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

Abstract For the discrete-time or the continuous-time Markov spin models for image generation when each pixel <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> can take only two values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> to extract the large deviation 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.