Denoising Is Stein: Moment Access and a Resource Hierarchy for Learning Hidden Structure in Diffusion Models
Abstract
We study score learning from denoising pairs when the data contain a hidden non-Gaussian component. If covariance reveals the component, its subspace can be recovered directly. If covariance is isotropic, the first distinguishing moment has order k∗, and the learner must extract that moment from clean points and their noisy views. Correlations in the denoising loss are Stein discrepancies against a baseline; for polynomial fields they determine exactly the corresponding moment differences. Averaging fresh noisy views conditional on a clean point gives an exact moment query on the clean distribution. In the covariance-matched regime, orientation-independent dictionaries satisfying an L4/L2 condition need order dk∗−1 features, and the stated dot-product kernel ridge models need the same power in samples. Small-step online SGD on one unit has this exponent up to logarithms. A two-stage learner uses the curvature of a smoothed denoising loss to recover the hidden direction, then sharpens it and fits a scalar readout. For every even k∗, it obtains a fixed score gain over the Gaussian baseline from O(dk∗/2log d) clean points. In the covariance-visible regime, the frozen-feature threshold occurs at dJ/J!, while projection onto principal components makes the fitted-model, score-risk and generated-law distributions independent of ambient dimension. In the covariance-matched experiments, the measured partial-trace overlaps are consistent with the proposed spectral transition. A width-8 network trained with clean-point reuse shows a d2-like training-set threshold over d=64--$192$, with fixed validation and calibration overhead.
Related
- Connections between the Föllmer process and the denoising diffusion probabilistic model
- Exact Global MCMC with Denoising Diffusion
- Denoising as Projection: Constrained Optimization with Gradient-Guided Diffusion
- Spatio-temporal Latent Denoising Diffusion Probabilistic Models for Reduced-order Modeling of Parametrized Dynamical Systems
- Why we should condition denoising diffusion generative models on windows of past observations
BibTeX
@misc{wang2026denoising,
title = {{Denoising Is Stein: Moment Access and a Resource Hierarchy for Learning Hidden Structure in Diffusion Models}},
author = {Rui Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04043},
eprint = {2611.04043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04043},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Denoising Is Stein: Moment Access and a Resource Hierarchy for Learning Hidden Structure in Diffusion Models
AU - Rui Wang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.04043
UR - https://arxiv.org/abs/2611.04043
AB - We study score learning from denoising pairs when the data contain a hidden non-Gaussian component. If covariance reveals the component, its subspace can be recovered directly. If covariance is isotropic, the first distinguishing moment has order k^∗, and the learner must extract that moment from clean points and their noisy views. Correlations in the denoising loss are Stein discrepancies against a baseline; for polynomial fields they determine exactly the corresponding moment differences. Averaging fresh noisy views conditional on a clean point gives an exact moment query on the clean distribution. In the covariance-matched regime, orientation-independent dictionaries satisfying an L^4/L^2 condition need order d^k^∗−1 features, and the stated dot-product kernel ridge models need the same power in samples. Small-step online SGD on one unit has this exponent up to logarithms. A two-stage learner uses the curvature of a smoothed denoising loss to recover the hidden direction, then sharpens it and fits a scalar readout. For every even k^∗, it obtains a fixed score gain over the Gaussian baseline from O(d^k^∗/2log d) clean points. In the covariance-visible regime, the frozen-feature threshold occurs at d^J/J!, while projection onto principal components makes the fitted-model, score-risk and generated-law distributions independent of ambient dimension. In the covariance-matched experiments, the measured partial-trace overlaps are consistent with the proposed spectral transition. A width-8 network trained with clean-point reuse shows a d^2-like training-set threshold over d=64--$192$, with fixed validation and calibration overhead.
ER -