直线上带距离核的差异函数的Wasserstein梯度流
Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line
AI总结：
本文利用Wasserstein空间到Hilbert空间的等距嵌入，研究实直线上带负距离核的最大均值差异泛函的凸性，并对Dirac测度终点情形解析求出了Wasserstein梯度流。
摘要：
本文给出了实直线上测度之间Wasserstein梯度流的结果。利用Wasserstein空间 $\mathcal P_2(\mathbb R)$ 到Hilbert空间 $L_2((0,1))$ 的等距嵌入，$\mathcal P_2(\mathbb R)$ 上泛函的Wasserstein梯度流可以被刻画为 $L_2((0,1))$ 上相应泛函的次梯度流。对于具有非光滑负距离核 $K(x,y) = -|x-y|$ 的最大均值差异泛函 $\mathcal F_ν:= \mathcal D^2_K(\cdot, ν)$，我们推导出相应泛函的公式。该泛函似乎是凸的，并且我们证明 $\mathcal F_ν$ 沿（广义）测地线是凸的。对于作为流终点的Dirac测度 $ν= δ_q$，$q \in \mathbb R$，这使我们能够解析地确定Wasserstein梯度流。文中给出了各种Wasserstein梯度流的例子以作说明。
英文摘要：
This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space $\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient flows of functionals on $\mathcal P_2(\mathbb R)$ can be characterized as subgradient flows of associated functionals on $L_2((0,1))$. For the maximum mean discrepancy functional $\mathcal F_ν:= \mathcal D^2_K(\cdot, ν)$ 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 $\mathcal F_ν$ is convex along (generalized) geodesics. For the Dirac measure $ν= δ_q$, $q \in \mathbb R$ 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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