Maximum Likelihood Line Spectral Estimation in the Signal Domain: A Rank-Constrained Structured Matrix Recovery Approach

Maximum likelihood estimation (MLE) provides a well-known benchmark for line spectral estimation and has been extensively studied in the parameter domain using a variety of optimization algorithms. To overcome the sensitivity of these algorithms to parameter initialization, in this paper we study the MLE in the signal domain. We formulate the MLE as an equivalent rank-constrained structured matrix recovery problem that admits a unique matrix solution containing the signal, from which the parameters of interest are uniquely retrieved. The alternating direction method of multipliers (ADMM) is used to solve the rank-constrained problem and it is shown to have a good convergence behavior. The proposed approach is generalized to the case of missing data and arbitrary-dimensional line spectral estimation. Extensive numerical results are provided that corroborate our analysis and confirm that the proposed approach globally solves the MLE problem and outperforms state-of-the-art algorithms.