Extreme Eigenvalues Research Articles

On the ratio of extremal eigenvalues of 𝛽-Laguerre ensembles

Classical β \beta -Laguerre ensembles consist of three special matrix models taking the form X X T \mathbf {X}\mathbf {X}^T with X \mathbf {X} denoting a random matrix having i.i.d. entries being real ( β = 1 \beta =1 ), complex ( β = 2 \beta =2 ) or quaternion ( β = 4 \beta =4 ) normal distribution. It had been actually believed that no other choice of β > 0 \beta >0 (besides 1 , 2 1,2 and 4 4 ) would correspond to a matrix model X β X β T \mathbf {X}_\beta \mathbf {X}_\beta ^T which can be constructed with entries from a classical distribution until the work done by Dumitriu and Edelman in 2002. Since then the spectral properties of general β \beta -Laguerre ensembles have been extensively studied dealing with both the bulk case (involving all the eigenvalues) and the extremal case (addressing the (first few) largest and smallest eigenvalues). However, the ratio of the extremal eigenvalues (equivalently the condition number of X β \mathbf {X}_\beta ) has not been well explored in the literature. In this paper, we study such ratio in terms of large deviations.

On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: A Riemannian optimization interpretation

In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for finding eigenspaces of extreme eigenvalues without explicitly computing orthogonal vectors in each iteration. For the top p eigenvalues, the simplest orthogonalization-free method is to find the best rank-p approximation to a positive semi-definite Hermitian matrix by algorithms solving the unconstrained Burer–Monteiro formulation. We show that the nonlinear conjugate gradient method for the unconstrained Burer–Monteiro formulation is equivalent to a Riemannian conjugate gradient method on a quotient manifold with the Bures–Wasserstein metric, thus its global convergence to a stationary point can be proven. Numerical tests suggest that it is efficient for computing the largest k eigenvalues for large-scale matrices if the largest k eigenvalues are nearly distributed uniformly.

Approximating Hessian matrices using Bayesian inference: a new approach for quasi-Newton methods in stochastic optimization

Using quasi-Newton methods in stochastic optimization is not a trivial task given the difficulty of extracting curvature information from the noisy gradients. Moreover, pre-conditioning noisy gradient observations tend to amplify the noise. We propose a Bayesian approach to obtain a Hessian matrix approximation for stochastic optimization that minimizes the secant equations residue while retaining the extreme eigenvalues between a specified range. Thus, the proposed approach assists stochastic gradient descent to converge to local minima without augmenting gradient noise. We propose maximizing the log posterior using the Newton-CG method. Numerical results on a stochastic quadratic function and an ℓ 2 -regularized logistic regression problem are presented. In all the cases tested, our approach improves the convergence of stochastic gradient descent, compensating for the overhead of solving the log posterior maximization. In particular, pre-conditioning the stochastic gradient with the inverse of our Hessian approximation becomes more advantageous the larger the condition number of the problem is.

Complete signed graphs with largest maximum or smallest minimum eigenvalue

In this paper, we deal with extremal eigenvalues of the adjacency matrices of complete signed graphs. The complete signed graphs with maximal index (i.e. the largest eigenvalue) with n vertices and m≤⌊n2/4⌋ negative edges have been already determined. We address the remaining case by characterizing those with m>⌊n2/4⌋ negative edges. We also identify the unique signed graph with maximal index among complete signed graphs whose negative edges induce a tree of diameter at least d for any given d. This extends a recent result by Li, Lin, and Meng [Discrete Math. 346 (2023), 113250] who established the same result for d=2. Finally, we prove that the smallest minimum eigenvalue of complete signed graphs with n vertices whose negative edges induce a tree is −n2−1−1+O(1n).

Lyapunov exponents and extensivity of strongly coupled chaotic maps in regular graphs

In Thermodynamics and Statistical Physics, a system’s property is extensive when it grows with the system size. When it happens, the system can be decomposed into separate components, which has been done in many systems with weakly interacting components, such as for various gas models. Similarly, Ruelle conjectured 40 years ago that the Lyapunov exponents (LEs) of some sufficiently large chaotic systems are extensive, which led to study the extensivity properties of chaotic systems with strong interactions. Because of the complexities in these systems, most results achieved so far are restricted to numerical simulations. Here, we derive closed-form expressions for the LEs and entropy rate of coupled maps in finite- and infinite-sized regular graphs, according to the coupling strength, map’s chaoticity, and graph’s spectral properties. We show that this type of system has either 4 or 5 cases for the LEs, depending on the graph’s extreme Laplacian eigenvalues. These cases represent qualitatively different collective behaviours emerging in parameter space, including chaotic synchronisation (N−1 negative LEs) and incoherent chaos (N positive LEs). From the entropy rate, we show that the ring and complete graphs (nearest-neighbour and all-to-all couplings, respectively) are extensive in all parameter regions outside the chaotic synchronisation region. Although our derivations are restricted to one-dimensional maps with constant positive derivative (i.e., chaotic), our approach can be used to find LE and entropy rates for other regular graphs (such as for cyclic graphs) or be the basis for tackling small world graphs via perturbative methods.

Shell extremal eigenvalues of tridiagonal Toeplitz Matrices

The shell of a complex tridiagonal Toeplitz matrix is studied. Closed formulas for all quantities involved in its equation are presented. Necessary and sufficient conditions for a Toeplitz tridiagonal matrix to have shell extremal eigenvalues are given. Several, recently introduced, geometric quantities related to the shell are studied as measures of non-normality of these extremal eigenvalues of such matrices. These quantities are also proposed as measures of non-normality for the matrix itself.

Sequential stabilizing spline algorithm for linear systems: Eigenvalue approximation and polishing

The sequential stabilizing spline (SSS) algorithm is a remarkable algorithm for identifying linear dynamical systems. It can guarantee the stability of an estimated model by polishing the maximum modulus roots of an equation. Therefore, the root structure of an equation plays an important role in the SSS algorithm. Although the traditional power iterative method can be applied to approximate the extremal eigenvalues which have the largest modulus, it has two limitations: (1) the extremal eigenvalues must be real numbers, and (2) the structures of the extremal eigenvalues should be known a priori. In this paper, a novel power iterative method is proposed to approximate the extremal eigenvalues. Compared with the traditional power iterative method, the method in this paper can (1) determine the types of the extremal eigenvalues without prior knowledge of the matrix; (2) approximate the true values of the extremal eigenvalues regardless of their type; (3) become a worthy addition to SSS algorithm and gradient descent algorithm. Simulation examples demonstrate the effectiveness of the proposed method.

Eigenvalue asymptotic expansion for non-Hermitian tetradiagonal Toeplitz matrices with real spectrum

In this paper we consider a family of tetradiagonal (= four non-zero diagonals) Toeplitz matrices with a limiting set consisting in one analytic arc only and obtain individual asymptotic expansions for all the eigenvalues, as the matrix size goes to infinity. Additionally, we provide specific expansions for the extreme eigenvalues which are the eigenvalues approaching the extreme points of the limiting set. In contrast to previous related works, we study non-Hermitian Toeplitz matrices having non-canonical distribution and a real limiting set. The considered family does not belong to the so-called simple-loop class, nevertheless we manage to extend the theory to this case. The achieved formulas reveal the fine details of the eigenvalue structure and allow us to directly calculate high accuracy eigenvalues, even for matrices of relatively small size.

Spectrum of random d‐regular graphs up to the edge

AbstractConsider the normalized adjacency matrices of random d‐regular graphs on N vertices with fixed degree . We prove that, with probability for any , the following two properties hold as provided that : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N, that is, . (ii) All eigenvectors of random d‐regular graphs are completely delocalized.

Correction: Extreme eigenvalues of principal minors of random matrices with moment conditions

Correction: Extreme eigenvalues of principal minors of random matrices with moment conditions

Iteration Complexity of Fixed-Step Methods by Nesterov and Polyak for Convex Quadratic Functions

This note considers the momentum method by Polyak and the accelerated gradient method by Nesterov, both without line search but with fixed step length applied to strongly convex quadratic functions assuming that exact gradients are used and appropriate upper and lower bounds for the extreme eigenvalues of the Hessian matrix are known. Simple 2-d-examples show that the Euclidean distance of the iterates to the optimal solution is non-monotone. In this context, an explicit bound is derived on the number of iterations needed to guarantee a reduction of the Euclidean distance to the optimal solution by a factor ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}. For both methods, the bound is optimal up to a constant factor, it complements earlier asymptotically optimal results for the momentum method, and it establishes another link of the momentum method and Nesterov’s accelerated gradient method.

Extreme eigenvalues of principal minors of random matrices with moment conditions

Extreme eigenvalues of principal minors of random matrices with moment conditions

Local Assortativity Affects the Synchronizability of Scale-Free Network

Synchronization is critical for system-level behavior in physical, chemical, biological, and social systems. Empirical evidence has shown that the network topology strongly impacts the synchronizability of the system, and the analysis of their relationship remains an open challenge. We know that the eigenvalue distribution determines a network's synchronizability, but analytical expressions that connect network topology and all relevant eigenvalues (e.g., the extreme values) remain elusive. Here, we accurately determine its synchronizability by proposing an analytical method to estimate the extreme eigenvalues using perturbation theory. Our analytical method exposes the role that global and local topology combine to influence synchronizability. We show that the smallest nonzero eigenvalue <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lambda ^{(2)}$</tex-math></inline-formula> , which determines synchronizability, is estimated by the smallest degree augmented by the inverse degree difference in the least connected nodes. From this, we can conclude that there exists a clear negative relationship between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lambda ^{(2)}$</tex-math></inline-formula> and the local assortativity of nodes with the smallest degree value. We validate the accuracy of our framework within the setting of a scale-free network and can be driven by commonly used ordinary differential equations (e.g., 3-D Rosler dynamics or Hindmarsh–Rose neuronal circuit). From the results, we demonstrate that the synchronizability of the network can be tuned by rewiring the connections of these particular nodes while maintaining the general degree profile of the network.

Exact and efficient Lanczos method on a quantum computer

We present an algorithm that uses block encoding on a quantum computer to exactly construct a Krylov space, which can be used as the basis for the Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the classical Lanczos method has exponential cost in the system size to represent the Krylov states for quantum systems, our efficient quantum algorithm achieves this in polynomial time and memory. The construction presented is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise. This is possible because, unlike previous quantum Krylov methods, our algorithm does not require simulating real or imaginary time evolution. We provide an explicit error bound for the resulting ground state energy estimate in the presence of noise. For our method to be successful efficiently, the only requirement on the input problem is that the overlap of the initial state with the true ground state must be &amp;#x03A9;(1/poly(n)) for n qubits.

Large deviations of extremal eigenvalues of sample covariance matrices

AbstractLarge deviations of the largest and smallest eigenvalues of $\mathbf{X}\mathbf{X}^\top/n$ are studied in this note, where $\mathbf{X}_{p\times n}$ is a $p\times n$ random matrix with independent and identically distributed (i.i.d.) sub-Gaussian entries. The assumption imposed on the dimension size p and the sample size n is $p=p(n)\rightarrow\infty$ with $p(n)={\mathrm{o}}(n)$ . This study generalizes one result obtained in [3].

A smooth transition towards a Tracy–Widom distribution for the largest eigenvalue of interacting k-body fermionic embedded Gaussian ensembles

In spite of its simplicity, the central limit theorem captures one of the most outstanding phenomena in mathematical physics, that of universality. While this classical result is well understood, it is still not very clear what happens to this universal behavior when the random variables become correlated. A fruitful mathematically laboratory to investigate the rising of new universal properties is offered by the set of eigenvalues of random matrices. In this regard, a lot of work has been done using the standard random matrix ensembles and focusing on the distribution of extreme eigenvalues. In this case, the distribution of the largest or smallest eigenvalue departs from the Fisher–Tippett–Gnedenko theorem yielding the celebrated Tracy–Widom distribution. One may wonder, yet again, how robust this new universal behavior captured by the Tracy–Widom distribution is when the correlation among eigenvalues changes. Few answers have been provided to this poignant question, and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we study numerically the probability distribution for the normalized largest eigenvalue of the interacting k-body fermionic orthogonal and unitary embedded Gaussian ensembles in the diluted limit. We find a smooth transition from a slightly asymmetric Gaussian-like distribution, for small , to the Tracy–Widom distribution as , where k is the rank of the interaction and m is the number of fermions. Correlations at the edge of the spectrum are stronger for small values of and are independent of the number of particles considered. Our results indicate that subtle correlations toward the edge of the spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the dilute limit, from those expected for the standard random matrix ensembles.

Reducing the Dimensionality of SPD Matrices with Neural Networks in BCI

In brain–computer interface (BCI)-based motor imagery, the symmetric positive definite (SPD) covariance matrices of electroencephalogram (EEG) signals with discriminative information features lie on a Riemannian manifold, which is currently attracting increasing attention. Under a Riemannian manifold perspective, we propose a non-linear dimensionality reduction algorithm based on neural networks to construct a more discriminative low-dimensional SPD manifold. To this end, we design a novel non-linear shrinkage layer to modify the extreme eigenvalues of the SPD matrix properly, then combine the traditional bilinear mapping to non-linearly reduce the dimensionality of SPD matrices from manifold to manifold. Further, we build the SPD manifold network on a Siamese architecture which can learn the similarity metric from the data. Subsequently, the effective signal classification method named minimum distance to Riemannian mean (MDRM) can be implemented directly on the low-dimensional manifold. Finally, a regularization layer is proposed to perform subject-to-subject transfer by exploiting the geometric relationships of multi-subject. Numerical experiments for synthetic data and EEG signal datasets indicate the effectiveness of the proposed manifold network.

On the shell and the shell-extremal eigenvalues of a square matrix

The shell of a matrix is a cubic curve that provides interesting spectral localization results. We study its geometry in the case where the shell has a closed branch that surrounds a simple extremal eigenvalue of the matrix. Several quantities which are related to the closed branch are introduced and studied as measures of non-normality of that particular eigenvalue.

An alternating shifted inverse power method for the extremal eigenvalues of fourth-order partially symmetric tensors

Extremal M-eigenvalues of fourth-order partially symmetric tensors play an important role in the nonlinear elastic material analysis and the entanglement problem in quantum physics. In this paper, we propose an alternating shifted inverse power method for computing the extremal M-eigenvalues of fourth-order partially symmetric tensors. The proposed algorithm is simple to operate and easy to understand for convergence analysis. Numerical experiments show the effectiveness of the proposed method.

Sparse matrices: convergence of the characteristic polynomial seen from infinity

We prove that the reverse characteristic polynomial det(In−zAn) of a random n×n matrix An with iid Bernoulli(d∕n) entries converges in distribution towards the random infinite product ∏ℓ=1∞(1−zℓ)Yℓ where Yℓ are independent Poisson(dℓ∕ℓ) random variables. We show that this random function is a Poisson analog of more classical Gaussian objects such as the Gaussian holomorphic chaos. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erdős-Rényi digraphs: for every d>1, the greatest eigenvalue of An is close to d and the second greatest is smaller than d, a Ramanujan-like property for irregular digraphs. For d<1, the only non-zero eigenvalues of An converge to a Poisson multipoint process on the unit circle. Our results also extend to the semi-sparse regime where d is allowed to grow to ∞ with n, slower than no(1). We show that the reverse characteristic polynomial converges towards a more classical object written in terms of the exponential of a log-correlated real Gaussian field. In the semi-sparse regime, the empirical spectral distribution of An∕ dn converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.