An explicit formulation of the learned noise predictor εθ(bf xt, t) via the forward-process noise εt in denoising diffusion probabilistic models (DDPMs)
On the map: Diffusion models → Diffusion foundations and sampling
Abstract (arXiv)
In denoising diffusion probabilistic models (DDPMs), the learned noise predictor εθ ( bf xt , t) is trained to approximate the forward-process noise εt. The equality nabla_bf xt log q(bf xt) = -frac 1 √ 1- bar αt εθ ( bf xt , 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 εθ ( bf xt , t) in terms of the forward-process noise εt is derived. This result show how the forward-process noise εt contributes to the learned predictor εθ ( bf xt , 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.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 3 of its 6 references that are in the map. First posted on arXiv 6 Jul 2025.
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- 2021Diffusion Models Beat GANs on Image Synthesis11 May
- 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
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