扩散模型生成新数据，源于对训练数据的平滑插值。
On the Interpolation Effect of Score Smoothing in Diffusion Models
- 通过分析得分函数平滑与去噪动态的关系，揭示生成机制。
- 在单维子空间上，平滑得分可使生成样本沿子空间插值训练数据。
- 无需显式正则化，神经网络自然实现此类插值，适用于简单非线性流形。
扩散模型在多个领域取得显著进展，具备生成训练集中不存在的新数据的能力。本文研究这一创造力源于神经网络学习经验得分函数的平滑版本，该平滑引导去噪过程生成插值于训练数据的样本。聚焦于训练数据均匀分布在一维子空间的情形，通过解析解与数值实验阐明得分平滑与去噪动态之间的相互作用，证明得分平滑可使去噪后的样本沿子空间插值训练集。此外，我们提供理论与实证证据表明，无论是否采用显式正则化，神经网络学习得分函数时均能自然实现类似效果，即使数据属于简单的非线性流形。
原文摘要 · Abstract (English)
Diffusion models have achieved remarkable progress in various domains with an intriguing ability to produce new data that do not exist in the training set. In this work, we study the hypothesis that such creativity arises from the neural network backbone learning a smoothed version of the empirical score function, which guides the denoising dynamics to generate data points that interpolate the training data. Focusing mainly on settings where the training set lies uniformly in a one-dimensional subspace, we elucidate the interplay between score smoothing and the denoising dynamics with analytical solutions and numerical experiments, demonstrating how smoothing the score function can cause the denoised data samples to interpolate the training set along the subspace. Moreover, we present theoretical and empirical evidence that learning score functions with neural networks - either with or without explicit regularization - can naturally achieve a similar effect, including when the data belong to simple nonlinear manifolds.
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