A fast approach for overcomplete sparse decomposition based on smoothed L0 norm

In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined Sparse Component Analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the L1 norm using Linear Programming (LP) techniques, our algorithm tries to directly minimize the L0 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.Comment: Accepted in IEEE Transactions on Signal Processing. For MATLAB codes, see (http://ee.sharif.ir/~SLzero). File replaced, because Fig. 5 was missing erroneousl

Contribution of Statistical Tests to Sparseness-Based Blind Source Separation

International audienceWe address the problem of blind source separation in the underdetermined mixture case. Two statistical tests are proposed to reduce the number of empirical parameters involved in standard sparseness-based underdetermined blind source separation (UBSS) methods. The first test performs multisource selection of the suitable time-frequency points for source recovery and is full automatic. The second one is dedicated to autosource selection for mixing matrix estimation and requires fixing two parameters only, regardless of the instrumented SNRs. We experimentally show that the use of these tests incurs no performance loss and even improves the performance of standard weak-sparseness UBSS approaches

Underdetermined Blind Identification for kk-Sparse Component Analysis using RANSAC-based Orthogonal Subspace Search

Sparse component analysis is very popular in solving underdetermined blind source separation (UBSS) problem. Here, we propose a new underdetermined blind identification (UBI) approach for estimation of the mixing matrix in UBSS. Previous approaches either rely on single dominant component or consider $k \leq m-1$ active sources at each time instant, where $m$ is the number of mixtures, but impose constraint on the level of noise replacing inactive sources. Here, we propose an effective, computationally less complex, and more robust to noise UBI approach to tackle such restrictions when $k = m-1$ based on a two-step scenario: (1) estimating the orthogonal complement subspaces of the overall space and (2) identifying the mixing vectors. For this purpose, an integrated algorithm is presented to solve both steps based on Gram-Schmidt process and random sample consensus method. Experimental results using simulated data show more effectiveness of the proposed method compared with the existing algorithms

Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization

The l1/l2 ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works, in the context of blind deconvolution. Indeed, it benefits from a scale invariance property much desirable in the blind context. However, the l1/l2 function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such a penalty term in current restoration methods. In this paper, we propose a new penalty based on a smooth approximation to the l1/l2 function. In addition, we develop a proximal-based algorithm to solve variational problems involving this function and we derive theoretical convergence results. We demonstrate the effectiveness of our method through a comparison with a recent alternating optimization strategy dealing with the exact l1/l2 term, on an application to seismic data blind deconvolution.Comment: 5 page

Blind Source Separation: the Sparsity Revolution

International audienceOver the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation

One-stage blind source separation via a sparse autoencoder framework

Blind source separation (BSS) is the process of recovering individual source transmissions from a received mixture of co-channel signals without a priori knowledge of the channel mixing matrix or transmitted source signals. The received co-channel composite signal is considered to be captured across an antenna array or sensor network and is assumed to contain sparse transmissions, as users are active and inactive aperiodically over time. An unsupervised machine learning approach using an artificial feedforward neural network sparse autoencoder with one hidden layer is formulated for blindly recovering the channel matrix and source activity of co-channel transmissions. The BSS sparse autoencoder provides one-stage learning using the receive signal data only, which solves for the channel matrix and signal sources simultaneously. The recovered co-channel source signals are produced at the encoded output of the sparse autoencoder hidden layer. A complex-valued soft-threshold operator is used as the activation function at the hidden layer to preserve the ordered pairs of real and imaginary components. Once the weights of the sparse autoencoder are learned, the latent signals are recovered at the hidden layer without requiring any additional optimization steps. The generalization performance on future received data demonstrates the ability to recover signal transmissions on untrained data and outperform the two-stage BSS process