提出平滑距离核，兼顾计算效率与理论可导性。
Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows
- 用分数阶积分平滑绝对值函数，构造新核函数。
- 保持负距离核的线性增长与切片结构优势。
- 支持水车梯度流理论分析，适合需可导性的研究者。
负距离核 $K(x,y) := - \|x-y\|$ 在统计学中的最大均值差异（MMD）定义中被广泛应用，其无参结构在高维核求和的切片方法中表现优异。然而，由于在 $x=y$ 处不可微，经典理论（如对应 MMD 泛函的水车梯度流）不再成立。本文提出一种新核函数，既保持负距离核的一阶条件正定性、近似线性增长至无穷及简洁的切片结构，又具备 Lipschitz 可导性。构造基于一维绝对值函数的简单光滑化与 Riemann-Liouville 分数阶积分变换。数值实验表明，该新核在梯度下降方法中性能与负距离核相当，且具备理论保证。
原文摘要 · Abstract (English)
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 functional do not longer hold true. In this paper, we propose a new kernel which keeps the favorable properties of the negative distance kernel as being conditionally positive definite of order one with a nearly linear increase towards infinity and a simple slicing structure, but is Lipschitz differentiable now. Our construction is based on a simple 1D smoothing procedure of the absolute value function followed by a Riemann-Liouville fractional integral transform. Numerical results demonstrate that the new kernel performs similarly well as the negative distance kernel in gradient descent methods, but now with theoretical guarantees.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。