# Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples

arXiv: https://arxiv.org/abs/2605.07119 · HTML: https://mlsanity.com/paper/2605.07119/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 45 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 30 of 31)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Deep Clustering for Unsupervised Learning of Visual Features](https://mlsanity.com/paper/1807.05520/) (2018)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [SCAN: Learning to Classify Images without Labels](https://mlsanity.com/paper/2005.12320/) (2020)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [A Comprehensive Survey on Deep Clustering: Taxonomy, Challenges, and Future Directions](https://mlsanity.com/paper/2206.07579/) (2022)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [A cost function for similarity-based hierarchical clustering](https://mlsanity.com/paper/1510.05043/) (2015)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [On the Optimal Memorization Power of ReLU Neural Networks](https://mlsanity.com/paper/2110.03187/) (2021)
- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization](https://mlsanity.com/paper/2309.04810/) (2023)
- [Minimum width for universal approximation using ReLU networks on compact domain](https://mlsanity.com/paper/2309.10402/) (2023)
- [Hierarchical Clustering via Spreading Metrics](https://mlsanity.com/paper/1610.09269/) (2016)
- [Differentiable Clustering with Perturbed Spanning Forests](https://mlsanity.com/paper/2305.16358/) (2023)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [Tree Variational Autoencoders](https://mlsanity.com/paper/2306.08984/) (2023)
- [Consistent procedures for cluster tree estimation and pruning](https://mlsanity.com/paper/1406.1546/) (2014)
- [Hierarchical Clustering: O(1)-Approximation for Well-Clustered Graphs](https://mlsanity.com/paper/2112.09055/) (2021)
- [Nearly-Optimal Hierarchical Clustering for Well-Clustered Graphs](https://mlsanity.com/paper/2306.09950/) (2023)
- [An Objective for Hierarchical Clustering in Euclidean Space and its Connection to Bisecting K-means](https://mlsanity.com/paper/2008.13235/) (2020)
- [Self-Similarity Priors: Neural Collages as Differentiable Fractal Representations](https://mlsanity.com/paper/2204.07673/) (2022)
- [Neural Snowflakes: Universal Latent Graph Inference via Trainable Latent Geometries](https://mlsanity.com/paper/2310.15003/) (2023)
- [Learning-Augmented Hierarchical Clustering](https://mlsanity.com/paper/2506.05495/) (2025)
- [The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation](https://mlsanity.com/paper/2110.03136/) (2021)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)
- [Deep Taxonomic Networks for Unsupervised Hierarchical Prototype Discovery](https://mlsanity.com/paper/2509.23602/) (2025)
- [An Axiomatic Definition of Hierarchical Clustering](https://mlsanity.com/paper/2407.03574/) (2024)
- [Joint Hierarchical Representation Learning of Samples and Features via Informed Tree-Wasserstein Distance](https://mlsanity.com/paper/2501.03627/) (2025)

## Abstract (arXiv)

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 ℝd 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.