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)

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.