Robust Generalized Schr\"{o}dinger Bridge via Sparse Variational Gaussian Processes

ICLR 2026 Conference Submission18943 Authors

Keywords: Gaussian processes, Distribution matching, Diffusion models, Bayesian statistics

TL;DR: We propose a generalized Schr\"{o}dinger Bridge matching algorithm that is robust to uncertain/noisy stage costs, by formulating a Gaussian process inference problem on the marginal probability paths.

Abstract: The famous Schr\"{o}dinger bridge (SB) has gained renewed attention in the generative machine learning field these days for its successful applications in various areas including unsupervised image-to-image translation and particle crowd modeling. Recently, a promising algorithm dubbed GSBM was proposed to solve the generalized SB (GSB) problem, an extension of SB to deal with additional path constraints. Therein the SB is formulated as a minimal kinetic energy conditional flow matching problem, and an additional task-specific stage cost is introduced as the conditional stochastic optimal control (CondSOC) problem. The GSB is a new emerging problem with considerable room for research contributions, and we introduce a novel Gaussian process pinned marginal path posterior inference as a meaningful contribution in this area. Our main motivation is that the stage cost in GSBM, typically representing task-specific obstacles in the particle paths and other congestion penalties, can be potentially noisy and uncertain. Whereas the current GSBM approach regards this stage cost as a noise-free deterministic quantity in the CondSOC optimization, we instead model it as a stochastic quantity. Specifically, we impose a Gaussian process (GP) prior on the pinned marginal path, view the CondSOC objective as a (noisy) likelihood function, and infer the posterior path via sparse variational free-energy GP approximate inference. The main benefit is more flexible marginal path modeling that takes into account the uncertainty in the stage cost such as more realistic noisy observations. On some image-to-image translation and crowd navigation problems under noisy scenarios, we show that our proposed GP-based method yields more robust solutions than the original GSBM.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Submission Number: 18943