## 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

is an*A*`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 g_{1}, g_{2}, … are groups of coordinates and w_{1},w_{2}, … are their weights.

Joint sparsity is a special case of group sparsity for recovering X = [x_{1},x_{2}, …, 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]