Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

2026-03-13

AI Summary

This work uncovers the intrinsic geometric mechanism underlying denoising diffusion models in image generation and establishes, for the first time, a theoretical connection between these models and partitioned iterated function systems (PIFS). By adopting a fractal geometry perspective, the deterministic reverse process of DDIM is formulated as a PIFS, enabling the derivation of three geometric quantities—based on the Lyapunov spectrum, Kaplan–Yorke dimension, and Moran equation—that can be computed without model evaluation. Corresponding optimal design principles are then proposed. This framework unifies and theoretically explains several empirical practices, including cosine schedule shift, resolution-dependent logSNR translation, Min-SNR loss weighting, and Align Your Steps sampling, interpreting them as approximate solutions to an underlying geometric optimization problem. The results thereby reveal the geometric foundations of prevailing methodologies and demonstrate the theory’s practical guidance for model design.

Technology Category

Computer Vision: Diffusion Models for VisionMachine Learning: Learning with ManifoldsSearch and Optimization: Sampling/Simulation-based Search

Application Category

Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionUser Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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.

Problem

• diffusion models
• iterated function systems
• fractal geometry
• denoising dynamics
• generative modeling

Innovation

• Partitioned Iterated Function Systems
• Denoising Diffusion Models
• Fractal Geometry
• Kaplan-Yorke Dimension
• Geometric Optimization