# Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport

arXiv: https://arxiv.org/abs/2305.14777 · HTML: https://mlsanity.com/paper/2305.14777/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 84 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 30 of 59)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Adversarial Latent Autoencoders](https://mlsanity.com/paper/2004.04467/) (2020)
- [StyleSwin: Transformer-based GAN for High-resolution Image Generation](https://mlsanity.com/paper/2112.10762/) (2021)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)

## Built on by (most cited first, 30 of 41)

- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation](https://mlsanity.com/paper/2311.15100/) (2023)
- [Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport](https://mlsanity.com/paper/2402.05443/) (2024)
- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Analyzing and Improving Optimal-Transport-based Adversarial Networks](https://mlsanity.com/paper/2310.02611/) (2023)
- [A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions](https://mlsanity.com/paper/2405.03664/) (2024)
- [Improving Neural Optimal Transport via Displacement Interpolation](https://mlsanity.com/paper/2410.03783/) (2024)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [A High-Quality Robust Diffusion Framework for Corrupted Dataset](https://mlsanity.com/paper/2311.17101/) (2023)
- [Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan](https://mlsanity.com/paper/2502.04583/) (2025)
- [Scalable Simulation-free Entropic Unbalanced Optimal Transport](https://mlsanity.com/paper/2410.02656/) (2024)
- [QUATRO: Query-Adaptive Trust Region Policy Optimization for LLM Fine-tuning](https://mlsanity.com/paper/2602.04620/) (2026)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (2025)
- [Fast and Large-Scale Unbalanced Optimal Transport via its Semi-Dual and Adaptive Gradient Methods](https://mlsanity.com/paper/2602.10697/) (2026)
- [Unlearning for One-Step Generative Models via Unbalanced Optimal Transport](https://mlsanity.com/paper/2603.16489/) (2026)
- [Generative Conditional Distributions by Neural (Entropic) Optimal Transport](https://mlsanity.com/paper/2406.02317/) (2024)
- [Unsupervised Point Cloud Completion through Unbalanced Optimal Transport](https://mlsanity.com/paper/2410.02671/) (2024)
- [Reweighted Flow Matching via Unbalanced OT for Label-free Long-tailed Generation](https://mlsanity.com/paper/2509.25713/) (2025)
- [Experts-Guided Unbalanced Optimal Transport for ISP Learning from Unpaired and/or Paired Data](https://mlsanity.com/paper/2512.05635/) (2025)
- [Rate-Optimal Noise Annealing in Semi-Dual Neural Optimal Transport: Tangential Identifiability, Off-Manifold Ambiguity, and Guaranteed Recovery](https://mlsanity.com/paper/2602.04110/) (2026)
- [Reward-guided Fine-Tuning of One-Step Generative Models via Wasserstein Gradient Flow](https://mlsanity.com/paper/2608.29647/) (2026)
- [Safe Screening for Unbalanced Optimal Transport](https://mlsanity.com/paper/2307.00247/) (2023)
- [Exploring bidirectional bounds for minimax-training of Energy-based models](https://mlsanity.com/paper/2506.04609/) (2025)
- [Federated Flow Matching](https://mlsanity.com/paper/2509.21250/) (2025)
- [Multivariate Time Series Data Imputation via Distributionally Robust Regularization](https://mlsanity.com/paper/2602.00844/) (2026)
- [Rethinking the Flow-Based Gradual Domain Adaptation: A Semi-Dual Optimal Transport Perspective](https://mlsanity.com/paper/2602.01179/) (2026)

## Abstract (arXiv)

Optimal Transport (OT) problem investigates a transport map that bridges two distributions while minimizing a given cost function. In this regard, OT between tractable prior distribution and data has been utilized for generative modeling tasks. However, OT-based methods are susceptible to outliers and face optimization challenges during training. In this paper, we propose a novel generative model based on the semi-dual formulation of Unbalanced Optimal Transport (UOT). Unlike OT, UOT relaxes the hard constraint on distribution matching. This approach provides better robustness against outliers, stability during training, and faster convergence. We validate these properties empirically through experiments. Moreover, we study the theoretical upper-bound of divergence between distributions in UOT. Our model outperforms existing OT-based generative models, achieving FID scores of 2.97 on CIFAR-10 and 6.36 on CelebA-HQ-256. The code is available at urlhttps://github.com/Jae-Moo/UOTM.