用拓扑维数刻画极值概念类的可复现性，连接了数学与机器学习。
Simplicial covering dimension of extremal concept classes
- 将拓扑覆盖维数引入概念类，构建其分布空间上的单纯形结构。
- 证明有限概念类的该维数精确等于列表可复现数，即全局稳定性。
- 为极值概念类提供精确计算可复现性的新方法，适合理论研究者。
维数理论是拓扑学中研究几何与拓扑空间维数的分支，仅用纯拓扑语言定义和分析。本文将经典拓扑维数（勒贝格覆盖）概念推广至二元概念类。与概念类自然关联的拓扑空间是其可实现分布的空间。损失函数与概念类本身在此空间上诱导出单纯形结构，据此定义了单纯形覆盖维数。我们证明，对有限概念类而言，该单纯形覆盖维数恰好刻画了列表可复现数（等价于全局稳定性）。这一联系使得我们可以应用经典维数理论工具，精确计算一大类极值概念类的列表可复现数。
原文摘要 · Abstract (English)
Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.
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