Forward Reverse Kernel Regression for the Schr"odinger bridge problem

In this paper, we study the Schr"odinger Bridge Problem (SBP), which is central to entropic optimal transport. For general reference processes and begin--endpoint distributions, we propose a forward-reverse iterative Monte Carlo procedure to approximate the Schr"odinger potentials in a nonparametric way. In particular, we use kernel based Monte Carlo regression in the context of Picard iteration of a corresponding fixed point problem. By preserving in the iteration positivity and contractivity in a Hilbert metric sense, we develop a provably convergent algorithm. Furthermore, we provide convergence rates for the potential estimates and prove their optimality. Finally, as an application, we propose a non-nested Monte Carlo procedure for the final dimensional distributions of the Schr"odinger Bridge process, based on the constructed potentials and the forward-reverse simulation method for conditional diffusions.