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Computer Science > Machine Learning

arXiv:2510.21890 (cs)
[Submitted on 24 Oct 2025]

Title:The Principles of Diffusion Models

Authors:Chieh-Hsin Lai, Yang Song, Dongjun Kim, Yuki Mitsufuji,
Stefano Ermon
View a PDF of the paper titled The Principles of Diffusion Models, by
Chieh-Hsin Lai and 4 other authors
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    Abstract:This monograph presents the core principles that have
    guided the development of diffusion models, tracing their origins
    and showing how diverse formulations arise from shared
    mathematical ideas. Diffusion modeling starts by defining a
    forward process that gradually corrupts data into noise, linking
    the data distribution to a simple prior through a continuum of
    intermediate distributions. The goal is to learn a reverse
    process that transforms noise back into data while recovering the
    same intermediates. We describe three complementary views. The
    variational view, inspired by variational autoencoders, sees
    diffusion as learning to remove noise step by step. The
    score-based view, rooted in energy-based modeling, learns the
    gradient of the evolving data distribution, indicating how to
    nudge samples toward more likely regions. The flow-based view,
    related to normalizing flows, treats generation as following a
    smooth path that moves samples from noise to data under a learned
    velocity field. These perspectives share a common backbone: a
    time-dependent velocity field whose flow transports a simple
    prior to the data. Sampling then amounts to solving a
    differential equation that evolves noise into data along a
    continuous trajectory. On this foundation, the monograph
    discusses guidance for controllable generation, efficient
    numerical solvers, and diffusion-motivated flow-map models that
    learn direct mappings between arbitrary times. It provides a
    conceptual and mathematically grounded understanding of diffusion
    models for readers with basic deep-learning knowledge.

Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI);
          Graphics (cs.GR)
Cite as:  arXiv:2510.21890 [cs.LG]
          (or arXiv:2510.21890v1 [cs.LG] for this version)
          https://doi.org/10.48550/arXiv.2510.21890
          Focus to learn more
          arXiv-issued DOI via DataCite

Submission history

From: Chieh-Hsin Lai [view email]
[v1] Fri, 24 Oct 2025 02:29:02 UTC (11,870 KB)
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