
% \vspace{0.02in}
\section{Conclusion}
\vspace{-0.05in}
% Thresholding techniques are widely used to tackle constrained generative modeling in real-world data. 


Reflected diffusion models, which are motivated by thresholding techniques, introduce explicit score-matching loss through reflected Brownian motion. Traditionally, these models are {applied} to hypercube-related domains {and necessitate specific diffeomorphic mappings for extension to other domains}. To enhance generality with optimal transport guarantees, we introduce the Reflected Schr\"odinger Bridge, which employs reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions. We establish connections between dynamic and static IPF algorithms in both primal and dual formulations. Additionally, we provide an approximate linear convergence analysis of the dual, potential, and couplings to deepen our understanding of the dynamic IPF algorithm. Empirically, our algorithm can be applied to any smooth domains using RVE-SDE and RVP-SDE. We evaluate its performance on 2D synthetic examples and standard image benchmarks, underscoring its competitiveness in constrained generative modeling. In future research, our focus includes integrating nonlinear diffusion based on importance sampling \citep{icsgld, awsgld} to further expedite the generation process.


\section*{Acknowledgements}
Q. Feng is partially supported by the National Science Foundation under grant DMS-2306769. 
