通过统一编码器与损失函数的双曲几何，显著提升点云补全性能。
Hyper^2: Unleashing Hyperbolic Geometry's Full Potential via Dual-Space Consistency
- 设计双空间一致性框架，使编码器与损失函数共享同一双曲度量
- 在ShapeNet-55上实现22.9%的Chamfer距离降低，超越单一分支效果之和
- 指标验证：仅当编码器与损失均为双曲时，监督信号利用率才显著提升
HyperbolicCD首次将双曲几何引入点云补全，用arcosh(1+alpha||x-y||^2)替代欧氏Chamfer距离，但增益有限（在PCN和ShapeNet-55上对SeedFormer、PointAttN和PMP-Net等骨干网络仅减少3-7% Chamfer）。我们指出瓶颈在于：损失为双曲而编码器为欧氏，导致位置相关监督在反向传播中被链式法则平均。为此提出双模型无关指标r_FL（特征-损失相关性）和u_G（有效梯度利用率），在仅用HyperbolicCD损失训练SVDFormer时测得(0.68, 39%)。提出Hyper^2框架，将相同的arcosh(1+alpha d^2)形式作为细化注意力的位置偏置（双曲距离编码），与HyperbolicCD损失共享单一曲率alpha。两者均为欧氏距离上的O(N log N)标量非线性，总额外计算量仅增加约1.6% FLOPs。Hyper^2在ShapeNet-55上相较SVDFormer实现-22.9% Chamfer（远超-12.0%损失单独项与-1.2%编码单独项之和的13.2%），在21个未见的ShapeNet-34类别上达-37.5%。两个指标在单空间配置下基本不变，仅当编码器与损失均为双曲时同时跃升至(0.95, 87%)，证实几何一致性才是双曲监督成功的关键。代码已公开于https://github.com/Ethan-Zheng136/Hyper-2。
原文摘要 · Abstract (English)
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.
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