Random Matrix Theory and its Applications II - IMS

Random Matrix Theory and its Applications II
(18 Jun - 15 Aug 2012)

~ Abstracts ~

Random matrix theory and Toeplitz determinants
Estelle Basor, American Institute of Mathematics, USA

Lecture 1 (Part 1 & Part 2):

Szegő Limit Theorems and Generalizations

This talk will describe the classical Szegő theorems, both the weak and strong versions. The proof of the strong version will be via an exact formula for Toeplitz determinants. It will include generalizations to other classes of operators and also describe results about canonical distributions of eigenvalues of Toeplitz matrices.

Lecture 2:

Extensions of Szegő theorems to the singular cases

This talk will focus on the extensions of the theorem to cases where the symbols are singular, in particular, on Fisher-Hartwig symbols. Outlines of proofs will be given and also results to other similar classes of operators. For example, results for Wiener-Hopf operators and Bessel operators will be presented.

Lecture 3:

Applications of the Szegő Theorems

This lecture will focus on the applications of the Szegő theorems to random matrix theory, in particular to the problem of computing the distributions of linear statistics. The classical case is for CUE, but other situations will be discussed via the extensions of the theorems.

Universality in chaotic quantum transport: the concordance between random matrix and semiclassical theories
Gregory Berkolaiko, Texas A&M University, USA

Electronic transport through chaotic quantum dots exhibits universal, system independent properties which are consistent with random matrix theory. The observable quantities can be expressed, via the semiclassical approximation, as sums over the classical scattering trajectories. Correlations between such trajectories are organized diagrammatically and have been shown to yield universal answers for some observables.

We develop a general combinatorial treatment of the semiclassical diagrams by casting them as unicellular maps (graphs embedded on surfaces) and relating them to factorizations of permutations. The expansion of transport quantities in inverse channel number corresponds to a genus expansion of the combinatorial generating function. Taking previously calculated answers (Heusler et al, 2006) for the contribution of a given diagram, we prove agreement between the semiclassical and random matrix approaches to moments of the transmission amplitudes. The proof covers all orders, all moments (including nonlinear), and systems with or without time reversal symmetry. It explains the mathematics behind the applicability of random matrix theory to chaotic quantum transport. The streamlined calculation could also pave the way for inclusion of non-universal effects.

Based on joint work with Jack Kuipers (Regensburg)

Random matrix models and their applications to statistical mechanics
Pavel Bleher, Indiana University - Purdue University Indianapolis, USA

We will review the Riemann-Hilbert approach to the large N asymptotics in random matrix models and its
applications to statistical mechanics. This will include the following topics:
(1) Random matrix models and orthogonal polynomials.
(2) The Riemann-Hilbert approach to the semiclassical asymptotic analysis of orthogonal polynomials and the Deift-Zhou nonlinear steepest descent method.
(3) The universal laws of the distribution of eigenvalues in random matrix models.
(4) The large N asymptotic behavior of the partition function and the topological expansion.
(5) Applications of random matrix models to the exact solution of the six-vertex model of statistical mechanics.

Prerequisites include basics of complex analysis, linear algebra, measure theory, and probability theory.

References:

1. P. Bleher, Lectures on Random Matrix Models. The Riemann-Hilbert Approach. In: Random Matrices, Random Processes, and Integrable Systems. CRM Series in Mathematical Physics. Ed. J. Harnad. Springer, 2011, 251-349.
2. P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, 3. Amer. Math. Soc., Providence RI, 1999.

Coherence of random matrices: limiting laws, phase transition, and statistical applications
Tony Cai, University of Pennsylvania, USA

The coherence of a random matrix, which is defined to be the largest magnitude of the correlations between the columns of the random matrix, is an important quantity for a wide range of applications including high-dimensional statistics and signal processing. In this talk I will discuss the limiting laws of the coherence of an $n\times p$ random matrix for a full range of the dimension p with a special focus on the ultra high-dimensional setting where $p$ can be much larger than $n$. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension $p$ grows as a function of $n$. Applications to statistics and compressed sensing in the ultra high-dimensional setting will also be discussed.

Around the circular law
Djalil Chafaï, Université Paris-Est Marne-la-Vallée, France

In random matrix theory, the circular law is one of the simplest results to state and hardest to prove. This talk will give an accessible state of the art on this subject, in connection with recent progresses. Some open problems will be also presented.

Introductory lectures to random matrices
Yang Chen, Imperial College London, UK

Lecture 1. Introduction to random matrices, in particular to the \beta=2 or unitary ensembles, where we introduce polynomials orthogonal to positive weights and the associated (non-standard) ladder operators.

Lecture 2. Gap Probability. A fundamental quantity that describes the probability that an interval (a, b) is free of eigenvalues, when b tends to infinity this gives the Largest eigenvalue distribution of Gaussian Unitary Ensembles described by the Tracy-Widom soft edge Law. What is remarkable the probability distribution is a particular Painleve II transcendent amogst the six discovered by Painleve and Gambier around 1905.

Lecture 3. A situation analogues to that described in Lecture 2 is the smallest eigenvalue distribution of the Unitary Laguerre Ensembles where one finds Tracy-Widom hard-edge law which is a particular Painleve III transcendent (after scaling).

Lecture 4. An application of what was described to the Shannon capacity (an entropy) of a single user Multi-Input-Multi-Output wireless communication mode; here ones a particular Painleve V.

RMT based design of massive dense networks
Merouane Debbah, SUPELEC, France

Wireless networks are inherently limited by their own interference. Therefore, a lot of research focuses on interference reduction techniques, such as mutiuser MIMO, interference alignment, interference coordination or multi-cell processing. Although these techniques might lead to considerable performance gains, it is unlikely that they will be able to meet the demand for wireless data traffic in the future. Therefore, a significant network densification, i.e., increasing the number of antennas per unit area, is inevitable. One way of densifying the network consists in cell-size shrinking, such as the deployment of femto or small cells, which comes at the cost of additional equipments and increased interference. Another much simpler, but also less explored, option is the use of massively more antennas at each base station (BS). In this talk, we will discuss the challenges of small cell versus massive MIMO networks and show how Random Matrices provides the ideal framework to model, design and optimize beyond LTE (Long Term Evolution) networks

RMT deterministic equivalent tools and applications: how complexity leads to simplicity
Merouane Debbah, SUPELEC, France

Multiple-input multiple-output (MIMO) techniques are key for advanced wireless communication systems and their theoretically predicted benefits are nowadays widely confirmed in practice. Future communication systems are consequently expected to rely even more on these technologies, e.g. via distributed antennas systems, multicell processing, virtual cells or large antenna arrays. However, with a growing system complexity, also the channel models for theoretical studies need to evolve and the effects of path loss, interference, imperfect channel state information and antenna correlation must be accounted for. Although such channel models become quickly intractable by exact analysis, random matrix theory (RMT) allows us to derive tight approximations of key performance criteria, such as the mutual information, outage probability and signal-to-noise-plus-interference-ratio (SINR) of linear detectors/precoders. These approximations are often analytically manipulable and provide insights about the most relevant system parameters. Moreover, they lend themselves to be successfully applied to optimization problems, e.g. precoder design, power allocation, optimal channel training, user scheduling, etc., which would be otherwise intractable. RMT is a very powerful and versatile tool, but it is today not yet commonly used in wireless communications research. The aim of this tutorial is, thus, to provide an application-oriented, accessible but rigorous introduction to the most important results of RMT, with a special focus on the co-called RMT deterministic equivalents.

Optimal designs, orthogonal polynomials and random matrices
Holger Dette, Ruhr-Universität Bochum, Germany

The talk explains several relations between different areas of mathematics: Mathematical statistics, random matrices and special functions. We give a careful introduction in the theory of optimal designs, which are used to improve the accuracy of statistical inference without performing additional experiments. It is demonstrated that for certain regression models orthogonal polynomials play an important role in the construction of optimal designs. In the next step these results are connected with some classical facts from random matrix theory.

In the third part of this talk we discusss some new results on special fumctions and random matrices. In particular we analyse random band matrices, which generalize the classical Gaußschen ensemble. We show that the random eigenvalues of such matrices behave similarly as the deterministic roots of matrix orthogonal polynomials with varying recurrence coefficnets. We study the asymptotic zero distribution of such polynomials and demonstrate that these results can be used to find the asymptotic proporties of the spectrum of random band matrices.

Progress in the method of ghosts and shadows for beta ensembles
Alan Edelman, Massachusetts Institute of Technology, USA

In this talk, we will recall how the threefold way of real, complex, quaternion is not very special anymore, and how one can work with a continuum of beta. We will show how this method has yielded an algorithm that is matching the statistics for singular values of an m by n ghost Gaussian by a real diagonal.

Signal reconstruction from the magnitude of subspace components
Martin Ehler, Helmholtz Center Munich, Germany

We consider signal reconstruction from the norms of subspace components. If the weighted linear subspaces form a so-called tight p-fusion frame and a cubature formula of strength 4, then even in case that p of the subspace norms are erased, we find a finite list of potential signals, one of which is the correct one. Moreover, we present a computationally feasible algorithm to determine this list. Alternatively, we use semi-definite programming and random subspaces to decrease the required number of subspace components for reconstruction.
(joint work with Prof Christine Bachoc, Bordeaux)

Deformed Laguerre weights and the amplify and forward MIMO system
Namuz Haq, Imperial College London, UK

We study the amplify and forward or the dual hop model of wireless communications. Let $M$, $N_R$ and $N_D$ be respectively the number of antennas at the source, relay and destination terminals in a dual hop MIMO communication system. We study the instantaneous signal to noise ratio at the destination terminal characterizing it as the random variable $\gamma$. It has been shown that the moment generating function of $\gamma$ may be expressed as a multiple integral, which we show to have a Hankel determinant representation, $D_n$, generated by a deformation of the classical Laguerre weight.

We apply two different methods to characterize the Hankel determinant. First, we use the theory of ladder operators of the corresponding monic orthogonal polynomials to give an exact characterization of the Hankel determinants in terms of partial differential equations that reduce to Painlevé differential
equations. In the general case, we find a PDE that reduces to Painlevé V. We also employ Coulomb fluid methods to derive new closed-form approximations for the Hankel determinants, and show that both approaches agree in the large antenna limit.

Convergence of the empirical spectral distribution of eigenvalues of Beta matrices
Jiang Hu, Northeast Normal University, China

Let B_n=X_nX_n^*(X_nX_n^*+a_nT_N)^{-1}, where X_n is a p×p matrix with independent and identically distributed random variables and T_N is a p×p Hermitian matrix which is independent of X_n. In this talk, we focus on the limiting of empirical spectral distribution function and the central limit theorem of linear eigenvalue statistics of B_n. Especially, we do not need X_nX_n^* or T_N to be reversible. Namely, we can deal with the case that p > n and (X_nX_n^* +\a_n T_N) is invertible.

The Beta matrix is well used in multivariate statistic analysis, such as when p (the dimension of the sample covariance matrices) is large, testing the hypothesis that k (k >=2) covariance matrices are equal, multivariate analysis of variance, testing independence of sets of variates, canonical correlation analysis and so on.

A Riemann-Hilbert approach to the boundary problems for linear PDEs. The elastodynamic equation in the quarter plane. A case study.
Alexander Its, Indiana University - Purdue University Indianapolis, USA

The Riemann-Hilbert method was originated in the theory of integrable nonlinear PDEs. In the 90s, the method was extended to random matrices and orthogonal polynomials, and since then it has played an important role in these areas. In the talk, we will present some recent developments in the Riemann-Hilbert approach obtained back in the PDE theory. This time, the Riemann-Hilbert techniques is applied to linear problems but in the domains which do not allow a direct separation of variables. We will focus on the solution of the boundary value problem for the elastodynamic equation in the quarter plane. We shall show that the problem is reduced to a matrix Riemann-Hilbert problem with a shift posed on a torus. This is a joint work with Elizabeth Its and Julius Kaplunov.

Painleve III and a singular linear statistics in Hermitian random matrix ensembles
Alexander Its, Indiana University - Purdue University Indianapolis, USA

We study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the Laguerre weight $x^{a}e^{-x}$ perturbed by a multiplicative factor $e^{-s/x},$ which induces an infinitely strong zero at the origin. We show that the probability density function of this linear statistics is expressed in terms of a particular solution of the third Painleve equation. We also study the large $n$ asymptotics of the polynomials and the corresponding recurrence coefficients. A special emphasis in the talk will be made to the double scaling limit, $n \to \infty, s \to 0, sn = O(1)$ when yet another third Painleve transcendence appears. This is the join work with Y. Chen.

Expository talk on Onatski et al.
Iain Johnstone, Stanford University, USA

In the rank one spiked Wishart model, it is known that, below a phase transition threshold, the largest sample eigenvalue follows the same limiting Tracy-Widom distribution as under the null hypothesis. A recent preprint by Onatski-Moreira-Hallin shows that there is information in the remaining eigenvalues that allows a likelihood ratio test with reasonable power to be made below the threshold. This informal and expository talk will attempt to describe their results.

Roy's Largest root test under rank one alternatives
Iain Johnstone, Stanford University, USA

Roy's largest root test appears in a variety of multivariate problems, including MANOVA, signal detection in noise, etc. In this work, assuming multivariate Gaussian observations, we derive a simple yet accurate approximation for the distribution of the largest eigenvalue in certain settings of "concentrated non-centrality", in which the signal or difference between groups is concentrated in a single direction. The results allow relatively simple power calculations for Roy's test. (Joint work with Boaz Nadler).

Towards a unified characterization of MIMO mutual Information distribution: an in-depth study involving painleve equations, Edgeworth and Saddlepoint methods
Matthew Mckay, Hong Kong University of Science and Technology, Hong Kong

[Note: This talk presents further progress and generalizations of the work discussed in the previous talk "Random Matrices, Painleve Transcendents and the Information Theory of MIMO Wireless Communication Systems", which focused on single-user MIMO systems with equal antenna numbers at the transmitter and receiver. Relevant background from the previous talk will be reviewed where necessary.]

Multi-input and multi-output (MIMO) technology has been one of the most influential breakthroughs in wireless communications in the last two decades. However, despite the plethora of existing work in this field, some very fundamental problems still remain unsolved, even for the simplest point-to-point MIMO communication. One of the most important is the characterization of the achievable transmitting rate over quasi-static channels, which is essential for delay constrained systems. This requires the distribution of the mutual information between the transmitter and the receiver, which is very difficult to characterize.

This work builds upon a recent result of the authors which computed the moment generating function of the MIMO mutual information exactly in terms of Painleve differential equations---a Painleve V for single-user MIMO systems, a Painleve VI for multi-user MIMO. By exploiting these key analytical tools, an in-depth characterization of the mutual information distribution is provided for sufficiently large (but finite) antenna numbers. In particular, new systematic recursive expansions for the high order cumulants are derived, providing closed-form expressions for the leading-order and finite-antenna correction terms. These results yield considerable new insight, such as providing a technical explanation as to WHY the well-known Gaussian approximation is quite robust to large SNR for the case of unequal antenna arrays, whilst it deviates strongly for equal antenna arrays. In addition, by drawing upon these new high order cumulant expansions, the Edgeworth expansion gives a refined Gaussian approximation which is shown to give a very accurate closed-form characterization of the mutual information distribution, both around the mean and for moderate deviations into the tails (where the Gaussian approximation fails remarkably). For stronger deviations where the Edgeworth expansion becomes unwieldy, saddle point methods and asymptotic integration tools are employed to establish new analytical characterizations which are shown to be very simple and accurate. Based on these results we also recover key well established properties of the tail distribution, including the so-called diversity-multiplexing tradeoff.

If time permits, some determinantal-based problems relating to the ordered eigenvalue distributions of random matrix models arising in wireless communications will also be introduced and discussed.

Random matrices, painleve transcendents and the information theory of MIMO wireless communication systems
Matthew Mckay, Hong Kong University of Science and Technology, Hong Kong

[Joint work with Yang Chen]: MIMO systems form the foundation of virtually all emerging wireless communication standards and will dominate the market in the future. We show in this talk that basic fundamental questions regarding the achievable capacity limits of such systems still remain unsolved. Specifically, the plethora of existing work focusing on MIMO capacity typically deals with obtaining (i) exact theoretical characterizations, which are usually too complicated to yield engineering insight, or (ii) asymptotic characterizations, which are often too approximate to capture the key features of the wireless system. To address this problem, this talk will introduce powerful new approaches from statistical physics which provide fundamental representations for the distributional properties of the MIMO capacity. A key feature of this analysis is a deep underlying connection which we establish with classical Painleve differential equations, obtained using the theory of random matrices and orthogonal polynomials. This yields important insight into the effect of the number of antennas and the signal to noise ratio; for example, explaining when and why Gaussian approximations are valid, which is not identifiable with previous methods. We also propose new closed-form approximations for the capacity distribution which are extremely accurate. The proposed methods are very general with potential applications to many other problems in communications and signal processing, which will be briefly discussed if time permits.

Spectral multiplication law for strictly non-hermitian random matrix models
Maciej Andrzej Nowak , Jagiellonian University, Poland

We derive a multiplication law for free non-hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define the corresponding non-hermitian S transform being a natural generalization of the Voiculescu S transform. In addition we extend the classical hermitian S transform approach to deal with the situation when the random matrix ensemble factors have vanishing mean including the case when both of them are centered. We use planar diagrammatic techniques to derive these results.

Corrections of some likelihood statistics in a high-dimensional strict factor model
Damien Passemier, Universite de Rennes 1, France

Factors models appear in many areas, such that economics or signal processing. If the factors and errors are Gaussian, a likelihood-based theory is well-known since Lawley (1940). However, these results are obtained in the classical scheme where the data dimension p is kept fixed while the sample size n tends to infinity. This point of view is no more valid for large-dimensional data, and usual statistics need to be modify. In this talk, we consider the strict factor model. First, we give the asymptotic bias of the estimator of the noise variance. Then we present a corrected LRT of the hypothesis that the factor model fits. Finally, we define a test of the equality of the norm of two vectors of factors scores.

On control of false discovery rate under dependence
Qiman Shao, Hong Kong University of Science and Technology, Hong Kong

It has been shown that the Benjamini-Hochberg method of controlling false discovery rate remains valid under various dependence structures. It is also often assumed that the p-values are known and the number of true alternative hypothesis is of the same order as the number of tests. However, this idealized assumption is hard to meet in practice because the population distribution is usually unknown and the signals in many applications may be sparse. In this talk we propose a robust control of false discovery rate under dependence. It not only allows the sparse alternatives but also is robust against the tails of the underlying distributions and the dependence structure. Only finite fourth moment of the null distribution is required to achieve asymptotic-level accuracy of large scale tests in the ultra high dimension. The method is applied to gene selection, shape analysis of brain structures and periodic patterns in gene expression data. The method also shares favorable numerical performance on both the simulated data and a real breast cancer data. To get a more accurate approximation for the null distribution, a computation efficient bootstrap procedure is also developed.
This talk is based on a joint work with Weidong Liu.

The central limit theorems for eigenvalues of random matrices
Zhonggen Su, Zhejiang University, China

There have been a lot of remarkable advances around the study of random matrices in the past decades. In this tutorial talk I shall mainly focus on the central limit theorems for eigenvalue statistics of random matrices. As examples I will talk about linear eigenvalue statistics, counting functions of eigenvalues, order statistics of eigenvalues. The CLT tell us what the asymptotic distributions is. This will be very helpful in various applications of random matrices to multivariate statistical analysis, wireless communications, financial data analysis. In addition to applications, many interesting methods and techniques have been particularly developed to establish the CLT since the eigenvalues are dependent. I will brie
y review some of methods such as moment methods, Stieltjes transform, Stein methods, comparison approaches.

Recent results on the asymmetric simple exclusion process
Craig Tracy, University of California, Davis, USA

This describes joint work with Craig A. Tracy. In the asymmetric simple exclusion process N particles are at integer sites; each particle waits exponential time, then with probability p it moves one step to the right if the site is unoccupied, otherwise it stays put; and with probability q it moves one step to the left if the site is unoccupied, otherwise it stays put. A configuration is the set of N occupied sites. We explain how we had earlier found a formula, given an initial configuration, for the probability distribution on the set of configurations at any future time. Then we explain how the argument can be extended to variations of the original model.
(Joint work with Harold Widom, University of California Santa Cruz, USA)

Some recent progresses in the theory of random matrices
Van Vu, Yale University, USA

I am going to give a survey on some recent progresses concerning the universality phenomenon in random matrix theory, which asserts that limiting spectral distributions usually do not depend on the microscopic structures of the entries of the matrix.

Hermitian matrix model with equispaced external source
Dong Wang, National University of Singapore

In this talk I will discuss the on-going research by Tom Claeys and me on Hermitian matrix model with equispaced external source. I will show how the Hermitian matrix model with equispaced external source, a special case of the external source model with general potential, is related a two dimensional vector-valued Riemann-Hilbert problem, and the solution to that RH problem.

Independence test for high dimensional data based on regularized canonical correlation coefficients
Yanrong Yang, Nanyang Technological University

This paper proposes a new statistic to test independence between two high dimensional random vectors x : p1 ×1 and y : p2 ×1. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of x and y. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when p1 and p2 are both comparable to the sample size n. As applications of the developed dependence test, various types of dependence, such as factor models, ARCH models and a general uncorrelated but dependent case etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily closed stock prices between different sections in New York Stock Exchange (NYSE) is detected by the proposed test.

On estimation of the population spectrum from large dimensional covariance matrices
Jianfeng Yao, The University of Hong Kong, Hong Kong

For large-dimensional data, sample covariance matrices significantly deviate from the population covariance matrix. For various inference problem, it is then crucial to recover population characteristics, e.g. distribution of eigenvalues of the population covariance matrix from the sample covariance matrices. First we will give a review of existing methods for estimation of this distribution. Then recent advances on this topic using contour-integral based methods or extended Stieldjes transform will be presented. In particular advantages and weakness of these methods will be discussed and compared.

In the context of time series, these methods can be applied to series of returns that are widely accepted as uncorrelated in time. The discussed methods are therefore intended to a analysis of the correlation structure within different stock prices.

On the estimation of integrated covariance matrices of high dimensional diffusion processes
Xinghua Zheng, Hong Kong University of Science and Technology, Hong Kong

We consider the estimation of integrated covariance (ICV) matrices of high dimensional diffusion processes based on high frequency observations. We start by studying the most commonly used estimator, the realized covariance (RCV) matrix. We show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate, the limiting spectral distribution (LSD) of RCV depends on the covolatility processnot only through the targeting ICV, but also on how the covolatility process varies in time. We establish a Marcenko-Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marcenko-Pastur type theorem for RCV for a class C of diffusion processes. The results explicitly demonstrate how the time variability of the covolatility process affects the LSD of RCV. We further propose an alternative estimator, the time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in class C, the TVARCV possesses the desirable property that its LSD depends solely on that of the targeting ICV through the Marcenko-Pastur equation, and hence, in particular, the TVARCV can be used to recover the empirical spectral distribution of the ICV by using existing algorithms.

Based on joint work with Yingying Li.

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