Robust Spectral Compressed Sensing via Structured Matrix Completion

TL;DR

This paper introduces EMaC, a novel structured matrix completion algorithm for spectral compressed sensing that achieves perfect recovery of spectrally sparse signals from limited samples, even with noise or corruptions.

Contribution

We develop EMaC, a new convex optimization-based method that overcomes basis mismatch and does not require prior model order knowledge, enabling robust super-resolution recovery.

Findings

EMaC achieves perfect recovery with sample complexity of O(r log^4 n).

The method is stable under bounded noise and resilient to sample corruptions.

Numerical experiments validate the effectiveness and applicability of EMaC to super-resolution tasks.

Abstract

The paper explores the problem of \emph{spectral compressed sensing}, which aims to recover a spectrally sparse signal from a small random subset of its $n$ time domain samples. The signal of interest is assumed to be a superposition of $r$ multi-dimensional complex sinusoids, while the underlying frequencies can assume any \emph{continuous} values in the normalized frequency domain. Conventional compressed sensing paradigms suffer from the basis mismatch issue when imposing a discrete dictionary on the Fourier representation. To address this issue, we develop a novel algorithm, called \emph{Enhanced Matrix Completion (EMaC)}, based on structured matrix completion that does not require prior knowledge of the model order. The algorithm starts by arranging the data into a low-rank enhanced form exhibiting multi-fold Hankel structure, and then attempts recovery via nuclear norm minimization. Under mild incoherence conditions, EMaC allows perfect recovery as soon as the number of samples exceeds the order of $r\log^{4}n$, and is stable against bounded noise. Even if a constant portion of samples are corrupted with arbitrary magnitude, EMaC still allows exact recovery, provided that the sample complexity exceeds the order of $r^{2}\log^{3}n$. Along the way, our results demonstrate the power of convex relaxation in completing a low-rank multi-fold Hankel or Toeplitz matrix from minimal observed entries. The performance of our algorithm and its applicability to super resolution are further validated by numerical experiments.