Statistical estimation in the presence of group actions

Abstract

Imagine we want to recover an unknown vector given many noisy copies of it, except each copy is cyclically shifted by an unknown offset (this is "multi-reference alignment"). Or imagine we want to reconstruct an unknown 3D structure (e.g. a molecule) given many noisy pictures of it taken from different unknown angles (this is "cryo-EM"). These problems (and many others) involve the action of unknown group elements drawn randomly from a compact group such as Z/p or SO(3). In this thesis we study two statistical models for estimation in the presence of group actions. The first is the synchronization model in which we attempt to learn an unknown collection of group elements based on noisy pairwise comparisons. The second is the orbit recovery model in which we observe noisy copies of a hidden signal, each of which is acted upon by a random group element. For both of these models, we explore the fundamental statistical limits as well as the fundamental computational limits (i.e. how well can a polynomial-time algorithm perform?). We use methods from a wide variety of areas, including statistical physics, approximate message passing, representation theory, contiguity and the associated second moment method, invariant theory, algebraic geometry, and the sum-of-squares hierarchy..