Tail bounds for all eigenvalues of a sum of random matrices

This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in "User-friendly tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both upper and lower bounds on each eigenvalue of a sum of random self-adjoint matrices. This machinery is used to derive eigenvalue analogues of the classical Chernoff, Bennett, and Bernstein bounds. Two examples demonstrate the efficacy of the minimax Laplace transform. The first concerns the effects of column sparsification on the spectrum of a matrix with orthonormal rows. Here, the behavior of the singular values can be described in terms of coherence-like quantities. The second example addresses the question of relative accuracy in the estimation of eigenvalues of the covariance matrix of a random process. Standard results on the convergence of sample covariance matrices provide bounds on the number of samples needed to obtain relative accuracy in the spectral norm, but these results only guarantee relative accuracy in the estimate of the maximum eigenvalue. The minimax Laplace transform argument establishes that if the lowest eigenvalues decay sufficiently fast, on the order of (K^2*r*log(p))/eps^2 samples, where K is the condition number of an optimal rank-r approximation to C, are sufficient to ensure that the dominant r eigenvalues of the covariance matrix of a N(0, C) random vector are estimated to within a factor of 1+-eps with high probability.Comment: 20 pages, 1 figure, see also arXiv:1004.4389v

Cooperative Spectrum Sensing based on the Limiting Eigenvalue Ratio Distribution in Wishart Matrices

Recent advances in random matrix theory have spurred the adoption of eigenvalue-based detection techniques for cooperative spectrum sensing in cognitive radio. Most of such techniques use the ratio between the largest and the smallest eigenvalues of the received signal covariance matrix to infer the presence or absence of the primary signal. The results derived so far in this field are based on asymptotical assumptions, due to the difficulties in characterizing the exact distribution of the eigenvalues ratio. By exploiting a recent result on the limiting distribution of the smallest eigenvalue in complex Wishart matrices, in this paper we derive an expression for the limiting eigenvalue ratio distribution, which turns out to be much more accurate than the previous approximations also in the non-asymptotical region. This result is then straightforwardly applied to calculate the decision threshold as a function of a target probability of false alarm. Numerical simulations show that the proposed detection rule provides a substantial performance improvement compared to the other eigenvalue-based algorithms.Comment: 7 pages, 2 figures, submitted to IEEE Communications Letter