On positive definite thresholding of correlation matrices
2026-03-11 • Information Theory
Information Theory
AI summaryⓘ
The authors study how to adjust correlation matrices by setting some values to zero (thresholding) without breaking their mathematical properties, which is usually tricky. They find special functions to keep the matrix valid after thresholding and give rules to measure how well this works. They also show that for matrices with fixed rank, any soft-thresholding method that keeps the matrix valid shrinks the data representation in a way that limits how much signal can be recovered. This means unbiased soft-thresholding methods inherently lose information.
correlation matrixpositive semidefinitenessthresholdingpositive definite functionsGegenbauer expansionDelsarte's methodsoft-thresholdingrank of a matrixfaithfulness constantfeature space
Authors
Sujit Sakharam Damase, James Eldred Pascoe
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.