A statistical theory for blind denoising: minimax estimation of the noise level from a single sample

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.