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

2024-12-31

arXiv.org

AI Summary

Semi-supervised classification suffers from slow convergence and limited robustness in conventional gradient-flow-based label propagation methods, which rely on parabolic PDEs and lack momentum acceleration. Method: This work proposes a momentum-accelerated minimization framework for the Ginzburg–Landau (GL) functional, constructing a damped hyperbolic dynamical system—both in Euclidean space and on graphs. It introduces, for the first time, momentum optimization into diffusion-interface modeling, employing a FISTA-type temporal discretization coupled with convex–concave splitting, and integrating graph signal processing with continuous functional discretization techniques. Contribution/Results: Theoretically and empirically, the method exhibits super-convergence under large step sizes; crucially, typical hyperbolic singularities (e.g., loss of solution regularity) vanish in practice—bypassing stringent PDE regularity constraints. It significantly improves label propagation efficiency and robustness, demonstrating both the theoretical validity and practical efficacy of momentum mechanisms in optimizing diffusive energy functionals.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Minimizing Ginzburg-Landau functional using momentum-based methods
• Comparing gradient flow and momentum methods for PDE solutions
• Analyzing convergence and singular limits in phase field models

Innovation

• Momentum-based minimization of Ginzburg-Landau functional
• Convex-concave splitting FISTA-type time discretization
• Singular limit analysis via formal expansions