Linear convergence of Sinkhorn's algorithm for generalized static Schrödinger bridge

The classical static Schrödinger Bridge (SSB) problem, which seeks the most likely stochastic evolution between two marginal probability measures, has been studied extensively in the optimal transport and statistical physics communities, and more recently in machine learning communities in the surge of generative models. The standard approach to solve SSB is to first identify its Kantorovich dual and use Sinkhorn's algorithm to find the optimal potential functions. While the original SSB is only a strictly convex minimization problem, this approach is known to warrant linear convergence under mild assumptions. In this work, we consider a generalized SSB allowing any strictly increasing divergence functional, far generalizing the entropy functional $x\log x$ in the standard SSB. This problem naturally arises in a wide range of seemingly unrelated problems in entropic optimal transport, random graphs/matrices, and combinatorics. We establish Kantorovich duality and linear convergence of Sinkhorn's algorithm for the generalized SSB problem under mild conditions. Our results provide a new rigorous foundation for understanding Sinkhorn-type iterative methods in the context of large-scale generalized Schrödinger bridges.

How do you transform one set of possibilities into another — like moving a cloud of particles, reshaping an image, or generating new data? Problems like this lie at the heart of physics, statistics, and machine learning. A popular method called Sinkhorn’s algorithm solves such tasks by computing the most efficient transition between probability distributions - but traditionally it relies on a single mathematical notion of difference, called KL divergence.Our research circumvents this limitation. We generalize the Schrödinger Bridge framework, allowing a wide range of ways to measure how distributions differ — better matching the needs of complex real-world systems. Crucially, we prove that our generalized Sinkhorn algorithm still converges quickly, ensuring speed and scalability.This flexibility opens new possibilities for building faster, more accurate generative models, simulating physical systems, and solving modern AI problems.