Stable generative modeling using diffusion maps
Georg Gottwald Fengyi Li Youssef Marzouk Sebastian Reich
We consider the problem of sampling from an unknown distribution for which
only a sufficiently large number of training samples are available. Such
settings have recently drawn considerable interest in the context of generative
modelling. In this paper, we propose a generative model combining diffusion
maps and Langevin dynamics. Diffusion maps are used to approximate the drift
term from the available training samples, which is then implemented in a
discrete-time Langevin sampler to generate new samples. By setting the kernel
bandwidth to match the time step size used in the unadjusted Langevin
algorithm, our method effectively circumvents any stability issues typically
associated with time-stepping stiff stochastic differential equations. More
precisely, we introduce a novel split-step scheme, ensuring that the generated
samples remain within the convex hull of the training samples. Our framework
can be naturally extended to generate conditional samples. We demonstrate the
performance of our proposed scheme through experiments on synthetic datasets
with increasing dimensions and on a stochastic subgrid-scale parametrization
conditional sampling problem.