Global Optimization with Orthogonality Constraints via Stochastic Diffusion on Manifold

Orthogonality constrained optimization is widely used in applications from science and engineering. Due to the non-convex orthogonality constraints, many numerical algorithms often can hardly achieve the global optimality. We aim at establishing an efficient scheme for finding global minimizers under one or more orthogonality constraints. The main concept is based on the noisy gradient flow constructed from stochastic differential equations (SDEs) on the Stiefel manifold, the differential geometric characterization of orthogonality constraints. We derive an explicit representation of SDE on the Stiefel manifold endowed with a canonical metric and propose a numerically efficient scheme to simulate this SDE based on Cayley transformation with a theoretical convergence guarantee. The convergence to global optimizers is proved under second-order continuity. The effectiveness and efficiency of the proposed algorithms are demonstrated on a variety of problems including homogeneous polynomial optimization, bi-quadratic optimization, stability number computation, and 3D structure determination from common lines in Cryo-EM.