On Failing Sets of the Interval-Passing Algorithm for Compressed Sensing

TL;DR

This paper provides a graph-theoretic analysis of the failure modes of the interval-passing algorithm in compressed sensing, introducing termatiko sets as key to understanding when recovery fails.

Contribution

It introduces the concept of termatiko sets and characterizes the failing sets of the IPA completely using graph theory.

Findings

Failure of IPA is linked to the presence of termatiko sets in the support.

Termatiko sets can be smaller than the stopping distance, affecting recovery.

Numerical results show many termatiko sets are smaller than the stopping distance.

Abstract

In this work, we analyze the failing sets of the interval-passing algorithm (IPA) for compressed sensing. The IPA is an efficient iterative algorithm for reconstructing a k-sparse nonnegative n-dimensional real signal x from a small number of linear measurements y. In particular, we show that the IPA fails to recover x from y if and only if it fails to recover a corresponding binary vector of the same support, and also that only positions of nonzero values in the measurement matrix are of importance for success of recovery. Based on this observation, we introduce termatiko sets and show that the IPA fails to fully recover x if and only if the support of x contains a nonempty termatiko set, thus giving a complete (graph-theoretic) description of the failing sets of the IPA. Finally, we present an extensive numerical study showing that in many cases there exist termatiko sets of size strictly smaller than the stopping distance of the binary measurement matrix; even as low as half the stopping distance in some cases.