Abstract:

Sparse Component Analysis represents an overlap of two problems (and methods) of Statistics/Computer Science/Electrical Engineering/Applied Mathematics: Independent Component Analysis (ICA), and Sparse Representations. Originally, the ICA problem is looking for decomposing a random d-vector into a linear compoisition of exactly d independent random variables: x = A.s , where A is dxd unknown mixing matrix, and s is the d-vector of independent components. The Blind Source Separation (BSS) problem is very much similar to ICA, except that A may be a matrix of (convolutive) operators. In practice, people applied these solutions to different type of signals. In particular audio (speech) signals gave rise to what is also known as "the cocktail party problem". Interesting algorithms were also obtained on images, bio-medical signals (e.g. EEG, ERP, fMRI). Independent of this, the Sparse Representation problem tries to decompose a vector x into a linear combination of (possibly redundant) frame vectors using a smallest number of coefficients. My talk uses sparse representation hypotheses in order to solve a convolutive BSS, including estimating the number of source signals. If time permitting, I would also like to comment on a standard result in ICA that says that x=A.s can be identified only if at most two independent components of 's' are Gaussian.