Computing Sparse Solutions to Linear Inverse Problems

In this talk, we will consider the problem of computing sparse solutions to undetermined linear inverse problems which has potential applications in DOA estimation/source localization, feature selection, and biomedical inverse problems that arise in MEG/EEG analysis. Unfortunately, the required optimization problem is computationally complex (NP-hard) motivating the search for suboptimal algorithms which offer a reasonable compromise between complexity and performance. Of particular interest in this talk are methods based on minimizing diversity measures such as the FOCUSS algorithm and sparse Bayesian learning (SBL) techniques. In addition to discussing the computational algorithms, we will examine the ability of the algorithms to identify the true sparse solution. For the SBL method we derive necessary conditions for local minima to occur and empirically demonstrate that there are typically many fewer for general problems of interest.