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

TL;DR

The paper analyzes discrete-time Markov spin models for image generation, formulating forward noising and backward reconstructive dynamics with binary pixels and an extensive overlap Q as the key sufficient statistic. It provides exact finite-N expressions and large-N large-deviation results for the joint distribution of the overlap Q and magnetization M, along with generating functions and Legendre relations, and connects backward generation to Doob conditioning. Three initial-condition scenarios are explored: a single target image, a two-image mixture, and Curie-Weiss-based initial ensembles, revealing how backward dynamics reweights trajectories and produces generated images with analytically controlled probabilities and asymptotic fluctuations. The results yield explicit rate functions and scaled cumulant generating functions, clarifying the interplay between forward degradation of information and backward reconstruction under manifold-hypothesis-inspired initial data. The work offers a rigorous statistical-physics foundation for understanding generative Markov models on discrete data and paves the way for extensions to multi-color images and data-driven manifold learning perspectives.

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.