Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples

2026-05-11 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This paper introduces "classification fields," a novel metric-geometric model for infinite-depth hierarchical clustering, moving beyond traditional finite dendrograms. Unlike classical methods that partition fixed data, classification fields model clusters as recursively refinable geometric objects generated by a local parent-to-child refinement rule, termed a classification field generator (CFG). The CFG maps each parent center to an ordered tuple of child residuals, which, along with a root and scale factor, recursively generates cluster centers and Voronoi cells, forming a metric Directed Acyclic Graph (DAG). The authors demonstrate that a classification field predictor (CFP), learned from a finite prefix of such a hierarchy, can approximate the CFG and be rolled out to generate unseen, deeper levels. Theoretical results include exponential truncation convergence in the completed cell metric and ReLU realizability with specific width and depth bounds. Experimental validation on CFG-generated hierarchies, IFS fractals, and image-induced recursive clustering (using CLIP embeddings of CIFAR datasets) shows that CFPs preserve ordered child slots, unordered geometry, and hierarchy-level path metrics under recursive rollout, outperforming simpler baselines.

Key takeaway

For AI Scientists and Machine Learning Engineers working with hierarchical data, this research offers a new paradigm for modeling and extending cluster structures. Instead of fitting finite dendrograms, you should consider learning local parent-to-child refinement rules to generate arbitrarily deep hierarchies. This approach allows for extrapolation to unseen depths, preserving geometric and path metrics, which is crucial for applications where the underlying structure is inherently recursive and extends beyond observed data.

Key insights

Classification fields enable learning infinite-depth hierarchical cluster structures from finite observations via local refinement rules.

Principles

• Hierarchies can be modeled as recursively generated metric DAGs.
• Finite observations can reveal local refinement rules for deeper structures.
• ReLU networks can realize finite refinement windows for infinite-depth fields.

Method

A classification field predictor (CFP) is trained to approximate a classification field generator (CFG) by minimizing the L2 norm between predicted and ground-truth child residuals, then recursively rolled out to generate deeper hierarchy levels.

In practice

• Apply CFPs to extrapolate hierarchical structures beyond observed data.
• Use CLIP embeddings for image-induced recursive clustering.
• Evaluate rollout stability using MSE, PI-CD, and dpt-distortion metrics.

Topics

• Classification Fields
• Hierarchical Clustering
• Recursive Refinement Rules
• Classification Field Predictor
• ReLU Neural Networks

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.