Non-unique games over compact groups and orientation estimation in cryo-EM

TL;DR

This paper introduces a semidefinite programming relaxation for the Non-Unique Games problem over compact groups, enabling efficient solutions for orientation estimation in cryo-EM and related tasks.

Contribution

It generalizes existing problems like the little Grothendieck and Unique Games, providing a unified SDP framework for orientation estimation and registration tasks.

Findings

Efficient SDP relaxation for NUG over compact groups.

Application to cryo-EM orientation estimation.

Unified framework for related registration problems.

Abstract

Let $\mathcal{G}$ be a compact group and let $f_{ij} \in L^2(\mathcal{G})$. We define the Non-Unique Games (NUG) problem as finding $g_1,\dots,g_n \in \mathcal{G}$ to minimize $\sum_{i,j=1}^n f_{ij} \left( g_i g_j^{-1}\right)$. We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of $f_{ij}$ over $\mathcal{G}$, which can then be solved efficiently. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the Unique Games problem and includes many practically relevant problems, such as the maximum likelihood estimator} to registering bandlimited functions over the unit sphere in $d$-dimensions and orientation estimation in cryo-Electron Microscopy.