00569v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='AC]  2 Jan 2023  ELIAS IDEALS  HAILONG DAO  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let (R, m) be a one dimensional local Cohen-Macaulay ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' An m-primary  ideal I of R is Elias if the types of I and of R/I are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Canonical and principal ideals  are Elias, and Elias ideals are closed under inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We give multiple characterizations  of Elias ideals and concrete criteria to identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We connect Elias ideals to other  well-studied definitions: Ulrich, m-full, integrally closed, trace ideals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Applications are  given regarding canonical ideals, conductors and the Auslander index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Introduction  Let (R, m) be a local Cohen-Macaulay ring of dimension one and I be an m-primary ideal  of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We say that I is Elias if the Cohen-Macaulay types of I and R/I coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' From  standard facts, principal ideals or canonical ideals are Elias, and we will soon see that this  property begets a rather rich and interesting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Our work is heavily influenced by a nice result in [7], where Elias proves that any ideal  ω that lies inside a high enough power of m and such that R/ω is Gorenstein must be a  canonical ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Although not stated explicitly there, the proof showed that any ideal that  lies in a high enough power of m is Elias, in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Another inspiration for the present  work is [2], where De Stefani studies, in our language, powers of m that are Elias in a  Gorenstein local ring, and gives a counter-example to a conjecture by Ding (see Section 4  for the precise connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  In this note, we study Elias ideals in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' They admit many different characterizations,  and enjoy rather useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For instance, they are closed under inclusion, and  principal or canonical ideals are Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' On the other hand, conductor ideals or regular trace  ideals are not Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' When R is Gorenstein, they are precisely ideals such that the Auslander  index δ(R/I) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  We are able to obtain many criteria to check whether an ideal is Elias, using very accessible  information such as the minimal number or valuations of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Combining them  immediately gives sharp bounds and information on conductor or canonical ideals, which  can be tricky to obtain otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  There are several obvious ways to extend the present definitions and results to higher  dimension rings or to modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' However, we choose to focus on the ideals in dimension one  case here as they are already interesting enough, and also to keep the paper short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We hope  to address the more general theory in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  We now describe briefly the structure and key results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  In section 1 we give the formal definition of Elias ideals and prove several key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2 contains several equivalent characterizations of Elias ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3 collects important consequences, for instance that Elias ideals are closed under  ideal containment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Also, criteria for Elias ideals using colon ideals are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Next,  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Primary: 13D02, 13H10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Secondary: 14B99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  1  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4 establishes the fundamental change of rings result that are used  frequently in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Section 2 connects Elias ideals to several well-studied class of ideals: Ulrich ideals,  m-full ideals, full ideals, integrally closed ideals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' After some basic observations,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5), we give Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='7 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='14, which contain concrete  ways to recognize Elias ideals using basic information such as number of generators  or valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We also derive that conductor ideals or regular trace ideals are not  Elias (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  This indicates one of the useful application: if we know,  for instance, that m2 is Elias, then the conductor or any regular trace ideal must  contains an element of m-adic order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Given the previous section, it is natural to study the Elias index eli(R), namely  the first power of m that is Elias, and we do so in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The first main result  here is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2, connecting this index to the generalized L¨oewy length and the  regularity of the associated graded ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Next, in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3, we characterize rings  with small indexes: eli(R) = 1 if and only if R is regular, and eli(R) = 2 plus R is  Gorenstein is equivalent to e(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We give a large class of non-Gorenstein rings  with Elias index 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Lastly, in Section 4 we focus on the special case of Gorenstein rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' In such situation,  we observe that Elias ideals are precisely ones whose quotient has Auslander δ-  invariant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' This immediately allows us to apply what we have to recover old  results about the Auslander invariant and Auslander index in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We give  a counter-example to a Theorem by Ding and also revisit a counter-example to a  conjecture by Ding given in [2] (Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Acknowledgements: It is a pleasure to thank Juan Elias and Alessandro Di Stefani for  helpful comments and encouragements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The author is partially supported by the Simons  Collaboration Grant FND0077558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Elias ideals: definitions and basic results  Throughout the paper, let (R, m, k) be Cohen-Macaulay local ring of dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For  a module M, set typeR(M) = dimk Extdim M  R  (k, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Set Q = Q(R) to be the total ring of  fractions of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Set e = e(R), the Hilbert-Samuel multiplicity of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For an element x ∈ R,  the m-adic order of x, denoted ord(x) is the smallest a such that x ∈ ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The order of an  ideal I, denoted ord(I), is the minimum order of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We say that a m-primary ideal I is an Elias ideal if it satisfies type(I) =  type(R/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We always have type(I) ≥ type(R/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (1) type(I) = type(R/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) For any NZD x ∈ m, xI : m ⊆ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3) For any NZD x ∈ m, xI : m = x(I : m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (4) For some NZD x ∈ m, xI : m ⊆ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (5) For some NZD x ∈ m, xI : m = x(I : m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (6) I :Q m ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (7) K ⊆ m(K :Q I) (assuming R admits a canonical ideal K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let x be a NZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then  type(I) = type(I/xI) = dimk  xI : m  xI  ≥ dimk  x(I : m)  xI  = dimk  I : m  I  = type(R/I)  2  Thus, type(I) = type(R/I) if and only if xI : m = x(I : m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Now, xI : m ⊆ (x) is equivalent  to xI : m = xJ for some ideal J, as x is a NZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Rewriting it as xJm ⊆ xI, which is  equivalent to Jm ⊆ I, we get J ⊆ I : m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' On the other hand x(I : m) ⊆ xI : m, thus  J = I : m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' That establishes the equivalence of first five items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Note that for any NZD x ∈ m, xI : m = x(I :Q m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thus, (6) is equivalent to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Let K be a canonical ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Apply HomR(−, K) to the sequence 0 → I → R →  R/I → 0, and indentifying HomR(I, K) with K :Q I, we get 0 → K → K :Q I →  Ext1  R(R/I, K) = ωR/I → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Since type(I) = µ(K :Q I) and type(R/I) = µ(ωR/I), the  equivalence of (7) and (1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We have:  (1) If I is isomorphic to R or the canonical module of R (assuming its existence), then  I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) If I is Elias, then so is J for any ideal J ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' (being Elias is closed under inclusion)  (3) Let K be a canonical ideal of R and I be an ideal containing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then I is Elias if  and only if K ⊆ m(K :R I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (4) Let K be a canonical ideal of R and I be an ideal such that K ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then K : I is  Elias if and only if K ⊆ mI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (5) Suppose that I contains a canonical ideal K such that ord(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then I is Elias  if and only if I = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For the first claim, I :Q m ⊂ I :Q I = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For the second claim, we have J :Q m ⊂  I :Q m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For (3), first note that K :Q I ⊂ K :Q K = R, so K :Q I = K :R I, and we can use  part (7) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  For part (4), note that K : (K : I) = I hence we can apply part (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  For part (5), we again apply part (3): if K ⊊ I, then m(K :R I) ⊆ m2, contradicting  ord(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  The following change of rings result would be used frequently in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let (R, m) → (S, n) be a local, flat rings extension such that dim S = 1  and S is Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then I is an Elias ideal of R if and only if IS is an Elias ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Under the assumption we have typeR(M) typeS/mS(S/mS) = typeS(M ⊗R S) for any  finitely generated R-module M (see for instance [11]), thus the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Elias ideals and other special ideals  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let I be an m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  I is called Ulrich (as an R-module) if µ(I) = e(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Assuming k is infinite, then I is  Ulrich if and only if xI = mI for some x ∈ m (equivalently, for any x ∈ m such that  ℓ(R/xR) = e(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  I is called m-full if Im : x = I for some x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  I is called full (or basically full) if Im : m = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' When the definition of special ideals such as Ulrich or m-full ones involves  an element x, we say that the property is witnessed by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Note that being such x is a  Zariski-open condition (for the image of x in the vector space m/m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For more on these  ideals, see [3, 10, 9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  3  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let I be an m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let e be the Hilbert-Samuel multiplicity of  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (1) I is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) type(I) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We can assume k is infinite by making the flat extension R → R[t](m,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let x ∈ m be  such that ℓ(R/xR) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then ℓ(I/xI) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Note that type(I) = ℓ(soc(I/xI)) ≤ ℓ(I/xI) =  e, and equality happens precisely when m(I/xI) = 0, in other words, I is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let I be an m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (1) Suppose k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If I is Ulrich, then it is m-full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) Suppose k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If I is integrally closed, then it is m-full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3) If I is m-full, then it is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' (1): We can find a NZD x such that Ix = Im, so Im : x = Ix : x = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2): see [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3): We have I ⊆ Im : m ⊆ Im : x, from which the assertion is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If I is m-full, witnessed by a NZD x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The following are equivalent:  (1) I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) I = xJ for some Ulrich ideal J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Assume I is Elias, witnessed by a NZD x, so Im : x = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  We will show that  I ⊆ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If not, then I contains an element s whose image in R/(x) is in the socle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thus  sm ⊂ Im ∩ (x) = x(Im : x) = xI, so s ∈ xI : m ⊂ (x), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Since I ⊆ (x) we must have I = xJ for some J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We have Jx = I = Im : x = Jxm : x =  Jm, so J is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Assume (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then I is Ulrich and also full by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4, so xI : m = mI : m = I = xJ ⊂ (x),  thus I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If e = 2 and k is infinite, then I is Elias if and only if I ⊆ (x) for some  NZD x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Since e = 2, any ideal is either principal or Ulrich, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4 together with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5 give  what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The following hold for an m primary ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (1) If µ(I) < e and type(R/I) ≥ e − 1, then I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) Assume µ(mI) ≤ µ(I) = e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then Im is Elias and Im : m = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3) Furthermore, assume R = S/(f) is a hypersurface, here S is a regular local ring of  dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let J be an S ideal minimally generated by e elements, one of them is  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then JR is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' By the inequality type(I) ≥ type(R/I), we must have type(I) is e or e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' But if  type(I) = e, then µ(I) = e by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Next, we have:  type(R/Im) = dimk  Im : m  Im  ≥ dimk  I  Im = µ(I) ≥ e − 1  and Im is not Ulrich by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' So Im is Elias and type(Im) = e − 1, which by the  chain above implies that Im : m = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  4  For the last part, let I = JR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then µR(I) = e − 1 and type(R/I) = type(S/J) = e − 1,  and we can apply the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 − ac, c2 − a3b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then m2 is  Elias: one can check directly or note that µ(m) = µ(m2) = 3 = e(R) − 1 and use 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' But  m2 is not contained in (x) for any (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let R = k[[t6, t7, t15]] ∼= k[[a, b, c]]/(a5 − c2, b3 − ac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Then the Hilbert  function is {1, 3, 4, 5, 5, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' }, thus m4 is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' In this case, m4 ⊆ (a), so m4 is trivially  Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Let R ⊂ S be a finite birational extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We recall that the conductor of S in R,  denoted cR(S), is R :Q(R) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let R ⊂ S be a finite birational extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If IS = I (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='e, I is an  S-module) and I is Elias, then I : m ⊆ cR(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let Q = Q(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We have R ⊃ I :Q m = IS :Q mS ⊃ (I : m)S, so I : m ⊆ R :Q S =  cR(S) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Note that if IS = I, then trace(I) ⊆ cR(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' So naturally, one can ask to extend 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='10 as  follows:  Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If I is Elias, do we have I : m ⊆ trace(I)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  The answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='8 above, Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 −  ac, c2 − a3b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' One can check that trace(m2) = (a2, ab, b2, c) while m2 : m = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Suppose m2 is Elias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=', if R has minimal multiplicity) and is integrally  closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If m2 ⊆ cR(R) then m ⊆ cR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Apply 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='10 to I = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Assume that the integral closure R is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then the conductor of R in  R is not Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' A regular trace ideal is not Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let c = cR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then c is a R-module, so if it is Elias we would have c : m ⊆ c,  absurd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Any regular trace ideal must contain c, see for instance [3], so it can not be Elias  either by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  The following is simple but quite useful for constructing Elias ideals from minimal gener-  ators of Ulrich ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' See the examples that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let I ⊂ J be regular ideals with J Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let x ∈ m be a minimal  reduction of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Assume that my ̸⊆ xI for any minimal generator of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The assumption implies that xI : m ⊆ mJ = xJ ⊂ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let R = k[[a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' , an]]/(aiaj)1≤i<j≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Apply 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='14 with J = m, x = a1 +  a2 + · · · + an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Note that each element f ∈ m has the form f = � αiasi  i where αis are units  or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then aif = αiasi+1  i  and xf = � αiasi+1  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' It follows easily then that the condition  my ̸⊆ xI for any minimal generator y of m is equivalent to a2  i /∈ xI for each i, which is  equivalent to ai /∈ I for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  For instance, if R = Q[[a, b, c]]/(ab, bc, ca), I = (a − b, b − c) is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Since R/I =  Q[[a]]/(a2) is Gorenstein, I is a canonical ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  5  One can use valuations to construct Elias ideals from part of a minimal generating set of  some Ulrich ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let R = k[[tn, tn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' , t2n−1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let I = (tn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' , t2n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Apply 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='14 with  J = m, x = tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let ν be the t-adic valuation on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Note that for any minimal generator of  y ∈ J = m, 3n − 2 ∈ ν(ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' On the other hand 3n − 2 /∈ ν(xI), so ym ̸⊆ xI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' It follows  that I, and any ideal contained in I, is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Note that again, since R/I is Gorenstein, I is  actually a canonical ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Elias index  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' One defines the following:  Let the Elias index of R, denoted by eli(R) be the smallest s such that ms is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Let the generalized L¨oewy length of R, denoted by gll(R), be the infimum of s such  that ms ⊆ (x) for some x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Let the Ulrich index of R, denoted by ulr(R) be the smallest s such that ms is Ulrich,  that is µ(ms) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We have:  (1) eli(R) ≤ gll(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) gll(R) ≤ ulr(R) + 1, if the residue field k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3) Suppose that the associated graded ring grm(R) is Cohen-Macaulay and the residue  field k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then eli(R) = gll(R) = ulr(R) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If ms ⊆ (x) then x must be a NZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thus ms is Elias by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The second statement  follows from definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The condition that grm(R) is Cohen-Macaulay implies that ms is  m-full for all s > 0, so the last assertion follows from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We have:  (1) eli(R) = 1 if and only if R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) Assume R is Gorenstein, then eli(R) = 2 if and only if e(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3) Let (A, n) be a Gorenstein local ring of dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Suppose that R = n :Q(A) n  is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then eli(R) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' (1): Assume m is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  To show that R is regular, we can make the extension  R → R[t](m,t) and assume k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Choose a NZD x ∈ m − m2, we have m2 : x = m,  that is m is m-full witnessed by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5 shows that m ⊂ (x), thus m is principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2): We can assume again by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4 that k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If e = 2, then m2 ⊂ (x) for a minimal  reduction x of m, thus m2 is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Now, suppose m2 is Elias and e ≥ 3, and we need a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We first claim that any Ulrich ideal I of R must lie in m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Take any minimal  reduction x of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then Im = xI ⊆ (x), so I ⊂ (x) : m ⊆ (x) + m2 (otherwise the socle of  R′ = R/xR has order 1, impossible as R′ is Gorenstein of length at least 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' As x is general,  working inside the vector space m/m2, we see that I ⊆ m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  The set of m-primary Ulrich ideals in R is not empty, as it contains high enough powers  of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thus, we can pick an element I in this set maximal with respect to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' By the  last claim, I ⊆ m2, and hence I is also Elias by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Now 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5 imply that I = xJ for  some NZD x ∈ m, so J is an Ulrich ideal strictly containing I, and that’s the contradiction  we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3): If R = A, then n is Elias by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2, hence A is regular by part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thus R is also  regular, and eli(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If R strictly contains A, then cA(R) = A :Q(A) R = n, hence  6  n ∼= HomA(R, A) ∼= ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' So n is a canonical ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' On the other hand, as A is not  regular, µA(R) = 2 (dualize the exact sequence 0 → n → A → A/n → 0 and identify R with  n∗ = HomA(n, A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thus ℓA(R/n) = 2, so ℓR(R/n) ≤ 2, which forces m2 ⊂ n, and since n is  Elias, so is m2 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We give some examples of item (3) in the previous Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  First let  A = R[[t, it]] with i2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then R = C[[t]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Next, let H = ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' , an⟩ be any symmetric semigroup and b be the Frobenius number  of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let A = k[[H]] be the complete Gorenstein numerical semigroup ring of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then  R = k[[⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' , an, b⟩]] has Elias index 2, unless if H = ⟨2, 3⟩, in which case eli(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Examples are R = k[[te, te+1, te2−e−1]] for e ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For such ring we have type(R) = 2,  e(R) = e, gll(R) = e − 1, ulr(R) = e − 1, yet eli(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' These examples show that one can  not hope to get upper bounds for gll(R) or ulr(R) just using eli(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Elias ideals in Gorenstein rings and Auslander index  In this section we focus on Gorenstein rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Throughout this section, let (R, m, k) be  a local Gorenstein ring of dimension one and I ⊂ R an m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Recall that for  a finitely generated module M, the Auslander δ invariant of M, δ(M) is defined to be the  smallese number s such that there is a surjection Rs ⊕ N → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' The first s such that  δ(R/ms) = 1 is called the Auslander index of R, denoted index(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  It turns out that Elias ideals are precisely those who quotient has Auslander invariant  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We collect here this fact and a few others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' They are mostly known or can be deduced  easily from results in previous sections, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let (R, m, k) be a local Gorenstein ring of dimension one and I ⊂ R an  m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We have:  (1) δ(R/I) = 1 if and only if I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) Suppose R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Then I is Elias if and only if for each NZD x ∈ I,  x ∈ m(x : I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3) Suppose R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For a NZD x ∈ I, x : I is Elias if and only if x ∈ mI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' In  particular, if x ∈ m2, then x : m is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (4) I is Elias if and only if 1 ∈ mI−1, where I−1 = R :Q I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If I is Elias, then I ⊆  m trace(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Part (1) is a special case of a result by Ding, [6, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2] and our definition  of Elias ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Part (2) and (3) are special cases of (3) and (4) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3, as in that case (x) is  isomorphic to the canonical module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Part (4) is [6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5], and also follows easily from results above: the first assertion is  just a rewriting of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For the second assertion, it follows from the first that I ⊆ mII−1 =  m trace(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  There have been considerable interest in the following question:  Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Given an ideal I with δ(R/I) = 1, when can one say that I ⊂ (x) for some  NZD x ∈ m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  For instance, a conjecture of Ding asks whether index(R) = gll(R) always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' From our  point of view, this is of course just a question about Elias ideals and Elias index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thus, one  immediately obtains the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  7  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let (R, m, k) be a local Gorenstein ring of dimension one and I ⊂ R an  m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (1) If I contains a NZD x of order 1, then I is Elias if and only if I = (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (2) index(R) = eli(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  (3) index(R) = gll(R) = ulr(R) + 1 if k is infinite and grm(R) is Cohen-Macaulay (this  happens for instance if R is standard graded or if R is a hypersurface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For part (1), we apply (5) of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Part (2) is trivial from part (1) of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Part (3) is [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1], [2, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='11], and is also a consequence of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  □  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' (Counter-examples to a result by Ding) In this example, we construct ex-  amples of homogenous Elias ideals that are not inside principal ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Let S = k[[x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' , xn]], and J be a homogenous ideal such that R = S/J is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Let f ∈ S be an irreducible element of degree at least 2 but lower than the initial degree of  J, and such that the image of f in R is a NZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Then I = fR : m is Elias by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1 but I is  not inside any principal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For by the irreducibility of f, we must have fR : m = (f),  absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  This class of examples contradicts Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1 in [6], which claims that for I homogenous  in a graded Gorenstein R, δ(R/I) = 1 (equivalently, I is Elias) if and only if I ⊆ (x) for  some x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  For concrete examples, one can take S = Q[[a, b]], J = (a3 − b3), and f = a2 + b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If one  wants algebraically closed field, one can take S = C[[a, b, c]], J is a complete intersection of  two general cubics, and f = a2 + b2 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  The mistake in [6, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='1] is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' First, one derives that 1 = � ziyi  xi  with  zi ∈ m and yi  xi ∈ I−1 and hence there is i such that deg(ziyi) = deg(xi), which is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Then Ding claimed that there is u ∈ k such that ziyi = uxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' But this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' In the  first example above we have z1 = y1 = a, z2 = y2 = b, x1 = x2 = a2 + b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' (De Stefani’s counter-example to a conjecture of Ding, revisited) As men-  tioned above, Ding conjectured that index(R) = gll(R) always when R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' De Ste-  fani gives a clever counter-example in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Let S = k[x, y, z](x,y,z), I = (x2−y5, xy2+yz3−z5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Then index(R) = 5 but gll(R) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We now show how some parts of the proof in [2], which  is quite involved, can be shortened using our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  We note that since the Hilbert functions of R are (1, 3, 5, 6, 7, 7, 8, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=') and e(R) = 8, we  get that m5 is Elias by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' To conclude we need to show that m5 is not contained  in (y) for any NZD y ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Note that m6 is Ulrich by Hilbert functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' We first show one  can assume ord(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Assume m5 ⊂ (y), m5 = yI, then m5 ∼= I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' If ord(y) ≥ 2, then  ym3 ⊂ m5 = yI, so m3 ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' But as mI ∼= m6 is Ulrich, we get m2I ⊂ (x) for some minimal  reduction of m, thus m5 ⊂ m2I ⊂ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' For the rest, one can follow [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  References  [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Bruns and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39,  Cambridge, Cambridge University Press, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' De Stefani, A counterexample to conjecture of Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Algebra, 452, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' 324–337, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Dao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Maitra, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Sridhar, On reflexive and I-Ulrich modules over curves, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='02641, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  of Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=', to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Dao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Kobayashi and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Takahashi, Burch ideals and Burch rings, Algebra Number Theory,  Algebra Number Theory 14 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' 8, 2121–2150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  8  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Ding, The associated graded ring and the index of a Gorenstein local ring, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=',  120 (4) (1994),1029–1033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Ding, Auslander’s δ-invariants of Gorenstein local rings, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=', 122 (3) (1994),  649–656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Elias, On the canonical ideals of one-dimensional Cohen-Macaulay local rings, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Edinb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' (2) 59 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' 1, 77–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Goto, Integral closedness of complete-intersection ideals, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Algebra 108 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' 1, 151–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Heinzer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='J Ratliff and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Rush, Basically full ideals in local rings, Journal of Algebra 250  (2002), 371–396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Huneke and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Swanson, Integral closures of ideals, rings and modules, London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Society Lecture  Note Series 336, Cambridge University Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [11] H-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Foxby and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Thorup, Minimal injective resolutions under flat base change, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=', 67 (1): 27–31, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' Watanabe, m-full ideals, Nagoya Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=' 106 (1987), 101–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='  Hailong Dao, Department of Mathematics, University of Kansas, 405 Snow Hall, 1460  Jayhawk Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content=', Lawrence, KS 66045  Email address: hdao@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}
+page_content='edu  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'}