Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows
Nicolaj Rux and Michael Quellmalz and Gabriele Steidl
负距离内核 K(x,y):= - x-y 用于统计中最大均值差异(MMD)的定义,并导致各种应用中有利的数值结果。 特别是,处理高维内核求和的所谓切片技术从距离内核的简单无参数结构中获利。 然而,由于其在x=y中的不平滑,大多数经典理论结果,例如,在Wasserstein梯度流的相应MMD功能不再成立。 在本文中,我们提出了一个新的内核,它保持负距离内核的有利属性,作为条件正确定的顺序,具有接近线性的无限和简单的切片结构,但现在是Lipschitz可微分的。 我们的构造基于绝对值函数的简单1D平滑过程,然后是黎曼 - 廖维尔分数积分变换。 数字结果表明,新内核在梯度下降方法中的表现与负距离内核相似,但现在具有理论保证。
Negative distance kernels K(x,y) := - x-y were used in the definition of maximum mean discrepancies (MMDs) in statistics and lead to favorable numerical results in various applications. In particular, so-called slicing techniques for handling high-dimensional kernel summations profit from the simple parameter-free structure of the distance kernel. However, due to its non-smoothness in x=y, most of the classical theoretical results, e.g. on Wasserstein gradient flows of the corresponding MMD func...