标题： 基于动量的欧几里得空间和图上的吉尼茨堡-朗道泛函最小化 显示英文标题

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

摘要： 我们研究在欧几里得空间和图上基于动量的扩散周长泛函最小化，并将其应用于机器学习中的半监督分类任务。 尽管该任务中的梯度流是一个抛物型偏微分方程，但动量方法对应于一个阻尼双曲型偏微分方程，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，我们实证表明，如果时间步长较大但不过大，动量可以导致更快的收敛。 在较大的时间步长下，PDE分析对解的几何行为提供的见解有限，典型的双曲现象如正则性的丧失在样本模拟中未被观察到。 通过形式展开，我们得到当相场长度参数趋于零时演化方程的奇异极限，并在二维圆上数值验证了其有效性。 我们的分析通过平面曲线、三维空间中的曲面以及图上的半监督学习任务的数值实验得到补充。 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. 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.