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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2512.05092 |
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| _version_ | 1866911302275825664 |
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| author | Pauline, Vincent Höppe, Tobias Neklyudov, Kirill Tong, Alexander Bauer, Stefan Dittadi, Andrea |
| author_facet | Pauline, Vincent Höppe, Tobias Neklyudov, Kirill Tong, Alexander Bauer, Stefan Dittadi, Andrea |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction Pauline, Vincent Höppe, Tobias Neklyudov, Kirill Tong, Alexander Bauer, Stefan Dittadi, Andrea Machine Learning 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. |
| title | Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2512.05092 |