Forward Reverse Kernel Regression for the Schr\"{o}dinger bridge problem

Denis Belomestny; John. Schoenmakers

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arXiv · 2507.00640

Forward Reverse Kernel Regression for the Schr\"{o}dinger bridge problem

Denis Belomestny·John. Schoenmakers

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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.

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Denis Belomestny, John. Schoenmakers. 2025-07-01. Forward Reverse Kernel Regression for the Schr\"{o}dinger bridge problem. https://arxiv.org/abs/2507.00640

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