Hello, and welcome to the Paper of the Day (Po’D): Signal Processing Edition. Today’s paper comes from our colleagues Mads Christensen, Jan Østergaard, and Søren Holdt Jensen: On Compressed Sensing and Its Application to Speech and Audio Signals, in Proc. Asilomar Conf. Signals, Systems, and Computers, 2009.
In this paper, the authors look at the contribution of compressed sensing to finding sparse approximations of audio and speech signals using redundant dictionaries. The main reason this work is exciting is because it demonstrates how we can significantly reduce the dimensionality of the problem of finding sparse and efficient representations for high-dimensional data, such as sampled audio. This means that I can keep working with large and redundant dictionaries, use an optimization method to build a representation based on minimizing the \ell_1-norm of the solution, and still be home in time for dinner. I do not need to rely only on greedy short-sighted methods of sparse approximation. The magic here is in how one can use compressed sensing to reduce the number of constraints involved in solving the problem — but only as long as the signal being decomposed is sparse in the given dictionary. And there is the rub: since the sparsity of the signal is unknown a priori, we must guess it to ensure our measurement matrix will permit recovery of the solution. However, it appears that in the world of sparse approximation, even if we misjudge the sparsity, the solution will not be significantly worse than otherwise. That is good news!