MAXIMUM LIKELIHOOD FOR HIGH-NOISE GROUP ORBIT ESTIMATION AND CRYO-EM
ZHOU FAN – YALE UNIVERSITY
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 https://arxiv.org/abs/2004.00041 and https://arxiv.org/abs/2107.01305.