Forward Reverse Kernel Regression for the Schrödinger bridge problem
This provides a method for entropic optimal transport, but it appears incremental as it builds on existing iterative and kernel-based techniques.
The paper tackles the Schrödinger Bridge Problem by proposing a forward-reverse iterative Monte Carlo procedure with kernel regression to approximate potentials nonparametrically, resulting in a provably convergent algorithm with optimal convergence rates.
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.