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Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions
Richard Duong, Viktor Stein, Robert Beinert, Johannes Hertrich, Gabriele Steidl
We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals ℱ_ν := MMD_K²(·, ν) towards given target measures ν 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 𝒞(0,1) ⊂L₂(0,1) of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on L₂(0,1). Based on the construction of an appropriate counterpart of ℱ_ν on L₂(0,1) and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures ν, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of 𝒞(0,1). For certain ℱ_ν-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 ν, also the explicit Euler scheme can be employed, although with limited convergence guarantees.
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