Nuclear Norm Optimization and Its Application to Observation Model Specification

Abstract

Optimization problems involving the minimization of the rank of a matrix subject to certain constraints are pervasive in a broad range of disciples, such as control theory [6, 26, 31, 62], signal processing [25], and machine learning [3, 77, 89]. However, solving such rank minimization problems is usually very difficult as they are NP-hard in general [65, 75]. The nuclear norm of a matrix, as the tightest convex surrogate of the matrix rank, has fueled much of the recent research and has proved to be a powerful tool in many areas. In this chapter, we aim to provide a brief review of some of the state-of-the-art in nuclear norm optimization algorithms as they relate to applications. We then propose a novel application of the nuclear norm to the linear model recovery problem, as well as a viable algorithm for solution of the recovery problem. Preliminary numerical results presented here motivates further investigation of the proposed idea.