## Tuesday, June 21, 2011

### Two Solvers: R1Magic and YALL1 Group

I just came across a CS solver written in R as shown in this presentation: Compressive Sampling with R: A Tutorial by Mehmet Suzen. The R1Magic package is here. A reference manual is here.

YALL1 now include a group sparsity solving capability. Let us look at what he can do. From the webpage:

YALL1 package now includes:
• YALL1 Basic
• , a solver for sparse reconstruction: Version 1.3, Apr 07, 2011.
• YALL1 Group
• , a solver for group/joint sparse reconstruction: Version 1.0, June 09, 2011. Wiki
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 and 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
• 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 sumi wi ||x_gi||2 s.t. Ax = b
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, ..., xl] where the xi‘s share a common sparse support.
(JointBP) min sumi wi ||Xi,:||2 s.t. AX = B.
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).
Tech report
YALL1 Basic: Alternating Direction Algorithms for L1 Problems in Compressive Sensing, Rice CAAM Report TR09-37, 2009.
YALL1 Group/Joint Sparsity: Group Sparse Optimization by Alternating Direction Method, Rice CAAM Report TR11-06, 2011.