标题： 欧氏空间和图上的金兹堡-朗道函数的基于动量的最小化 显示英文标题

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

摘要： 我们研究了欧氏空间和图上的弥散周长泛函的基于动量的最小化问题，并将其应用于机器学习中的半监督分类任务。虽然当前任务中的梯度流是一个抛物线型偏微分方程，但动量方法对应于一个阻尼双曲型偏微分方程，从而导致定性和定量上不同的轨迹。使用基于凸凹分裂的FISTA型时间离散化，我们通过经验证明，如果时间步长较大但不过大，动量可以导致更快的收敛。 当时间步长较大时，偏微分方程分析对解的几何行为的洞察有限，并且在样本模拟中未观察到典型的双曲现象，例如失去规律性。 显示英文摘要

摘要： 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.