mlsanityData to 7 Oct 2026

Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry

arXiv 2307.12868 ↗First posted 24 Jul 2023cs.CVcited by 121

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space xt ∈ X, we analyze them from a geometrical perspective. Our approach involves deriving the local latent basis within X by leveraging the pullback metric associated with their encoding feature maps. Remarkably, our discovered local latent basis enables image editing capabilities by moving xt, the latent space of DMs, along the basis vector at specific timesteps. We further analyze how the geometric structure of DMs evolves over diffusion timesteps and differs across different text conditions. This confirms the known phenomenon of coarse-to-fine generation, as well as reveals novel insights such as the discrepancy between xt across timesteps, the effect of dataset complexity, and the time-varying influence of text prompts. To the best of our knowledge, this paper is the first to present image editing through x-space traversal, editing only once at specific timestep t without any additional training, and providing thorough analyses of the latent structure of DMs. The code to reproduce our experiments can be found at https://github.com/enkeejunior1/Diffusion-Pullback.

Key facts. Cited by 121 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 54 of its 62 references that are in the map. First posted on arXiv 24 Jul 2023.

Builds on

54 of 62 references.

Closest ancestors · 14

Same direction, newest first

  1. 2023Discovering Interpretable Directions in the Semantic Latent Space of Diffusion Models20 Mar
  2. 2022Scalable Diffusion Models with Transformers19 Dec
  3. 2022Diffusion Models already have a Semantic Latent Space20 Oct
  4. 2022gDDIM: Generalized denoising diffusion implicit models11 Jun
  5. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  6. 2022Perception Prioritized Training of Diffusion Models1 Apr
  7. 2022Generating High Fidelity Data from Low-density Regions using Diffusion Models31 Mar
  8. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  9. 2021ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models6 Aug
  10. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  11. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  12. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
2 more
  1. 2020Denoising Diffusion Implicit Models6 Oct
  2. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 40

  1. 2021Learning Transferable Visual Models From Natural Language Supervisioncited by 33,456
  2. 2014Microsoft COCO: Common Objects in Contextcited by 17,498
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  4. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  5. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 5,986
  6. 2021GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Modelscited by 3,595
  7. 2017Progressive Growing of GANs for Improved Quality, Stability, and Variationcited by 2,967
  8. 2022InstructPix2Pix: Learning to Follow Image Editing Instructionscited by 2,591
  9. 2022Prompt-to-Prompt Image Editing with Cross Attention Controlcited by 2,409
  10. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  11. 2022RePaint: Inpainting using Denoising Diffusion Probabilistic Modelscited by 1,419
  12. 2015LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loopcited by 1,350
28 more
  1. 2022Null-text Inversion for Editing Real Images using Guided Diffusion Modelscited by 1,168
  2. 2019StarGAN v2: Diverse Image Synthesis for Multiple Domainscited by 1,095
  3. 2022Human Motion Diffusion Modelcited by 1,062
  4. 2021StyleCLIP: Text-Driven Manipulation of StyleGAN Imagerycited by 990
  5. 2021Blended Diffusion for Text-driven Editing of Natural Imagescited by 924
  6. 2022eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoiserscited by 921
  7. 2022Plug-and-Play Diffusion Features for Text-Driven Image-to-Image Translationcited by 897
  8. 2023Attend-and-Excite: Attention-Based Semantic Guidance for Text-to-Image Diffusion Modelscited by 696
  9. 2020GANSpace: Discovering Interpretable GAN Controlscited by 612
  10. 2023Zero-shot Image-to-Image Translationcited by 503
  11. 2022Improving Diffusion Models for Inverse Problems using Manifold Constraintscited by 501
  12. 2021Label-Efficient Semantic Segmentation with Diffusion Modelscited by 495
  13. 2020Closed-Form Factorization of Latent Semantics in GANscited by 407
  14. 2023Open-Vocabulary Panoptic Segmentation with Text-to-Image Diffusion Modelscited by 371
  15. 2020StyleFlow: Attribute-conditioned Exploration of StyleGAN-Generated Images using Conditional Continuous Normalizing Flowscited by 348
  16. 2023Drag Your GAN: Interactive Point-based Manipulation on the Generative Image Manifoldcited by 218
  17. 2021More Control for Free! Image Synthesis with Semantic Diffusion Guidancecited by 211
  18. 2017Latent Space Oddity: on the Curvature of Deep Generative Modelscited by 169
  19. 2022Uncovering the Disentanglement Capability in Text-to-Image Diffusion Modelscited by 116
  20. 2017The Riemannian Geometry of Deep Generative Modelscited by 81
  21. 2017Metrics for Deep Generative Modelscited by 53
  22. 2023DiffusionSeg: Adapting Diffusion Towards Unsupervised Object Discoverycited by 53
  23. 2021LatentCLR: A Contrastive Learning Approach for Unsupervised Discovery of Interpretable Directionscited by 52
  24. 2021Low-Rank Subspaces in GANscited by 52
  25. 2020Geometrically Enriched Latent Spacescited by 45
  26. 2022Multiresolution Textual Inversioncited by 33
  27. 2018A Spectral Regularizer for Unsupervised Disentanglementcited by 30
  28. 2021Do Not Escape From the Manifold: Discovering the Local Coordinates on the Latent Space of GANscited by 26

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 4; 2024: 28; 2025: 57; 2026: 32.

121 papers.

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  1. 2023Concept Sliders: LoRA Adaptors for Precise Control in Diffusion Modelscited by 148
  2. 2025SAeUron: Interpretable Concept Unlearning in Diffusion Models with Sparse Autoencoderscited by 75
  3. 2023Self-Discovering Interpretable Diffusion Latent Directions for Responsible Text-to-Image Generationcited by 69
  4. 2024Smoothed Energy Guidance: Guiding Diffusion Models with Reduced Energy Curvature of Attentioncited by 55
  5. 2023NoiseCLR: A Contrastive Learning Approach for Unsupervised Discovery of Interpretable Directions in Diffusion Modelscited by 46
  6. 2024Interpreting the Weight Space of Customized Diffusion Modelscited by 35
  7. 2024Open-Vocabulary Attention Maps with Token Optimization for Semantic Segmentation in Diffusion Modelscited by 29
  8. 2026Diamond Maps: Efficient Reward Alignment via Stochastic Flow Mapscited by 26
  9. 2024Isometric Representation Learning for Disentangled Latent Space of Diffusion Modelscited by 22
  10. 2025Drag Your Gaussian: Effective Drag-Based Editing with Score Distillation for 3D Gaussian Splattingcited by 21
  11. 2025Emergence and Evolution of Interpretable Concepts in Diffusion Modelscited by 20
  12. 2024Beta Sampling is All You Need: Efficient Image Generation Strategy for Diffusion Models using Stepwise Spectral Analysiscited by 19
38 more
  1. 2024Learning Discrete Concepts in Latent Hierarchical Modelscited by 17
  2. 2024DiffusionPID: Interpreting Diffusion via Partial Information Decompositioncited by 16
  3. 2025TIDE : Temporal-Aware Sparse Autoencoders for Interpretable Diffusion Transformers in Image Generationcited by 16
  4. 2024Global Counterfactual Directionscited by 15
  5. 2024Exploring Low-Dimensional Subspaces in Diffusion Models for Controllable Image Editingcited by 15
  6. 2024Prompt Sliders for Fine-Grained Control, Editing and Erasing of Concepts in Diffusion Modelscited by 14
  7. 2024Counterfactual Explanations via Riemannian Latent Space Traversalcited by 9
  8. 2025Hessian Geometry of Latent Space in Generative Modelscited by 9
  9. 2024Linear combinations of latents in generative models: subspaces and beyondcited by 8
  10. 2024HARIVO: Harnessing Text-to-Image Models for Video Generationcited by 8
  11. 2025Responsible Diffusion Models via Constraining Text Embeddings within Safe Regionscited by 8
  12. 2025Diffusion Counterfactual Generation with Semantic Abductioncited by 8
  13. 2025Unleashing the Potential of the Semantic Latent Space in Diffusion Models for Image Dehazingcited by 8
  14. 2025SAEmnesia: Erasing Concepts in Diffusion Models with Supervised Sparse Autoencoderscited by 8
  15. 2025Bures-Wasserstein Flow Matching for Graph Generationcited by 7
  16. 2024Learn Once, Edit Anywhere: Visual Direction Transfer for Diffusion Modelscited by 6
  17. 2024Joint Diffusion models in Continual Learningcited by 6
  18. 2025Diffusion Generative Modeling on Lie Group Representationscited by 6
  19. 2025What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifoldcited by 6
  20. 2025The Spacetime of Diffusion Models: An Information Geometry Perspectivecited by 6
  21. 2025StyleKeeper: Prevent Content Leakage using Negative Visual Query Guidancecited by 6
  22. 2025Riemann2: Learning Riemannian Submanifolds from Riemannian Datacited by 5
  23. 2025Causally Steered Diffusion for Automated Video Counterfactual Generationcited by 5
  24. 2025Text Slider: Efficient and Plug-and-Play Continuous Concept Control for Image/Video Synthesis via LoRA Adapterscited by 5
  25. 2024Latent Abstractions in Generative Diffusion Modelscited by 4
  26. 2025Masked Generative Nested Transformers with Decode Time Scalingcited by 4
  27. 2025Progressive Prompt Detailing for Improved Alignment in Text-to-Image Generative Modelscited by 4
  28. 2025Image Interpolation with Score-based Riemannian Metrics of Diffusion Modelscited by 4
  29. 2025Deeper Diffusion Models Amplify Biascited by 4
  30. 2025Nonparametric Data Attribution for Diffusion Modelscited by 4
  31. 2024EvolvED: Evolutionary Embeddings to Understand the Generation Process of Diffusion Modelscited by 3
  32. 2024Enabling Local Editing in Diffusion Models by Joint and Individual Component Analysiscited by 3
  33. 2025Exploring the latent space of diffusion models directly through singular value decompositioncited by 3
  34. 2025What's in a Latent? Leveraging Diffusion Latent Space for Domain Generalizationcited by 3
  35. 2025Manipulating 3D Molecules in a Fixed-Dimensional E(3)-Equivariant Latent Spacecited by 3
  36. 2025From Cradle to Cane: A Two-Pass Framework for High-Fidelity Lifespan Face Agingcited by 3
  37. 2025Concept-TRAK: Understanding how diffusion models learn concepts through concept-level attributioncited by 3
  38. 2025Asynchronous Denoising Diffusion Models for Aligning Text-to-Image Generationcited by 3

50 most cited of 121 All 121 (JSON)

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  2. 2026Intrinsic-Hybrid Latent Diffusion Models for Generative Modeling on Unknown Manifolds
  3. 2023Smooth Diffusion: Crafting Smooth Latent Spaces in Diffusion Models
  4. 2026Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold
  5. 2026Understanding Latent Diffusability via Fisher Geometry
  6. 2022Unifying Diffusion Models' Latent Space, with Applications to CycleDiffusion and Guidance
  7. 2024Do Diffusion Models Learn Semantically Meaningful and Efficient Representations?
  8. 2024Hyperbolic Geometric Latent Diffusion Model for Graph Generation