Distribution Of Random Matrices Research Articles

The limiting spectral distribution of large random permutation matrices

Abstract We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let $\textbf X = (\textbf x_1,\ldots,\textbf x_n)$ $\in \mathbb{C} ^{m \times n}$ be an $m \times n$ data matrix after self-normalization (n samples and m features), where $\textbf x_j = (x_{1j}^{*},\ldots, x_{mj}^{*} )^{*}$ . Specifically, we generate a permutation matrix $\textbf X_\pi$ by permuting the entries of $\textbf x_j$ $(j=1,\ldots,n)$ and demonstrate that the empirical spectral distribution of $\textbf {B}_n = ({m}/{n})\textbf{U} _{n} \textbf{X} _\pi \textbf{X} _\pi^{*} \textbf{U} _{n}^{*}$ weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where $\textbf{U} _n$ is a sequence of $p \times m$ non-random complex matrices. The conditions we require are $p/n \to c >0$ and $m/n \to \gamma > 0$ .

Stieltjes Transforms and R-Transforms Associated with Two-Parameter Lambert-Tsallis Functions.

In this paper, we study a two-parameter family of Stieltjes transformations related to holomorphic Lambert-Tsallis functions, which are a two-parameter generalization of the Lambert function. Such Stieltjes transformations appear in the study of eigenvalue distributions of random matrices associated with some growing statistically sparse models. A necessary and sufficient condition on the parameters is given for the corresponding functions being Stieltjes transformations of probabilistic measures. We also give an explicit formula of the corresponding R-transformations.

Singular value distribution of dense random matrices with block Markovian dependence

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is Θ(n2) with n the size of the state space.The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependent entries called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to the singular value distributions associated with the block Markov chain.

Robust H2 Analysis of Discrete-Time Linear Systems Characterized by Random Polytopes and Time-Varying Parameters

This paper deals with discrete-time linear stochastic systems characterized by random polytopes and time-varying parameters. Random polytopes and time-varying parameters are introduced so that they can represent a sort of temporal variations in the distributions of coefficient random matrices of the systems. An H2 norm is defined for associated stochastic systems, and H2 performance analysis is discussed. In particular, we derive a numerically tractable linear matrix inequality condition for such analysis, as an extension of our earlier results about H2 control for stochastic systems without random polytopes.

On Cramér–von Mises statistic for the spectral distribution of random matrices

Let FN and F be the empirical and limiting spectral distributions of an N×N Wigner matrix. The Cramér–von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between FN and F in L2-norm. In this paper, we consider a mesoscopic approximation of the CvM statistic for Wigner matrices, and derive its limiting distribution. In the Appendix, we also give the limiting distribution of the CvM statistic (without approximation) for the toy model CUE.

Eigenvalue distribution of large random matrices arising in deep neural networks: Orthogonal case

This paper deals with the distribution of singular values of the input–output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem has been considered in the several recent studies of the field for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Based on a free probability argument, it was claimed in those papers that, in the limit of infinite width (matrix size), the singular value distribution of the Jacobian coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. In this paper, we justify the claim for random Haar distributed weight matrices and Gaussian biases. This, in particular, justifies the validity of the mean field approximation in the infinite width limit for the deep untrained neural networks and extends the macroscopic universality of random matrix theory to this new class of random matrices.

Spectral distribution of random matrices from mutually unbiased bases

<p style='text-indent:20px;'>We consider the random matrix obtained by picking vectors randomly from a large collection of mutually unbiased bases of <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{C}^n $\end{document}</tex-math></inline-formula>, and prove that the spectral distribution converges to the Marchenko-Pastur law. This shows that vectors in mutually unbiased bases behave like random vectors. This phenomenon is similar to that of binary linear codes of dual distance at least 5 ([<xref ref-type="bibr" rid="b33">33</xref>]), which was studied in previous work.</p>

Kernel Density Estimates in a Non-standard Situation

Kernel density estimate is an integral part of the statistical tool box. It has been widely studied and is very well understood in situations where the observations $$\{x_i\}$$ are i.i.d., or is a stationary process with some weak dependence. However, there are situations where these conditions do not hold. For instance, while the eigenvalue distribution of large-dimensional random matrices converges, the eigenvalues themselves are highly correlated for most common random matrix models. Suppose $$\{F_n\}$$ is a sequence of empirical distribution functions (usually random) which converges weakly to a non-random distribution function F with density f in some probabilistic sense. We show that under mild conditions on the kernel K and the limit density f, the kernel density estimate $$\hat{f}$$ based on $$F_n$$ converges to f in suitable probabilistic senses. This demonstrates the robustness of the kernel density estimate. We show how the rate of convergence of $$\hat{f}$$ to f can be linked to the rate of convergence of $$F_n$$ and $${\mathrm{E}}(F_n)$$ to F. Using the above general results, we establish the consistency of the kernel density estimates, including upper bounds on the rate of convergence, for two popular random matrix models. We also provide a few simulations to demonstrate these results and conclude with a few open questions.

Extreme-value statistics of stochastic transport processes

We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. For one-dimensional biased random walks, we derive exact results which tighten bounds for entropy production extrema obtained with martingale theory and reveal a symmetry between the distribution of the maxima and minima of entropy production. Furthermore, we show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Marčenko–Pastur distribution of random-matrix theory. Using this result, we obtain efficient estimates for the extreme-value statistics of stochastic transport processes from the eigenvalue distributions of suitable Wishart and Laguerre random matrices. We confirm our results with numerical simulations of stochastic models of molecular motors.

Convergence Rate of Empirical Spectral Distribution of Random Matrices From Linear Codes

It is known that the empirical spectral distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this paper, we prove that the convergence in probability is at least of the order $n^{-1/4}$ where $n$ is the length of the code.

Rate of convergence to the Circular Law via smoothing inequalities for log-potentials

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

The almost sure semicircle law for random band matrices with dependent entries

We analyze the empirical spectral distribution of random periodic band matrices with correlated entries. The correlation structure we study was first introduced in Hochstättler et al. (2015) by Hochstättler, Kirsch and Warzel, who named their setup almost uncorrelated and showed convergence to the semicircle distribution in probability. We strengthen their results which turn out to be also valid almost surely. Moreover, we extend them to band matrices. Sufficient conditions for convergence to the semicircle law both in probability and almost surely are provided. In contrast to convergence in probability, almost sure convergence seems to require a minimal growth rate for the bandwidth in the correlated case. Examples that fit our general setup include Curie–Weiss distributed, correlated Gaussian, and as a special case, independent entries.

Traffic Distributions and Independence: Permutation Invariant Random Matrices and the Three Notions of Independence

Voiculescu’s notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability. We first establish this construction for random matrices. We define the traffic distribution of random matrices, which is richer than the ∗ ^* -distribution of free probability. The knowledge of the individual traffic distributions of independent permutation invariant families of matrices is sufficient to compute the limiting distribution of the joint family. Under a factorization assumption, we call traffic independence the asymptotic rule that plays the role of independence with respect to traffic distributions. Wigner matrices, Haar unitary matrices and uniform permutation matrices converge in traffic distributions, a fact which yields new results on the limiting ∗ ^* -distributions of several matrices we can construct from them. Then we define the abstract traffic spaces as non-commutative probability spaces with more structure. We prove that at an algebraic level, traffic independence in some sense unifies the three canonical notions of tensor, free and Boolean independence. A central limiting theorem is stated in this context, interpolating between the tensor, free and Boolean central limit theorems.

Limiting spectral distribution of large dimensional random matrices of linear processes

The limiting spectral distribution (LSD) of large sample radom matrices is derived under dependence conditions. We consider the matrices \(X_{N}T_{N}X_{N}^{\prime}\) , where \(X_{N}\) is a matrix (\(N \times n(N)\)) where the column vectors are modeled as linear processes, and \(T_{N}\) is a real symmetric matrix whose LSD exists. Under some conditions we show that, the LSD of \(X_{N}T_{N}X_{N}^{\prime}\) exists almost surely, as \(N \rightarrow \infty\) and \(n(N)/N \rightarrow c > 0\). Numerical simulations are also provided with the intention to study the convergence of the empirical density estimator of the spectral density of \(X_{N}T_{N}X_{N}^{\prime}\).

A result on the limiting spectral distribution of random matrices with unequal variance entries

A classical result in random matrix theory reveals that the limiting spectral distribution of a Wigner matrix whose entries have a common variance and satisfy other regular assumptions almost surely converges to the semicircular law. In the paper, we will relax the assumption of uniform covariance of each entry, when the average of the normalized sums of the variances in each row of the data matrix converges to a constant, we prove that the same limiting spectral distribution holds. A similar result on a sample covariance matrix is also established. The proofs mainly depend on the Stein equation and the generalized Stein equation of independent random variables.

Identifiability of parametric random matrix models

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

Estimation of Deviation for Random Covariance Matrices

We give a deviation estimate for the empirical spectral distribution of random covariance matrices whose entries are independent random variables with mean 0, variance 1, and controlled fourth moments. We also give some new properties of Laguerre polynomials.

MIXANDMIX: numerical techniques for the computation of empirical spectral distributions of population mixtures

The MIXANDMIX (mixtures by Anderson mixing) tool for the computation of the empirical spectral distribution of random matrices generated by mixtures of populations is described. Within the population mixture model the mapping between the population distributions and the limiting spectral distribution can be obtained by solving a set of systems of non-linear equations, for which an efficient implementation is provided. The contributions include a method for accelerated fixed point convergence, a homotopy continuation strategy to prevent convergence to non-admissible solutions, a blind non-uniform grid construction for effective distribution support detection and approximation, and a parallel computing architecture. Comparisons are performed with available packages for the single population case and with results obtained by simulation for the more general model implemented here. Results show competitive performance and improved flexibility.

Matrix Optimization on Universal Unitary Photonic Devices

Universal unitary photonic devices can apply arbitrary unitary transformations to a vector of input modes and provide a promising hardware platform for fast and energy-efficient machine learning using light. We simulate the gradient-based optimization of random unitary matrices on universal photonic devices composed of imperfect tunable interferometers. If device components are initialized uniform-randomly, the locally-interacting nature of the mesh components biases the optimization search space towards banded unitary matrices, limiting convergence to random unitary matrices. We detail a procedure for initializing the device by sampling from the distribution of random unitary matrices and show that this greatly improves convergence speed. We also explore mesh architecture improvements such as adding extra tunable beamsplitters or permuting waveguide layers to further improve the training speed and scalability of these devices.

Spectrum Concentration in Deep Residual Learning: A Free Probability Approach

We revisit the weight initialization of deep residual networks (ResNets) by introducing a novel analytical tool in free probability to the community of deep learning. This tool deals with the limiting spectral distribution of non-Hermitian random matrices, rather than their conventional Hermitian counterparts in the literature. This new tool enables us to evaluate the singular value spectrum of the input-output Jacobian of a fully connected deep ResNet in both linear and nonlinear cases. With the powerful tool of free probability, we conduct an asymptotic analysis of the (limiting) spectrum on the single-layer case, and then extend this analysis to the multi-layer case of an arbitrary number of layers. The asymptotic analysis illustrates the necessity and university of rescaling the classical random initialization by the number of residual units L, so that the squared singular value of the associated Jacobian remains of order O(1), when compared with the large width and depth of the network. We empirically demonstrate that the proposed initialization scheme learns at a speed of orders of magnitudes faster than the classical ones, and thus attests a strong practical relevance of this investigation.