Dimensional Random Matrices Research Articles

Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard product

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices [Formula: see text], studied in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. [Formula: see text] (Kluwer Academic Publishers, Dordrecht, 2001)]. Here, [Formula: see text] is an [Formula: see text] random matrix consisting of independent complex standardized random variables, [Formula: see text], [Formula: see text], has nonnegative entries, and ∘ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of [Formula: see text] and [Formula: see text] which are different from those in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)], which include a Lindeberg condition on the entries of [Formula: see text], as well as a bound on the average of the rows and columns of [Formula: see text]. The present paper separates the assumptions needed on [Formula: see text] and [Formula: see text]. It assumes a Lindeberg condition on the entries of [Formula: see text], along with a tightness-like condition on the entries of [Formula: see text].

Weak convergence of a collection of random functions defined by the eigenvectors of large dimensional random matrices

For each n, let [Formula: see text] be Haar distributed on the group of [Formula: see text] unitary matrices. Let [Formula: see text] denote orthogonal nonrandom unit vectors in [Formula: see text] and let [Formula: see text], [Formula: see text]. Define the following functions on [Formula: see text]: [Formula: see text], [Formula: see text], [Formula: see text]. Then it is proven that [Formula: see text], [Formula: see text], considered as random processes in [Formula: see text], converge weakly, as [Formula: see text], to [Formula: see text] independent copies of Brownian bridge. The same result holds for the [Formula: see text] processes in the real case, where [Formula: see text] is real orthogonal Haar distributed and [Formula: see text], with [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text] replaced with [Formula: see text] and [Formula: see text], respectively. This latter result will be shown to hold for the matrix of eigenvectors of [Formula: see text] where [Formula: see text] is [Formula: see text] consisting of the entries of [Formula: see text], i.i.d. standardized and symmetrically distributed, with each [Formula: see text] and [Formula: see text] as [Formula: see text]. This result extends the result in [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix [Formula: see text] is studied where [Formula: see text] is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or [Formula: see text], [Formula: see text] nonrandom and [Formula: see text] is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to [Formula: see text] with the eigenvector associated with the largest eigenvalue of [Formula: see text]

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $\log$-H\"older continuous.

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 \(X_{N}T_{N}X_{N}^{\prime}\).

On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.

Large Deviations for Products of Random Two Dimensional Matrices

We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and require no irreducibility assumptions. In consequence, we obtain a uniform local modulus of continuity for the corresponding Lyapunov exponent regarded as a function of the support of the distribution. This in turn has consequences on the modulus of continuity of the integrated density of states and on the localization properties of random Jacobi operators.

Asymptotic performance of ZF and MMSE crosstalk cancelers for DSL systems

We present asymptotic expressions for user throughput in a multi-user digital subscriber line system (DSL) with a linear decoder, in increasingly large system sizes. This analysis can be seen as a generalization of results obtained for wireless communication. The features of the diagonal elements of the wireline DSL channel matrices make wireless asymptotic analyses inapplicable for wireline systems. Further, direct application of results from random matrix theory (RMT) yields a trivial lower bound. This paper presents a novel approach to asymptotic analysis, where an alternative sequence of systems is constructed that includes the system of interest in order to approximate the spectral efficiency of the linear zero-forcing (ZF) and minimum mean squared error (MMSE) crosstalk cancelers. Using works in the field of large dimensional random matrices, we show that the user rate in this sequence converges to a non-zero rate. The approximation of the user rate for both the ZF and MMSE cancelers are very simple to evaluate and does not need to take specific channel realizations into account. The analysis reveals the intricate behavior of the throughput as a function of the transmission power and the channel crosstalk. This unique behavior has not been observed for linear decoders in other systems. The approximation presented here is much more useful for the next generation G.fast wireline system than earlier DSL systems as previously computed performance bounds, which are strictly larger than zero only at low frequencies. We also provide a numerical performance analysis over measured and simulated DSL channels which show that the approximation is accurate even for relatively low dimensional systems and is useful for many scenarios in practical DSL systems.

Asymptotic distributions of Wishart type products of random matrices

We study asymptotic distributions of large dimensional random matrices of the form $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using free probability and combinatorics of colored labeled noncrossing partitions. These matrices are taken from the set of off-diagonal blocks of the family $\mathcal{Y}$ of independent Hermitian random matrices which are asymptotically free, asymptotically free against the family of deterministic diagonal matrices, and whose norms are uniformly bounded almost surely. This class includes unitarily invariant Hermitian random matrices with limit distributions given by compactly supported probability measures $\nu$ on the real line. We express the limit moments in terms of colored labeled noncrossing pair partitions, to which we assign weights depending on even free cumulants of $\nu$ and on asymptotic dimensions of blocks (Gaussianization). For products of $p$ independent blocks, we show that the limit moments are linear combinations of a new family of polynomials called generalized multivariate Fuss-Narayana polynomials. In turn, the product of two blocks of the same matrix leads to an example with rescaled Raney numbers.

A shortcut through the Coulomb gas method for spectral linear statistics on random matrices

In the last decade, spectral linear statistics on large dimensional random matrices have attracted significant attention. Within the physics community, a privileged role has been played by invariant matrix ensembles for which a two-dimensional Coulomb gas analogy is available. We present a critical revision of the Coulomb gas method in random matrix theory (RMT) borrowing language and tools from large deviations theory. This allows us to formalize an equivalent, but more effective and quicker route toward RMT free energy calculations. Moreover, we argue that this more modern viewpoint is likely to shed further light on the interesting issues of weak phase transitions and evaporation phenomena recently observed in RMT.

Outliers in the Single Ring Theorem

This text is about spiked models of non-Hermitian random matrices. More specifically, we consider matrices of the type \({\mathbf {A}}+{\mathbf {P}}\), where the rank of \({\mathbf {P}}\) stays bounded as the dimension goes to infinity and where the matrix \({\mathbf {A}}\) is a non-Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if \({\mathbf {P}}\) has some eigenvalues out of the maximal circle of the single ring, then \({\mathbf {A}}+{\mathbf {P}}\) has some eigenvalues (called outliers) in the neighborhood of those of \({\mathbf {P}}\), which is not the case for the eigenvalues of \({\mathbf {P}}\) in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of \({\mathbf {A}}\) around the eigenvalues of \({\mathbf {P}}\) and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such kind of fluctuations had already been shown for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of \({\mathbf {P}}\)) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in (Ann Probab 42:1980–2031, 2014) in the Hermitian case, but only for non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a result by Tao proved specifically for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.

On two simple and effective procedures for high dimensional classification of general populations

In this paper, we generalize two criteria, the determinant-based and trace-based criteria proposed by Saranadasa (1993), to general populations for high dimensional classification. These two criteria compare some distances between a new observation and several different known groups. The determinant-based criterion performs well for correlated variables by integrating the covariance structure and is competitive to many other existing rules. The criterion however requires the measurement dimension be smaller than the sample size. The trace-based criterion in contrast, is an independence rule and effective in the "large dimension-small sample size" scenario. An appealing property of these two criteria is that their implementation is straightforward and there is no need for preliminary variable selection or use of turning parameters. Their asymptotic misclassification probabilities are derived using the theory of large dimensional random matrices. Their competitive performances are illustrated by intensive Monte Carlo experiments and a real data analysis.

Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices

Let $\mathbf {x}_1,\ldots,\mathbf {x}_n$ be a random sample from a $p$-dimensional population distribution, where $p=p_n\to\infty$ and $\log p=o(n^{\beta})$ for some $0<\beta\leq1$, and let $L_n$ be the coherence of the sample correlation matrix. In this paper it is proved that $\sqrt{n/\log p}L_n\to2$ in probability if and only if $Ee^{t_0|x_{11}|^{\alpha}}<\infty$ for some $t_0>0$, where $\alpha$ satisfies $\beta=\alpha/(4-\alpha)$. Asymptotic distributions of $L_n$ are also proved under the same sufficient condition. Similar results remain valid for $m$-coherence when the variables of the population are $m$ dependent. The proofs are based on self-normalized moderate deviations, the Stein-Chen method and a newly developed randomized concentration inequality.

CONVERGENCE OF A CLASS OF TOEPLITZ TYPE MATRICES

We use the method of moments to study the spectral properties in the bulk for finite diagonal large dimensional random and non-random Toeplitz type matrices via the joint convergence of matrices in an appropriate sense. As a consequence we revisit the famous limit theorem of Szegö for non-random symmetric Toeplitz matrices.

On correlation function of high moments of large Wigner random matrices

We consider the Wigner ensemble of Hermitian n -dimensional random matrices with elements and study the asymptotic behavior of the expression in the limit such that and are the values of the order . Assuming that the random variables have a symmetric probability distribution such that all of its moments are of the sub-gaussian form, we prove that the limit of exists and does not depend on the particular values of , . The proof is based on a combination of the arguments by Ya. Sinai and A. Soshnikov with the detailed study of a moment analog of the Green's function representation of the Inverse Participation Ratio (IPR) considered for Gaussian Unitary Invariant Ensemble of random matrices (GUE).

Spectral Analysis of Large Dimensional Random Matrices, 2nd edn

Spectral Analysis of Large Dimensional Random Matrices, 2nd edn

Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels Under Limited Feedback

In this paper, we study the sum rate performance of zero-forcing (ZF) and regularized ZF (RZF) precoding in large MISO broadcast systems under the assumptions of imperfect channel state information at the transmitter and per-user channel transmit correlation. Our analysis assumes that the number of transmit antennas $M$ and the number of single-antenna users $K$ are large while their ratio remains bounded. We derive deterministic approximations of the empirical signal-to-interference plus noise ratio (SINR) at the receivers, which are tight as $M,K\to\infty$. In the course of this derivation, the per-user channel correlation model requires the development of a novel deterministic equivalent of the empirical Stieltjes transform of large dimensional random matrices with generalized variance profile. The deterministic SINR approximations enable us to solve various practical optimization problems. Under sum rate maximization, we derive (i) for RZF the optimal regularization parameter, (ii) for ZF the optimal number of users, (iii) for ZF and RZF the optimal power allocation scheme and (iv) the optimal amount of feedback in large FDD/TDD multi-user systems. Numerical simulations suggest that the deterministic approximations are accurate even for small $M,K$.

A Deterministic Equivalent for the Analysis of Correlated MIMO Multiple Access Channels

In this article, novel deterministic equivalents for the Stieltjes transform and the Shannon transform of a class of large dimensional random matrices are provided. These results are used to characterize the ergodic rate region of multiple antenna multiple access channels, when each point-to-point propagation channel is modelled according to the Kronecker model. Specifically, an approximation of all rates achieved within the ergodic rate region is derived and an approximation of the linear precoders that achieve the boundary of the rate region as well as an iterative water-filling algorithm to obtain these precoders are provided. An original feature of this work is that the proposed deterministic equivalents are proved valid even for strong correlation patterns at both communication sides. The above results are validated by Monte Carlo simulations.

Limiting spectral distribution of XX' matrices

Une des méthodes pour obtenir la limite des distributions spectrales (LSD) des grandes matrices aléatoires est la fameuse méthode des moments, basée sur la formule des traces. Son succès a été clairement établi pour différents types de matrices telles que les matrices de Wigner et les matrices de covariance. Dans un article récent, Bryc, Dembo et Jiang [Ann. Probab. 34 (2006) 1–38] ont obtenu la LSD pour des matrices de Toeplitz et de Hankel en utilisant cette méthode. Ils arrivent à estimer les traces des moments de telles matrices en séparant les différents termes par classes d’équivalence et en reliant les asymptotiques des dénombrements afférents avec les calculs de certains volumes. Bose et Sen [Electron. J. Probab. 13 (2008) 588–628] ont développé cette idée et ont donné un cadre général pour traiter de matrices symmétriques dont les entrées viennent d’une suite indépendante. Dans cet article, nous généralisons cette approche pour considérer des matrices de la form $A_{p}=\frac{1}{n}XX'$ où X est une matrice p×n avec des entrées réelles. Nous démontrons un résultat général d’existence de la LSD de telles matrices, correctement recentrées et rééchelonnées, quand p et n tendent vers l’infini de telle façon que p/n tende vers y∈(0, ∞). Par exemple, nous montrons l’existence de la LSD quand X est la matrice asymétrique de Hankel, de Toeplitz, circulante ou circulante inverse. En particulier, quand y=0, les limites correspondent à celles obtenues par Bryc, Dembo et Jiang [Ann. Probab. 34 (2006) 1–38]. Sinon, nous obtenons de nouvelles lois limites pour lesquelles aucune expression explicite n’est connue. Nous étudions ces lois par quelques simulations.

Another look at the moment method for large dimensional random matrices

The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample variance covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) establish the LSD for the random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalent classes and relating the limits of the counts to certain volume calculations. We build on their work and present a unified approach. This helps provide relatively short and easy proofs for the LSD of common matrices while at the same time providing insight into the nature of different LSD and their interrelations. By extending these methods we are also able to deal with matrices with appropriate dependent entries.

The limiting spectral distribution function of large dimensional random matrices of sample covariance type

Results on the analytic behavior of the limiting spectral distribution of matrices of sample covariance type, studied in Marcěnko and Pastur [2], are derived. Using the Stieltjes transform, it is shown that the limiting distrbution has a continuous derivative away from zero, the derivative being analytic whenever it is positive, and the behavior of it resembles the behavior of a square root function near the boundary of its support.