mlsanityData to 14 Oct 2026

Latent Space Oddity: on the Curvature of Deep Generative Models

arXiv 1710.11379 ↗First posted 31 Oct 2017stat.MLcited by 174

Not on the map. Why?

Cited by 25 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

Deep generative models provide a systematic way to learn nonlinear data distributions, through a set of latent variables and a nonlinear "generator" function that maps latent points into the input space. The nonlinearity of the generator imply that the latent space gives a distorted view of the input space. Under mild conditions, we show that this distortion can be characterized by a stochastic Riemannian metric, and demonstrate that distances and interpolants are significantly improved under this metric. This in turn improves probability distributions, sampling algorithms and clustering in the latent space. Our geometric analysis further reveals that current generators provide poor variance estimates and we propose a new generator architecture with vastly improved variance estimates. Results are demonstrated on convolutional and fully connected variational autoencoders, but the formalism easily generalize to other deep generative models.

Key facts. Cited by 174 papers in the map (data through 14 Oct 2026). First posted on arXiv 31 Oct 2017.

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Citing papers per year: 2021: 24; 2022: 23; 2023: 30; 2024: 22; 2025: 29; 2026: 46.

174 papers.

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  1. 2023Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometrycited by 125
  2. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 77
  3. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 70
  4. 2023Norm-guided latent space exploration for text-to-image generationcited by 43
  5. 2021Qimera: Data-free Quantization with Synthetic Boundary Supporting Samplescited by 40
  6. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  7. 2021The Geometry of Deep Generative Image Models and its Applicationscited by 35
  8. 2021Smoothing the Disentangled Latent Style Space for Unsupervised Image-to-Image Translationcited by 35
  9. 2022Indeterminacy in Generative Models: Characterization and Strong Identifiabilitycited by 33
  10. 2023Directional Connectivity-based Segmentation of Medical Imagescited by 30
  11. 2023Unsupervised Discovery of Semantic Latent Directions in Diffusion Modelscited by 28
  12. 2021Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencodercited by 27
38 more
  1. 2022Pythae: Unifying Generative Autoencoders in Python -- A Benchmarking Use Casecited by 26
  2. 2022A Geometric Perspective on Variational Autoencoderscited by 24
  3. 2021Learning Riemannian Manifolds for Geodesic Motion Skillscited by 23
  4. 2021Pulling back information geometrycited by 23
  5. 2024Isometric Representation Learning for Disentangled Latent Space of Diffusion Modelscited by 22
  6. 2024Manifold Integrated Gradients: Riemannian Geometry for Feature Attributioncited by 21
  7. 2025Action-Minimization Meets Generative Modeling: Efficient Transition Path Sampling with the Onsager-Machlup Functionalcited by 20
  8. 2023Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimizationcited by 18
  9. 2021Multi-Asset Spot and Option Market Simulationcited by 17
  10. 2023Latent Traversals in Generative Models as Potential Flowscited by 17
  11. 2021Data Augmentation with Variational Autoencoders and Manifold Samplingcited by 16
  12. 2023The Geometry of Neural Nets' Parameter Spaces Under Reparametrizationcited by 16
  13. 2023Riemannian Laplace approximations for Bayesian neural networkscited by 15
  14. 2021DeepBlur: A Simple and Effective Method for Natural Image Obfuscationcited by 13
  15. 2021A prior-based approximate latent Riemannian metriccited by 11
  16. 2022Lagrangian Manifold Monte Carlo on Monge Patchescited by 11
  17. 2023Data Representations' Study of Latent Image Manifoldscited by 11
  18. 2026The Information Geometry of Softmax: Probing and Steeringcited by 11
  19. 2026Curveball Steering: The Right Direction To Steer Isn't Always Linearcited by 11
  20. 2021Uncertainty Estimation Using Riemannian Model Dynamics for Offline Reinforcement Learningcited by 10
  21. 2021Exploring the Latent Space of Autoencoders with Interventional Assayscited by 10
  22. 2023VTAE: Variational Transformer Autoencoder with Manifolds Learningcited by 10
  23. 2021GeomCA: Geometric Evaluation of Data Representationscited by 9
  24. 2024Embedding-Aligned Language Modelscited by 9
  25. 2024Words in Motion: Extracting Interpretable Control Vectors for Motion Transformerscited by 9
  26. 2024Counterfactual Explanations via Riemannian Latent Space Traversalcited by 9
  27. 2025Hessian Geometry of Latent Space in Generative Modelscited by 9
  28. 2021On Linear Interpolation in the Latent Space of Deep Generative Modelscited by 8
  29. 2023A Geometric Perspective on Autoencoderscited by 8
  30. 2024Vocabulary-Defined Semantics: Latent Space Clustering for Improving In-Context Learningcited by 8
  31. 2024Pulling back symmetric Riemannian geometry for data analysiscited by 8
  32. 2024Pullback Flow Matching on Data Manifoldscited by 8
  33. 2025Riemannian-Geometric Fingerprints of Generative Modelscited by 8
  34. 2022GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductionscited by 7
  35. 2023Normalizing flow sampling with Langevin dynamics in the latent spacecited by 7
  36. 2023RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representationscited by 7
  37. 2023On convex decision regions in deep network representationscited by 7
  38. 2024Tilt your Head: Activating the Hidden Spatial-Invariance of Classifierscited by 7

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