Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

2026/03/13 by Ann Dooms · 1 voice
Computer Science · Mathematics · Neuroscience · #Attractor #Context (archaeology) #Diagonal #Dimension (graph theory) #Fractal #Generative Adversarial Networks and Image Synthesis #Iterated function #Iterated function system #Medical Image Segmentation Techniques #Neural dynamics and brain function #Noise (video) #Noise reduction #cs.CV #cs.IT #cs.LG #math.DS

paper · pdf · open access · doi:10.48550/arxiv.2603.13069

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2026/03/13 · arxiv published 2026/03/13 · arxiv updated 2026/03/13 · openalex created_date 2026/03/17 · openalex updated_date 2026/07/28

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.

Citations

• Align Your Steps: Optimizing Sampling Schedules in Diffusion Models
• Spontaneous Symmetry Breaking in Generative Diffusion Models
• Efficient Diffusion Training via Min-SNR Weighting Strategy
• Simple diffusion: End-to-end diffusion for high resolution images
• Scalable Diffusion Models with Transformers
• Flow Matching for Generative Modeling
• Building Normalizing Flows with Stochastic Interpolants
• Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
• Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow
• Variational Diffusion Models
• Improved Denoising Diffusion Probabilistic Models
• Score-Based Generative Modeling through Stochastic Differential Equations
• Denoising Diffusion Implicit Models
• Denoising Diffusion Probabilistic Models
• A Connection Between Score Matching and Denoising Autoencoders

Discussions

• Diffusion Models as Fractal Geometry: PIFS Unifies Architecture - By arXiv (2026-03-16) https://arxiv.org/abs/2603.13069 Artificial Intelligence, Machine Learning, Computational Geometry, Mathematical [bsky, 0 points, 0 comments]

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