# Polarity Sampling: Quality and Diversity Control of Pre-Trained Generative Networks via Singular Values

arXiv: https://arxiv.org/abs/2203.01993 · HTML: https://mlsanity.com/paper/2203.01993/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 56 references that are in the map.
- First posted on arXiv 3 Mar 2022.

## Builds on (most cited first, 30 of 44)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [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)
- [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)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [WaveGlow: A Flow-based Generative Network for Speech Synthesis](https://mlsanity.com/paper/1811.00002/) (2018)
- [Generating Images with Sparse Representations](https://mlsanity.com/paper/2103.03841/) (2021)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)
- [Projected GANs Converge Faster](https://mlsanity.com/paper/2111.01007/) (2021)
- [VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning](https://mlsanity.com/paper/1705.07761/) (2017)
- [Seeing What a GAN Cannot Generate](https://mlsanity.com/paper/1910.11626/) (2019)
- [Instance-Conditioned GAN](https://mlsanity.com/paper/2109.05070/) (2021)
- [Your GAN is Secretly an Energy-based Model and You Should use Discriminator Driven Latent Sampling](https://mlsanity.com/paper/2003.06060/) (2020)
- [Bias Correction of Learned Generative Models using Likelihood-Free Importance Weighting](https://mlsanity.com/paper/1906.09531/) (2019)

## Built on by (most cited first, 21 of 21)

- [Self-Consuming Generative Models Go MAD](https://mlsanity.com/paper/2307.01850/) (2023)
- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [Self-Improving Diffusion Models with Synthetic Data](https://mlsanity.com/paper/2408.16333/) (2024)
- [SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries](https://mlsanity.com/paper/2302.12828/) (2023)
- [POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks](https://mlsanity.com/paper/2211.01340/) (2022)
- [Characterizing Large Language Model Geometry Helps Solve Toxicity Detection and Generation](https://mlsanity.com/paper/2312.01648/) (2023)
- [Fair Sampling in Diffusion Models through Switching Mechanism](https://mlsanity.com/paper/2401.03140/) (2024)
- [Generator Knows What Discriminator Should Learn in Unconditional GANs](https://mlsanity.com/paper/2207.13320/) (2022)
- [What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Models](https://mlsanity.com/paper/2408.08307/) (2024)
- [Fast and Exact Enumeration of Deep Networks Partitions Regions](https://mlsanity.com/paper/2401.11188/) (2024)
- [GLeaD: Improving GANs with A Generator-Leading Task](https://mlsanity.com/paper/2212.03752/) (2022)
- [Improving the Effectiveness of Deep Generative Data](https://mlsanity.com/paper/2311.03959/) (2023)
- [Stylized Face Sketch Extraction via Generative Prior with Limited Data](https://mlsanity.com/paper/2403.11263/) (2024)
- [Unveiling the Latent Space Geometry of Push-Forward Generative Models](https://mlsanity.com/paper/2207.10541/) (2022)
- [Training Dynamics of Deep Network Linear Regions](https://mlsanity.com/paper/2310.12977/) (2023)
- [Controlling the Fidelity and Diversity of Deep Generative Models via Pseudo Density](https://mlsanity.com/paper/2407.08659/) (2024)
- [Spider GAN: Leveraging Friendly Neighbors to Accelerate GAN Training](https://mlsanity.com/paper/2305.07613/) (2023)
- [Mitigating over-exploration in latent space optimization using LES](https://mlsanity.com/paper/2406.09657/) (2024)
- [Rethinking Image Skip Connections in StyleGAN2](https://mlsanity.com/paper/2407.05527/) (2024)
- [On the Geometry of Deep Learning](https://mlsanity.com/paper/2408.04809/) (2024)
- [Complexity of Linear Regions in Self-supervised Deep ReLU Networks](https://mlsanity.com/paper/2604.24393/) (2026)

## Abstract (arXiv)

We present Polarity Sampling, a theoretically justified plug-and-play method for controlling the generation quality and diversity of pre-trained deep generative networks DGNs). Leveraging the fact that DGNs are, or can be approximated by, continuous piecewise affine splines, we derive the analytical DGN output space distribution as a function of the product of the DGN's Jacobian singular values raised to a power ρ. We dub ρ the polarity parameter and prove that ρ focuses the DGN sampling on the modes (ρ < 0) or anti-modes (ρ > 0) of the DGN output-space distribution. We demonstrate that nonzero polarity values achieve a better precision-recall (quality-diversity) Pareto frontier than standard methods, such as truncation, for a number of state-of-the-art DGNs. We also present quantitative and qualitative results on the improvement of overall generation quality (e.g., in terms of the Frechet Inception Distance) for a number of state-of-the-art DGNs, including StyleGAN3, BigGAN-deep, NVAE, for different conditional and unconditional image generation tasks. In particular, Polarity Sampling redefines the state-of-the-art for StyleGAN2 on the FFHQ Dataset to FID 2.57, StyleGAN2 on the LSUN Car Dataset to FID 2.27 and StyleGAN3 on the AFHQv2 Dataset to FID 3.95. Demo: bit.ly/polarity-samp