Schrödinger bridge with transport relaxation

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.

💡 Research Summary

The paper addresses a fundamental limitation of the classical Schrödinger bridge problem when applied to modern machine learning tasks: the marginal distributions are often empirical (discrete) measures obtained from a finite sample set, while the reference measure (typically a Brownian motion law) is absolutely continuous. In such a semi‑discrete setting the feasibility condition for the classical bridge (finite relative entropy with respect to the reference) fails, making the problem ill‑posed.

To overcome this, the authors propose a transport‑relaxed Schrödinger bridge. Instead of enforcing exact marginal constraints, they penalize the squared‑Euclidean transport cost between the bridge’s marginals and the prescribed empirical measures. The relaxed problem reads

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