Convergence of a Sparse Representations Algorithm Applicable to Real or Complex Data

Sparse representations has become an important topic in years. It consists in representing, say, a signal (vector) as a linear combination of as few as possible components (vectors) from a redundant basis (of the vector space). This is usually performed, either iteratively (adding a component at a time), or globally (selecting simultaneously all the needed components). We consider a specific algorithm, that we obtain as a fixed point algorithm, but that can also be seen as an iteratively reweighted least-squares algorithm. We analyze it thoroughly and show that it converges to the global optimum. We detail the proof in the real case and indicate how to extend it to the complex case. We illustrate the result with some easily reproducible toy simulations, that further illustrate the potential tracking properties of the proposed algorithm.