LOWER BOUNDS FOR THE SMALLEST SINGULAR VALUE OF STRUCTURED RANDOM MATRICES

• 成果类型: Article
• 署名作者: Cook, Nicholas
• 署名单位: California State University System; California State University Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1251
• 发表日期: 2018
• 页码: 3442-3500
• 关键词: sample covariance-matrix; circular law; LARGEST EIGENVALUE; condition numbers; INVERTIBILITY; geometry; inverse; SPACES; NORM

摘要

We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider n x n matrices with complex entries of the form M = A o X + B = (a(ij) xi(ij) + b(ij)), where X = (xi(ij)) has i.i.d. centered entries of unit variance and A and B are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of M for the case that A has bounded (possibly zero) entries, and B = Z root n where Z is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations B under additional hypotheses on A, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemeredi's regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.