Hyper^2: Unleashing Hyperbolic Geometry's Full Potential via Dual-Space Consistency

2026-08-23

AI Summary

本文针对点云补全中欧几里得编码器与双曲损失函数不匹配的问题，提出了一种名为Hyper^2的双空间一致性框架，通过在编码和损失中使用相同的双曲距离公式来提高模型性能。

Abstract

HyperbolicCD pioneered hyperbolic geometry for point cloud completion by replacing the Euclidean Chamfer distance with arcosh(1+alpha||x-y||^2), but the reported gains are modest (3-7% Chamfer reduction across SeedFormer, PointAttN and PMP-Net backbones on PCN and ShapeNet-55). We argue the bottleneck lies elsewhere: the loss is hyperbolic but the encoder it back-propagates through is Euclidean, so the position-dependent supervision of the loss is averaged away by the chain rule before it reaches the parameters. We call this a cross-geometry mismatch, and make it testable through two model-agnostic indicators, feature-loss correlation r_FL and effective gradient utilisation u_G. On an SVDFormer backbone trained with HyperbolicCD's loss alone we measure (r_FL, u_G) = (0.68, 39%). We propose Hyper^2, a dual-space consistency framework that extends HyperbolicCD by reusing the identical arcosh(1+alpha d^2) functional form as a positional bias on the refinement attention (a hyperbolic distance encoding), paired with HyperbolicCD's hyperbolic Chamfer loss under a single shared curvature alpha. Both operators are O(N log N) scalar non-linearities on Euclidean distances and together add only ~1.6% FLOPs over SVDFormer. Hyper^2 delivers -22.9% Chamfer on ShapeNet-55 over SVDFormer (well above the 13.2% linear sum of the -12.0% loss-only and -1.2% encoding-only single-space ablations) and -37.5% on the 21 unseen ShapeNet-34 categories. The two indicators remain essentially flat for any single-space configuration but jump together to (0.95, 87%) only when both encoder and loss are hyperbolic, supporting the claim that geometric consistency across encoder and loss, rather than either operator alone, is what enables hyperbolic supervision in point cloud completion. Code is available at https://github.com/Ethan-Zheng136/Hyper-2.

Problem

• hyperbolic geometry
• point cloud completion
• cross-geometry mismatch
• encoder
• loss function

Innovation

• Dual-Space Consistency
• Hyperbolic Geometry
• Cross-Geometry Mismatch
• Positional Bias
• Geometric Consistency