Aug 2026· International Journal For Multidisciplinary Research· Vol 8· 0 citations· 5 references
Abstract
In the era of high-dimensional data, the classical assumption that the number of observations n vastly exceeds the number of variables p is frequently violated. When p and n grow proportionally (p/n → c > 0), the sample covariance matrix becomes severely distorted by sampling noise. Its eigenvalues are systematically biased: large population variances are overestimated, and small ones are underestimated. This phenomenon, governed by the Marchenko–Pastur law of Random Matrix Theory (RMT), renders standard statistical procedures highly unstable. This paper provides a comprehensive, mathematically rigorous treatment of spectral shrinkage, the optimal remedy for this distortion. We transition from the theoretical foundations of the Stieltjes transform to the practical implementation of rotationally invariant estimators. By combining formal proofs, geometrical interpretations, and reproducible R simulations with explicit console outputs, we demonstrate why spectral shrinkage is not merely a heuristic regularization technique, but a mathematically undeniable necessity for modern high-dimensional statistics.
This paper investigates the asymptotic behavior of the out-of-sample prediction risk of the high-dimensional ridgeless least-squares estimator when the feature dimension $p$ and the sample size $n$ grow proportionally. We consider a generalized spiked population covariance model with multiple latent factors, where the...
We investigate the geometric fluctuations of principal subspaces for high-dimensional covariance matrices through the squared Frobenius $\sin\Theta$ distance between the sample and population eigenspaces associated with the $r_p$ largest eigenvalues. An explicit first-order expansion and a central limit theorem are est...
We investigate the least squares linear regression problem with random partial Discrete Fourier Transform (DFT) matrices, providing a rigorous analysis of the model's generalization error. By leveraging tools from random matrix theory, we derive exact non-asymptotic bounds for the risk of the Moore-Penrose estimator, w...
Active subspaces identify low-dimensional linear structure in high-dimensional parameter-to-output maps by estimating the dominant eigenspace of a gradient covariance operator. In practice this covariance is replaced by a Monte Carlo estimator built from a limited number of gradient evaluations. Classical analyses base...
Fabio Nobile, Matteo Raviola, R. Tempone· 0 citations
The proposed methodology delivers interpretable, simultaneous hyper-rectangular confidence regions that are statistically robust, memory-efficient, and strictly scalable for high-dimensional inference.
We study identity testing for high-dimensional covariance matrices against dense alternatives of unknown direction, with $p/n \to \gamma$. Along a globally positive quadratic precision path, mixing Gaussian alternatives over a Gaussian Orthogonal Ensemble direction yields a contiguous experiment whose log likelihood re...
P. Hansen, W. Ploberger, Tong Chen· 0 citations
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