Home  | Publications | PHN+25

Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction

MCML Authors

Link to Profile Stefan Bauer PI Matchmaking

Stefan Bauer

Prof. Dr.

Principal Investigator

Algorithmic Machine Learning & Explainable AI

Abstract

Although diffusion models now occupy a central place in generative modeling, introductory treatments commonly assume Euclidean data and seldom clarify their connection to discrete-state analogues. This article is a self-contained primer on diffusion over general state spaces, unifying continuous domains and discrete/categorical structures under one lens. We develop the discrete-time view (forward noising via Markov kernels and learned reverse dynamics) alongside its continuous-time limits -- stochastic differential equations (SDEs) in ℝd and continuous-time Markov chains (CTMCs) on finite alphabets -- and derive the associated Fokker--Planck and master equations. A common variational treatment yields the ELBO that underpins standard training losses. We make explicit how forward corruption choices -- Gaussian processes in continuous spaces and structured categorical transition kernels (uniform, masking/absorbing and more) in discrete spaces -- shape reverse dynamics and the ELBO. The presentation is layered for three audiences: newcomers seeking a self-contained intuitive introduction; diffusion practitioners wanting a global theoretical synthesis; and continuous-diffusion experts looking for an analogy-first path into discrete diffusion. The result is a unified roadmap to modern diffusion methodology across continuous domains and discrete sequences, highlighting a compact set of reusable proofs, identities, and core theoretical principles.

misc PHN+25

Preprint

Dec. 2025

Authors

V. Pauline • T. Höppe • K. Neklyudov • A. Tong • S. Bauer • A. Dittadi

Links

arXiv

Research Area

 A1 | Statistical Foundations & Explainability

BibTeXKey: PHN+25

#p-bauer-s
Back to Top