# SAN: Inducing Metrizability of GAN with Discriminative Normalized Linear Layer

arXiv: https://arxiv.org/abs/2301.12811 · HTML: https://mlsanity.com/paper/2301.12811/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 67 of its 94 references that are in the map.
- First posted on arXiv 30 Jan 2023.

## Builds on (most cited first, 30 of 67)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [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)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [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)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [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)
- [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)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [HiFi-GAN: Generative Adversarial Networks for Efficient and High Fidelity Speech Synthesis](https://mlsanity.com/paper/2010.05646/) (2020)
- [Self-Attention Generative Adversarial Networks](https://mlsanity.com/paper/1805.08318/) (2018)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [MoCoGAN: Decomposing Motion and Content for Video Generation](https://mlsanity.com/paper/1707.04993/) (2017)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [StyleGAN-XL: Scaling StyleGAN to Large Diverse Datasets](https://mlsanity.com/paper/2202.00273/) (2022)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [MelGAN: Generative Adversarial Networks for Conditional Waveform Synthesis](https://mlsanity.com/paper/1910.06711/) (2019)
- [cGANs with Projection Discriminator](https://mlsanity.com/paper/1802.05637/) (2018)

## Built on by (most cited first, 14 of 14)

- [Community Forensics: Using Thousands of Generators to Train Fake Image Detectors](https://mlsanity.com/paper/2411.04125/) (2024)
- [Flow Generator Matching](https://mlsanity.com/paper/2410.19310/) (2024)
- [MoLA: Motion Generation and Editing with Latent Diffusion Enhanced by Adversarial Training](https://mlsanity.com/paper/2406.01867/) (2024)
- [Diffusion Models with Deterministic Normalizing Flow Priors](https://mlsanity.com/paper/2309.01274/) (2023)
- [BigVSAN: Enhancing GAN-based Neural Vocoders with Slicing Adversarial Network](https://mlsanity.com/paper/2309.02836/) (2023)
- [Universal Inverse Distillation for Matching Models with Real-Data Supervision (No GANs)](https://mlsanity.com/paper/2509.22459/) (2025)
- [Neon: Negative Extrapolation From Self-Training Improves Image Generation](https://mlsanity.com/paper/2510.03597/) (2025)
- [Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transport](https://mlsanity.com/paper/2410.13431/) (2024)
- [Enhancing Semantic Communication with Deep Generative Models -- An ICASSP Special Session Overview](https://mlsanity.com/paper/2309.02478/) (2023)
- [Efficiency without Compromise: CLIP-aided Text-to-Image GANs with Increased Diversity](https://mlsanity.com/paper/2506.01493/) (2025)
- [VITON-DRR: Details Retention Virtual Try-on via Non-rigid Registration](https://mlsanity.com/paper/2505.23439/) (2025)
- [SONA: Learning Conditional, Unconditional, and Mismatching-Aware Discriminator](https://mlsanity.com/paper/2510.04576/) (2025)
- [Pairing Regularization for Mitigating Many-to-One Collapse in GANs](https://mlsanity.com/paper/2604.20130/) (2026)
- [Improving Generative Adversarial Networks with Self-Distillation](https://mlsanity.com/paper/2605.08577/) (2026)

## Abstract (arXiv)

Generative adversarial networks (GANs) learn a target probability distribution by optimizing a generator and a discriminator with minimax objectives. This paper addresses the question of whether such optimization actually provides the generator with gradients that make its distribution close to the target distribution. We derive metrizable conditions, sufficient conditions for the discriminator to serve as the distance between the distributions by connecting the GAN formulation with the concept of sliced optimal transport. Furthermore, by leveraging these theoretical results, we propose a novel GAN training scheme, called slicing adversarial network (SAN). With only simple modifications, a broad class of existing GANs can be converted to SANs. Experiments on synthetic and image datasets support our theoretical results and the SAN's effectiveness as compared to usual GANs. Furthermore, we also apply SAN to StyleGAN-XL, which leads to state-of-the-art FID score amongst GANs for class conditional generation on ImageNet 256×256. Our implementation is available on https://ytakida.github.io/san.