# Reflected Schr"odinger Bridge for Constrained Generative Modeling arXiv: https://arxiv.org/abs/2401.03228 ยท HTML: https://mlsanity.com/paper/2401.03228/ ## Key facts - Cited by 12 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 42 of its 67 references that are in the map. - First posted on arXiv 6 Jan 2024. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) ## Builds on (most cited first, 30 of 42) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021) - [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022) - [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022) - [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022) - [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021) - [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021) - [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021) - [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017) - [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022) - [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013) - [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022) - [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021) - [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022) - [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021) - [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022) - [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023) - [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023) - [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022) - [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022) - [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023) - [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018) - [Sampling from a log-concave distribution with Projected Langevin Monte Carlo](https://mlsanity.com/paper/1507.02564/) (2015) - [Simulating Diffusion Bridges with Score Matching](https://mlsanity.com/paper/2111.07243/) (2021) - [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019) - [Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo](https://mlsanity.com/paper/1705.08964/) (2017) - [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023) - [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019) - [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015) ## Built on by (most cited first, 12 of 12) - [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024) - [Variational Schr"odinger Diffusion Models](https://mlsanity.com/paper/2405.04795/) (2024) - [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024) - [Neural Shr"odinger Bridge Matching for Pansharpening](https://mlsanity.com/paper/2404.11416/) (2024) - [Recurrent Interpolants for Probabilistic Time Series Prediction](https://mlsanity.com/paper/2409.11684/) (2024) - [Optimal Stochastic Trace Estimation in Generative Modeling](https://mlsanity.com/paper/2502.18808/) (2025) - [Score-based constrained generative modeling via Langevin diffusions with boundary conditions](https://mlsanity.com/paper/2510.23985/) (2025) - [Pathwise Learning of Stochastic Dynamical Systems with Partial Observations](https://mlsanity.com/paper/2601.21860/) (2026) - [Intrinsic-Noise Consolidation: A Doob-Barrier-Conditioned Diffusion Turns Analog Device Noise into a Continual-Learning Resource](https://mlsanity.com/paper/2607.06924/) (2026) - [Inconsistency-aware Multimodal Schr"odinger Bridge for Deepfake Localization](https://mlsanity.com/paper/2605.23113/) (2026) - [Reflected Schr"odinger Bridge Matching](https://mlsanity.com/paper/2607.03626/) (2026) - [Constrained Flow Matching via Lagrangian Dual Flows](https://mlsanity.com/paper/2607.04513/) (2026) ## Abstract (arXiv) 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 techniques for boundary enforcement. Reflected diffusion models (Lou23) aim to enhance generalizability by generating the data distribution through a backward process governed by reflected Brownian motion. However, reflected diffusion models may not easily adapt to diverse domains without the derivation of proper diffeomorphic mappings and do not guarantee optimal transport properties. To overcome these limitations, we introduce the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive elegant reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and explore natural connections to entropic optimal transport for the study of approximate linear convergence - a valuable insight for practical training. Our algorithm yields robust generative modeling in diverse domains, and its scalability is demonstrated in real-world constrained generative modeling through standard image benchmarks.