Gaussian entropic optimal transport: Schrödinger bridges and the Sinkhorn algorithm
File(s)
Accepted version
Author(s)
Akyildiz, O Deniz
Del Moral, Pierre
Miguez, Joaquín
Type
Journal Article
Abstract
Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved using Sinkhorn algorithm (a.k.a. iterative proportional fitting procedure). However, in more general settings the Sinkhorn iterations are based on nonlinear conditional/conjugate transformations and exact finite-dimensional solutions cannot be computed.
This article presents a finite-dimensional recursive formulation of the iterative proportional fitting procedure for general Gaussian multivariate models. As expected, this recursive formulation is closely related to the celebrated Kalman filter and related Riccati matrix difference equations, and it yields algorithms that can be implemented in practical settings without further approximations. We extend this filtering methodology to develop a refined and self-contained convergence analysis of Gaussian Sinkhorn algorithms, including closed form expressions of entropic transport maps and Schrödinger bridges.
This article presents a finite-dimensional recursive formulation of the iterative proportional fitting procedure for general Gaussian multivariate models. As expected, this recursive formulation is closely related to the celebrated Kalman filter and related Riccati matrix difference equations, and it yields algorithms that can be implemented in practical settings without further approximations. We extend this filtering methodology to develop a refined and self-contained convergence analysis of Gaussian Sinkhorn algorithms, including closed form expressions of entropic transport maps and Schrödinger bridges.
Date Issued
2026-12-01
Date Acceptance
2025-11-08
Citation
Foundations of Data Science, 2026, 11, pp.50-120
ISSN
2639-8001
Publisher
American Institute of Mathematical Sciences
Start Page
50
End Page
120
Journal / Book Title
Foundations of Data Science
Volume
11
Issue
0
Copyright Statement
© 2026 American Institute of Mathematical Sciences. v This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Subjects
2020 Mathematics Subject Classification. Primary 49N05
49Q22
94A17
62J99
60J20; secondary 62C10
35Q49 Entropic optimal transport
Sinkhorn's algorithm
iterative proportional fitting procedure
Gaussian processes
Schrödinger bridges
Monge maps
Riccati matrix difference equations
Publication Status
Published
Date Publish Online
2026-01-05