Probabilistic Modeling on Riemannian Manifolds: A Unified Framework for Geometric Data Analysis

We present a comprehensive framework for probabilistic modeling on Riemannian manifolds, encompassing diffusion processes, continuous normalizing flows, energy-based models, and information-theoretic measures adapted to curved geometries. Our unified approach extends classical probabilistic methods from Euclidean spaces to arbitrary Riemannian manifolds, providing principled tools for modeling data with inherent geometric structure. We develop complete mathematical foundations including forward and reverse stochastic differential equations, probability-flow ordinary differential equations, intrinsic Langevin dynamics, and manifold-aware information measures. The framework is demonstrated on canonical manifolds including spheres, rotation groups SO(3), symmetric positive definite matrices, and hyperbolic spaces, with applications spanning computer vision, robotics, neuroscience, and network analysis.