The generalized method of moments for multi-reference alignment

TL;DR

This paper demonstrates that the generalized method of moments (GMM) is asymptotically optimal for multi-reference alignment (MRA) and significantly outperforms classical methods, with applications to cryo-electron microscopy.

Contribution

It proves the asymptotic optimality of GMM for models with group symmetry and applies it to 3D molecular structure estimation in cryo-EM.

Findings

GMM outperforms classical method of moments in MRA.

GMM maintains asymptotic optimality for symmetric models.

Numerical results show effectiveness in cryo-EM simulations.

Abstract

This paper studies the application of the generalized method of moments (GMM) to multi-reference alignment (MRA): the problem of estimating a signal from its circularly-translated and noisy copies. We begin by proving that the GMM estimator maintains its asymptotic optimality for statistical models with group symmetry, including MRA. Then, we conduct a comprehensive numerical study and show that the GMM substantially outperforms the classical method of moments, whose application to MRA has been studied thoroughly in the literature. We also formulate the GMM to estimate a three-dimensional molecular structure using cryo-electron microscopy and present numerical results on simulated data.