The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach

TL;DR

This paper provides an elementary proof of a key inequality that bounds the expected spectral norm of a sum of independent random matrices, highlighting its dependence on the matrices' moments and dimension.

Contribution

It offers a complete, elementary proof of an important inequality relating the expectation of the spectral norm to moments of the summands, which was previously underappreciated.

Findings

The inequality bounds the expected spectral norm using the norm of the expected square.

The bound includes a weak dependence on the matrix dimension.

The proof simplifies understanding of spectral norm bounds for random matrices.

Abstract

In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled by the norm of the expected square of the random matrix and the expectation of the maximum squared norm achieved by one of the summands; there is also a weak dependence on the dimension of the random matrix. The purpose of this paper is to give a complete, elementary proof of this important, but underappreciated, inequality.