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

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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

The following weighted LaTeX: \ell_{2,1}-regularization is known to facilitate group/joint sparsity in the solution.

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

                  Minimize     LaTeX: \|x\|_{w,2,1}:=\sum_{i=1}^s w_i\|x_{g_i}\|_2   
                  subject to   LaTeX: Ax=b\,
                               LaTeX: x\geq0 (optional)

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

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

                  Minimize     LaTeX: \|X\|_{w,2,1}:=\sum_{i=1}^n w_i\|x^i\|_2
                  subject to   LaTeX: AX=B\,
                               LaTeX: X\geq0 (optional)

where LaTeX: A\in \mathbb{R}^{m\times n}\,(m<n), LaTeX: B\in \mathbb{R}^{m\times l}, LaTeX: x^i denotes the LaTeX: i-th row of matrix LaTeX: X, and LaTeX: w_i\geq0 is the weight for the LaTeX: i-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 LaTeX: A*x;
2) A.trans (required): a function handle for LaTeX: A^T*x;
3) A.invIpAAt: a function handle for LaTeX: (\beta_1I_m+\beta_2AA^T)^{-1}*x;
4) A.invAAt: a function handle for LaTeX: (AA^T)^{-1}*x.

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 LaTeX: x 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 LaTeX: c\,(c\,>\,1) if LaTeX: \|R\|_2 > \alpha\|Rp\|_2, where 0< LaTeX: \alpha <1 is a parameter, LaTeX: R and LaTeX: Rp 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 LaTeX: \alpha (0 < LaTeX: \alpha <1) in the continuation scheme.
'ContFactor' scalar greater than 1 The factor LaTeX: c\,(c\,>\,1) 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

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