We consider the problem of estimating the noise level σ² in a Gaussian linear model Y = Xβ+σξ, where ξ ∈ ℝⁿ is a standard discrete white Gaussian noise and β ∈ ℝ^p an unknown nuisance vector. It is assumed that X is a known ill-conditioned n × p matrix with n ≥ p and with large dimension p. In this situation the vector β 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 − Xβ||²/n.

中文翻译：

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高维线性模型中的噪声水平估计

我们认为估计噪声电平的问题σ ²中的高斯线性模型Ŷ = Xβ + σξ，其中ξ＆Element;ℝ ^Ñ是一个标准的离散的白高斯噪声和β＆Element;ℝ ^p未知滋扰矢量。据推测，X是一个已知的病态Ñ × p与矩阵Ñ ≥ p，并用大尺寸p。在这种情况下，向量β借助于最大似然估计的频谱正则化来估计噪声估计，并且借助于二次预测误差的自适应（即，数据驱动）归一化来计算噪声电平估计。对于此估计，我们围绕伪估计||计算其集中度。Ÿ - Xβ || ² / n。