The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding

Abstract

We are concerned with the small ball behavior of the smallest singular value of random matrices. Often, establishing such results involves, in some capacity, a discretization of the unit sphere. This requires bounds on the norm of the matrix, and the latter bounds require strong assumptions on the distribution of the entries, such as bounded fourth moments (for a weak estimate), sub-gaussian tails (for a strong estimate), and structural assumptions such as mean zero and variance one. Recently, Rebrova and Tikhomirov developed a discretization procedure which does not rely on strong tail assumptions for the entries. However, their argument still required the structural assumptions of mean zero, variance one i.i.d. entries. In this paper, we discuss an efficient discretization of the unit sphere, which works with exponentially high probability, does not require any such structural assumptions, and, furthermore, does not require independence of the rows of the matrix. We show the existence of nets near the sphere, which compare values of any (deterministic) random matrix on the sphere and on the net via the regularized the Hilbert-Schmidt norm, which we introduce. Such refinement is a form of averaging, and enjoys strong large deviation properties. As a consequence we show sharp small ball estimates for the smallest singular value of square random matrices under mild assumptions, and for the random matrices with arbitrary aspect ratio.