PreprintCC BY 4.0arXiv
A statistical theory for blind denoising: minimax estimation of the noise level from a single sample
Abstract
Motivated by blind denoising in diffusion models, we study estimation of an unknown Gaussian noise level from a single high-dimensional observation, assuming the signal law P is known. We characterize the minimax mean-squared error under two structural assumptions on P. For signals with covering complexity k, the minimax rate is Θ̃Λ(min{ΔΛ2,d−1+k2d−2}), and the constrained MLE attains it up to logarithmic factors. For α-strongly log-concave signals, the rate is ΘΛ(min{ΔΛ2,(1+α−1)2d−1}), attained up to constants by a P-centered norm estimator. These results show that the structure of the signal law determines both the difficulty of blind noise estimation and the appropriate estimator.
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BibTeX
@misc{shen2026statistical,
title = {{A statistical theory for blind denoising: minimax estimation of the noise level from a single sample}},
author = {Zhuoer Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33008},
eprint = {2609.33008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33008},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A statistical theory for blind denoising: minimax estimation of the noise level from a single sample
AU - Zhuoer Shen
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.33008
UR - https://arxiv.org/abs/2609.33008
AB - Motivated by blind denoising in diffusion models, we study estimation of an unknown Gaussian noise level from a single high-dimensional observation, assuming the signal law P is known. We characterize the minimax mean-squared error under two structural assumptions on P. For signals with covering complexity k, the minimax rate is Θ̃_Λ(min{Δ_Λ^2,d^−1+k^2d^−2}), and the constrained MLE attains it up to logarithmic factors. For α-strongly log-concave signals, the rate is Θ_Λ(min{Δ_Λ^2,(1+α^−1)^2d^−1}), attained up to constants by a P-centered norm estimator. These results show that the structure of the signal law determines both the difficulty of blind noise estimation and the appropriate estimator.
ER -