浅层扩散模型能自动捕捉低维结构，突破高维分布学习的维度诅咒。
Shallow diffusion networks provably learn hidden low-dimensional structure
- 基于Barron空间分析，浅层网络可自适应低维结构
- 证明扩散模型在低维结构下样本复杂度不随维度增长
- 无需特殊架构，适用于多种隐含结构分布
基于扩散的生成模型在图像、视频等高维信号上表现卓越，但经典理论指出分布恢复面临维度诅咒。本文通过分析单层神经网络构成的Barron空间上的扩散模型，证明此类浅层模型能有效适应低维结构，避免维度诅咒。结合最新采样分析，给出了从结构化分布中学习采样的端到端样本复杂度界。关键在于利用Barron空间的低指标结构，无需为特定潜变量结构设计专用架构，具有广泛适用性。
原文摘要 · Abstract (English)
Diffusion-based generative models provide a powerful framework for learning to sample from a complex target distribution. The remarkable empirical success of these models applied to high-dimensional signals, including images and video, stands in stark contrast to classical results highlighting the curse of dimensionality for distribution recovery. In this work, we take a step towards understanding this gap through a careful analysis of learning diffusion models over the Barron space of single layer neural networks. In particular, we show that these shallow models provably adapt to simple forms of low dimensional structure, thereby avoiding the curse of dimensionality. We combine our results with recent analyses of sampling with diffusion models to provide an end-to-end sample complexity bound for learning to sample from structured distributions. Importantly, our results do not require specialized architectures tailored to particular latent structures, and instead rely on the low-index structure of the Barron space to adapt to the underlying distribution.
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