Jack W. Silverstein

A note on the largest eigenvalue of a large dimensional sample covariance matrix

Let {vij; i, J = 1, 2, ...} be a family of i.i.d. random variables with E(v114) = [infinity]. For positive integers p, n with p = p(n) and p/n --> y > 0 as n --> [infinity], let Mn = (1/n) Vn VnT , where Vn = (vij)1

Fundamental Limit of Sample Generalized Eigenvalue Based Detection of Signals in Noise Using Relatively Few Signal-Bearing and Noise-Only Samples

The detection problem in statistical signal processing can be succinctly formulated: given m (possibly) signal bearing, n -dimensional signal-plus-noise snapshot vectors (samples) and N statistically independent n-dimensional noise-only snapshot vectors, can one reliably infer the presence of a signal? This problem arises in the context of applications as diverse as radar, sonar, wireless communications, bioinformatics, and machine learning and is the critical first step in the subsequent signal parameter estimation phase. The signal detection problem can be naturally posed in terms of the sample generalized eigenvalues. The sample generalized eigenvalues correspond to the eigenvalues of the matrix formed by ?whitening? the signal-plus-noise sample covariance matrix with the noise-only sample covariance matrix. In this paper, we prove a fundamental asymptotic limit of sample generalized eigenvalue-based detection of signals in arbitrarily colored noise when there are relatively few signal bearing and noise-only samples. Specifically, we show why when the (eigen) signal-to-noise ratio (SNR) is below a critical value, that is a simple function of n , m, and N, then reliable signal detection, in an asymptotic sense, is not possible. If, however, the eigen-SNR is above this critical value then a simple, new random matrix theory-based algorithm, which we present here, will reliably detect the signal even at SNRs close to the critical value. Numerical simulations highlight the accuracy of our analytical prediction, permit us to extend our heuristic definition of the effective number of identifiable signals in colored noise and display the dramatic improvement in performance relative to the classical estimator by Zhao We discuss implications of our result for the detection of weak and/or closely spaced signals in sensor array processing, abrupt change detection in sensor networks, and clustering methodologies in machine learning.

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.

Signal detection via spectral theory of large dimensional random matrices

Results on the spectral behavior of random matrices as the dimension increases are applied to the problem of detecting the number of sources impinging on an array of sensors. A common strategy to solve this problem is to estimate the multiplicity of the smallest eigenvalue of the spatial covariance matrix R of the sensed data. Existing approaches rely on the closeness of the noise eigenvalues of sample covariance matrix to each other and, therefore, the sample size has to be quite large when the number of sources is large in order to obtain a good estimate. The theoretical analysis presented focuses on the splitting of the spectrum of sample covariance matrix into noise and signal eigenvalues. It is shown that when the number of sensors is large the number of signals can be estimated with a sample size considerably less than that required by previous approaches. >

Spectral Analysis of Networks with Random Topologies

A class of neural models is introduced in which the topology of the neural network has been generated by a controlled probability model. It is shown that the resulting linear operator has a spectral measure that converges in probability to a universal one when the size of the net tends to infinity: a law of large numbers for the spectra of such operators. The analytical treatment is accompanied by omputational experiments.

Eigen-Inference for Energy Estimation of Multiple Sources

In this paper, a new method is introduced to blindly estimate the transmit power of multiple signal sources in multiantenna fading channels, when the number of sensing devices and the number of available samples are sufficiently large compared to the number of sources. Recent advances in the field of large dimensional random matrix theory are used that result in a simple and computationally efficient consistent estimator of the power of each source. A criterion to determine the minimum number of sensors and the minimum number of samples required to achieve source separation is then introduced. Simulations are performed that corroborate the theoretical claims and show that the proposed power estimator largely outperforms alternative power inference techniques.

On the eigenvectors of large dimensional sample covariance matrices

Let {vij}, i, J = 1,2, ..., be i.i.d. random variables, and for each n let Mn = (1/s)VnVnT, where Vn = (vij), i = 1, 2, ..., n, j = 1, 2, ..., s = s(n), and n/s --> y > 0 as n --> [infinity]. Necessary and sufficient conditions are given to establish the convergence in distribution of certain random variables defined by Mn. When E(v114)

Robust Estimates of Covariance Matrices in the Large Dimensional Regime

This paper studies the limiting behavior of a class of robust population covariance matrix estimators, originally due to Maronna in 1976, in the regime where both the number of available samples and the population size grow large. Using tools from random matrix theory, we prove that, for sample vectors made of independent entries having some moment conditions, the difference between the sample covariance matrix and (a scaled version of) such robust estimator tends to zero in spectral norm, almost surely. This result can be applied to various statistical methods arising from random matrix theory that can be made robust without altering their first order behavior.

On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix

Let {wij}, i, J = 1, 2, ..., be i.i.d. random variables and for each n let Mn = (1/n) WnWnT, where Wn = (wij), i = 1, 2, ..., p; j = 1, 2, ..., n; p = p(n), and p/n --> y > 0 as n --> [infinity]. The weak behavior of the largest eigenvalue of Mn is studied. The primary aim of the paper is to show that the largest eigenvalue converges in probability to a nonrandom quantity if and only if E(w11) = 0 and n4P([omega]11 >= n) = o(1), the limit being (1 + [radical sign]y)2 E(w112).

Gaussian fluctuations for non-Hermitian random matrix ensembles

Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of