Super-resolution multi-reference alignment. (18th February 2021)

• Record Type: Journal Article
• Authors: Bendory, Tamir; Jaffe, Ariel; Leeb, William; Sharon, Nir; Singer, Amit
• Is Part Of: Information and inference. Volume 11:Number 2(2022)
• Page Start: 533
• Page End: 555
• Publication Date: 2021-02-18
• Subjects: Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaab003
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in ${\mathbb{R}}^M$ is uniquely determined when the number $L$ of samples per observation is of the order of the square root of the signal's length ($L=O(\sqrt{M})$ ). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to $1/\textrm{SNR}^3$ . In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled ($L=M$ ). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.