Gaussian entropic optimal transport: Schr""odinger bridges and the Sinkhorn algorithm

2024-12-24

AI Summary

In entropy-regularized optimal transport under Gaussian distributions, the Sinkhorn algorithm struggles to solve nonlinear transformations exactly. Method: This paper proposes a finite-dimensional recursive Sinkhorn algorithm. It establishes, for the first time, an explicit recursive analytical form of Sinkhorn iterations in the Gaussian setting, deeply coupling iterative scaling with the Kalman filter and Riccati matrix difference equation frameworks. Contributions/Results: We derive closed-form solutions for both the entropic transport map and the Schrödinger bridge. Moreover, we provide the first complete convergence analysis, rigorously proving linear convergence. The method enables exact, efficient, and analytically tractable numerical computation for multivariate Gaussian settings—without approximation. By unifying optimal transport, Schrödinger bridges, and filtering theory, it offers a novel tool for probabilistic modeling and dynamic inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

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 Schr""odinger bridges.

Problem

• Optimal Transport
• Gaussian Entropy
• Sinkhorn Algorithm

Innovation

• Gaussian optimal transport
• Kalman filter analogy
• Gaussian Sinkhorn algorithm analysis