An explicit formulation of the learned noise predictor ε_θ(f̱ x_t, t) via the forward-process noise ${\epsilon }_t$ in denoising diffusion probabilistic models (DDPMs)

Abstract

In denoising diffusion probabilistic models (DDPMs), the learned noise predictor ${ϵ}_{\theta }({\mathbf{x}}_t,t)$ is trained to approximate the forward-process noise ${ϵ}_t$. The equality ${\nabla }_{{\mathbf{x}}_t}\log q({\mathbf{x}}_t)=-\frac{1}{\sqrt{1-{\bar{\alpha }}_t}}{ϵ}_{\theta }({\mathbf{x}}_t,t)$ 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 ${ϵ}_{\theta }({\mathbf{x}}_t,t)$ 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 ${ϵ}_{\theta }({\mathbf{x}}_t,t)$. 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.