# YALL1-Group: A solver for group/joint sparse reconstruction

### From Wikimization

YALL1-Group is a MATLAB software package for group/joint sparse reconstruction, written by Wei Deng, Wotao Yin and Yin Zhang at Rice University. Download

## Contents |

## Introduction

In the last few years, finding sparse solutions to underdetermined linear systems has become an active research topic, particularly in the area of *compressive sensing*, *statistics* and *machine learning*. Sparsity allows us to reconstruct high dimensional data with only a small number of samples. In order to further enhance the recoverability, recent studies propose to go beyond sparsity and take into account additional information about the underlying structure of the solutions.

In practice, a wide class of solutions are known to have **group sparsity** structure. Namely, the solution has a natural grouping of its components, and the components within a group are likely to be either all zeros or all nonzeros. **Joint sparsity** is an interesting special case of the group sparsity structure. Joint sparse solutions consist of multiple sparse solutions that share a common nonzero support. Encoding the group/joint sparsity structure can reduce the degrees of freedom in the solution, thereby leading to better recovery performance.

## Model

-based minimizatoin is one of the approaches for group or joint sparse reconstruction.

- YALL1-Group solves models (1) and (2), and its future versions will support extensions of (1) and (2).

(1) Group-sparse basis pursuit model with or without nonnegativity constraint:

Minimize subject to (optional)

where

- ;
- ;
- is the weight for the -th group;
- denotes the index set of the -th group;
- the groups may overlap.

(2) Joint-sparse basis pursuit model with or without nonnegativity constraint:

Minimize subject to or , for j=1,...,l (optional)

where

- the sensing matrix can be the same for each channel (column) of X, or can be different for each channel;
- ;
- and denote the i-th row and j-th column of matrix , respectively;
- is the weight for the -th row.

## Syntax

- [x,Out] = YALL1_group(A,b,groups,'param1',value1,'param2',value2,...);

## Technical Report

The description and theory of the YALL1-Group algorithm can be found in

- Wei Deng, Wotao Yin, and Yin Zhang, Group Sparse Optimization by Alternating Direction Method. (TR11-06, Department of Computational and Applied Mathematics, Rice University, 2011)