# The generalized method of moments for multi-reference alignment

arXiv: https://arxiv.org/abs/2103.02215 · HTML: https://mlsanity.com/paper/2103.02215/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2021.

## Built on by (most cited first, 4 of 4)

- [An accelerated expectation-maximization algorithm for multi-reference alignment](https://mlsanity.com/paper/2105.07372/) (2021)
- [Unrolled algorithms for group synchronization](https://mlsanity.com/paper/2207.09418/) (2022)
- [Diagonally-Weighted Generalized Method of Moments Estimation for Gaussian Mixture Modeling](https://mlsanity.com/paper/2507.20459/) (2025)
- [If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics](https://mlsanity.com/paper/2510.09902/) (2025)

## Abstract (arXiv)

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.