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Transporting between arbitrary distributions is a fundamental goal in generative modeling. Recently proposed diffusion bridge models provide a potential solution, but they rely on a joint distribution that is difficult to obtain in…

Machine Learning · Computer Science 2025-03-03 Jun Hyeong Kim , Seonghwan Kim , Seokhyun Moon , Hyeongwoo Kim , Jeheon Woo , Woo Youn Kim

Schr\"odinger Bridge (SB) is an entropy-regularized optimal transport problem that has received increasing attention in deep generative modeling for its mathematical flexibility compared to the Scored-based Generative Model (SGM). However,…

Machine Learning · Statistics 2023-04-04 Tianrong Chen , Guan-Horng Liu , Evangelos A. Theodorou

Denoising diffusion models have recently emerged as a powerful class of generative models. They provide state-of-the-art results, not only for unconditional simulation, but also when used to solve conditional simulation problems arising in…

Machine Learning · Statistics 2022-06-28 Yuyang Shi , Valentin De Bortoli , George Deligiannidis , Arnaud Doucet

This paper aims to unify Score-based Generative Models (SGMs), also known as Diffusion models, and the Schr\"odinger Bridge (SB) problem through three reparameterization techniques: Iterative Proportional Mean-Matching (IPMM), Iterative…

Computer Vision and Pattern Recognition · Computer Science 2025-08-26 Zhicong Tang , Tiankai Hang , Shuyang Gu , Dong Chen , Baining Guo

The Quantum Schr\"odinger Bridge Problem (QSBP) describes the evolution of a stochastic process between two arbitrary probability distributions, where the dynamics are governed by the Schr\"odinger equation rather than by the traditional…

Machine Learning · Computer Science 2025-10-01 Mykola Bordyuh , Djork-Arné Clevert , Marco Bertolini

The Schr\"odinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. Most SB-related research focuses on continuous data space $\mathbb{R}^{D}$ and leaves open theoretical and…

Machine Learning · Computer Science 2025-08-19 Grigoriy Ksenofontov , Alexander Korotin

We consider the Schr\"odinger bridge problem which, given ensemble measurements of the initial and final configurations of a stochastic dynamical system and some prior knowledge on the dynamics, aims to reconstruct the "most likely"…

Machine Learning · Statistics 2026-02-04 Stephen Y. Zhang , Michael P H Stumpf

Predicting single-cell perturbation outcomes directly advances gene function analysis and facilitates drug candidate selection, making it a key driver of both basic and translational biomedical research. However, a major bottleneck in this…

Machine Learning · Computer Science 2025-11-18 Changxi Chi , Yufei Huang , Jun Xia , Jiangbin Zheng , Yunfan Liu , Zelin Zang , Stan Z. Li

Inferring cellular trajectories from destructive snapshots is complicated by the challenges of stochasticity and non-conservative mass dynamics such as cell proliferation and apoptosis. Existing unbalanced Optimal Transport (OT) methods…

Machine Learning · Computer Science 2026-05-04 Junda Ying , Yuxuan Wang , Bowen Yang , Peijie Zhou , Lei Zhang

Given two boundary distributions, the Schr\"odinger Bridge (SB) problem seeks the ``most likely`` random evolution between them with respect to a reference process. It has revealed rich connections to recent machine learning methods for…

Machine Learning · Computer Science 2025-06-03 Maosheng Yang

We investigate the generative capabilities of the Schr\"odinger Bridge (SB) approach for time series. The SB framework formulates time series synthesis as an entropic optimal interpolation transport problem between a reference probability…

Machine Learning · Computer Science 2025-10-27 Alexandre Alouadi , Baptiste Barreau , Laurent Carlier , Huyên Pham

The Schr\"odinger Bridge (SB) problem offers a powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem is the Iterative Markovian Fitting (IMF) procedure, which alternates…

Machine Learning · Computer Science 2024-11-06 Nikita Gushchin , Daniil Selikhanovych , Sergei Kholkin , Evgeny Burnaev , Alexander Korotin

In this work we explore a new framework for approximate Bayesian inference in large datasets based on stochastic control (i.e. Schr\"odinger bridges). We advocate stochastic control as a finite time and low variance alternative to popular…

In this work, we revisit the discrete-time Schr\"{o}dinger Bridge (SB) and Density Steering (DS) problems for Gaussian mixture model (GMM) boundary distributions. Building on the existing literature, we construct a set of feasible Markovian…

Systems and Control · Electrical Eng. & Systems 2026-04-02 George Rapakoulias , Fengjiao Liu , Panagiotis Tsiotras

Simulating trajectories of multi-particle systems on complex energy landscapes is a central task in molecular dynamics (MD) and drug discovery, but remains challenging at scale due to computationally expensive and long simulations. Previous…

Machine Learning · Computer Science 2025-11-11 Sophia Tang , Yinuo Zhang , Pranam Chatterjee

Computational methods for learning to sample from the Boltzmann distribution -- where the target distribution is known only up to an unnormalized energy function -- have advanced significantly recently. Due to the lack of explicit target…

Machine Learning · Statistics 2025-11-26 Guan-Horng Liu , Jaemoo Choi , Yongxin Chen , Benjamin Kurt Miller , Ricky T. Q. Chen

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal…

Machine Learning · Computer Science 2024-03-20 Alexander Korotin , Nikita Gushchin , Evgeny Burnaev

Schr\"odinger bridge (SB) has emerged as the go-to method for optimizing transportation plans in diffusion models. However, SB requires estimating the intractable forward score functions, inevitably resulting in the costly implicit training…

Machine Learning · Computer Science 2025-05-27 Wei Deng , Weijian Luo , Yixin Tan , Marin Biloš , Yu Chen , Yuriy Nevmyvaka , Ricky T. Q. Chen

We study generative modeling for time series using entropic optimal transport and the Schr\"odinger bridge (SB) framework, with a focus on applications in finance and energy modeling. Extending the diffusion-based approach of Hamdouche,…

Mathematical Finance · Quantitative Finance 2026-02-24 Stefano De Marco , Huyên Pham , Davide Zanni

Schr\"{o}dinger bridge is a stochastic optimal control problem to steer a given initial state density to another, subject to controlled diffusion and deadline constraints. A popular method to numerically solve the Schr\"{o}dinger bridge…

Optimization and Control · Mathematics 2023-09-14 Alexis M. H. Teter , Yongxin Chen , Abhishek Halder