mlsanityData to 14 Oct 2026

On the Mathematics of Diffusion Models

arXiv 2301.11108 ↗First posted 25 Jan 2023cs.LG cs.AI math.PRcited by 10

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

This paper gives direct derivations of the differential equations and likelihood formulas of diffusion models assuming only knowledge of Gaussian distributions. A VAE analysis derives both forward and backward stochastic differential equations (SDEs) as well as non-variational integral expressions for likelihood formulas. A score-matching analysis derives the reverse diffusion ordinary differential equation (ODE) and a family of reverse-diffusion SDEs parameterized by noise level. The paper presents the mathematics directly with attributions saved for a final section.

Key facts. Cited by 10 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 5 of its 12 references that are in the map. First posted on arXiv 25 Jan 2023.

Builds on

5 of 12 references.

Closest ancestors · 4

Same direction, newest first

  1. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  2. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  4. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 1

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058

Built on by

Cited by year

Citing papers per year: 2023: 4; 2024: 5; 2025: 1.

10 papers.

  1. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 60
  2. 2023Interpretable Diffusion via Information Decompositioncited by 37
  3. 2023Memory in Plain Sight: Surveying the Uncanny Resemblances of Associative Memories and Diffusion Modelscited by 13
  4. 2024Tackling the Singularities at the Endpoints of Time Intervals in Diffusion Modelscited by 13
  5. 2025Anomaly Detection and Generation with Diffusion Models: A Surveycited by 8
  6. 2024Neural Entropycited by 7
  7. 2023Diffusion Posterior Illumination for Ambiguity-aware Inverse Renderingcited by 6
  8. 2024Your Diffusion Model is Secretly a Noise Classifier and Benefits from Contrastive Trainingcited by 5
  9. 2024Comparison of Generative Learning Methods for Turbulence Surrogatescited by 1
  10. 2024Combinatorial Complex Score-based Diffusion Modelling through Stochastic Differential Equationscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  2. 2025Diffusion Models under Alternative Noise: Simplified Analysis and Sensitivity
  3. 2022A Flexible Diffusion Model
  4. 2024Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors
  5. 2026Rethinking the Diffusion Model from a Langevin Perspective
  6. 2025Generative diffusion models from a PDE perspective
  7. 2023A Geometric Perspective on Diffusion Models
  8. 2023Gaussian Mixture Solvers for Diffusion Models