Near minimax line spectral estimation

Line spectral estimation is a classical signal processing problem involving estimation of frequencies and amplitudes from noisy equispaced samples of a sparse combination of complex sinusoids. We view this as a sparse recovery problem with a continuous, infinite dictionary, and employ tools from convex optimization for estimation. In this paper, we establish that using atomic norm soft thresholding (AST), we can achieve near minimax optimal prediction error rate for line spectral estimation, in spite of having a highly coherent dictionary corresponding to arbitrarily close frequencies. We also derive guarantees on the frequency localization performance of AST.