# A Geometric View of Optimal Transportation and Generative Model

arXiv: https://arxiv.org/abs/1710.05488 · HTML: https://mlsanity.com/paper/1710.05488/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2017.

## Built on by (most cited first, 30 of 31)

- [A Survey of Machine Unlearning](https://mlsanity.com/paper/2209.02299/) (2022)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [On the Principles of Parsimony and Self-Consistency for the Emergence of Intelligence](https://mlsanity.com/paper/2207.04630/) (2022)
- [Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance)](https://mlsanity.com/paper/2103.01678/) (2021)
- [A Survey on Optimal Transport for Machine Learning: Theory and Applications](https://mlsanity.com/paper/2106.01963/) (2021)
- [Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Scheme](https://mlsanity.com/paper/2211.16757/) (2022)
- [Accelerating Sinkhorn Algorithm with Sparse Newton Iterations](https://mlsanity.com/paper/2401.12253/) (2024)
- [What's the Situation with Intelligent Mesh Generation: A Survey and Perspectives](https://mlsanity.com/paper/2211.06009/) (2022)
- [Score Mismatching for Generative Modeling](https://mlsanity.com/paper/2309.11043/) (2023)
- [A scalable deep learning approach for solving high-dimensional dynamic optimal transport](https://mlsanity.com/paper/2205.07521/) (2022)
- [Deep Learning as Ricci Flow](https://mlsanity.com/paper/2404.14265/) (2024)
- [Manifold Matching via Deep Metric Learning for Generative Modeling](https://mlsanity.com/paper/2106.10777/) (2021)
- [Machine Learning Algebraic Geometry for Physics](https://mlsanity.com/paper/2204.10334/) (2022)
- [Aligning Logits Generatively for Principled Black-Box Knowledge Distillation](https://mlsanity.com/paper/2205.10490/) (2022)
- [Monotone Generative Modeling via a Gromov-Monge Embedding](https://mlsanity.com/paper/2311.01375/) (2023)
- [A Sinkhorn-type Algorithm for Constrained Optimal Transport](https://mlsanity.com/paper/2403.05054/) (2024)
- [Point Cloud Compression via Constrained Optimal Transport](https://mlsanity.com/paper/2403.08236/) (2024)
- [Continual Learning of Generative Models with Limited Data: From Wasserstein-1 Barycenter to Adaptive Coalescence](https://mlsanity.com/paper/2101.09225/) (2021)
- [Generative Adversarial Network: Some Analytical Perspectives](https://mlsanity.com/paper/2104.12210/) (2021)
- [A new method for determining Wasserstein 1 optimal transport maps from Kantorovich potentials, with deep learning applications](https://mlsanity.com/paper/2211.00820/) (2022)
- [OT-Net: A Reusable Neural Optimal Transport Solver](https://mlsanity.com/paper/2306.08233/) (2023)
- [Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transport](https://mlsanity.com/paper/2410.13431/) (2024)
- [Convex Smoothed Autoencoder-Optimal Transport model](https://mlsanity.com/paper/2101.05679/) (2021)
- [Theoretical Insights into CycleGAN: Analyzing Approximation and Estimation Errors in Unpaired Data Generation](https://mlsanity.com/paper/2407.11678/) (2024)
- [HOTS3D: Hyper-Spherical Optimal Transport for Semantic Alignment of Text-to-3D Generation](https://mlsanity.com/paper/2407.14419/) (2024)
- [General Autonomous Cybersecurity Defense: Learning Robust Policies for Dynamic Topologies and Diverse Attackers](https://mlsanity.com/paper/2506.22706/) (2025)
- [ShuffleMix: Improving Representations via Channel-Wise Shuffle of Interpolated Hidden States](https://mlsanity.com/paper/2305.18684/) (2023)
- [Optimal Transport for Kernel Gaussian Mixture Models](https://mlsanity.com/paper/2310.18586/) (2023)
- [Motion Mapping Cognition: A Nondecomposable Primary Process in Human Vision](https://mlsanity.com/paper/2402.04275/) (2024)

## Abstract (arXiv)

In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel framework for generative models. By using the optimal transportation view of GAN model, we show that the discriminator computes the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. Therefore, it is sufficient to solely optimize the discriminator. This shows the adversarial competition can be avoided, and the computational architecture can be simplified. Preliminary experimental results show the geometric method outperforms WGAN for approximating probability measures with multiple clusters in low dimensional space.