# A Likely Geometry of Generative Models

arXiv: https://arxiv.org/abs/2510.26266 · HTML: https://mlsanity.com/paper/2510.26266/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 41 of its 79 references that are in the map.
- First posted on arXiv 30 Oct 2025.

## Builds on (most cited first, 30 of 41)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Brain Imaging Generation with Latent Diffusion Models](https://mlsanity.com/paper/2209.07162/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Noise-Free Score Distillation](https://mlsanity.com/paper/2310.17590/) (2023)
- [Neural Autoregressive Distribution Estimation](https://mlsanity.com/paper/1605.02226/) (2016)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [normflows: A PyTorch Package for Normalizing Flows](https://mlsanity.com/paper/2302.12014/) (2023)
- [Smooth Diffusion: Crafting Smooth Latent Spaces in Diffusion Models](https://mlsanity.com/paper/2312.04410/) (2023)
- [Interpolating between Images with Diffusion Models](https://mlsanity.com/paper/2307.12560/) (2023)
- [Variational Autoencoders with Riemannian Brownian Motion Priors](https://mlsanity.com/paper/2002.05227/) (2020)
- [Metrics for Probabilistic Geometries](https://mlsanity.com/paper/1411.7432/) (2014)
- [Fast and Robust Shortest Paths on Manifolds Learned from Data](https://mlsanity.com/paper/1901.07229/) (2019)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)

## Abstract (arXiv)

The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data dimension. In this paper, we construct a general geometry compatible with different metrics and probability distributions to analyze generative models that do not require additional training. We consider curves analogous to geodesics constrained to a suitable data distribution aimed at targeting high-density regions learned by generative models. We formulate this as a (pseudo)-metric and prove that this corresponds to a Newtonian system on a Riemannian manifold. We show that shortest paths in our framework can be characterized by a system of ordinary differential equations, which locally corresponds to geodesics under a suitable Riemannian metric. Numerically, we derive a novel algorithm to efficiently compute shortest paths and generalized Fr'echet means. Quantitatively, we show that curves using our metric traverse regions of higher density than baselines across a range of models and datasets.