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

arXiv: https://arxiv.org/abs/2404.03885 · HTML: https://mlsanity.com/paper/2404.03885/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2024.

## Built on by (most cited first, 1 of 1)

- [Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes](https://mlsanity.com/paper/2608.16126/) (2026)

## Abstract (arXiv)

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 O(1), but these analyses only show that the error ε decays at a slow rate ε=mathcaltildeO(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 ε = mathcaltildeO(n-3/2), exhibiting noisy super-resolution scaling beyond the Nyquist limit ε = 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.