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, up to logarithmic factors, $$\begin{gathered} Thet{a}_{ \\ Lambda} \\ left( \\ min \\ left \\ Delt{a}_{ \\ Lambda}^2,d^{-1}+k^2{d}^{-2} \\ right \\ right) \end{gathered}$$, and the constrained MLE attains this rate. For $alpha$-strongly log-concave signals, the rate is $$\begin{gathered} Thet{a}_{ \\ Lambda} \\ left( \\ min \\ left \\ Delt{a}_{ \\ Lambda}^2,(1+ \\ alph{a}^{-1}{)}^2{d}^{-1} \\ right \\ right) \end{gathered}$$, 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.