Second-Order Matrix Concentration Inequalities

TL;DR

This paper develops sharper matrix concentration inequalities by refining the matrix Khintchine inequality, reducing the dependence on dimension and providing tighter bounds for spectral-norm deviations of random matrices.

Contribution

It identifies the source of dimensional dependence in matrix concentration bounds and introduces refined inequalities that leverage additional information beyond matrix variance.

Findings

Refined matrix Khintchine inequalities with reduced dimensional dependence

Sharper bounds for spectral-norm deviations of random matrices

Enhanced understanding of sources of dimensional dependence in matrix inequalities

Abstract

Matrix concentration inequalities give bounds for the spectral-norm deviation of a random matrix from its expected value. These results have a weak dimensional dependence that is sometimes, but not always, necessary. This paper identifies one of the sources of the dimensional term and exploits this insight to develop sharper matrix concentration inequalities. In particular, this analysis delivers two refinements of the matrix Khintchine inequality that use information beyond the matrix variance to reduce or eliminate the dimensional dependence.