# Do Not Escape From the Manifold: Discovering the Local Coordinates on the Latent Space of GANs

arXiv: https://arxiv.org/abs/2106.06959 · HTML: https://mlsanity.com/paper/2106.06959/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 42 references that are in the map.
- First posted on arXiv 13 Jun 2021.

## Builds on (most cited first, 29 of 29)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [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)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [StyleCLIP: Text-Driven Manipulation of StyleGAN Imagery](https://mlsanity.com/paper/2103.17249/) (2021)
- [Interpreting the Latent Space of GANs for Semantic Face Editing](https://mlsanity.com/paper/1907.10786/) (2019)
- [Image2StyleGAN: How to Embed Images Into the StyleGAN Latent Space?](https://mlsanity.com/paper/1904.03189/) (2019)
- [GANSpace: Discovering Interpretable GAN Controls](https://mlsanity.com/paper/2004.02546/) (2020)
- [Designing an Encoder for StyleGAN Image Manipulation](https://mlsanity.com/paper/2102.02766/) (2021)
- [Closed-Form Factorization of Latent Semantics in GANs](https://mlsanity.com/paper/2007.06600/) (2020)
- [StyleFlow: Attribute-conditioned Exploration of StyleGAN-Generated Images using Conditional Continuous Normalizing Flows](https://mlsanity.com/paper/2008.02401/) (2020)
- [StyleSpace Analysis: Disentangled Controls for StyleGAN Image Generation](https://mlsanity.com/paper/2011.12799/) (2020)
- [Unsupervised Discovery of Interpretable Directions in the GAN Latent Space](https://mlsanity.com/paper/2002.03754/) (2020)
- [On the "steerability" of generative adversarial networks](https://mlsanity.com/paper/1907.07171/) (2019)
- [GANalyze: Toward Visual Definitions of Cognitive Image Properties](https://mlsanity.com/paper/1906.10112/) (2019)
- [Semantic Hierarchy Emerges in Deep Generative Representations for Scene Synthesis](https://mlsanity.com/paper/1911.09267/) (2019)
- [On Disentangled Representations Learned From Correlated Data](https://mlsanity.com/paper/2006.07886/) (2020)
- [Controlling generative models with continuous factors of variations](https://mlsanity.com/paper/2001.10238/) (2020)
- [Deep Feature Interpolation for Image Content Changes](https://mlsanity.com/paper/1611.05507/) (2016)
- [Low-Rank Subspaces in GANs](https://mlsanity.com/paper/2106.04488/) (2021)
- [The Geometry of Deep Generative Image Models and its Applications](https://mlsanity.com/paper/2101.06006/) (2021)
- [A Spectral Regularizer for Unsupervised Disentanglement](https://mlsanity.com/paper/1812.01161/) (2018)
- [Disentangling by Subspace Diffusion](https://mlsanity.com/paper/2006.12982/) (2020)

## Built on by (most cited first, 26 of 26)

- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Unsupervised Discovery of Semantic Latent Directions in Diffusion Models](https://mlsanity.com/paper/2302.12469/) (2023)
- [The Euclidean Space is Evil: Hyperbolic Attribute Editing for Few-shot Image Generation](https://mlsanity.com/paper/2211.12347/) (2022)
- [Isometric Representation Learning for Disentangled Latent Space of Diffusion Models](https://mlsanity.com/paper/2407.11451/) (2024)
- [Latent Traversals in Generative Models as Potential Flows](https://mlsanity.com/paper/2304.12944/) (2023)
- [Latent Diffusion Models with Masked AutoEncoders](https://mlsanity.com/paper/2507.09984/) (2025)
- [Self-Conditioned Generative Adversarial Networks for Image Editing](https://mlsanity.com/paper/2202.04040/) (2022)
- [Householder Projector for Unsupervised Latent Semantics Discovery](https://mlsanity.com/paper/2307.08012/) (2023)
- [Is There a Better Source Distribution than Gaussian? Exploring Source Distributions for Image Flow Matching](https://mlsanity.com/paper/2512.18184/) (2025)
- [Analyzing the Latent Space of GAN through Local Dimension Estimation](https://mlsanity.com/paper/2205.13182/) (2022)
- [Local Manifold Augmentation for Multiview Semantic Consistency](https://mlsanity.com/paper/2211.02798/) (2022)
- [Deep Curvilinear Editing: Commutative and Nonlinear Image Manipulation for Pretrained Deep Generative Model](https://mlsanity.com/paper/2211.14573/) (2022)
- [Compressing Image-to-Image Translation GANs Using Local Density Structures on Their Learned Manifold](https://mlsanity.com/paper/2312.14776/) (2023)
- [Enabling Local Editing in Diffusion Models by Joint and Individual Component Analysis](https://mlsanity.com/paper/2408.16845/) (2024)
- [Exploring the latent space of diffusion models directly through singular value decomposition](https://mlsanity.com/paper/2502.02225/) (2025)
- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [Learning Input-agnostic Manipulation Directions in StyleGAN with Text Guidance](https://mlsanity.com/paper/2302.13331/) (2023)
- [Revisiting Latent Space of GAN Inversion for Real Image Editing](https://mlsanity.com/paper/2307.08995/) (2023)
- [Finding the global semantic representation in GAN through Frechet Mean](https://mlsanity.com/paper/2210.05509/) (2022)
- [Example-Based Framework for Perceptually Guided Audio Texture Generation](https://mlsanity.com/paper/2308.11859/) (2023)
- [(Deep) Generative Geodesics](https://mlsanity.com/paper/2407.11244/) (2024)
- [ColorwAI: Generative Colorways of Textiles through GAN and Diffusion Disentanglement](https://mlsanity.com/paper/2407.11514/) (2024)
- [Trained Latent Space Navigation to Prevent Lack of Photorealism in Generated Images on Style-based Models](https://mlsanity.com/paper/2310.00936/) (2023)
- [U-Face: An Efficient and Generalizable Framework for Unsupervised Facial Attribute Editing via Subspace Learning](https://mlsanity.com/paper/2603.14004/) (2026)
- [MD-Face: MoE-Enhanced Label-Free Disentangled Representation for Interactive Facial Attribute Editing](https://mlsanity.com/paper/2604.20317/) (2026)
- [The Data Manifold under the Microscope](https://mlsanity.com/paper/2606.15760/) (2026)

## Abstract (arXiv)

The discovery of the disentanglement properties of the latent space in GANs motivated a lot of research to find the semantically meaningful directions on it. In this paper, we suggest that the disentanglement property is closely related to the geometry of the latent space. In this regard, we propose an unsupervised method for finding the semantic-factorizing directions on the intermediate latent space of GANs based on the local geometry. Intuitively, our proposed method, called Local Basis, finds the principal variation of the latent space in the neighborhood of the base latent variable. Experimental results show that the local principal variation corresponds to the semantic factorization and traversing along it provides strong robustness to image traversal. Moreover, we suggest an explanation for the limited success in finding the global traversal directions in the latent space, especially W-space of StyleGAN2. We show that W-space is warped globally by comparing the local geometry, discovered from Local Basis, through the metric on Grassmannian Manifold. The global warpage implies that the latent space is not well-aligned globally and therefore the global traversal directions are bound to show limited success on it.