# Hyperbolic Residual Quantization: Discrete Representations for Data with Latent Hierarchies

arXiv: https://arxiv.org/abs/2505.12404 · HTML: https://mlsanity.com/paper/2505.12404/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 56 references that are in the map.
- First posted on arXiv 18 May 2025.

## Builds on (most cited first, 30 of 36)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Self-Attentive Sequential Recommendation](https://mlsanity.com/paper/1808.09781/) (2018)
- [MPNet: Masked and Permuted Pre-training for Language Understanding](https://mlsanity.com/paper/2004.09297/) (2020)
- [SoundStream: An End-to-End Neural Audio Codec](https://mlsanity.com/paper/2107.03312/) (2021)
- [Autoregressive Image Generation using Residual Quantization](https://mlsanity.com/paper/2203.01941/) (2022)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Vector-quantized Image Modeling with Improved VQGAN](https://mlsanity.com/paper/2110.04627/) (2021)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Recommender Systems with Generative Retrieval](https://mlsanity.com/paper/2305.05065/) (2023)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [How to Index Item IDs for Recommendation Foundation Models](https://mlsanity.com/paper/2305.06569/) (2023)
- [HiFi-Codec: Group-residual Vector quantization for High Fidelity Audio Codec](https://mlsanity.com/paper/2305.02765/) (2023)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Modeling Scale-free Graphs with Hyperbolic Geometry for Knowledge-aware Recommendation](https://mlsanity.com/paper/2108.06468/) (2021)
- [Hyperbolic Deep Reinforcement Learning](https://mlsanity.com/paper/2210.01542/) (2022)
- [Generative Sequential Recommendation with GPTRec](https://mlsanity.com/paper/2306.11114/) (2023)

## Built on by (most cited first, 1 of 1)

- [Geometry-Aware Hyperbolic Residual-Quantized Variational Autoencoders](https://mlsanity.com/paper/2609.26342/) (2026)

## Abstract (arXiv)

Hierarchical data arise in countless domains, from biological taxonomies and organizational charts to legal codes and knowledge graphs. Residual Quantization (RQ) is widely used to generate discrete, multitoken representations for such data by iteratively quantizing residuals in a multilevel codebook. However, its reliance on Euclidean geometry can introduce fundamental mismatches that hinder modeling of hierarchical branching, necessary for faithful representation of hierarchical data. In this work, we propose Hyperbolic Residual Quantization (HRQ), which embeds data natively in a hyperbolic manifold and performs residual quantization using hyperbolic operations and distance metrics. By adapting the embedding network, residual computation, and distance metric to hyperbolic geometry, HRQ imparts an inductive bias that aligns naturally with hierarchical branching. We claim that HRQ in comparison to RQ can generate more useful for downstream tasks discrete hierarchical representations for data with latent hierarchies. We evaluate HRQ on two tasks: supervised hierarchy modeling using WordNet hypernym trees, where the model is supervised to learn the latent hierarchy - and hierarchy discovery, where, while latent hierarchy exists in the data, the model is not directly trained or evaluated on a task related to the hierarchy. Across both scenarios, HRQ hierarchical tokens yield better performance on downstream tasks compared to Euclidean RQ with gains of up to 20% for the hierarchy modeling task. Our results demonstrate that integrating hyperbolic geometry into discrete representation learning substantially enhances the ability to capture latent hierarchies.