Noise level estimation in high-dimensional linear models

We consider the problem of estimating the noise level $\sigma^2$ in a Gaussian linear model $Y=X\beta+\sigma \xi$, where $\xi\in\mathbb{R}^n$ is a standard discrete white Gaussian noise and $\beta\in\mathbb{R}^p$ an unknown nuisance vector. It is assumed that $X$ is a known ill-conditioned $n\times p$ matrix with $n\ge p$ and with large dimension $p$. In this situation the vector $\beta$ is estimated with the help of spectral regularization of the maximum likelihood estimate, and the noise level estimate is computed with the help of adaptive (i.e., data-driven) normalization of the quadratic prediction error. For this estimate, we compute its concentration rate around the pseudo-estimate $\|Y-X\beta\|^2/n$.