Robust spectral compressive sensing via vanilla gradient descent

TL;DR

This paper introduces a hyperparameter-free vanilla gradient descent method for robust spectral compressive sensing, achieving near-optimal sample complexity and outperforming existing approaches in noisy settings.

Contribution

Proposes a novel, hyperparameter-free vanilla gradient descent algorithm for spectral compressive sensing with theoretical guarantees under noise.

Findings

VGD recovers spectrally sparse signals from noisy samples with near-optimal sample complexity.

Theoretical analysis confirms robustness of VGD under mild conditions.

Numerical results demonstrate VGD's superior performance over existing methods.

Abstract

This paper investigates the recovery of a spectrally sparse signal from its partially revealed noisy entries within the framework of spectral compressive sensing. Nonconvex optimization approaches have recently been proposed based on low-rank Hankel matrix completion and projected gradient descent (PGD). The PGD however involves unknown tuning parameters and its theoretical analysis is available only in the absence of noise. In this paper, we propose a hyperparameter-free, vanilla gradient descent (VGD) algorithm and prove that the VGD enables robust recovery of an $N$-dimensional $K$-spectrally-sparse signal from order $K^2 log^2N$ number of noisy samples under coherence and other mild conditions. The above sample complexity increases by factor $logN$ as compared with PGD without noise. Numerical simulations are provided that corroborate our analysis and show advantageous performances of VGD.