The Generalized Method of Moments for Multi-Reference Alignment

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.

Automated summary

This paper studies the application of the generalized method of moments (GMM) to multi-reference alignment (MRA). The paper proves the asymptotic optimality of the GMM estimator for statistical models with group symmetry, including MRA, and demonstrates the significant improvement of GMM compared to the classical method of moments in MRA through comprehensive numerical experiments. The paper also presents the formulation of GMM for estimating a three-dimensional molecular structure using cryo-electron microscopy and provides numerical results on simulated data.