从少量样本中学习递归聚类规则，生成无限深度的层次结构。
Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples
- 基于局部父子更新规则，构建无限深度的聚类场结构。
- 在有限观测下实现指数级收敛，支持任意深度展开。
- 适用于图像、分形等具有精细递归结构的数据建模。
传统聚类方法通常只返回有限划分或有限层级的树状结构，难以刻画持续细化的递归几何层次。本文提出分类场（Classification Fields）：在ℝᵈ上由局部父子细化规则生成的无限深度层次聚类结构。分类场生成器将每个父中心映射为有序、有界且分离的子残差元组，结合根节点与尺度因子，递归生成聚类中心、Voronoi单元及编码层次的度量有向无环图（DAG）。给定该层次的有限前缀，我们训练一个分类场预测器以逼近生成器，并可扩展至未见深度。证明在完成单元度量下具有指数截断收敛性，且使用ReLU网络时宽度为O(ε⁻γ)，深度为~O(ε⁻³γ/₂)，其中γ = log K / (-log s)，忽略有限窗口的纵横比因子。近似效果在诱导紧致度量结构上以完成单元度量的Hausdorff距离衡量。实验验证了在匹配CFG生成的层次、IFS分形以及图像驱动的递归聚类层次上，学习到的预测器能保持有序子槽、无序几何和层次路径度量。结果表明，有限层次观测可揭示生成更深层分类场的局部细化规则。
原文摘要 · Abstract (English)
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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