Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples

2026-05-07

AI Summary

Traditional clustering methods struggle to model the infinitely fine recursive structures inherent in real-world geometric hierarchies. This work proposes Classification Fields, a novel framework that recursively generates hierarchical clusterings of unbounded depth through local parent–child refinement rules, and for the first time enables learning a generator from finite observations that extrapolates to arbitrary depths. The approach integrates a residual tuple recursion mechanism, Voronoi cell construction, and metric DAG encoding, and is implemented via ReLU networks with provable theoretical convergence. Experiments demonstrate that the method effectively preserves child ordering, geometric structure, and path metrics across synthetic context-free grammar (CFG) hierarchies, iterated function system (IFS) fractals, and image-induced clustering tasks, exhibiting strong generalization capabilities.

Abstract

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.

Problem

• hierarchical clustering
• recursive refinement
• infinite-depth hierarchy
• classification fields
• few-shot learning

Innovation

• classification fields
• recursive hierarchical clustering
• infinite-depth hierarchy
• local refinement rule
• ReLU realizability