# Mirror Diffusion Models for Constrained and Watermarked Generation

arXiv: https://arxiv.org/abs/2310.01236 · HTML: https://mlsanity.com/paper/2310.01236/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 48 of its 79 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)
- LLM safety and security / [Watermarking generated content and models](https://mlsanity.com/safety/watermarking-generated-content-and-models/)

## Builds on (most cited first, 30 of 48)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Magic3D: High-Resolution Text-to-3D Content Creation](https://mlsanity.com/paper/2211.10440/) (2022)
- [HiFi-GAN: Generative Adversarial Networks for Efficient and High Fidelity Speech Synthesis](https://mlsanity.com/paper/2010.05646/) (2020)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [A Watermark for Large Language Models](https://mlsanity.com/paper/2301.10226/) (2023)
- [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)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)

## Built on by (most cited first, 30 of 42)

- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Diffusion Models as Constrained Samplers for Optimization with Unknown Constraints](https://mlsanity.com/paper/2402.18012/) (2024)
- [Continuously Augmented Discrete Diffusion model for Categorical Generative Modeling](https://mlsanity.com/paper/2510.01329/) (2025)
- [Constrained Diffusion Models via Dual Training](https://mlsanity.com/paper/2408.15094/) (2024)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [CDI: Copyrighted Data Identification in Diffusion Models](https://mlsanity.com/paper/2411.12858/) (2024)
- [Reflected Flow Matching](https://mlsanity.com/paper/2405.16577/) (2024)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Neural Approximate Mirror Maps for Constrained Diffusion Models](https://mlsanity.com/paper/2406.12816/) (2024)
- [Quantum State Generation with Structure-Preserving Diffusion Model](https://mlsanity.com/paper/2404.06336/) (2024)
- [Constrained Generative Modeling with Manually Bridged Diffusion Models](https://mlsanity.com/paper/2502.20371/) (2025)
- [Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains](https://mlsanity.com/paper/2510.08929/) (2025)
- [Constrained Posterior Sampling: Time Series Generation with Hard Constraints](https://mlsanity.com/paper/2410.12652/) (2024)
- [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025)
- [Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevin](https://mlsanity.com/paper/2505.24222/) (2025)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Neural Shr"odinger Bridge Matching for Pansharpening](https://mlsanity.com/paper/2404.11416/) (2024)
- [Diffusion-based Planning with Learned Viability Filters](https://mlsanity.com/paper/2502.19564/) (2025)
- [Constraint-Aware Flow Matching via Randomized Exploration](https://mlsanity.com/paper/2508.13316/) (2025)
- [Logic-Guided Vector Fields for Constrained Generative Modeling](https://mlsanity.com/paper/2602.02009/) (2026)
- [Improved Constrained Generation by Bridging Pretrained Generative Models](https://mlsanity.com/paper/2603.06742/) (2026)
- [Constrained Diffusion Models with Primal-Dual Inference](https://mlsanity.com/paper/2606.17192/) (2026)
- [Physics-aligned Schr"odinger bridge](https://mlsanity.com/paper/2409.17825/) (2024)
- [Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations](https://mlsanity.com/paper/2512.14980/) (2025)
- [Data Provenance for Image Auto-Regressive Generation](https://mlsanity.com/paper/2606.28386/) (2026)
- [Generative modeling assisted simulation of measurement-altered quantum criticality](https://mlsanity.com/paper/2412.01513/) (2024)
- [When Diffusion Breaks Constraints: Sequential Autoregressive Generation with RL and MCTS](https://mlsanity.com/paper/2512.01242/) (2025)
- [Q-Tag: Watermarking Quantum Circuit Generative Models](https://mlsanity.com/paper/2602.23085/) (2026)
- [Gaussian Shannon: High-Precision Diffusion Model Watermarking Based on Communication](https://mlsanity.com/paper/2603.26167/) (2026)

## Abstract (arXiv)

Modern successes of diffusion models in learning complex, high-dimensional data distributions are attributed, in part, to their capability to construct diffusion processes with analytic transition kernels and score functions. The tractability results in a simulation-free framework with stable regression losses, from which reversed, generative processes can be learned at scale. However, when data is confined to a constrained set as opposed to a standard Euclidean space, these desirable characteristics appear to be lost based on prior attempts. In this work, we propose Mirror Diffusion Models (MDM), a new class of diffusion models that generate data on convex constrained sets without losing any tractability. This is achieved by learning diffusion processes in a dual space constructed from a mirror map, which, crucially, is a standard Euclidean space. We derive efficient computation of mirror maps for popular constrained sets, such as simplices and ℓ2-balls, showing significantly improved performance of MDM over existing methods. For safety and privacy purposes, we also explore constrained sets as a new mechanism to embed invisible but quantitative information (i.e., watermarks) in generated data, for which MDM serves as a compelling approach. Our work brings new algorithmic opportunities for learning tractable diffusion on complex domains. Our code is available at https://github.com/ghliu/mdm