Invertibility via distance for noncentered random matrices with continuous distributions

Abstract

Let A be an n×n random matrix with independent rows R1(A),…,Rn(A), and assume that for any i ≤ n and any three‐dimensional linear subspace the orthogonal projection of Ri(A) onto F has distribution density satisfying (x∈F) for some constant C1>0. We show that for any fixed n×n real matrix M we have urn:x-wiley:rsa:media:rsa20920:rsa20920-math-0004 where C′>0 is a universal constant. In particular, the above result holds if the rows of A are independent centered log‐concave random vectors with identity covariance matrices. Our method is free from any use of covering arguments, and is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar‐Spielman‐Teng for noncentered Gaussian matrices.

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

“…Note that Ea ij = 0, but Ea 2 ij = ∞, and therefore the result of Rebrova and Tikhomirov [25] is not applicable to estimate the smallest singular value of A. Further, the density of a ij is unbounded, and hence the result of Tikhomirov [44] (about random matrices whose entries have bounded density) is not applicable either. However, Corollary 2 asserts that…”

“…Recently, Tikhomirov [44] found a sharp small ball estimate for square random matrices whose entries have bounded density, which does not depend on moments. Further, in regards to matrices whose entries are not i.i.d., Cook [6] obtained a general estimate for "structured" random matrices.…”

Matrix anti-concentration inequalities with applications

“…Better bounds are derived under stronger conditions such as x is subgaussian [8] and that A has Ω(n) singular values which are O(n) [17]. The smoothed analysis of matrices with independent rows was conducted [43] by Tikhomirov. Farrell and Vershynin [13] studied the smoothed analysis of symmetric random matrices.…”

Untitled

“…The precise distribution of 1 .XX / was shown in [Tao and Vu 2010a] to coincide with the Gaussian case (3) under a bounded high moment condition and with an O.N c / error term; see also [Che and Lopatto 2019a; for more general ensembles. In the case of general A 0 , lower bounds on 1 .AA / in the non-Gaussian setting have been obtained in [Tao and Vu 2007;2010b], albeit not uniformly in A 0 ; see also [Cook 2018;Tikhomirov 2020] beyond the i.i.d. case.…”