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ScreeNOT：相关噪声下精确 MSE 最优的奇异值阈值化AI 解读

We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattell’s ever-popular but vague scree plot heuristic from 1966. ScreeNOT has a surprising…

尖峰协方差模型中的特征值最优收缩AI 解读

We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the spiked covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker $\eta$ that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible…

奇异值阈值矩阵去噪的极小极大风险AI 解读

An unknown $m$ by $n$ matrix $X_{0}$ is to be estimated from noisy measurements $Y=X_{0}+Z$, where the noise matrix $Z$ has i.i.d. Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem $\operatorname{min}_{X}\|Y-X\|_{F}^{2}/2+\lambda\|X\|_{*}$, where $\|X\|_{*}$ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of $\ell_{1}$ penalization in the vector case. It has been empirically observed that if $X_{0}$…

稀疏指数分布数据的错误发现率阈值估计的渐近极小极大性AI 解读

We apply FDR thresholding to a non-Gaussian vector whose coordinates Xi, i=1, …, n, are independent exponential with individual means μi. The vector μ=(μi) is thought to be sparse, with most coordinates 1 but a small fraction significantly larger than 1; roughly, most coordinates are simply ‘noise,’ but a small fraction contain ‘signal.’ We measure risk by per-coordinate mean-squared error in recovering log(μi), and study minimax estimation over parameter spaces defined by…

用于检测稀疏异质混合的 Higher Criticism 检验AI 解读

Higher criticism, or second-level significance testing, is a multiple-comparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested comparing the fraction of observed significances at a given α-level to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming a…