Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

TL;DR

This paper investigates the geometric covering properties of random ellipsoids generated by matrices with i.i.d. heavy-tailed entries and establishes bounds on the smallest singular value, extending prior subgaussian results.

Contribution

It introduces new covering bounds for random ellipsoids and generalizes invertibility results to matrices with heavy-tailed entries, beyond subgaussian assumptions.

Findings

Covering of the image of the unit ball by a small net with controlled radius.

Bound on the probability that the smallest singular value is very small.

Extension of invertibility results to heavy-tailed distributions.

Abstract

Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{E} a_{11} = 0$ and $\mathbb{E} {a_{11}}^2 = 1$. We prove that for any $\delta>0$ there is $L>0$ depending only on $\delta$, and a subset $\mathcal{N}$ of $B_2^n$ of cardinality at most $\exp(\delta n)$ such that with probability very close to one we have $$A(B_2^n)\subset \bigcup_{y\in A(\mathcal{N})}\bigl(y+L\sqrt{n}B_2^n\bigr).$$ As an application, we show that for some $L'>0$ and $u\in[0,1)$ depending only on the distribution law of $a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies $\mathbb{P}\{s_n(A)\le \varepsilon n^{-1/2}\}\le L'\varepsilon+u^n$ for all $\varepsilon>0$. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.