Orthogonal Trace-Sum Maximization: Tightness of the Semidefinite Relaxation and Guarantee of Locally Optimal Solutions

TL;DR

This paper demonstrates that a semidefinite relaxation accurately solves a generalized orthogonal trace-sum maximization problem with high probability under small noise, and that a specific nonconvex algorithm also finds the global solution under similar conditions.

Contribution

It establishes the tightness of the semidefinite relaxation and guarantees global optimality of a nonconvex algorithm for a generalized synchronization problem.

Findings

Semidefinite relaxation solves the nonconvex problem exactly with high probability.

The nonconvex algorithm's solution is globally optimal under small noise.

Results generalize phase synchronization to broader orthogonal trace-sum problems.

Abstract

This paper studies an optimization problem on the sum of traces of matrix quadratic forms in $m$ semi-orthogonal matrices, which can be considered as a generalization of the synchronization of rotations. While the problem is nonconvex, the paper shows that its semidefinite programming relaxation solves the original nonconvex problems exactly with high probability, under an additive noise model with small noise in the order of $O(m^{1/4})$. In addition, it shows that the solution of a nonconvex algorithm considered in Won, Zhou, and Lange [SIAM J. Matrix Anal. Appl., 2 (2021), pp. 859-882] is also its global solution with high probability under similar conditions. These results can be considered as a generalization of existing results on phase synchronization.