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{{Machine learning|Artificial neural network}}
In [[machine learning]], '''diffusion models''', also known as '''diffusion-based generative models''' or '''score-based generative models''', are a class of [[latent variable model|latent variable]] [[generative model|generative]] models. A diffusion model consists of two major components: the forward diffusion process, and the reverse sampling process. The goal of diffusion models is to learn a [[diffusion process]] for a given dataset, such that the process can generate new elements that are distributed similarly as the original dataset. A diffusion model models data as generated by a diffusion process, whereby a new datum performs a [[Wiener process|random walk with drift]] through the space of all possible data.<ref name="song"/> A trained diffusion model can be sampled in many ways, with different efficiency and quality. The sampling is generally achieved by numerically solving a deterministic or stoachstic differential equation.<ref name="song"/> In particular, the sampling trajectory appears strong regularity for deterministic sampling.<ref>{{Cite journal |last1=Chen |first1=Defang |last2=Zhou |first2=Zhenyu |last3=Wang |first3=Can |last4=Shen |first4=Chunhua |last5=Lyu |first5=Siwei |date=2024-05-18 |title=On the Trajectory Regularity of ODE-based Diffusion Sampling | url=https://openreview.net/pdf?id=H86WzfH5N1 |journal=Proceedings of the 41st International Conference on Machine Learning |language=en |publisher=PMLR | volume=235 |pages=7905-7934|arxiv=2405.11326}}</ref>
There are various equivalent formalisms, including [[Markov chain]]s, denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations.<ref>{{cite journal |last1=Croitoru |first1=Florinel-Alin |last2=Hondru |first2=Vlad |last3=Ionescu |first3=Radu Tudor |last4=Shah |first4=Mubarak |date=2023 |title=Diffusion Models in Vision: A Survey |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=45 |issue=9 |pages=10850–10869 |arxiv=2209.04747 |doi=10.1109/TPAMI.2023.3261988 |pmid=37030794 |s2cid=252199918}}</ref> They are typically trained using [[Variational Bayesian methods|variational inference]].<ref name="ho" /> The model responsible for denoising is typically called its "[[#Choice of architecture|backbone]]". The backbone may be of any kind, but they are typically [[U-Net|U-nets]] or [[Transformer (deep learning architecture)|transformers]].
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<math display="block">\min_{\theta} \int_0^1 \mathbb{E}_{\pi_0, \pi_1, p_t}\left [\lVert{(x_1-x_0) - v_t(x_t)}\rVert^2\right] \,\mathrm{d}t.</math>
The data pair <math>(x_0, x_1)</math> can be any coupling of <math>\pi_0</math> and <math>\pi_1</math>, typically independent (i.e., <math>(x_0,x_1) \sim \pi_0 \times \pi_1</math>) obtained by randomly combining observations from <math>\pi_0</math> and <math>\pi_1</math>. This process ensures that the trajectories closely mirror the density map of <math>x_t</math> trajectories but ''reroute'' at intersections to ensure causality. This rectifying process is also known as Flow Matching,<ref>{{cite arXiv |last1=Lipman |first1=Yaron |title=Flow Matching for Generative Modeling |date=2023-02-08 |eprint=2210.02747 |last2=Chen |first2=Ricky T. Q. |last3=Ben-Hamu |first3=Heli |last4=Nickel |first4=Maximilian |last5=Le |first5=Matt|class=cs.LG }}</ref> Stochastic Interpolation,<ref>{{cite arXiv |last1=Albergo |first1=Michael S. |title=Building Normalizing Flows with Stochastic Interpolants |date=2023-03-09 |eprint=2209.15571 |last2=Vanden-Eijnden |first2=Eric|class=cs.LG }}</ref> Inversion by Direct Iteration (InDI),<ref>{{Citation |last=Delbracio |first=Mauricio |title=Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration |date=2024-02-02 |url=http://arxiv.org/abs/2303.11435 |access-date=2025-05-30 |publisher=arXiv |doi=10.48550/arXiv.2303.11435 |id=arXiv:2303.11435 |last2=Milanfar |first2=Peyman}}</ref> and alpha-(de)blending.<ref>{{cite arXiv |last1=Heitz |first1=Eric |title=Iterative α-(de)Blending: a Minimalist Deterministic Diffusion Model |date=2023-05-05 |eprint=2305.03486 |last2=Belcour |first2=Laurent |last3=Chambon | first3=Thomas |class=cs.GR}}</ref>
[[File:Reflow Illustration.png|thumb|390px|The reflow process<ref name=":0"/>]]
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