Generative Learning of Densities on Manifolds
This work addresses the challenge of generative modeling for high-dimensional data constrained to manifolds, which is incremental as it builds on existing diffusion and manifold learning techniques.
The authors tackled the problem of sampling data densities on unknown low-dimensional manifolds in high-dimensional spaces by combining diffusion models with manifold learning, achieving efficient sampling as demonstrated on a benchmark problem and a multiscale material.
A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an Itô stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps, a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.