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

Abstract

We show the existence of a net near the sphere, such that the values of any matrix on the sphere and on the net are compared via a regularized Hilbert-Schmidt norm, which we introduce. This allows to construct an efficient net which controls the length of Ax for any random matrix A with independent columns (no other assumptions are required).

As a consequence we show that the smallest singular value σ_n (A) of an N × n random matrix A with i.i.d. mean zero, variance one entries enjoys the following small ball estimate, for any ϵ > 0

$$P({\sigma _n}(A) < \epsilon (\sqrt {N + 1} - \sqrt n )) \le {(C\epsilon \,\log \,1/\epsilon )^{N - n + 1}} + {e^{ - cN}}.$$

The proof of this result requires working with matrices whose rows are not independent, and, therefore, the fact that the theorem about discretization works for matrices with dependent rows, is crucial.

Furthermore, in the case of the square n×n matrix A with independent entries having concentration function separated from 1, i.i.d. rows, and such that $\mathbb{E}\left\| A \right\|_{HS}^2 \le c{n^2}$, one has

$$P({\sigma _n}(A) < {\epsilon \over {\sqrt n }} \le C\epsilon + {e^{ - cn}},$$

for any ϵ > 0. In addition, for $\epsilon > {c \over {\sqrt n }}$ the assumption of i.i.d. rows is not required. Our estimates generalize the previous results of Rudelson and Vershynin [29], [30], which required the sub-gaussian mean zero variance one assumptions, as well as the work of Rebrova and Tikhomirov [25], where mean zero variance 1 and i.i.d. entries were required.