An Optimization View of MUSIC and Its Extension to Missing Data

One of the classical approaches for estimating the frequencies and damping factors in a spectrally sparse signal is the MUltiple SIgnal Classification (MUSIC) algorithm, which exploits the low-rank structure of an autocorrelation matrix. Low-rank matrices have also received considerable attention recently in the context of optimization algorithms with partial observations. In this work, we offer a novel optimization-based perspective on the classical MUSIC algorithm that could lead to future developments and understanding. In particular, we propose an algorithm for spectral estimation that involves searching for the peaks of the dual polynomial corresponding to a certain nuclear norm minimization (NNM) problem, and we show that this algorithm is in fact equivalent to MUSIC itself. Building on this connection, we also extend the classical MUSIC algorithm to the missing data case. We provide exact recovery guarantees for our proposed algorithms and quantify how the sample complexity depends on the true spectral parameters. Simulation results also indicate that the proposed algorithms significantly outperform some relevant existing methods in frequency estimation of damped exponentials.