Slicing: Nonsingular Estimation of High Dimensional Covariance Matrices Using Multiway Kronecker Delta Covariance Structures

Nonsingular estimation of high dimensional covariance matrices is an important step in many statistical procedures like classification, clustering, variable selection an future extraction. After a review of the essential background material, this paper introduces a technique we call slicing for obtaining a nonsingular covariance matrix of high dimensional data. Slicing is essentially assuming that the data has Kronecker delta covariance structure. Finally, we discuss the implications of the results in this paper and provide an example of classification for high dimensional gene expression data.

💡 Research Summary

The paper tackles a fundamental obstacle in modern multivariate statistics: estimating a nonsingular covariance matrix when the dimensionality of the data far exceeds the number of observations. In such “large‑p, small‑n” settings the ordinary sample covariance matrix is almost surely singular, precluding the use of any method that requires an inverse (e.g., linear discriminant analysis, Mahalanobis distance‑based clustering, regularized regression). Traditional remedies—ridge regularization, shrinkage toward a target, factor models, or banded/Töeplitz constraints—either introduce substantial bias or rely on structural assumptions that are rarely justified for complex biological or imaging data.

The authors propose a novel approach called slicing. The central idea is to assume that the true covariance matrix possesses a multi‑way Kronecker‑delta structure:

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