Semidefinite Representations of Gauge Functions for Structured Low-Rank Matrix Decomposition

Abstract

This paper presents generalizations of semidefinite programming formulations of 1-norm optimization problems over infinite dictionaries of vectors of complex exponentials, which were recently proposed for superresolution, gridless compressed sensing, and other applications in signal processing. Results related to the generalized Kalman--Yakubovich--Popov lemma in linear system theory provide simple, constructive proofs of the semidefinite representations of the penalty functions used in these applications. The connection leads to several extensions to gauge functions and atomic norms for sets of vectors parameterized via the nullspace of matrix pencils. The techniques are illustrated with examples of low-rank matrix approximation problems arising in spectral estimation and array processing.