# A gradual, semi-discrete approach to generative network training via explicit Wasserstein minimization

arXiv: https://arxiv.org/abs/1906.03471 · HTML: https://mlsanity.com/paper/1906.03471/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2019.

## Built on by (most cited first, 13 of 13)

- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Fair and Optimal Classification via Post-Processing](https://mlsanity.com/paper/2211.01528/) (2022)
- [Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance)](https://mlsanity.com/paper/2103.01678/) (2021)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport](https://mlsanity.com/paper/2403.03777/) (2024)
- [On the Existence of Optimal Transport Gradient for Learning Generative Models](https://mlsanity.com/paper/2102.05542/) (2021)
- [OT-Net: A Reusable Neural Optimal Transport Solver](https://mlsanity.com/paper/2306.08233/) (2023)
- [HOTS3D: Hyper-Spherical Optimal Transport for Semantic Alignment of Text-to-3D Generation](https://mlsanity.com/paper/2407.14419/) (2024)
- [On optimal solutions of classical and sliced Wasserstein GANs with non-Gaussian data](https://mlsanity.com/paper/2509.06505/) (2025)
- [Stochastic Optimization in Semi-Discrete Optimal Transport: Convergence Analysis and Minimax Rate](https://mlsanity.com/paper/2510.25287/) (2025)
- [Decreasing Entropic Regularization Averaged Gradient for Semi-Discrete Optimal Transport](https://mlsanity.com/paper/2510.27340/) (2025)
- [Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimation](https://mlsanity.com/paper/2602.04335/) (2026)
- [MaDeL: Manifold-Decomposed Feature Losses for Generative Modeling](https://mlsanity.com/paper/2610.04419/) (2026)

## Abstract (arXiv)

This paper provides a simple procedure to fit generative networks to target distributions, with the goal of a small Wasserstein distance (or other optimal transport costs). The approach is based on two principles: (a) if the source randomness of the network is a continuous distribution (the "semi-discrete" setting), then the Wasserstein distance is realized by a deterministic optimal transport mapping; (b) given an optimal transport mapping between a generator network and a target distribution, the Wasserstein distance may be decreased via a regression between the generated data and the mapped target points. The procedure here therefore alternates these two steps, forming an optimal transport and regressing against it, gradually adjusting the generator network towards the target distribution. Mathematically, this approach is shown to minimize the Wasserstein distance to both the empirical target distribution, and also its underlying population counterpart. Empirically, good performance is demonstrated on the training and testing sets of the MNIST and Thin-8 data. The paper closes with a discussion of the unsuitability of the Wasserstein distance for certain tasks, as has been identified in prior work [Arora et al., 2017, Huang et al., 2017].