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.
Figures & tables
Figure 1: Hyper 2 framework overview. Partial input Pin is encoded by a self-view backbone (point-level and pixel-level aggregation) and passed to the refinement network on the right. The two “Hyper Embedding” blocks mark the two places where Hyper 2 injects the arcosh(1+αd2) distance encoding as a positional bias on the refinement attention (Eq. 4 ). The three output stages P0 (Coarse), P1 (Refined) and P2 (Output) are each supervised by the hyperbolic Chamfer Distance LHyper2 (Eq. 6 ), so both encoder and loss are governed by the same scalar curvature α . The two operators thus share α , implementing end-to-end geometric consistency.
Figure 2: HyperEmbedding refinement module. Inside every refinement stage we keep the SVDFormer backbone [ Zhu et al.(2023a)Zhu, Chen, He, Wang, Qin, and Wei ] (feature embedding, self-attention, cross-attention, decoder) and replace only its positional bias. Our Hyperbolic Bias ei=arcosh(1+αdi2) ( red highlight ) takes the place of the original linear bias di/γ , with di=miny∈Pin∥xi−y∥ being each point’s distance to the observed partial input. The right panel zooms in on the computation chain: distance lookup →arcosh(1+αdi2)→ sinusoidal embedding. The new ingredient is sub-linear in di and γ -free, so far points are softly log-compressed instead of dominating attention.
Figure 3: arcosh(1+αd2) is sub-linear, not exponential. Solid: function value (left axis); dashed: gradient magnitude (right axis). The value tracks 2αd near d=0 and log(2αd2) for large d (log-compressed); the gradient is bounded by 2α and decays as 2/d . A single shared α thus governs both encoder and loss (“no-vanish, no-explode”).
Figure 4: Why dual-space consistency wins. Each panel draws encoder (left) and loss (right) as the two halves of one 2D manifold joined at a vertical seam; the flow arrow parallels the local curvature, and the (rFL,uG) values are real ShapeNet-55 measurements (Tab. 4.5 ). (a) Euc+Euc [baseline]: flat seam, flat flow; matching still dominated by wrong (red) links. (b) Euc+Hyp [ Lin et al.(2023)Lin, Yue, Hou, Yu, Xu, Yamada, and Zhang ] : flat meets curved, the seam shows a geometric kink and the flow is forced to step across it. (c) Hyp+Hyp (Hyper 2 , ours): one continuous curved manifold; smooth seam, the flow follows the arc, and (rFL,uG) jump to (0.95,87%) .
Figure 5: Qualitative comparison on ShapeNet-55 (hard difficulty). Six representative shapes (rows) reconstructed under the four loss/encoder configurations of Tab. 4.5 (columns 2–5), bracketed by the partial input (left) and the ground truth (right). Notation Loss + Enc. : Euc. + Euc. is the vanilla SVDFormer [ Zhu et al.(2023a)Zhu, Chen, He, Wang, Qin, and Wei ] baseline, Euc. + Hyp. replaces only the encoder positional bias, Hyp. + Euc. corresponds to a faithful re-implementation of HyperbolicCD [ Lin et al.(2023)Lin, Yue, Hou, Yu, Xu, Yamada, and Zhang ] ’s loss on the SVDFormer backbone, and Hyp. + Hyp. is our dual-space Hyper 2 . All four columns share the GT camera/origin per row to make the geometric differences directly comparable. Quantitative counterparts of these configurations are the four rows of Tab. 4.5 , with ShapeNet-55 averages in Tab. 1 .
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.
Alessio Colombo, Melika Ayoughi
Universiteit van Amsterdam, Amsterdam, Netherlands
Point cloud completion is inherently ill-posed due to severe sparsity and ambiguity in partial observations. Existing multi-view methods alleviate this by incorporating 2D semantics, but often rely on learned attention and fixed fusion, which lack geometric consistency and adaptability. We propose ProjFormer, a cross-modal framework that enforces geometry-consistent 2D-3D interaction through explicit projection and adaptive feature routing. A Projective Guided View Attention module aligns 3D points with multi-view features via deterministic projection, enabling efficient and geometrically consistent aggregation. Building on this, a geometry-aware routing network performs point-wise adaptive fusion of structural and observation-driven features for progressive refinement. Experiments show that, under a lightweight design, ProjFormer delivers competitive performance with improved structural completeness.
High-curvature regions in 3D point clouds encapsulate critical fine-grained geometric semantics yet exhibit a distinct long-tail sparsity in their spatial distribution. The inherent limitations of polynomial volume growth in Euclidean space frequently render these intricate geometric features challenging to adequately resolve within a uniform-scale feature space. Consequently, these regions are frequently overshadowed by smooth global features dominated by low-curvature regions, thereby limiting the discriminative capacity of the network. To address this issue, we propose PointCHR, a curvature-aware hyperbolic rectification (CHR) for point cloud analysis. Utilising the property of exponential volume expansion in the vicinity of hyperbolic manifolds, CHR presents a learnable curvature-guided radial rectification mechanism. By adaptively projecting high-curvature points towards boundary regions endowed with larger effective embedding capacities, PointCHR effectively mitigates the representation crowding problem inherent in Euclidean settings. Extensive experimentation has demonstrated that PointCHR significantly enhances the ability of backbone to capture fine-grained geometric details, achieving state-of-the-art performance across multiple benchmarks.
Xinxing Yu, Liying Yang, Hao Mo +4
School of Computer Science and Engineering, Macau University of Science and Technology, Macau, China · School of Computing and Information Technology, Great Bay University, Dongguan, China · Shenzhen International Graduate School, Tsinghua University, Shenzhen, China +1