Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

TL;DR

This paper reveals that diffusion models function as Partitioned Iterated Function Systems (PIFS), providing a geometric framework that explains their behavior and guides optimal design choices without requiring model evaluation.

Contribution

It introduces PIFS as a unifying geometric framework for diffusion models, deriving key quantities and explaining their two-regime behavior, leading to practical design criteria.

Findings

Diffusion models operate as PIFS with quantifiable geometric properties.

Self-attention is identified as the natural primitive for PIFS contraction.

Optimal design criteria are derived from fractal geometry analysis.

Abstract

What is a diffusion model actually doing when it turns noise into a photograph? We show that the deterministic DDIM reverse chain operates as a Partitioned Iterated Function System (PIFS) and that this framework serves as a unified design language for denoising diffusion model schedules, architectures, and training objectives. From the PIFS structure we derive three computable geometric quantities: a per-step contraction threshold $L^*_t$, a diagonal expansion function $f_t(\lambda)$ and a global expansion threshold $\lambda^{**}$. These quantities require no model evaluation and fully characterize the denoising dynamics. They structurally explain the two-regime behavior of diffusion models: global context assembly at high noise via diffuse cross-patch attention and fine-detail synthesis at low noise via patch-by-patch suppression release in strict variance order. Self-attention emerges as the natural primitive for PIFS contraction. The Kaplan-Yorke dimension of the PIFS attractor is determined analytically through a discrete Moran equation on the Lyapunov spectrum. Through the study of the fractal geometry of the PIFS, we derive three optimal design criteria and show that four prominent empirical design choices (the cosine schedule offset, resolution-dependent logSNR shift, Min-SNR loss weighting, and Align Your Steps sampling) each arise as approximate solutions to our explicit geometric optimization problems tuning theory into practice.