Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion

TL;DR

This paper presents fast, provably effective algorithms for reconstructing spectrally sparse signals from limited samples by leveraging low-rank Hankel matrix completion, with theoretical guarantees and empirical validation.

Contribution

Introduction of IHT and FIHT algorithms with proven recovery guarantees for spectrally sparse signals using low-rank Hankel matrix completion.

Findings

FIHT guarantees exact recovery with O(r^2 log^2(n)) samples.

Algorithms outperform existing methods in computational efficiency.

Demonstrated success on large, high-dimensional data sets.

Abstract

A spectrally sparse signal of order $r$ is a mixture of $r$ damped or undamped complex sinusoids. This paper investigates the problem of reconstructing spectrally sparse signals from a random subset of $n$ regular time domain samples, which can be reformulated as a low rank Hankel matrix completion problem. We introduce an iterative hard thresholding (IHT) algorithm and a fast iterative hard thresholding (FIHT) algorithm for efficient reconstruction of spectrally sparse signals via low rank Hankel matrix completion. Theoretical recovery guarantees have been established for FIHT, showing that $O(r^2\log^2(n))$ number of samples are sufficient for exact recovery with high probability. Empirical performance comparisons establish significant computational advantages for IHT and FIHT. In particular, numerical simulations on $3$D arrays demonstrate the capability of FIHT on handling large and high-dimensional real data.