Constructive Regularization of the Random Matrix Norm

Abstract

We show a simple local norm regularization algorithm that works with high probability. Namely, we prove that if the entries of a n × n matrix A are i.i.d. symmetrically distributed and have finite second moment, it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms in order to bring the operator norm of A to the almost optimal order O( √ log log n · n). As a corollary, we also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same goal.This work is a natural continuation of our recent work with R. Vershynin, where we have shown that the norm of A can be reduced to the optimal order O( √ n) by zeroing out just a small submatrix of A, but did not provide a constructive procedure to find this small submatrix.Our current approach extends the norm regularization techniques developed for the graph adjacency (Bernoulli) matrices in the works of Feige and Ofek, and Le, Levina and Vershynin to the considerably broader class of matrices.√ n with high probability. If we are concerned to get an explicit (non-asymptotic) probability estimate for all large enough n, an application of Bernstein's inequality (see, for example, in [18,19]) gives P{ A ≤ t √ n} ≥ 1 − e −c 0 t 2 n for t ≥ C 0 Affiliation:

Outliers in spectrum of sparse Wigner matrices

“…It seems interesting to understand if such a connection could be elaborated. We remark that there has been several works recently concerned with regularizations of random graphs/matrices, that is, procedures designed to reduce the norm of random matrices by changing a few of its entries (see [20,33,42,43]).…”

Sparse random tensors: Concentration, regularization and applications

“…A different regularization analysis was given in [40] by decomposing the adjacency matrix into several parts and modify a small submatrix. This method was later generalized to other random matrices in [52,51]. In this section we consider the regularization for Bernoulli random tensors.…”

Optimal and algorithmic norm regularization of random matrices

“…In [7,Section 11], Rebrova and Vershynin asked whether one can obtain an explicit description of an ǫn × ǫn matrix whose removal regularizes the norm. This question was the focus of the work of Rebrova [6] who showed [6,Corollary 1.3] that for an n × n random matrix A with i.i.d. entries having a symmetric distribution and unit variance, for any ǫ ∈ (0, 1/2], and for any r ≥ 1, there is a deterministic, polynomial time algorithm to zero out an ǫn × ǫn sub-matrix in order to obtain A satisfying…”