Hankel Matrix Nuclear Norm Regularized Tensor Completion for $N$-dimensional Exponential Signals

TL;DR

This paper proposes a novel tensor completion method using Hankel matrix nuclear norm regularization to recover high-dimensional exponential signals from limited data, improving efficiency and robustness.

Contribution

It introduces a low-rank tensor completion framework that exploits exponential and CANDECOMP/PARAFAC structures with Hankel nuclear norm regularization for N-dimensional signals.

Findings

Successfully recovers signals from limited samples

Robust to tensor rank estimation

Effective on simulated and real data

Abstract

Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. But existing approaches can not efficiently recover $N$-dimensional exponential signals with $N\geq 3$. In this paper, we study the problem of recovering N-dimensional (particularly $N\geq 3$) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.