Stein Transport for Generative Modeling

We propose a nonparametric, kernel-based approach to generative modeling based on Stein operators and particle transport. Our method constructs a continuous-time transport from a simple reference distribution (typically, a standard Gaussian) to a target distribution using a Stein formulation of the continuity equation. Unlike alternative Stein transport methods, which require access to the target density or its gradient, we focus on the sample-only setting and introduce a tractable surrogate objective based on an Ornstein--Uhlenbeck (OU) probability path. This construction allows the velocity field driving the transport to be learned using only samples from the target distribution and Gaussian noise. By restricting the velocity field to a reproducing kernel Hilbert space, we obtain closed-form solutions via kernel ridge regression at each time step. The resulting method is particularly well suited to small data regimes, where heavily parameterised neural generative models may be difficult to train or overfit. We demonstrate the usefulness of the approach for sample-based generative modeling in the small data regime (generative data augmentation), and for posterior emulation in Bayesian inverse problems, where expensive Monte Carlo samplers produce limited sets of posterior samples that must be efficiently augmented.