The LiMapS Algorithm

This algorithm results in a new regularization method for sparse representation based on a fixed-point iteration schema which combines two Lipschitzian-type mappings, a nonlinear one aimed to uniformly enhance the sparseness level of a candidate solution and a linear one which projects back into the feasible space of solutions. It is shown that this strategy locally minimizes a problem whose objective function falls into the class of the $\ell_p$-norm and represents an efficient approximation of the intractable problem focusing on the $\ell_0$-norm. Numerical experiments on randomly generated signals using classical stochastic models show better performances of the proposed technique with respect to a wide collection of well known algorithms for sparse representation, as shown in the following figures.

Performances

Fig. 1 Averages SNR of the algorithms vs. sparsity, expressed in percentage of the number of equations n. The algorithms SL0, LARS, LASSO, MP, OMP, STOMP are the most representative in the area.


Errors

Fig. 2 Relative number (in %) of correctly recovered atoms not equal to zero. An atom is considered correctly reconstructed if the deviation from the true value of the estimated value is less than 5%.


Times

Fig. 3 Averages computational times of the algorithms vs. sparsity, expressed in percentage of the number of equations n.



The MATLAB code of LiMapS and related resurses are available under GPL.

limaps-v1.0.zip (MATLAB code, tests and examples)

limaps_slides (slide presentation at ISPITT '11)


The PrunICA Algorithm

The purpose of PrunICA is to look at the performance of the FastICA algorithm when a controlled random pruning of the input mixtures is done, both on the entire mixture available and when they are segmented into fixed-size blocks. Naturally, it is likely to speed up FastICA preserving, at the same time, the demixture quality. The method consists in randomly select a sufficiently small subset of sample data in such a way that its sample kurtosis is not too far from those of the entire observations. In other words, we perform an analysis of the kurtosis estimator on the subsample with the purpose to find the maximum reduction which guarantees a narrow confidence interval with high confidence level.

The MATLAB code of PrunICA and related resurses are available under GPL.

PrunedFastICA.tgz (MATLAB code, tests and examples).


LVAICA10.pdf, ICA07.pdf (papers)