# Can Push-forward Generative Models Fit Multimodal Distributions?

arXiv: https://arxiv.org/abs/2206.14476 · HTML: https://mlsanity.com/paper/2206.14476/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 74 references that are in the map.
- First posted on arXiv 29 Jun 2022.

## Builds on (most cited first, 30 of 54)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [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)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Self-Attention Generative Adversarial Networks](https://mlsanity.com/paper/1805.08318/) (2018)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)

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

- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [Fast ODE-based Sampling for Diffusion Models in Around 5 Steps](https://mlsanity.com/paper/2312.00094/) (2023)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Closed-Form Diffusion Models](https://mlsanity.com/paper/2310.12395/) (2023)
- [Simple ReFlow: Improved Techniques for Fast Flow Models](https://mlsanity.com/paper/2410.07815/) (2024)
- [On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates](https://mlsanity.com/paper/2311.13584/) (2023)
- [GOPlan: Goal-conditioned Offline Reinforcement Learning by Planning with Learned Models](https://mlsanity.com/paper/2310.20025/) (2023)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [CUNSB-RFIE: Context-aware Unpaired Neural Schr"odinger Bridge in Retinal Fundus Image Enhancement](https://mlsanity.com/paper/2409.10966/) (2024)
- [ReDi: Efficient Learning-Free Diffusion Inference via Trajectory Retrieval](https://mlsanity.com/paper/2302.02285/) (2023)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [Analyzing and Improving Optimal-Transport-based Adversarial Networks](https://mlsanity.com/paper/2310.02611/) (2023)
- [Particle Semi-Implicit Variational Inference](https://mlsanity.com/paper/2407.00649/) (2024)
- [The Generation Phases of Flow Matching: a Denoising Perspective](https://mlsanity.com/paper/2510.24830/) (2025)
- [Stable Training of Normalizing Flows for High-dimensional Variational Inference](https://mlsanity.com/paper/2402.16408/) (2024)
- [Parallel Simulation for Log-concave Sampling and Score-based Diffusion Models](https://mlsanity.com/paper/2412.07435/) (2024)
- [Proximal Residual Flows for Bayesian Inverse Problems](https://mlsanity.com/paper/2211.17158/) (2022)
- [Neural Solver Selection for Combinatorial Optimization](https://mlsanity.com/paper/2410.09693/) (2024)
- [Action Hallucination in Generative Vision-Language-Action Models](https://mlsanity.com/paper/2602.06339/) (2026)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [One-Line-of-Code Data Mollification Improves Optimization of Likelihood-based Generative Models](https://mlsanity.com/paper/2305.18900/) (2023)
- [Balanced Training of Energy-Based Models with Adaptive Flow Sampling](https://mlsanity.com/paper/2306.00684/) (2023)
- [Dark Noise Diffusion: Noise Synthesis for Low-Light Image Denoising](https://mlsanity.com/paper/2503.11262/) (2025)
- [Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems](https://mlsanity.com/paper/2509.03340/) (2025)
- [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)
- [TR0N: Translator Networks for 0-Shot Plug-and-Play Conditional Generation](https://mlsanity.com/paper/2304.13742/) (2023)
- [Magnitude Distance: A Geometric Measure of Dataset Similarity](https://mlsanity.com/paper/2602.08859/) (2026)

## Abstract (arXiv)

Many generative models synthesize data by transforming a standard Gaussian random variable using a deterministic neural network. Among these models are the Variational Autoencoders and the Generative Adversarial Networks. In this work, we call them "push-forward" models and study their expressivity. We show that the Lipschitz constant of these generative networks has to be large in order to fit multimodal distributions. More precisely, we show that the total variation distance and the Kullback-Leibler divergence between the generated and the data distribution are bounded from below by a constant depending on the mode separation and the Lipschitz constant. Since constraining the Lipschitz constants of neural networks is a common way to stabilize generative models, there is a provable trade-off between the ability of push-forward models to approximate multimodal distributions and the stability of their training. We validate our findings on one-dimensional and image datasets and empirically show that generative models consisting of stacked networks with stochastic input at each step, such as diffusion models do not suffer of such limitations.