Analysis of Basis Pursuit Via Capacity Sets

TL;DR

This paper introduces 'capacity sets' as a new analytical tool to evaluate the success of Basis Pursuit in finding sparse solutions, improving upon previous coherence-based methods.

Contribution

It proposes capacity sets as an alternative to mutual coherence for analyzing Basis Pursuit, providing better bounds and performance predictions.

Findings

Capacity sets lead to improved success bounds.

Theoretical analysis using capacity sets enhances understanding of Basis Pursuit.

Numerical experiments confirm the effectiveness of capacity sets in performance prediction.

Abstract

Finding the sparsest solution $\alpha$ for an under-determined linear system of equations $D\alpha=s$ is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of $\alpha$ that allow its recovery using L1-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of $D$ called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given $D$, called the "capacity sets". We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of $D\alpha=s$.