The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912087318462464 |
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| author | Ding, Zhiyan Epperly, Ethan N. Lin, Lin Zhang, Ruizhe |
| author_facet | Ding, Zhiyan Epperly, Ethan N. Lin, Lin Zhang, Ruizhe |
| contents | 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 $ε$ decays at a slow rate $ε=\mathcal{\tilde{O}}(n^{-1/2})$ 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 $ε= \mathcal{\tilde{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling beyond the Nyquist limit $ε= \mathcal{O}(n^{-1})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03885 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution Ding, Zhiyan Epperly, Ethan N. Lin, Lin Zhang, Ruizhe Information Theory Data Structures and Algorithms Signal Processing Statistics Theory 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 $ε$ decays at a slow rate $ε=\mathcal{\tilde{O}}(n^{-1/2})$ 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 $ε= \mathcal{\tilde{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling beyond the Nyquist limit $ε= \mathcal{O}(n^{-1})$ 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. |
| title | The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution |
| topic | Information Theory Data Structures and Algorithms Signal Processing Statistics Theory |
| url | https://arxiv.org/abs/2404.03885 |