The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution

TL;DR

This paper demonstrates that the ESPRIT algorithm can achieve near-super-resolution error scaling of (n^{-3/2}) under high noise, surpassing the classical Nyquist limit, and establishes this as an optimal bound.

Contribution

The work proves improved error scaling for ESPRIT under high noise and introduces new matrix perturbation results, extending understanding of super-resolution capabilities.

Findings

ESPRIT attains (n^{-3/2}) error scaling under high noise.

This scaling surpasses the Nyquist limit of (n^{-1}) for spectral estimation.

The (n^{-3/2}) scaling is proven to be optimal.

Abstract

Subspace-based signal processing techniques, such as the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular methods for spectral estimation. These algorithms can achieve the so-called super-resolution scaling under low noise conditions, surpassing the well-known Nyquist limit. However, the performance of these algorithms under high-noise conditions is not as well understood. Existing state-of-the-art analysis indicates that ESPRIT and related algorithms can be resilient even for signals where each observation is corrupted by statistically independent, mean-zero noise of size $\mathcal{O}(1)$, but these analyses only show that the error $\epsilon$ decays at a slow rate $\epsilon=\mathcal{\tilde{O}}(n^{-1/2})$ with respect to the cutoff frequency $n$ (i.e., the maximum frequency of the measurements). In this work, we prove that under certain assumptions, the ESPRIT algorithm can attain a significantly improved error scaling $\epsilon = \mathcal{\tilde{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling beyond the Nyquist limit $\epsilon = \mathcal{O}(n^{-1})$ given by the Nyquist-Shannon sampling theorem. We further establish a theoretical lower bound and show that this scaling is optimal. Our analysis introduces novel matrix perturbation results, which could be of independent interest.