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