  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.
