新理论让扩散模型在复杂数据结构上更高效，突破了传统方法的维度瓶颈。
Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures
- 提出基于离散混合的新建模方式，实现对任意低维支撑集分布的得分逼近。
- 神经网络复杂度仅随低维数d指数增长，避免了高维空间的计算灾难。
- 适用于具有奇异点、边界和离散簇的真实数据，解释了扩散模型在实际生成中的表现。
得分驱动的扩散模型取得了显著成功，但现有得分近似复杂度分析依赖于利普希茨连续密度或光滑流形支撑等严格假设，这些在真实感知数据中常因奇点、锐边界和分离聚类而被违反。本文建立了一个通用得分逼近定理，适用于任意紧致集上支持、上闵可夫斯基维数为d的分布。通过一种新颖的离散混合建模方法，证明了得分函数可由仅随d指数增长的ReLU网络逼近，从而打破环境维度的指数诅咒。结合现有关于任意紧致分布后向扩散SDE的精确求解理论，本工作表明扩散模型可自然适应不规则、非光滑的数据结构，解释了其在真实生成任务中的强大能力。
原文摘要 · Abstract (English)
The remarkable success of score-based diffusion models has spurred significant efforts to establish their theoretical foundations. However, existing complexity bounds for score approximation rely heavily on restrictive assumptions like Lipschitz continuous densities or smooth manifold supports, which are routinely violated by the singularities, sharp boundaries, and disjoint clusters inherent to real-world perceptual data. This work establishes a universal score approximation theorem that works for any distribution supported on any compact set of upper Minkowski dimension $d$. Using a novel discrete-mixture formulation, we prove that the score function can be approximated with a ReLU network whose complexity grows exponentially only with $d$, thus breaking the exponential curse of ambient dimensionality. Combined with existing theories on accurately solving the backward diffusion SDE for arbitrary compact distributions, our work shows that diffusion models readily adapt to irregular, non-smooth data structures, explaining their competence in real-world generative tasks.
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