Convergence properties of Markov models for image generation with applications to spin-flip dynamics and to diffusion processes

TL;DR

The paper develops a unified spectral framework to analyze convergence in generative Markov models for images, focusing on how forward noising drives data toward noise and how a time-reversed, reconstructive backward dynamics can recover initial structure. By decomposing the forward propagator into eigenmodes of the generator, it derives explicit expressions for the global propagator, the Kemeny convergence time, and the backward reconstruction probabilities, with the leading relaxation set by the first excited energy $e_1$ and the backward convergence governed by $e^{-2e_1 t}$. It applies the formalism to two main classes: binary spin-flip models (Ising-like) and diffusion-based Pearson processes, including detailed toy models (Ising chain) and diffusion examples (Ornstein–Uhlenbeck, Gamma, Beta) to illustrate how magnetizations, correlations, and moments evolve under forward and backward dynamics. The work clarifies how initial data memorization and correlation structures influence reconstruction quality and time scales, offering analytic insight into the behavior of forward-backward generative schemes and informing design choices for diffusion and spin-based generative models.

Abstract

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.