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.
The literature on the existence and universality of LSDs is massive; we mention only the highlights. It is now known that the semicircular law is universal for Wigner matrices. Suppose that @math is a sequence of independent @math Wigner matrices. Grenander established that if all the moments are finite, then the ESD of @math converges weakly to the semicircular law in probability @cite_73 . Arnold showed that, assuming a finite fourth moment, the ESD almost surely converges weakly to the semicircular law @cite_40 . Around the same time, Mar c enko and Pastur determined the form of the limiting spectral distribution of sample covariance matrices @cite_25 . More recently, Tao and Vu confirmed the long-conjectured circular law hypothesis. Let @math be a sequence of independent @math matrices whose entries are i.i.d. and have unit variance. Then the ESD of @math converges weakly to the uniform measure on the unit disk, both in probability and almost surely @cite_17 .
@cite_40 @cite_73 @cite_25 @cite_17
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Given an n x n complex matrix A, let mu(A)(x, y) := 1 n vertical bar 1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1 root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1 root n A(n) - zI for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that mu(1 root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean. In the study of the spectra of power-law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law, whereas the other predicts that the eigenvalues follow a power-law distribution. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power-law graph follow the semicircle law, whereas the spectrum of the adjacency matrix of a power-law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent β, where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as, for example, randomized algorithms that involve rapidly mixing Markov chains. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 We consider @math Hermitian matrices with i.i.d. entries @math whose tail probabilities @math behave like @math for some @math and @math . We establish a large deviation principle for the empirical spectral measure of @math with speed @math with a good rate function @math that is finite only if @math is of the form @math for some probability measure @math on @math , where @math denotes the free convolution and @math is Wigner's semicircle law. We obtain explicit expressions for @math in terms of the @math th moment of @math . The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.