PROX_L21 - Proximal operator with L21 norm

Usage

sol=prox_l21(x, gamma, param)
[sol,info] = prox_l21(x, gamma, param)

Input parameters

x Input signal.
gamma Regularization parameter.
param Structure of parameters.

Output parameters

sol Solution.
info Structure summarizing informations at convergence

Description

prox_L21(x, gamma, param) solves:

\begin{equation*} sol = \min_{z} \frac{1}{2} \|x - z\|_2^2 + \gamma \| x\|_{2,1} \end{equation*}

where

\begin{equation*} \| x \|_{2,1} = \sum_j \left| \sum_i |x(i,j)|^2 \right|^{1/2} \end{equation*}

The easiest way to use this proximal operator is to give a matrix \(x\) as input. In this case, the \(l_{2,1}\) norm is computed like in the expression above.

param is a Matlab structure containing the following fields:

  • param.weights1 : weights for a weighted L21-norm works on the norm L1 (default = 1) (Experimental)

  • param.weights2 : weights for a weighted L21-norm works on the L2 norm (default = 1) (Experimental)

  • param.g_d, param.g_t are the group vectors. If you give a matrix, do not set those parameters.

    param.g_d contains the indices of the elements to be grouped and param.g_t the size of the different groups.

    Warning: param.g_d and param.g_t have to be row vector!

    Example: suppose x=[x1 x2 x3 x4 x5 x6]
    and Group 1: [x1 x2 x4 x5]

    group 2: [x3 x6]

    In matlab:

    param.g_d = [1 2 4 5 3 6]; param.g_t=[4 2];

    Also this is also possible:

    param.g_d = [4 5 3 6 1 2]; param.g_t=[2 4];

  • param.multi_group: in order to group component in a not disjoint manner, it is possible to use the multi_group option. param.multi_group is now set automatically by the function.

    Overlaping group: In order to make overlapping group just give a vector of g_d, g_b and g_t. Example:

    param.g_d=[g_d1; g_d2; ...; g_dn];
param.g_t=[g_t1; g_t2; ...; g_tn];

    Warning! There must be no overlap in g_d1, g_d2,... g_dn

info is a Matlab structure containing the following fields:

  • info.algo : Algorithm used
  • info.iter : Number of iteration
  • info.time : Time of exectution of the function in sec.
  • info.final_eval : Final evaluation of the function
  • info.crit : Stopping critterion used

References:

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski. Optimization with sparsity-inducing penalties. arXiv preprint arXiv:1108.0775, 2011.

M. Kowalski, K. Siedenburg, and M. Dorfler. Social sparsity! neighborhood systems enrich structured shrinkage operators. Signal Processing, IEEE Transactions on, 61(10):2498--2511, 2013.

M. Kowalski. Sparse regression using mixed norms. Applied and Computational Harmonic Analysis, 27(3):303--324, 2009.

M. Kowalski and B. Torresani. Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients. Signal, image and video processing, 3(3):251--264, 2009.