Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs

TL;DR

This paper explores a momentum-based approach to minimize the Ginzburg-Landau functional on Euclidean spaces and graphs, demonstrating faster convergence and analyzing the behavior of solutions through PDE analysis and numerical experiments.

Contribution

It introduces a momentum method for the Ginzburg-Landau functional minimization, analyzes its PDE dynamics, and validates the approach with numerical experiments on curves, surfaces, and graph-based learning.

Findings

Momentum accelerates convergence with appropriate step sizes.

Large step sizes lead to hyperbolic PDE behavior and loss of regularity.

Numerical validation confirms the singular limit for circles and applications to semi-supervised learning.

Abstract

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations. We obtain the singular limit of the evolution equations as the length parameter of the phase fields tends to zero by formal expansions and numerically confirm its validity for circles in two dimensions. Our analysis is complemented by numerical experiments for planar curves, surfaces in three-dimensional space, and semi-supervised learning tasks on graphs.