{"ID":2897784,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.04203","arxiv_id":"2507.04203","title":"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)","abstract":"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.","short_abstract":"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 an...","url_abs":"https://arxiv.org/abs/2507.04203","url_pdf":"https://arxiv.org/pdf/2507.04203v1","authors":"[\"KiHyun Yun\"]","published":"2025-07-06T01:16:16Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.AP\"]","methods":"[\"Diffusion Model\"]","has_code":false}