Asymptotic Distribution Of Eigenvalues Research Articles

Inference on the eigenvalues of the normalized precision matrix

Recent developments in the spectral theory of Bayesian Networks has led to a need for a developed theory of estimation and inference on the eigenvalues of the normalized precision matrix, Ω. In this paper, working under conditions where n→∞ and p remains fixed, we provide multivariate normal asymptotic distributions of the sample eigenvalues of Ω under general conditions and under normal populations, a formula for second-order bias correction of these sample eigenvalues, and a Stein-type shrinkage estimator of the eigenvalues. Numerical simulations are performed which demonstrate under what generative conditions each estimation technique is most effective. When the largest eigenvalue of Ω is small the simulations show that the second order bias-corrected eigenvalue was considerably less biased than the sample eigenvalue, whereas the smallest eigenvalue was estimated with less bias using either the sample eigenvalue or the proposed shrinkage method.

Algebra preconditionings for 2D Riesz distributed‐order space‐fractional diffusion equations on convex domains

AbstractWhen dealing with the discretization of differential equations on non‐rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two‐dimensional constant coefficient distributed‐order space‐fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume‐penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume‐penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2‐level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill‐conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and ‐preconditioners and of the generalized minimal residual with split circulant and ‐preconditioners and conduct related numerical experiments.

The free field: Realization via unbounded operators and Atiyah property

Let X1,…,Xn be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the noncommutative distribution of X1,…,Xn. More precisely, X1,…,Xn generate the free skew field if and only if there exist no non-zero finite rank operators T1,…,Tn such that ∑i[Ti,Xi]=0. Sufficient conditions for this are the maximality of the free entropy dimension or the existence of a dual system of X1,…,Xn. Our general theory is not restricted to selfadjoint operators and thus does also include and recover the result of Linnell that the generators of the free group give the free skew field.We give also consequences of our result for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices. This solves in particular a conjecture of Charlesworth and Shlyakhtenko.

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups.

Semi‐classical edge states for the Robin Laplacian

Motivated by the study of high-energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two-dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative eigenvalues, and we prove that the corresponding eigenfunctions decay away from the boundary, for all dimensions.

A dynamical version of the SYK model and theq-Brownian motion

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our results for the sparse SYK random matrices resembles the formulas for higher order freeness for ordinary GUE random matrices.

Symbol-based preconditioning for Riesz distributed-order space-fractional diffusion equations

In this work, we examine the numerical solution of a 1D distributed-order space-fractional diffusion equation. Discretizing the given problem by means of an implicit finite difference scheme based on the shifted Grünwald-Letnikov formula, the resulting linear systems show a Toeplitz structure. Then, by using well-known spectral tools for Toeplitz sequences, we determine the corresponding symbol describing its asymptotic eigenvalue distribution as the matrix size diverges. The spectral analysis is performed under different assumptions with the aim of estimating the intrinsic asymptotic ill-conditioning of the involved matrices. The obtained results suggest to precondition the involved linear systems with either a Laplacian-like preconditioner or with more general τ-preconditioners. Due to the symmetric positive definite nature of the coefficient matrices, we opt for the preconditioned conjugate gradient method, and we compare the performances of our proposal with a Strang circulant alternative given in the literature.

Eigenvalue distribution of some nonlinear models of random matrices

This paper is concerned with the asymptotic empirical eigenvalue distribution of a non linear random matrix ensemble. More precisely we consider $M= \frac{1}{m} YY^*$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. The function $f$ is applied pointwise and can be seen as an activation function in (random) neural networks. We compute the asymptotic empirical distribution of this ensemble in the case where $W$ and $X$ have sub-Gaussian tails and $f$ is real analytic. This extends a previous result where the case of Gaussian matrices $W$ and $X$ is considered. We also investigate the same questions in the multi-layer case, regarding neural network applications.

Стохастичні моделі в задачах штучного інтелекту

In this paper, we consider some properties of stochastic random matrices of large dimensions under conditions of independence of matrix elements or under conditions of independence of rows (columns). The main properties of stochastic random matrices spectrum are analyzed and the result of convergence to 0 is proved of almost all eigenvalues. Also, the application of these results to clustering problems and selection of the optimal number of clusters is considered. Note that the results obtained in this work are consistent with the Marchenko - Pastur theorem on the asymptotic distribution of eigenvalues of random matrices with independent elements. The results proved in this paper can be interpreted as a law of large numbers and will be used in the study of the asymptotic behavior of the maximum.

Asymptotic joint distribution of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model

This paper studies the joint limiting behavior of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model, where the asymptotic regime is such that the dimension and sample size grow proportionally. The form of the joint limiting distribution is applied to conduct Johnson–Graybill-type tests, a family of approaches testing for signals in a statistical model. For this, higher order correction is further made, helping alleviate the impact of finite-sample bias. The proof rests on determining the joint asymptotic behavior of two classes of spectral processes, corresponding to the extreme and linear spectral statistics, respectively.

Explicit descriptions of spectral properties of Laplacians on spheres $${\mathbb {S}}^{N}\,(N\ge 1)$$: a review

In their remarkable paper, Minakshisundaram and Pleijel established by using the parametrix for the heat equation the asymptotic expansion of the heat kernel on compact Riemannian manifolds. The result has since been extensively used in the spectral analysis of the Laplace-Beltrami operator, and in particular, in proving Weyl’s law for the asymptotic distribution of eigenvalues and various direct and inverse problems in spectral geometry. However, the question of describing the explicit values of the corresponding heat trace coefficients associated with an arbitrary compact Riemannian manifold has remained an interesting task. In this paper, we review results on Minakshisundaram-Pleijel coefficients associated with the Laplacian on spheres $${\mathbb {S}}^{N}$$ ( $$N\ge 1$$ ) and other associated spectral invariants, namely, the Minakshisundaram-Pleijel zeta functions & their residues, and the zeta-regularised determinants of the Laplacian on spheres. The results reviewed deal mainly with closed-form formulae for the afore-mentioned spectral invariants and the explicit values of the first few of these spectral invariants are given.

Eigenvalues and eigenfunctions for the two dimensional Schrödinger operator with strong magnetic field

We study the eigenvalues of the two-dimensional Schrödinger operator with a large constant magnetic field perturbed by a decaying scalar potential. For each Landau level, we give the precise asymptotic distribution of eigenvalues created by the minimum, maximum and the closed energy curve of the potential. Normal form reduction, WKB construction and pseudodifferential calculus are applied to the effective Hamiltonian.

Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information

This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of the considered polynomial. For linear polynomials, we reach in the case of finite non-microstates free Fisher information the optimal Hölder exponent 23, and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy, which partially settles a conjecture of Charlesworth and Shlyakhtenko [8].We further provide a very general criterion that gives for weak approximations of measures having Hölder continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance.Finally, we combine these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws. For Gibbs laws, this extends the corresponding result obtained in [21] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [23,24] in terms of the Kolmogorov distance.

On the direct and inverse transmission eigenvalue problems for the Schrödinger operator on the half line

For the direct problem, we give the asymptotic distribution of the (real and nonreal) transmission eigenvalues for the Schrödinger operator on the half line. For the inverse problem, we prove that the potential can be uniquely determined by all transmission eigenvalues, the parameter in the boundary condition, and some nonzero value related to the potential (whereas without the certain constant γ). The reconstruction algorithms are also revealed.

Channel Capacity of Massive MIMO With Selected Multi-Stream Transmission in Spatially Correlated Fading Environments

This study analyzed the channel capacity of multiple-input multiple-output (MIMO) reception by large-scale arrays at both ends in spatially correlated fading environments. The environments considered were Rayleigh fading for representing the non-line-of-sight condition and Nakagami-Rice fading for the line-of-sight condition. The transmission characteristics of massive MIMO were evaluated using the asymptotic eigenvalue distribution of a Wishart matrix composed of the channel matrix. The transmission scheme adopted here was eigen-mode multi-stream transmission that selected L from N streams of N-by-M (M ≥ N) massive MIMO. From the analysis, we determined that the effect of spatial correlation for the case of L/N being less than approximately 20% enhances the channel capacity, unlike spatial diversity operation. In this way, we clarified the effect of spatial correlation on channel capacity of massive MIMO quantitatively through asymptotic eigenvalue analysis.

A Sturm-Liouville Type Problem with Retarded Argument Which Contains a Spectral Parameter in the Boundary Condition

The aim of this study is to investigate a discontinuous problem of the Sturm-Liouville type with a retarded argument containing a spectral parameter in the boundary condition and two additional conditions of transmission at the two discontinuity points. Also, eigenparameter dependent boundary conditions and obtains asymptotic formulas for the eigenvalues and eigenfunctions. And the spectral parameter is real. And the real valued function is continuous on that interval and it has a finite limit. The goal of this article is to obtain asymptotic formulas for eigenvalues of eigenfunctions for problem of the form : with boundary conditions and transmission conditions To this aim, first, the principal term of asymptotic distribution of eigenvalues and eigenfunctions of given problem was obtained up to but, afterwards under some additional conditions, we improve these formulas up to Thus, when the number of points of discontinuity is more than one, we see how the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem with retarded argument which contains a spectral parameter in the boundary conditions change.

Spectral analysis of hypoelliptic Višik–Ventcel’ boundary value problems

This paper is devoted to the study of a class of hypoelliptic Visik–Ventcel’ boundary value problems for second order, uniformly elliptic differential operators. Our boundary conditions are supposed to correspond to the diffusion phenomenon along the boundary, the absorption and reflection phenomena at the boundary in probability. If the absorbing boundary portion is not a trap for Markovian particles, then we can prove two existence and uniqueness theorems of the non-homogeneous Visik–Ventcel’ boundary value problem in the framework of $$L^{2}$$ Sobolev spaces. Moreover, if the absorbing boundary portion is empty, then we can prove a generation theorem of analytic semigroups for the closed realization of the uniformly elliptic differential operator associated with the hypoelliptic Visik–Ventcel’ boundary condition in the $$L^{2}$$ topology. As a by-product, this paper is the first time to prove the angular distribution of eigenvalues, the asymptotic eigenvalue distribution and the completeness of generalized eigenfunctions of the closed realization, similar to the elliptic (non-degenerate) case.

Asymptotic distribution of negative eigenvalues for three-body systems in two dimensions: Efimov effect in the antisymmetric space

The [Formula: see text]-wave resonances induce an infinite number of negative eigenvalues accumulating at the origin for the system of three identical particles in two dimensions, provided that the energy operator is restricted on the subspace of wave functions which are antisymmetric with respect to the permutations. This quantum phenomenon is called the super Efimov effect and corresponds to the Efimov effect in three dimensions. It has been predicted in physics literature [ 10 ] and a mathematical proof has been given by Gridnev [ 5 ]. In this paper, we prove this effect for a wider class of pair interactions and improve the results obtained by [ 5 ]. We do not necessarily assume that the interactions are radially symmetric, have a definite signature and fall off exponentially at infinity. The essence of the proof is put in the asymptotic analysis of the singular behavior at low energy for resolvents of the Schrödinger operators with [Formula: see text]-wave resonances at zero energy.

Spectral properties of exterior transmission problem for spherically stratified anisotropic media

ABSTRACT The exterior transmission eigenvalue problem for anisotropic media with spherical symmetry assumptions is considered. First, we discuss the existence and the asymptotic distribution of exterior transmission eigenvalues. Then, we investigate the inverse problem of determining the index of refraction from a knowledge of two sets of spectral data under appropriate conditions.

Extreme eigenvalue distributions of Jacobi ensembles: New exact representations, asymptotics and finite size corrections

Let W1 and W2 be independent n×n complex central Wishart matrices with m1 and m2 degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices (W1+W2)−1W1, which are analogous to those of F matrices W1W2−1 and those of the Jacobi unitary ensemble (JUE). Defining α1=m1−n and α2=m2−n with m1, m2≥n, we derive new exact distribution formulas in terms of (α1+α2)-dimensional matrix determinants, with entries involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with α1 and α2 fixed (i.e., under the so-called “hard-edge” scaling limit). The analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as n→∞ in terms of α1- and α2-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-n corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations and properties of Legendre polynomials, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.