Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction

TL;DR

This paper presents a method for robustly recovering complex exponential signals from limited Gaussian measurements by reconstructing a low-rank Hankel matrix, without requiring frequency separation conditions, and demonstrates its effectiveness through theoretical analysis and experiments.

Contribution

It introduces a nuclear norm minimization approach for low-rank Hankel matrix recovery that does not need frequency separation, improving measurement bounds for spectral compressed sensing.

Findings

Recovery guaranteed with O(R log^2 N) measurements

No incoherence or separation conditions needed

Effective in spectral compressed sensing and NMR sampling

Abstract

This paper explores robust recovery of a superposition of $R$ distinct complex exponential functions from a few random Gaussian projections. We assume that the signal of interest is of $2N-1$ dimensional and $R<<2N-1$. This framework covers a large class of signals arising from real applications in biology, automation, imaging science, etc. To reconstruct such a signal, our algorithm is to seek a low-rank Hankel matrix of the signal by minimizing its nuclear norm subject to the consistency on the sampled data. Our theoretical results show that a robust recovery is possible as long as the number of projections exceeds $O(R\ln^2N)$. No incoherence or separation condition is required in our proof. Our method can be applied to spectral compressed sensing where the signal of interest is a superposition of $R$ complex sinusoids. Compared to existing results, our result here does not need any separation condition on the frequencies, while achieving better or comparable bounds on the number of measurements. Furthermore, our method provides theoretical guidance on how many samples are required in the state-of-the-art non-uniform sampling in NMR spectroscopy. The performance of our algorithm is further demonstrated by numerical experiments.