An Ultra-Fast MLE for Low SNR Multi-Reference Alignment

TL;DR

This paper presents a rapid, low-complexity algorithm for multi-reference alignment in low SNR conditions, inspired by cryo-electron microscopy, which outperforms traditional EM in speed and provides good initial estimates.

Contribution

The authors introduce a novel Taylor expansion-based method for MRA over SO(2), significantly reducing computational cost and improving initialization in low-SNR scenarios.

Findings

Achieves high accuracy in low-SNR environments

Requires only one data pass, unlike EM

Provides effective initialization for EM refinement

Abstract

Motivated by single-particle cryo-electron microscopy, multi-reference alignment (MRA) models the task of recovering an unknown signal from multiple noisy observations corrupted by random rotations. The standard approach, expectation-maximization (EM), often becomes computationally prohibitive, particularly in low signal-to-noise ratio (SNR) settings. We introduce an alternative, ultra-fast algorithm for MRA over the special orthogonal group $\mathrm{SO}(2)$. By performing a Taylor expansion of the log-likelihood in the low-SNR regime, we estimate the signal by sequentially computing data-driven averages of observations. Our method requires only one pass over the data, dramatically reducing computational cost compared to EM. Numerical experiments show that the proposed approach achieves high accuracy in low-SNR environments and provides an excellent initialization for subsequent EM refinement.