mlsanityData to 7 Oct 2026

Can Push-forward Generative Models Fit Multimodal Distributions?

arXiv 2206.14476 ↗First posted 29 Jun 2022stat.ML cs.LGcited by 41

Not on the map. Why?

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.

Key facts. Cited by 41 papers in the map (data through 7 Oct 2026). Builds on 54 of its 74 references that are in the map. First posted on arXiv 29 Jun 2022.

Builds on

54 of 74 references.

References · 54

  1. 2017Attention Is All You Needcited by 59,932
  2. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  3. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  4. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  5. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  6. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  7. 2014ImageNet Large Scale Visual Recognition Challengecited by 8,441
  8. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  9. 2014Explaining and Harnessing Adversarial Examplescited by 7,287
  10. 2016Densely Connected Convolutional Networkscited by 6,691
  11. 2016Image-to-Image Translation with Conditional Adversarial Networkscited by 4,510
  12. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
38 more
  1. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  2. 2019Analyzing and Improving the Image Quality of StyleGANcited by 3,580
  3. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,609
  4. 2017Non-local Neural Networkscited by 2,290
  5. 2017Improved Training of Wasserstein GANscited by 2,223
  6. 2015Variational Inference with Normalizing Flowscited by 2,161
  7. 2015Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networkscited by 2,048
  8. 2016Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Networkcited by 1,898
  9. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  10. 2018Group Normalizationcited by 1,514
  11. 2016Variational Inference: A Review for Statisticianscited by 1,466
  12. 2017Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learningcited by 935
  13. 2019Improved Precision and Recall Metric for Assessing Generative Modelscited by 925
  14. 2018Self-Attention Generative Adversarial Networkscited by 809
  15. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 636
  16. 2017Towards Principled Methods for Training Generative Adversarial Networkscited by 460
  17. 2018Lipschitz regularity of deep neural networks: analysis and efficient estimationcited by 413
  18. 2018Assessing Generative Models via Precision and Recallcited by 401
  19. 2021Skillful Precipitation Nowcasting using Deep Generative Models of Radarcited by 308
  20. 2021The Intrinsic Dimension of Images and Its Impact on Learningcited by 307
  21. 2019Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networkscited by 290
  22. 2015Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulationscited by 285
  23. 2017Geometric GANcited by 232
  24. 2016Unrolled Generative Adversarial Networkscited by 208
  25. 2020Stochastic Normalizing Flowscited by 156
  26. 2017Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practicecited by 132
  27. 2017DeLiGAN : Generative Adversarial Networks for Diverse and Limited Datacited by 77
  28. 2020Understanding and Mitigating Exploding Inverses in Invertible Neural Networkscited by 71
  29. 2017Many Paths to Equilibrium: GANs Do Not Need to Decrease a Divergence At Every Stepcited by 45
  30. 2020On Implicit Regularization in β-VAEscited by 34
  31. 2020Blindness of score-based methods to isolated components and mixing proportionscited by 33
  32. 2020Towards GAN Benchmarks Which Require Generalizationcited by 31
  33. 2020Stabilizing Invertible Neural Networks Using Mixture Modelscited by 30
  34. 2021Implicit Normalizing Flowscited by 28
  35. 2018Disconnected Manifold Learning for Generative Adversarial Networkscited by 27
  36. 2017Learning Wasserstein Embeddingscited by 25
  37. 2020Learning disconnected manifolds: a no GANs landcited by 25
  38. 2017Improving Neural Machine Translation with Conditional Sequence Generative Adversarial Netscited by 20

50 most cited of 54 All 54 (JSON)

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 3; 2023: 14; 2024: 12; 2025: 8; 2026: 4.

41 papers.

Sort:

By area:

  1. 2023Unpaired Image-to-Image Translation via Neural Schr"odinger Bridgecited by 96
  2. 2023Fast ODE-based Sampling for Diffusion Models in Around 5 Stepscited by 83
  3. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  4. 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 62
  5. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  6. 2023Closed-Form Diffusion Modelscited by 46
  7. 2024Simple ReFlow: Improved Techniques for Fast Flow Modelscited by 34
  8. 2023On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimatescited by 24
  9. 2023GOPlan: Goal-conditioned Offline Reinforcement Learning by Planning with Learned Modelscited by 20
  10. 2024CUNSB-RFIE: Context-aware Unpaired Neural Schr"odinger Bridge in Retinal Fundus Image Enhancementcited by 13
  11. 2023Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problemscited by 12
  12. 2023ReDi: Efficient Learning-Free Diffusion Inference via Trajectory Retrievalcited by 11
29 more
  1. 2023Learning from small data sets: Patch-based regularizers in inverse problems for image reconstructioncited by 9
  2. 2023Analyzing and Improving Optimal-Transport-based Adversarial Networkscited by 8
  3. 2024Particle Semi-Implicit Variational Inferencecited by 7
  4. 2024Parallel Simulation for Log-concave Sampling and Score-based Diffusion Modelscited by 6
  5. 2025The Generation Phases of Flow Matching: a Denoising Perspectivecited by 6
  6. 2022Proximal Residual Flows for Bayesian Inverse Problemscited by 5
  7. 2024Stable Training of Normalizing Flows for High-dimensional Variational Inferencecited by 5
  8. 2024Neural Solver Selection for Combinatorial Optimizationcited by 5
  9. 2026Action Hallucination in Generative Vision-Language-Action Modelscited by 5
  10. 2023NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problemscited by 4
  11. 2023One-Line-of-Code Data Mollification Improves Optimization of Likelihood-based Generative Modelscited by 4
  12. 2023Balanced Training of Energy-Based Models with Adaptive Flow Samplingcited by 4
  13. 2025Dark Noise Diffusion: Noise Synthesis for Low-Light Image Denoisingcited by 4
  14. 2022Unveiling the Latent Space Geometry of Push-Forward Generative Modelscited by 3
  15. 2023Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problemscited by 3
  16. 2025Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problemscited by 3
  17. 2023TR0N: Translator Networks for 0-Shot Plug-and-Play Conditional Generationcited by 2
  18. 2026Magnitude Distance: A Geometric Measure of Dataset Similaritycited by 2
  19. 2024Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overviewcited by 1
  20. 2024On the Nonconvexity of Push-Forward Constraints and Its Consequences in Machine Learningcited by 1
  21. 2024Learning Quantized Adaptive Conditions for Diffusion Modelscited by 1
  22. 2024Preferential Normalizing Flowscited by 1
  23. 2025Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinementcited by 1
  24. 2025LatentStealth: Unnoticeable and Efficient Adversarial Attacks on Expressive Human Pose and Shape Estimationcited by 0
  25. 2025Time-reversed Flow Matching with Worst Transport in High-dimensional Latent Space for Image Anomaly Detectioncited by 0
  26. 2025Adapting Noise to Data: Generative Flows from 1D Processescited by 0
  27. 2025A Non-Adversarial Approach to Idempotent Generative Modellingcited by 0
  28. 2026Bridging Restoration and Diagnosis: A Comprehensive Benchmark for Retinal Fundus Enhancementcited by 0
  29. 2026Why Do Few-Step Text Latents Fail When Image Latents Work? Non-Commitment at Sharp Categorical Readoutscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025On the Statistical Capacity of Deep Generative Models
  2. 2026Memorisation, convergence and generalisation in generative models
  3. 2021Variational Diffusion Models
  4. 2025Generative Distribution Prediction: A Unified Approach to Multimodal Learning
  5. 2024Diffusion Models For Multi-Modal Generative Modeling
  6. 2024Towards Theoretical Understandings of Self-Consuming Generative Models
  7. 2025Sparse-to-Sparse Training of Diffusion Models
  8. 2025Diffusion-based supervised learning of generative models for efficient sampling of multimodal distributions