Homotopy continuation for sparse signal representation

We explore the application of a homotopy continuation-based method for sparse signal representation in overcomplete dictionaries. Our problem setup is based on the basis pursuit framework, which involves a convex optimization problem consisting of terms enforcing data fidelity and sparsity, balanced by a regularization parameter. Choosing a good regularization parameter in this framework is a challenging task. We describe a homotopy continuation-based algorithm to find and trace efficiently all solutions of basis pursuit as a function of the regularization parameter. In addition to providing an attractive alternative to existing optimization methods for solving the basis pursuit problem, this algorithm can also be used to provide an automatic choice for the regularization parameter, based on prior information about the desired number of non-zero components in the sparse representation. Our numerical examples demonstrate the effectiveness of this algorithm in accurately and efficiently generating entire solution paths for basis pursuit, as well as producing reasonable regularization parameter choices. Furthermore, exploring the resulting solution paths in various operating conditions reveals insights about the nature of basis pursuit solutions.