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Consider a spectrally sparse signal x that consists of r complex sinusoids with or without damping. We study the robust recovery problem for the spectrally sparse signal under the fully observed setting, which is about recovering x and a sparse corruption vector s from their sum z = x + s. In this paper, we exploit the low-rank property of the Hankel matrix formed by x, and formulate the problem as the robust recovery of a corrupted low-rank Hankel matrix. We develop a highly efficient nonconvex algorithm, coined accelerated structured alternating projections (ASAP). The high computational efficiency and low space complexity of ASAP are achieved by fast computations involving structured matrices, and a subspace projection method for accelerated low-rank approximation. Theoretical recovery guarantee with a linear convergence rate has been established for ASAP, under some mild assumptions on x and s. Empirical performance comparisons on both synthetic and real-world data confirm the advantages of ASAP, in terms of computational efficiency and robustness aspects.

Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery