Moment Constraints and Phase Recovery for Multireference Alignment

TL;DR

This paper introduces a novel method for multireference alignment that combines moment constraints with phase recovery, leading to faster and more accurate signal reconstruction from noisy, shifted samples.

Contribution

The authors propose a new approach that integrates moment constraints with a gradient-based likelihood maximization on a manifold, improving speed and accuracy over existing methods.

Findings

Faster reconstruction compared to EM algorithms.

Improved accuracy over bispectrum-based methods.

Effective phase recovery using moment constraints.

Abstract

Multireference alignment (MRA) refers to the problem of recovering a signal from noisy samples subject to random circular shifts. Expectation--maximization (EM) and variational approaches use statistical modeling to achieve high accuracy at the cost of solving computationally expensive optimization problems. The method of moments, instead, achieves fast reconstructions by utilizing the power spectrum and bispectrum to determine the signal up to shift. Our approach combines the two philosophies by viewing the power spectrum as a manifold on which to constrain the signal. We then maximize the data likelihood function on this manifold with a gradient-based approach to estimate the true signal. Algorithmically, our method involves iterating between template alignment and projections onto the manifold. The method offers increased speed compared to EM and demonstrates improved accuracy over bispectrum-based methods.