Maximum likelihood for high-noise group orbit estimation and cryo-EM

Abstract: Motivated by applications to single-particle cryo-electron microscopy, we study a problem of group orbit estimation where samples of an unknown signal are observed under uniform random rotations from a rotational group. In high-noise settings, we show that geometric properties of the log-likelihood function are closely related to algebraic properties of the invariant algebra of the group action. Eigenvalues of the Fisher information matrix are stratified according to a sequence of transcendence degrees in this invariant algebra, and critical points of the log-likelihood optimization landscape are in correspondence with those of a sequence of polynomial optimization problems. I will discuss the implications of this theory in several examples, including a simplified model of cryo-EM.

The talk will be based on the papers arxiv.org/abs/2004.00041 and arxiv.org/abs/2107.01305, joint work with Roy Lederman, Yi Sun, Tianhao Wang, Yihong Wu, and Sheng Xu.

Bio: Zhou Fan is an Assistant Professor in the Department of Statistics and Data Science at Yale University. His research is driven by the goals of understanding high-dimensional phenomena in statistics and machine learning, developing computationally efficient algorithms for high-dimensional inference problems with structure, and bridging theoretical advances in these areas with scientific applications. Recent interests include random matrix theory and free probability, statistical physics and high-dimensional Bayesian inference, and inferential problems arising in cryo-electron microscopy and statistical genetics.