AI 中文总结
本文构建了Z-GW空间等三类关联空间的统一度量几何理论，明确了其间的次淹没映射结构，证明了不同p值下的测地性关系，刻画了Z-GW空间的测地线并分类了其Alexandrov曲率界。
AI 中文摘要
度量空间Z会衍生出三类与Z相关的无穷维度量空间：Z上概率测度的p-Wasserstein空间、Z值映射的非线性勒贝格L^p空间，以及Z值核的p-Gromov-Wasserstein空间。后一类空间被称为Z-Gromov-Wasserstein（Z-GW）空间，它将经典的Gromov-Wasserstein框架从度量测度空间拓展到更一般的、可能带属性的类网络结构，统一了诸多如今在度量几何、数据科学和机器学习领域发挥重要作用的GW型距离。本文针对这三类空间构建了统一的度量几何理论，尤其聚焦于Z-GW空间。我们的首个主要结果明确了关联这三类空间的基础次淹没映射结构：非线性勒贝格空间通过一次淹没映射映射到Z-GW空间，而Z-GW空间又通过一次淹没映射映射到Wasserstein空间。该结构为在三类空间间传递几何信息提供了机制。我们将该框架应用于测地线与Alexandrov曲率：对于1<p<∞，我们证明Z的测地性等价于三类关联空间各自的测地性；在端点情形p=1时，即便Z不具备测地性，三类关联空间均为测地空间。我们还将Z-GW空间中的测地线刻画为广义插值，拓展了Sturm在经典情形下的已知刻画。最后，我们基于Z的曲率，对这些空间的Alexandrov曲率界给出了完整分类。因此，本文的核心重点是构建Z-GW空间的新度量几何理论，同时该次淹没映射框架也拓展了Wasserstein空间与Gromov-Wasserstein空间的经典定理，并为非线性勒贝格空间带来了新的几何推论。
英文摘要
A metric space $Z$ gives rise to three natural classes of infinite-dimensional metric spaces associated to $Z$: $p$-Wasserstein spaces of probability measures on $Z$, nonlinear Lebesgue $L^p$-spaces of $Z$-valued maps, and $p$-Gromov-Wasserstein spaces of $Z$-valued kernels. The latter class, referred to as $Z$-Gromov-Wasserstein ($Z$-GW) spaces, extends the classical Gromov-Wasserstein framework from metric measure spaces to more general, possibly attributed, network-like structures, and unifies many GW-type distances that nowadays play a significant role in metric geometry, data science and machine learning. In this article we develop a unified metric-geometric theory of these three classes of spaces, with a particular focus on the $Z$-GW spaces. Our first main result identifies a fundamental submetry structure linking them: the nonlinear Lebesgue space maps via a submetry onto the $Z$-GW space, which in turn maps via a submetry onto the Wasserstein space. This structure provides a mechanism for transferring geometric information among the three spaces. We apply this framework to geodesics and Alexandrov curvature. For $1<p<\infty$, we prove that geodesicity of $Z$ is equivalent to geodesicity of each of the three associated spaces; in the endpoint case $p=1$, all three associated spaces are geodesic, even when $Z$ is not. We also characterize geodesics in the $Z$-GW space as generalized interpolations, extending a known characterization in the classical setting due to Sturm. Finally, we give a complete classification of Alexandrov curvature bounds for these spaces in terms of the curvature of $Z$. Thus, while the main focus of the paper is a new metric-geometric theory of $Z$-GW spaces, the submetry framework also extends classical theorems for Wasserstein and Gromov-Wasserstein spaces and yields new geometric consequences for nonlinear Lebesgue spaces.
Comments48 pages, 3 figures
