{"ID":2899589,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.00640","arxiv_id":"2507.00640","title":"Forward Reverse Kernel Regression for the Schrödinger bridge problem","abstract":"In this paper, we study the Schrödinger 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 Schrödinger 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 Schrödinger Bridge process, based on the constructed potentials and the forward-reverse simulation method for conditional diffusions.","short_abstract":"In this paper, we study the Schrödinger 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 Schrödinger potentials in a nonparametric way. In particular, w...","url_abs":"https://arxiv.org/abs/2507.00640","url_pdf":"https://arxiv.org/pdf/2507.00640v1","authors":"[\"Denis Belomestny\",\"John. Schoenmakers\"]","published":"2025-07-01T10:32:36Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"math.NA\"]","methods":"[\"Diffusion Model\"]","has_code":false}