# Discrete Optimization Methods for Group Model Selection in Compressed Sensing

## TL;DR

This paper advances algorithms for recovering group-sparse signals in compressed sensing by developing new projection methods and analyzing their performance with Gaussian and expander sensing matrices.

## Contribution

It introduces improved algorithms for group model projections, including exact and approximate methods, and extends analysis to more general group models beyond existing approaches.

## Key findings

- Exact projection algorithms based on dynamic programming and Benders' Decomposition.
- Approximate projections using greedy methods and LP-rounding.
- Successful recovery demonstrated with Gaussian and expander sensing matrices.

## 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.