Global Optimization with Orthogonality Constraints via Stochastic Diffusion on Manifold
This work provides a theoretically grounded global optimization approach for orthogonality-constrained problems, which are common in science and engineering, but the novelty is incremental as it combines existing SDE and manifold techniques.
The paper proposes a stochastic diffusion method on the Stiefel manifold for global optimization under orthogonality constraints, achieving convergence to global minimizers with theoretical guarantees. The method is demonstrated on problems like homogeneous polynomial optimization and Cryo-EM 3D structure determination.
Orthogonality constrained optimization is widely used in applications from science and engineering. Due to the nonconvex 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 noisy gradient flow constructed from stochastic differential equations (SDE) 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 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, computation of stability number, and 3D structure determination from Common Lines in Cryo-EM.