The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution
<|reference_start|>The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution: 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 $n$. In this work, we prove that under certain assumptions of bias and high noise, 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. 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.<|reference_end|>
@article{ding2024the,
    title={The ESPRIT algorithm under high noise: Optimal error scaling and noisy
  super-resolution},
    author={Zhiyan Ding, Ethan N. Epperly, Lin Lin, Ruizhe Zhang},
    journal={arXiv preprint arXiv:2404.03885},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.03885},
    primaryClass={cs.IT cs.DS eess.SP math.IT math.ST stat.TH}
}