# HRCF: Enhancing Collaborative Filtering via Hyperbolic Geometric Regularization

arXiv: https://arxiv.org/abs/2204.08176 · HTML: https://mlsanity.com/paper/2204.08176/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 57 references that are in the map.
- First posted on arXiv 18 Apr 2022.

## Builds on (most cited first, 30 of 34)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation](https://mlsanity.com/paper/2002.02126/) (2020)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Neural Graph Collaborative Filtering](https://mlsanity.com/paper/1905.08108/) (2019)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Variational Autoencoders for Collaborative Filtering](https://mlsanity.com/paper/1802.05814/) (2018)
- [MAGNN: Metapath Aggregated Graph Neural Network for Heterogeneous Graph Embedding](https://mlsanity.com/paper/2002.01680/) (2020)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [GaAN: Gated Attention Networks for Learning on Large and Spatiotemporal Graphs](https://mlsanity.com/paper/1803.07294/) (2018)
- [Unifying Graph Convolutional Neural Networks and Label Propagation](https://mlsanity.com/paper/2002.06755/) (2020)
- [Graph-based Semi-supervised Learning: A Comprehensive Review](https://mlsanity.com/paper/2102.13303/) (2021)
- [Hyperbolic Graph Attention Network](https://mlsanity.com/paper/1912.03046/) (2019)
- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [Discrete-time Temporal Network Embedding via Implicit Hierarchical Learning in Hyperbolic Space](https://mlsanity.com/paper/2107.03767/) (2021)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [STAR-GCN: Stacked and Reconstructed Graph Convolutional Networks for Recommender Systems](https://mlsanity.com/paper/1905.13129/) (2019)
- [HyperML: A Boosting Metric Learning Approach in Hyperbolic Space for Recommender Systems](https://mlsanity.com/paper/1809.01703/) (2018)
- [Latent Relational Metric Learning via Memory-based Attention for Collaborative Ranking](https://mlsanity.com/paper/1707.05176/) (2017)
- [Modeling Scale-free Graphs with Hyperbolic Geometry for Knowledge-aware Recommendation](https://mlsanity.com/paper/2108.06468/) (2021)
- [Where are we in embedding spaces? A Comprehensive Analysis on Network Embedding Approaches for Recommender Systems](https://mlsanity.com/paper/2105.08908/) (2021)
- [AutoGraph: Automated Graph Neural Network](https://mlsanity.com/paper/2011.11288/) (2020)
- [Enhancing Hyperbolic Graph Embeddings via Contrastive Learning](https://mlsanity.com/paper/2201.08554/) (2022)

## Built on by (most cited first, 30 of 34)

- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [COSTA: Covariance-Preserving Feature Augmentation for Graph Contrastive Learning](https://mlsanity.com/paper/2206.04726/) (2022)
- [Hyperbolic Fine-Tuning for Large Language Models](https://mlsanity.com/paper/2410.04010/) (2024)
- [HICF: Hyperbolic Informative Collaborative Filtering](https://mlsanity.com/paper/2207.09051/) (2022)
- [Hyperbolic Representation Learning: Revisiting and Advancing](https://mlsanity.com/paper/2306.09118/) (2023)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Lorentzian Residual Neural Networks](https://mlsanity.com/paper/2412.14695/) (2024)
- [Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning](https://mlsanity.com/paper/2309.04082/) (2023)
- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [HGWaveNet: A Hyperbolic Graph Neural Network for Temporal Link Prediction](https://mlsanity.com/paper/2304.07302/) (2023)
- [SEFraud: Graph-based Self-Explainable Fraud Detection via Interpretative Mask Learning](https://mlsanity.com/paper/2406.11389/) (2024)
- [HyperCore: The Core Framework for Building Hyperbolic Foundation Models with Comprehensive Modules](https://mlsanity.com/paper/2504.08912/) (2025)
- [EasyRec: Simple yet Effective Language Models for Recommendation](https://mlsanity.com/paper/2408.08821/) (2024)
- [Understanding and Mitigating Hyperbolic Dimensional Collapse in Graph Contrastive Learning](https://mlsanity.com/paper/2310.18209/) (2023)
- [Toward a Better Understanding of Loss Functions for Collaborative Filtering](https://mlsanity.com/paper/2308.06091/) (2023)
- [Hyperbolic Graph Diffusion Model](https://mlsanity.com/paper/2306.07618/) (2023)
- [HC-GLAD: Dual Hyperbolic Contrastive Learning for Unsupervised Graph-Level Anomaly Detection](https://mlsanity.com/paper/2407.02057/) (2024)
- [DeepHGCN: Toward Deeper Hyperbolic Graph Convolutional Networks](https://mlsanity.com/paper/2310.02027/) (2023)
- [Challenging the Myth of Graph Collaborative Filtering: a Reasoned and Reproducibility-driven Analysis](https://mlsanity.com/paper/2308.00404/) (2023)
- [Mitigating Semantic Confusion from Hostile Neighborhood for Graph Active Learning](https://mlsanity.com/paper/2308.08823/) (2023)
- [Foundations and Frontiers of Graph Learning Theory](https://mlsanity.com/paper/2407.03125/) (2024)
- [Hyperbolic Graph Representation Learning: A Tutorial](https://mlsanity.com/paper/2211.04050/) (2022)
- [Hgformer: Hyperbolic Graph Transformer for Recommendation](https://mlsanity.com/paper/2502.15693/) (2024)
- [Large Language Models Enhanced Hyperbolic Space Recommender Systems](https://mlsanity.com/paper/2504.05694/) (2025)
- [Hyperbolic Multimodal Continual Learning](https://mlsanity.com/paper/2608.09572/) (2026)
- [Lorentz Equivariant Model for Knowledge-Enhanced Hyperbolic Collaborative Filtering](https://mlsanity.com/paper/2302.04545/) (2023)
- [FedHGN: A Federated Framework for Heterogeneous Graph Neural Networks](https://mlsanity.com/paper/2305.09729/) (2023)
- [Breaking Information Cocoons: A Hyperbolic Framework for Balancing Exploration and Exploitation in Recommender Systems](https://mlsanity.com/paper/2411.13865/) (2024)

## Abstract (arXiv)

In large-scale recommender systems, the user-item networks are generally scale-free or expand exponentially. The latent features (also known as embeddings) used to describe the user and item are determined by how well the embedding space fits the data distribution. Hyperbolic space offers a spacious room to learn embeddings with its negative curvature and metric properties, which can well fit data with tree-like structures. Recently, several hyperbolic approaches have been proposed to learn high-quality representations for the users and items. However, most of them concentrate on developing the hyperbolic similitude by designing appropriate projection operations, whereas many advantageous and exciting geometric properties of hyperbolic space have not been explicitly explored. For example, one of the most notable properties of hyperbolic space is that its capacity space increases exponentially with the radius, which indicates the area far away from the hyperbolic origin is much more embeddable. Regarding the geometric properties of hyperbolic space, we bring up a Hyperbolic Regularization powered Collaborative Filtering(HRCF) and design a geometric-aware hyperbolic regularizer. Specifically, the proposal boosts optimization procedure via the root alignment and origin-aware penalty, which is simple yet impressively effective. Through theoretical analysis, we further show that our proposal is able to tackle the over-smoothing problem caused by hyperbolic aggregation and also brings the models a better discriminative ability. We conduct extensive empirical analysis, comparing our proposal against a large set of baselines on several public benchmarks. The empirical results show that our approach achieves highly competitive performance and surpasses both the leading Euclidean and hyperbolic baselines by considerable margins.