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.

翻译：标准的相关系数矩阵阈值处理方法往往会破坏半正定性。本文研究了在特定集合$K \subseteq [-1,1)$上取零值的正定函数的构造方法，以确保阈值处理后的矩阵仍保持为有效的相关系数矩阵。我们建立了存在性结果，并基于归一化盖根鲍尔展开的线性系数定义了保真度准则（该方法与编码理论中的Delsarte方法具有类比关系），同时给出了单点及点对阈值处理的界。我们证明：对于秩为$n$的相关矩阵，任何保持半正定性的软阈值算子必然导致特征空间的几何坍缩，这通过保真度常数的$\mathcal{O}(1/n)$上界得以量化。该结果表明，几何无偏的软阈值处理会限制可恢复信号的范围。