On positive definite thresholding of correlation matrices
This addresses a fundamental issue in statistics and machine learning for researchers and practitioners dealing with correlation-based methods, though it appears incremental as it builds on existing thresholding techniques and Delsarte's method.
The paper tackles the problem of thresholding correlation matrices while preserving positive semidefiniteness, proving that any soft-thresholding operator that does so induces a geometric collapse in the feature space, with an O(1/n) bound on faithfulness for rank n matrices.
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.