# Learning disconnected manifolds: a no GANs land

arXiv: https://arxiv.org/abs/2006.04596 · HTML: https://mlsanity.com/paper/2006.04596/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2020.

## Built on by (most cited first, 25 of 25)

- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [MG-GAN: A Multi-Generator Model Preventing Out-of-Distribution Samples in Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2108.09274/) (2021)
- [Can Push-forward Generative Models Fit Multimodal Distributions?](https://mlsanity.com/paper/2206.14476/) (2022)
- [Quo Vadis: Is Trajectory Forecasting the Key Towards Long-Term Multi-Object Tracking?](https://mlsanity.com/paper/2210.07681/) (2022)
- [Polarity Sampling: Quality and Diversity Control of Pre-Trained Generative Networks via Singular Values](https://mlsanity.com/paper/2203.01993/) (2022)
- [Unifying GANs and Score-Based Diffusion as Generative Particle Models](https://mlsanity.com/paper/2305.16150/) (2023)
- [FairStyle: Debiasing StyleGAN2 with Style Channel Manipulations](https://mlsanity.com/paper/2202.06240/) (2022)
- [Textual Manifold-based Defense Against Natural Language Adversarial Examples](https://mlsanity.com/paper/2211.02878/) (2022)
- [On the expressivity of bi-Lipschitz normalizing flows](https://mlsanity.com/paper/2107.07232/) (2021)
- [Out-of-Distribution Detection with Reconstruction Error and Typicality-based Penalty](https://mlsanity.com/paper/2212.12641/) (2022)
- [Partition-Guided GANs](https://mlsanity.com/paper/2104.00816/) (2021)
- [Precision-Recall Divergence Optimization for Generative Modeling with GANs and Normalizing Flows](https://mlsanity.com/paper/2305.18910/) (2023)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [Latent reweighting, an almost free improvement for GANs](https://mlsanity.com/paper/2110.09803/) (2021)
- [Action Hallucination in Generative Vision-Language-Action Models](https://mlsanity.com/paper/2602.06339/) (2026)
- [Fast Jacobian-Vector Product for Deep Networks](https://mlsanity.com/paper/2104.00219/) (2021)
- [Multi-level Latent Space Structuring for Generative Control](https://mlsanity.com/paper/2202.05910/) (2022)
- [Optimal 1-Wasserstein Distance for WGANs](https://mlsanity.com/paper/2201.02824/) (2022)
- [Unveiling the Latent Space Geometry of Push-Forward Generative Models](https://mlsanity.com/paper/2207.10541/) (2022)
- [Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems](https://mlsanity.com/paper/2309.12128/) (2023)
- [Generalization of GANs and overparameterized models under Lipschitz continuity](https://mlsanity.com/paper/2104.02388/) (2021)
- [Smooth image-to-image translations with latent space interpolations](https://mlsanity.com/paper/2210.00841/) (2022)
- [Spider GAN: Leveraging Friendly Neighbors to Accelerate GAN Training](https://mlsanity.com/paper/2305.07613/) (2023)
- [Provably Valid and Diverse Mutations of Real-World Media Data for DNN Testing](https://mlsanity.com/paper/2112.01956/) (2021)
- [Training Normalizing Flows with the Precision-Recall Divergence](https://mlsanity.com/paper/2302.00628/) (2023)

## Abstract (arXiv)

Typical architectures of Generative AdversarialNetworks make use of a unimodal latent distribution transformed by a continuous generator. Consequently, the modeled distribution always has connected support which is cumbersome when learning a disconnected set of manifolds. We formalize this problem by establishing a no free lunch theorem for the disconnected manifold learning stating an upper bound on the precision of the targeted distribution. This is done by building on the necessary existence of a low-quality region where the generator continuously samples data between two disconnected modes. Finally, we derive a rejection sampling method based on the norm of generators Jacobian and show its efficiency on several generators including BigGAN.