Diffusion Processes on Implicit Manifolds

Simulating diffusion processes on a manifold typically requires access to tangent spaces and projection operators, which are usually unavailable when the manifold is known only through samples. We introduce Implicit Manifold-valued Diffusions (IMDs): a data-driven framework for constructing continuous stochastic dynamics on such implicit manifolds. Rather than reconstructing the geometry explicitly, IMDs estimate a manifold diffusion operator from the point cloud and use it to define the drift and diffusion coefficients of an ambient-valued SDE. The point cloud therefore serves only to estimate the diffusion operator. The induced process evolves continuously in the ambient space and becomes manifold-valued in the large-sample limit. We prove that strong-resolvent convergence of the estimated operator is sufficient for weak convergence of the induced stochastic process on path space. As a concrete realization, we construct a family of graph-based estimators whose induced IMDs range from density-biased Langevin dynamics to manifold Brownian motion. Experiments validate the predicted path space convergence and demonstrate geodesic recovery, density-debiased manifold exploration, and stochastic interpolation on image data.