Parameter estimation for mixture models via convex optimization

Many applications encounter signals that are a linear combination of multiple components, where each component represents a low-resolution observation of a point source model captured through a low-pass point spread function. This paper proposes a convex optimization algorithm to simultaneously separate and identify the point source models of each component from a noisy observation corrupted by possibly adversarial noise, by leveraging the recently proposed atomic norm framework. The proposed algorithm can be solved efficiently via semidefinite programming, where locations of the point sources can be identified via the constructed dual polynomials without estimating the model orders a priori. Stability of the proposed algorithm is established under certain conditions of the point source models and the point spread functions in the presence of bounded noise. Furthermore, numerical examples are provided to corroborate the theoretical analysis, with comparisons against the Cramèr-Rao bound for parameter estimation.