Forward Reverse Kernel Regression for the Schr”{o}dinger bridge problem
arXiv:2507.00640v1 Announce Type: new
Abstract: 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.
Abstract: 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.
Denis Belomestny, John. Schoenmakers
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