Vahid Shahverdi: Moment Constraints and Phase Recovery for Multireference Alignment

Abstract

Multireference alignment (MRA) involves recovering a signal from noisy samples subject to random circular shifts and is often considered a simplified model for single-particle cryo-electron microscopy (cryo-EM). In cryo-EM, the goal is to estimate a three-dimensional structure, such as a molecule, from two-dimensional noisy projections taken from unknown angles, analogous to the unknown shifts in MRA. In this talk, I will present the fundamental frameworks of both MRA and cryo-EM and briefly discuss current approaches, including Expectation Maximization and the Method of Moments. I will then introduce a novel method for MRA reconstruction with potential extensions to cryo-EM. Our approach focuses on optimizing a distance function defined on the phase manifold, a space of signals constrained by a fixed power spectrum. We analyze the properties of this function and characterize critical points that include the clean signal. Finally, using Morse theory, we investigate the nature of these critical points to gain deeper insights into the reconstruction process. This is joint work with Emanuel Ström and Joakim Andén.