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This paper presents generalizations of semidefinite programming formulationsof 1-norm optimization problems over infinite dictionaries of vectors ofcomplex 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 linearsystem theory provide simple, constructive proofs of the semidefiniterepresentations of the penalty functions used in these applications. Theconnection leads to several extensions to gauge functions and atomic norms forsets of vectors parameterized via the nullspace of matrix pencils. Thetechniques are illustrated with examples of low-rank matrix approximationproblems arising in spectral estimation and array processing.

Semidefinite representations of gauge functions for structured low-rank matrix decomposition