用分形几何解释扩散模型去噪过程，揭示其内在设计原理。
Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems
- 将去噪过程建模为分段迭代函数系统（PIFS），揭示其几何本质。
- 提出三个可计算的几何量，精确刻画去噪动态行为。
- 解释主流设计选择为何有效，为模型优化提供理论依据。
扩散模型在从噪声生成图像时实际在做什么？我们发现确定性DDIM反向过程本质上是分段迭代函数系统（PIFS），该框架可统一描述去噪扩散模型的调度、架构与训练目标。基于PIFS结构，我们推导出三个可计算的几何量：每步收缩阈值 $L^*_t$、对角展开函数 $f_t(λ)$ 和全局展开阈值 $λ^{**}$。这些量无需模型评估即可完全表征去噪动态。它们结构化解释了扩散模型的双阶段行为：高噪声下通过广域跨块注意力进行全局上下文组装，低噪声下通过逐块抑制释放按严格方差顺序合成细节。自注意力自然成为PIFS收缩的原始操作。通过离散莫兰方程分析李雅普诺夫谱，可解析求解PIFS吸引子的卡普兰-约克维数。通过对PIFS分形几何的研究，我们提出三项最优设计准则，并证明四种主流经验设计（余弦调度偏移、分辨率相关logSNR偏移、最小信噪比损失加权、对齐你的步骤采样）均为显式几何优化问题的近似解，实现理论到实践的转化。
原文摘要 · Abstract (English)
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.
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