Learning with Recurrence Geometric AI in Spatial Transcriptomics

Cross-domain analysis of spatial transcriptomics is challenging because tissues from different organs, diseases and experimental platforms exhibit distinct cellular compositions, spatial organisations and technical biases, making direct comparison of tissue states difficult. Existing methods primarily focus on domain integration or batch correction but generally do not explicitly model the intrinsic geometry underlying tissue-state organisation across biological systems. This paper presents recurrence geometric artificial intelligence (RGAI), a geometric deep-learning framework for discovering and aligning latent tissue states across heterogeneous spatial transcriptomic domains. RGAI first learns domain-specific latent representations using variational graph autoencoders while simultaneously estimating a Riemannian metric tensor that captures the local geometry of each latent manifold. Geodesic distances induced by the learned metric are used to construct multiscale recurrence graphs that characterise intrinsic tissue-state organisation independently of the original measurement space. Cross-domain manifold correspondence is then established through entropy-regularised Gromov--Wasserstein alignment, after which fuzzy clustering identifies latent tissue states and optimal transport aligns tissue-state signatures across domains. Evaluation on six human spatial transcriptomic datasets spanning wound healing, periodontitis, oral squamous cell carcinoma, head and neck squamous cell carcinoma, cardiac tissue and colorectal cancer shows that RGAI automatically determines biologically meaningful latent tissue-state complexity and identifies coherent recurrence-based tissue states within each domain. The learned geometric representations enable cross-domain alignment of latent manifolds while preserving biologically interpretable tissue-state correspondences despite substantial differences in cellular composition and tissue architecture, demonstrating that integrating learned Riemannian geometry, recurrence analysis and optimal transport provides a robust and interpretable framework for cross-domain tissue-state discovery and comparison in spatial transcriptomics.