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 Suzen. 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

• 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 w_i ||x_g_i||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, ..., x_l] where the x_i‘s share a common sparse support.
(JointBP) min sum_i w_i ||X_i,:||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.
Download
YALL1 is now open-source. It is distributed under the terms of the GNU General Public License. [Proceed to the download page]

Credit Video: ESA/DLR/FU Berlin (G. Neukum), via the Planetary Society blog.
Phobos ( a moon of Mars) slips past Jupiter, On June 1 2011, Mars express watched as Phobos (the inner and larger of Mars' two moons) slipped past distant Jupiter. Mars Express is studying Phobos to help the Russian Phobos-Grunt mission prepare to land on the moon and grab a sample for return to Earth. Credit: ESA/DLR/FU Berlin (G. Neukum)