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

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

2009-07-13

arxiv.org/abs/0907.2083v1

SIAM J. Scientific Computing, vol. 31, no. 5, pp. 4533-4579, January 2010

Computer Science

Numerical Analysis

36 pages; manuscript unchanged from July 21, 2008, except for updated references; content appears in September 2008 PhD thesis

Scientific paper

10.1137/080730822

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.

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