论文标题
关于沃斯坦斯坦空间上的公制投影操作员的非专业性
On nonexpansiveness of metric projection operators on Wasserstein spaces
论文作者
论文摘要
在本文中，我们研究了度量预测的特性，这些属性在特定的封闭式和地理上凸出的瓦斯坦斯坦空间的适当子集$（\ nathcal {p} _p（\ mathbf {r}^d），w_p），w_p），w_p），$ d = 1 $时具有希尔伯特结构的平面空间的同构，相应的投影算子有望具有非专业性。我们直接证明了这一事实，依靠固有分析，这也意味着在较高维度的某些特殊情况下，也没有专业。当$ d> 1 $时，我们在两个制度中显示了该物业的失败：当$ p> 1 $的小度或足够大时。 Finally, we prove some positive curvature properties of Wasserstein spaces $(\mathcal{P}_p(\mathbf{R}^d),W_p)$ when $d\ge 2$ and $p\in(1,+\infty)$ are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called投影标准。作为前者的推论，它们具有非负上的Alexandrov曲率，我们在这里定义了一种确切的意义。在我们的分析中，特定的概率度量的特定子集具有密度在上面的密度均匀界定的特殊作用。
In this paper we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces $(\mathcal{P}_p(\mathbf{R}^d),W_p).$ When $d=1$, as $(\mathcal{P}_2(\mathbf{R}),W_2)$ is isometrically isomorphic to a flat space with a Hilbertian structure, the corresponding projection operators are expected to be nonexpansive. We give a direct proof of this fact, relying on intrinsic analysis, which also implies nonexpansiveness in certain special cases in higher dimensions. When $d>1$, we show the failure of this property in two regimes: when $p>1$ is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces $(\mathcal{P}_p(\mathbf{R}^d),W_p)$ when $d\ge 2$ and $p\in(1,+\infty)$ are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called projection criterion. As a corollary of the former, they have nonnegative upper Alexandrov curvature, in a precise sense that we define here. In our analysis a particular subset of probability measures having densities uniformly bounded above by a given constant plays a special role.
