Geometric Optimal Transport for Cross-Modal Medical Manifold Alignment: A Differential Approach to Multimodal Diagnosis

We present a novel theoretical framework for multimodal medical diagnosis that formulates heterogeneous data integration as a problem of aligning Riemannian manifolds through differential geometric principles. Medical data modalities—ranging from imaging to clinical parameters—naturally reside on distinct manifolds with inherent topological structures that conventional fusion approaches fail to preserve. Our method, Twin-Topology Heteromodal Contrastive Alignment (TTHCA), establishes rigorous mathematical foundations for this alignment through three key innovations: 1) formulation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>-isometric embeddings that provably preserve geodesic distances during cross-modal mapping with theoretical error bounds; 2) development of a curvature-aware attention mechanism that adapts to local manifold geometry and approximates diffusion processes with convergence guarantees; and 3) introduction of a structure-preserving optimal transport framework that minimizes cross-modal distributional discrepancies while maintaining topological consistency. Empirically, we validate TTHCA on Parkinsonian neuroimaging and dermatological datasets, demonstrating significant improvements over state-of-the-art approaches. Through ablation studies and theoretical analyses, we show that our framework’s superiority stems from its ability to preserve intrinsic geometric structures during cross-modal learning. Our approach establishes a mathematically rigorous bridge between differential geometry and transfer learning in medical diagnostics, offering both theoretical contributions to representation learning and practical advances in early disease detection capabilities.