Geometry-Aware Hyperbolic Residual Quantization

2026-09-22

AI Summary

本文提出了一种几何感知的双曲残差量化方法，解决了双曲空间中残差聚合和梯度估计的几何不一致性问题，适用于层次化数据表示。

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.

Problem

• Residual Vector Quantization
• Hyperbolic Geometry
• Geometric Inconsistencies

Innovation

• Geometry-Aware Hyperbolic Residual Quantization
• Hyperbolic Residual Aggregation
• Discounted Hyperbolic Straight-Through Estimator