发表机构
BIFOLD; Technical University of Berlin; Shanghai Jiao Tong University; The University of Sydney; University of Trento; Max Planck Institute for Intelligent Systems; RIKEN AIP（柏林智能数据与机器学习研究所; 柏林工业大学; 上海交通大学; 悉尼大学; 特伦托大学; 马克斯·普朗克智能系统研究所; 日本理化学研究所先进智能研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出常曲率切片Gromov-Wasserstein（CCSGW）散度，用于对齐异质常曲率空间上的分布，通过测地线投影实现高效比较，并在图异常检测、节点分类和多模态学习中取得一致性能提升。
AI 中文摘要
表示学习的最新进展凸显了常曲率模型（如双曲空间和球面空间）在建模复杂数据中的实用性。混合曲率模型通过整合多个常曲率组件进一步增强了这一能力。然而，由于不同曲率的空间本质上是异质的且缺乏统一度量，这些模型通常独立地学习每个组件空间。因此，它们缺乏在不同空间之间强制几何一致性的显式机制。此外，跨混合曲率空间比较概率分布的问题仍未得到探索。为了比较异质空间上的分布，Gromov-Wasserstein（GW）距离通过对齐其空间内几何提供了一种原则性框架。基于此，我们提出了常曲率切片Gromov-Wasserstein（CCSGW），一种用于对齐支持在异质常曲率空间上的分布的新型散度。我们首先引入了球面空间缺失的基于测地线的一维投影，然后将切片GW扩展到常曲率空间，从而实现了跨不同曲率流形的高效且原则性的比较。该公式保留了内在几何关系，同时避免了高计算成本。我们提供了理论分析，表明CCSGW控制跨异质空间的内在几何差异，促进了分布级几何一致性。通过将CCSGW集成到现有的混合曲率学习任务中，包括图异常检测、图节点分类和多模态学习，我们在多种设置下观察到一致的性能提升。
英文摘要
Recent advances in representation learning have highlighted the utility of constant-curvature models, such as hyperbolic and spherical spaces, for modeling complex data. Mixed-curvature models further enhance this by integrating multiple constant-curvature components. However, these models typically learn each component space independently because spaces with different curvatures are inherently heterogeneous and lack a unified metric. Consequently, they lack explicit mechanisms to enforce geometric consistency across various spaces. Moreover, the problem of comparing probability distributions across mixed-curvature spaces remains unexplored. To compare distributions on heterogeneous spaces, Gromov-Wasserstein (GW) distances provide a principled framework by aligning their intra-space geometries. Building on this, we propose constant-curvature sliced Gromov-Wasserstein (CCSGW), a novel divergence for aligning distributions supported on heterogeneous constant-curvature spaces. We first introduce the missing geodesic-based one-dimensional projections for spherical spaces, and then extend sliced GW to constant-curvature spaces, enabling efficient and principled comparison across manifolds with different curvatures. This formulation preserves intrinsic geometric relationships while avoiding the high computational cost. We provide theoretical analysis showing that CCSGW controls intrinsic geometric discrepancy across heterogeneous spaces, promoting distribution-level geometric consistency. By integrating CCSGW into existing mixed-curvature learning tasks, including graph anomaly detection, graph node classification, and multimodal learning, we observe consistent performance gains across diverse settings.
