Convergence Analysis of the Discrete Constrained Saddle Dynamics and Their Momentum Variants

TL;DR

This paper analyzes the convergence of discrete constrained saddle dynamics and introduces momentum variants to accelerate convergence, especially in ill-conditioned problems, with theoretical proofs and numerical experiments on various applications.

Contribution

It provides the first convergence analysis of discrete constrained saddle dynamics, introduces momentum-based variants, and reduces computational costs by eliminating the need for exact eigenvector computations.

Findings

Momentum accelerates convergence in ill-conditioned problems.

Single-step eigenvector updates suffice for convergence.

Numerical experiments validate theoretical results.

Abstract

We study the discrete constrained saddle dynamics and their momentum variants for locating saddle points on manifolds. Under the assumption of exact unstable eigenvectors, we establish a local linear convergence of the discrete constrained saddle dynamics and show that the convergence rate depends on the condition number of the Riemannian Hessian. To mitigate this dependence, we introduce a momentum-based constrained saddle dynamics and prove local convergence of the continuous-time dynamics as well as the corresponding discrete scheme, which further demonstrates that momentum accelerates convergence, particularly in ill-conditioned settings. In addition, we show that a single-step eigenvector update is sufficient to guarantee local convergence; thus, the assumption of exact unstable eigenvectors is not necessary, which substantially reduces the computational cost. Finally, numerical experiments, including applications to the Thomson problem, the Rayleigh quotient on the Stiefel manifold, and the energy functional of Bose-Einstein condensates, are presented to complement the theoretical analysis.