Vandermonde Factorization of Hankel Matrix for Complex Exponential Signal Recovery -- Application in Fast NMR Spectroscopy

TL;DR

This paper introduces a Vandermonde factorization approach for Hankel matrix completion to recover exponential signals from limited samples, significantly improving performance in applications like fast NMR spectroscopy.

Contribution

It proposes a novel Hankel matrix completion method with Vandermonde factorization (HVaF) that outperforms existing techniques in signal recovery tasks.

Findings

HVaF succeeds over a wider regime than nuclear-normminimization methods.

HVaF has fewer restrictions on frequency separation compared to atomic norm minimization.

HVaF is validated on biological magnetic resonance spectroscopy data.

Abstract

Many signals are modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. This paper studies the problem of recovering exponential signals from a random subset of samples. We exploit the Vandermonde structure of the Hankel matrix formed by the exponential signal and formulate signal recovery as Hankel matrix completion with Vandermonde factorization (HVaF). A numerical algorithm is developed to solve the proposed model and its sequence convergence is analyzed theoretically. Experiments on synthetic data demonstrate that HVaF succeeds over a wider regime than the state-of-the-art nuclear-normminimization-based Hankel matrix completion method, while has a less restriction on frequency separation than the state-of-the-art atomic norm minimization and fast iterative hard thresholding methods. The effectiveness of HVaF is further validated on biological magnetic resonance spectroscopy data.