Schrödinger bridge with transport relaxation

TL;DR

This work develops a transport-relaxed Schrödinger bridge to accommodate empirical marginals by penalizing transport between the bridge marginals and prescribed marginals. It derives a strong, finite-dimensional dual formulation in the semi-discrete setting, proving existence and uniqueness of primal and dual optimizers, and analyzes a blow-up limit in which the relaxed problem converges to a discrete Schrödinger bridge with a leading divergence of $-d\ln\varepsilon$. Two scalable numerical solvers are proposed: a gradient ascent method on the dual and a Sinkhorn-type algorithm, both shown to enjoy linear convergence. Numerical experiments in low dimensions validate the blow-up behavior and the effectiveness of the two algorithms, demonstrating interpretable space partitions via the learned dual potentials. The results provide a practical, data-driven framework for entropy-regularized transport with empirical marginals, with implications for diffusion-based generative modeling and robust transport tasks.

Abstract

Motivated by modern machine learning applications where we only have access to empirical measures constructed from finite samples, we relax the marginal constraints of the classical Schrödinger bridge problem by penalizing the transport cost between the bridge's marginals and the prescribed marginals. We derive a duality formula for this transport-relaxed bridge and demonstrate that it reduces to a finite-dimensional concave optimization problem when the prescribed marginals are discrete and the reference distribution is absolutely continuous. We establish the existence and uniqueness of solutions for both the primal and dual problems. Moreover, as the penalty blows up, we characterize the limiting bridge as the solution to a discrete Schrödinger bridge problem and identify a leading-order logarithmic divergence. Finally, we propose gradient ascent and Sinkhorn-type algorithms to numerically solve the transport-relaxed Schrödinger bridge, establishing a linear convergence rate for both algorithms.

Figures (5)

• Figure 7.1: Scaling of $I^\varepsilon(\pi^{*,\varepsilon})$ with respect to $\varepsilon$.
• Figure 7.2: Convergence of $\|(\alpha^t,\beta^t)-(\alpha^T,\beta^T)\|_{l_{\oplus}^2}$.
• Figure 7.3: Space partition and empirical mass.
• Figure 7.4: Convergence of $\|(\alpha^t,\beta^t)-(\alpha^T,\beta^T)\|_{l_{\oplus}^\infty}$.
• Figure 7.5: Space partition and empirical mass.

Theorems & Definitions (30)

• Definition 2.1
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• Remark 2.2
• Proposition 2.5: Analytic selection theorem
• Remark 3.1
• Remark 3.2