# An explicit formulation of the learned noise predictor εθ(bf xt, t) via the forward-process noise εt in denoising diffusion probabilistic models (DDPMs)

arXiv: https://arxiv.org/abs/2507.04203 · HTML: https://mlsanity.com/paper/2507.04203/

## 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.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 3 of 3)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)

## 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.