Random matrices acting on sets: Independent columns

TL;DR

This paper investigates how random matrices with independent subgaussian columns can serve as near-isometries on subsets of their domain, revealing a linear relationship between distortion and the subgaussian norm, which improves embedding guarantees.

Contribution

It establishes new conditions under which such matrices act as near-isometries, highlighting a linear dependence on the subgaussian norm, unlike previous superlinear dependencies.

Findings

Maximum distortion is proportional to Gaussian complexity scaled by subgaussian norm.

Normalizing columns of sparse matrices enhances embedding guarantees.

Linear dependence on subgaussian norm is a novel phenomenon.

Abstract

We study random matrices with independent subgaussian columns. Assuming each column has a fixed Euclidean norm, we establish conditions under which such matrices act as near-isometries when restricted to a given subset of their domain. We show that, with high probability, the maximum distortion caused by such a matrix is proportional to the Gaussian complexity of the subset, scaled by the subgaussian norm of the matrix columns. This linear dependence on the subgaussian norm is a new phenomenon, as random matrices with independent rows or independent entries typically exhibit superlinear dependence. As a consequence, normalizing the columns of random sparse matrices leads to stronger embedding guarantees.