Sparsity: What is it good for ?

The solution of an underdetermined system of linear equations is generally too large (i.e. it generally is an entire subspace as opposed to a unique solution). When you have too many unknowns and not enough equations, experience tells you to wait for additional equations or constraints to obtain a potentially unique solution. In Compressed Sensing, sparsity of the solution is the element that provides an ability to put additional constraints in order to find a unique solution.

But sparsity can be also used for other purpose as shown by this excellent study by Jerome Bobin, Jean-Luc Starck, Jalal Fadili and Yassir Moudden in Sparsity and Morphological Diversity in Blind Source Separation where their approach "takes advantage of this “morphological diversity” to differentiate between the sources with accuracy". Evidently, the reconstruction techniques parallels the ones found in Compressed Sensing. The abstract reads:

Over 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 new and 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 introduces a new 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. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.

The GMCALab matlab code for the implementation of this algorithm can be found here.