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Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples

NeurIPS 2026 · Learning theory and algorithms › Clustering · approach: Clustering approximation algorithms · 0 citations (OpenAlex)

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Introduces classification fields, infinite-depth recursive hierarchical clusters generated by a learned local parent-to-child refinement rule, with truncation and ReLU realizability theory and experiments on CFG, IFS, and CIFAR-CLIP hierarchies.

Problem

Standard hierarchical clustering returns finite partitions or dendrograms over observed samples and cannot describe recursive fine-scale refinements beyond the observed levels; the paper asks whether a local refinement rule can be learned from a finite hierarchy prefix and rolled out to deeper unobs

Approach

Model clusters as infinite-depth recursively generated metric DAGs defined by a classification field generator mapping parent centres to ordered K-tuples of child residuals (with root and scale); learn a ReLU predictor from a finite prefix, prove exponential truncation convergence in the completed cell metric, and roll it out recursively.

Key results

• On out-of-family nonlinear IFS hierarchies, CFP reaches macro-average MSE 1.56e-3 versus 11.84e-3 for the affine residual predictor.
• On CLIP-induced CIFAR hierarchies at level 6, CFP obtains macro-average MSE 3.19e-3 versus 5.55e-3 for the affine baseline, and PI-CD 963.69 versus 3325.16.
• Theory: ReLU predictor realizes depth-L rollout with width O(eps^-gamma) and depth Otilde(eps^-3gamma/2) under polynomially bounded aspect ratios.

Limitations

• Image experiment uses an approximate, finite-depth clustering-induced target rather than an exact generator, with child-slot canonicalization needed.
• Theory assumes boundedness, separation, and regularity of the generator; noisy, variable-branching hierarchies are left to future work.
• Out-of-family IFS results are not claims of optimal IFS identification.

Experimental setup

Datasets: CFG-generated hierarchies nonlinear IFS fractals (Sierpinski, Cantor dust, Koch, random) CIFAR-10 CIFAR-100

Benchmarks: CFG-generated hierarchies nonlinear IFS hierarchies CIFAR-10 CLIP-induced hierarchy CIFAR-100 CLIP-induced hierarchy

Metrics: MSE PI-CD dpt-distortion

Baselines: Affine Residual Predictor Learnable Const Avg Residual

Builds on: Ward's method Dasgupta's cost Matryoshka representation learning CLIP ReLU memorization results

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