Quantitative Resolvent Calculus for Sample Covariance Matrices
Abstract
In this work, we develop a quantitative resolvent calculus for sample covariance matrices tailored to applications in high-dimensional statistics. We treat single resolvents, their derivatives, and products of two resolvents with different spectral parameters in the proportional-dimensional regime. Our bounds hold uniformly over compact subsets of ℂ∖ℝ≥0 with approximation errors of order Oℙ(|Θn|F/√n) for deterministic test matrices Θn. These bounds require only uniformly bounded fourth moments and a uniformly bounded population covariance operator norm. Neither regular spectral edges nor convergence of the population spectral distribution is required. As a key technical ingredient, we develop an averaging argument for centered quadratic forms involving one or two leave-one-out resolvents. Applications of our results include precision-matrix shrinkage, prediction risk in ridge regression, and generalized cross-validation. For these problems, we obtain quantitative guarantees for estimation and data-driven parameter selection.
Related
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BibTeX
@misc{stark2026quantitative,
title = {{Quantitative Resolvent Calculus for Sample Covariance Matrices}},
author = {Thomas Stark and Nina Dörnemann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33087},
eprint = {2610.33087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33087},
}
RIS
TY - UNPB
TI - Quantitative Resolvent Calculus for Sample Covariance Matrices
AU - Thomas Stark
AU - Nina Dörnemann
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.33087
UR - https://arxiv.org/abs/2610.33087
AB - In this work, we develop a quantitative resolvent calculus for sample covariance matrices tailored to applications in high-dimensional statistics. We treat single resolvents, their derivatives, and products of two resolvents with different spectral parameters in the proportional-dimensional regime. Our bounds hold uniformly over compact subsets of ℂ∖ℝ_≥0 with approximation errors of order O_ℙ(|Θ_n|_F/√n) for deterministic test matrices Θ_n. These bounds require only uniformly bounded fourth moments and a uniformly bounded population covariance operator norm. Neither regular spectral edges nor convergence of the population spectral distribution is required. As a key technical ingredient, we develop an averaging argument for centered quadratic forms involving one or two leave-one-out resolvents. Applications of our results include precision-matrix shrinkage, prediction risk in ridge regression, and generalized cross-validation. For these problems, we obtain quantitative guarantees for estimation and data-driven parameter selection.
ER -