Disentangling Orthogonal Matrices

TL;DR

This paper introduces a semi-definite programming approach to solve linear systems involving two unknown orthogonal matrices, improving over naive methods and extending to multiple matrices, with theoretical guarantees and empirical validation.

Contribution

It presents a novel SDP-based algorithm for solving systems with multiple unknown orthogonal matrices, generalizing the orthogonal Procrustes problem with performance guarantees.

Findings

The SDP relaxation outperforms naive solutions in accuracy.

The algorithm is effective for systems with more than two orthogonal matrices.

Theoretical analysis confirms the method's performance bounds.

Abstract

Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal Procrustes problem. We propose an algorithm based on a semi-definite programming (SDP) relaxation, and give a theoretical guarantee for its performance. Both theoretically and empirically, the proposed algorithm performs better than the na\"{i}ve approach of solving the linear system directly without the orthogonal constraints. We also consider the generalization to linear systems with more than two unknown orthogonal matrices.