Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery

TL;DR

This paper introduces ASAP, an efficient non-convex algorithm for robustly recovering spectrally sparse signals and sparse corruptions by exploiting low-rank Hankel matrix structures, with proven linear convergence.

Contribution

The paper develops ASAP, a novel accelerated structured alternating projections algorithm that offers high efficiency and theoretical guarantees for robust spectral sparse signal recovery.

Findings

ASAP achieves faster recovery than existing methods.

Theoretical linear convergence of ASAP is established.

Empirical results show robustness and efficiency on synthetic and real data.

Abstract

Consider a spectrally sparse signal $\boldsymbol{x}$ 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 $\boldsymbol{x}$ and a sparse corruption vector $\boldsymbol{s}$ from their sum $\boldsymbol{z}=\boldsymbol{x}+\boldsymbol{s}$. In this paper, we exploit the low-rank property of the Hankel matrix formed by $\boldsymbol{x}$, and formulate the problem as the robust recovery of a corrupted low-rank Hankel matrix. We develop a highly efficient non-convex 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 $\boldsymbol{x}$ and $\boldsymbol{s}$. Empirical performance comparisons on both synthetic and real-world data confirm the advantages of ASAP, in terms of computational efficiency and robustness aspects.