Multi-reference alignment in high dimensions: sample complexity and phase transition

TL;DR

This paper investigates the sample complexity of multi-reference alignment in high dimensions, revealing a phase transition at a critical parameter value that drastically changes the difficulty of estimating signals from noisy, shifted copies.

Contribution

It introduces a high-dimensional analysis of the problem, identifying a phase transition in sample complexity governed by the dimension and noise level, extending prior finite-dimensional studies.

Findings

For lpha > 2, sample complexity matches that of standard Gaussian noise estimation.

For lpha , sample complexity increases significantly, indicating a harder estimation problem.

A phase transition phenomenon is characterized by the parameter lpha = L/(^2  log L).

Abstract

Multi-reference alignment entails estimating a signal in $\mathbb{R}^L$ from its circularly-shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finite-dimensional setting (fixed $L$). Motivated by single-particle cryo-electron microscopy, we analyze the sample complexity of the problem in the high-dimensional regime $L\to\infty$. Our analysis uncovers a phase transition phenomenon governed by the parameter $\alpha = L/(\sigma^2\log L)$, where $\sigma^2$ is the variance of the noise. When $\alpha>2$, the impact of the unknown circular shifts on the sample complexity is minor. Namely, the number of measurements required to achieve a desired accuracy $\varepsilon$ approaches $\sigma^2/\varepsilon$ for small $\varepsilon$; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when $\alpha\leq 2$, the problem is significantly harder and the sample complexity grows substantially quicker with $\sigma^2$.