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 $\widetildeΘ_Λ(\min\{Δ_Λ^2,d^{-1}+k^2d^{-2}\})$, 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.