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12
The polynomial method for random matrices
, 2007
"... We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions a ..."
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We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the MarčenkoPastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability ” theory. We hope that the tools developed allow researchers to finally harness the power of the infinite random matrix theory.
Joint Beamforming for Multiaccess MIMO Systems with Finite Rate Feedback
"... Abstract—We consider multiaccess multipleinput multipleoutput (MIMO) systems with finite rate feedback with the aim of understanding how to efficiently employ the given feedback resource to maximize the sum rate. A joint quantization and feedback strategy is proposed: the base station selects the s ..."
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Abstract—We consider multiaccess multipleinput multipleoutput (MIMO) systems with finite rate feedback with the aim of understanding how to efficiently employ the given feedback resource to maximize the sum rate. A joint quantization and feedback strategy is proposed: the base station selects the strongest users, jointly quantizes their strongest eigenchannel vectors and broadcasts a common feedback to all the users. This joint strategy differs from an individual strategy in which quantization and feedback are performed independently across users, and it improves upon the individual strategy in the same way that vector quantization improves upon scalar quantization. To analyze the proposed strategy, the effect of user selection is described by extreme order statistics, while the effect of joint quantization is quantified through what we term “the composite Grassmann manifold”. The achievable sum rate is then estimated using random matrix theory providing an analytic benchmark for the performance. Index Terms—Beamforming, Grassmann manifold, limited feedback, MIMO, multiaccess channels.
THE STRONG ASYMPTOTIC FREENESS OF HAAR AND DETERMINISTIC MATRICES
, 2011
"... Abstract. In this paper, we are interested in sequences of qtuple of N ×N random matrices having a strong limiting distribution (i.e. given any noncommutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having ..."
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Abstract. In this paper, we are interested in sequences of qtuple of N ×N random matrices having a strong limiting distribution (i.e. given any noncommutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the qtuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in noncommutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjørnsen. We also show that aptuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz.
1 The eigenvalue distribution function
"... For an N × N matrix AN, the eigenvalue distribution function 1 (e.d.f.) F AN (x) is defined as ..."
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For an N × N matrix AN, the eigenvalue distribution function 1 (e.d.f.) F AN (x) is defined as
Certified by....................................................................
, 2006
"... The first part of the dissertation investigates the application of the theory of large random matrices to highdimensional inference problems when the samples are drawn from a multivariate normal distribution. A longstanding problem in sensor array processing is addressed by designing an estimator f ..."
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The first part of the dissertation investigates the application of the theory of large random matrices to highdimensional inference problems when the samples are drawn from a multivariate normal distribution. A longstanding problem in sensor array processing is addressed by designing an estimator for the number of signals in white noise that dramatically outperforms that proposed by Wax and Kailath. This methodology is extended to develop new parametric techniques for testing and estimation. Unlike techniques found in the literature, these exhibit robustness to highdimensionality, sample size constraints and eigenvector misspecification.
for Wireless Network Design Approved by:
"... to learn and grow not only technically but also as an individual with better discipline, more patience, and perseverance. Her support and encouragement motivated me to continue my research even when I moved out of Georgia Tech and started working before finishing my Ph.D. Also, I would like to thank ..."
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to learn and grow not only technically but also as an individual with better discipline, more patience, and perseverance. Her support and encouragement motivated me to continue my research even when I moved out of Georgia Tech and started working before finishing my Ph.D. Also, I would like to thank my committee members: Prof. Raghupathy Sivakumar, Prof. John Copeland, Prof. Henry Owen, and Prof. Alfred Andrew. I especially appreciate the efforts Professors Sivakumar and Copeland for reading this thesis. I would like to thank my lab members, Sudhanshu Gaur, Vikram Anreddy, Lu Dong, and Guillermo Acosta for making the work environment in the lab very enjoyable. I extend special thanks to Sudhanshu and Guru for technical discussions that helped me get some nice ideas for my thesis. Ms. Cordai Farrar was of immense help in distributing copies of my thesis to the committee members. My friends Masoodur Rahman and Abhinav Saxena, were always encouraging and motivating me to focus and work hard on my thesis while I was working at ArrayComm. Their encouragement certainly helped me finish my thesis sooner. Todd
THE SHANNON TRANSFORM IN RANDOM MATRIX THEORY
"... and Pastur (1967), were motivated to a large extent by their applications. In this paper we report on a new transform motivated by the application of random matrices to various problems in the information theory of noisy communication channels. The Shannon transform of a nonnegative random variable ..."
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and Pastur (1967), were motivated to a large extent by their applications. In this paper we report on a new transform motivated by the application of random matrices to various problems in the information theory of noisy communication channels. The Shannon transform of a nonnegative random variable X is defined as VX(γ) = E[log(1 + γX)]. (1) where γ is a nonnegative real number. Originally introduced in [12], its applications to random matrix theory and engineering applications have been developed in [3]. In this paper we give a summary of its main properties and applications in random matrix theory. As is well known since the work of Mar˘cenko and Pastur [4], it is rare the case that the limiting empirical distribution of the squared singular values of random matrices (whose aspect ratio converges to a constant) admit closedform expressions. However, [4] showed a very general result where the characterization of the solution is accomplished through a fixedpoint equation involving the Stieltjes transform. Also motivated by applications, [2] introduced the ηtransform which is very related to both the Stieltjes and Shannon transforms and leads to compact definitions of other transforms used in random matrix theory such as the Stransform [5]. In applications in information theory, the Shannon transform is directly of interest as it gives the capacity of