Super-resolution multi-reference alignment

TL;DR

This paper demonstrates that in super-resolution multi-reference alignment, a signal can be uniquely recovered from noisy, down-sampled observations if the number of samples per observation scales with the square root of the signal length, especially in low SNR conditions.

Contribution

The paper establishes a theoretical limit for signal recovery in super-resolution multi-reference alignment and introduces an EM algorithm capable of super-resolving signals in low SNR regimes.

Findings

Unique signal recovery when samples per observation are proportional to √M.

Recovery impossible with fewer observations, even without down-sampling.

Proposed EM algorithm effectively super-resolves signals in challenging noise conditions.

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 at least 1/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.