🤖 AI Summary
To address the challenge of density modeling and sampling on low-dimensional manifolds embedded in high-dimensional data, this paper proposes a dual-path framework integrating manifold learning with diffusion-based generation. First, Diffusion Maps are employed to uncover the intrinsic low-dimensional manifold structure. Second, a score-matching diffusion model is constructed—or equivalently, an Itô stochastic differential equation (SDE) is solved—directly on the learned manifold for latent-space sampling. Finally, Double Diffusion Maps—novelly applied here for unsupervised, differentiable high-dimensional reconstruction rather than dynamical dimensionality reduction—are used to lift samples back to the ambient space. The method avoids explicit manifold parameterization and enables end-to-end density modeling on unknown manifolds. Evaluated on benchmark tasks and multiscale materials datasets, it achieves significant improvements in sampling fidelity and manifold consistency, while maintaining both accuracy and computational efficiency.
📝 Abstract
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 It^o 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.