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We study the problem of generating synthetic time series that reproduce both marginal distributions and temporal dynamics, a central challenge in financial machine learning. Existing approaches typically fail to jointly model drift and…

Machine Learning · Computer Science 2026-04-10 Alexandre Alouadi , Grégoire Loeper , Célian Marsala , Othmane Mazhar , Huyên Pham

Homotopy approaches to Bayesian inference have found widespread use especially if the Kullback-Leibler divergence between the prior and the posterior distribution is large. Here we extend one of these homotopy approach to include an…

Numerical Analysis · Mathematics 2022-11-04 Sebastian Reich

This paper aims to conduct a comprehensive theoretical analysis of current diffusion models. We introduce a novel generative learning methodology utilizing the Schr{\"o}dinger bridge diffusion model in latent space as the framework for…

Machine Learning · Statistics 2024-12-24 Yuling Jiao , Lican Kang , Huazhen Lin , Jin Liu , Heng Zuo

The Schr\"odinger bridge problem (SBP) is gaining increasing attention in generative modeling and showing promising potential even in comparison with the score-based generative models (SGMs). SBP can be interpreted as an entropy-regularized…

A Schr\"{o}dinger bridge establishes a dynamic transport map between two target distributions via a reference process, simultaneously solving an associated entropic optimal transport problem. We consider the setting where samples from the…

Machine Learning · Computer Science 2025-01-22 Stefano Peluchetti

We study a stochastic optimal control problem for jump-diffusion systems whose drift coefficient is piecewise Lipschitz continuous and exhibits threshold-induced discontinuities. Such dynamics naturally arise in applications with…

Optimization and Control · Mathematics 2026-05-08 Antoine-Marie Bogso , Edward Fuituh Kameh , Olivier Menoukeu-Pamen , Felix Shu

The Schr\"odinger bridge problem seeks the optimal stochastic process that connects two given probability distributions with minimal energy modification. While the Sinkhorn algorithm is widely used to solve the static optimal transport…

Machine Learning · Statistics 2025-10-28 Ibuki Maeda , Rentian Yao , Atsushi Nitanda

We reassess the concept of transition at minimum work in classical stochastic finite-time thermodynamics, when the system dynamics is modelled by a diffusion process. We show that a well-posed formulation of the optimal control problem…

Statistical Mechanics · Physics 2026-03-13 Paolo Muratore-Ginanneschi , Julia Sanders

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 the Schr\"odinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schr\"odinger potentials via Sinkhorn iterations on empirical measures and then…

Machine Learning · Statistics 2026-02-10 Denis Belomestny , Alexey Naumov , Nikita Puchkin , Denis Suchkov

Diffusion models serve as a powerful generative framework for solving inverse problems. However, they still face two key challenges: 1) the distortion-perception tradeoff, where improving perceptual quality often degrades reconstruction…

Machine Learning · Computer Science 2025-11-20 Qing Yao , Lijian Gao , Qirong Mao , Ming Dong

We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs), such as recently developed diffusion probabilistic models. In particular, we derive a…

Machine Learning · Computer Science 2024-03-27 Julius Berner , Lorenz Richter , Karen Ullrich

We study a generalization of the Brownian bridge as a stochastic process that models the position and velocity of inertial particles between the two end-points of a time interval. The particles experience random acceleration and are assumed…

Systems and Control · Computer Science 2014-07-15 Yongxin Chen , Tryphon Georgiou

Diffusion models have become the go-to method for large-scale generative models in real-world applications. These applications often involve data distributions confined within bounded domains, typically requiring ad-hoc thresholding…

Machine Learning · Statistics 2024-01-09 Wei Deng , Yu Chen , Nicole Tianjiao Yang , Hengrong Du , Qi Feng , Ricky T. Q. Chen

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

Resampling from a target measure whose density is unknown is a fundamental problem in mathematical statistics and machine learning. A setting that dominates the machine learning literature consists of learning a map from an easy-to-sample…

Machine Learning · Computer Science 2026-02-24 Leticia Mattos Da Silva , Silvia Sellán , Francisco Vargas , Justin Solomon

We consider the problem of simulating diffusion bridges, which are diffusion processes that are conditioned to initialize and terminate at two given states. The simulation of diffusion bridges has applications in diverse scientific fields…

Computation · Statistics 2025-06-19 Jeremy Heng , Valentin De Bortoli , Arnaud Doucet , James Thornton

We present Path Integral Sampler~(PIS), a novel algorithm to draw samples from unnormalized probability density functions. The PIS is built on the Schr\"odinger bridge problem which aims to recover the most likely evolution of a diffusion…

Machine Learning · Computer Science 2022-03-11 Qinsheng Zhang , Yongxin Chen

We study a semimartingale optimal transport problem interpolating between the Schr\"odinger bridge and the stretched Brownian motion associated with the Bass solution of the Skorokhod embedding problem. The cost combines an entropy term on…

Probability · Mathematics 2026-03-31 Pierre Henry-Labordere , Grégoire Loeper , Othmane Mazhar , Huyên Pham , Nizar Touzi

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