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We propose a new Iteratively Reweighted Least Squares (IRLS) algorithm forthe problem of completing or denoising low-rank matrices that are structured,e.g., that possess a Hankel, Toeplitz or block-Hankel/Toeplitz structure. Thealgorithm optimizes an objective based on a non-convex surrogate of the rank bysolving a sequence of quadratic problems. Our strategy combines computationalefficiency, as it operates on a lower dimensional generator space of thestructured matrices, with high statistical accuracy which can be observed inexperiments on hard estimation and completion tasks. Our experiments show thatthe proposed algorithm StrucHMIRLS exhibits an empirical recovery probabilityclose to 1 from fewer samples than the state-of-the-art in a Hankel matrixcompletion task arising from the problem of spectral super-resolution of badlyseparated frequencies. Furthermore, we explain how the proposed algorithm forstructured low-rank recovery can be used as preprocessing step for improvedrobustness in frequency or line spectrum estimation problems.

Denoising and Completion of Structured Low-Rank Matrices via Iteratively Reweighted Least Squares