Submatrices with non-uniformly selected random supports and insights into sparse approximation

TL;DR

This paper derives probabilistic bounds for the norms of non-uniformly supported random submatrices and applies these to analyze and improve sparse approximation methods and sensing dictionaries.

Contribution

It introduces tail bounds for non-uniform random supports and applies them to enhance sparse approximation techniques and sensing dictionary design.

Findings

Tail bounds for non-uniform random submatrices

Improved average-case analysis of sparse approximation algorithms

Numerical validation of sensing dictionary performance

Abstract

In this paper we derive tail bounds on the norms of random submatrices with non-uniformly distributed supports. We apply these results to sparse approximation and conduct an analysis of the average case performance of thresholding, Orthogonal Matching Pursuit and Basis Pursuit. As an application of these results we characterise sensing dictionaries to improve average performance in the non-uniform case and test their performance numerically.