mlsanityData to 14 Oct 2026

Connections between the F"ollmer process and the denoising diffusion probabilistic model

arXiv 2605.18040 ↗First posted 18 May 2026stat.ML cs.LG math.PRcited by 0

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

The F"ollmer process is a Brownian motion conditioned to have a pre-specified distribution at time 1. This process can be interpreted as an ``augmented'' time-compressed version of the reverse stochastic differential equation (SDE) corresponding to the denoising diffusion probabilistic model (DDPM). While this fact has been indirectly used to analyze DDPM sampling errors via discretization of the reverse SDE, the connection between direct discretization of the F"ollmer process and the DDPM sampler has not yet been fully explored. This paper clarifies this point while surveying relevant results from the literature. We show that discretized F"ollmer processes give natural hyper-parameter settings of the DDPM sampler while accommodating a broader class of variance schedules than discretized reverse SDEs. Moreover, this allows us to systematically recover state-of-the-art results on DDPM sampling error bounds, along with slight improvements.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 28 of its 63 references that are in the map. First posted on arXiv 18 May 2026.

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28 of 63 references.

Closest ancestors · 14

Same direction, newest first

  1. 2025Optimal Convergence Analysis of DDPM for General Distributions31 Oct
  2. 2025A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptions22 Aug
  3. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality24 Oct
  4. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions27 Sep
  5. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models23 May
  6. 2024Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory18 Mar
  7. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptions23 Aug
  8. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  9. 2023Diffusion Models are Minimax Optimal Distribution Estimators3 Mar
  10. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  11. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  12. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
2 more
  1. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  2. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 14

  1. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  2. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 219
  3. 2021Path Integral Sampler: a stochastic control approach for samplingcited by 181
  4. 2021Deep Generative Learning via Schr"odinger Bridgecited by 104
  5. 2019Theoretical guarantees for sampling and inference in generative models with latent diffusionscited by 97
  6. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  7. 2024Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processescited by 66
  8. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  9. 2023Sampling, Diffusions, and Stochastic Localizationcited by 40
  10. 2021Global Optimization via Schr"odinger-F"ollmer Diffusioncited by 4
  11. 2022Unbiased Estimation using a Class of Diffusion Processescited by 4
  12. 2025Perspectives on Stochastic Localizationcited by 2
2 more
  1. 2025Path Integral Optimiser: Global Optimisation via Neural Schr"odinger-F"ollmer Diffusioncited by 1
  2. 2026Variational Optimality of F"ollmer Processes in Generative Diffusionscited by 1

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  1. 2024Structural and Convergence Analysis of Discrete-Time Denoising Diffusion Probabilistic Models
  2. 2023Fast Diffusion Probabilistic Model Sampling through the lens of Backward Error Analysis
  3. 2025Sublinear iterations can suffice even for DDPMs
  4. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
  5. 2024Denoising Diffusion Probabilistic Models in Six Simple Steps
  6. 2024Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat
  7. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers
  8. 2023To smooth a cloud or to pin it down: Guarantees and Insights on Score Matching in Denoising Diffusion Models