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.
Infrared small target detection (IRSTD) has achieved substantial progress under domain-consistent evaluation, yet detector performance often degrades markedly when generalizing to unseen infrared domains. Existing methods primarily improve detection by enhancing target responses and suppressing background interference. However, when trained on only a limited set of source domains, their learned decision rules are inevitably established from a restricted range of source-domain target-background relation patterns. We formulate this cross-domain failure as target-background relation shift: unseen domains may exhibit relation patterns that are not observed during training, thereby weakening the discriminative capability learned from the source domains. To address this problem, we propose HyTBE, a Hyperbolic Target-Background Expert model that expands source-domain relation patterns and adaptively adjusts visual representations using explicit relation cues. The Target-Background Relation Intervention selectively perturbs either targets or backgrounds, broadening the observable relation patterns during training while maintaining valid supervision. Subsequently, the Hyperbolic Relation Modeling maps multi-scale visual cues into a Poincaré ball and characterizes the target-background relation of each feature token according to its relative distances to the target and background anchors. The Hyperbolic-guided MoE Adapter further uses these hyperbolic relation representations to calibrate multi-scale visual features and aggregate expert-specific feature corrections for different relation patterns. Leave-one-domain-out experiments on NUAA-SIRST, NUDT-SIRST, and IRSTD-1K demonstrate that HyTBE achieves stronger cross-domain generalization than competitive baselines.
Planar tiled diffusion denoises overlapping windows of one rectangular canvas. The hyperbolic plane has no such canvas, and its area grows exponentially with radius. We introduce HyperbolicDiffusion, a training-free method for generating finite visual fields directly on the hyperbolic plane H2. Our Hyperbolic Blooming Cover reduces window placement to a compact dynamic program that runs in seconds while providing strong theoretical guarantees. Permanent surface IDs form a shared latent canvas: a standard diffusion model denoises local windows, whose predictions are fused back onto H2. Because curvature causes residual disagreement and blur at multi-window junctions, a geometry-derived second stage re-noises and repairs precisely those regions. The resulting fields are sharp, reprojectable, and consistent across viewpoints, providing a prompt-driven generative counterpart to Escher's Circle Limit series.
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.
While recent advancements like the Poincaré ResNet have demonstrated the potential of learning visual representations directly in hyperbolic space, their optimisation remains hampered by the computationally intensive nature of Riemannian gradients and the strict boundaries of the manifold. Furthermore, standard hyperbolic networks treat spatial transformations of the same object as distinct hierarchical concepts, leading to redundant parameter usage and vanishing signals. We propose Equivariant Poincaré ResNets, combining hyperbolic geometry with discrete symmetry groups ($C_4$ and $D_4$). We identify critical roadblocks in applying Euclidean equivariance to hyperbolic space and propose geometrically safe tensor reshaping, left-regular permutations for hyperbolic group convolutions, and joint-orientation Poincaré Midpoint Batch normalisation. Empirically, embedding equivariance drastically reduces the optimisation space, accelerating convergence while accelerating convergence while respecting the boundary constraints of the Poincaré ball and preserving spatial-group equivariance.
Skeleton-based action recognition aims to understand human behaviors from body joint sequences and is especially challenging in the one-shot setting, where only a single labeled exemplar is available for each novel action. A key challenge is learning representations that capture the hierarchical and compositional structure of human motion while aligning effectively with high-level action semantics under extreme data scarcity. Existing approaches, largely based on Euclidean embeddings and low-level motion cues, struggle to model the tree-like organization of skeleton data, limiting cross-modal alignment and generalization to unseen action categories. We propose SkelHCC, a unified skeleton hyperbolic CLIP-driven cache adaptation framework for one-shot skeleton-based action recognition. SkelHCC introduces an Explicitly Hierarchical Hyperbolic CLIP (EH-HCLIP) module that embeds skeleton sequences and action language into a shared hyperbolic space. By leveraging the negative curvature and exponential volume growth of hyperbolic geometry, EH-HCLIP naturally encodes the joint-part-body hierarchy of human anatomy and yields structurally consistent cross-modal representations. To support efficient one-shot adaptation, SkelHCC further integrates a training-free LLM-guided Multi-granularity Voting Cache (LMV-Cache) for context-aware inference. Experiments on NTU RGB+D 60, NTU RGB+D 120, and PKU-MMD demonstrate that SkelHCC consistently outperforms state-of-the-art methods.