Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions

2024-08-14

AI Summary

This work investigates the gradient flow of the Maximum Mean Discrepancy (MMD) functional under a negative distance kernel with respect to the target measure in the Wasserstein-2 space, focusing on the one-dimensional setting. Methodologically, it leverages the isometric embedding via quantile functions to recast the Wasserstein gradient flow as a Cauchy problem in (L^2(0,1)), and constructs, for the first time, the corresponding MMD functional and its subdifferential structure in this embedded space. Theoretically, it derives an analytical piecewise-linear solution for discrete targets and rigorously proves that point-mass initial distributions instantaneously evolve into absolutely continuous ones. Algorithmically, it proposes an efficient numerical solver combining implicit/explicit Euler schemes with bisection. Experiments confirm the flow’s invariance, smoothness, convergence, and stability.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Web measurements

Abstract

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $mathcal F_ u := ext{MMD}_K^2(cdot, u)$ towards given target measures $ u$ on the real line, where we focus on the negative distance kernel $K(x,y) := -|x-y|$. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone $mathcal C(0,1) subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $mathcal F_ u$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $ u$, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of $mathcal C(0,1)$. For certain $mathcal F_ u$-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme, which is easily computable by a bisection algorithm. For continuous targets $ u$, also the explicit Euler scheme can be employed, although with limited convergence guarantees.

Problem

• Characterize Wasserstein gradient flows of MMD functionals with distance kernel
• Solve Cauchy problems on quantile functions in L2 space
• Analyze smoothing properties and behavior of flows for discrete measures

Innovation

• Uses Wasserstein gradient flows for MMD functionals
• Employs negative distance kernel in one dimension
• Solves Cauchy problem on quantile functions