  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 and Bayesian inference. In this paper, we propose a generative model
combining Schr\"odinger bridges and Langevin dynamics. Schr\"odinger bridges
over an appropriate reversible reference process are used to approximate the
conditional transition probability from the available training samples, which
is then implemented in a discrete-time reversible Langevin sampler to generate
new samples. By setting the kernel bandwidth in the reference process to match
the time step size used in the unadjusted Langevin algorithm, our method
effectively circumvents any stability issues typically associated with the
time-stepping of stiff stochastic differential equations. Moreover, 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 and to Bayesian inference problems. 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.
