This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in [Tro11c] 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 analogs 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, Ω(e^(-2)κ^2_l l log p) samples, where κ_l = λ_1(C) λ_l(C), are sufficient to ensure that the dominant l eigenvalues of the covariance matrix of a N(0,C) random vector are estimated to within a factor of 1 ± e with high probability.
More generally, bounds on extremal eigenvalues can be obtained from knowledge of the moments of the entries. For example, the smallest singular value of a square matrix with i.i.d. zero-mean subgaussian entries with unit variance is O( @math ) with high probability @cite_74 . Concentration of measure results, such as Talagrand's concentration inequality for product spaces @cite_19 , have also contributed greatly to the nonasymptotic theory. We mention in particular the work of Achlioptas and McSherry on randomized sparsification of matrices @cite_76 @cite_77 , that of Meckes on the norms of random matrices @cite_35 , and that of Alon, Krivelevich and Vu @cite_46 on the concentration of the largest eigenvalues of random symmetric matrices, all of which are applications of Talagrand's inequality. In cases where geometric information on the distribution of the random matrices is available, the tools of empirical process theory---such as the generic chaining, also due to Talagrand @cite_78 ---can be used to convert this geometric information into information on the spectra. One natural example of such a case consists of matrices whose rows are independently drawn from a log-concave distribution @cite_31 @cite_66 .
@cite_35 @cite_31 @cite_78 @cite_19 @cite_77 @cite_74 @cite_46 @cite_76 @cite_66
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The paper considers random matrices with independent subgaussian columns and provides a new elementary proof of the Uniform Uncertainty Principle for such matrices. The Principle was introduced by Candes, Romberg and Tao in 2004; for subgaussian random matrices it was carlier proved by the present authors, as a consequence of a general result based on a generic chaining method of Talagrand. The present proof combines a simple measure concentration and a covering argument, which are standard tools of high-dimensional convexity. It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e − t 232 s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces. [1] conjectured the existence of a sharp threshold on model parameters for community detection in sparse random graphs drawn from the stochastic block model. Mossel, Neeman and Sly [2] established the negative part of the conjecture, proving impossibility of non-trivial reconstruction below the threshold. In this work we solve the positive part of the conjecture. To that end we introduce a modified adjacency matrix B which counts self-avoiding paths of a given length e between pairs of nodes. We then prove that for logarithmic length e, the leading eigenvectors of this modified matrix provide a non-trivial reconstruction of the underlying structure, thereby settling the conjecture. A key step in the proof consists in establishing a weak Ramanujan property of the constructed matrix B. Namely, the spectrum of B consists in two leading eigenvalues ρ(B), λ2 and n -- 2 eigenvalues of a lower order O(ne √ρ(B) for all e 0, ρ(B) denoting B's spectral radius. We prove concentration inequalities for some classes of Markov chains and (F-mixing processes, with constants independent of the size of the sample, that extend the inequalities for product measures of Talagrand. The method is based on information inequalities put forward by Marton in case of contracting Markov chains. Using a simple duality argument on entropy, our results also include the family of logarithmic Sobolev inequalities for convex functions. Applications to bounds on supremum of dependent empirical processes complete this work.