Semi-metric optimal transport enables robust spatial multi-omics integration

Integrating heterogeneous spatial multi-omics data to uncover underlying biological and pathological mechanisms remains a major challenge, while existing approaches generally lack mathematically justified distances for comparing complex spatial tissues. We introduce Spatial Optimal Transport (SpaOT), a spatial multi-omics integration framework based on a theoretically grounded semi-metric formulation of unbalanced optimal transport. Utilizing a total-variation-relaxed Fused Partial Gromov-Wasserstein strategy, SpaOT establishes a mathematically consistent distance geometry that enables stable biological comparison, representative tissue barycenters, and robust downstream analyses while accommodating differences in cellular composition, measurement technologies, spatial resolution, and noise. Across multiple studies with diverse spatial omics, SpaOT enables robust and accurate alignment, integration, and stratifications across samples, resolutions, and omics modalities in complex disease-related biological systems. These capabilities facilitate the identification of biologically meaningful cellular organization, molecular signatures, and spatial relationships, establishing SpaOT as a general foundation for spatial multi-omics integration and comparative analysis.