局部Urysohn宽度：分类问题的拓扑复杂性度量
The Metric Slingshot: Navigational Reuse as Width-Optimal Structural Decoupling in Continual Learning
AI总结：
本文提出局部Urysohn宽度，用于度量度量空间上的分类问题复杂性，通过拓扑-几何复杂性分析分类问题本身，证明了拓扑复杂性与分类器复杂性的严格层次定理、拓扑-几何缩放定律、VC维的双向分离以及样本复杂性下界。
摘要：
我们引入局部Urysohn宽度，一种用于度量度量空间上分类问题的复杂性度量。不同于VC维、肥刺穿维和Rademacher复杂度，这些度量刻画假设类的丰富性，而Urysohn宽度刻画分类问题本身的拓扑-几何复杂性：即正确分类所有点在安全区域内的最小连接、直径有界的局部专家数量。我们证明了四个主要结果。首先，一个严格层次定理：对于每个整数w≥1，存在一个在连通紧度量空间（由环形结构组成，第一贝蒼数β₁=w）上的分类问题，其Urysohn宽度恰好为w，证明了输入空间的拓扑复杂性迫使分类器复杂性。其次，一个拓扑×几何缩放定律：宽度随Ω(w·L/D₀)缩放，其中w计独立环数，L/D₀是环周长与局部性尺度的比值。第三，一个VC维的双向分离：存在问题族中宽度无界而VC维有界，反之亦然。第四，一个样本复杂性下界：任何必须正确分类宽度-w问题安全区域的所有点的学习者需要Ω(w log w)样本，与VC维无关。
英文摘要：
The mammalian brain, most extensively studied in rodents and bats, solves an enormous variety of non-spatial cognitive tasks using neural circuitry, including grid cells, place cells, and hippocampal indexing, that originally evolved for physical navigation. We formalize the above observation within the local Urysohn width (LUW) framework for continual learning. The central construct is the \emph{metric slingshot}: a learned embedding $ϕ: X \to Z$ that maps an arbitrary learning problem into a navigational latent space $Z$ where pre-evolved contraction maps (grid cells) already provide the metric machinery, so that only the topological indexing subproblem must be solved de novo. We prove three results. First, the optimal spacing of multi-scale grid cell modules is a geometric series whose ratio is determined by the $Ω(w \log w)$ sample complexity bound of the LUW framework; for ecologically plausible parameters, optimality yields $r^* \approx 1.4$--$1.7$, matching electrophysiological measurements in rodent medial entorhinal cortex. Second, the slingshot preserves the width hierarchy with a Lipschitz-controlled transfer bound: a contractive embedding into a fine-resolution navigational space reduces the effective number of contexts the learner must discover. Third, the anatomical separation of the ventral (``what'') and dorsal (``where'') visual streams achieves the structural decoupling required by Metric-Topology Factorization (MTF) \emph{by architecture}, without gradient-routing mechanisms. We demonstrate that such metric slingshot is applicable to both perception cognition and motor control. Together, these results provide a unified, complexity-theoretic account of navigation in non-spatial domain, grid cell multi-modularity, and hippocampal-neocortical complementary learning as consequences of a single exaptation principle: metric slingshot.
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