Optimal and algorithmic norm regularization of random matrices

Abstract

Let A be an n × n random matrix whose entries are i.i.d. with mean 0 and variance 1. We present a deterministic polynomial time algorithm which, with probability at least 1 − 2 exp(−Ω(ǫn)) in the choice of A, finds an ǫn × ǫn sub-matrix such that zeroing it out results in A with A = O( n/ǫ). Our result is optimal up to a constant factor and improves previous results of Rebrova and Vershynin, and Rebrova. We also prove an analogous result for A a symmetric n × n random matrix whose upper-diagonal entries are i.i.d. with mean 0 and variance 1.