# Structure-preserving GANs

arXiv: https://arxiv.org/abs/2202.01129 · HTML: https://mlsanity.com/paper/2202.01129/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 81 references that are in the map.
- First posted on arXiv 2 Feb 2022.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Self-Attention Generative Adversarial Networks](https://mlsanity.com/paper/1805.08318/) (2018)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Differentiable Augmentation for Data-Efficient GAN Training](https://mlsanity.com/paper/2006.10738/) (2020)
- [DM-GAN: Dynamic Memory Generative Adversarial Networks for Text-to-Image Synthesis](https://mlsanity.com/paper/1904.01310/) (2019)
- [Generative Adversarial Network in Medical Imaging: A Review](https://mlsanity.com/paper/1809.07294/) (2018)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [The relativistic discriminator: a key element missing from standard GAN](https://mlsanity.com/paper/1807.00734/) (2018)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Scalable Normalizing Flows for Permutation Invariant Densities](https://mlsanity.com/paper/2010.03242/) (2020)
- [Variational Representations and Neural Network Estimation of R'enyi Divergences](https://mlsanity.com/paper/2007.03814/) (2020)
- [Group Equivariant Generative Adversarial Networks](https://mlsanity.com/paper/2005.01683/) (2020)
- [CosmoGAN: creating high-fidelity weak lensing convergence maps using Generative Adversarial Networks](https://mlsanity.com/paper/1706.02390/) (2017)

## Built on by (most cited first, 17 of 17)

- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce data](https://mlsanity.com/paper/2210.17230/) (2022)
- [Function-space regularized R'enyi divergences](https://mlsanity.com/paper/2210.04974/) (2022)
- [Statistical Guarantees of Group-Invariant GANs](https://mlsanity.com/paper/2305.13517/) (2023)
- [Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributions](https://mlsanity.com/paper/2405.13962/) (2024)
- [Combining Wasserstein-1 and Wasserstein-2 proximals: robust manifold learning via well-posed generative flows](https://mlsanity.com/paper/2407.11901/) (2024)
- [LYSTO: The Lymphocyte Assessment Hackathon and Benchmark Dataset](https://mlsanity.com/paper/2301.06304/) (2023)
- [Monotone Generative Modeling via a Gromov-Monge Embedding](https://mlsanity.com/paper/2311.01375/) (2023)
- [Structure Preserving Diffusion Models](https://mlsanity.com/paper/2402.19369/) (2024)
- [Nonlinear denoising score matching for enhanced learning of structured distributions](https://mlsanity.com/paper/2405.15625/) (2024)
- [Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences](https://mlsanity.com/paper/2410.01244/) (2024)
- [Proximal optimal transport divergences](https://mlsanity.com/paper/2505.12097/) (2025)
- [Statistical Error Bounds for GANs with Nonlinear Objective Functionals](https://mlsanity.com/paper/2406.16834/) (2024)
- [Deep Invertible Approximation of Topologically Rich Maps between Manifolds](https://mlsanity.com/paper/2210.00577/) (2022)
- [MediQ-GAN: Quantum-Inspired GAN for High Resolution Medical Image Generation](https://mlsanity.com/paper/2506.21015/) (2025)
- [Graph-Informed Adversarial Modeling: Infimal Subadditivity of Interpolative Divergences](https://mlsanity.com/paper/2603.20025/) (2026)

## Abstract (arXiv)

Generative adversarial networks (GANs), a class of distribution-learning methods based on a two-player game between a generator and a discriminator, can generally be formulated as a minmax problem based on the variational representation of a divergence between the unknown and the generated distributions. We introduce structure-preserving GANs as a data-efficient framework for learning distributions with additional structure such as group symmetry, by developing new variational representations for divergences. Our theory shows that we can reduce the discriminator space to its projection on the invariant discriminator space, using the conditional expectation with respect to the sigma-algebra associated to the underlying structure. In addition, we prove that the discriminator space reduction must be accompanied by a careful design of structured generators, as flawed designs may easily lead to a catastrophic "mode collapse" of the learned distribution. We contextualize our framework by building symmetry-preserving GANs for distributions with intrinsic group symmetry, and demonstrate that both players, namely the equivariant generator and invariant discriminator, play important but distinct roles in the learning process. Empirical experiments and ablation studies across a broad range of data sets, including real-world medical imaging, validate our theory, and show our proposed methods achieve significantly improved sample fidelity and diversity -- almost an order of magnitude measured in Fr'echet Inception Distance -- especially in the small data regime.