Sparseness by Iterative Projections Onto Spheres

Many interesting signals share the property of being sparsely active. The search for such sparse components within a data set commonly involves a linear or nonlinear projection step in order to fulfill the sparseness constraints. In addition to the proximity measure used for the projection, the result of course is also intimately connected with the actual definition of the sparseness criterion. In this work, we introduce a novel sparseness measure and apply it to the problem of finding a sparse projection of a given signal. Here, sparseness is defined as the fixed ratio of p-over 2-norm, and existence and uniqueness of the projection holds. This framework extends previous work by Hoyer in the case of p = 1, where it is easy to give a deterministic, more or less closed-form solution. This is not possible for p ≠ 1, so we introduce an algorithm based on alternating projections onto spheres (POSH), which is similar to the projection onto convex sets (OCS). Although the assumption of convexity does not hold in our setting, we observe not only convergence of the algorithm, but also convergence to the correct minimal distance solution. Indications for a proof of this surprising property are given. Simulations confirm these results.