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

TL;DR

This paper analyzes the convergence of backward Markov dynamics used in image generation, focusing on spectral properties of forward dynamics for binary and continuous pixel models, with applications to spin-flip and diffusion processes.

Contribution

It provides a spectral analysis framework for understanding the convergence of reconstructive backward Markov processes in image generation.

Findings

Convergence depends on spectral properties of forward dynamics.

Analysis applies to binary spin models and continuous diffusion models.

Results inform the design of image generation algorithms.

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.