Fast and Provable Hankel Tensor Completion for Multi-measurement Spectral Compressed Sensing

TL;DR

This paper presents a fast, provably accurate low-rank Hankel tensor completion method for multi-measurement spectral compressed sensing, improving efficiency and recovery guarantees over previous approaches.

Contribution

Introduces ScalHT, a scalable gradient descent algorithm with theoretical guarantees for low-rank Hankel tensor completion in spectral compressed sensing.

Findings

Achieves up to $O( ext{min}\{s,n\ imes ext{folding} ext{ in storage and computation}

Demonstrates superior recovery performance in simulations

Provides the first theoretical guarantees for this tensor completion approach

Abstract

In this paper, we introduce a novel low-rank Hankel tensor completion approach to address the problem of multi-measurement spectral compressed sensing. By lifting the multiple signals to a Hankel tensor, we reformulate this problem into a low-rank Hankel tensor completion task, exploiting the spectral sparsity via the low multilinear rankness of the tensor. Furthermore, we design a scaled gradient descent algorithm for Hankel tensor completion (ScalHT), which integrates the low-rank Tucker decomposition with the Hankel structure. Crucially, we derive novel fast computational formulations that leverage the interaction between these two structures, achieving up to an $O(\min\{s,n\})$-fold improvement in storage and computational efficiency compared to the existing algorithms, where $n$ is the length of signal, $s$ is the number of measurement vectors. Beyond its practical efficiency, ScalHT is backed by rigorous theoretical guarantees: we establish both recovery and linear convergence guarantees, which, to the best of our knowledge, are the first of their kind for low-rank Hankel tensor completion. Numerical simulations show that our method exhibits significantly lower computational and storage costs while delivering superior recovery performance compared to prior arts.