Discrete optimization methods for group model selection in compressed sensing

Mathematical Programming, vol. 190, pp. 171–220

Abstract

Abstract In this article we study the problem of signal recovery for group models. More precisely for a given set of groups, each containing a small subset of indices, and for given linear sketches of the true signal vector which is known to be group-sparse in the sense that its support is contained in the union of a small number of these groups, we study algorithms which successfully recover the true signal just by the knowledge of its linear sketches. We derive model projection complexity results and algorithms for more general group models than the state-of-the-art. We consider two versions of the classical iterative hard thresholding algorithm (IHT). The classical version iteratively calculates the exact projection of a vector onto the group model, while the approximate version (AM-IHT) uses a head- and a tail-approximation iteratively. We apply both variants to group models and analyse the two cases where the sensing matrix is a Gaussian matrix and a model expander matrix. To solve the exact projection problem on the group model, which is known to be equivalent to the maximum weight coverage problem, we use discrete optimization methods based on dynamic programming and Benders’ decomposition. The head- and tail-approximations are derived by a classical greedy-method and LP-rounding, respectively.

Authors 3

1. Bubacarr Bah

   African Institute for Mathematical Sciences · Stellenbosch University

   Affiliation as printed

   AIMS, Cape Town, South Africa

   Stellenbosch University, 6 Melrose Road, Muizenberg, Cape Town, 7945, South Africa

   Stellenbosch University, Cape Town, South Africa

2. Jannis Kurtz corresponding Aachen Chair for Mathematics of Information Processing

   RWTH Aachen University

   Affiliation as printed

   Chair for Mathematics of Information Processing, RWTH Aachen University, Pontdriesch 10, 52062, Aachen, Germany

   Chair for Mathematics of Information Processing RWTH Aachen University Aachen Germany

3. Oliver Schaudt Aachen Chair for Mathematics of Information Processing

   RWTH Aachen University

   Affiliation as printed

   Chair for Mathematics of Information Processing, RWTH Aachen University, Pontdriesch 10, 52062, Aachen, Germany

   Chair for Mathematics of Information Processing RWTH Aachen University Aachen Germany