# Hyperbolic Continuous Structural Entropy for Hierarchical Clustering

arXiv: https://arxiv.org/abs/2512.00524 · HTML: https://mlsanity.com/paper/2512.00524/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 59 references that are in the map.
- First posted on arXiv 29 Nov 2025.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Cluster-GCN: An Efficient Algorithm for Training Deep and Large Graph Convolutional Networks](https://mlsanity.com/paper/1905.07953/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Towards Unsupervised Deep Graph Structure Learning](https://mlsanity.com/paper/2201.06367/) (2022)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Structural Entropy Guided Graph Hierarchical Pooling](https://mlsanity.com/paper/2206.13510/) (2022)
- [Hyperbolic Contrastive Learning for Visual Representations beyond Objects](https://mlsanity.com/paper/2212.00653/) (2022)
- [RiemannGFM: Learning a Graph Foundation Model from Riemannian Geometry](https://mlsanity.com/paper/2502.03251/) (2025)
- [LSEnet: Lorentz Structural Entropy Neural Network for Deep Graph Clustering](https://mlsanity.com/paper/2405.11801/) (2024)
- [Contrastive Multi-view Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2205.02618/) (2022)
- [Hierarchical State Abstraction Based on Structural Information Principles](https://mlsanity.com/paper/2304.12000/) (2023)
- [Ultrametric Fitting by Gradient Descent](https://mlsanity.com/paper/1905.10566/) (2019)
- [An Improved Cost Function for Hierarchical Cluster Trees](https://mlsanity.com/paper/1812.02715/) (2018)
- [Nearly-Optimal Hierarchical Clustering for Well-Clustered Graphs](https://mlsanity.com/paper/2306.09950/) (2023)
- [Deeper with Riemannian Geometry: Overcoming Oversmoothing and Oversquashing for Graph Foundation Models](https://mlsanity.com/paper/2510.17457/) (2025)
- [An Information-theoretic Perspective of Hierarchical Clustering](https://mlsanity.com/paper/2108.06036/) (2021)
- [Differentially-Private Hierarchical Clustering with Provable Approximation Guarantees](https://mlsanity.com/paper/2302.00037/) (2023)
- [Supervised Hierarchical Clustering with Exponential Linkage](https://mlsanity.com/paper/1906.07859/) (2019)

## Built on by (most cited first, 3 of 3)

- [KAT-Coder-V2.5 Technical Report](https://mlsanity.com/paper/2607.05471/) (2026)
- [Semi-Supervised Hyperbolic Hierarchical Clustering with Set-Level Structural Priors](https://mlsanity.com/paper/2606.01525/) (2026)
- [FinInvest-GTCN: Explainable Graph-Temporal-Causal Modeling for Risk-Aware Investment Decision Optimization](https://mlsanity.com/paper/2606.28933/) (2026)

## Abstract (arXiv)

Hierarchical clustering is a fundamental machine-learning technique for grouping data points into dendrograms. However, existing hierarchical clustering methods encounter two primary challenges: 1) Most methods specify dendrograms without a global objective. 2) Graph-based methods often neglect the significance of graph structure, optimizing objectives on complete or static predefined graphs. In this work, we propose Hyperbolic Continuous Structural Entropy neural networks, namely HypCSE, for structure-enhanced continuous hierarchical clustering. Our key idea is to map data points in the hyperbolic space and minimize the relaxed continuous structural entropy (SE) on structure-enhanced graphs. Specifically, we encode graph vertices in hyperbolic space using hyperbolic graph neural networks and minimize approximate SE defined on graph embeddings. To make the SE objective differentiable for optimization, we reformulate it into a function using the lowest common ancestor (LCA) on trees and then relax it into continuous SE (CSE) by the analogy of hyperbolic graph embeddings and partitioning trees. To ensure a graph structure that effectively captures the hierarchy of data points for CSE calculation, we employ a graph structure learning (GSL) strategy that updates the graph structure during training. Extensive experiments on seven datasets demonstrate the superior performance of HypCSE.