# Mode Collapse and Regularity of Optimal Transportation Maps

arXiv: https://arxiv.org/abs/1902.02934 · HTML: https://mlsanity.com/paper/1902.02934/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Feb 2019.

## Built on by (most cited first, 5 of 5)

- [Large-scale optimal transport map estimation using projection pursuit](https://mlsanity.com/paper/2106.05838/) (2021)
- [Construct Informative Triplet with Two-stage Hard-sample Generation](https://mlsanity.com/paper/2112.02259/) (2021)
- [Convex Smoothed Autoencoder-Optimal Transport model](https://mlsanity.com/paper/2101.05679/) (2021)
- [DPOT: A DeepParticle method for Computation of Optimal Transport with convergence guarantee](https://mlsanity.com/paper/2506.23429/) (2025)
- [LPIPS-AttnWav2Lip: Generic Audio-Driven lip synchronization for Talking Head Generation in the Wild](https://mlsanity.com/paper/2602.00189/) (2026)

## Abstract (arXiv)

This work builds the connection between the regularity theory of optimal transportation map, Monge-Amp`ere equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. According to the regularity theory of Monge-Amp`ere equation, if the support of the target measure is disconnected or just non-convex, the optimal transportation mapping is discontinuous. General DNNs can only approximate continuous mappings. This intrinsic conflict leads to the convergence difficulty and mode collapse in GANs. We test our hypothesis that the supports of real data distribution are in general non-convex, therefore the discontinuity is unavoidable using an Autoencoder combined with discrete optimal transportation map (AE-OT framework) on the CelebA data set. The testing result is positive. Furthermore, we propose to approximate the continuous Brenier potential directly based on discrete Brenier theory to tackle mode collapse. Comparing with existing method, this method is more accurate and effective.