Constant curvature sliced gromov-wasserstein aligns diverse curved spaces
Constant-Curvature Sliced Gromov-Wasserstein for Heterogeneous Cross-Curvature Alignment
Machine Learning
Summary
Comparing and aligning data that live in different curved spaces, like spheres and hyperbolic spaces, is difficult because these spaces have different geometric properties. The authors developed a new way to measure differences between data distributions across these varied curved spaces efficiently, preserving their intrinsic geometry. They introduced novel mathematical projections for spheres and extended an existing method to handle multiple types of curved spaces together. Their approach helps maintain geometric consistency and improves performance in tasks like detecting anomalies in graphs and classifying graph nodes.
What this means in practice
- •For graph data engineers: Improve graph anomaly detection and node classification by aligning data distributions across diverse curved geometric spaces.
- •For multimodal machine learning teams: Enhance multimodal data integration by comparing distributions from different geometric spaces with preserved geometric consistency.
Authors
Shanglin Li, Wenjing Lu, Muyang Li, Nicu Sebe, Ziheng Chen
Abstract
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.