mlsanityData to 7 Oct 2026

A Mathematical Introduction to Diffusion Models

arXiv 2607.01693 ↗First posted 2 Jul 2026cs.LG math.PRcited by 1

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

These notes give a proof-oriented introduction to diffusion models from the viewpoint of sampling, tracing a single arc from classical sampling dynamics to modern diffusion samplers, their error analysis, and inference-time control. Throughout, the material is layered into core definitions and identities proved in full, representative estimates proved under simplifying assumptions, and research-level theorems stated with a proof roadmap. The intended audience is beginning graduate students with a background in probability but no prior exposure to stochastic differential equations, stochastic numerics, or diffusion models.

Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 32 of its 52 references that are in the map. First posted on arXiv 2 Jul 2026.

Builds on

32 of 52 references.

Closest ancestors · 14

Same direction, newest first

  1. 2026High-accuracy sampling for diffusion models and log-concave distributions1 Feb
  2. 2025Parallel Sampling via Autospeculation11 Nov
  3. 2025Discrete Diffusion Models: Novel Analysis and New Sampler Guarantees20 Sep
  4. 2024Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformization12 Feb
  5. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptions23 Aug
  6. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  7. 2023Consistency Models2 Mar
  8. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  9. 2022Classifier-Free Diffusion Guidance26 Jul
  10. 2022A Continuous Time Framework for Discrete Denoising Models30 May
  11. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  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 · 18

  1. 2022Flow Matching for Generative Modelingcited by 5,426
  2. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  3. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  4. 2023Discrete Diffusion Modeling by Estimating the Ratios of the Data Distributioncited by 712
  5. 2025Mean Flows for One-step Generative Modelingcited by 524
  6. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 224
  7. 2023Practical and Asymptotically Exact Conditional Sampling in Diffusion Modelscited by 218
  8. 2021Autoregressive Diffusion Modelscited by 208
  9. 2024Unlocking Guidance for Discrete State-Space Diffusion and Flow Modelscited by 124
  10. 2013A Deep and Tractable Density Estimatorcited by 67
  11. 2021Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Samplingcited by 44
  12. 2023Faster high-accuracy log-concave sampling via algorithmic warm startscited by 44
6 more
  1. 2023Sampling, Diffusions, and Stochastic Localizationcited by 38
  2. 2022Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chainscited by 16
  3. 2016Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixingcited by 13
  4. 2025Error Bounds and Optimal Schedules for Masked Diffusions with Factorized Approximationscited by 13
  5. 2023A Simple Proof of the Mixing of Metropolis-Adjusted Langevin Algorithm under Smoothness and Isoperimetrycited by 7
  6. 2024Parallel Sampling via Countingcited by 7

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Parallelism, critical windows, and separations among diffusion language modelscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Rethinking the Diffusion Model from a Langevin Perspective
  2. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  3. 2025On the Collapse Errors Induced by the Deterministic Sampler for Diffusion Models
  4. 2023A Geometric Perspective on Diffusion Models
  5. 2023On the Design Fundamentals of Diffusion Models: A Survey
  6. 2022An optimal control perspective on diffusion-based generative modeling
  7. 2025Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
  8. 2023On the Equivalence of Consistency-Type Models: Consistency Models, Consistent Diffusion Models, and Fokker-Planck Regularization