Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line

01/11/2023

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by   Johannes Hertrich, et al.

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This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space 𝒫_2(ℝ) into the Hilbert space L_2((0,1)), Wasserstein gradient flows of functionals on 𝒫_2(ℝ) can be characterized as subgradient flows of associated functionals on L_2((0,1)). For the maximum mean discrepancy functional ℱ_ν := 𝒟^2_K(·, ν) with the non-smooth negative distance kernel K(x,y) = -|x-y|, we deduce a formula for the associated functional. This functional appears to be convex, and we show that ℱ_ν is convex along (generalized) geodesics. For the Dirac measure ν = δ_q, q ∈ℝ as end point of the flow, this enables us to determine the Wasserstein gradient flows analytically. Various examples of Wasserstein gradient flows are given for illustration.