YALL1: Your ALgorithms for L1

Yall是一个能求解6种(或者更多)不同最小化L1问题的matlab软件包。里面有详细的使用说明和算法求解的基本思路。
(It is a Matlab solver that at present can be applied to the six L1-minimization models)

Welcome

 

YALL1 package now includes:

• YALL1 Basic, a solver for sparse reconstruction: Version 1.4, July 15, 2011.
• YALL1 Group, a solver for group/joint sparse reconstruction: Version 1.2, June 28, 2011.

YALL1 Basic

solves the following L1-minimization problems:

   (BP)          min ||Wx||w,1,   s.t. Ax = b

   (L1/L1)       min ||Wx||w,1 + (1/ν)||Ax - b||1

   (L1/L2)       min ||Wx||w,1 + (1/2ρ)||Ax - b||22

   (L1/L2con)    min ||Wx||w,1,   s.t. ||Ax - b||2 <= δ

   (BP+)         min ||x||w,1,   s.t. Ax = b,  x >= 0

   (L1/L1+)      min ||x||w,1 + (1/ν)||Ax - b||1,   s.t. x >= 0

   (L1/L2+)      min ||x||w,1 + (1/2ρ)||Ax - b||22, s.t. x >= 0

   (L1/L2con+)   min ||x||w,1,  s.t. ||Ax - b||2 <= δ,  x >= 0

where

• A is an m-by-n matrix with m << n,
• the solution x (or its representation Wx) is supposed to be (approximately) sparse,
• the data and solution can be real or complex, (If complex, then no non-negativity constraint is allowed)
• a unitary sparsifying basis W is optional,
• the 1-norm can be optionally weighted by a nonnegative vector w.

Go to discussions and Q&As.

Supported Features

• Multiple types of A
  • explicit matrix
  • ensembles of fast transforms such as FFT, DCT, wavelets
  • ensembles of convolution or circulant matrices
• Both real and complex data
• Both sparse and compressible signals
• Non-negative signals

YALL1 Group

The group sparsity code solves the following model

   (GroupBP)          min sum_i wi ||x_gi||2 s.t. Ax = b

   (GroupBP+)         min sum_i wi ||x_gi||2 s.t. Ax = b, x >= 0

where g₁, g₂, … are groups of coordinates and w₁,w₂, … are their weights.

Joint sparsity is a special case of group sparsity for recovering X = [x₁,x₂, ..., x_l] where the x_i‘s share a common sparse support.

   (JointBP)          min sum_i wi ||X_i,:||2 s.t. AX = B.

   (JointBP+)         min sum_i wi ||X_i,:||2 s.t. AX = B, x>= 0.

Supported Features

• Multiple types of A
  • explicit matrix
  • ensembles of fast transforms such as FFT, DCT, wavelets
  • general function handle
• Groups can overlap
• The union of groups does not need to cover all coordinates
• Easy to modify for the support of complex numbers and your signals

Contributors

Yin Zhang*, Wei Deng, Junfeng Yang, and Wotao Yin.

* The original author of YALL1 (beta 1 – 6).

Citation:

YALL1 basic models and tests: J. Yang and Y. Zhang. Alternating direction algorithms for L1-problems in compressive sensing, SIAM Journal on Scientific Computing, 33, 1-2, 250-278, 2011. [arXiv]

YALL1 basic solver code: Y. Zhang, J. Yang, and W. Yin. YALL1: Your ALgorithms for L1, online at yall1.blogs.rice.edu, 2011.

YALL1 group/joint sparsity: W. Deng, W. Yin, and Y. Zhang, Group Sparse Optimization by Alternating Direction Method, Rice CAAM Report TR11-06, 2011. [pdf]

Download

YALL1 is now open-source. It is distributed under the terms of the GNU General Public License. [Proceed to the download page]

http://yall1.blogs.rice.edu/

做一个sparse representation的model，需要求解一个weighted sparse coding问题，看到工具包，可以直接对以下6种问题进行求解：

其中，W表示对L1正则的加权，选择想要的某些模型，有一部分进行推导之后，转化为不同的公式。

Official website

参考文献
Alternating direction algorithms for L1-problems in compressive sensing