We develop a unified framework for nonlinear subdivision schemes on complete metric spaces (CMS). We begin with CMS preliminaries and formalize refinement in CMS, retaining key structural properties, such as locality. We prove a convergence theorem under contractivity and demonstrate its applicability. To address schemes where contractivity is unknown, we introduce two notions of proximity. Our proximity methods relate a nonlinear scheme to another nonlinear scheme with known contractivity, rather than to a linear scheme, as in much of the literature. Specifically, the first type proximity compares the two schemes after a single refinement step and, as in the classical theory, yields convergence from sufficiently dense initial data. The proximity of the second type monitors alignment across all refinement levels and provides strong convergence without density assumptions. We formulate and prove the corresponding theorems, and illustrate them with various examples, such as schemes over metric spaces of compact sets in $\R^n$ and schemes over the Wasserstein space, as well as a geometric Hermite metric space. These results extend subdivision theory beyond Euclidean and manifold-valued data for data in metric spaces.

翻译：我们为完备度量空间(CMS)上的非线性细分格式建立了一个统一框架。我们从CMS预备知识出发，在CMS中形式化细化过程，保留了诸如局部性等关键结构性质。我们证明了压缩性条件下的收敛定理，并论证了其适用性。针对压缩性未知的格式，我们引入了两种邻近性概念。与文献中多数方法将非线性格式与线性格式相关联不同，我们的邻近性方法将非线性格式与另一个具有已知压缩性的非线性格式相联系。具体而言，第一类邻近性在单次细化步骤后比较两个格式，并如经典理论所示，从足够稠密的初始数据中产生收敛性。第二类邻近性监控所有细化层级间的对齐关系，无需密度假设即可提供强收敛性。我们建立并证明了相应定理，并通过多种示例加以阐释，例如$\R^n$中紧集度量空间上的格式、Wasserstein空间上的格式以及几何Hermite度量空间。这些成果将细分理论从欧几里得数据和流形值数据推广至度量空间中的数据。