Strong Duality of Sparse Functional Optimization

Abstract

Signal processing is rich in inherently continuous applications, such as radar, MRI, and source localization, in which sparsity priors play a key role in obtaining state-of-the-art results. To cope with the infinite dimensionality and non-convexity of these estimation problems, they are typically discretized and solved by means of convex relaxations, e.g., using atomic norms. Although successful, this approach is not without issues. Discretization often leads to high dimensional, potentially ill-conditioned optimization problems. Moreover, due to grid mismatch and other coherence issues, a sparse signal in the continuous domain may no longer be sparse when discretized. Finally, performance guarantees for atomic norm relaxations hold under assumptions that may be hard to meet in practice. We address these issues by directly tackling the continuous problem cast as a sparse functional optimization program. We prove that these problems have no duality gap and show that they can be solved efficiently using duality and a stochastic gradient ascent-type algorithm. We illustrate the performance of this new approach on a line spectral estimation problem.