Random walks that stay on the data manifold
High-dimensional data are often assumed to lie on lower-dimensional manifolds. We study how to construct diffusion processes on such a data manifold using only point-cloud samples, without access to charts, projections or other geometric primitives.
Implicit manifold-valued diffusions
We introduce Implicit Manifold-valued Diffusions (IMDs), a data-driven mathematical formalism for defining stochastic differential equations in the original high-dimensional space that describe drifting Brownian particles evolving intrinsically on the underlying manifold. Our construction approximates the infinitesimal generator of the diffusion process using a proximity graph over the data. It then uses the carré du champ of the generator, which encodes the local tangent spaces of the manifold, to lift the intrinsic process into ambient coordinates.
We show that as the number of samples grows, our discrete diffusion process converges in law, on the space of probability paths, to its smooth manifold counterpart. We also present an Euler-Maruyama scheme for the numerical integration of IMDs.
Results
Numerical experiments on synthetic manifolds and on the MNIST data manifold show that IMDs remain confined to the manifold and enable its guided exploration. This work provides the mathematical foundation and practical implementations of diffusion processes on data manifolds, opening new avenues for manifold-aware sampling, exploration and generative modelling.
See also carré du champ flow matching and Riemannian metric matching.