Certain inverse problems can be solved quite efficiently if the solution is known to have a sparse atomic decomposition with respect to some basis or frame in a Hilbert space. One particular example of such an inverse problem is the so-called cocktail party (or blind source separation) problem: Suppose we use a few microphones to record several people speaking simultaneously. How can we separate individual speech signals from these mixtures?

In this talk, I will describe an algorithm adressing the blind source separation problem when the number of speakers is larger than the number of available mixtures. The algorithm is based on the key observation that Gabor expansions of speech signals are sparse. The separation is done in two stages: First, the "mixing matrix" A is estimated via clustering. Next, the Gabor coefficients of individual sources are computed by solving many q-norm minimization problems of type {min ||x||_q subject to Ax=b}. Several choices for the value of q will be compared.