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

### From Wikimization

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== Model == | == Model == | ||

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

Minimize <math>\|x\|_{w,2,1}:=\sum_{i=1}^s w_i\|x_{g_i}\|_2,</math> | Minimize <math>\|x\|_{w,2,1}:=\sum_{i=1}^s w_i\|x_{g_i}\|_2,</math> | ||

- | subject to <math>Ax=b,</math> | + | subject to <math>Ax=b</math>, |

+ | <math>x\geq0</math> (optional), | ||

where <math>A\in \mathbb{R}^{m\times n}\,(m<n)</math>, <math>b\in \mathbb{R}^m</math>, <math>g_i</math> denotes the index set of the <math>i</math>-th group, and <math>w_i\geq0</math> is the weight for the <math>i</math>-th group. | where <math>A\in \mathbb{R}^{m\times n}\,(m<n)</math>, <math>b\in \mathbb{R}^m</math>, <math>g_i</math> denotes the index set of the <math>i</math>-th group, and <math>w_i\geq0</math> is the weight for the <math>i</math>-th group. | ||

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

Minimize <math>\|X\|_{w,2,1}:=\sum_{i=1}^n w_i\|x^i\|_2,</math> | Minimize <math>\|X\|_{w,2,1}:=\sum_{i=1}^n w_i\|x^i\|_2,</math> | ||

subject to <math>AX=B,</math> | subject to <math>AX=B,</math> | ||

+ | <math>X\geq0</math> (optional), | ||

where <math>A\in \mathbb{R}^{m\times n}\,(m<n)</math>, <math>B\in \mathbb{R}^{m\times l}</math>, <math>x^i</math> denotes the <math>i</math>-th row of matrix <math>X</math>, and <math>w_i\geq0</math> is the weight for the <math>i</math>-th row. | where <math>A\in \mathbb{R}^{m\times n}\,(m<n)</math>, <math>B\in \mathbb{R}^{m\times l}</math>, <math>x^i</math> denotes the <math>i</math>-th row of matrix <math>X</math>, and <math>w_i\geq0</math> is the weight for the <math>i</math>-th row. | ||

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|- | |- | ||

| ''''GrpWeights'''' || nonnegetive n-vector || Weights for the groups/rows. | | ''''GrpWeights'''' || nonnegetive n-vector || Weights for the groups/rows. | ||

+ | |- | ||

+ | | ''''Nonnegative'''' || true or false || True for imposing nonnegativity constraints. | ||

|- | |- | ||

| ''''nonorthA'''' || true or false || Specify if matrix A has non-orthonormal rows (true) or orthonormal rows (false). | | ''''nonorthA'''' || true or false || Specify if matrix A has non-orthonormal rows (true) or orthonormal rows (false). |

## Revision as of 18:05, 28 June 2011

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 |

## Model

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

Minimize subject to , (optional),

where , , denotes the index set of the -th group, and is the weight for the -th group.

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

Minimize subject to (optional),

where , , denotes the -th row of matrix , and is the weight for the -th row.

## Syntax

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

## Input Arguments

**A**: an m-by-n matrix with m < n, or a structure with the following fields:

- 1)
**A.times**(required): a function handle for ; - 2)
**A.trans**(required): a function handle for ; - 3)
**A.invIpAAt**: a function handle for ; - 4)
**A.invAAt**: a function handle for .

Note: Field **A.invIpAAt** is only required when (a) primal solver is to be used, and b) A is non-orthonormal, and (c) exact linear system solving is to be performed. Field **A.invAAt** is only required when (a) dual solver is to be used, and b) A is non-orthonormal, and (c) exact linear system solving is to be performed.

**b**: an m-vector for the group-sparse model or an m-by-l matrix for the joint-sparse model.

**groups**: an n-vector containing the group number of the corresponding component of for the group-sparse model, or [] for the joint-sparse model.

- Optional input arguments:

Parameter Name | Value | Description |
---|---|---|

'StopTolerance' | positive scalar | Stopping tolerance value. |

'GrpWeights' | nonnegetive n-vector | Weights for the groups/rows. |

'Nonnegative' | true or false | True for imposing nonnegativity constraints. |

'nonorthA' | true or false | Specify if matrix A has non-orthonormal rows (true) or orthonormal rows (false). |

'ExactLinSolve' | true or false | Specify if linear systems are to be solve exactly (true) or approximately by taking a gradient descent step (false). |

'QuadPenaltyPar' | nonnegative 2-vector for primal solver or nonnegative scalar for dual solver | Penalty parameters. |

'StepLength' | nonnegative 2-vector for primal solver or nonnegative scalar for dual solver | Step lengths for updating the multipliers. |

'maxIter' | positive integer | Maximum number of iterations allowed. |

'xInitial' | an n-vector for group-sparse model or an n-by-l matrix for joint-sparse model | An initial estimate of the solution. |

'Solver' | 1 or 2 | Specify which solver to use: 1 for primal solver; 2 for dual solver. |

'Continuation' | true or false | Specify if continuation on the penalty parameters is to be used (true) or not (false). The continuation scheme is as follows: multiply the penalty parameters by a factor if , where 0< <1 is a parameter, and denote the constraint violations at the current and previous iterations, respectively. Continuation allows small initial penalty parameters for
constraint violations, which lead to faster initial convergence, and it increases those parameters whenever the violation reduction slows down. It leads to overall speedups in most cases. |

'ContParameter' | scalar between 0 and 1 | The parameter (0 < <1) in the continuation scheme. |

'ContFactor' | scalar greater than 1 | The factor in the continuation scheme. |

**Note**: the parameter names are not case-sensitive.

## Output Arguments

**x**: last iterate (hopefully an approximate solution).**Out**: a structure with fields:- Out.status—exit information;
- Out.iter—number of iterations taken;
- Out.cputime—solver CPU time.

## Examples

Please see the demo files.

## 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)