# Metrics for Deep Generative Models

arXiv: https://arxiv.org/abs/1711.01204 · HTML: https://mlsanity.com/paper/1711.01204/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2017.

## Built on by (most cited first, 30 of 53)

- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [The Geometry of Deep Generative Image Models and its Applications](https://mlsanity.com/paper/2101.06006/) (2021)
- [Unsupervised Discovery of Semantic Latent Directions in Diffusion Models](https://mlsanity.com/paper/2302.12469/) (2023)
- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [Pythae: Unifying Generative Autoencoders in Python -- A Benchmarking Use Case](https://mlsanity.com/paper/2206.08309/) (2022)
- [A Geometric Perspective on Variational Autoencoders](https://mlsanity.com/paper/2209.07370/) (2022)
- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Computationally-Efficient Neural Image Compression with Shallow Decoders](https://mlsanity.com/paper/2304.06244/) (2023)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Amortized Variational Inference: A Systematic Review](https://mlsanity.com/paper/2209.10888/) (2022)
- [DeepBlur: A Simple and Effective Method for Natural Image Obfuscation](https://mlsanity.com/paper/2104.02655/) (2021)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [Uncertainty Estimation Using Riemannian Model Dynamics for Offline Reinforcement Learning](https://mlsanity.com/paper/2102.11327/) (2021)
- [Exploring the Latent Space of Autoencoders with Interventional Assays](https://mlsanity.com/paper/2106.16091/) (2021)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Unsupervised Discovery of Interpretable Directions in h-space of Pre-trained Diffusion Models](https://mlsanity.com/paper/2310.09912/) (2023)
- [Atlas Generative Models and Geodesic Interpolation](https://mlsanity.com/paper/2102.00264/) (2021)
- [Mario Plays on a Manifold: Generating Functional Content in Latent Space through Differential Geometry](https://mlsanity.com/paper/2206.00106/) (2022)
- [Learning Identity-Preserving Transformations on Data Manifolds](https://mlsanity.com/paper/2106.12096/) (2021)
- [Adversarial robustness of VAEs through the lens of local geometry](https://mlsanity.com/paper/2208.03923/) (2022)
- [Convergent autoencoder approximation of low bending and low distortion manifold embeddings](https://mlsanity.com/paper/2208.10193/) (2022)
- [Image Interpolation with Score-based Riemannian Metrics of Diffusion Models](https://mlsanity.com/paper/2504.20288/) (2025)
- [Continuous Conditional Generative Adversarial Networks (cGAN) with Generator Regularization](https://mlsanity.com/paper/2103.14884/) (2021)
- [Unveiling the Latent Space Geometry of Push-Forward Generative Models](https://mlsanity.com/paper/2207.10541/) (2022)
- [Deep Curvilinear Editing: Commutative and Nonlinear Image Manipulation for Pretrained Deep Generative Model](https://mlsanity.com/paper/2211.14573/) (2022)

## Abstract (arXiv)

Neural samplers such as variational autoencoders (VAEs) or generative adversarial networks (GANs) approximate distributions by transforming samples from a simple random source---the latent space---to samples from a more complex distribution represented by a dataset. While the manifold hypothesis implies that the density induced by a dataset contains large regions of low density, the training criterions of VAEs and GANs will make the latent space densely covered. Consequently points that are separated by low-density regions in observation space will be pushed together in latent space, making stationary distances poor proxies for similarity. We transfer ideas from Riemannian geometry to this setting, letting the distance between two points be the shortest path on a Riemannian manifold induced by the transformation. The method yields a principled distance measure, provides a tool for visual inspection of deep generative models, and an alternative to linear interpolation in latent space. In addition, it can be applied for robot movement generalization using previously learned skills. The method is evaluated on a synthetic dataset with known ground truth; on a simulated robot arm dataset; on human motion capture data; and on a generative model of handwritten digits.