Convergence properties of Markov models for image generation with applications to spin-flip dynamics and to diffusion processes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909469073473536 |
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| author | Monthus, Cecile |
| author_facet | Monthus, Cecile |
| contents | In the field of Markov models for image generation, the main idea is to learn how non-trivial images are gradually destroyed by a trivial forward Markov dynamics over the large time window $[0,t]$ converging towards pure noise for $t \to + \infty$, and to implement the non-trivial backward time-dependent Markov dynamics over the same time window $[0,t]$ starting from pure noise at $t$ in order to generate new images at time $0$. The goal of the present paper is to analyze the convergence properties of this reconstructive backward dynamics as a function of the time $t$ using the spectral properties of the trivial continuous-time forward dynamics for the $N$ pixels $n=1,..,N$. The general framework is applied to two cases : (i) when each pixel $n$ has only two states $S_n=\pm 1$ with Markov jumps between them; (ii) when each pixel $n$ is characterized by a continuous variable $x_n$ that diffuses on an interval $]x_-,x_+[$ that can be either finite or infinite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence properties of Markov models for image generation with applications to spin-flip dynamics and to diffusion processes Monthus, Cecile Statistical Mechanics Probability In the field of Markov models for image generation, the main idea is to learn how non-trivial images are gradually destroyed by a trivial forward Markov dynamics over the large time window $[0,t]$ converging towards pure noise for $t \to + \infty$, and to implement the non-trivial backward time-dependent Markov dynamics over the same time window $[0,t]$ starting from pure noise at $t$ in order to generate new images at time $0$. The goal of the present paper is to analyze the convergence properties of this reconstructive backward dynamics as a function of the time $t$ using the spectral properties of the trivial continuous-time forward dynamics for the $N$ pixels $n=1,..,N$. The general framework is applied to two cases : (i) when each pixel $n$ has only two states $S_n=\pm 1$ with Markov jumps between them; (ii) when each pixel $n$ is characterized by a continuous variable $x_n$ that diffuses on an interval $]x_-,x_+[$ that can be either finite or infinite. |
| title | Convergence properties of Markov models for image generation with applications to spin-flip dynamics and to diffusion processes |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2410.10255 |