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The dynamic Schr\"odinger bridge problem seeks a stochastic process that defines a transport between two target probability measures, while optimally satisfying the criteria of being closest, in terms of Kullback-Leibler divergence, to a…

Machine Learning · Statistics 2023-12-25 Stefano Peluchetti

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

The classical Schrodinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and…

Mathematical Physics · Physics 2015-06-19 Tryphon T. Georgiou , Michele Pavon

Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE),…

Machine Learning · Statistics 2023-04-06 Valentin De Bortoli , James Thornton , Jeremy Heng , Arnaud Doucet

Schrodinger Bridges (SBs) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been…

Machine Learning · Computer Science 2025-10-15 George Rapakoulias , Ali Reza Pedram , Fengjiao Liu , Lingjiong Zhu , Panagiotis Tsiotras

The solution of the path structured multimarginal Schr\"{o}dinger bridge problem (MSBP) is the most-likely measure-valued trajectory consistent with a sequence of observed probability measures or distributional snapshots. We leverage recent…

Systems and Control · Electrical Eng. & Systems 2023-10-05 Georgiy A. Bondar , Robert Gifford , Linh Thi Xuan Phan , Abhishek Halder

Schr\"odinger bridges (SBs) provide an elegant framework for modeling the temporal evolution of populations in physical, chemical, or biological systems. Such natural processes are commonly subject to changes in population size over time…

Machine Learning · Computer Science 2023-06-16 Matteo Pariset , Ya-Ping Hsieh , Charlotte Bunne , Andreas Krause , Valentin De Bortoli

Consider a reference Markov process with initial distribution $\pi_{0}$ and transition kernels $\{M_{t}\}_{t\in[1:T]}$, for some $T\in\mathbb{N}$. Assume that you are given distribution $\pi_{T}$, which is not equal to the marginal…

Computation · Statistics 2020-01-01 Espen Bernton , Jeremy Heng , Arnaud Doucet , Pierre E. Jacob

Schr\"odinger Bridges (SB) have recently gained the attention of the ML community as a promising extension of classic diffusion models which is also interconnected to the Entropic Optimal Transport (EOT). Recent solvers for SB exploit the…

Machine Learning · Computer Science 2024-07-31 Nikita Gushchin , Sergei Kholkin , Evgeny Burnaev , Alexander Korotin

In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of large deviations of the empirical distribution and an early example of the so-called maximum entropy inference method. This so-called Schroedinger bridge problem…

Optimization and Control · Mathematics 2020-11-30 Yongxin Chen , Tryphon T. Georgiou , Michele Pavon

We consider the problem to steer a linear dynamical system with full state observation from an initial gaussian distribution in state-space to a final one with minimum energy control. The system is stochastically driven through the control…

Systems and Control · Computer Science 2014-08-12 Yongxin Chen , Tryphon Georgiou , Michele Pavon

This work studies the Schr\"odinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while…

Optimization and Control · Mathematics 2026-03-23 Hamza Mahmood , Abhishek Halder , Adeel Akhtar

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

We study the Schr\"{o}dinger bridge problem (SBP) with nonlinear prior dynamics. In control-theoretic language, this is a problem of minimum effort steering of a given joint state probability density function (PDF) to another over a finite…

Optimization and Control · Mathematics 2021-03-17 Kenneth F. Caluya , Abhishek Halder

In this paper, we investigate the multi-marginal Schrodinger bridge (MSB) problem whose marginal constraints are marginal distributions of a stochastic differential equation (SDE) with a constant diffusion coefficient, and with time…

Probability · Mathematics 2025-07-15 Rentian Yao , Young--Heon Kim , Geoffrey Schiebinger

The dynamic Schr\"odinger bridge problem provides an appealing setting for solving constrained time-series data generation tasks posed as optimal transport problems. It consists of learning non-linear diffusion processes using efficient…

Machine Learning · Computer Science 2023-11-27 Ella Tamir , Martin Trapp , Arno Solin

The problem of reconciling a prior probability law on paths with data was introduced by E. Schr\"odinger in 1931/32. It represents an early formulation of a maximum likelihood problem. This specific formulation can also be seen as the…

Systems and Control · Electrical Eng. & Systems 2024-12-13 Asmaa Eldesoukey , Tryphon T. Georgiou

We consider a Schr\"odinger bridge problem where the Markov process is subject to parameter perturbations, forming an ensemble of systems. Our objective is to steer this ensemble from the initial distribution to the final distribution using…

Optimization and Control · Mathematics 2024-12-05 Daniel Owusu Adu , Yongxin Chen

Recent diffusion probabilistic models (DPM) in the field of pansharpening have been gradually gaining attention and have achieved state-of-the-art (SOTA) performance. In this paper, we identify shortcomings in directly applying DPMs to the…

Computer Vision and Pattern Recognition · Computer Science 2024-04-18 Zihan Cao , Xiao Wu , Liang-Jian Deng

Population dynamics is the study of temporal and spatial variation in the size of populations of organisms and is a major part of population ecology. One of the main difficulties in analyzing population dynamics is that we can only obtain…

Machine Learning · Computer Science 2023-02-28 Takeshi Koshizuka , Issei Sato