分类场：从少量示例中进行任意精细的递归分层聚类
Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples
AI总结：
本文提出分类场，一种通过局部父到子细化规则生成无限深度分层聚类结构的方法，证明了其在完成单元度量下的指数截断收敛性，并在实验中验证了其在递归展开中的有效性。
摘要：
经典聚类方法通常返回观测数据的有限划分或有限树状图。当感兴趣的层次结构是递归几何对象且具有细粒度细化时，有限样本观点不足。我们引入分类场：在R^d上由局部父到子细化规则生成的无限深度分层聚类结构。分类场生成器将每个父中心映射到一个有序、有界且分离的子残差元组。结合根和缩放因子，该规则递归生成聚类中心、Voronoi单元和度量DAG编码的层次结构。仅给定此类层次结构的有限前缀，我们学习一个分类场预测器，可以近似生成器并展开到未见深度。我们证明了在完成单元度量下的指数截断收敛性，并在ReLU可实现性上具有宽度O(ε^{-γ})和深度O(ε^{-3γ/2})，其中γ=log K/(-log s)，考虑到有限窗口纵横比因素。近似在诱导的紧度量结构层面成立，以完成单元度量Hausdorff距离衡量。在匹配CFG生成的层次结构、IFS分形和图像诱导的递归聚类层次结构上的实验验证显示，学习的预测器在递归展开中保持了有序的子槽、无序的几何和层次级路径度量。这些结果支持了有限层次观察可以揭示局部细化规则，从而生成更深层次分类场的主张。
英文摘要：
Classical clustering methods usually return either a finite partition of the observed data or a finite dendrogram over it. This finite-sample view is inadequate when the hierarchy of interest is a recursive geometric object with fine-scale refinements that continue beyond the levels directly observed. We introduce classification fields: infinite-depth hierarchical cluster structures on $\mathbb{R}^d$ generated by a local parent-to-child refinement rule. A classification field generator maps each parent centre to an ordered, bounded, and separated tuple of child residuals. Together with a root and a scale factor, this rule recursively generates cluster centres, Voronoi cells, and a metric DAG encoding the hierarchy. Given only a finite prefix of such a hierarchy, we learn a classification field predictor that approximates the generator and can be rolled out to unseen depths. We prove exponential truncation convergence in the completed cell metric and ReLU realizability with width $O(\varepsilon^{-γ})$ and depth $\widetilde O(\varepsilon^{-3γ/2})$, where $γ=\log K/(-\log s)$, up to finite-window aspect-ratio factors. The approximation holds at the level of the induced compact metric structures, measured in the completed cell-metric Hausdorff distance. Experimental validation on matched CFG-generated hierarchies, IFS fractals, and image-induced recursive clustering hierarchies shows that learned predictors preserve ordered child slots, unordered geometry, and hierarchy-level path metrics under recursive rollout. These results support the claim that finite hierarchical observations can reveal local refinement rules capable of generating substantially deeper classification fields.
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