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

arXiv: https://arxiv.org/abs/2501.00389 · HTML: https://mlsanity.com/paper/2501.00389/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 76 references that are in the map.
- First posted on arXiv 31 Dec 2024.

## Builds on (most cited first, 14 of 14)

- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)
- [Convergence of stochastic gradient descent schemes for Lojasiewicz-landscapes](https://mlsanity.com/paper/2102.09385/) (2021)
- [Properly-weighted graph Laplacian for semi-supervised learning](https://mlsanity.com/paper/1810.04351/) (2018)
- [Provable non-accelerations of the heavy-ball method](https://mlsanity.com/paper/2307.11291/) (2023)
- [Nesterov acceleration in benignly non-convex landscapes](https://mlsanity.com/paper/2410.08395/) (2024)
- [A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces](https://mlsanity.com/paper/2310.17610/) (2023)
- [Study of the behaviour of Nesterov Accelerated Gradient in a non convex setting: the strongly quasar convex case](https://mlsanity.com/paper/2405.19809/) (2024)
- [Generalizing diffuse interface methods on graphs: non-smooth potentials and hypergraphs](https://mlsanity.com/paper/1611.06094/) (2016)
- [PDE Acceleration: A convergence rate analysis and applications to obstacle problems](https://mlsanity.com/paper/1810.01066/) (2018)
- [Joint reconstruction-segmentation on graphs](https://mlsanity.com/paper/2208.05834/) (2022)
- [Models for information propagation on graphs](https://mlsanity.com/paper/2201.07577/) (2022)
- [BDF schemes for accelerated gradient flows in projection-free approximation of nonconvex constrained variational minimization](https://mlsanity.com/paper/2409.14670/) (2024)

## Abstract (arXiv)

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.