# Latent Space Non-Linear Statistics

arXiv: https://arxiv.org/abs/1805.07632 · HTML: https://mlsanity.com/paper/1805.07632/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 May 2018.

## Built on by (most cited first, 12 of 12)

- [GAN Inversion: A Survey](https://mlsanity.com/paper/2101.05278/) (2021)
- [DeepBlur: A Simple and Effective Method for Natural Image Obfuscation](https://mlsanity.com/paper/2104.02655/) (2021)
- [medXGAN: Visual Explanations for Medical Classifiers through a Generative Latent Space](https://mlsanity.com/paper/2204.05376/) (2022)
- [What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Models](https://mlsanity.com/paper/2408.08307/) (2024)
- [Atlas Generative Models and Geodesic Interpolation](https://mlsanity.com/paper/2102.00264/) (2021)
- [Tool- and Domain-Agnostic Parameterization of Style Transfer Effects Leveraging Pretrained Perceptual Metrics](https://mlsanity.com/paper/2105.09207/) (2021)
- [Rayleigh EigenDirections (REDs): GAN latent space traversals for multidimensional features](https://mlsanity.com/paper/2201.10423/) (2022)
- [Force-in-domain GAN inversion](https://mlsanity.com/paper/2107.06050/) (2021)
- [3D GANs and Latent Space: A comprehensive survey](https://mlsanity.com/paper/2304.03932/) (2023)
- [Discriminating Against Unrealistic Interpolations in Generative Adversarial Networks](https://mlsanity.com/paper/2203.01035/) (2022)
- [Video2StyleGAN: Encoding Video in Latent Space for Manipulation](https://mlsanity.com/paper/2206.13078/) (2022)
- [Prime Convolutional Model: Breaking the Ground for Theoretical Explainability](https://mlsanity.com/paper/2503.02773/) (2025)

## Abstract (arXiv)

Given data, deep generative models, such as variational autoencoders (VAE) and generative adversarial networks (GAN), train a lower dimensional latent representation of the data space. The linear Euclidean geometry of data space pulls back to a nonlinear Riemannian geometry on the latent space. The latent space thus provides a low-dimensional nonlinear representation of data and classical linear statistical techniques are no longer applicable. In this paper we show how statistics of data in their latent space representation can be performed using techniques from the field of nonlinear manifold statistics. Nonlinear manifold statistics provide generalizations of Euclidean statistical notions including means, principal component analysis, and maximum likelihood fits of parametric probability distributions. We develop new techniques for maximum likelihood inference in latent space, and adress the computational complexity of using geometric algorithms with high-dimensional data by training a separate neural network to approximate the Riemannian metric and cometric tensor capturing the shape of the learned data manifold.