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Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

Ann Dooms

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 $f_t(λ)$ and a global expansion threshold $λ^{**}$. 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.

Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

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 , a diagonal expansion function and a global expansion threshold . 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.
Paper Structure (65 sections, 29 theorems, 107 equations, 1 figure, 11 tables)

This paper contains 65 sections, 29 theorems, 107 equations, 1 figure, 11 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Score-Based Diffusion Models
  4. Partitioned Iterated Function Systems
  5. Contraction Structure of the Denoising Operator
  6. Contraction in the Euclidean Norm
  7. Contraction in the Block-Max Norm
  8. Learning the Geometry: Training as Collage Minimisation
  9. The Training Objective as Collage Error
  10. Collage-Contraction Bridges
  11. PIFS Collage Regulariser and Enforcement of (PC)
  12. Two-Regime Structure of the Denoising Chain
  13. Regime I: Global Context Assembly
  14. Deviation of the Trained Score from the Gaussian Score
  15. Regime II: Fine-Detail Synthesis
  16. ...and 50 more sections

Key Result

Theorem 2

If $s_\theta$ is differentiable and certain regularity conditions hold, then

Figures (1)

  • Figure 1: Paired-difference proxy $\mathbb{E}[\|\widehat{\Delta}_t\|_2]$ for DDPM CIFAR-10 ($N=200$ images); $\widehat{\Delta}_t \approx \Delta_t$ at high noise and overestimates $\Delta_t$ at lower noise (see text). Left: proxy vs. timestep $t$; the dashed line marks the high-noise boundary at $t=800$. Right: log-log plot with OLS fit over the high-noise region ($t\geq800$, dark markers); slope $0.95$ is consistent with $O(\sqrt{\bar{\alpha}_t})$ scaling of $\|\Delta_t\|_2$ (Theorem \ref{['thm:high_noise']}). Mid-noise points (light markers) lie above the extrapolated line.

Theorems & Definitions (37)

  • Definition 1
  • Theorem 2: Score matching
  • Proposition 3: Sign of $b_t$
  • Theorem 4: Banach Fixed Point Theorem banach1922
  • Theorem 5: Collage Theorem barnsley1988
  • Theorem 6: Euclidean Contraction
  • Definition 7: (EC): Euclidean Contraction
  • Corollary 8
  • Theorem 9: Block-Max Contraction
  • Definition 10: (PC): Block-Max Contraction
  • ...and 27 more