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

• URL: http://arxiv.org/abs/2501.00389v1
• Date: Tue, 31 Dec 2024 11:05:49 GMT
• Title: Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs
• Authors: Oluwatosin Akande, Patrick Dondl, Kanan Gupta, Akwum Onwunta, Stephan Wojtowytsch,
• Abstract summary: 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. We demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large.
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• 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.

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