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

• URL: http://arxiv.org/abs/2512.05092v1
• Date: Thu, 04 Dec 2025 18:55:36 GMT
• Title: Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
• Authors: Vincent Pauline, Tobias Höppe, Kirill Neklyudov, Alexander Tong, Stefan Bauer, Andrea Dittadi,
• Abstract summary: This article is a self-contained primer on diffusion over general state spaces.<n>We develop the discrete-time view (forward noising via Markov kernels and learned reverse dynamics) alongside its continuous-time limits.<n>A common variational treatment yields the ELBO that underpins standard training losses.
• Score: 54.95522167029998
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\mathbb{R}^d$ 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.

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