揭示扩散模型中噪声预测器与前向噪声的显式关系
An explicit formulation of the learned noise predictor $ε_θ({\bf x}_t, t)$ via the forward-process noise $ε_{t}$ in denoising diffusion probabilistic models (DDPMs)
- 从前向过程噪声推导出噪声预测器的显式表达式
- 严格证明了扩散模型核心公式的数学来源
- 为理解扩散模型结构提供新理论视角
在去噪扩散概率模型（DDPMs）中，学习到的噪声预测器 $ ε_θ ( {f x}_t , t)$ 被训练以逼近前向过程噪声 $ε_t$。关键等式 $\nabla_{{\bf x}_t} \log q({\bf x}_t) = -\frac 1 {\sqrt {1- {\bar α}_t} } ε_θ ( {f x}_t , t)$ 在理论分析与算法设计中具有基础性作用，广泛应用于各类基于扩散的生成模型。本文推导出 $ ε_θ ( {f x}_t , t)$ 关于前向过程噪声 $ε_t$ 的显式表达式，揭示了 $ε_t$ 如何贡献于学习到的预测器。进一步地，基于该表达式，给出了上述核心等式的全新、严谨的数学证明，澄清了其起源，并为扩散模型的结构提供了新的理论洞见。
原文摘要 · Abstract (English)
In denoising diffusion probabilistic models (DDPMs), the learned noise predictor $ ε_θ ( {\bf x}_t , t)$ is trained to approximate the forward-process noise $ε_t$. The equality $\nabla_{{\bf x}_t} \log q({\bf x}_t) = -\frac 1 {\sqrt {1- {\bar α}_t} } ε_θ ( {\bf x}_t , t)$ plays a fundamental role in both theoretical analyses and algorithmic design, and thus is frequently employed across diffusion-based generative models. In this paper, an explicit formulation of $ ε_θ ( {\bf x}_t , t)$ in terms of the forward-process noise $ε_t$ is derived. This result show how the forward-process noise $ε_t$ contributes to the learned predictor $ ε_θ ( {\bf x}_t , t)$. Furthermore, based on this formulation, we present a novel and mathematically rigorous proof of the fundamental equality above, clarifying its origin and providing new theoretical insight into the structure of diffusion models.
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