aiqbench analysis
This paper introduces a geometry-aware hyperbolic residual quantization method that resolves geometric inconsistencies in hierarchical latent spaces, offering improved stability for tasks like image tokenization and recommendation systems. The approach distinguishes itself by restoring telescoping behavior in the forward pass and utilizing a discounted gradient estimator to prevent unstable recursive transport during training. While Euclidean methods remain optimal for pure compression, this technique provides a robust framework for organizing discrete tokens within non-Euclidean hierarchies.
Geometry-Aware Hyperbolic Residual Quantization
arXiv:2609.26342v1 Announce Type: cross Abstract: Residual Vector Quantization turns continuous representations into discrete, multi-level token sequences. Yet most methods operate in Euclidean space, despite the coarse-to-fine structure of the resulting codes and the latent hierarchies present in many data domains. Hyperbolic geometry offers a natural alternative for hierarchical representations, but naive hyperbolic extensions introduce geometric inconsistencies: non-associative hyperbolic addition prevents consistent residual aggregation, while standard straight-through gradient estimation ignores the geometry of the latent space. We propose a geometry-aware hyperbolic residual quantization that addresses these issues in both the forward and backward passes. In the forward pass, Hyperbolic Residual Aggregation restores the telescoping behavior of residual quantization on the Poincare ball. In the backward pass, a discounted Hyperbolic Straight-Through Estimator routes the reconstruction gradient through the quantizer as a single geometric block, avoiding unstable recursive gradient transport across residual stages. Evaluations on hierarchical prediction, recommendation, image tokenization, and neural audio coding tasks show that our method improves the stability and structural organization of hyperbolic residual codes over naive hyperbolic baselines. At the same time, we observe a clear structure-compression trade-off: Euclidean residual quantization remains preferable for pure compression, while geometry-aware hyperbolic quantization is most useful for hierarchically organized discrete latent spaces.
Abstract:Residual Vector Quantization turns continuous representations into discrete, multi-level token sequences. Yet most methods operate in Euclidean space, despite the coarse-to-fine structure of the resulting codes and the latent hierarchies present in many data domains. Hyperbolic geometry offers a natural alternative for hierarchical representations, but naive hyperbolic extensions introduce geometric inconsistencies: non-associative hyperbolic addition prevents consistent residual aggregation, while standard straight-through gradient estimation ignores the geometry of the latent space. We propose a geometry-aware hyperbolic residual quantization that addresses these issues in both the forward and backward passes. In the forward pass, Hyperbolic Residual Aggregation restores the telescoping behavior of residual quantization on the Poincare ball. In the backward pass, a discounted Hyperbolic Straight-Through Estimator routes the reconstruction gradient through the quantizer as a single geometric block, avoiding unstable recursive gradient transport across residual stages. Evaluations on hierarchical prediction, recommendation, image tokenization, and neural audio coding tasks show that our method improves the stability and structural organization of hyperbolic residual codes over naive hyperbolic baselines. At the same time, we observe a clear structure-compression trade-off: Euclidean residual quantization remains preferable for pure compression, while geometry-aware hyperbolic quantization is most useful for hierarchically organized discrete latent spaces.
14-page main paper (30 pages total with references and appendix), 3 figures, 8 tables. Accepted at the Beyond Euclidean Workshop, ECCV 2026 (Oral)
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2609.26342 [cs.LG]
(or arXiv:2609.26342v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.26342 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 316 words
Related coverage
GeoRVQ: Decoder-aware geometry for residual-token prediction in physiological signals
arXiv · Sep 24, 2026
When Does Low-Bit Quantization Preserve the Decisions of Vector Search?
arXiv · Sep 10, 2026
Qift: Shift-Friendly No-Zero W2 Post-Training Quantization for Rotated W2A4/KV4 LLM Inference
arXiv · Jun 3, 2026
Hyperbolic codes could quadruple efficiency of quantum error correction
Quantum Zeitgeist · Oct 13, 2026