On positive definite thresholding of correlation matrices

TL;DR

This paper explores methods to threshold correlation matrices while maintaining positive semidefiniteness, introducing new functions and bounds that reveal limitations of soft-thresholding in preserving geometric features.

Contribution

It develops positive definite functions that vanish on specific sets, establishes faithfulness criteria via Gegenbauer expansions, and shows soft-thresholding's geometric limitations for rank-$n$ matrices.

Findings

Existence of positive definite functions vanishing on sets

Bounds for thresholding at points and pairs

Soft-thresholding induces geometric collapse with an $ ext{O}(1/n)$} faithfulness bound

Abstract

Standard thresholding techniques for correlation matrices often destroy positive semidefiniteness. We investigate the construction of positive definite functions that vanish on specific sets $K \subseteq [-1,1)$, ensuring that the thresholded matrix remains a valid correlation matrix. We establish existence results, define a criterion for faithfulness based on the linear coefficient of the normalized Gegenbauer expansion in analogy with Delsarte's method in coding theory, and provide bounds for thresholding at single points and pairs of points. We prove that for correlation matrices of rank $n$, any soft-thresholding operator that preserves positive semidefiniteness necessarily induces a geometric collapse of the feature space, as quantified by an $\mathcal{O}(1/n)$ bound on the faithfulness constant. Such demonstrates that geometrically unbiased soft-thresholding limits the recoverable signal.