Markov spin models for image generation : explicit large deviations with respect to the number of pixels

TL;DR

This paper analyzes Markov spin models for image generation, deriving explicit large deviations for pixel configurations and exploring implications for reconstructive dynamics under various initial conditions.

Contribution

It introduces a framework to compute large deviations in Markov spin models for images, linking microscopic configurations to macroscopic overlap variables.

Findings

Explicit large deviations formulas for pixel overlap and magnetization.

Analysis of reconstructive dynamics for different initial image conditions.

Connections to the manifold hypothesis in generative diffusion models.

Abstract

For the discrete-time or the continuous-time Markov spin models for image generation when each pixel $n=1,..,N$ can take only two values $S_n=\pm 1$, the finite-time forward propagator depends on the initial and on the final configurations of the $N$ spins only via a single global variable, namely the extensive overlap that counts the number of spins that have the same value or not in the two configurations. The joint probability distribution of the overlap and of the magnetization during the forward noising dynamics can be written for any finite number $N$ of pixels and in the limit $N \to + \infty$ to extract the large deviations properties. The consequences for the backward reconstructive dynamics are then analyzed for various initial conditions, namely (i) a single image (ii) a mixture of two images (iii) when the initial condition corresponds to the Curie-Weiss mean-field ferromagnetic model in the microcanonical ensemble, as a simple analog of the manifold-hypothesis concerning continuous generative diffusion models.