Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector

TL;DR

This paper introduces the MSSO simultaneous sparsity problem, proposing seven algorithms to approximate solutions for this NP-hard problem, and evaluates their performance in various scenarios including MRI pulse design.

Contribution

It formulates the novel MSSO simultaneous sparsity problem and develops seven algorithms, including greedy and convex relaxation methods, to solve it.

Findings

Algorithms effectively recover sparsity profiles in noiseless scenarios.

Convex relaxation methods perform well in noisy conditions.

Application to MRI pulse design demonstrates practical utility.

Abstract

A linear inverse problem is proposed that requires the determination of multiple unknown signal vectors. Each unknown vector passes through a different system matrix and the results are added to yield a single observation vector. Given the matrices and lone observation, the objective is to find a simultaneously sparse set of unknown vectors that solves the system. We will refer to this as the multiple-system single-output (MSSO) simultaneous sparsity problem. This manuscript contrasts the MSSO problem with other simultaneous sparsity problems and conducts a thorough initial exploration of algorithms with which to solve it. Seven algorithms are formulated that approximately solve this NP-Hard problem. Three greedy techniques are developed (matching pursuit, orthogonal matching pursuit, and least squares matching pursuit) along with four methods based on a convex relaxation (iteratively reweighted least squares, two forms of iterative shrinkage, and formulation as a second-order cone program). The algorithms are evaluated across three experiments: the first and second involve sparsity profile recovery in noiseless and noisy scenarios, respectively, while the third deals with magnetic resonance imaging radio-frequency excitation pulse design.