Constrained dynamics for searching saddle points on general Riemannian manifolds

Abstract: Finding constrained saddle points on Riemannian manifolds is significant for analyzing energy landscapes arising in physics and chemistry. Existing works have been limited to special manifolds that admit global regular level-set representations, excluding applications such as electronic excited-state calculations. In this paper, we develop a constrained saddle dynamics applicable to smooth functions on general Riemannian manifolds. Our dynamics is formulated compactly over the Grassmann bundle of the tangent bundle. By analyzing the Grassmann bundle geometry, we achieve universality via incorporating the second fundamental form, which captures variations of tangent spaces along the trajectory. We rigorously establish the local linear stability of the dynamics and the local linear convergence of the resulting algorithms. Remarkably, our analysis provides the first convergence guarantees for discretized saddle-search algorithms in manifold settings. Moreover, by respecting the intrinsic quotient structure, we remove unnecessary nondegeneracy assumptions on the eigenvalues of the Riemannian Hessian that are present in existing works. We also point out that locating saddle points can be more ill-conditioning than finding local minimizers, and requires using nonredundant parametrizations. Finally, numerical experiments on linear eigenvalue problems and electronic excited-state calculations showcase the effectiveness of the proposed algorithms and corroborate the established local theory.