# Super-resolution multi-reference alignment

Tamir Bendory, Ariel Jaffe, William Leeb, Nir Sharon, Amit Singer

Research output: Contribution to journalArticlepeer-review

## 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.

Original language English 533-555 23 Information and Inference 11 2 https://doi.org/10.1093/imaiai/iaab003 Published - 1 Jun 2022

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