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

TL;DR

The paper analyzes constrained saddle dynamics on manifolds for locating index-$k$ saddle points, establishing local convergence of discrete CSD under the exact unstable-eigenvector assumption with a rate governed by the Riemannian Hessian condition number $\kappa$. To address ill-conditioning, it introduces momentum-based constrained saddle dynamics (MCSD) and proves local convergence for both the continuous-time and discrete schemes, showing that momentum reduces sensitivity to $\kappa$ and accelerates convergence. It further proves that a one-step eigenvector update suffices, removing the need for exact eigenvectors at every iteration and drastically reducing computational cost. Numerical experiments on the Thomson problem, the Rayleigh quotient on the Stiefel manifold, and the Bose–Einstein condensate energy functional corroborate the theory and demonstrate practical acceleration, especially in ill-conditioned scenarios. Overall, the work provides a rigorous framework for efficient saddle-point computation under equality constraints with momentum-based acceleration and minimal eigenvector updating requirements.

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.

Figures (5)

• Figure 1: (a) The energy landscape of $f(x,y,z)=(x^2-1)^2+ay^2+2az^2$ with $a=2$ defined on a unit sphere, and the trajectory of the discrete CSD without momentum. (b) Error plots for the discrete CSD without momentum under different condition numbers. As $a$ decreases, the condition number of $\mathrm{Hess} f(x^{*})$ increases, which in turn yields a slower linear convergence rate. (c) For (a=0.1) (ill-conditioned), trajectories of the discrete CSD without momentum ($\gamma=0$, black) and MCSD with momentum ($\gamma=0.9$, light sky blue). (d) For $a=0.1$, error plots for the discrete CSD without momentum ($\gamma=0$) and MCSD with momentum ($\gamma=0.9$) for the step size $\Delta t=0.01$.
• Figure 2: (a) The energy landscape of $f(x,y,z)=-y^2-0.05z^2$ defined on a cylinder, and trajectories of the discrete CSD without momentum ($\gamma=0$, black) and MCSD with momentum ($\gamma=0.9$, light sky blue). (d) Error plots for the discrete CSD without momentum ($\gamma=0$) and MCSD with momentum ($\gamma=0.5,0.9,0.99$) for the step size $\Delta t=0.01$.
• Figure 3: Error plot for the discrete CSD without momentum ($\gamma= 0$) and MCSD with momentum ($\gamma = 0.5, 0.9, 0.99$) applied on the Thomson problem with particle number $M=5$ and step size $\Delta t=0.001$. Each red ball represents a particle on the sphere.
• Figure 4: Error plot for the discrete CSD ($\gamma= 0$) and MCSD ($\gamma = 0.5, 0.9, 0.99$) applied on the Rayleigh quotient problem with $n=100,p=2$. The step size is chosen as $\Delta t=0.01$.
• Figure 5: Error plot for the discrete CSD without momentum ($\gamma= 0$) and MCSD with momentum ($\gamma = 0.9, 0.99, 0.995$) applied on the BEC functional. The step size is chosen as $\Delta t=0.001$. The color bar represents the probability density $|\phi|^2$ for the state. The subscript $\gamma=0,n=6000$ corresponds to the configuration obtained after $n=6000$ iterations of the discrete CSD without momentum ($\gamma=0$).

Theorems & Definitions (34)

• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3: Local convergence of the MCSD
• proof
• Remark 3.4
• Remark 3.5
• Remark 4.1
• Remark 4.3
• Lemma 4.4: The smoothness of the reflection operator